MISSILE AUTOPILOT DESIGN
BY PROJECTIVE CONTROL THEORY
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY
BY
RESAT ÖZGÜR DORUK
IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
IN
THE DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
AUGUST 2003
Approval of the Graduate School of Natural and Applied Sciences
_____________________
Prof. Dr. Canan Özgen Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science.
_____________________
Prof. Dr. Mübeccel Demirekler Head of Department
This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science.
_____________________
Prof. Dr. Erol Kocaoglan Supervisor
Examining Committee Members
Prof. Dr. Mübeccel Demirekler (Chairperson) _____________________
Prof. Dr. Erol Kocaoglan _____________________
Prof. Dr. Kemal Leblebicioglu _____________________
Assist. Prof. Dr. Yakup Özkazanç _____________________
Dr. Ayse Pinar Koyaz _____________________
iii
ABSTRACT
MISSILE AUTOPILOT DESIGN BY PROJECTIVE CONTROL THEORY
Doruk, Resat Özgür
M.Sc, Department of Electrical and Electronics Engineering
Supervisor: Prof.Dr. Erol Kocaoglan
August 2003, 151 pages
In this thesis, autopilots are developed for missiles with moderate dynamics and
stationary targets. The aim is to use the designs in real applications. Since the real
missile model is nonlinear, a linearization process is required to get use of systematic
linear controller design techniques. In the scope of this thesis, the linear quadratic
full state feedback approach is applied for developing missile autopilots. However,
the limitations of measurement systems on the missiles restrict the availability of all
the states required for feedback. Because of this fact, the linear quadratic design will
be approximated by the use of projective control theory. This method enables the
designer to use preferably static feedback or if necessary a controller plus a low
order compensator combination to approximate the full state feedback reference.
Autopilots are checked for the validity of linearization, robust stability against
aerodynamic, mechanical and measurement uncertainties.
Keywords: Guided Missile, Autopilot, Linear Quadratic Design, Full State Feedback
Projective Control, Static Feedback, Compensator, Robust Stability
iv
ÖZ
IZDÜSÜM KONTROL YÖNTEMIYLE FÜZE OTOPILOTU TASARIMI
Doruk, Resat Özgür
Yüksek Lisans, Elektrik ve Elektronik Mühendisligi Bölümü
Tez Yöneticisi : Prof.Dr. Erol Kocaoglan
Agustos 2003, 151 sayfa
Bu çalismada hizli dinamigi olmayan, sabit hedefli füzeler için gerçek
uygulamalarda da kullanilabilecek füze otopilotlari tasarimlari yapilmistir. Gerçek
füze modelinin dogrusal olmamasi, sistematik kontrol tasarim yöntemlerinden
faydalanabilmek için bir dogrusallastirma sürecini gerektirmektedir. Bu tez
kapsaminda, dogrusal karesel tam durum geribesleme yöntemi kullanilarak otopilot
tasarimi gerçeklestirilmistir. Ancak füzelerdeki ölçüm sisteminin getirdigi
kisitlamalar tam durum geribeslemesine engel olmaktadir. Bu nedenle izdüsüm
kontrol yönteminden faydalanilarak dogrusal karesel yöntemle yapilan tasarimi
yaklasik olarak gerçeklestirmeye çalisilmistir. Bu yöntem ile basta statik geri besleme
olmak üzere gerektirdigi hallerde de denetleç ve düsük dereceli düzeltici bileskesi
kullanilarak dogrusal karesel tasarim yaklasik olarak gerçeklestirilebilir. Yapilan
tüm otopilotlar, dogrusallastirmanin geçerliligi ve tüm sistemin aerodinamik ve
mekanik belirsizlikler ile ölçüm gürültüsüne karsi kararliligi yönünden incelenerek
gerçek uygulamalarda kullanilip kullanilamayacagi belirlenmeye çalisilmistir.
Anahtar Kelimeler: Füze, Otopilot, Dogrusal Karesel Tasarim, Tam Durum
Geribesleme, Izdüsüm Kontrol, Statik Geribesleme, Düzeltici, Gürbüz Kararlilik
v
ACKNOWLEDGEMENTS
I would like to express great thanks to my supervisor Prof. Dr. Erol Kocaoglan for
his help and guidance throughout this work. Thanks also go to Asst. Prof. Dr. Yakup
Özkazanç for his valuable recommendations about the study.
My sincere thanks go to Burak Kaygisiz, Kadriye Tiryaki, Çaglar Karasu, Dr. Pinar
Koyaz, Alper Ünver and my other colleagues in TÜBITAK – SAGE for their
suggestions and contributions to this study. The support of TÜBITAK – SAGE
throughout this study is greatly acknowledged.
Finally, I would like to offer my sincere thanks to my family for their great support
and understanding.
vi
TABLE OF CONTENTS
ABSTRACT .............................................................................................................iii
ÖZ ............................................................................................................................iv
ACKNOWLEDGEMENTS .......................................................................................v
TABLE OF CONTENTS ..........................................................................................vi
TABLE OF FIGURES ...............................................................................................ix
LIST OF SYMBOLS ...............................................................................................xiv
CHAPTER
1 INTRODUCTION..................................................................................................1
2 PROJECTIVE CONTROL THEORY......................................................................6
2.1 . Introduction to Theory ..................................................................................6
2.2 . Linear Quadratic Full State Feedback Design ...............................................6
2.3 . Projective Control..........................................................................................9
3 MODELLING OF AERODYNAMIC MISSILES ..................................................19
3.1 . Introduction .................................................................................................19
3.2 . Aerodynamic Missile Modeling..................................................................20
3.2.1 . The Equation of Motion.........................................................................20
3.3 . Introduction to Missile Aerodynamics........................................................24
3.3.1 . The Expression of Forces and Moments ...............................................24
3.3.2 . The Aerodynamic Coefficients..............................................................25
3.3.3 . The Linearization of the Missile Model.................................................27
3.4 . Modeling of Other Effects............................................................................31
3.4.1 . Modeling of Thrust and Its Effects........................................................31
3.4.2 . The Uncertainty in the Center of Gravity Function..............................34
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4 AUTOPILOT DESIGN BY STATIC PROJECTIVE CONTROL...........................35
4.1 . Introduction .................................................................................................35
4.2 . Autopilot Design by Linear Quadratic Control Theory..............................36
4.2.1 . Mathematical Framework for Linear Quadratic Controller Design .....36
4.2.2 . Design Example.....................................................................................38
4.3 . Autopilot Design by Static Projective Control ............................................49
4.4 . Non – Linear Simulations............................................................................55
4.4.1 . Introduction...........................................................................................55
4.4.2 . The Results of Simulations....................................................................56
5 AUTOPILOT DESIGN BY DYNAMIC PROJECTIVE CONTROL......................62
5.1 . Reasons of Dynamic Projective Control......................................................62
5.2 . Rate Autopilot Design Using Dynamic Projective Control.........................62
5.3 . Dynamic Acceleration Autopilot Design ....................................................71
5.4 . Nonlinear Flight Simulations......................................................................76
5.4.1 . Simulation of Dynamically Compensated Projective Rate Autopilot ..76
5.4.2 . Dynamically Compensated Acceleration Autopilot Simulation ..........80
6 ROBUSTNESS ANALYSIS...................................................................................85
6.1 . Importance of Robustness Analysis ............................................................85
6.2 . Singular Value Robustness Tests.................................................................85
6.3 . Linear Fractional Transformations..............................................................92
6.4 . The Robustness Analysis of the Static Rate Autopilot ................................97
6.5 . The Robustness Analysis of Dynamic Rate Autopilot ..............................104
6.6 . Robustness Analysis of Dynamic Acceleration Autopilot ........................107
6.7 . Overall Analysis of the Autopilots............................................................112
6.7.1 . Overall Results for Static Rate Autopilot ............................................113
6.7.2 . Overall Results for Dynamic Rate Autopilot ......................................114
6.7.3 . Overall Results for Dynamic Acceleration Autopilot.........................115
7 REAL ENVIRONMENT SIMULATIONS..........................................................116
7.1 . Introduction ...............................................................................................116
7.2 . Implementation of Uncertainties in Simulations ......................................116
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7.2.1 . Implementation of the Aerodynamic Uncertainties ...........................117
7.2.2 . The Modeling of Inertial Sensor Noise................................................117
7.2.3 . The Modeling Thrust Misalignment ...................................................117
7.2.4 . The Misalignment in Center of Gravity of the Missile........................118
7.2.5 . The Initial Rolling of the Missile .........................................................118
7.2.6 . The Deviation in the Natural Frequency of the Actuator...................118
7.2.7 . The Side Wind Effects .........................................................................119
7.3 . The Real Environment Simulations...........................................................119
7.3.1 . Simulations of the System Including Static Rate Autopilot................120
7.3.2 . Simulation of the System Including Dynamic Rate Autopilot............125
8 CONCLUSIONS.................................................................................................130
APPENDIX A CONTROL ACTUATION SYSTEM............................................134
APPENDIX B ROLL CONTROL SYSTEM ........................................................138
B.1 . Design of the Roll Autopilot .....................................................................138
B.2 . The Structured Singular Value Analysis of Roll Autopilot ......................142
REFERENCES .....................................................................................................148
ix
TABLE OF FIGURES
FIGURE
1-1 The general guided missile configuration............................................................. 1
2-1 General block diagram for full state feedback design ........................................... 7
3-1 Missile coordinate system including system variables ....................................... 20
3-2 The forces and moments acting on the missile body........................................... 21
3-3 The angle of attack and sideslip........................................................................... 25
3-4 The variation of thrust with time......................................................................... 32
3-5 The thrust misalignment phenomenon ............................................................... 33
4-1 The block diagram form of full state feedback rate autopilot ............................. 42
4-2. The block diagram of the full state feedback acceleration autopilot.................. 42
4-3. Settling time variation with quadratic weight.................................................... 44
4-4. Phase margin variation with quadratic weight.................................................. 45
4-5. The step response of the rate autopilot............................................................... 45
4-6 The OL frequency response characteristics of the rate autopilot ........................ 46
4-7 The variation of the settling time for acceleration autopilot design.................... 46
4-8 The variation of phase margin in acceleration autopilot design ......................... 47
4-9 The step response of the full state feedback acceleration autopilot .................... 48
4-10 The open loop frequency response of the acceleration autopilot...................... 48
4-11 The block diagram of the static projection rate autopilot.................................. 53
4-12 Step responses of full state feedback and static projective autopilots............... 54
4-13 Open loop frequency response of the static projective autopilot...................... 54
4-14 The variation of autopilot gain phK% agains altitude and velocity....................... 56
4-15 Missile range (in x – direction)........................................................................... 57
4-16 Lateral range of the missile (in y – direction) .................................................... 57
4-17 Angle of attack variation................................................................................... 57
x
4-18 Elevator (pitch plane control surface) deflection............................................... 58
4-19 Angle of sideslip variation ................................................................................. 58
4-20 Rudder (yaw control surface) deflection .......................................................... 58
4-21 Pitch rate command and output ........................................................................ 59
4-22 Yaw rate command and output......................................................................... 59
4-23 Mach number variation...................................................................................... 59
4-24 Roll angle variation ............................................................................................ 60
4-25 Roll rate variation............................................................................................... 60
4-26 Aileron (roll plane control surface) deflection................................................... 60
5-1 The step response of dynamic rate autopilot ( )10prt = −reA ................................... 69
5-2 The step response of dynamic rate autopilot ( )1prt = −reA ..................................... 69
5-3 The OL frequency response of dynamic rate autopilot ( )10prt = −reA .................... 70
5-4 The OL frequency response of dynamic rate autopilot ( )1prt = −reA ...................... 70
5-5 The step response of the dynamic acceleration autopilot ................................... 75
5-6 The OL frequency response of the dynamic acceleration autopilot.................... 75
5-7 Missile Range (x – direction)................................................................................ 76
5-8 Lateral range of the missile (y- direction)............................................................ 76
5-9 Angle of attack variation...................................................................................... 77
5-10 Elevator (pitch control surface) deflection......................................................... 77
5-11 Angle of sideslip variation ................................................................................. 77
5-12 Rudder (yaw control surface) deflection ........................................................... 78
5-13 Pitch rate command and output ........................................................................ 78
5-14 Yaw rate command and output......................................................................... 78
5-15 Mach number variation...................................................................................... 79
5-16 Roll angle variation ............................................................................................ 79
5-17 Roll rate variation............................................................................................... 79
5-18 Aileron (roll plane control surface) deflection................................................... 80
5-19 Missile range (x – direction)............................................................................... 80
5-20 Lateral range of the missile (y – direction) ........................................................ 80
xi
5-21 Angle of attack variation.................................................................................... 81
5-22 Elevator (pitch plane control surface) deflection............................................... 81
5-23 Angle of sideslip variation ................................................................................. 81
5-24 Rudder (yaw control surface) deflection ........................................................... 82
5-25 Pitch acceleration command and output........................................................... 82
5-26 Yaw acceleration command and output............................................................ 82
5-27 Mach number variation...................................................................................... 83
5-28 Roll angle variation ............................................................................................ 83
5-29 Roll rate variation............................................................................................... 83
5-30 Aileron (roll plane control surface) deflection................................................... 84
6-1 A conventional control system ............................................................................ 86
6-2 The feedback interconnection structure for singular value analysis................... 86
6-3 Complex System Matrix Relating Input and Output .......................................... 93
6-4 The partitioned form of the system matrix and inputs. ...................................... 93
6-5 The Linear Fractional Transformation System Connection................................. 94
6-6 System Connection as an Upper Linear Fractional Transformation................... 94
6-7 µ - analysis framework for the analysis of the rate autopilot ............................ 99
6-8 Rate autopilot constructed for the robustness analysis framework .................... 99
6-9 The rate autopilot LFT block form......................................................................100
6-10 The overall combination of the LFT formed system.........................................101
6-11 The variation of upper bound of µ with frequency ........................................102
6-12 The step response of the perturbated system. ..................................................103
6-13 The converted form of the autopilot for LFT integration.................................104
6-14 The variation of singular value for dynamic rate autopilot .............................105
6-15 The time response of perturbed dynamic rate autopilot..................................106
6-16 The plant block diagram including uncertainties.............................................109
6-17 The acceleration autopilot structure for singular value analysis......................109
6-18 The LFT block structure of the autopilot ..........................................................109
6-19 The closed loop interconnection of the acceleration control system................110
xii
6-20 Plot of upper bound versus frequency for acceleration autopilot....................111
6-21 The step response of the perturbed acceleration autopilot...............................112
6-22 Variation of perturbation bound for static rate autopilot.................................113
6-23 Variation of perturbation bound for dynamic rate autopilot...........................114
6-24 Variation of perturbation boundfor dynamic acceleration autopilot...............114
7-1 Angle of attack variation.....................................................................................120
7-2 Elevator (pitch control surface) deflection..........................................................121
7-3 Pitch plane guidance command..........................................................................121
7-4 Pitch rate .............................................................................................................121
7-5 Angle of sideslip .................................................................................................122
7-6 Rudder (yaw control surface) deflection............................................................122
7-7 Yaw rate guidance command .............................................................................122
7-8 Yaw rate variation...............................................................................................123
7-9 Aileron (roll control surface) deflction ...............................................................123
7-10 Roll angle variation ...........................................................................................123
7-11 Roll rate variation..............................................................................................124
7-12 Velocity variation..............................................................................................124
7-13 Angle of attack variation...................................................................................125
7-14 Elevator deflection ............................................................................................125
7-15 Pitch plane guidance commands ......................................................................125
7-16 Pitch rate variation............................................................................................126
7-17 Sideslip angle variation.....................................................................................126
7-18 Rudder deflection..............................................................................................126
7-19 Yaw plane guidance command.........................................................................127
7-20 Yaw rate variation.............................................................................................127
7-21 Aileron deflection..............................................................................................127
7-22 Roll angle variation ...........................................................................................128
7-23 Roll rate variation..............................................................................................128
7-24 Velocity variation..............................................................................................128
A-1 The nonlinear control actuation system ............................................................134
xiii
A-2 The linear model of a control actuation system.................................................136
A-3 The linear fractional transformed actuator model.............................................137
A-4 The LFT block form of the control actuation system.........................................137
B-1 The roll autopilot................................................................................................139
B-2 The step response of the roll autopilot...............................................................141
B-3 The roll controller structure................................................................................142
B-4 The schematic of the roll model .........................................................................142
B-5 The LFT framework of the roll model for analysis ............................................143
B-6 Closed loop interconnection of the LFT formed roll autopilot..........................143
B-7 The closed loop interconnection for the roll autopilot analysis .........................144
B-8 The structural singular value variation of roll autopilot ...................................145
B-9 The perturbed roll system’s step response.........................................................146
B-10 The overall robustness analysis of roll autopilot..............................................147
xiv
LIST OF SYMBOLS
x general notation for a state vector
y general output for a single input and output system
u general input for a single input and output system
A,B state space system and input matrices (determines property)
C,D output matrices of a state space system definition (depends on the
output definition)
yε the integral of the control error (between output and reference)
which is defined as a an additional state variable.
yr reference (set point) input to the control system.
( )a ∞ the value of a general variable ( )a t at steady state.
ex vector of errors of state variables in x according to steady state.
ue error of system input variable u according to steady state
x augmented state vector (the state vector x is augmented by error of
output integral yε )
y The vector of states that are available for feedback (output feedback
vector)
e vector of errors of elements of x according to steady state.
ˆ ˆ,A B augmented state space system matrices (applies on x or e )
F closed loop system matrix
,Q R quadratic weight matrices on states and input respectively
K Full state feedback control vector (including feedback from integral
of control error)
qε Quadratic weight on the integral of the error
xv
? Closed loop spectrum (matrix containing all the eigenvalues of the
closed loop along the diagonal) of full state feedback design
X Matrix containing all of the eigenvectors corresponding to
? (eigenmatrix)
r? Subspectrum containing the eigenvalues to be retained by projective
control
rX Matrix containing eigenvectors corresponding to r? in separated form.
It is a subset of X and called as sub-eigenmatrix.
P Projection matrix from full state space to the space of available states.
rC Output matrix relating the full state vector to available state vector
proK Projective control gain vector
,p p? X Unretained spectrum and corresponding sub-eigenmatrix.
z General notation for dynamic compensator state vector.
H,D General notation for compensator system and input matrices.
,y zK K Control gain vectors in controller + compensator combination.
iI Identity matrix of order i.
ex Error vector augmented by the compensator state
ey Output vector including available and compensator states.
,e eA B System and input matrices of the state space definition composed of
compensator and error dynamics.
,e eQ R Quadratic weights on ex and eu
eF Closed loop system matrix of compensated system.
eK Full state feedback control gain vector of dynamically compensated
system (from ex ).
eC Output matrix that selects the available plant and compensator states
from ex .
eX Eigenmatrix of the closed loop of the compensated system (full state
feedback).
xvi
eP Projection matrix from the state space of compensated system to the
subspace of compensator and available plant states.
reA Residual spectrum (additional poles coming from the compensator)
, ,u v w Body axis velocities in x, y, z directions.
, ,x y z Missile position defined according to the projection of the initial
position on the earth.
, ,p q r Angular velocities of the missile body in x, y, z directions.
, ,φ θ ψ Euler angles defining the missiles orientation
, ,x y zF F F Total forces in x, y, z directions.
, ,x y zM M M Total moments in x, y, z directions.
, ,ax ay azF F F Aerodynamic forces in x, y, z directions.
, ,L M N Aerodynamic moments in x, y, z directions.
,A AF M Total aerodynamic force and moment applied on the missile body.
, TT M Force and additional moments caused by thrust
,E EF M Forces and moments caused by external disturbances.
, ,x y zI I I Moments of inertia
m Mass of the missile body
, ,x y za a a Body accelerations in x, y, z directions.
A Cross sectional area
dQ Dynamic pressure
d Missile diameter
, ,x y zC C C Dimensionless aerodynamic force coefficients.
, ,l m nC C C Dimensionless aerodynamic moment coefficients.
,α β Angle of attack and sideslip
Mach Velocity of the missile in mach numbers.
a Speed of sound
,k R Specific heat ratio and universal gas constant
T Ambient temperature
xvii
, ,a e rδ δ δ Aileron, elevator and rudder deflections (control surface deflections
of roll, pitch and yaw planes)
,p pi ia b Aerodynamic entries of linearized system matrices (pitch plane)
,y yi ia b Aerodynamic entries of linearized system matrices (yaw plane)
,pL Lδ Aerodynamic coefficients of linearized roll model.
( )S t Impulse
totalS Total Impulse
( )T t Instantaneous thrust
0 1,m m Mass of the missile in the beginning and end of the thrust phase.
0 1,i iI I Moment of inertia (in any axis) of the missile in the beginning and
end of the thrust phase. 0 1,cg cgx x Position of the center of gravity in the beginning and end of the thrust
phase (measured from the nose).
,M N′ ′ Aerodynamic moments compensated for thrust.
1 2,δ δ Thrust misalignment angles
, ,x y zT T T Components of the thrust force due to misalignment
,t tM N Components of the additional pitch and yaw moments caused by
thrust misalignment.
cgδ Deviation in center of gravity due to production uncertainties
,nom actcg cgx x Nominal and deviated values of center of gravity
, , ,r r zr ryq r a a Guidance commands passed to the autopilots
, , ,z yq r a aε ε ε ε Integral of the control error in rate and acceleration autopilots
, ,w q aG G G Transfer functions relating downward velocity, pitch rate and
acceleration to elevator deflection.
qwG Transfer function relating pitch rate to downward velocity
aqG Transfer function between pitch acceleration and rate
xviii
, ,yv r aG G G Transfer functions relating lateral velocity, yaw rate and acceleration
to rudder deflection.
rvG Transfer function relating yaw rate to lateral velocity
arG Transfer function between yaw acceleration and rate
,prate paccOL OLG G Open loop transfer functions of pitch rate and acceleration autopilots
,yrate yaccOL OLG G Open loop transfer functions of yaw rate and acceleration autopilots
ˆ ˆ,p ya aK K Full state feedback control gain vector for acceleration autopilots
ˆ ˆ,p yr rK K Full state feedback control gain vector for rate autopilots
,r aJ J Performance indices for linear quadratic rate and acceleration
autopilots
,s rt t Settling and rise times
. , .G M P M Gain and phase margins
ˆ ˆ ˆ ˆ, , ,rt rt ac acA B A B Linear system matrices of design models for rate and acceleration
autopilots.
,rt ac? ? Closed loop spectrum of full state feedback turn rate and acceleration
autopilots.
,rt acX X Closed loop eigenmatrices (all eigenvectors columnwise) of full state
feedback turn rate and acceleration autopilots.
,rt acr rX X Eigenmatrices containing the eigenvectors corresponding to the
retained eigenvalues in projective controlled rate and acceleration
autopilot (subeigenmatrices)
,rt acP P Projection matrices (from full state space to available subspace) used
in design of turn rate and acceleration autopilots.
,prt pacpro proK K Control gain vectors obtained from projection (for pitch rate and
acceleration autopilots)
,prt pacpro pro? ? Closed loop spectrum of uncompensated (static) projective pitch rate
and acceleration autopilot
, , ,p p y yh q h qK K K K Control gains of pitch and yaw rate autopilots (on feedback lines)
xix
( , )f M h Function representing autopilot gain tables (as a function of Mach
number and height)
0 0,V h Initial missile velocity and height (releasing from aircraft)
,f fx y Position of the stationary target
,prt prtc cH D Compensator system matrices (pitch rate autopilot)
,prt prtKω ωK Controller gains in dynamically compensated pitch rate autopilot
,pac pacc cH D Compensator system matrices (pitch acceleration autopilot)
,pac pacKξXK Controller gains in dynamically compensated pitch acceleration
autopilot
,prt pacp pX X Eigenmatrices corresponding to unretained eigenvalues (during static
projective control)
,prt pacF F Closed loop system matrices for pitch rate and acceleration autopilots
,prt pacre reA A Residual spectrum (additional eigenvalues due to compensator) of
dynamically compensated pitch rate and acceleration autopilots
,prt pac0 0P P Free parameter that has direct effect on residual eigenvalues
( ). + Pseudo inversion operation
,,
prt pac
prt pacw ξ
T TP PP P
Projection matrices for obtaining controller gains in dynamically
compensated pitch rate and acceleration autopilot
,,
prt pac
prt pacK Kω ξ
T TK K Controller gains of the dynamically compensated pitch rate and
acceleration autopilots
? Perturbation matrix (a matrix consists of deviations of each
parameter)
M Complex matrix representing the nominal system (frequency
response)
1....nσ Singular values obtained from singular value decomposition (SVD)
[ ] [ ],σ σA A Maximum and minimum singular values of matrix A
xx
2A 2l norm of the matrix A
( )µ∆ M Structured singular value (highest singular value of perturbation
matrix ? that makes −I M? singular)
( )ρ M Spectral radius of matrix M where 1( ) max ( )i n iρ λ≤ ≤=M M
iδ Perturbation on each specific uncertain parameter (elements of ? )
,S F The number of repeated scalar and full blocks
,l uβ β The peaks of the lower and upper bounds of the structured singular
value across frequency
c Nominal value of uncertain parameter c
( , )LF M ? Lower linear fractional transformation for −M ? combination
( , )UF M ? Upper linear fractional transformation for −M ? combination
E Shorthand definition for compact system expression
E Nominal value of system matrices (compact form)
iE The weight of each perturbation iδ on linear system matrices in
compact system definition
,i iα β Row – Column decomposition of each iE
, ,m m mA B C State space triple for defining matrix M after linear fractional
transformation of linear plant definition nη Gaussian distributed noise signal for modeling inertial sensor noise
ω Noise free (nominal) inertial sensor output nω Noisy inertial sensor output
( )0 0 0, ,T
wx wy wzV V V Velocity of wind on the sea level
( ), ,T
wx wy wzV V V Velocity of wind on a specific altitude
0wK Decaying level of exponential wind profile
, ,w w wu v w The wind velocity components on the body axes of the missile
1
CHAPTER 1
INTRODUCTION
The developing technology brings many innovations to the field of aeronautics as of
many others. One of these is the guidance and control of the missiles which increase
the serviceability of the guided weapons. A guided missile is composed of four main
systems which are the guidance system, the autopilot, the control actuation system
and the missile body. The last of them, the missile body is an aerodynamic
mechanical system that is desired to be controlled (plant). The guidance system uses
the missile’s position and target information and commands the missile according to
the desired target position. To do that, the guidance system produces either an
acceleration command or turn rate command which depends on the application. The
realization of the guidance commands are performed by the autopilot system. It is a
control system in reality and compares the actual acceleration (or rate) with the
commands produced by the guidance system to generate a fin deflection command.
The missile is directed by the fins on its body. The actuation of the fins is performed
by an electromechanical or pneumatic actuator according to the deflection
commands coming from the autopilot. The actuator is called as control actuation
system in missile control literature. In Figure 1-1 the above structure of the guided
missile is converted into a block diagram in a sequential manner.
Figure 1-1 The general guided missile configuration
2
The aim of this study is to design autopilots that have possible uses in practical
applications. The control system should only get feedback from available
information. It should be possible for the designer to use systematic design
techniques without the need of too complex computational algorithms.
In the studies so far, there are classical missile autopilot design techniques
developed that applies eigenvalue placement directly from the transfer functions of
the missile models (Atesoglu (1997), Blackelock (1991), Mahmutyazicioglu (1994) and
Tiryaki (2002)). These designs are relatively simple; however they have some
restrictions due to the difficulty of making decision about the place of the resultant
eigenvalues. In addition, the solution is found through the closed loop transfer
function and a solution can not be found for several configurations. In some studies
(Blackelock (1991)), the classical control design techniques such as root locus and
frequency response are applied but it is generally successful for low order models,
and in some cases stability problems are encountered.
To solve the stated deficiencies, modern systematic control techniques based on state
space approach are developed for single input - single output systems with full state
feedback. A theoretical development is given by Ogata (1997) for general control
system design. Those can be applied both through standard eigenvalue placement
and optimal control methods. There are numerous studies about the application of
full state feedback systematic methods to missile autopilots (Wise (1990, 2001))
which utilizes linear quadratic full state feedback optimal control designs. Good
performance and robustness can be obtained from linear quadratic approaches
provided that all states are available for implementation.
In actual missiles, full state feedback control is not possible. Because only few of the
states are measured directly by inertial sensor systems (Wise (1991)) and the others
are computed by the navigation system from the sensor outputs. Because of that
many studies focus on developing control systems based on output feedback.
3
In Lewis (1992), an output feedback using the linear quadratic approach is
developed through an iterative algorithm. This may be a solution to the problem
however it requires too much iteration which in turn brings convergence problems.
Medanic (1983, 1985), proposes a different output feedback design approach by
using the full state feedback solution as a reference. The output feedback solution is
obtained through a projection from full state space to the available state space. This
approach is called as static projective control since there are no dynamic elements.
However, this method can also be utilized to obtain low order dynamic
compensators to be used with projective controllers. In that case, the procedure is
named as dynamic projective control. An application of static and dynamic
projective control to missile autopilot design is presented in Wise (1991).
Missile modeling is an important step in autopilot design, and various studies are
available on modeling. For the case of aerodynamic missiles Atesoglu (1997),
Blackelock (1991), Mahmutyazicioglu (1994), McLean (1990) and Tiryaki (2002)
present systematic development of models suitable for design purposes including
various uncertainties of real missiles. In addition to missile modeling, there are
approaches presented for checking the validity of the designed autopilots for
practical use.
Aeronautical systems are operating in environments including several uncertainties.
Because of that, a flight controller should be robust to changes caused by
uncertainties. There are numerous studies about analysis of control systems against
parameter uncertainties in theoretical basis. In one of them, Garloff (1985), develops
a simple approach to analyze robustness under perturbations in closed loop
characteristic polynomial. This may be a good approach for the beginning however
the coefficients of polynomial may be a non – linear function of uncertain parameters
(aerodynamic coefficients in the case of a missile). In another study (Vardulakis
(1987)), a necessary and sufficient condition is derived for a nominal plant with a
4
perturbation defined by a known matrix. By this way the maximum perturbation
level can be computed.
There are also systematic robustness analysis methods based on singular value
theories. These methods separate unperturbed plant and the coefficient
perturbations into two interconnected systems, and these systems are expressed by
complex matrices. The result obtained is generally a bound where the closed loop is
always stable. The most commonly used one is the structured singular value theory
studied by Packard (1988, 1993) and Zhou (1998) in detail. Some applications of the
analysis method to missile control systems are presented by Doyle (1987), Hewer
(1988) and Wise (1991). The application is performed through utilization of
algorithms of Fan (1991), Packard (1988) and Young (1992).
There are also special robustness algorithms developed especially for real parameter
perturbations. Some of those utilize mapping theorem (De Gaston (1988) and Wedell
(1991)) to form real boundaries of perturbations for stability. Although the
algorithms give almost exact results, their level of complexity makes implementation
very difficult.
Wise (1992), compares the results of six different robustness analysis techniques
applied on a missile autopilot. The methods are, singular value analysis, stability
bound computation by mapping theorem, Monte Carlo analysis, Kharitonov’s
theorem and Lyapunov methods. The advantages and disadvantages are stated to
give insight to the designer.
In this study, the projective control method is used to design missile autopilots
operating with feedback from the physically available states (including the output)
of the plant. The objective of the autopilots is to control either the turn rate or
acceleration of the missile. To make use of the projective control theory the non –
linear actual missile model is linearized. Then a linear quadratic full state feedback
design is made in order to obtain a reference controller. The projective control theory
5
is applied to get an output feedback for each autopilot. To verify that the autopilots
are operating successfully for practical considerations their stability robustness
against aerodynamic uncertainties is analyzed by structured singular value theory of
Packard (1993). The MATLAB µ - Analysis and Synthesis Toolbox by Balas (1998) is
used to implement the algorithms of Fan (1991), Packard (1988) and Young (1992).
The level of success of the operation of the missile in realistic environments is also
verified by Monte Carlo simulations including many uncertainties.
The second chapter of this study is reserved for a detailed presentation of the
projective control theory with its mathematical basis. Chapter three introduces the
equations of motion and the aerodynamic properties of tactical missiles. This chapter
constructs non – linear and linear models for use in the design of the autopilots and
flight simulation. The fourth chapter starts with the types of the autopilots and the
guidelines necessary for autopilot design, and then presents the design of turn rate
and acceleration autopilots using static projective control approach. For each
autopilot a non – linear flight simulation is presented for checking its validity. Fifth
chapter is organized in a similar way to chapter four, and presents autopilot design
using dynamic projective control. Chapter six presents a robust stability analysis
against the aerodynamic uncertainties with the structured singular value algorithm.
This chapter makes a linear analysis of the autopilot designs against perturbations in
the aerodynamic and mechanical properties of the actuator for stability of the overall
system. Chapter seven is reserved for complete simulations including thrust
misalignment, aerodynamic uncertainties, sensor noises and wind effects. Each
simulation corresponding to a specific autopilot is repeated 100 times to check that
the operation of the autopilot is reliable in a realistic medium. There are two
appendices, the first one (Appendix A) presents various models of the actuator used
in the simulations and robustness analysis studies. The second one (Appendix B)
presents a design of roll autopilot and makes a robust stability analysis for it. The
design is also used in all flight simulations throughout this thesis.
6
CHAPTER 2
PROJECTIVE CONTROL THEORY
2.1. Introduction to Theory
The performance requirements of modern missile control system design necessitate
the usage of linear quadratic control techniques where a specific control cost
function is minimized. Recent autopilot designs involve full state feedback controller
implementations. However, for the missile control design the usage of a full state
feedback controller is not recommended due to the practical limitations. The most
important limitation is the scarcity of the directly available data. The linear quadratic
techniques provide good performance and stability margins and its results can be
used as a reference for the subsequent designs. The unavailable states can be
estimated by dynamic compensators. However all compensator design methods do
not provide low order products with sufficiently satisfactory results. In the last 20
years, there are studies conducted for solving the problem related with
unavailability of states and approximating the linear quadratic full state feedback
controller design using only output feedback. The purpose of those studies is to
realize the design, at first, without using any dynamic compensators. If this design
does not provide satisfactory results then, a low order dynamic compensator design
is proposed.
2.2. Linear Quadratic Full State Feedback Design
The linear quadratic full state (LQSF) feedback design is used to obtain a reference
solution (also called as eigenstructure) for projective control design method. The
quadratic design is in fact a simple optimization procedure in which the cost is
defined as a function of state and input vectors. Before going into the linear
7
quadratic theory it is convenient to represent the plant as a linear state space system
as shown below:
1 1
, , ,, ,
n n n
n n
uy u
y u×
× ×
= += +
∈ ∈ ∈ ∈
∈ ∈ ∈
x Ax BCx D
x AB C D
&
¡ ¡ ¡ ¡¡ ¡ ¡
(2.1)
In order to control the output y , a feedback must be formed from the output to the
reference input as shown in Figure 2-1. If the plant transfer function does not
contain any integrator, then it is a common practice to integrate the error signal
between the command and the plant output. If one denotes the command yr and
control error ye as in Figure 2-1, then the following can be written (Ogata (1997)):
.y yrε = − −Cx Du& (2.2)
The above differential equation is also a state equation and can be augmented into
(2.1) as shown below (Ogata (1997)):
0 0.
0 1 yy y
u rε ε
= + + −
x xA B-C D
&& (2.3)
1s fK
K
= += +
x Ax Buy Cx Du&y
r y
x
uye yε
Figure 2-1 General block diagram for full state feedback design
Full state feedback controllers are designed such that the errors of the states at
steady state are zero. To achieve this, the error state equation at the steady state is
8
necessary and it can be derived from the general state equation (2.3) as follows
(Ogata (1997)):
( )( ) ( ) ( ) ( ) 1
( ) ( ) ( ( ))( ) ( ) ( ) ( )0 0 y y
y y y y
t td u t u r rt tdt ε ε ε ε
− ∞ − ∞ = + − ∞ + − ∞ − ∞ − ∞− −
x x x xA 0 BC D
(2.4)
The reference (command input) yr is taken as a step input in the design. For the
missile autopilot the control conditions are continuously changing (in Chapter 3
these conditions are explained in detail) so the assumption of constant reference
input is acceptable provided that the design is sufficiently fast. Since the reference of
the controller is constant the steady state error of yr is zero. Then the equation (2.4)
reduces to:
( )
( )
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )0
.0
y y y y
ee ey y
t td u t ut tdt
du
dt
ε ε ε ε
ε ε
− ∞ − ∞ = + − ∞ − ∞ − ∞− −
= + − −
e e
x x x xA 0 BC D
A 0 Bx xC D
(2.5)
The second expression in (2.5) is the equation of design in linear quadratic theory.
For the sake of simplicity, the error dynamics equation is rewritten accepting that
( )ˆ eyε= ee x . So the resulting equation is:
ˆ ˆˆ ˆ ˆ= + ee Ae Bu& (2.6)
According to Figure 2-1 the control gain vector is defined to be ( )ˆfK= −K K and the
closed loop error state equation is then defined as (Ogata (1997)):
1 1 1 1 1 1 1 1
ˆ ˆ ˆˆ ˆ( )
ˆ ˆˆ ˆ ˆ, , ,n n n n n n+ × + + × × + + × +
= −
=
∈ ∈ ∈ ∈
e A BK e
e Fe
A B K F
&&
¡ ¡ ¡ ¡ (2.7)
The feedback structure (connections and numerical values of the control gains)
applied to the system in (2.6) will be the same as of the feedback applied to the
9
original system in (2.3). Because of that there is no need to modify the control system
obtained from the design using (2.6).
The philosophy of the linear quadratic control design is basically an optimization
problem, in other words, minimization of a cost which is a function of the states and
inputs of the system in consideration. In the case of linear quadratic control the cost
is a quadratic equation defined as follows:
( )0
1 1
ˆ ˆ
,n n
J dt∞
+ × +
= +
∈ ∈
∫ T Te Qe u Ru
Q R¡ ¡ (2.8)
The minimization of this cost function is performed by solving the algebraic
equation which is defined as follows (Ogata (1997), Wise (1991)):
ˆ ˆ ˆ ˆ ˆ
ˆˆ ˆˆˆ ˆ
−
−
+ − + =
= −
= −
T 1 T
e 1 T
e
A M MA MBR B M Q 0
u R B Mx
u Kx
(2.9)
The design of the autopilot is generally done by assigning weights to artificial state
which is the integral of the control error as defined in (2.2). For the input a unity
weight is generally used. Shortly, a quadratic performance index of the following
form is used:
( )2 2
0
.J q u dtε ε∞
= +∫ (2.10)
2.3. Projective Control
The solution obtained from the linear quadratic design namely the K matrix in
equation (2.9) is used as a reference for the projective control design. The reference
solution is characterized by its eigenstructure which is defined by a diagonal matrix
( )? F consisting of the eigenvalues of the closed loop system i.e. F and an
eigenmatrix X (columns are the eigenvectors corresponding to each unique
10
eigenvalue). The eigenvalue matrix is also called as spectrum (Medanic (1985)).
Mathematically,
1 1 1 1,n n n n+ × + + × +
=
∈ ∈
FX X?X ?¡ ¡ (2.11)
The projective control approach, as it can be understood from its name, is based on
the orthogonal projection concept. The projection of the elements of the full state
space onto the available state space is generated. In projective control not all of the
eigenvalues of the original linear quadratic full state feedback system can not be
retained. The number of retainable eigenvalues is equal to the number of available
outputs for feedback ( )r . So to perform the projection operation the subset of the full
state spectrum ? (F) denoted by r? is needed. If rX represents the corresponding
eigenvector matrix, then the following is valid (Medanic (1985)):
1 ,n r r r+ × ×∈ ∈r r r
r r
FX = X ?X ?¡ ¡ (2.12)
The projection is performed using the projection equation below (Medanic (1985)),
-1r r r rP = X ( C X ) C (2.13)
and the projected control gain vector is computed as:
ˆproK =KP (2.14)
In (2.13) the matrix 1r n× +∈rC ¡ relates available state vector to full state vector in(2.6).
The output equation ˆˆ = ry C e should give the output vector consisting of the available
states for feedback.
The result obtained from the above equation has non – zero entries corresponding to
the available states. The other entries of the projected control vector are zero. The
result that is obtained from the design may not give the closed loop structure of the
original full state feedback design. However, the obtained result may be acceptable if
11
it satisfies the requirements. This type of design, in other words the design using
directly the result of the projective control theory, is also called as static projective
control in the literature (Wise (1991)) since there is no dynamic controller involved.
In some cases the implementation of the static projective control results in unstable
closed loop system. Before going further, it is useful to present some guidelines
about the selection of the sub – spectrum eigenmatrix rX .
The eigenvalues and eigenvectors of the full state feedback design are generally in
complex conjugate form. Sometimes, all or part of the eigenvalues (and thus
eigenvectors) may be pure real. The projective control approach retains some of
those eigenvalues the number of which is equal to the available outputs. The choice
of the retainable eigenvalues depends on their values and the number of complex
conjugate poles. For the real valued eigenvalues (to be retained), the corresponding
eigenvector is taken directly into the eigenmatrix rX . On the other hand, if the set of
eigenvalues to be retained have complex conjugate elements then the real and
imaginary parts of the eigenvector corresponding to the element of the considered
complex conjugate pair with negative imaginary part are selected for the sub –
spectrum eigenmatrix. The selected eigenvectors are formed as columns of the
eigenmatrix and they should be in the same order as the retained eigenvalues in the
spectrum of the full state feedback design.
It is important to state that the static projective control approach could result in an
unstable closed loop. If such a case is encountered and nothing can be achieved in
the sense of stability by changing the design, then the designer should utilize low
order dynamic compensators for achieving the closed loop spectrum of the full state
feedback design (Wise (1991)). A compensator of the following form is proposed:
1
ˆˆ
, ,p p rr r
u
× ××
= − −
= +
∈ ∈ ∈
y zK y K zz Hz Dyz H D
&¡ ¡ ¡
(2.15)
12
where z is the compensator state and p is the number of unavailable states. If the
above compensator state equation is augmented to the original system then the
following can be obtained:
ˆ ˆˆˆ
ddt
= = + =
re e e e
pe e
r
z H DC 0zx u A x + B u
ee 0 A BI 0
y = x0 C
&& &
(2.16)
The output vector ey in (2.16) is defined as:
ˆ
=
e
zy
y (2.17)
If linear quadratic full state feedback design is applied to the system defined in (2.16)
then the control gain vector will be obtained from the Riccati equation just as in(2.9).
The obtained control gain will minimize the following cost function:
eJ∞
= ∫ T Te e e e
0
(x Q x + u R u). (2.18)
This cost function are equal to the original cost function in (2.8) if eQ and eR
matrices of equation (2.18) are equal to (Medanic (1983)):
, .
= =
e e
0 0Q R R
0 Q (2.19)
According to the above definitions, the minimal solution to (2.18) is given by:
ˆ .
T -1 Te e e e e e e e e
-1 Te e e
A M + M A - M B R B M = 0u = - R B M e
(2.20)
The structures of the eQ and eR matrices lead to the solution of the Riccati equation
to be in the form:
13
=
e
0 0M
0 M (2.21)
The closed loop system matrix for the system in (2.16) is defined as:
, , =-1 Te e e e e e e e e e eF = A - B K K = R B M x F x& (2.22)
Substituting the solutions obtained, one gets:
( ) = −
-1 T
e
H DC 0 0 0F R 0 B
0 A B 0 M (2.23)
Further simplification gives,
.ˆ ˆ ˆ
= = − e -1 T
H DC H DCF 0 F0 A BR B M
(2.24)
The eigenstructure expansion of F in (2.11) can be repeated for eF and an expansion
of the following form is obtained (Medanic (1983)):
=
=
e e e e
C C Ce
F X X ?W W W W ? 0
F 0 X 0 X 0 ? (2.25)
where ( 1) ( 1 ) ( 1) ( 1) ( 1), , ,p p p n n n n n× × + + × + + × +∈ ∈ ∈ ∈CW W X ?¡ ¡ ¡ ¡ , ( 1) ( )n p r p+ + × +∈eX ¡ . If projection
equation of (2.13) is applied to retain the eigenvalues in ? which is the original
closed loop spectrum of linear quadratic solution in (2.8) and(2.9). In order to apply
projective control to the system of (2.16), the subspectrum relation which is shown
below is used:
where,
,
=
=
= =
pe
r
e er er
per
r
W W ? 0F
X X 0 ?F X X ?
W ? 0X ?
X 0 ?
(2.26)
14
In the above equations, p? and r? are disjoint subspaces of spectrum ? . For each of
the spectrums the following equations can be proposed (Medanic (1985)):
=r r r
p p p
FX X ?FX = X ? (2.27)
If the extended sub – spectrum is used for controller design by means of projective
control theory; the projection matrix can be defined as:
Where, =
=
-1e er er er er
per
r
P X (C X ) C
I 0C
0 C
(2.28)
where erC is a relation such that ˆ =e er ex C x , in which ˆ ex is the state vector consists of
compensator and available states from original system in (2.6). In Medanic (1985) it
is stated that the projection matrix eP is obtained in the following form:
=
p
e r
p r
I 0 0P 0 I 0
N N 0 (2.29)
where ( 1)n r p− + ×∈pN ¡ and ( - 1)n r r+ ×=rN ¡ . The projection in (2.28) is applied to the linear
quadratic solution of the system in (2.16) as shown below:
= − = −-1 Te e e e e e e e eu = - R B M P x K P x K x% (2.30)
Above solution is decomposed according to the expression of u in (2.15), as:
ˆ-1 T -1 Tz yu = - R B M P z - R B MP y (2.31)
In Medanic (1983) the matrices yP and zP are evaluated to be:
, = =
rz y
p r
0 IP P
M N (2.32)
15
The eigenspectrum of the system matrix eF can be further processed as follows. First
of all, the eigenmatrix decompositions in (2.27) are divided into two sub – matrices
as follows:
,
, r pr r ××
= =
∈ ∈
p1r1r p
p2r2
r1 p1
XXX X XX
X X¡ ¡ (2.33)
Combining and expanding equation (2.27) the following can be obtained,
, r r×
= ∈
p r p rp
11 12 p1 r1 p1 r1 11r
21 22 p2 r2 p2 r2
H D 0 W W W W? 0
0 F F X X X X F0 ?
0 F F X X X X¡ (2.34)
From equation(2.34), it is easy to obtain:
p p1 p p
r r1 r r
HW +DX = W ?HW +DX = W ?
(2.35)
The linear system of matrix equations obtained above can be solved for compensator
matrix H and D to have the following:
1−− -1 -1 -1
p p r r1 p1 r1 p1 p
-1p r p r1 p1
H = W (? L? X X )(I-LX X ) W
D = W (L? - ? L)(X - X L) (2.36)
where -1p rL = W W . For the sake of simplicity, the following shorthand notations are
used (Medanic (1985)), in the latter derivations:
-10 r2 r1
-10 r1 p1
0 p2 0 p1
N = X XP =L(X - X L)B = X - N X
(2.37)
The dynamic projective control retains r p+ dimensional subspace of the reference
solution. Mathematically this feature is explained as follows:
16
, = −ce e e e ce e e eA X = X ? A A B K% (2.38)
where spectrum of ceA is expressed as follows:
( ) ( )= ∪ ∪
= +ce r p re
re r 0 0 12
r 22 0 12
? A ? ? ? AA A B P AA = A - N A
(2.39)
The proof of the above fact is given in Medanic (1985). In addition to ones given in
Eq.(2.35), from Eq.(2.34) the relations given below are also obtained:
11 p1 12 p2 p1 p
11 r1 12 r2 r1 r
F X +F X = X ?F X + F X = X ?
(2.40)
Together with (2.37) the above relations lead to reduction of the solution of (2.35) to
(Medanic (1985)):
=
-1 -1p 0 p1 p 0 p1 p r p1 0 p1 p
-1p 0 p1 0 r p 0 12 0 0
-1r r1 r r1
H W ( I + P X ) ? + P ( X ? - F X ) ( I + P X )W
D = W ( I + P X ) P F - ( ? + P F B )P
F = X ? X
(2.41)
The system matrix rF can also be defined as r 11 12 0F = F +F N and from (2.40) the
following definition is valid:
p1 p r p1 12 0X ? - F X = F B (2.42)
Using (2.41) and the above result the compensator system matrices are solved to give
the following (Medanic (1985)):
1−′ ′
′
′
0 p 0 12 0
0 0 r 0 0
p 0 p
p 0
-1p p 0 p1
H = ? + P F BD = P F - H PH = W H WD = W D
W = W ( I + P X )
(2.43)
17
The matrix ′pW can be chosen arbitrarily and Medanic (1985), suggests that it can be
an identity matrix. The feedback parameters are defined as:
ˆ ,
ˆ ,
= = =
r ry y y
r 0 0 0
z z z0
I IK =KP P
N N - B P0
K =KP PB
(2.44)
The free parameter matrix 0P is important in projective control theory. In Wise (1991)
it is advised to select the value of 0P to have a stable residual dynamics which is
defined by(2.39). The reA matrix should have eigenvalues with negative real parts
with the selected free parameter matrix. Providing the stated fact, the 0P matrix can
be arbitrarily chosen.
In the design by projective control theory, the first step is to decide the structure of
the control system which includes the controlled variables and the available states
for feedback. Next a full state feedback controller is designed to have the reference
solution. Static projective control approach is applied to the reference solution to
obtain the output feedback control elements. If the achieved closed loop
performance is adequate then there is no need of a dynamic compensator. If the
closed loop is unstable or unsatisfactory in the view of design requirements and
performance then a low order dynamic compensator should be designed by means
of the theoretical basis given in this chapter. In the methodology, it is advisable to
retain the dominant poles of the reference solution in static projective control design.
This fact is also valid for dynamically compensated version. The residual spectrum is
the additional eigenvalues obtained as a result of order increase due to compensator.
The places of the eigenvalues are not very critical (any stable combination is
sufficient) however they can affect the final characteristics of the design and
magnitude of the controller and compensator gains.
18
In the Chapter 4 and Chapter 5 theory presented in this chapter is applied to missile
autopilot design. The design will be supported by detailed numerical examples and
flight simulations are also given for verification of the design.
19
CHAPTER 3
MODELLING OF AERODYNAMIC MISSILES
3.1. Introduction
In this section, a brief introduction to the missile model to which the projective
control theory is applied is given. The modeling starts with the derivations of
kinematical and dynamical equations of motion. These equations of motion are all
non – linear so they should be linearized. To perform linearization the aerodynamic
properties of a generic missile should be known. A summary of aerodynamic
modeling will also be given in this chapter. Lastly, the linearized model will be
presented. The considered structure is an aerodynamic missile, the motions of which
are controlled through the deflection of small fins connected to either ahead of the
center of gravity (near to nose of the missile) or at the tail of the missile. If the fins are
constructed near to the nose the missile is called as canard controlled. The other one
is called tail controlled missile. The missile modeled in this thesis is a canard
controlled system.
20
3.2. Aerodynamic Missile Modeling
3.2.1. The Equation of Motion
The kinematical and dynamical equations of motion of a missile are derived from
basic principles of mechanics. The dynamical part of the equations is generally
derived from Newton’s Second Law of Motion. The kinematical equations based on
the basic coordinate transformations. The detailed derivation of the equations of
motion is presented in Atesoglu (1997), Blackelock (1991) and Mahmutyazicioglu
(1994). So only the results of those derivations will be presented in this chapter.
Before going further, the reference coordinate axis system is given in Figure 3-1.
Figure 3-1 Missile coordinate system including system variables
21
Figure 3-2 The forces and moments acting on the missile body
The aerodynamic missile equations can be presented in two main parts namely the
kinematical and dynamical equations. Those equations in turn can be also divided
into two sub sections, rotational and translational. According to this classification the
equations of motion can be presented as given below:
Translational Dynamic Equations:
s
s c
c c
x
y
z
Fu g rv qw
mF
v g ru pwmF
w g qu pvm
θ
φ θ
φ θ
= − + −
= + − +
= + + −
&
&
&
(3.1)
where the triple ( ), ,u v w represents the velocity of the missile in three directions on
the coordinate axis system mounted on the center of gravity of the missile body
(forward, lateral and downward velocities).
22
Rotational Dynamic Equations:
( )
( )
( )
y z
x x
z x
y y
x z
z z
I ILp qrI I
I IMq prI I
I INr pqI I
−= +
−= +
−= +
&
&
&
(3.2)
where ( ), ,p q r are the angular rates of the missile in three directions (roll, pitch and
yaw rates) on the body axis system. They can be measured directly by inertial
sensors.
Translational Kinematical Equations:
( ) ( )( ) ( )
x uc c v s s c c s w s s c s sy uc s v s s s c c w c s s s cz us vs c wc c
θ ψ φ θ ψ φ ψ φ θ ψ φ ψθ ψ φ θ ψ φ ψ φ θ ψ φ ψθ φ θ φ θ
= + − + += + + + −= − + +
&&&
(3.3)
where ( ), ,x y z is the position of the missile referenced to the projection of the point
where the missile is released from the aircraft on the earth (reference coordinate
system).
Rotational Kinematical Equations:
p qs t rc t
qc rss c
q rc c
φ φ θ φ θ
θ φ φφ φ
ψθ θ
= + +
= −
= +
&&
& (3.4)
In the four sets of equations above, the abbreviation ( )s θ and ( )c θ stands for ( )sin θ
and ( )cos θ respectively. The triple ( ), ,φ θ ψ is the orientation of the missile according
to the reference coordinate system (roll, pitch and yaw angles). Figure 3-1 explains
the conventions of angular positions (attitudes), angular velocities, translational
positions and velocities together with the missile coordinates. The Figure 3-2,
presents the directions of moments and forces applied on the body of the missile.
23
These forces and moments are composed of both the external aerodynamic effects
and the thrust. The forces and moments are assumed to be applied on the center of
gravity of the missile. The coordinate system is also attached on the center of gravity
of the missile and is assumed to be moving with it. The equations above constitute
the 12 state equations representing the motion of a short range missile. In equations
(3.1) and (3.2) the triples ( , , )x y zF F F and ( , , )L M N are the applied external forces. They
include aerodynamics, thrust and other external effects. Generally, this expansion is
expressed as follows:
==
A E
A T E
F F + T + FM M +M + M
(3.5)
In the above equations, the terms AF and AM represents the aerodynamic forces and
moments, T and TM denotes the thrust force and moments (coming from the
misalignment effects), lastly the terms EF and EM stands for other external effec ts the
most important of which is the uncertainty in the position of the center of gravity.
This misalignment comes from the fabrication errors and is modeled as an
uncertainty in this thesis. The acceleration variable is accepted as an output variable
and the forward, lateral and downward accelerations are expressed in terms of the
total forces as shown below:
, ,yx zx y z
FF Fa a am m m
= = = (3.6)
This assumption excludes the effect of gravity. As a result of this fact, the effect of
gravity should be compensated if the acceleration is selected as a controlled variable.
This is generally performed in the acceleration commands entering into the autopilot.
24
3.3. Introduction to Missile Aerodynamics
The modeling of the motion of the missile in the real atmosphere requires the
concept of aerodynamics. This is also very important in relating the effect of the
control action on the missile. As it is known the control action is given to the missile
by the fins attached on the its body and they have aerodynamic properties.
3.3.1. The Expression of Forces and Moments
In the aerodynamic modeling of aerodynamic forces and moments it is convenient to
make them non – dimensional (Atesoglu (1997)). To do that, the forces and moments
are expressed as a product of dynamic pressure, area and diameter as given below:
ax x
ay d y
az z
l
d m
n
F CF Q A CF C
L CM Q Ad CN C
= =
(3.7)
The left hand side of the equations are the vectors AF and AM in expanded form. The
triples ( , , )x y zC C C and ( , , )l m nC C C are non – dimensional form of the aerodynamic
forces and moments. The term dQ is the dynamic pressure which is expressed as:
212dQ Vρ= (3.8)
The term ρ is the air density and in this study its variation is taken according to the
ICAO standards as explained in Atesoglu (1997). The same standard will be used for
expressing the temperature variation.
In aerodynamic analysis of missile systems there are two important angles defined
between velocity vectors which are used in expressing aerodynamic forces and
moments. These are the angle of attack ( )α and sideslip ( )β and they are defined as
shown below:
25
1
1
tan
sin
wuvV
α
β
−
−
= =
(3.9)
The geometric interpretation of the angles is shown in Figure 3-3. The detailed
explanation of the aerodynamic meanings is given in Atesoglu (1997) and McLean
(1990).
Figure 3-3 The angle of attack and sideslip
3.3.2. The Aerodynamic Coefficients
The non – dimensional aerodynamic coefficients are functions of motion variables,
aerodynamic angles and the fin deflection angles (also called as control surface
deflection angles). One of the important variables from the category of motion
variables is the Mach number and is defined as:
VMacha
= (3.10)
where a is the speed of sound:
a kRT= (3.11)
26
where k is the specific heat ratio, R is the universal gas constant and T is the
ambient temperature.
The non-dimensional forces and moments are functions of angle of attack and
sideslip, angular velocities and control surface deflections (related to the total effect
of the fin deflections on the motion about each axis). The relationship is derived
through Taylor series expansion as shown below:
. .
i i i ii e r
e r
i i i ia
a
C C C CC
C C C Cp q r H O Tp q r
α β δ δα β δ δ
δδ
∂ ∂ ∂ ∂= + + + +
∂ ∂ ∂ ∂∂ ∂ ∂ ∂
+ + + +∂ ∂ ∂ ∂
(3.12)
For each of the aerodynamic moment and force coefficients, some of the terms in
series expansion are omitted since their effect is considerably smaller than the others.
In (3.12) each partial derivative represents a special meaning in aerodynamics and
written in a shorthand notation according to those meanings. The detailed
explanation of the aerodynamic coefficients is presented in Atesoglu (1997),
Blackelock (1991), Mahmutyazicioglu (1994) and McLean (1990). As a result, the
relevant expansion comes out to be:
2
2
2
2
2
r r
q e
p a
q e
r r
x d
y y y y r
z z z z e
l l l a
m m m m e
n n n n r
C Cd
C C C r CVdC C C q CV
dC C p C
VdC C C q CV
dC C C r C
V
β δ
α δ
δ
α δ
β δ
β δ
α δ
δ
α δ
β δ
=
= + +
= + +
= +
= + +
= + +
(3.13)
The coefficients seen in (3.13) are evaluated at several points in the operating range
of the missile. For each point the nonlinear missile model is linearized, to have a
model applicable to controller design methods in scope of this study.
27
3.3.3. The Linearization of the Missile Model
The practical controller design in aerospace studies involves linearization of the 12
state non – linear differential equations. The linearization procedure is based on
certain assumptions which are stated below:
1. For each point of linearization, a constant ambient temperature ( )T ,
ambient density ( )ρ and velocity (Mach Number) is assumed.
2. Angle of attack and sideslip ( ),α β and the fin deflections ( ), ,a e rδ δ δ are
assumed to be small ( ), ,15 , 15 , 15e a rα β δ< < <o o o .
3. Rolling motion is constant and very small ( 5 , 5 /secpφ ≤ ≤o o ).
4. Components of the gravitational acceleration are assumed to be external
disturbances.
Since the angle of attack and sideslip is very small the trigonometric identities are
dropped to give the following:
,w vu V
α β≅ ≅ (3.14)
As it can be understood from (3.14), downward and lateral velocities ( ),v w are very
small compared to the total velocity ( )V and to the forward velocity( )u respectively
(assumption 1). That is, ,v u w u= = . Remembering that 2 2 2V u v w= + + , and using
the above result one can easily conclude that V u= .
One of the assumptions was the constancy of the total velocity V for each instant of
linearization. Since u V= is constant, u& vanishes and hence the first equation in (3.1)
drops. As the gravitational acceleration is accepted as a disturbance (assumption 4),
the remaining translational dynamic equations will then simplify to:
y
z
Fv ru pw
mF
w qu pvm
= − +
= + −
&
& (3.15)
28
Using the assumption ( )0, 0pφ ≅ ≅ in the above equations one obtains the following:
y
z
Fv ru
mF
w qum
= −
= +
&
& (3.16)
The missile is mechanically symmetric, so the lateral moment of inertias are taken to
be equal i.e. y zI I= . Using this fact together with the assumption of zero roll rate, the
rotational dynamics in (3.2) will be reduces to:
x
y
z
Lp
IM
qINrI
=
=
=
&
&
&
(3.17)
The smallness of the roll angle enables one to assume that sin 0,cos 1φ φ φ= ≅ ≅ and
thus pitch and yaw components of (3.4) simplifies as:
qrc
θ
ψθ
=
=
&
& (3.18)
In the roll component of (3.4), the product tanr θ is neglected as its magnitude is
assumed to be small. So the overall rotational kinematics is obtained as shown
below:
cos
p
qr
φ
θ
ψθ
=
=
=
&&
& (3.19)
To continue linearization, the expansion of aerodynamic force and moment
coefficients are substituted into (3.7) and the following equations are obtained:
29
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
2
2
2
2
2
2
2
q e
r r
p a
q e
r r
z d z d z d z e
y d y d y d y r
d l d l a
dd m m d m e
dd n n d n r
dF Q AC Q AC q Q ACVdF Q AC Q AC r Q ACV
dL Q AdC p Q AdCV
Q AdM Q AdC C q Q AdCV
Q AdN Q AdC C r Q AdCV
α δ
β δ
δ
α δ
β δ
α δ
β δ
δ
α δ
β δ
= + + = + +
= +
= + +
= + +
(3.20)
The above procedure leads to:
( )
( )2
2
q
e
r
r
d zd zz d z e
d y d yy d y r
Q AC dQ ACF w q Q AC
u u
Q AC Q AC dF v r Q AC
u u
α
δ
β
δ
δ
δ
= + +
= + +
(3.21)
for the aerodynamic forces, proceeding in a similar way, one obtains the following
for the aerodynamic moments:
( )
( )
( )
2
2
2
2
2
2
p
a
q e
r r
d ld l a
d m dm d m e
d n dn d n r
Q Ad CL p Q AdC
u
Q AdC Q AdM w C q Q AdCu u
Q AdC Q AdN v C r Q AdC
u u
δ
α
δ
β
δ
δ
δ
δ
= +
= + +
= + +
(3.22)
Substituting the force and m oment expansions in (3.21) and (3.22) to (3.16) and (3.17)
together with the assumption of decoupling (very small roll motion), the linear state
equations are grouped as:
30
Pitch Plane State Equations:
2
2
2
q e
e
q
d z d zd z
ed md m d
myy y
Q AC d Q ACQ ACu
um um mw wq q Q AdCQ AdC Q Ad C
IuI uI
δα
δα
δ
+
= +
&& (3.23)
Yaw Plane State Equations:
2
2
2
rr
r
r
d yd y d y
rd n d nd
nz z z
Q ACQ AC Q AC du
mum umv vr rQ AdC Q AdCQ Ad C
uI uI I
δβ
β δ
δ
− = +
&& (3.24)
Roll Plane State Equations:
2
0 1 0
02
p ad ld l a
xx
pQ AdCQ Ad C
pIuI
δ
φδ
φ
= +
&& (3.25)
The three second order state equations presented above are used in linear autopilot
designs. The shorthand representations of the equations, will then be:
Pitch Plane State Equations:
( )1 2 1
3 4 2
, ,p p p
Te ep p p
w a a w bw q uq qa a b
δ δ
= + = = x
&& (3.26)
Yaw Plane State Equations:
( )1 2 1
3 4 2
, ,y y y
Tr ry y y
v a a v bv r ur ra a b
δ δ
= + = = x
&& (3.27)
Roll Plane State Equations:
( )0 1 0
, ,0
Ta a
pp u
L p Lp δ
φφδ φ δ
= + = = x
&& (3.28)
31
3.4. Modeling of Other Effects
In the previous sections the aerodynamic behavior of the flying munitions is
investigated. The information up to now may be adequate for the design purposes
but there are some important facts to be considered when simulating. First of all the
missile used in the application of the theory has a motor for supplying adequate
energy to the missile body in order to reach high speeds. The application of thrust
brings important effects that should be modeled and used in the simulations. In
addition to thrust, the manufacturing errors lead to a misalignment in the center of
gravity. This may seem an unimportant effect but it modifies the moments acting on
the missile and can make the missile unstable.
3.4.1. Modeling of Thrust and Its Effects
The thrust data is generally obtained from the measurement of the force produced
by the motor of the missile. The thrust used in this application is continuously
varying and lasts for 2.71 seconds and has a variation as shown in the Figure 3-4. The
fuel is a considerable add on the missile’s weight. Since the fuel is continuously
decreasing the mechanical equilibrium on the missile causes the position of the
center of gravity of the missile to change, until the fuel is finished. Together with this,
the inertia of the missile also changes in a similar way. The change is directly related
to the impulse ( )S t which is the integral of the thrust force ( )T t . Mathematically,
0
( ) ( )t
S t T t dt= ∫ (3.29)
32
Figure 3-4 The variation of thrust with time
The center of gravity, mass and inertia properties of the missile changes with the
thrust according to the following mathematical identities,
0 10
0 10
0 10
( ) ( )
( ) ( )
( ) ( )
total
i ii i total
cg cgcg cg total
m mm t m S tS
I II t I S tSx x
x t x S tS
−= −
−= −
−= −
(3.30)
In the above expressions, the position of center of gravity cgx is measured from the
nose. The term iI represents general moment of inertia. The change in the center of
gravity of the missile also affects the aerodynamic moments acting on the missile.
The mathematical expression of this effect is (Atesoglu (1997)):
( )
( )
1
1
cg cg az
cg cg ay
M M x x F
N N x x F
′ = + −
′ = − − (3.31)
The other important fact about the application of thrust is the misalignment of the
thrust force. In ideal conditions the thrust force is accepted to be perfectly aligned
with the x- axis of the body coordinate system. However, in real application the
thrust gas jet has some dispersion and because of that thrust force has a
33
misalignment. This phenomenon can be described graphically as in Figure 3-5. The
thrust misalignment is modeled by defining two angles as shown in Figure 3-5.
Figure 3-5 The thrust misalignment phenomenon
As it can be seen from the interpretation the thrust vector T makes an angle of 1δ
with the x – axis. The vector has a projection on the y – z plane denoted by ′T and
from the figure its mathematical representation is ( )1sinT T δ′ = . According to those
the components of the thrust vector along the three axes is defined as follows:
1
1 2
1 2
cos( )sin( )sin( )sin( )cos( )
x
y
z
T TT TT T
δδ δ
δ δ
== −
= −
(3.32)
These components also causes additional moments on the missile body and they are
expressed as:
34
(1 )
(1 )t z cg
t y cg
M T xN T x
= − −
= − (3.33)
The above moments are added to the moments defined in (3.31) which are produced
due to the change in center of gravity position as fuel is consumed.
The application of thrust misalignment to the missile flight simulation is performed
by giving a normally distributed random angle in the beginning. The example
application will be given in chapter seven.
3.4.2. The Uncertainty in the Center of Gravity Function
The mechanical manufacturing techniques can be as precise as possible however this
is not enough for having a 100% percent certainty in the position of center of gravity
of the missile body. There are non – symmetrical parts in the missile, the mechanical
structure may not be uniform and also the fuel’s mass may be different to a small
extent. Those and similar other manufacturing faults cause the center of gravity of
the missile to be different from the ideal one. Since the design and evaluation of
aerodynamic coefficient is performed assuming the center of gravity is in its ideal
position the effect of this uncertainty must be taken into account in the simulations.
This effect is modeled as a uniformly distributed error cgδ added to the nominal
(ideal) position nomc gx of the center of gravity
(1 )act nomcg cg c gx x δ= + (3.34)
The deviation in center of gravity position is randomly given in the beginning of the
simulation. Since the nominal value (value with no uncertainty) of the center of
gravity position is constant after thrust phase, the initially assigned deviation is used
as a constant perturbation in center of gravity position throughout the simulation.
35
CHAPTER 4
AUTOPILOT DESIGN BY STATIC PROJECTIVE CONTROL
4.1. Introduction
The theory presented in Chapter 2 can be used in autopilot design to the missile
model presented in Chapter 3. The autopilot can be commanded to maintain either
the desired accelerations or the angular velocities according to the commands
coming from the guidance system. There are two approaches in missile control, the
first of which is the Bank – To – Turn (BTT) control which takes the roll and yaw
channels coupled and separate the pitch system. This type of autopilots causes the
missile to roll while turning about its z – axis. The other type is called as Skid – To –
Turn (STT) control which decouples all the channels (roll, pitch and yaw). In Skid –
To – Turn approach the roll motion should be appreciably small in order to have
decoupled system of equations of motion as stated in Chapter 3. The equations
presented in that chapter will be used in developing the control system. In this
chapter two types of autopilots (rate and acceleration controller) are tried to be
designed by using static projective control method presented in Chapter 2. The
success of the method in both types of autopilots is investigated in this chapter.
To design any control system by using projective control theory a reference control
structure should be designed. In this study, the reference control structure is the
linear quadratic full state feedback control.
A reference solution is obtained and it is utilized in both static and dynamic
projective control design structures.
36
4.2. Autopilot Design by Linear Quadratic Control Theory
4.2.1. Mathematical Framework for Linear Quadratic Controller Design
The theoretical background of the linear quadratic full state feedback control is
presented in Chapter 2. The linear quadratic control provides optimal and robust
results if all the states are available for feedback. Linear quadratic design brings
integral control to the autopilot and either the acceleration or the rate command
error is integrated according to the design requirements. A general interpretation
was given in Figure 2-1.
According to the framework presented in Figure 2-1, the state equations given in
(3.26) and (3.27) can be augmented to obtain state equations accepting the integral of
the error as an additional state variable. Taking yr and y of Figure 2-1 as rq and q ,
the equation (2.2) is rewritten for the pitch rate autopilot as shown below:
q rq qε = −& (4.1)
and accordingly the augmented design state equations are obtained in the form of
equation (2.3).
For the pitch autopilot:
( )
4 3 2
2 1 1
0 1 0 0 10 0
00ˆˆ ˆ
,
q qp p p
e r
p p p
prt prt prt prt prt prtr
Tprt prtq e
q a a q b qw wa a b
u q
q w u
ε εδ
ε δ
− = + +
= + +
= =
x A x B G
x
&&&
& (4.2)
the resemblance in the pitch and yaw models enables one to form the yaw plane
design equation just like equation (4.2) as presented below:
37
( )
4 3 2
2 1 1
0 1 0 0 10 0
00ˆˆ ˆ
,
r ry y y
r r
y y y
yrt yrt yrt yrt yrt yrtr
Tyrt yrtr r
r a a r b rv va a b
u r
r v u
ε εδ
ε δ
− = + +
= + +
= =
x A x B G
x
&&&
& (4.3)
in the design the reference (set point) of the autopilot is accepted as a constant
valued input and in this study it is chosen to be a unit step.
Similar to the case of pitch rate autopilot the equation (2.2) can be rewritten taking
the pitch plane acceleration za as a controlled variable as shown below:
za zr za aε = −& (4.4)
From the general state equation and acceleration definition of (3.6), the linear
acceleration equations can be derived neglecting the gravity, as:
y
y
zz
Fa v ur
mF
a w uqm
= = +
= = −
&
& (4.5)
Substituting v& and w& from the linearized state equation the design state equations
can be formed for pitch and yaw planes as shown in below,
( )
2 1 1
4 3 2
2 1 1
2 1
4
0 10 0
00
ˆˆ ˆ
,
0
0
z z
z
y
p p pa a
p p pe zr
p p p
pac pac pac pac pac paczr
Tpac paca e
y ya
y
a u a bq a a q b aw wa a b
u a
q w u
a u a
r av
ε εδ
ε δ
ε
− + − − = + +
= + +
= =
− − −
=
x A x B G
x
&&&
&
&&&
( )
1
3 2
2 1 1
1000
ˆˆ ˆ
,
y
y
ya
y yr yr
y y y
yac yac yac yac yac yacyr
Tyac yaca r
b
a r b ava a b
u a
r v u
ε
δ
ε δ
− + +
= + +
= =
x A x B G
x
&
(4.6)
38
In Chapter 2 it was stated that the linear quadratic design is performed using the
error dynamics equation in (2.5), so the reference input in equations (4.2), (4.3) and
(4.6) will vanish since it is assumed to be constant. Then, one obtains the following,
for the rate autopilot:
whereas,
4 3 2
2 1 1
4 3 2
2 1 1
0 1 0 0
00
0 1 0 00
0
z
q qp p p
e
p p p
r ry y y
ry y y
a
q a a q bw wa a b
r a a r bv va a b
qw
ε εδ
ε εδ
ε
− ∆ ∆ ∆ = ∆ + ∆ ∆ ∆
− ∆ ∆ ∆ = ∆ + ∆ ∆ ∆
∆ ∆ ∆
e e e
e e e
&&&&&&
&&&
2 1 1
4 3 2
2 1 1
2 1 1
4 3 2
2 1 1
000
00
0
z
y y
p p pa
p p pe
p p p
y y ya a
y y yr
y y y
a u a ba a q b
wa a b
a u a br a a r bv va a b
εδ
ε ε
δ
− + − −∆ = ∆ + ∆ ∆
∆ ∆ − − − − ∆ = ∆ + ∆ ∆ ∆
e e e
e e e
&&&
(4.7)
for the acceleration autopilot. The subscript e indicates that the state vector contains
the error variables as defined in (2.5) and ∆ term is used such that ( ) ( )i i itδ δ δ∆ = − ∞ .
The control gains for full state feedback reference solution are computed using the
system equations of(4.7). A specific example will be given for one aerodynamic
condition in order to demonstrate the linear quadratic design for the missile model
discussed in Chapter 3.
4.2.2. Design Example
In this section a linear quadratic controller design will be performed with numerical
and graphical illustrations. The design will be performed for both rate and
acceleration autopilots and graphical results will be presented. The design will be
performed at a single aerodynamic condition where 0.86Mach = and 5000h m= . The
aerodynamic coefficients defined in (3.13) will have the following values:
39
10.39220.266
1764.7
17.1493.2355
185.87
0.41871.449
0.77464
17.1493.2355
185.8710.93220.266
1764.7
e
q
e
q
p
a
r
r
r
r
m
m
m
z
z
z
d
l
l
y
y
y
n
n
n
CC
C
CC
C
CCC
CC
CCC
C
α
δ
α
δ
δ
β
δ
β
δ
= −
= −
= −
= −
=
= −
= −= −
=
= −
=
=
=
=
= −
(4.8)
Having the information that 27961dQ = from equation (3.8) and that the diameter of
the cross section of the missile is 0.227d m= , the elements of the state space equations
can be evaluated as follows:
For the pitch plane:
1
2
3
4
1
2
0.41362
275.120.0571491.0471
21.5129.201
p
p
p
p
p
p
a
aaa
bb
= −
=
= −
= −
=
= −
(4.9)
and the elements of the yaw plane state equations are:
1
2
3
4
1
2
0.41362
275.120.057149
1.0471
21.5129.201
y
y
y
y
y
y
a
aaa
bb
= −
= −
=
= −
=
=
(4.10)
40
The numerical forms of the state equations in (4.7) are evaluated as shown below:
0 1 0 00 1.0471 0.057148 29.2010 275.12 0.41362 21.51
0 1 00 1.0471 0.0571490 275.12 0.41362
q q
e
r r
q qw w
r rv v
ε εδ
ε ε
∆ − ∆ ∆ = − − ∆ + − ∆ ∆ − ∆
∆ − ∆ ∆ = − ∆ + ∆ − − ∆
e e
e e
&&&&&&
029.20121.51
0 0.50883 0.41362 21.510 1.0471 0.057149 29.2010 275.12 0.41362 21.51
0 0.50883 0.413620 1.
z z
y
r
a a
e
a
q qw w
rv
δ
ε εδ
ε
∆
∆ − ∆ − ∆ = − − ∆ + − ∆ ∆ − ∆
∆ − ∆ = − ∆
e e
e
&&&&&&
21.510471 0.057149 29.201
0 275.12 0.41362 21.51
ya
rrv
ε
δ
∆ − ∆ + ∆
− − ∆ e
(4.11)
The controller design and analysis requires the computation of the open and closed
loop transfer functions. For general system representation of (2.6), the plant transfer
function is computed as:
ˆ ˆ
( )( )
w
q
s sG s
sG s
=
=
-1G( ) ( I - A) B
G( ) (4.12)
When the above is applied to the system equations in (3.26) and (3.27) considering
the numerical values in (4.9) and (4.10) the following plant transfer functions will be
obtained:
2
2
2
2
( ) 21.51 8011( )
( ) 1.461 16.16( ) 29.2 13.31
( )( ) 1.461 16.16( ) 21.51 37.38 3668( )( ) 1.461 16.16y
we
qe
za
e
w s sG s
s s sq s s
G ss s s
a s s sG ss s s
δ
δ
δ
−= =
+ +− −
= =+ +
+ += =+ +
(4.13)
for pitch, and
41
2
2
2
2
( ) 21.51 8011( )( ) 1.461 16.16
( ) 29.2 13.31( )( ) 1.461 16.16( ) 21.51 37.38 3668( )( ) 1.461 16.16y
vr
rr
ya
r
v s sG ss s s
r s sG ss s s
a s s sG ss s s
δ
δ
δ
−= =
+ ++= =
+ +
+ += =+ +
(4.14)
for the yaw planes.
In Figure 4-1 and Figure 4-2, the cascaded form of the full state feedback pitch rate
and acceleration autopilots are presented. The symbol ∆ is not used here because the
block diagrams present the operation of the autopilots. The error variables are used
for design purposes only. There are additional transfer functions required for
deriving the open loop transfer functions:
2
2
( ) 1.358 0.6187( )( ) 372.4( ) 0.7366 1.28 125.6( )( ) 0.4557( ) 1.358 0.6187( )( ) 372.4( ) 0.7366 1.28 125.6( )
( ) 0.4557
qw
zaq
rv
yar
q s sG sw s sa s s sG sq s sr s sG sv s sa s s sG sr s s
− −= =−
− − −= =+
+= =−
+ += =
+
(4.15)
The open loop transfer functions defined below are calculated using the above
transfer functions in Figure 4-1 and Figure 4-2 with the control coefficients as
variables:
( )
( )
( )
( )
( )( ) 1
( )( ) 1( )( ) 1
( )( ) 1
z
y
f qw wprateOL
q w w w qw q
f rv vyrateOL
r v v v rv r
f qw w aqpacc zOL
a w w w qw q
y f rv v aryaccOL
a v v v rv r
K G Gq sG
e s s G K G G KK G Gr sG
e s s G K G G KK G G Ga s
Ge s s G K G G K
a s K G G GG
e s s G K G G K
= =+ +
= =+ +
= =+ +
= =+ +
(4.16)
42
1s fK
wK
( )( )e
w ssδ
( )( )
q sw s
rq
qK
qweδqε&
Figure 4-1 The block diagram form of full state feedback rate autopilot
1s
fK
wK
( )( )e
w ssδ
( )( )
q sw s
zra
qK
za( )( )
za sq s
qweδzaε&
Figure 4-2. The block diagram of the full state feedback acceleration autopilot
Lastly the closed loop transfer functions of the systems in Figure 4-1 and Figure 4-2,
and defined below:
1
1
1
1
prateprate OLCL prate
OLyrate
yrate OLCL yrate
OLpacc
pacc OLCL pacc
OLyacc
yacc OLCL yacc
OL
GG
GG
GG
GG
GG
GG
=+
=+
=+
=+
(4.17)
43
The solution of Riccati equation gives a vector of control coefficients which is
denoted generally by K . The single control coefficients in Figure 4-1 & 4-2 are
expanded as follows:
ˆ ˆ,
ˆ ˆ,
fq fap yr q r r
w v
fa fap ya q a r
w v
K KK KK K
K KK KK K
− − = =
− − = =
K K
K K
(4.18)
The numerical state equations in (4.11) are used to minimize the linear quadratic
performance index in (2.10) which is repeated in (4.19) for convenience. This
approach makes the design process simpler and faster also since there is only one
tunable parameter one may develop an automation scheme to complete the design
without the necessity of any manual computation.
2 2, ,
0
2 2, ,
0
( )
( )z y
r r q r e r
a a a a e r
J q dt
J q dt
ε δ
ε δ
∞
∞
= ∆ + ∆
= ∆ + ∆
∫
∫ (4.19)
The design will be performed according to the settling time and basic frequency
response characteristics which are gain and phase margins. The margins and settling
time are plotted against the weighting term ( rq or aq ) in (4.19), and the value of the
weighting term satisfying the design criteria will be selected. The numerical values
of the design criteria for the case of rate autopilots were taken as:
0.3 0.4
. 10
. 50
stG M dBP M
< <
>
> o (4.20)
and, for the case of acceleration autopilots, they were:
44
0.6 0.8
. 10
. 50
stG M dBP M
< <
>
> o (4.21)
In the prescribed aerodynamic design conditions, the quadratic state weight ( rq ) is
varying in the range:
0.1,0.2,.....,100rq = (4.22)
for the rate autopilot, while for the other case, namely the acceleration autopilot the
quadratic weight ( aq ) variation is:
0.1,0.2,....,5aq = (4.23)
The variation of settling time and phase margins against the quadratic weights in
(4.22) are presented in Figure 4-3 and Figure 4-4 where gain margin is infinity for the
whole range.
Figure 4-3. Settling time variation with quadratic weight
45
Figure 4-4. Phase margin variation with quadratic weight
The gain margin of the system occurs to be infinite so it will not be a constraint in the
design. From the figures it can be understood that a very large portion of the
variation curves ( )30 90qε≤ ≤ satisfy the design criteria. So for the rate autopilot, a
selection of 65rq = is suitable. According to this selection the control vector for the
pitch rate autopilot is obtained as shown in equation (4.24).
( )ˆ 8.0623 0.6838 0.0018892
8.0623
0.6838
0.0018892
prtrf
rq
rw
K
K
K
= −
= −
= −
=
K
(4.24)
Figure 4-5. The step response of the rate autopilot
46
Figure 4-6 The OL frequency response characteristics of the rate autopilot
The step response and the open loop frequency response characteristics for the
chosen parameter values are presented in Figure 4-5 and Figure 4-6 respectively.
The full state acceleration autopilot is designed in the same way as the rate autopilot.
The variation of the settling time and open loop frequency response with respect to
the quadratic weight aq is shown in Figure 4-7 and Figure 4-8 respectively. In this
case also, the gain margin always goes to infinity.
Figure 4-7 The variation of the settling time for acceleration autopilot design
47
Figure 4-8 The variation of phase margin in acceleration autopilot design
From the variation curve a value of 2.5qε = can satisfy the design requirements with
a settling time about 0.7 seconds, and a phase margin of 77 degrees. With this
selection the control gains are evaluated as given below:
( )ˆ 1.5811 0.23205 0.028137
1.5811
0.23205
0.028137
pacaf
aq
aw
K
K
K
= − − −
=
= −
= −
K
(4.25)
The obtained closed loop system has the step response and the open loop frequency
response characteristics shown in Figure 4-9 and Figure 4-10 respectively. The
obtained results will be utilized in order to achieve static and dynamically
compensated projective control designs. In subsequent sections and chapters,
numerical examples and simulation results will be presented.
48
Figure 4-9 The step response of the full state feedback acceleration autopilot
Figure 4-10 The open loop frequency response of the acceleration autopilot.
49
As it can be understood from(4.11), the symmetry properties of the missile is
reflected to linear quadratic design equations (open loop state equations). All entries
of the system equations in(4.11) are equal for pitch and yaw planes in magnitude
and there is only a sign difference. This feature enables the designer to make only a
single design for yaw autopilot (for one linearization point) to determine the signs of
the yaw autopilot gains. For all other linearization points, the yaw autopilot gains
are directly obtained from pitch autopilot design. For the yaw rate autopilot the
following result is obtained:
( )ˆ 8.0623 0.6838 0.0018892
8.0623
0.68380.0018892
yrt
rf
rrrv
K
KK
= −
=
=
=
K
(4.26)
while, for the yaw acceleration autopilot design, one gets:
( )ˆ 1.5811 0.23205 0.028137
1.5811
0.23205
0.028137
yac
yf
yryv
K
K
K
= − −
=
=
= −
K
(4.27)
4.3. Autopilot Design by Static Projective Control
In the previous section a rate and acceleration autopilot is designed by means of full
state feedback linear quadratic controller design theory in order to have a reference
solution to the projective control problem.
In missiles, all system variables can not be measured due to the practical limitations.
Most important one is the lack of direct availability of all state variables of the design
models in equation (4.7). The only available system variables are the angular body
rates and accelerations ( , , , , ,x y zp q r a a a ) through inertial sensors. The others (body
velocities and positions) are computed in the navigation computer to be used by
guidance systems. Due to the complex computation algorithms, the noise level of the
signals produced by the navigation computer is comparably higher than the inertial
50
sensor signals. Because of that, in autopilots the outputs of navigation computer are
not used for feedback purposes. So in conjunction with the state equations presented
in (4.11) the available states are ( )q qε or ( )r rε for the pitch and yaw rate
autopilots and ( )za qε or ( )ya rε for the pitch and yaw acceleration autopilots. This
results in a common output matrix for all autopilots to be expressed as follows:
1 0 00 1 0
=
rC (4.28)
After determining the available outputs, the next step in the procedure is to
determine the subset of the eigenvectors of the closed loop full state feedback design
that is denoted by rX in (2.13).
The first step in determining that sub – eigenmatrix is to examine the eigenvalues of
the closed loop system. The application here will only be given for the pitch
autopilot as the result is the same for yaw autopilot due to the symmetry properties.
For the rate autopilot design the closed loop eigenvalues (spectrum) are found to be:
0.45462 0 0
0 10.507 11.207 00 0 10.507 11.207
rt jj
− = − + − −
? (4.29)
For the case of acceleration autopilot design, the closed loop spectrum is, found as:
37.665 0 0
0 1.9884 12.248 00 0 1.9884 12.248
ac jj
− = − + − −
? (4.30)
The corresponding eigenvectors (or eigenmatrix) are computed to be:
6
6 5 5
1 0.9986 0.99863.986 10 0.0367 0.0386 0.0367 0.03868.768 10 0.003467 2.486 10 0.003467 2.486 10
rt j jj j
−
− − −
− = − × − + − − − × − − × − + ×
X (4.31)
for the rate autopilot, and
51
0.09744 0.00642 0.003722 0.00642 0.0037220.12167 0.006995 0.04413 0.006995 0.04413
0.9878 0.999 0.999
ac
j jj j
− − − + = − + − − −
X (4.32)
for the acceleration autopilot.
In the spectrum of the two autopilots two of the eigenvalues are in form of complex
conjugate pairs, and the other is a negative real number. In this occurrence the
retained eigenvalues should be the complex conjugate pairs. Then the retained
eigenvalues for the rate autopilot is ( )10.507 11.207 10.507 11.207j j− + − − while
( )1.9884 12.248 1.9884 12.248j j− + − − for the acceleration autopilot. In Chapter 2 it was
explained that for the complex conjugate eigenvalues to be retained, the eigenvectors
corresponding to the eigenvalues having negative imaginary parts are taken as the
columns of the sub- eigenmatrix rX . It is formed by splitting the real and imaginary
parts of the eigenvalue 1.9884 12.248j− − (for acceleration autopilot) or
10.507 11.207j− − (for rate autopilot). Numerically, this results in the following:
and
5
0.9986 00.03671 0.03860.00347 2.4862 10
0.006422 0.00372160.006995 0.044133
0.999 0
rt
acr
−
= − − − ×
− = − −
rX
X
(4.33)
respectively.
These results are then applied to the projection equations given below:
ˆ
ˆ
rt rt rt
prt p rtpro rt
ac ac ac
pac p acpro ac
=
=
-1r r r r
-1r r r r
P = X ( C X ) C
K K P
P = X ( C X ) C
K K P
(4.34)
52
The numerical result for the case of the rate autopilot is:
( )7.5219 0.68415 0prtpro = −K (4.35)
whereas, for the case of acceleration autopilots the projective control gain vector
results in:
( )2.4275 0.10598 0pacpro =K (4.36)
In Chapter 2 it was stated that the static method of projective control does not
guarantee that the resultant closed loop system to be stable. Because of this fact it is a
must to check whether the closed loop poles are in stable locations or not. To
perform this, one should look at the eigenvalues of the matrices:
and
ˆ ˆ ˆ
ˆ ˆ ˆ
prtrt rtpro
pacac acpro
−
−
A B K
A B K
(4.37)
respectively.
The closed loop spectrum of the static projective rate and acceleration autopilots are
found to be:
and
10.507 11.207 0 00 10.507 11.207 00 0 0.42415
1.9884 12.248 0 00 1.9884 12.248 00 0 57.827
prtpro
pacpro
jj
jj
− + = − − −
− + = − −
?
?
(4.38)
respectively.
The pre – determined closed loop poles (for the case of the complex conjugate
eigenvalues) are retained successfully. It is clearly understood that the static
projection controlled acceleration autopilot has unstable closed loop poles. So this
53
shows that static projection control approach is not useful for a Skid – To - Turn
acceleration autopilot design to the aerodynamic missile considered in this study.
The rate autopilot design shows successful results for stability and closeness of the
poles to the reference design (LQSF rate autopilot). The closed loop poles of the static
rate autopilot are almost the same as that of its full state feedback reference. The
design of the acceleration autopilot requires the usage of a dynamic compensator.
For visualization, the block diagram of the rate autopilot is shown in Figure 4-11.
The block diagram of the static projective acceleration autopilot is not presented
since it is not available for practical use.
ε& εd t∫
Figure 4-11 The block diagram of the static projection rate autopilot
In Figure 4-11 the control parameters have the values given in (4.35) as in below
7.5219
0.68415
phpq
K
K
= −
= − (4.39)
Up to here, the given design example is mainly based on the pitch autopilots. By
using the symmetry property of the missile model the yaw autopilot gains are
directly computed as follows:
7.5219
0.68415
yhyq
K
K
=
= (4.40)
The results of the autopilot applied to the linearized system are shown as step and
open loop (OL) frequency responses in Figure 4-12 and 4-13 respectively. The step
54
response of the static projective rate autopilot (SPRA) is superimposed on the full
state feedback (LQSF) step response for comparison purposes.
Figure 4-12 Step responses of full state feedback and static projective autopilots.
Figure 4-13 Open loop frequency response of the static projective autopilot.
55
The time response characteristic of the designed autopilot has a small discrepancy
from the original full state feedback design. This is a normal outcome because the
projective controlled autopilot (SPRA) and its reference (LQSF) are not exactly the
same even the closed loop spectrums are almost equal. The zero dynamics of the
projective autopilot is different to some extent since no manipulation can be
performed on it. The open loop frequency response characteristics satisfy the design
requirements. In general, considering the design criteria, it can be said that the
design is successful.
4.4. Non – Linear Simulations
4.4.1. Introduction
In this section a simulation for the static projective rate autopilot is done. The
simulations model the flight of the missile released from a fighter aircraft. The
simulation in this section is only for verification purposes. The program used for this
purpose contains neither actuator models nor any uncertainties. The simulation
program tabulates the aerodynamic coefficients for each linearization point as
functions of total missile velocity. It is interpolated according to the changing
velocity to evaluate the instantaneous values of the aerodynamic coefficients
throughout the simulation.
Just like the aerodynamic coefficients, the autopilot gains are also tabulated for each
linearization point as functions of velocity and altitude of the missile as shown
below:
( , )
( , )
pq
ph
K f M h
K f M h
=
=
%% (4.41)
In this study there are 108 linearization points and the autopilot gains are linearly
interpolated. In Figure 4-14 a 3-D plot of variation of one of the autopilot gains ( )phK%
is plotted against changing altitude h and velocity M .
56
Figure 4-14 The variation of autopilot gain phK% agains altitude and velocity
The instantaneous values of the rate autopilot gains are evaluated through
interpolation on the gain tables in order to make the autopilots to produce the
necessary commands according to the changing flight conditions. This approach is
called as gain scheduling in control literature. The initial and final conditions of the
missile in this simulation are listed below:
0
0
0.940000 121926000020000
f
f
V Machh ft mx my m
== ==
=
(4.42)
The roll angle control system (roll autopilot) used in the following and other
simulations are presented in Appendix B. The guidance system is based on the
proportional navigation algorithm presented by Tiryaki (2002).
4.4.2. The Results of Simulations
In this section, the results of the flight simulation of the static projective rate
autopilot are presented as figures.
57
Figure 4-15 Missile range (in x – direction)
Figure 4-16 Lateral range of the missile (in y – direction)
Figure 4-17 Angle of attack variation
58
Figure 4-18 Elevator (pitch plane control surface) deflection
Figure 4-19 Angle of sideslip variation
Figure 4-20 Rudder (yaw control surface) deflection
59
Figure 4-21 Pitch rate command and output
Figure 4-22 Yaw rate command and output
Figure 4-23 Mach number variation
60
Figure 4-24 Roll angle variation
Figure 4-25 Roll rate variation
Figure 4-26 Aileron (roll plane control surface) deflection
61
From the nonlinear simulation results it can be clearly seen that the assumptions are
almost satisfied. First of all, the decoupling assumption which requires the value of
roll motion to be small ( 5 , 5 /secpφ ≤ ≤o o ) is perfectly satisfied. The second
assumption was about the angle of attack and sideslip definitions in (3.9) the
linearization of which requires their values to be small enough to drop the
trigonometric functions. Numerically this requirement is 15α < o and 15β < o . The
graphs show that the variations of those variables fall much below the
recommended limits. Same limitations are also required for the fin deflection
commands generated by the autopilot, and simulation results also provide those
requirements. So, it can be concluded that the linearization approach of Section 3.3.3
is perfectly applicable to static projective rate autopilot design and gives satisfactory
results.
The static projective controlled rate autopilot is successful in keeping its reference.
The closed loop poles are almost the same with the linear quadratic rate autopilot
design. The acceleration autopilot came out to be unstable. This is not a surprising
result because one of the eigenvalue is free and sits in right hand plane. Because of
that, in the next chapter a dynamic compensator will be designed to make it a stable
autopilot.
62
CHAPTER 5
AUTOPILOT DESIGN BY DYNAMIC PROJECTIVE CONTROL
5.1. Reasons of Dynamic Projective Control
In the previous chapter the design of missile autopilots by means of uncompensated
method of projective control, in other words, the static projective control is
introduced. An uncompensated controller is always preferred because its
computational complexity is less than the other one. However, it is an unfortunate
fact that the static projective control is not always successful in obtaining a stable
product, just as in the case of acceleration autopilot design in the last chapter.
Because of that, dynamic compensators are necessary to stabilize the closed loop
design. In this chapter, the unstable acceleration autopilot of Chapter 4 is to be
stabilized by adding dynamic compensators as an application of the theory
presented in Chapter 2. For comparison purposes, a dynamically compensated rate
autopilot is also designed.
5.2. Rate Autopilot Design Using Dynamic Projective Control
The design of a dynamically compensated rate autopilot design has similar
procedure like in the static projective control design. From the theory, the difference
is the selection of a free parameter which affects the additional pole brought to the
closed loop spectrum. In static design, the eigenvalues of the reference spectrum
other than the retained ones by projec tion are placed by the dynamic compensators
as discussed in Chapter 2.
As it can be recalled from the previous chapter the retained eigenvalues are two
complex conjugate pairs so 2r = . Than there is only a single real valued eigenvalue
for placement using dynamic compensation so the value of p in (2.15) is one. So a
63
first order dynamic compensator of the following form is proposed for rate autopilot
design,
( )( )
( )
1 2
1 2
2 1 1 2 2 1, ,
prt prtc
prt prte
prt prt prt
prt prt prt
T
q
prt prt
H
K
d d
K K
q
ω
ω ω
δ ω
ε× × ×
= +
= − −
=
=
=
∈ ∈ ∈
T
T
T
c
c
c
D T
K T
D
K
T
D T K
&
¡ ¡ ¡
(5.1)
In the above system representation, the variable ω represents the compensator state,
the vector T is the available states from the missile model (it is the compensator
system input also), prtT
K and prtKω are the compensator control output gains.
As in the static projective control design, the results are performed for again pitch
plane autopilots and the results of the yaw autopilots are only given in order to note
the sign changes in the autopilot gains.
The first step of the procedure for the design of dynamic compensators is to
determine the second disjoint sub - spectrum p? stated in (2.26) which corresponds
to the non – retained subset of the full state feedback reference spectrum. In addition
to this, the eigenvectors pX (sub – eigenmatrix) corresponding to p? is computed.
This is performed just like in the case of static projective control (the selection of the
matrix rX ). Since the remaining eigenvalue of the reference spectrum in (4.29) is real,
the ultimate solution is 0.45462prt = −p? and the corresponding eigenmatrix consists of
single eigenvector column and it is the first column of (4.31) then:
6
6
1 13.986 10 08.768 10 0
prt −
−
= − × = − ×
pX (5.2)
the prt index is added in order to notify that the computation is performed for the
rate autopilot.
64
In Chapter 2 the derivation requires that the two sub – eigenmatrices should be
partitioned into two other sub – matrices as in (2.33) and the outcome of this division
considering 2r = is:
( )
1, 0
00.9986 0
, 0.00347 00.03671 0.0386
prt prt
prt prt
= = = = − − −
p1 p2
r1 r2
X X
X X (5.3)
The partition of the closed loop system matrix ˆ ˆ ˆ= −F A BK is done according to (2.34)
as shown below:
( )
0 1 0235.42 21.015 0.0019823173.42 289.83 0.45426
0 1235.42 21.015
00.0019823
173.42 289.83
0.45426
prt
prt
prt
prt
prt
− = − − − −
− = − = −
= −
= −
11
12
21
22
F
F
F
F
F
(5.4)
Substituting the sub – eigenmatrices in (5.3) into (2.37), the matrices prt0N and prt
0B are
computed as:
( )286.04 0.18426
1.0025
prt
prt
= − −
=0
0
N
B (5.5)
Continuing the procedure, equation (2.39) requires the augmented plant system
matrix A to be partitioned just as the closed loop system matrix F , as shown below:
65
( )
0 1 0 ˆ ˆˆ 0 1.0471 0.057149 ˆ ˆ
0 275.12 0.413620 1ˆ0 1.0471
0ˆ0.057149
ˆ 0 275.12ˆ 0.41362
prt prtprt
prt prt
prt
prt
prt
prt
− = − − = − − = −
= −
=
= −
11 12
21 22
11
12
21
22
A AA
A A
A
A
A
A
(5.6)
In equation (2.39) there was a free parameter denoted by 0P (in pitch rate autopilot
design it will be denoted as prt0P ) which effects the residual closed loop spectrum of
the system as defined in (2.39). The residual spectrum consists of eigenvalues
generated due to the order increase with the addition of the dynamic compensator.
In equation (2.39) the additional eigenvalues are denoted by a third spectrum ( )re? A .
The free parameter prt0P can be chosen arbitrarily, provided that it makes the residual
system matrix defined in (2.39) ( reA ) stable. However, for a high order compensator,
it is a difficult to find a free parameter to stabilize the residual spectrum. In this
study, since the dimension of prtp? is one, the dimension of rA is one and this leads
to the dimension of prtreA to be one as seen from equation(2.39). This enables the
designer to have the advantage of pseudo inversion theory which brings an
approximate inversion by defining directly the residual pole. Mathematically, this
fact is explained as follows:
( ) ( )( )
prt prt prt prt prt
prt prt prt prt prt+ += −
re r 0 0 12
0 0 re r 12
A = A +B P A
P B A A A (5.7)
The + symbol, as a superscript on the matrices given above, represents the operation
of pseudo – inversion and mathematically it is calculated by the following formula:
( ) ( ) ( ) ( )1T Tprt prt prt prt
−+ = 12 12 12 12A A A A (5.8)
66
The above operation can be performed by MATLAB® command pinv(A) which
simplifies the operation. The pseudo inverse of the matrix prt0B is not given here since
it is only a constant as seen from (5.5). When prt12A matrix is applied to (5.8), the
following result is obtained:
( )
1( ) 0.9975
1.0025( ) 0 17.498
prt
prt
+
+
= =
= −
0
12
B
A (5.9)
The prtrA matrix is computed from equation (2.39) as:
prt prt prt prt= −r 22 0 12A A N A (5.10)
and numerically:
0.42415prt = −rA (5.11)
Notice that, the outcome is again a scalar. Those outcomes simplify the design of the
autopilot appreciably.
The selection of the free parameter prt0P (or the extra spectrum by the direct pseudo
inversion of prtreA ) is not very critical, however its effects are important from practical
aspects of consideration. The most important point in this concept is the magnitude
of the gains obtained from the design computations. The design should be
performed carefully not to have large gains since their microprocessor based
implementation is very difficult and robustness is adversely affected. The
proceeding example will show this fact. The design will be performed by selecting
two different extra pole locations, the first one is far from the imaginary axis whereas,
the other is near to the imaginary axis. Both of the chosen poles are on the real axis.
First of all, the solution of the pseudo – inversion equation in (5.8) ( prt0P ) for
10prt = −reA is:
( )0 167.14prt =0P (5.12)
67
whereas, if prtreA is selected as -1 then the solution is obtained as:
( )0 10.051prt =0P (5.13)
The value of the matrix rF in equation (2.41) which is independent of free parameter
prt0P , has the numerical value given below:
0 1
235.99 21.014prt −
= − rF (5.14)
The compensator system gains are obtained from equation (2.43) for the first case,
i.e. 10prt = −reA , as follows:
( )
0.7867739444 3380.8
prtcprt
H = −
= −cD (5.15)
and for the other situation, i.e. 1prt = −reA ,
( )
0.47462372 206.45
prtcprt
H = −
= −cD (5.16)
The above values are computed by the equations in (2.43) taking the matrix ′pW as
identity as stated in Chapter 2. There are also the control system gains which are
defined in (2.44) through the two projection matrices yP and zP as, computed below:
1 00 1
286.04 167.74
00
1.0025
prt
prt
= − − =
T
?
P
P
(5.17)
The matrices computed above are for the first situation, where as for the other case,
the same terms are computed as follows:
68
1 00 1
286.04 10.261
00
1.0025
prt
prt
= − − =
T
?
P
P
(5.18)
Having the projection matrices computed, then one can calculate the control gains
using equation(2.44):
For 10prt = −reA :
( )7.5219 1.0007
0.001894
prt
prtKω
= −
=TK (5.19)
and for 1prt = −reA ,
( )7.5219 0.70319
0.001894
prt
prtKω
= −
=TK (5.20)
The closed loop poles of the two designs then are obtained as:
1010.507 11.207100.4572
prt
j= −
− ±−−
reA
(5.21)
110.507 11.20710.4572
prt
j= −
− ±−−
reA
(5.22)
Up to here it can be observed that, the poles of the reference autopilot (LQSF design)
are successfully preserved by using a dynamic compensator. However, a quite
important point arises here, that one of the gains has a very large magnitude which
is practically undesirable. The first element of the prtcD vector has a magnitude of
39444 in the first case ( )10prt = −reA which is extremely high for use in a practical
69
implementation. Because of these occurrences, the second case is more applicable for
analog or digital implementation. The step responses and open loop (OL) frequency
characteristics of dynamic projective rate autopilot (DPRA) are shown in the
proceeding figures.
Figure 5-1 The step response of dynamic rate autopilot ( )10prt = −reA
Figure 5-2 The step response of dynamic rate autopilot ( )1prt = −reA
70
Figure 5-3 The OL frequency response of dynamic rate autopilot ( )10prt = −reA
Figure 5-4 The OL frequency response of dynamic rate autopilot ( )1prt = −reA
71
The time response curves show that the result of the first case is almost the same
with the original whereas the other one have a small discrepancy. However,
practical consequences are very important in application. The stated discrepancy is
not too critical to satisfy the necessary response in the application.
The design of the dynamically compensated rate autopilot in the yaw plane results
the same magnitude of control gains with the pitch autopilot design. When the yaw
plane model is invoked in the procedure explained up to here, the design gives the
following numerical results:
( )( )
0.4746
2372 206.45
7.5219 0.70319
0.001894
prtcprt
prt
prt
H
Kω
=
= −
= −
=
c
T
D
K (5.23)
5.3. Dynamic Acceleration Autopilot Design
In the chapter about design of autopilots by static projection method, it is found that
the acceleration controller design can not stabilize the closed loop for the considered
missile model. Because of that, it is necessary to design dynamic compensators to
have a stable system. In the design of a dynamic projective acceleration autopilot
(DPAA), a compensator of the form like (5.1) is proposed. Only difference is in the
compensator input, so it is repeated here with the necessary changes.
( )( )
( )
1 2
1 2
2 1 1 2 2 1, ,z
pac pacc
pac pace
pac pac pac
pac pac pac
T
a
pac pac
HK
d d
K K
q
ξ
ξ ξ
δ ξ
ε× × ×
= +
= − −
=
=
=
∈ ∈ ∈
?
?
?
c
c
c
D ?K ?
D
K
?
D ? K
&
¡ ¡ ¡
(5.24)
The design procedure is just the same with that of the dynamically compensated
projection rate autopilot presented in the previous section. The sub-eigenmatrix
pX in equation (2.27) (denoted by pacpX for the current case), is exactly the eigenvector
72
corresponding to the closed loop eigenvalue of the LQSF acceleration autopilot at
37.665pac = −p? which is the following:
0.0974390.121670.98778
pac
= −
pX (5.25)
where the superscript pac stands for pitch acceleration. The sub – eigenmatrices pacpX and pac
rX are partitioned considering 2r = as shown below:
( )
0.0064221 0.00372160.006995 0.044133
0.99897 00.0974390.12167
0.98778
pac
pac
pac
pac
− = − − =
=
= −
r1
r2
p1
p2
X
X
X
X
(5.26)
The partitions of A and F matrices are performed according to equation (2.34) and
(2.39), in (5.27) and (5.28). The F matrix is computed according to the gain vector in
(4.25). Then ˆ,F A and their partitions are found as:
( )
34.011 4.4827 0.1916246.17 7.8232 0.87878
34.011 280.12 0.1916234.011 4.482746.17 7.82320.191620.87878
34.011 280.12
pac pacpac
pac pac
pac
pac
pac
− − − = − − − =
− − = − − − = −
=
11 12
21 22
11
12
21
2
F FF
F F
F
F
F
F 0.19162pac =2
(5.27)
and,
73
( )
0 0.50883 0.41362 ˆ ˆˆ 0 1.0471 0.057149 ˆ ˆ
0 275.12 0.413620 0.50883ˆ0 1.0471
0.41362ˆ0.057149
ˆ 0 275.12ˆ 0.41362
pac pacpac
pac pac
pac
pac
pac
pac
= − − = − = −
= −
=
= −
11 12
21 22
11
12
21
22
A AA
A A
A
A
A
A
(5.28)
The terms in (2.37) is computed and the following numerical results are obtained
( )142.47 12.014
14.356
pac
pac
= − −
=0
0
N
B (5.29)
As done in design of dynamic projective rate autopilot of Section 5.2, the free
parameter 0P in equation (2.39) is computed directly from the residual eigenvalue
( )pacreA through pseudo-inversion. The term rA in (2.39) is computed for this case by
taking 3pac = −reA :
57.827pac =rA (5.30)
Knowing the value of pacrA and applying the inversion rules in (5.7) and (5.8) to the
acceleration autopilot the following free parameter pac0P matrix is obtained:
( )10.052 1.3889pac = −0P (5.31)
For the acceleration autopilot, the rF matrix in equation (2.41) is evaluated as
follows:
6.7112 2.1806
79.026 2.7343pac − −
=
rF (5.32)
Knowing all the necessary parameters, the compensator and controller gains are
evaluated for the acceleration autopilot (the matrices in(5.24)) as:
74
( )
( )
27.53599.565 63.96
1 00 1
1.8383 31.592
00
14.356
1.6329 0.66699
0.40393
pacc
pac
pac
pac
pac
pac
H
Kξ
= −
= −
= − =
= −
= −
c
?
?
?
D
P
P
K
(5.33)
The closed loop poles of the designed acceleration autopilot are obtained as:
1.9884 12.24817.6653
j− ±−−
(5.34)
From the above result, it can be said that the closed loop poles of the original
reference system in Chapter 5 is successfully placed with the additional pole at -3
which comes from the order increase due to the presence of a compensator. Also the
magnitude of the controller and compensator gains is appreciable. So it is not
necessary to try another residual pole location. The step and open loop frequency
responses are shown in the proceeding figures. Just like in the previous result
presentations, the time response curves are superimposed on the full state feedback
equivalents in order to have a means of comparison.
75
Figure 5-5 The step response of the dynamic acceleration autopilot
Figure 5-6 The OL frequency response of the dynamic acceleration autopilot.
The figures show that the design of dynamically compensated projective acceleration
autopilots has results close to the reference autopilot designed by linear quadratic
full state feedback methods.
76
5.4. Nonlinear Flight Simulations
The autopilot design is to be verified for the assumptions of linearization stated in
Chapter 3. The simulation is performed with the same initial conditions stated in the
Section 4.4 of Chapter 4 and results are again presented in the proceeding figures.
The controller and compensator gains are tabulated as discussed in Section 4.4, and
linearly interpolated as the simulation proceeds.
5.4.1. Simulation of Dynamically Compensated Projective Rate Autopilot
Figure 5-7 Missile Range (x – direction)
Figure 5-8 Lateral range of the missile (y- direction)
77
Figure 5-9 Angle of attack variation
Figure 5-10 Elevator (pitch control surface) deflection
Figure 5-11 Angle of sideslip variation
78
Figure 5-12 Rudder (yaw control surface) deflection
Figure 5-13 Pitch rate command and output
Figure 5-14 Yaw rate command and output
79
Figure 5-15 Mach number variation
Figure 5-16 Roll angle variation
Figure 5-17 Roll rate variation
80
Figure 5-18 Aileron (roll plane control surface) deflection
5.4.2. Dynamically Compensated Acceleration Autopilot Simulation
Figure 5-19 Missile range (x – direction)
Figure 5-20 Lateral range of the missile (y – direction)
81
Figure 5-21 Angle of attack variation
Figure 5-22 Elevator (pitch plane control surface) deflection
Figure 5-23 Angle of sideslip variation
82
Figure 5-24 Rudder (yaw control surface) deflection
Figure 5-25 Pitch acceleration command and output
Figure 5-26 Yaw acceleration command and output
83
Figure 5-27 Mach number variation
Figure 5-28 Roll angle variation
Figure 5-29 Roll rate variation
84
Figure 5-30 Aileron (roll plane control surface) deflection
With the aid of the dynamic compensator, a stable acceleration autopilot is obtained.
The exploration of the graphs shows that the dynamically compensated autopilots
designs are valid (considering linearization assumptions) according to the
recommendations in Chapter 4. The obtained results of static and dynamic rate
autopilots are similar which should be due to the low effectiveness of the
unavailable states ( ,v w ) in control action.
85
CHAPTER 6
ROBUSTNESS ANALYSIS
6.1. Importance of Robustness Analysis
Robustness is a measure of strength of the control system to external disturbances
and uncertainties in the plant model. The more robust the controller the higher the
stability of the overall system against external effects. The classical frequency
response parameters which are the gain and phase margins are the most common
measures of robustness in linear design. However, they reflect a general point of
view about robustness but they do not give direct information about plant parameter
deviations. In missile control applications the aerodynamic coefficient uncertainty is
the most important parameter that affect robustness to the stated fact. In this chapter
analysis of the autopilot robustness against aerodynamic parameter uncertainty will
be investigated in detail.
6.2. Singular Value Robustness Tests
One of the most common group of robustness analysis methods against plant
parameter perturbations is the group of methods based on the singular value theory
which are discussed in Balas (1998), Doyle (1987), Fan (1991), Packard (1988, 1993),
Zhou (1998) in detail. There are also other robustness analysis methods based on
polynomial coefficient uncertainties such as Garloff (1985), Vardulakis (1987), and
other ones which are based on the mapping theorem De Gaston (1988), Wedell
(1991). However the computational implementation of the methods based on
mapping theorem is very difficult. In this study the structured singular value theory
will be implemented for the analysis of the designs performed in Chapter 4 and
Chapter 5.
86
In the singular value analysis, the uncertain and the known part of the system are
separated and converted to a feedback interconnection form. In Figure 6-1 a
conventional control system block diagram is shown in which ( )cG s represents the
controller and ( )G s represents the plant which contains uncertain parameters.
Figure 6-1 A conventional control system
∆
M
Figure 6-2 The feedback interconnection structure for singular value analysis
This generic control system structure is divided into its uncertain and known (or
nominal) parts. This type of structure is shown in Figure 6-2. The matrix M
represents the nominal or known part of the system. The matrix ? is the uncertain
part of the closed loop. ? contains the perturbations on the nominal values of the
uncertain parameters of the plant. It may be full or block diagonal. If the
uncertainties are unstructured then ? is a full matrix, whereas if the uncertainties are
structured then it is in block diagonal form. In the analysis, the set point yr is taken
as zero. In this research, the modeled uncertainties will be structured and thus
structured singular value method will be applied. The analysis method was
87
developed by Packard (1993) and has various applications on flight control systems
(Doyle (1987), Hewer (1988), Wise (1991), and Wise (1992)).
According to Zhou (1998) the closed loop system in Figure 6-2 is stable if the
following is satisfied:
[ ]det 0+ ≠I M? (6.1)
The purpose of the analysis is to find a bound on ? where the stability criterion is
satisfied. Empirically the bound can be calculated from the singular value theory.
Before going into details, it is convenient to review the singular value definition.
Given any matrix A , it can be decomposed as:
= *A USV (6.2)
where
1 2
1 2
1
1
1
, ,.......,
, ,.......,0
0
00 0
0
0
0
n
n
n
u u u
v v v
σ
σ
= =
=
=
U
VS
S0
S O
(6.3)
the terms iσ ’s are called as singular values and the notations σ and σ stands for
maximum and minimum singular values, respectively. Those two parameters have
an alternative definition in norm basis and shown below (Wise (1992)):
[ ]
[ ]
22
2
21
2 2
max
1min
σ
σ−
= =
= =
AxA A
xAx
Ax A
(6.4)
Singular values also have properties which are important in developing the robust
stability theory. One of which is (Wise (1992)):
88
( ) ( ) ( )σ σ σ≤M? M ? (6.5)
If one proposes + = +I M? A B , then the instability condition gives +A B to be
singular and thus rank deficient. This result in a variable x which gives ( )+ =A B x 0 ,
so = −Ax Bx . However the norms will be equal i.e. 2 2
=Ax Bx (Wise (1992)). The
definitions of the singular values will give some important inequalities presented
below (Wise (1992)):
[ ] [ ]2 2 2σ σ≤ = ≤ =A Ax Bx B B (6.6)
The reverse of the above inequality or ( ) ( )σ σ≥A B implies that, +A B thus +I M?
(taking ,= =A I B M? ) being non – singular.
By using the inverse inequality above, one can obtain an important relationship for
non-singularity of +I M? as shown below:
( ) ( )1 ( )σ σ
σ>
>I M?
M? (6.7)
If the right hand side of the property in (6.5) is assumed to be less than one, then the
same boundary can be set on the left hand side due to the in equality.
Mathematically this is shown below:
( ) ( ) ( )( )
11
σ σ σ
σ
≤ <
<
M? M ?M?
(6.8)
If the ( ) ( ) 1σ σ <M ? then the matrix +I M? will be non-singular and by dividing both
sides by ( )σ M a sufficient condition on the stability of the system in Figure 6-2 can
be derived as shown below:
1( )( )
σσ
<?M
(6.9)
89
The above equation is the basic criterion for robust stability and it is called as small
gain theorem (SGT). It is a sufficient condition for stability and if the criterion is not
satisfied the system may still be stable. Wise (1992) recommends that the small gain
theorem should be used in cases where the uncertainties are unstructured ( ? is a full
matrix). However, there are several cases where the plant uncertainties can only be
modeled as structured uncertainties. If theorem of (6.9) is applied to those situations,
the obtained robust stability bound is considerably narrowed. This phenomenon is
called as conservatism (Wise (1992)). In order to reduce the conservatism of SGT,
Packard (1993) developed the µ - analysis technique which is applicable to situations
in which there are structured uncertainties.
The system in Figure 6-2 is stable if the following condition is satisfied:
det( ) 0+ ≠I M? (6.10)
It is also desired to find the minimum value of the largest singular value of the
perturbation matrix ? such that the closed loop system is unstable. Namely, (Zhou
(1998)):
( ){ }min inf :det( ) 0 , r rα σ ×= + = ∈? I M? ? £ (6.11)
For a more easy looking criterion, the maximum singular value term can be replaced
by the variable α as shown below (Zhou (1998)):
{ }min inf :det( ) 0 , r rIα α α ×= + = ∈M? ? £ (6.12)
In this research the parameters are assumed to deviate maximum 100%± from their
nominal values so ( ) 1σ ≤? . The minimum value of the destabilizing variable α is
related to the reciprocal of the spectral radius ( )( ).ρ of the open loop system matrix
M? of the system in Figure 6-2 (Zhou (1998)). Namely:
min1 , ( ) 1
max ( )α σ
ρ= ≤?
M? (6.13)
90
The reciprocal of the minimum destabilizing variable α is termed as structured
singular value and a new variable ( )µ∆ M is defined as follows:
{ }
1( )min ( ),det( )
µσ∆ =
+ =M
? I M? 0 (6.14)
If no ? makes +I M? singular, then ( ) 0µ∆ =M . Before going further it is useful to
present the structure of the uncertainty matrix.
To define the uncertainty structure, one needs the following three information;
namely the total number of blocks, the size and type of each block. In the analysis,
the uncertainties can be both, complex and real. The uncertainty matrix may contain
repeated scalar blocks or full blocks. In the structured singular value literature
(Packard (1993), Zhou (1998)) generally S and F denote the number of repeated scalar
blocks and the number of full blocks along the diagonal of perturbation matrix,
respectively. Mathematically this definition is expressed as (Packard (1993)):
{ }11 1,...., , ,.....,
, ,1 ,1S
i j
r S r S S F
m mi S j
diag I I
i S j F
δ δ
δ
+ +
×+
= ∆ ∆
∈ ∆ ∈ ≤ ≤ ≤ ≤
?
£ £ (6.15)
The sum of dimensions of the repeated scalar and full blocks should give the number
of uncertain parameters n .
1 1
S F
i ji j
r m n= =
+ =∑ ∑ (6.16)
Packard (1993) derives some useful relationships which are presented below:
• If 1, ( 1, 0, )S F r nδ δ= ∈ = = =? I £ then the structural singular value is equal to the spectral radius of the known system matrix namely ( ) ( )µ ρ∆ =M M
• If ( 0, 1, )n niS F m n×= = = =? £ then ( ) ( )µ σ∆ =M M
Combining the two properties mentioned above, following bounds can be obtained
on the structured singular value (Packard (1993)):
91
( ) ( ) ( )ρ µ σ∆≤ ≤M M M (6.17)
In the above relation, the bounds may give satisfactory information on stability if the
gap is small; however in some situations, this is not the case. To prevent this
problem, Packard (1993) proposes a transformation which does not affect the
singular value but solves the problem. To do so, two subsets of n n×£ are defined as
shown below:
{ }{ }
*
*1 1
:
,...., , ,...., : , 0, , 0i ir rS S S j i i i S j S j
Q Q Q
diag D D d d D D D d d×+ + + +
= ∈ =
= ∈ = > ∈ > 1 F
n
m m
Q ? I
D I I £ ¡(6.18)
These transformations have the following properties (Packard (1993)):
( ) ( ) ( )
*
1 / 2 1/2
, ,Q Q QQ Q
D Dσ σ σ
∈ ∆ ∈ ∆ ∈
∆ = ∆ = ∆
∆ = ∆
Q ? ? (6.19)
When the above properties are compiled together, the bounds in (6.17) takes the
following form (Packard (1993)):
1/2 1 / 2max ( ) ( ) inf ( )Q D MDρ µ σ −∆≤ ≤M M (6.20)
Matrix D can be solved by optimization technique which is generally known as D
matrix optimization. This algorithm is used for computing the upper bound of the
structured singular value, and can be implemented for computer use. There are
numerous variants of this algorithm for software implementation. One of them is
presented in Fan (1991) and Young (1992), and its implementation is also integrated
into MATLAB® “ µ - Analysis and Synthesis Toolbox” by Balas (1998). In this study
this software will be used for implementing the analysis algorithms.
As it can be understood from (6.20), the structured singular value has both lower
and upper bounds. The upper bound is a safety bound and its reciprocal gives a
safety margin. The usage of the analysis is explained in the next paragraph.
92
For system definitions, the analysis is performed by taking the frequency response of
the nominal portion of the system. The bounds of the structured singular value of
each frequency response point are calculated and both of the bounds are plotted. For
each plot, the position and value of the peak should be detected, and noted. One can
call the peak of the lower bound as lβ and the upper bound as uβ . If all
perturbations satisfy the following:
( ) 1maxu
jσ ωβ
∆ < (6.21)
then the system will be stable and,
( ) 1maxi
jσ ωβ
∆ = (6.22)
gives a particular perturbation matrix where the system becomes unstable. The
criterion in (6.21) gives a spherical region on parameter space in which no
perturbation makes the closed loop interconnection in Figure 6-2 unstable. This
region can also be called as robust stability region . The lower bound can be
calculated by the power algorithm stated in Packard (1993), whereas the algorithm
developed by Fan (1991) and Young (1992) is used for the computation of the upper
bound. The upper bound can be calculated in presence of both real and complex
perturbations. However, the power algorithm does not work in presence of pure real
perturbations but the upper bound is enough since it gives the guaranteed region of
stability.
6.3. Linear Fractional Transformations
In the previous section the theory of the structured singular value (the so called µ -
analysis procedure) is presented briefly. In this section the framework necessary for
implementation of the theory to the considered missile autopilot design problem is
developed. In the analysis framework, first of all the perturbation form is produced
from general control system structure (like the one in Figure 6-1) , and it is converted
93
to the form in Figure 6-2. To perform that, a method called Linear Fractional
Transformation (LFT) is developed and presented in Balas (1998), Packard (1993)
and Zhou (1998). The application of the analysis in MATLAB® is based on linear
fractional structures.
To develop the framework, it is convenient to consider a multi input and multi
output system defined by a complex matrix M as shown in the Figure 6-3.
rv
Figure 6-3 Complex System Matrix Relating Input and Output
The system inputs and output vectors can be divided into two parts ( 1 2 1 2, ,r r v v−
respectively). To relate them the system matrix should be partitioned accordingly as
shown in the Figure 6-4.
11 12
21 22
M M
M M1r
2r
1v
2v
2v 2r
Figure 6-4 The partitioned form of the system matrix and inputs.
The second block in the figure relates the second part of the output vector to the
second part of the input vector by a matrix ? which has a block diagonal form. The
linear fractional transformation connects those two elements as shown in Figure 6-5.
In Figure 6-4, the inputs and outputs are related by:
1
= += +
1 11 1 12 2
2 21 22 2
v M r M rv M r M r
(6.23)
94
11 12
21 22
M MM M
1r1v
Figure 6-5 The Linear Fractional Transformation System Connection
From (6.23) the relationship between 1r and 1v in Figure 6-5 can be found as :
1( ( ) )
( , )
−= + −=
1 11 12 22 21 1
1 L 1
v M M ? I M ? M rv F M ? r
(6.24)
The matrix ? can be connected from the upper ports to have an upper linear
fractional transformation as shown in Figure 6-6.
11 12
21 22
M MM M
2r2v
Figure 6-6 System Connection as an Upper Linear Fractional Transformation
The mathematical expression relating input and output in Figure 6-6 is calculated as:
1( ( ) )
( , )
−= + −=
2 22 21 11 12 2
2 U 2
v M M ? I M ? M rv F M ? r
(6.25)
The application of the theory of the linear fractional transformations in robustness
analysis using µ – analysis techniques requires representation of each uncertainty as
a LFT structure. Before representation of the system in linear fractional form, the
95
system should be represented in a realization diagram. The uncertain parameters
should occur as gains in this realization diagram. If one assumes that the gain is
denoted by , it has a perturbed form as shown below:
(1 )c cc c c cδ δ= + = + (6.26)
where c represents the nominal value of the parameter and the symbol cδ stands for
the perturbation which has an assumed value between [ ]1,1− . When the linear
fractional transformation input – output relationship is equated to (6.26), the
following will be obtained for a lower LFT (Balas (1998)):
, , 1, 0c c= = = =11 12 21 22M M M M (6.27)
So whenever the parameter c is seen in the realization it is replaced by the lower LFT
structure seen in Figure 6-5 with the parameters in(6.27).
The preparation of the linear fractional transformation can be performed directly
from the state space representation of a linear time invariant system as illustrated in
the next section.
The linear fractional transformation procedure can be applied to all structures by
replacing the unknown parameters by any LFT structure presented in this section.
However, a faster procedure is developed in Balas (1998) and Wise (1992) for state
space representations which have entries with uncertain parameters.
To start the LFT procedure, a general state space representation is restated:
ˆ ˆˆ ˆ ˆˆ ˆˆ ˆˆ
= +
= +
x Ax Bu
y Cx Du
& (6.28)
The above system model can be converted into the following form which is also used
in developing linear fractional structure for MATLAB analysis framework.
96
ˆ ˆ ˆˆ
ˆˆ ˆˆ
=
A B xxuy C D
& (6.29)
or shortly,
=X EU (6.30)
The compact system equation in (6.29) can be decomposed to be written as a sum of
the nominal and perturbed uncertainties as,
1 2
ˆ ˆ...
ˆ ˆn n
nn n
δ δ δ = + + + +
1 1 2 2
1 1 2 2
A B A B A B A B A BC D C D C DC DC D
(6.31)
where the sign . is used for representing the nominal values of the parameters. Each
term iδ represents the uncertainty perturbation related with the thi parameter. The
conversion into LFT form is a modified version of decomposition presented in Wise
(1992). In this method each uncertainty weight matrix (or the matrix coefficients of
each uncertainty perturbation term) is decomposed into one column and one row
vector. Before illustrating this, equation (6.31) can be written shortly as:
1 2 .... nδ δ δ= + + + +1 2 nE E E E E (6.32)
Then, each matrix iE can be decomposed as defined in above expansion (Wise
(1992)):
=i i iE ß a (6.33)
where iß and ia are the column and row vectors, respectively. Then the overall
decomposition can be written as:
1
n
i=
= + ∑ i i iE E ß d a (6.34)
According to Wise (1992), the decomposition elements can form matrices which
represent certain part of the closed loop system. These are:
97
( )1
21 2, .... ,
:n
n
αα
β β β
α
= = =
m m mA E B C (6.35)
In order to be able to use the mu – analysis framework, these matrices should be
packed as:
=
m m
m
A BM
C 0 (6.36)
Up to here, the theoretical and practical background necessary for the structured
singular value robustness analysis of the autopilots designed in the Chapter 4 and
Chapter 5 are summarized. The proceeding sections are devoted for the numerical
applications of the material discussed.
6.4. The Robustness Analysis of the Static Rate Autopilot
In this section ,first,numerical application of the theory and tools discussed up to
now will be presented. This is the analysis of the static projective rate autopilot
against the aerodynamic uncertainties occuring in the real environment. In this
analysis, the uncertainties will be assumed to be on the entries of the state space
representation of the linear missile model which is repeated here for convenience:
1 2 1
3 4 2
p p p
ep p p
w a a w bq qa a b
δ
= +
&& (6.37)
In the above representation, the terms ,j ji ia b are linear functions of the aerodynamic
coefficients (weighted by the dynamic pressure), and involves the uncertainties
generally shown by the proceeding equation:
(1 )
(1 )
j ji i
j ji i
a a
b b
δ
δ
= +
= + (6.38)
98
Since the autopilot processes the integral of the error between the measured rate
variable (either q or r ) and the guidance rate command, the integral of the error
should be added to the linear state equation in (6.37). However, the integral of the
error is not directly included in the state equation since it is not a variable obtained
from the model directly, but it is generated arbitrarily. The system model in (6.37) is
converted into the compact form in (6.29) and written again as:
1 2 1
3 4 2
0 1 0
p p p
p p p
e
a a bw wq a a b qq δ
=
&& (6.39)
The compacted model in (6.39) is decomposed as proposed in (6.31) :
1 2 11 2 1 1 2
3 4 2 3 4 2 1 2
1
3 3 4 4 5 2
0 0 0 00 0 0 0 0 0
0 1 0 0 1 0 0 0 0 0 0 0
0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
p p pp p p p p
p p p p p p
p
p p p
a a ba a b a aa a b a a b
ba a b
δ δ
δ δ δ
= + + +
+ + +
6δ
(6.40)
This decomposition is rearranged and rewritten in terms of column and row vectors,
like in (6.33). The implementation to the considered case is written below,
( ) ( ) ( )
( ) ( ) ( )
1 2 1 1 2
3 4 2 1 2 3 3
1
4 4 5 2 6
00 1 0 0 0 0 1 0 1 0 0
0 1 0 0 0 0
00
0 1 0 0 0 0 1 0 0 10 0 0
p p p p p
p p p p
p
p p
a a b a aa a b a
ba b
δ δ δ
δ δ δ
+ + + +
+ +
(6.41)
The next step is to pack the expansion above into the form in (6.36) which results in:
99
1 2 1 1 2 1
3 4 2 3 4 2
1 1
2 2
3 3
4 4
5 5
6 6
0 0 0
0 0 00 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0
p p p p p p
p p p p p p
e
w a a b a a b wq qa a b a a bqz wz wz wz wz wz w
δ
=
&&
(6.42)
When combined with the control actuation system model given in Appendix A, the
general block from of the nominal model will be as follows:
1z
2z
3z
4z
6z5z
7z
8z
9z
q
1w
2w
3w
4w
5w
6w
7w
8w
9w
ecδ
Figure 6-7 µ - analysis framework for the analysis of the rate autopilot
In the following figure, the autopilot section of the analyzed system is presented.
1s
Kh
Kq
qc
q
eδ
Figure 6-8 Rate autopilot constructed for the robustness analysis framework
100
In Figure 6-7, the variables iz ’s represent the input to the perturbation block (or
matrix representing the perturbation as a system) and iw ’s represent the output of
the perturbation block which enters to the closed loop linear fractional transformed
system as perturbed aerodynamic coefficients.
ecδ
q qc
q
Figure 6-9 The rate autopilot LFT block form.
The block diagram of the autopilot composed for the LFT analysis is shown in Figure
6-9. The overall linear fractional transformed form of the system is presented in the
Figure 6-10. This structure is also suitable for analysis in MATLAB® MUTOOLS
toolbox.
The perturbation structure of the system is a 9 9× diagonal matrix as it can be seen
from Figure 6-10. The first three element of the diagonal matrix represents the same
uncertainty as it can be seen from Appendix A, so the perturbation block can be
shown by a 3 3× identity matrix multiplied by a single uncertainty.
The analysis by MATLAB® MUTOOLS software will compute the upper bound of
the structured singular value in (6.20) which is the safety bound for perturbations in
all directions. As it can be understood from equation (6.21) and (6.22), the structured
singular value is computed at a point of frequency response. Since the frequency
response is varying over a broad range of frequencies, the analysis is performed by
plotting the variation of the upper bound of the structured singular value versus the
varying frequency. The peak point of the variation curve represents the boundary of
the perturbation defined according to(6.21). The reciprocal of the peak point gives a
perturbation bound on the aerodynamic coefficients for stability. The analysis for the
101
rate autopilot designed in Chapter 5 will be performed at the conditions of the
design example. When the µ - analysis routine is invoked with the designed control
system, the variation of the upper bound of the structured singular value ( )µ is
obtained as in the Figure 6-11.
1
2
3
4
5
6
7
8
9
0 00 0 0 0 0 0 0 0
0 0
0
0
0
00
0
δδ
δ
δ
δ
δ
δδ
δ
0
0
1z
2z
3z
4z
5z
6z
7z
8z
9z
q
1w
2w
3w
4w
5w
6w
7w
8w
9w
ecδ
Figure 6-10 The overall combination of the LFT formed system.
102
The peak of the upper bound occurs at a frequency of 33.313 /secrad with the
value 1.8124 . This corresponds to a perturbation bound of 55.176 % . This is an
appreciable value for the design. The results given here are computed taking the
nominal value of the natural frequency as 0 90 /secradω = and damping ratio
as 0.6ζ = . The instability occurs at this perturbation when the signs of the
perturbations are suitable. In (6.43), the signs of the perturbations that make the
system in consideration unstable, can be seen. In (6.44), the closed loop poles are
given to show the instability result of the analyzed system. This is supported by a
step response also in Figure 6-12. The step response figure includes two plots. Both
of them shows the same result, one of them computed by the MATLAB MUTOOLS
transient response computation routines, and the other one is obtained from the
conventional step response commands. The two figures are superimposed for
verification.
Figure 6-11 The variation of upper bound of µ with frequency
103
1
2
3
4
5
6
7
8
9
0.560.560.560.56
0.560.56
0.560.560.56
δδδδδδδδδ
= −= −= −= −=== −==
(6.43)
0.028859 32.71628.13519.8630.22248
j±−−−
(6.44)
Figure 6-12 The step response of the perturbated system.
The obtained result shows that, a deviation of the aerodynamic coefficients from
their nominal values (the values obtained from the computer databases) less than
55.176 % in any direction, will not cause any destabilization. From practical point of
104
view this is a quite appreciable result for the considered aerodynamic conditions.
However it should be mentioned that, the analysis is performed for only one
condition, and the other aerodynamic operating points should be also investigated.
The variation of the perturbation bounds against the changing operating conditions
are presented in the last section of this chapter.
6.5. The Robustness Analysis of Dynamic Rate Autopilot
In this section the second version of the rate autopilot considered in this study will
be analyzed for robustness. Since all the variables processed by this autopilot is the
same with the static version no framework development will be presented. The only
difference is in the autopilot itself. So, only the converted form of the autopilot (for
integration with linear fractional transformed plant in the previous section) is
presented in Figure 6-13.
1s
1s
prtcH
ωK
1prtK
2prtK
1prtd
2prtd
rq
q
ecδ
q
Figure 6-13 The converted form of the autopilot for LFT integration
The form of autopilot presented here has the same block structure as in Figure 6-9.
So, the overall structure will be exactly the same as in Figure 6-10. When the
proposed model is invoked into the MATLAB structured singular value analysis and
105
synthesis software with the results obtained from the Chapter 5, the variation of the
singular value with the frequency is obtained as in Figure 6-14. From the figure, it
can be understood that the peak of the curve occurs at a frequency of 33.313 /secrad
with a value of 1.8406 . This corresponds to a perturbation ratio of 54.331% in
aerodynamic coefficients. Again the variations of signs are important in analyzing
destabilization. For this case, the sign structure that results in instability is shown
below:
1
2
3
4
5
6
7
8
9
0.5510.5510.5510.551
0.5510.551
0.5510.5510.551
δδδδδδδδδ
= −= −= −= −=== −==
(6.45)
Figure 6-14 The variation of singular value for dynamic rate autopilot
In (6.45), the perturbation magnitudes are chosen to be slightly larger than the peak
value of singular value variation in order to guarantee instability and the closed loop
poles of the perturbed system are found to be:
106
0.014703 33.29429.41818.9980.992720.24336
j±−−−−
(6.46)
The corresponding time response of the perturbed system is shown in Figure 6-15.
Figure 6-15 The time response of perturbed dynamic rate autopilot
The results of the analysis show that the static and dynamic projective rate autopilots
have comparable critical perturbation bounds for the design example condition.
However, it is important to note that overall analysis is necessary for a concrete
conclusion.
107
6.6. Robustness Analysis of Dynamic Acceleration Autopilot
The analysis of the dynamically compensated acceleration autopilot has a similar
procedure as the rate autopilots except some changes in the development of the
linear fractional transformation. The compact form of the plant (missile) model for
the acceleration autopilot design is written as:
1 2 1
3 4 2
1 2 1
0 1 0
p p p
p p p
p p pz
e
a a bww
q a a bq
a a a u bq
δ
= −
&&
(6.47)
The decomposition according to (6.31) then becomes:
1 2 11 2 1 1 2
3 4 2 3 4 21 2
1 21 2 1 1 2 1
3 43
0 0 0 00 0 0 0 0 0
0 0 0 00 0 0 0 0 00 1 0 0 1 0
0 0 0 0 0 0
0 0 0 00 0 0 00 0 0
p p pp p p p p
p p p p p p
p pp p p p p p
p p
a a ba a b a aa a b a a b
a aa a u b a a u b
a a
δ δ
δ
= + + + − −
+
1
24 5 6
1
0 0 00 00 0 0 0 0
0 0 0 0 00 00 0 0 0 0 00 0 0
p
p
p
bb
bδ δ δ
+ +
(6.48)
where,
1 2 1
3 4 2
1 2 1
0 1 0
p p p
p p p
p p p
a a b
a a b
a a u b
=
−
E (6.49)
If all matrices in the above decomposition are written as a product of one column
and one row vector, then:
108
( ) ( ) ( )
( ) ( ) ( )
1 2
31 2 3
1 2
1
4 24 5 6
1
00 0
1 0 0 0 1 0 1 0 0000 0
000
0 1 0 0 0 1 0 0 10 00 00
p p
p
p p
p
p p
p
a aa
a a
ba b
b
δ δ δ
δ δ δ
+ + + +
+ +
E
(6.50)
is obtained.
The µ - analysis framework of the acceleration autopilot is formed including the
inputs from and to the perturbation block as shown in (6.51).
1 2 1 1 2 1
3 4 2 3 4 2
1 2 1 1 2 1
1
2
3
4
5
6
0 0 0
0 0 0
0 0 00 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 1 0 0 0 0 0 0
p p p p p p
p p p p p p
p p p p p pz
a a b a a bwq a a b a a ba a a u b a a bqzzzzzz
− =
&&
1
2
3
4
5
6
e
wq
wwwwww
δ
(6.51)
The above representation has a block form for linear fractional transformation in
Figure 6-16. The acceleration autopilot in Figure 6-17 which has a block diagram in
Figure 6-18 is connected to the LFT form Figure 6-16. The overall system
interconnection, including the actuator of Appendix A, is presented in Figure 6-19.
The dynamically compensated acceleration controller framework is shown in the
next figure. This framework is designed in order to build µ – analysis framework for
MATLAB® software.
109
azq
1z
2z
3z
4z
5z
6z
eδ
1w
2w
3w
4w
5w
6w
Figure 6-16 The plant block diagram including uncertainties
1s
1s
prtcH
ξK
1pacK
2pacK
1pacd
2pacd
azr
az
ecδ
az
q
Figure 6-17 The acceleration autopilot structure for singular value analysis
ecδ
az
az
q
azc
Figure 6-18 The LFT block structure of the autopilot
110
The perturbation structure is just the same with the ones in the other autopilots in
this study as the uncertain parameters are same in all cases. The results of singular
value analysis are shown in Figure 6-20, at the operating points of the design
example in Chapter 5. The result is again shown as the variation of the upper bound
of the structured singular value against the changing frequency. The control
actuation system has again a natural frequency of 0 90 /secradω = , and a damping
ratio of 0.6ζ = .
1
2
3
4
5
6
7
8
9
0 00 0 0 0 0 0 0 0
0 0
0
0
0
00
0
δδ
δ
δ
δ
δ
δδ
δ
0
0
1z
2z
3z
4z
5z
6z
7z
8z
9z
azq
1w
2w
3w
4w
5w
6w
7w
8w
9w
ecδ
Figure 6-19 The closed loop interconnection of the acceleration control system
111
Figure 6-20 Plot of upper bound versus frequency for acceleration autopilot
As it can be observed from the figure, the variation of the upper bound has a very
high peak with a value 56.888 at a frequency of12.638 /secrad . The peak corresponds
to a 1.758% perturbation bound in the aerodynamic coefficients. The verification of
the instability at this amount of perturbation can be performed by moving the
aerodynamic coefficients by the given amount of perturbation like in the previous
analysis. The signs of the perturbation are all the same for each element of
perturbation matrix. At the specified value of perturbation, the closed loop poles of
the system, including the control actuation system, is obtained as:
43.359 71.20451.726
1.2595 11.782.9945
j
j
− ±−
±−
(6.52)
112
To demonstrate the occurrence of instability in the system the, unit step response of
the closed loop perturbed system is presented in Figure 6-21.
Figure 6-21 The step response of the perturbed acceleration autopilot
From the analysis it can be observed that the robustness of the missile is not very
appreciable from practical point of view. It is less than 2 % , and in the best case the
aerodynamic coefficients have a deviation of at least 5 10%− . So a minimum of 15%
robustness is required for obtaining a useful autopilot design.
6.7. Overall Analysis of the Autopilots
In this section, the singular vale analysis will be expanded to all of the operating
points in order to see the overall stability during the flight. The perturbation
measure of the autopilots will be plotted against the Mach number. The analysis will
be performed at the following ranges and points,
113
0,2500,5000,7500,10000,125000.53,...,3.00
h mMach
=
= (6.53)
The results of the analysis are presented as plots of reciprocal of the maximum
singular value bound plotted with respect to Mach number. Six plots will be given
on the same figure for each autopilot analyzed in this chapter.
6.7.1. Overall Results for Static Rate Autopilot
The variation of the perturbation bound (reciprocal of the peak of the upper bound)
for the rate autopilot designed by the static projection method (no observer) is
presented in Figure 6-22. From the results it can be seen that the allowable
perturbation bound have a variation changing in a fairly wide interval however the
magnitude of the bound do not fall below a certain value which is about25% . In real
applications, the aerodynamic coefficients obtained from an advanced aerodynamic
database have an uncertainty about15% , so the result of the system with the static
rate autopilot is acceptable.
Figure 6-22 Variation of perturbation bound for static rate autopilot
114
Figure 6-23 Variation of perturbation bound for dynamic rate autopilot
. Figure 6-24 Variation of perturbation boundfor dynamic acceleration autopilot
115
6.7.2. Overall Results for Dynamic Rate Autopilot
The variation of the perturbation bound for the dynamically compensated rate
autopilot is shown in Figure 6-23. The results of the analysis are approaching to that
of the static rate autopilot. The minimum value of the perturbation bound is
around25% . For the practical situation this result is also appreciable.
6.7.3. Overall Results for Dynamic Acceleration Autopilot
The overall analysis outcome for the dynamically compensated acceleration
autopilot is presented in the Figure 6-24. The analysis shows that the perturbation
bound possesses an abrupt change which is an undesirable behavior in real situation.
In addition, the value of the perturbation bound is very small in some of the
operating points, and hence not an appreciable result.
116
CHAPTER 7
REAL ENVIRONMENT NONLINEAR SIMULATIONS
7.1. Introduction
To verify that the autopilots designed for the missile are successful, one should
perform simulations in which the autopilots are integrated with guidance and non –
linear missile model (presented in Chapter 3). This simulation in fact reflects the
flight of the missile. Since the flight medium has many uncertainties, they should be
taken into consideration during the flight simulation. In Chapter 4 and 5, there are
non – linear flight simulations, however they are only performed for verifying the
assumptions done in Chapter 3 for autopilot design. Neither of them has any
uncertainties, nor is control actuation system used. In addition, their robustness
analyses are performed in Chapter 6 in order to have information about its reaction
to aerodynamic uncertainties. The aerodynamic uncertainties should be added to
flight simulation in order to verify the results of the robustness analysis. Another
important disturbing effect is the sensor noise which is unavoidable. It should be
also taken into account. This is something that is not performed in this study up to
now, and will be analyzed in the proceeding section. Shortly, the simulations of this
chapter will resemble very much to the real flight of the missile.
7.2. Implementation of Uncertainties in Simulations
The method of implementation or modeling of the uncertainties is important to have
meaningful information from real environment simulation. Since the uncertainties
are random occurrences they can be implemented in many ways. In this section, the
ways of implementation of uncertainties and disturbances considered in this
research.
117
7.2.1. Implementation of the Aerodynamic Uncertainties
The aerodynamic uncertainties are implemented as deviations at the beginning of
simulations. Uncertainty is added to an aerodynamic coefficient as defined by the
following equation:
(1 )i i iC C δ= + (7.1)
where the deviation factor iδ is a uniformly distributed variable the value of which
is determined once at the beginning of the simulation. Its value does not exceed
0.15± and remain constant throughout the simulation.
7.2.2. The Modeling of Inertial Sensor Noise
The measurement elements are highly noisy elements and their effect at high
frequencies is important. In non – linear simulations they are modeled as a normal
(Gaussian distribution) with zero mean and varianceϖ . The effect of the noise is
modeled by the following equation:
n nω ω η= + (7.2)
where nω is the output of the sensor, ω is the measured variable from the dynamic
model and nη is the Gaussian distributed noise. In this study, a variance of 410− is
selected in simulations. The noise is applied continuously throughout the simulation.
7.2.3. The Modeling Thrust Misalignment
The modeling of thrust misalignment is discussed in Chapter 3, and an equation is
given for this purpose in(3.32). The misalignment angles are denoted by 1δ and 2δ
which are uncertain variables. Their values are determined from Gaussian
distribution with zero mean. Just like the aerodynamic coefficients, the misalignment
angles are also assigned a random value at the beginning of the simulations and kept
constant throughout. Their variances are selected as:
118
1
2
0.0570
ϖ
ϖ
=
=
o
o (7.3)
and they are applied in the simulation as follows:
(0, )ni iGδ ϖ= (7.4)
where ( )0,niG ϖ represents a Gaussian distributed variable with zero mean and
variance equal to iϖ .
7.2.4. The Misalignment in Center of Gravity of the Missile
This phenomenon is due to the production errors in the missile body which is
explained in Chapter 3. The effect of this misalignment causes a change in the
position of the center of gravity of the missile in the longitudinal axis of the missile
as defined by equation(3.34). The misalignment factor cgδ is a uniformly distributed
uncertainty changing in the ranges 0.005± .
7.2.5. The Initial Rolling of the Missile
When released from the aircraft, the missile may roll to some angle because of some
disturbance effects such as wind. This phenomenon can be included in the
simulation by assigning a uniformly distributed random value to the initial rolling
angle in the range 5± o .
7.2.6. The Deviation in the Natural Frequency of the Actuator
There may be a deviation in the characteristics of the control actuation system. This
may be caused because of the difference between the modeling and actual
production of the actuator. This fact is also taken into account in robustness analysis
in the previous chapter. This deviation is implemented in the simulation just the
same as the aerodynamic coefficients within the range 0.10± .
119
7.2.7. The Side Wind Effects
As a natural fact, there is always wind affecting on the missile during its flight. There
are many ways to model the wind effect. In this study a simple wind model will be
applied to the simulation which takes side wind into consideration. The wind is
modeled by generating a wind profile which relates the velocity of the wind with the
altitude. A simple model can be generated using an exponential as shown below:
0
0
0
0
(1 )w
wx wxK z
wy wy
wz wz
V VV e VV V
−
= −
(7.5)
where ( )T
wx wy wzV V V triple represents the wind velocity in three directions at the
specified altitude and( )0 0 0 T
wx wy wzV V V represents the wind velocity on the ground or
sea level. This is the wind profile for this purpose of simulations. Equation (7.5)
calculates the wind velocities in earth frame only. To have their effects on the missile
body they should be transformed to body axis by the so called direction cosine
matrix (Atesoglu (1997)). This results in the following:
( ) ( )( ) ( )
w wx wy wz
w wx wy wz
w wx wy wz
u u V c c V c c V sv v V s s c c s V s s s c c V s cw w V s s c s s V c s s s s V c c
θ ψ θ ψ θφ θ ψ φ ψ φ θ ψ φ ψ φ θφ θ ψ φ ψ φ θ ψ φ ψ φ θ
+ − = − − + + +
+ + − +
(7.6)
( )Tw w wu v w is the velocity of the missile including the wind effect. This definition
of velocity will be effective in computation of angle of attack, and sideslip, and all
other places where the body velocities of the missile are taken into consideration.
7.3. The Real Environment Simulations
Real environment simulations are performed in presence of the effects discussed in
the previous section. To obtain reasonable information from the simulations, the
trials are repeated several times in order to have reasonable information. In this
study, each simulation corresponding to a specific autopilot is repeated 100 times
along the trajectory defined below,
120
0 40000 12192
5000020000
f
f
h ft mx my m
= ==
=
(7.7)
In what follows, the results of the real environment simulations will be presented by
figures. The simulations are performed for rate autopilots only. The reason for this is
the fact that the dynamic projective acceleration autopilot is not robust as was shown
in the related sections. The results of the real environment simulations will be given
mostly in form of figures, showing variations of all trials superimposed on the same
figure.
For each of the static and dynamic projective rate autopilots, the following results
will be presented in this chapter:
1. Angle of attack and sideslip variations
2. Elevator, rudder and aileron deflections
3. Guidance command and turn rate variations
4. Roll angle variations
5. Velocity variations in Mach number.
7.3.1. Simulations of the System Including Static Rate Autopilot
Figure 7-1 Angle of attack variation
121
Figure 7-2 Elevator (pitch control surface) deflection
Figure 7-3 Pitch plane guidance command
Figure 7-4 Pitch rate
122
Figure 7-5 Angle of sideslip
Figure 7-6 Rudder (yaw control surface) deflection
Figure 7-7 Yaw rate guidance command
123
Figure 7-8 Yaw rate variation
Figure 7-9 Aileron (roll control surface) deflction
Figure 7-10 Roll angle variation
124
Figure 7-11 Roll rate variation
Figure 7-12 Velocity variation
125
7.3.2. Simulation of the System Including Dynamic Rate Autopilot
Figure 7-13 Angle of attack variation
Figure 7-14 Elevator deflection
Figure 7-15 Pitch plane guidance commands
126
Figure 7-16 Pitch rate variation
Figure 7-17 Sideslip angle variation
Figure 7-18 Rudder deflection
127
Figure 7-19 Yaw plane guidance command
Figure 7-20 Yaw rate variation
Figure 7-21 Aileron deflection
128
Figure 7-22 Roll angle variation
Figure 7-23 Roll rate variation
Figure 7-24 Velocity variation
129
The graphical presentations show the variations of most important missile system
variables during flight in a medium having several uncertainties. As expected, the
curves occurred as a band since there are a large number of trials. For some variables
the bands are large, for some others they are very small like single line. The suitable
values for the angle of attack and sideslip, roll angle and rate are as of the level for
linearization assumptions which are stated in Chapter 4. The aerodynamics
properties of the missile body are affected from control surface deflections. Also the
mechanical restrictions of the actuator require a suitable level of deflection angles.
Practical experiences show that the control surface deflections should not exceed the
range of 10o - 15o for a satisfactory operation.
The guidance commands ( ,q rr r ) and associated controlled variables ( ,q r ) are
following each other which is very important for an autopilot to operate successfully.
Their values are smaller than 2 / so which is suitable for a practical application of a
missile with moderate dynamics. The missile does not have very fast maneuvers.
130
CHAPTER 8
CONCLUSIONS
In this study, the method of projective control is applied to approximate linear
quadratic missile autopilot designs by using output feedback. The applications of the
theory lead to turn rate and acceleration autopilots which are implemented on an
aerodynamic missile with moderate dynamics and a stationary target.
The projective controlled autopilots are developed in two steps. First only an output
feedback loop is formed for both turn rate and acceleration autopilots. For this case
only turn rate autopilot is obtained as a stable product. Secondly, a low order
dynamic compensator is involved in both autopilots which solve the instability
problem of acceleration autopilot.
The designs in the scope of this study are aimed to be applied on a real system. In
order to determine whether that requirement is satisfied, a group of analyses are
established on the designs. First of all, the validity of approach used in linearization
of the real (nonlinear) missile model is checked for validity. To do that the guidance-
autopilot combination is simulated with real missile model to check whether the
linearization assumptions stated in Section 3.3.3 are satisfied. As a result of the
analysis, it is determined that the linearization approach is valid.
The next procedure is the investigation of the system behavior against the parameter
uncertainties and measurement noise that occur during flight in a realistic medium.
The stability of the overall system in presence of deviations in aerodynamic
coefficients and actuator frequency are analyzed through structured singular value
analysis using linear models. Both the static and dynamic projective rate autopilots
support 25% of deviation whereas the dynamic projective acceleration autopilot can
131
support less than 1% of deviation in some of operating conditions. This results in the
elimination of the acceleration autopilot from the candidate group of autopilots
usable in a real application because there is no guarantee for any of the parameters
to have a deviation always less than 1%.
In real environment simulations, flight simulations of the first investigation are
repeated by taking the most important uncertainties and noise possible during a real
flight. The analysis is performed for only the rate autopilots since the robustness of
the system with acceleration autopilot is very low for practical purposes. The
uncertainties are modeled according to the data obtained from practical experiences.
For both the static and dynamical projective rate autopilots the simulations are
repeated 100 times because there are many uncertain parameters and noisy signals.
The results of the last analysis show that the missile does not possess any instability
until hitting the target. Effects of uncertainties and noise occur in the control surface
deflections and aerodynamic angles as thick bands staying in acceptable ranges as
stated in Chapter 7.
From overall analyses, it can be concluded that the designed rate autopilots can be
used for the missile considered in this study. The unapproved characteristics of the
acceleration autopilot for this particular missile do not prevent the designer to apply
projective control theory in acceleration autopilot design in a different model. This is
a result of the model dependency of the linear quadratic methods. Other missile
models can produce more robust autopilot designs.
The theory of projective control approximates the properties of full state feedback
control approaches by feedback from available states (feedback from outputs).This is
an advantage for the designer because many of the full state feedback design
methods have systematic procedures like in the linear quadratic approach used in
this study. The advantage of the linear quadratic approach comes from the ability of
the designer to select a very simple cost function. In classical control approaches the
designer could determine the places of the eigenvalues through several pole
132
distribution templates, but has no idea about their influences on the state variables.
In linear quadratic methods there is a means of control over the decay rates of the
state variables. In several cases, a single weight on the integral of error signal is
enough. The minimization criterion in Section 4.2 is an example to that and it is
proven to be enough for autopilot design to missile model in this study and others
having similar characteristics. Although the projective control theory brings a means
of implementing missile controllers through available states, it has a disadvantage
that one can end up with an unstable outcome as encountered in Chapter 4.
Because of that disadvantage, at first look, a usage of a reduced order observer to
approximate a full state feedback design may seem to be more advantageous. An
observer (or a dynamic compensator) can be designed by both classical (Ogata(1997))
and projective control (Medanic (1985) and (1983)) methods. For linear models
involved in this research, usage of a dynamic compensator is possible because their
order is practically low. In Chapter 5, both autopilots require only a first order
compensator to place the spectrum of the full state feedback design. However, for
high order linear models (such as Bank – To – Turn autopilots which may have an
order of 8 or 9) the case is not that easy. For low order models, the number of
controller and compensator (observer) gains are in a practically possible level. On
the other hand, the compensator (observer) designs for high order models result in a
very large set of gains. If one also consider that the design should be repeated for
each linearization point then the overall design gives a very huge set of gains to be
interpolated during flight. The implementation of such control scheme increases
computational complexity and may decrease robustness to an impractical level.
For the cases where the implementation of an observer or a dynamic compensator is
not possible due to the above reasons, the projective control method serves as a tool
which produces a control structure directly from the results of a full state feedback
design. By this way, the design task can be automated to perform similar
computations for each linearization point to satisfy the design criteria. The overall
133
autopilot gain sets ready for online interpolation can be obtained without any
human intervention.
For future research, a new acceleration autopilot structure can be formed by closing
an acceleration feedback loop around the rate autopilots designed in Chapter 4 and
Chapter 5. This new structure has an advantage that it can be used both as a rate and
an acceleration autopilot. In addition, its robustness properties can be improved by
testing different outer acceleration loops. Another study can be performed by using
reference control design methods other than linear quadratic theory. There are many
robust control techniques such as H∞ , 2H and µ - synthesis.
134
APPENDIX A
CONTROL ACTUATION SYSTEM
In the introduction of the subject, it was pointed that the fin deflection commands
generated by the autopilot according to the desired acceleration or turn rate
commanded by the guidance system are realized by the control actuation system.
Control actuation system is itself a control system which takes the fin deflection
values produced by the autopilot as a command. Generally it is modeled by a second
order system function and a more corrected model can be presented by using non –
linear elements. These are a rate limiter and position limiter as shown in the system
diagram Figure A-1.
20
ω 1s
1s
20
ζω
δci 1
δ
Figure A-1 The nonlinear control actuation system
In the above figure the input and output of the system are defined different from the
conventional definitions which are , ,e r aδ δ δ for pitch, yaw and roll planes,
respectively. The term iδ represents each of the four fins constructed on the body of
the missile which are expressed in terms of the three control surface deflections
, ,e r aδ δ δ as follows (Atesoglu (1997)):
135
1
2
3
4
a r
a e
a r
a e
δ δ δ
δ δ δδ δ δδ δ δ
= +
= += −= −
(A.1)
The definition done above, are given for a body which is not rolling, if the missile is
rolling during flight, which is of course the actual case, then the effect of the rolling
angle should be taken into consideration as given below (Mahmutyazicioglu (1994)):
cos( ) sin( )cos( ) sin( )
res c ce e r
res c cr r e
δ δ φ δ φ
δ δ φ δ φ
= −
= + (A.2)
the superscript res is used for indicating the roll resolved deflection commands. The
resolved commands are distributed to the four fins according to (A.1).
In Figure A-1, there are two non – linear elements, the position limit directly limits
the magnitude of the fin angle in order to protect the system from high torques. The
second one which is a rate limiter sets a limit to the time rate of change of the fin
angle. This limit comes from the physical restrictions of the actuator. In simulations,
often an additional limiter is placed in the fin deflection command with the same
amount of limit as the angle limit. In modeling, the deflections of the four fins on the
missile body are reflected to the dynamic model of the missile by recalculating the
three conventional control surface deflections from the control actuation system
outputs as shown below:
( )
( )
( )
2 4
1 3
1 2 3 4
121214
e
r
a
δ δ δ
δ δ δ
δ δ δ δ δ
= −
= −
= + + +
(A.3)
In linear modeling the non – linear elements are omitted and a linear structure is
obtained as shown in Figure A-2.
136
20
ω 1s
1s
20
ζω
δci
δi
Figure A-2 The linear model of a control actuation system.
The mathematical expression of Figure A-2 is shown by a second order transfer
function given as:
2
2 22i oc
i o os sδ ωδ ζω ω
=+ +
(A.4)
where oω is the natural frequency of the actuator. In linear analysis, the distribution
of the actuator effect over the four fins is not taken into consideration. So the
subscript i is replaced directly by either ,e r or a . Typically, the natural frequency of
the actuator can take a value 60 – 100. For simulation purposes a damping ratio of 0.6
is adequate.
In robustness analysis, the linear fractional transformed form of the linear actuator
model should be used. The uncertainty in control actuation system is generally the
natural frequency. In this study, only uncertainty in the natural frequency will be
taken into consideration. However, the squared terms in numerator and
denominator of the transfer function makes the framework development difficult.
The problem is solved by taking the squared term as a product of two natural
frequencies ( 2o o oω ω ω= ). Taking each oω as a linear fractional transformed uncertainty
the actuator part of the analysis framework is obtained as in Figure A-3.
137
1δ
2δ
3δ
0 0
1 0ω ω
0 0
1 0ω ω
0 02 21 0
ζω ζω
1s
1s
, ,δc
e r a , ,δe r a
1z
1w
2z
2w
3z
3w
Figure A-3 The linear fractional transformed actuator model
In the above figure the term oω represents the nominal value of the actuator natural
frequency. The three uncertainties 1 2 3, ,δ δ δ are all pointing the same uncertain
variable so their values are equal ( 1 2 3δ δ δ= = ) so they are indicated by a 3 3×
diagonal uncertainty with same diagonal elements.
1z
2z
3z
1w
2w
3w
ecδ
eδ
Figure A-4 The LFT block form of the control actuation system
138
APPENDIX B
ROLL CONTROL SYSTEM
B.1. Design of the Roll Autopilot
The Skid – To – Turn missile autopilot design makes an assumption that all the
planes (roll, pitch and yaw) are decoupled from each other. This requires that the
rolling angle φ should be kept around zero. Because of that, an autopilot should be
designed in order to regulate the rolling angle to zero. This can be performed by
remembering the roll plane aerodynamic model state equation from Chapter 3:
0 1 00 a
pL p Lp δ
φφδ
= +
&& (B.1)
In order to regulate the roll angle to zero, the feedback of the control system should
be performed directly from the variableφ , since the second variable p in (B.1) is
available directly from the inertial measurement unit (or more generally the sensors)
a full state feedback control system can be designed. To ease the design of the
autopilot, the full state feedback control can be performed by pole placement
holding the closed loop poles at a constant place. The equivalence of (B.1) in s –
domain is written as,
( )
( )( )a p
Lss s s L
δφδ
=−
(B.2)
It can be understood from the above representation that the roll system has a natural
integrator, this greatly simplifies the design since there will be no need of an
artificial integrator in the controller. The roll control system with no artificial
integrator can be seen in figure below.
139
Figure B-1 The roll autopilot
A design procedure for this type of control systems is presented in Ogata (1997),
where state space methods are used. In order to perform the design, the external
input (or the set point rφ in Figure B-1) is assumed to be constant which is very
suitable for this case. When the loop in the figure is closed the following is obtained:
a p
a
K K K rp
K r
φ φ φ
φ φ
φδ
δ
= − + = − +KF
(B.3)
In the above approach, the control actuation system is again neglected for
simplification purposes.
If one generalizes the missile model in (B.1) as aφ φδ= +F A F B& where ( )pφ=F , the
closed loop dynamics can be obtained as:
( ) K rφ φ φ φ= − +F A B K F B& (B.4)
The steady state values are dropped out to have an error dynamics state equation,
given by:
( )( ) ( )K r rφ φ φ φ φ∞ ∞ ∞− = − − + −F F A B K F F B& & (B.5)
140
Since the reference input is taken as constant, the error of the reference will be zero
so the steady state error equation in (B.5) reduces to:
( )( )φ φ∞ ∞− = − −F F A B K F F& & (B.6)
The above equation obtained can be shortly written as ( )φ φ φ= −e A B K e& which is a
general error dynamics equation. The closed loop poles of the system can now be
selected according to (B.6). The desired poles of the system are equated to the
eigenvalues of the matrix ( )φ φ−A B K or to the roots of the determinant of the
matrix s φ φ− +I A B K . That is,
2
0 1 00
0 11 0 000 1
( )
ap
pp
p p
L p Lp
s K KL L
s L L K s L K
δ
φδ
δ δ φ
φφδ
= +
∆ = − + ∆ = + − + +
&&
(B.7)
For the current study a natural frequency of 20 rad/sec and a damping ratio of 0.7 is
selected. The second order characteristic equation ( 2 22 n ns sζω ω+ + ) will
be 2 28 400s s+ + . For all aerodynamic conditions, the closed loop poles will be placed
at the same location and be given by the previous characteristic equation.
For the example analysis condition at 0.86 , 5000V Mach h m= = , the elements of the
linear state equation in (B.1) will be obtained:
0.10236
132.89pL
Lδ
= −
= (B.8)
The control coefficients according to the design procedure presented up to here, are:
3.010.24003p
KK
φ =
= (B.9)
141
The closed loop poles and step responses are also given for showing the properties
of the design. The poles are come out to be at:
16 12j− ± (B.10)
with the step response as in Figure B-2.
Up to here, the necessary procedure for designing a roll autopilot is finished. The
next step is to analyze this autopilot in robustness in order to determine that the
autopilot design is successful. This will be performed by structured singular value
analysis, just like the other autopilots.
Figure B-2 The step response of the roll autopilot.
The properties of the figure above shows that the roll autopilot has a settling time of
about 0.175secst = and rise time of about 0.16rt = which is fast enough to satisfy the
decoupled missile model assumption.
142
B.2. The Structured Singular Value Analysis of Roll Autopilot
The most important process in robustness analysis was the development of a linear
fractional transformed framework for the design. To do that, first, schematic
diagram of the autopilot will be reconstructed for linear fractional transformation as
in figure below:
Kφ
Kp
p
rφ
φ
aδ
Figure B-3 The roll controller structure
In Chapter 7, the development of a linear fractional transformed framework was
introduced in detail. In Figure B-3 the schematics of the roll autopilot is presented.
φp
δL
aδ 1
s1s
Lp
Figure B-4 The schematic of the roll model
It can be remembered from Chapter 7, that for a simple model the uncertain
parameters which are Lδ and pL the uncertainty can be expressed as,
1
2
(1 )
(1 )p p
L L
L Lδ δ δ
δ
= +
= + (B.11)
where the terms with a bar on top are the nominal values of the uncertain
parameters. Again from Chapter 7, each uncertainty is expressed by linear fractional
143
transformation and the schematic diagram in Figure B-4 is redrawn by considering
this fact.
φp
1 0
L Lδ δ
1 0
L Lp p
1
δ
2δ
aδ 1
s1s
Figure B-5 The LFT framework of the roll model for analysis
The LFT model in Figure B-5 is combined with the control actuation system
described in Appendix A and the result of this interconnection is given in Figure B-6.
1z
2z
3z
4z
5z
φ
1w
2w
3w
4w
5w
cφ
p
Figure B-6 Closed loop interconnection of the LFT formed roll autopilot
The perturbation matrix is a 5 5× diagonal matrix with the first three of the diagonal
entries belongs to the control actuation system uncertainties and their values are all
equal. The combination in Figure B-6 should be connected to autopilot structure in
Figure B-3 and the uncertainty matrix to give the closed loop interconnection is
144
given in Figure B-7. The analysis is performed at the design conditions presented in
the previous section and again the upper bound of the structured singular value ( µ )
is plotted against frequency. The next figure shows this variation of structural
singular value.
1z
2z
3z
4z
5z
p
1w
2w
3w
4w
5w
acδ
φ
1
2
3
4
5
00
0 00 00 0
δδ
δδ
δ
00
0 0
Figure B-7 The closed loop interconnection for the roll autopilot analysis
145
Figure B-8 The structural singular value variation of roll autopilot
Before analyzing the system for robustness it will be useful to give the nominal pole
locations:
30.106 60.02123.945 12.05
jj
− ±− ±
(B.12)
As it can be seen from the variation curve, the peak of the upper bound occurs at a
frequency of 43.976 /rad sn with the value 2.3784 . This means a perturbation of
42.05% can be applied to the system to make it unstable. As in the previous analyses,
the sign of perturbations are effective in detection of the instability point. If the signs
in (B.13)applied to the obtained perturbation bound then instability is occurred in
the closed loop system.
146
1
2
3
4
5
0.42050.42050.42050.42050.4205
δδδδδ
== −= −= −= −
(B.13)
To show the occurrence of instability in the system a the closed loop poles of the
perturbed system is given in below,
0.20668 44.297
45.89817.161
j±−−
(B.14)
the above poles have a positive real part in complex conjugate pair and this causes
an oscillatory instability in the closed loop system which is demonstrated by the step
response in Figure B-9.
Figure B-9 The perturbed roll system’s step response
147
The overall robustness analysis (the analysis for all the operating points) is given in
Figure B-10.
Figure B-10 The overall robustness analysis of roll autopilot
148
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