HELSINKI UNIVERSITY OF TECHNOLOGY

Department of Engineering Physics and Mathematics

Systems Analysis Laboratory

Mat-2.4108 Independent research projects in applied mathematics

Control Design for a Tactical Missile Autopilot

an Application of Robust Control

Lauri Kovanen

19.10.2007

1Abstra tThis study presents a holisti design pro edure for an autopilot ontrollerof a ta ti al SAM missile. First, the missile dynami s were modelled usingthe most realisti non-linear model available, and then a simpler linearmodel was onstru ted to be the basis of the ontrol design.An autopilot ontroller using the H∞ framework was designed, and itsorder redu ed from 6th order into 2nd order. The resulting ontroller'sperforman e was evaluated both in terms of theory and simulations.Both the autopilot and the guidan e system were in luded in the mostrealisti missile model, whi h was used in evaluating the performan e ofthe resulting ontrol system. This way, a holisti design for the autopilotwas a hieved, as well as the bene�ts of the design pro edure were proven.Also the most ru ial performan e riteria were s reened.

2Contents1 Introdu tion 41.1 Ta ti al missile systems and ontrol theory . . . . . . . . . . . . 41.2 The purpose of this study and ex lusions made . . . . . . . . . . 41.3 Overview of the missile system . . . . . . . . . . . . . . . . . . . 52 Models Used in the Study 62.1 The "Real Model" . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 The Equations of Motion . . . . . . . . . . . . . . . . . . 62.1.3 The State-spa e Model . . . . . . . . . . . . . . . . . . . . 92.1.4 Numeri al values . . . . . . . . . . . . . . . . . . . . . . . 102.2 The Control System Design Model . . . . . . . . . . . . . . . . . 102.3 The Guidan e Law Model . . . . . . . . . . . . . . . . . . . . . . 123 Control System Design 133.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 H∞ performan e and (sub)optimal ontrollers . . . . . . . . . . . 133.3 Weighing Fun tion Sele tion . . . . . . . . . . . . . . . . . . . . . 163.4 H∞ optimal solution . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Controller Performan e Analyzed . . . . . . . . . . . . . . . . . . 193.6 Model redu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.2 Balan ed Realization . . . . . . . . . . . . . . . . . . . . . 223.6.3 Redu ed Model Formulation . . . . . . . . . . . . . . . . 233.6.4 Model redu tion omparison . . . . . . . . . . . . . . . . 244 The Guidan e System 254.1 Pit h rate ommand . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Throttle ommand . . . . . . . . . . . . . . . . . . . . . . . . . . 26

35 Control System Performan e Study by Simulations 265.1 The Simulation Framework . . . . . . . . . . . . . . . . . . . . . 265.2 Simulations performed . . . . . . . . . . . . . . . . . . . . . . . . 275.2.1 General about Simulation Conditions . . . . . . . . . . . 275.2.2 E�e t of Model Order Redu tion . . . . . . . . . . . . . . 275.2.3 E�e t of Target Maneuvering . . . . . . . . . . . . . . . . 305.2.4 Reasons behind Failure Analyzed . . . . . . . . . . . . . . 386 Con lusions 39

41 Introdu tion1.1 Ta ti al missile systems and ontrol theoryTa ti al missiles o�er an interesting appli ation for ontrol systems. The missilesystems an be divided into two fun tional subsystems, whi h an again bestudied independently, the other half abstra�ed away. Problem division wasessential also in this study, where the autopilot system was designed by �rstde�ning the interfa e between the two subsystems.Aerodynami s of the missile result in highly omplex mathemati al models. This omplexity an be redu ed using simpler models and linearization, among othermethods of model redu tion. The ne essity of these redu tions pose on their partanother hallenge to the ontrol system, requiring a strong level of robustnessof the ontroller. The missile is also mu h a�e ted by external noise, espe iallywind onditions.Simulations be ome an important method in studying the missile ontrol sys-tem's performan e, sin e experiments in real onditions are ostly and alwaysin lude a military aspe t. Due to the aerodynami omplexity and oordinationtransformations, even the formulation of the simulation model is a demandingtask.1.2 The purpose of this study and ex lusions madeThe purpose of the study was to model the dynami s of a ta ti al missile, designa ontrol system using robust ontrol design methods and �nally implement this ontroller into a simulation model and study the ontroller performan e in real-isti tasks by simulations. Spe i� on ern was laid on the e�e t of various modelredu tions to the perfoman e of the ontrol design and the resulting missile. Thegoal of this study was, a ordingly, a rather holisti ontrol design, validatedwith extensive simulations made using the most realisti model available.The study was restri ted in the verti al plane, so the aerodynami s were de-s ribed only in two dimensions, and the rotational motion of the missile repre-sented by only the pit h rate. This approa h was hosen in order to maintainthe problem omplexity in a onvenient level. The study ould later be extendedinto three dimensions in a quite straightforward way.The missile was assumed to be a surfa e-to-air (SAM) missile apable to velo -ities faster than the target, and no de ision theoreti al aspe ts of the pursuitsituation was modeled. This was a ons ious simpli� ation, reating a sensibleframework for studying the ontroller performan e.In addition, the missile sensor and a tuator dynami s were either negle ted orhighly simpli�ed, thereby on entrating only on the performan e of the designed ontrol system.

51.3 Overview of the missile systemConsidering this study, the most important subsystems of the missile are theautopilot system and the ommand guidan e system. The purpose of the latter isto generate guidan e signals, most often in terms of desired normal a elerationor pit h rate, so that the �ight path of the missile is optimal. Optimality in this ase means most often minimum miss distan e, minimum �ight time or the twoof these ombined. The autopilot system gets the optimal ommand signal fromthe guidan e system as referen e, and tries to make the missile tra k it usingthe ontrol surfa es.A blo k diagram of the missile system is shown in �gure 1. Other importantsubsystems in lude the observation system and the warhead of the missile, butthese are not onsided in this study. Instead, the important state variables aresimply onsidered measurable, and target destru tion is assumed within a spe -i�ed distan e from the missile.

Figure 1: The missile system blo k diagram.The inputs of the guidan e system are the missile and target positions and ve-lo ities; the guidan e system is dis ussed in more detail in se tion 4. The outputof the guidan e system is a signal representing the optimal normal a eleration(or pit h rate).The autopilot system uses this referen e as its input, and produ es the ommandsignal to the a tuator surfa es. These surfa es ontrol the roll, pit h and yawrate, respe tively. Be ause this study is restri ted in the verti al plane, onlythe elevator surfa e ontrolling the pit h rate is taken into a ount. Thus, ourautopilot be omes a SISO system.

62 Models Used in the StudyThe study involves two di�erent mathemati al models of the missile dynami s,and yet one more model, whi h in ludes also the target motion and is used forthe guidan e system design.2.1 The "Real Model"2.1.1 GeneralThe most realisti model was based on a six degree-of-freedom model of themissile aerodynami s, presented in [1, p. 33℄. In addition to the missile's longi-tudinal motion, the rotational dynami s were modelled. This was done by takingthe full model of the missile dynami s presented and dropping out longitudinalmotion in the y axis dire tion. Rotational motion was des ribed only as pit hrate, that is, the rotation in the xz-plane.The lift and drag oe� ients of the aerodynami for e resultant were modelledas nonlinear. The amount of fuel de reases as a fun tion of the thrust for e, sothe mass of the missile does not remain onstant in our model.The airframe was, however, onsidered onstant, be ause in the operationalheight of the simulated missions (0-500 m) the e�e t of hange in the air densitywas quite minimal. The air density was hosen to be at the sea level a ordingto the ARDC 1 model atmosphere referred in [1, app. D℄.2.1.2 The Equations of MotionThe missile equations of motions were �rst des ribed in terms of the missilebody axis. In the simulation model, these state variables are then transformedto the world referen e axis to a hieve omparability with the target motion.The missile is a�e ted by the aerodynami for e, whi h an be divided into liftand drag oe� ients. In addition to these, it is a�e ted by the gravity and thethrust for e produ ed by the missile's ro ket engine. The missile and the for es(in relation to the earth referen e axis) are des ribed in �gure 2.The �ight path angle γ depi ts the angle between the positive earth referen edx axis and the missile velo ity relative to the airframe. Angle-of-atta k α isthe angle between the longitudinal axis of the missile and the velo ity, and thepit h angle θ is the angle between the longitudinal axis and the positive bodyaxis referen ed x axis, respe tively.The lift for e an be des ribed as a produ t

L = qSCL(α) , (1)1Air Resear h and Development Command, of the U.S. Air For e

7

Figure 2: For es a�e ting the missile with respe t to the earth referen ed axis.where q is the dynami pressure, de�ned asq =

1

2ρV 2 , (2)and S is the missile surfa e area. Lift oe� ient CL = CL(α) is a nonlinearfun tion of the angle-of-atta k.Drag for e is de�ned orrespondingly as

D = qSCD(α) , (3)where the drag oe� ient an be expressed in terms of the zero drag oe� ientand the lift oe� ient [1, p. 54-61℄:CD(α) = CD0

(V ) + KC2

L(α) . (4)Now referring to �gure 2 we an write the for es and moments a ting on themissile body. The for es along the body referen e axis are{

FL = T + L sin(α) − D cos(α) − G sin(θ)FN = −D sin(α) − L cos(α) + G cos(θ)

. (5)The moments in lude the moments resulting from angle-of-atta k and elevatorsurfa e angle (elevation angle), and also the hange in the position of the missile's

8 enter of gravity. The enter of gravity moves be ause the amount of fuel, andthis way the weight of the fuel tank, de reases as the thrust engine is used. Thefor e ausing this moment is the normal for e resultant, and an a ordingly beexpressed asFM = D sin(α) + L cos(α) − G cos(θ) . (6)The torque arm here is the di�eren e between the referen ed enter of gravityand the a tual enter of gravity that moves as a fun tion of time:

ccg = dcg(t) − drefcg . (7)Here d denotes the distan e from the tip of the missile, and is negative. dref

cg isthe initial enter of gravity. [1, p. 62-70℄The moment of inertia being denoted with Iy, the total moment due to hangein the enter of gravity be omesMcg =

1

Iy

(dcg − drefcg )(D sin(α) + L cos(α) − G cos(θ)) . (8)For determining the moment of inertia for rotation in the xz-plane, the missile an be onsidered as a uniform rod that has the axis of rotation at its enter.The moment of inertia then be omes

Iy =1

12ml2 , (9)where l is the length of the missile.The moments due to angle-of-atta k and elevation angle are more ompli atedto derive, and usually the nonlinear for es are determined using wind-tunneltests [1, p.63℄. For this study these moments an be simpli�ed using equations [2,p. 233-238℄

{

Mα = cαCα · α

Mδ = −cδCδ · δ, (10)where c is the torque arm, and C is a oe� ient representing the relation of theangle and the for e resulting from that angle.The missile is assumed to have been designed so that Mδ has the opposite signthan Mα. This an be a omplished by pla ing the enter of pressure betweenthe enter of gravity and the tail of the missile, and this ondition is a tually aprerequisite for a hieving a stable missile. [1, p. 80℄

92.1.3 The State-spa e ModelThe state variables an be hosen as the velo ity of the missile in dire tion ofthe longitudinal and lateral missile axis, the rotational (pit h) rate, and the orresponding angle. When simulating the missile �ight, these state variableshave to be transformed from body axis oordinates into earth referen e axis.This simulation pro edure is illustrated in �gure 3.

Figure 3: Flow hart of the simulation pro edure.The pit h rate has to be oupled into the body axis state-spa e equations. Usingthis oupling and ombining the prementioned equations for longitudinal androtational motion, we obtain [1, p.28-33℄

u = −Qw + 1

m(T + L sin(α) − D cos(α)) − g sin(θ)

w = Qu + 1

m(−D sin(α) − L cos(α)) + g cos(θ)

Q = qS[

Mα + Mδ + 1

Iy(dcg − dref

cg )(D sin(α) + L cos(α) − mg cos(θ))]

θ = Q

,(11)u representing the velo ity in the (body axis) x dire tion, w the velo ity in thez dire tion, Q the pit h rate and θ the pit h angle, respe tively.Note that the inputs to this body axis model in lude the airspeed Vair , theangle-of-atta k α and the ontrols (the throttle for e T and the elevation angleδ).In addition to the body axis model, we need dynami s for the a tuators andthe missile mass redu tion during the �ight for the model to be omplete. Thea tuators an be represented as �rst order dynami s:

{

T = 1

τTT (uT − T )

δ = 1

τδδ(uδ − δ) , (12)

10where τi is a onstant des ribing the settling time, and ui the input signal.The thrust for e output is assumed to be linearly related to the mass of the fuelburned at ea h unit of time. The dynami s for the missile mass now followm = KP T , (13)where KP is a onstant des ribing the fuel energy ontent. The mass dynami sis restri ted by inequality mempty < m(t) < mfull.Elevation angle is onstrained by |δ(t)| ≤ δmax, and the throttle for e by T (t) <

Tmax.2.1.4 Numeri al valuesModel oe� ients and their numeri al values were ombined from [1℄ and [3℄,and are presented in table 1. Ma h number M(v) in the lift and drag oe� ientsis de�ned as the relation of missile velo ity and the speed of sound:M(v) =

V

c0

. (14)2.2 The Control System Design ModelA linearized model of the missile dynami s was used in the ontrol system design.This model negle ts longitudinal dynami s, and assumes only rotational motion,sin e it is the key to ontrol the missile normal a eleration (or pit h rate) inorder to a hieve the optimal �ight path and eventually ollision.The linearized dynami s of the missile an be represented using the followingequations [4, p. 76-78℄:

α = q + Zα

Vα+ Zδ

Vδ

q = Mα · α + Mδ · δ

δ = 1

τδ(uδ − δ)

. (15)The elevator dynami s is now identi al to the "real model". Also the se ond stateequation of this model is quite similar to the more omplex model, negle tingonly the moment indu ed by the hange in the missile's mass. On the other hand,the nonlinearities of the drag and lift oe� ients are simpli�ed into onstantsZα and Zδ. Moreover, the missile velo ity is assumed onstant.The pit h rate was hosen to be the output variable, mainly be ause the feedba kloop using it was more easily onstru ted in the simulation model. Input of themodel is the ommand signal to the elevator, uδ.

11Des ription Coe� ient ValueAtmosphere and earthAir density ρ 1.2249 kg/m3A eleration due to gravity g 9.80665 m/s2Speed of sound c0 340.3 m/sMissile dimensionsEmpty missile mass mempty 280 kgInitial mass of fuel m0

fuel 250 kgMissile length l 1.80 mMissile surfa e area S 0.150 m2Missile rotational hara teristi sCenter of gravity of empty missile demptycg 0.38 lCenter of gravity of full fuel tank dfuelcg 0.63 lMissile enter of pressure dcp 0.625 lElevator surfa e enter of pressure dδ 0.92 lAngle-of-atta k moment oe� ient Cα -0.01Elevator moment oe� ient Cδ -0.5Missile lift and dragLift oe� ient CL(α) 2.93 + 0.34008M

+0.2615M2 + 0.0108M3Zero drag oe� ient CD00.45 − (0.04/3)MIndu ed lift oe� ient K 0.053Missile ontrol and fuel onsumptionMinimum throttle Tmin 0 NMaximum throttle Tmax 100630 NFuel energy oe� ient KP -0.00038 Ns/mThrust engine delay onstant τT 0.05Maximum elevation angle δmax 60◦Elevator delay onstant τδ 0.05Table 1: Numeri al values for the "real model".Throttle is not in luded in this model as an input. Constant velo ity does notimply onstant throttle; instead, one ould de�ne another ontrol system, thepurpose of whi h would be to keep the velo ity onstant. In this study we use amore simple approa h, where throttle is dependent on the measured distan e be-tween the missile and the target, resulting, naturally, in a time-varying velo ity(see hapter 4).The linear state-spa e representation of the model an be derived from equa-tion 15 as

12

αq

δ

=

Zα

V1 Zδ

V

Mα 0 Mδ

0 0 1

τδ

αqδ

+

001

τe

u

y =[

0 1 0]

αqδ

, (16)where, by repla ing numeri al values, we obtainA =

−111.7 1 −37.12−0.2393 0 −41.96

0 0 20.00

(17)B =

[

0 0 20.00]T (18)

C =[

0 1 0] (19)

D = 0 . (20)The transfer fun tion of the system now be omesG(s) =

−839.2 s − 9.358 · 104

s3 + 91.72 s2 − 2234 s − 4.787. (21)2.3 The Guidan e Law ModelThe third missile dynami s model used in the study formed the basis for theguidan e law being used. The purpose of this model is to des ribe the relation-ship between the missile and the target, making it possible to determine theoptimal a eleration of the missile in order to a hieve ollision.The guidan e law used in this study was proportional navigation. This law anbasi ally be derived from the following state-spa e model of the missile-targets heme [4, p. 367-368℄:

{

λ = 1

V T2 z

z = TaN. (22)Here λ is the line-of-sight to the target and z is the proje ted miss distan e,if λ remains un hangeable. V is the missile velo ity relative to the target, and

T = T − t is "time-to-go", a quantity assumed to be known, and des ribing thetime to ollision. The model input is aN , the missile normal a eleration.Both states are assumed to be measurable, and the target velo ity is assumed onstant. In spite of several of simpli� ations of this model, the guidan e lawderived from it has been proven to be very e�e tive. Another upside is also theease of its implementation. [4, p. 367-369℄

133 Control System Design3.1 GeneralControl design in this study was made using robust ontrol te hniques. Robust-ness of the resulting ontroller was essential be ause of the numerous un ertain-ties related to the task. First, the missile dynami s model was linearized for the ontrol system design, simultaneously making many simpli� ations about thequantity and onstantness of ertain state variables. Se ond, the �ying missileis exposed to di�erent external disturban es in form of sensor noise and wind.Even the target behaviour an be onsidered as external noise, sin e its a el-eration was not part of our model used in guidan e law design. Last, the use ofpit h rate as feedba k variable instead of normal a eleration an be onsideredas an un ertainty-in reasing fa tor.Espe ially important from the point of view of this study were the internaldisturban es resulting from simpli�ed modeling, sin e they are known to existand have a relatively large magnitude.3.2 H∞

performan e and (sub)optimal ontrollersRobust performan e of a ontroller is always a trade-o� between the two on-�i ting requirements: robustness and performan e. In the ontrol design problemof this study, the design obje tives are:• Minimize the pit h rate error with respe t to the referen e signal• Minimize the e�e t of disturban e to pit h rate• Keep the ontrol signal mostly inside ertain boundariesIt is obvious that �rst two obje tives are on�i ting. In addition to them, we havea state onstraint that again downgrades the ontroller performan e, limitingthe maximum ontroller gain.These requirements an be dealt e�e tively using the H∞ framework. The prin- iple of H∞ optimal design is to build a ontroller, whi h minimizes somesignals in the losed-loop system. The quantities of these signals an be rep-resented in terms of norms, thus transforming the H∞ ontrol design into anorm-minimizing task. [5℄The ∞-norm of a signal an be represented as the least upper bound of itsabsolute value: [5, p. 12℄

‖u‖∞ := supt

|u(t)| . (23)The original plant blo k diagram has to be augmented to in lude the importantsignals (the norms of whi h are to be minimized). This is done by in luding

14disturban e and noise as inputs, and adding spe i� weighing transfer fun tionsinto the model. The role of the weighing fun tions is both to normalize therequirements and give ertain frequen ies spe i� importan e; for example, wewant the system output to tra k the referen e signal only in low frequen ies,whereas high frequen y inputs are most probably noise, and an be negle ted. [6,p. 85-89℄The modi�ed open-loop system for the H∞ minimizing task is presented in �g-ure 4. P represents our plant, and there are three weighting fun tions in luded:We weighs the performan e of the ontroller, Wu weighs the ontrol signal quan-tity, and Wd weighs the external disturban e. It is worth pointing out that thetwo �rst weights di�er from last one: We and Wu weigh the important fre-quen ies of two system outputs, and are hen e shaping the output signals, thenorms of whi h are involved in the H∞ minimizing task. Wd, on the other hand,des ribes the frequen y ontent of the external disturban e.

Figure 4: Open-loop system for H∞ optimizing task.The feedba k loop is onstru ted from y to u, whereas signals u and e are theones to be minimized by the resulting ontroller.This extended MIMO system an be des ribed as a partitioned system matrixof the form [6℄G(s) =

A B1 B2

C1 0 D12

C2 D21 0

. (24)

15Assuming the following:i) (A, B1) is ontrollable and (C1, A) is observable;ii) (A, B2) is stabilizable and (C2, A) is dete table;iii) D∗

12

[

C D12

]

= [0I];iv) [

B1 D21

]

D∗

21= [0I] ,it has been shown [6, p. 270-277℄ that there exists a admissible ontroller su hthat the robust performan e norm asso iated with the open-loop system rea hesa level below γ, that is, ‖Tzw‖∞ < γ, if and only if the following onditions hold:i) H∞ ∈ dom(Ric) and X∞ := Ric(H∞) > 0 ;ii) J∞ ∈ dom(Ric) and Y∞ := Ric(J∞) > 0 ;iii) ρ(X∞Y∞) < γ2 ,where

H∞ :=

[

A γ−2B1B∗

1− B2B

∗

2

−C∗

1C1 −A∗

] (25)J∞ :=

[

A∗ γ−2C∗

1C1 − C∗

2C2

−B1B∗

1−A

]

. (26)These onditions holding, one su h ontroller is [6, p. 271℄Ksub(s) :=

[

A∞ −Z∞L∞

F∞ 0

]

, (27)whereA∞ := A + γ−2B1B

∗

1X∞ + B2F∞ + Z∞L∞C2, (28)F∞ := −B∗

2X∞, (29)

L∞ := −Y∞C∗

2 , (30)Z∞ := (I − γ−2Y∞X∞)−1. (31)Sin e it is virtually useless to ompute all the H∞ optimal ontrollers [6, p. 269-270℄, we shall on entrate on this realization of a suboptimal ontroller (27). Thesystem matrix G(s) de�ned, this ontroller an easily be solved with Matlab'shinfsyn ommand. Before entering this stage, we �rst have to take a loser lookat the weighing fun tions.

163.3 Weighing Fun tion Sele tionEquation (27) is guaranteed to result in an H∞ suboptimal ontroller, giventhe weights Wu, Wd and We. The question arising is ommon to all weightedoptimization problems: How to hoose the performan e weights to a hieve the"best among the optimal solutions"?The weighing fun tion sele tion is generally onsidered as a rather demandingtask [5, 6℄. The weight sele tion be omes, thus, a massive iteration pro ess,where the performan e of the resulting ontroller is evaluated after ea h weightadjustment. The �rst task onsidering the weights now be omes hoosing the orre t shape of the frequen y response magnitude plot.We is the performan e weight, and should be large in those frequen ies, where agood performan e is desired. Usually this frequen y band is the low frequen ies,as also in this ase. This way the resulting ontroller attempts to tra k lowfrequen y referen e signals, but high frequen y signals, whi h are most likelynoise, are negle ted. We has then the same shape than a low-pass �lter, themagnitude rolling o� at ertain frequen y.Wu des ribes the frequen ies where the ontrol signal has to be kept small. Sin ehigh-frequen y ontrols are those, whi h the ontroller is assumed to negle t to ertain extent, there is no reason to downgrade the performan e of the ontrollerby restri ting the ontrols also in these frequen ies. The ontrol weight Wu hasa ordingly the same shape than We: ontrols are restri ted, but only at lowfrequen ies.Transfer fun tions for We and Wu now have the following generalized form:

We,u(s) =µ

1 + κs(µ, κ ∈ R) . (32)The disturban e weight Wd is used to des ribe the frequen y ontent of thedisturban e. Usually this an be assumed to be high at high frequen ies, and rollo� at low frequen ies [5℄. The weight Wd thus be omes a high-pass �lter. Sin ethe algorithm used for the ontroller synthesis requires the weighing fun tionsto be stri tly proper, we add a far-away pole, a hieving

Wd(s) =µs + 1

ǫs + κ(µ, κ ∈ R) , (33)where ǫ ≪ max(µ, κ).After determining the shape of the magnitude plots of the weights, the parame-ters in their transfer fun tions were hosen after areful iteration. The obje tiveof the iteration was to gain a pit h rate of 1 rad/s in a time of less than 1 s,with a reasonable overshoot. Therefore, the performan e of the ontroller wasevaluated based on the response to a step input of 1 rad/s.After ea h weight adjustment the H∞ optimal ontroller was solved, and itsperforman e evaluated using step response and state traje tory plots. The best

17a hieved weights resulted in a su� ient ompromise between the performan e ofthe ontroller, in other words, the hara teristi s of the step response, and themagnitude of the elevation angle. The step response is shown in �gure 5, andthe orresponding ontrol signal and the elevation angle traje tory in �gure 6.

Figure 5: The losed-loop step response with the hosen ontroller.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Figure 6: Elevation angle traje tory in step response with the hosen ontroller.The weights resulting for the best performan e were

18We =

0.006s + 90

15s + 0.1(34)

Wu =0.72

0.45s + 1(35)

Wd =0.15s + 15

0.0001s + 10. (36)The frequen y response magnitude urves of these weights are shown in �gure 7.

10−4

10−2

100

102

104

106

−80

−60

−40

−20

0

20

40

60

80

Mag

nitu

de (

dB)

We

Wu

Wd

Frequency response

Frequency (rad/sec)Figure 7: Frequen y response magnitude plots for the performan e weights.3.4 H∞

optimal solutionConsidering the determined weights, (35-36) the transfer matrix of the weighedopen-loop system be omesG(s) =

−110.0 1.000 −37.00 0 0 0 0 0−0.240 0 −42.00 0 0 0 0 0 0

0 0 20.00 0 0 0 0 0 20.000 2.000 0 −0.0067 0 −18290 3000 0 −2.0000 0 0 0 −2.200 0 0 0 2.0000 0 0 0 0 −100000 16000 0 00 0.00040 0 3.000 0 −3.700 0.600 0 −0.00040 0 0 0 1.600 0 0 0 00 1.000 0 0 0 −9150 1500 0 0

.

(37)

19The H∞ ontroller was solved with Matlab's hinfsyn ommand, using toleran e0.01. The resulting ontroller wasC(s) =

P (s)

Q(s), (38)where

P (s) = s6 + 3.327 · 104 s5 + 1.051 · 109 s4 + 1.238 · 1011 s3

+7.6891011 s2 + 1.11 · 1012 s + 2.375 · 109 (39)Q(s) = s6 + 2.459 · 104 s5 + 1.344 · 108 s4 + 3.491 · 1010 s3

+2.2991012 s2 + 5.005 · 1012 s + 2.828 · 1010 (40)3.5 Controller Performan e AnalyzedWhile hoosing the weighing fun tions, the performan e of this ontroller wasexamined using intuitive step response and elevation angle traje tory plots.The performan e an more formally be studied by su h on epts as nominalperforman e, robust stability and robust performan e.The formal treatment of these on epts requires us to de�ne an un ertaintymodel for the plant transfer fun tion. Without onsidering, how these un er-tainties are generated, we an refer to the set of all possible plant transferfun tions with Π, dis ussed in more detail below.The relationships of the three performan e measures are shown in �gure 8. Nom-inal performan e means the ontroller's performan e with respe t to the nomi-nal plant, whereas robust stability indi ates losed-loop system stability for allplants Π. Last, robust performan e measures the ontroller's performan e withrespe t to the whole model set Π. Clearly, robust performan e is the stri testmeasure. [6, 5℄

Figure 8: Mutual relationships of the three performan e measures used.Furthermore, we need two transfer fun tions des ribing the losed-loop system:

20S(s) =

1

1 + P (s)C(s)(41)is the sensitivity fun tion, and

T (s) = 1 − S(s) =P (s)C(s)

1 + P (s)C(s)(42)is the omplementary sensitivity fun tion. Now these fun tions de�ned, we anstudy the robust performan e of the ontroller with respe t to some un ertaintymodel. Be ause of the omplexity of our model (see equation 15), the un ertaintyis best to be modeled as unstru tured. This means that the frequen y ontentsof the un ertainty are en losed inside a urve, whi h then represents the "worst ase un ertainty".For simpli ity, let us assume the un ertainty to be of multipli ative type. Thisis a safe assumption, sin e the performan e will eventually be studied also usingsimulations, and the results of this analysis are only suggestive. For a SISOsystem, multipli ative perturbation an be des ribed as

Π := (I + W2∆)P , (43)where W2 is a transfer fun tion des ribing the frequen y ontents of the pertur-bation, and ∆ is the normalized perturbation satisfying ‖∆‖ ≤ 1.The ontroller performan e in the sense of nominal performan e, robust stabilityand robust performan e an be measured in terms of weighed norms. For a SISOsystem and multipli ative perturbation model, these norms are as follows:Nominal performance : ‖W1S‖∞ (44)

Robust stability : ‖W2T ‖∞ (45)Robust performance : ‖|W1S| + |W2T |‖∞ (46)Performan e is guaranteed, if the orresponding norm has a value less than 1. [6,p. 147-149℄It is most onvenient to have the already de�ned performan e weight to weighthe nominal performan e norm, that is, W1 = We. Sin e our ontroller designwas made so that performan e orresponding this riterion was guaranteed, itis no surprise that the nominal performan e norm satis�es the performan e riterion:

‖WeS‖∞ = 0.65 . (47)

21For robust stability and robust performan e, the analysis be omes again a pur-suit after admissible weighing fun tions, with whi h the norms satisfy the perfor-man e riteria. For robust stability, the following weighing fun tion was found:WRS

2 =0.1s + 1.5

6, (48)for whi h the performan e norm has the value

‖WRS2

T ‖∞ = 0.96 . (49)Singular value plots for transfer fun tions WeS and WRS2

T are shown in �g-ure 9. There we see that for both transfer fun tions, the peak o urs at quitesimilar frequen ies. This implies that this frequen y range ould be importantin enhan ing the ontroller's performan e.

Figure 9: Singular value plots for nominal performan e and robust stabilitytransfer fun tions.Nominal performan e and robust stability are important features of the de-signed ontroller, but even together they don't imply robust performan e, thatis, performan e for the whole plant set Π. Indeed, for robust performan e, anadmissible weighing fun tion was not found in spite of extensive iterations. Evenfor a weighing fun tionWRP

2=

0.1s + 1

500(50)

22the robust performan e norm has a value of ‖ · ‖∞ = 3.06. The failure of the ontroller with regard to this riterion an be seen in the �gure 10, where thefrequen y response magnitudes of the perturbation weighing fun tions are il-lustrated. We see that for example at frequen y w = 10rad/s even as smallperturbation as of magnitude |W2∆| = 0.01 results in bad performan e. Onthe other hand, robust stability is guaranteed for relatively large perturbationmagnitudes.

Figure 10: Frequen y response magnitude plots for perturbation, for whi h ro-bust stability is a hieved. In addition, the magnitude is plotted for the pertur-bation, for whi h a robust performan e level of 3.06 is a hieved.3.6 Model redu tion3.6.1 GeneralSin e the resulting ontroller is of order 6, it is very probably bene� ial to tryredu ing its order. Lower-order ontroller is generally easier - and that way heaper - to implement than the full-order optimal ontroller.The algorithm used in the model redu tion is pra ti ally divided in two parts.First, the balan ed realization is al ulated, and based on the ontrollability andobservability matri es of this realization, the model order is further redu ed.3.6.2 Balan ed RealizationThe balan ed realization for a general, linear state-spa e model

23{

x = Ax + Buy = Cx + Du

(51)is formulated using a state variable transformation of the formx = Tx . (52)The algorithm used results in a transformed state variable x so that the balan edsystem has the observability and ontrollability matri es being diagonal andidenti al:

Wo = Wc = diag(g) . (53)The diagonal items gi an be used to delete nonsigni� ant states from the model.The algorithm for �nding the state transformation is des ribed in [7℄ and nottreated here.The diagonals for the balan ed realization of the H∞ optimal ontroller (38)were al ulated using Matlab, and are shown below:g =

[

4.0209 3.8902 0.5186 0.0689 0.0007 0.0004]

. (54)Clearly, the two last states an be deleted, sin e the orresponding diagonalsare signi� antly smaller ompared to the others. Being more on�dent, even thenext two states ould be deleted, resulting in a ontroller of only se ond order.This would de�nitely be a good goal, so we shall build both the two redu tionsand study their hara teristi s in the frequen y plane.3.6.3 Redu ed Model FormulationThe method used for model redu tion produ es a redu ed-order model with asteady-state (step) response mat hing to the original model. The prin iple ofthe method is to make the derivatives of the states to be deleted zero, and thento solve the remaining states [7℄. The model is �rst partitioned into state groupsx1 and x2, the former being preserved and the latter being deleted:

[

x1

x2

]

=

[

A11 A12

A21 A22

] [

x1

x2

]

+

[

B1

B2

]

u

y =[

C1 C2]

x + Du. (55)After setting x2 to zero, the redu ed-order model an be represented as

{

x1 = (A11 − A12A−1

22A21)x1 + (B1 − A12A

−1

22B2)u

y = [C1 − C2A−1

22A21]x + [D − C2A

−1

22B2]u

. (56)

24Using this method, the resulting redu ed-order ontrollers for (38) areG4(s) =

0.9978s4 + 3.314 · 104s3 + 1.039 · 109s2 + 4.382 · 109s + 9.384 · 106

s4 + 2.431 · 104s3 + 1.307 · 108s2 + 1.973 · 1010s + 1.117 · 108 (57)for the fourth-order ontroller, andG2(s) =

−0.1773s2 + 5.315 · 104s + 8.043 · 104

s2 + 6758s + 9.577 · 105(58)for the se ond-order ontroller, respe tively.3.6.4 Model redu tion omparisonThe bode diagrams of the original and redu ed-order ontrollers are shown in�gure 11. In the �gure we an see that the frequen y response of the fourth-ordermodel is virtually identi al to the original, full-order, ontroller. This impliesthat at least the two last states an be deleted.

−300

−250

−200

−150

−100

−50

0

50

100

Mag

nitu

de (

dB)

10−4

10−2

100

102

104

106

−180

−135

−90

−45

0

45

90

Pha

se (

deg)

G6(s)

G4(s)

G2(s)

Figure 11: Frequen y responses for the original and redu ed-order ontrollers.However, also the greater redu tion results in a very similar frequen y response.The di�eren e to the original ontroller model appears only in very high frequen- ies, whi h again implies that even model order redu tion of this level ould bea eptable. Sin e the ontroller is spe i� ally not meant to tra k high frequen yinputs, we de ide to hoose the se ond-order ontroller.

25

Figure 12: Nyquist plots for the original (dotted line) and se ond-order (solidline) ontrollers.As seen in the nyquist plots in �gure 12, the redu ed model has in fa t bettergain margin than the full-order model. This refers to better robustness, althoughmost possibly with the ost of lost performan e.4 The Guidan e System4.1 Pit h rate ommandThe missile-target model that was the basis of the guidan e law derivation waspresented in hapter 2.3. In prin iple, several di�erent feedba k guidan e laws an be derived from this model using dynami optimization.Whereas in the autopilot design a method was used to minimize an H∞ normof the system, the guidan e law derivation involves minimizing an H2 norm. Aspe ial ase of proportional navigation is a hieved, if the optimization riterionis hosen to beJ = k2z2(T ) +

∫ T

t

a2

N(τ)dτ , (59)where k des ribes the importan e of the miss distan e at �nal time T . Thederivation of the proportional navigation guidan e law in ludes setting k → ∞,whi h pra ti ally means that the target has to be hit at all osts. Solving theoptimal guidan e law with this addition results in

26aN (t) = −3V (t)λ(t) , (60)where λ is the line-of-sight angle and V the missile velo ity relative to the target.Repla ing the onstant with N gives us the general proportional navigationformula. Usually a value between 3 < N < 5 used. [4, p. 367-369℄The guidan e law used with the ontrol system derived in hapter 3 is a modi�edversion of this general proportional navigation law: the normal a eleration isrepla ed with the pit h rate. Intuitively, this orresponds to using normal a el-eration, but it requires us to �nd a suitable value for the stati gain onstant.After some simulations, a de ent value was found to be N = 0.015.4.2 Throttle ommandThe proportional navigation now derived onsiders only the pit h rate ommanddelivered to the autopilot. In addition, the desired throttle for e should be de-termined. Not mu h on ern was laid on this, but instead a simple approa hwas taken.A good starting point was to make the throttle dire tly proportional to thedistan e between the missile and the target. The throttle should not, however,go to zero when the target is "near", so the relation was made logarithmi .Again after some simulations, the following throttle feedba k law was found tobe good enough:

T (d(t)) = Tmax

log(d0)

log(d(t)), (61)where d0 is the initial distan e between the target and the missile. A plot of thethrottle ommand is shown in �gure 13.5 Control System Performan e Study by Simu-lations5.1 The Simulation FrameworkThe "real model" was built using Simulink, and it was equipped with the de-signed ontroller and the guidan e system in order to simulate the missile per-forman e in a target en ounter situation. A blo k diagram of the simulationmodel is shown in �gure 14.The missile body axis model has airspeed, angle-of-atta k, elevator angle andthrottle for e as its inputs. Outputs, missile velo ity and rotational motion, arethen transformed into earth referen e axis velo ity and orientation. The missile

27

0 500 1000 1500 2000 2500 30000

2

4

6

8

10

12x 10

4

d

T

Throttle command

T(d)T

max

Figure 13: The throttle ommand as a fun tion of distan e between the targetand the missile.a tuators are des ribed in a separate blo k getting its inputs from the guidan esystem. In this study, the observation system is a dummy blo k, observing themissile and target states with a perfe t a ura y.5.2 Simulations performed5.2.1 General about Simulation ConditionsIn all the simulations, the missile was laun hed at the origin of the xz-plane.The target initial height was 500 m in all the simulations. In the s enarios wherethe target was moving away, its initial x oordinate was 2000 m. In approa hings enarios, the target started at 5000 m, unless otherwise mentioned.Initial velo ity of the missile was 10 m/s, and of the target 200 m/s. The targetvelo ity was assumed onstant.5.2.2 E�e t of Model Order Redu tionFirst the e�e t of ontroller order redu tion to the performan e of the missilewas studied. The target was assumed to approa h the missile laun hing pointmaking sinusoidal movements. Missile and target traje tories using both thefull-order and redu ed-order ontrollers are shown in �gure 15.

28

Figure 14: The simulation model for the missile-target en ounter situation.We see that both ontrollers result in ollision. The original full-order ontrolleris slightly more e�e tive is steering the motion, resulting in the missile turningtowards the target more than its ounterpart.The error of pit h rate realization ompared to the referen e signal are shownin �gure 16 for both the redu ed and full-order ontroller. There we see thatneither one of the ontrollers tra ks the signal perfe tly, but full-order ontrollerrea hes a signi� antly better performan e.These simulation results are in line with the analysis presented in hapter 3.6.4:the redu ed-order ontroller has better robustness than its ounterpart, but losesin performan e.Based on these simulations, we an on lude that the redu ed-order ontroller isenough for our purposes. It appears, in fa t, that in the simulations performed,

29

Figure 15: Flight paths of missile with the original (solid) and redu ed-order(dashed) ontrollers.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Pitch rate error

t

e

Figure 16: Pit h rate error of the missile with the original and redu ed-order ontrollers.none of the non- ollision situations ould have been saved using the full-order ontroller, but the missile failure was due to other reasons.

305.2.3 E�e t of Target ManeuveringThe missile performan e was evaluated using simulations with respe t to di�er-ent kinds of target maneuvering. Flight paths resulting from di�erent kinds ofmaneuver patterns are shown in �gures 17-19. The target was assumed to bemoving away from the laun hing point of the missile.

Figure 17: Flight paths; onstant target velo ity.In the situations illustrated in these �gures the missile performan e is rathergood: ollision is a hieved. Pit h rate ompared to the referen e signal produ edby the guidan e system is illustrated for third of the simulation setups in �g-ure 20, and elevation angle traje tory in �gure 21.From the pit h rate plot we see that the tra king is a tually far from beinga urate. This seems to be enough, however, for rea hing ollision. Be ause ofthe high un ertainties and modeling errors, better tra king might easily resultin instability of the missile rotational motion.The elevation angle plot gives us a good measure for the e�e t of the ontrolweight Wu. The elevation angle gets saturated only in the very end of the missile�ight, a result of greater steering movements required when being very lose tothe target; this saturation is a hara teristi s of the guidan e law used, and ould be diminished with a di�erent guidan e law design.However, in reasing the frequen y of the target sinusoidal maneuvers results inthe failure of the missile. As we an see in �gure 22, the minimum distan e of themissile and the target is 155 meters, whi h implies a total failure of the ontrolsystem. This suggests that our ontrol system is not robust enough to damphigh-frequen y ommand signals e�e tively; this an be veri�ed by examining

31

Figure 18: Flight paths; target performing sinusoidal maneuvers.

Figure 19: Flight paths; target performing sinusoidal maneuvers.�gure 23, whi h represents the referen e signal and the pit h rate realization.The high-frequen y signals result in missile instability.To in rease missile robustness in order to a hieve better performan e in thissetup, the ontrol design ould be ondu ted again, having the performan eweight We bode magnitude plot adjusted more to the left, that is, having a

32

Figure 20: Pit h rate and referen e signal for a simulation.

0 1 2 3 4 5 6 7−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2Elevation deflection

t

δ

Figure 21: Elevation angle traje tory for a simulation.smaller roll-o� frequen y. The e�e t of this would be that the ontrol systemperforman e around the original roll-o� frequen y be omes less important, andthus the missile would not rea t so deli ately in inputs with this frequen y.Studies were also made with an approa hing target. Simulations shown in �g-ures 24 and 25 illustrate, how even a non-maneuvering target an be missed, if

33

Figure 22: Flight paths for a simulation, where ollision is avoided.

Figure 23: Pit h rate and referen e signal for a simulation, where ollision isavoided.the laun hing point is too lose to the target onsidering its velo ity.This is learly a result of greater derivative of hange in the line-of-sight an-gle, ompared to the the situation where the target is moving away. Based onthe guidan e law (60), this results in greater referen e signals to the ontrol

34system, and again saturation of the ontroller surfa e. Figure 26 illustrates thepit h angle traje tory and referen e orresponding to the se ond simulation, andelevation angle plot in �gure 27 show the saturation.

Figure 24: Flight paths; approa hing non-maneuvering target.

Figure 25: Flight paths; approa hing non-maneuvering target.

35

4.6 4.7 4.8 4.9 5 5.10

5

10

15

20

25

30Pitch rate

t

qReferencePitch rate

Figure 26: Pit h angle and referen e signal for a simulation.

4.6 4.7 4.8 4.9 5 5.1−1.5

−1

−0.5

0

0.5

1

1.5Elevation deflection

t

δ

Figure 27: Elevation angle traje tory for a simulation.

36One more simulation with an approa hing target is shown in �gure 28. Thetarget performs sinusoidal movements orresponding to the ase in �gure 19.This ase leads to ollision, whi h implies that missile nominal performan e isnot a�e ted by the target orientation itself.

Figure 28: Flight paths; target performing sinusoidal movements.

37Target velo ity (m/s) Loop radius (m) Target aN (m/s2) Minimum distan e (m)60 120 30 5.0060 75 48 5.0060 60 60 5.0060 40 90 9.20100 200 50 5.00100 100 100 5.00100 40 250 5.00100 30 330 7.24150 140 160 5.00150 45 500 5.00150 30 750 13.53Table 2: Simulation results with a dodging target.Last, simulations were made in order to �nd the boundaries of missile per-forman e in terms of G for es the target has to produ e to be able to avoidthe missile. In these simulations, the target was moving away, starting at 2000meters. When the missile was approximately 300-400 meters away, the targetperformed a single loop dodging maneuver lasting 0.2-1.0 se onds, dependingon the target velo ity.Simulation results and performed target normal a elerations are shown in ta-ble 2, as well as the radius of the target loop. A lose-up of the minimum-distan epoint of one simulation that ended up in target miss is shown in �gure 29.

Figure 29: A detail of �ight paths; target performing a dodging maneuver witha normal a eleration of 330 m/s2.These results show that our missile is very apable of hitting a �ghter airplane.

38The normal a elerations needed to avoid the missile ex eed greatly the toleran eof best �ghter pilots [8℄, indi ating, that a �ghter an't avoid a hunting missileby performing simple dodging maneuvers. Espe ially faster-moving targets �ndthe ne essary normal a elerations very high. Of the simulated ases the onlyone where the pilot might have a han e to survive the maneuver is when the�ghter moves at 60 m/s. There the normal a eleration is 90 m/s2, or 9.2 G.5.2.4 Reasons behind Failure AnalyzedIt is worth taking a deeper look at the reasons behind the missile failure inthe simulations performed. Three main auses of miss were found, two of whi hresult from the ontrol system's performan e.First of all, it is easy to see that the initial onditions of the missile have a greatin�uen e on the performan e. Figure 25 shows that even a non-maneuveringtarget an be missed, if the losing velo ity is too high. This is the ase usuallywhen the target is too lose to the missile laun hing point and approa hing.Further studies would be needed to determine, what distan e exa tly is "too lose". Always trying to hit an away-moving target might not be a lever strategyeither, sin e the missile has a limited amount of fuel, and the �ight an besustained only for a limited amount of time. From table 1 and equation (13) we an al ulate, that if the missile uses approximately 80% of its throttle apa ity,the thrust an be sustained only fort =

mfuel

0.80 · KP Tmax

= 8.2 seconds . (62)This viewpoint was negle ted in this study, assuming the target velo ity on-stant. A further study ould be done to determine the optimal laun hing pointand initial �ight path angle of the missile with respe t to maximizing the prob-ability of ollision.The problems with the ontroller design deal with either the la k of robustness,or the la k of performan e. The robustness problem appeared with a targetmaneuvering at a high frequen y (see �gure 22), the ontroller attempting totra k high-frequen y referen e signals, and eventually resulting in instability.The same ould also be ta kled by a more sophisti ated guidan e law design.Performan e problems are due to too high referen e signals: when the referen eis small enough, even a poorer performan e of the ontroller results in ollision inthe as ading overall feedba k loop, onsisting of the autopilot and the guidan esystem. But when the referen e is too high (see �gure 28), the tra king be omespoor, and the missile misses the target.These limits of robustness and performan e are essential onsidering the ontroldesign. Sin e both the robustness and performan e la k some e�e tiveness, andthe missile performan e in the simulations is "rather good", we an quite safely on lude that a su� ient trade-o� between these design goals is a hieved. Ifwe felt that the ontroller is biased towards either robustness or performan e,

39we ould repeat the design pro ess by adjusting the weights orrespondingly.Based on the simulations made, this design rea hes a good balan e between therequirements for the ontrol system.6 Con lusionsH∞ framework was an advantageous approa h for this ontrol system designtask, where the obje tives were on�i ting, and un ertainty of the plant wasfurther in reased by model redu tion. A ontroller, whi h was designed on asigni� antly simpler model resulted in an ex ellent in-a tion performan e insimulations with the more realisti model. We an therefore on lude that thesimpli�ed linear model of the missile dynami s was enough for ontroller design.The simulations show, that a de ent trade-o� between robustness and perfor-man e was a hieved. Robust performan e was not met, however, whi h impliesthat the ontrol design ould be further improved by other methods, like µsynthesis.Robust stability appears to be ru ial among the performan e riteria, be ausethe guidan e system ompensates poor performan e of the autopilot, but theloss of stability an't be orre ted. Be ause of the similar overall feedba k loopamong all guided missiles, the importan e of robust stability over robust per-forman e is likely to be universal.Further studies ould be made to �nd out the ontroller performan e under sen-sor noise and wind onditions. Be ause of the high variation in missile velo ity ompared to the onstant value used in ontrol design, the performan e underwindy onditions is likely to be good, however. The most important open ques-tion after this study is, if the missile model now used was physi ally realisti . Inany ase, the design pro edure ould be repeated for any missile data in orderto �nd the limits of that physi al missile's apability.Referen es[1℄ Siouris, George. Missile Guidan e and Control Systems. Springer. NewYork, USA, 2004. ISBN 0-387-00726-1.[2℄ Blakelo k, John. Automati Control of Air raft and Missiles. 2nd edition.John Wiley & Sons. USA, 1991. ISBN 0-471-50651-6.[3℄ Lin, Chun-Liang & Hung, Hao-Zhen & Chen, Yung-Yue & Chen, Bor-Sen. Development of an Integrated Fuzzy-Logi -Based Missile Guidan e LawAgainst High Speed Target. IEEE Transa tions on Fuzzy Systems, 2004.Vol.12:2. p. 157-169. ISSN 1063-6706.[4℄ Friedland, Bernard. Control System Design - An Introdu tion to State-Spa e Methods. M Graw-Hill. USA, 1987. ISBN 0-07-0022441-2.

40[5℄ Doyle, John & Fran is, Bru e & Tannenbaum, Allen. Feedba k Control The-ory. Ma millan Publishing Co. 1990. [Retrieved on 12.10.2007℄. Availableat http://www. ontrol.utoronto. a/people/profs/fran is/dft.html[6℄ Zhou, Kemin. Essentials of Robust Control. Prenti e-Hall. New Jersey,USA, 1998. ISBN 0-13-525833-2.[7℄ Mathworks, In . Control System Toolbox Refer-en e. [Retrieved on 12.10.2007℄. 2002. Available athttp://www.mathworks. om/a ess/helpdesk_r13/help/pdf_do / ontrol/referen e.pdf[8℄ Creer, Brent & Smedal, Harald & Wingrove, Rodney. Centrifuge Studyof Pilot Toleran e to A eleration and the E�e ts of A eleration on PilotPerforman e. NASA. Washington, USA, 1960. [Retrieved on 12.10.2007℄.Available at http://ntrs.nasa.gov/ar hive/nasa/ asi.ntrs.nasa.gov/19980223621_1998381731.pdf