Flight Control with Backstepping Part I - Preliminaries
Dr. Abhay Pashilkar Deputy Head, Flight Mechanics & Control Div.
National Aerospace Laboratories Bangalore
Part I - Overview
• Aircraft Degrees of Freedom • Aircraft as a Dynamic System • Review of Linear Control Systems Theory
– Controllable Canonical Form – Full State Feedback – Cascade Control System & Time Scale Separation – Anti-windup
2 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Overview
• Flight Control Challenge – Kinematic Coupling – Inertia Coupling – Gravity Vector Compensation – Control Decoupling & Redundancy – Handling Qualities
• Nonlinear Aircraft Dynamics & Kinematics • Alternative Formulations of EoM
3 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Aircraft Mechanical Degrees of Freedom
4 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Aircraft as a Dynamic System
5
𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑀𝑀 = 𝑀𝑀𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝑀𝑀𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹
�̇�𝑥 = 𝑓𝑓 𝑥𝑥,𝑢𝑢,𝑤𝑤 𝑧𝑧 = ℎ 𝑥𝑥, 𝑢𝑢,𝐹𝐹
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Analysis Equations of Motion - Nonlinear & Linear
6
�̇�𝑥 = 𝑓𝑓 𝑥𝑥(𝐹𝐹),𝑢𝑢(𝐹𝐹),𝑤𝑤(𝐹𝐹) �̇�𝑥 = 𝐴𝐴𝑥𝑥(𝐹𝐹) + 𝐵𝐵𝑢𝑢(𝐹𝐹) + 𝐿𝐿𝑤𝑤(𝐹𝐹)
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Review of Linear Systems
7
�̇�𝑥 = 𝐴𝐴𝑥𝑥 + 𝐵𝐵𝑢𝑢 + 𝐿𝐿𝑤𝑤 𝑧𝑧 = 𝐶𝐶𝑥𝑥 + 𝐷𝐷𝑢𝑢 + 𝐺𝐺𝐹𝐹
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Controllable Canonical Form
8
nnnn
nnn
ssssssG
sUsY
αααβββ++++
+++== −−
−−
2
21
1
22
11)(
)()(
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Block Diagram
This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state)
9
∫ ∫ ∫ ∫
1α−
2α−
nα−
cx1cx2
ncx)(ny …
)(tu
+ +
nβycnx )1( −
1−nβ
+
1β
2β
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Full State Feedback
10
�̇�𝑥 = 𝐴𝐴𝑥𝑥 + 𝐵𝐵𝑢𝑢 𝑢𝑢 = −𝐾𝐾𝑥𝑥
-
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Full State Feedback
• System needs to be controllable • With more than 1 actuator, we have more than n degrees of
freedom in the control → we can change the eigenvectors as desired, as well as the poles
• The real issue now is where to put the poles • If all states are not measurable → develop an estimator
11 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
State Feedback
• State feedback can place a pole at the location of a zero: pole-zero cancellation
• If the original system is controllable, the closed-loop system is controllable
• Pole-zero cancellation: closed-loop system can have unobservable modes even if the original system is completely observable
12 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Cascade Loops
13
GC2 GC1 GP Σ
Primary controller
Secondary controller Plant
Secondary loop
Primary loop
GFF
Feed Forward
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Cascade Control Loops
• Time scale separation: Secondary loop must be fast responding otherwise system will not settle
• Since secondary loop is fast, proportional action alone is sufficient, offset is not a problem in secondary loop
• Feedforward can help speed up the response if the relation ship between response and control variable is known
14 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Antiwindup
15 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
• Typical combat envelope – Mach range 0.5-0.9 – Altitude range 15,000 to
18,000ft • Elements of air combat
– Turns – Rolls – Accelerations & decelerations
Modern Combat Aircraft Design Klaus Huenecke, 1987
16
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Flight Control Challenge – Combat Envelope
1. Kinematic Coupling
Roll about velocity vector and suppress stability axis yaw rate
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 17
2. Inertia Coupling
Correct for Inertia Coupling
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 18
3. Gravity Vector Perturbation
Gravity
Compensate for Gravity Disturbance
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 19
Aerodynamic Control Effectors
20 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Control Decoupling & Redundancy
• Explicit Ganging
• Pseudo Control
• Pseudo Inverse
• Daisy Chain 21
𝑢𝑢 = 𝐺𝐺𝛿𝛿
min𝑢𝑢
12𝑢𝑢𝑇𝑇𝑊𝑊𝑢𝑢𝑢𝑢
𝐵𝐵𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑆𝑆𝑠𝑠B𝑆𝑆𝑟𝑟=𝑈𝑈Σ𝑉𝑉𝑇𝑇
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Handling Qualities
• Stability and control (short period, dutch roll): predictable aircraft response (damping and frequency, sideslip excursions) depending on pilot task – Landing and take-off – Tracking task (AAR, close formation, ground attack) – Terrain following
• Roll mode time constant and delay • Pilot control forces and control harmony • Pilot induced oscillation (avoid saturations within the control system!)
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 22
Aircraft Dynamics: Newton-Euler Equations
23
𝐹𝐹 = 𝑠𝑠𝑠𝑠𝑑𝑑
𝑚𝑚𝑉𝑉 = 𝑚𝑚�̇�𝑉 + ω × 𝑚𝑚V
𝑀𝑀 =𝑑𝑑𝑑𝑑𝐹𝐹
𝐼𝐼 ∙ 𝜔𝜔 = 𝐼𝐼 ∙ �̇�𝜔 + ω × 𝐼𝐼 ∙ 𝜔𝜔
𝐹𝐹 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑀𝑀 = 𝑀𝑀𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 + 𝑀𝑀𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Kinematic & Navigation Equations
24
Φ̇ = 𝑓𝑓𝑘𝑘 𝜔𝜔,Φ �̇�𝑋 = 𝑓𝑓𝑛𝑛 𝑉𝑉,Φ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
State Vector x – Alternate Formulations
25
𝑉𝑉 =𝑢𝑢𝐹𝐹𝑤𝑤
,𝜔𝜔 =𝑝𝑝𝑞𝑞𝐹𝐹
,Φ =𝜙𝜙𝜃𝜃𝜓𝜓
,𝑋𝑋 =𝑥𝑥𝐹𝐹𝑧𝑧
, x =
𝑢𝑢𝐹𝐹𝑤𝑤𝑝𝑝𝑞𝑞𝐹𝐹𝜙𝜙𝜃𝜃𝜓𝜓𝑥𝑥𝐹𝐹𝑧𝑧
α = tan−1𝑤𝑤𝑢𝑢 ,𝛽𝛽 = sin−1
𝐹𝐹𝑉𝑉 , x =
𝑉𝑉𝛽𝛽𝛼𝛼𝑝𝑝𝑞𝑞𝐹𝐹𝜙𝜙𝜃𝜃𝜓𝜓𝑥𝑥𝐹𝐹𝑧𝑧
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
State Vector
26
𝜇𝜇 = tan−1sin𝜃𝜃 cos𝛼𝛼 sin𝛽𝛽 + sin𝜙𝜙 cos𝜃𝜃 cos𝛽𝛽 − cos𝜙𝜙 cos𝜃𝜃 sin𝛼𝛼 sin𝛽𝛽
sin𝜃𝜃 sin𝛼𝛼 + cos𝜙𝜙 cos𝜃𝜃 cos𝛼𝛼 , 𝛾𝛾 = sin−1ℎ̇𝑉𝑉 ,𝜒𝜒 = tan−1
�̇�𝐹�̇�𝑥 , x =
𝑉𝑉𝛽𝛽𝛼𝛼𝑝𝑝𝑠𝑠𝑞𝑞𝐹𝐹𝑠𝑠𝜇𝜇𝛾𝛾𝜒𝜒𝑥𝑥𝐹𝐹ℎ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Flight Control with Backstepping Part II - Design Dr. Abhay Pashilkar
Deputy Head, Flight Mechanics & Control Div. National Aerospace Laboratories
Bangalore
Part II - Overview
• Nonlinear Control Design – Feedback Linearization or Nonlinear Dynamic Inversion – Backstepping – Simplified NDI with Backstepping – Diagonally Dominant Backstepping
• State Limiting and Anti-windup
28 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Feedback Control
29
Actuators
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
30
Nonlinear Flight Control Design • Conventional flight control designs assume linear aircraft dynamics
and schedule the gains • NDI and Integrator Backstepping offer more flexibility
• Equations motion to be grouped in blocks to get a cascaded controller – Block Backstepping
V
yh
rqp
thr
s
s
yaw
pitch
roll
→
→
→
→
→
δ
γχ
βαµ
δ
δδ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Feedback Linearization
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 31
�̇�𝑥 = 𝑓𝑓(𝑥𝑥) + 𝐹𝐹(𝑥𝑥)𝑢𝑢
𝐹𝐹 = ℎ(𝑥𝑥)
�̇�𝐹 =𝜕𝜕ℎ𝜕𝜕𝑥𝑥
�̇�𝑥 =𝜕𝜕ℎ𝜕𝜕𝑥𝑥 [𝑓𝑓(𝑥𝑥) + 𝐹𝐹(𝑥𝑥)𝑢𝑢]
=𝜕𝜕ℎ𝜕𝜕𝑥𝑥
𝑓𝑓(𝑥𝑥) +𝜕𝜕ℎ𝜕𝜕𝑥𝑥
𝐹𝐹(𝑥𝑥)𝑢𝑢
= 𝐿𝐿𝑓𝑓ℎ + (𝐿𝐿𝑔𝑔ℎ)𝑢𝑢
𝐹𝐹 𝑟𝑟 = 𝐿𝐿𝑓𝑓 𝑟𝑟 ℎ + 𝐿𝐿𝑔𝑔 𝐿𝐿𝑓𝑓 𝑟𝑟−1 ℎ 𝑢𝑢
.
.
.
If 𝐿𝐿𝑔𝑔ℎ = 0, continue differentiating till non–zero term appears
𝑢𝑢 =1
𝐿𝐿𝑔𝑔 𝐿𝐿𝑓𝑓 𝑟𝑟−1 ℎ−𝐿𝐿𝑓𝑓 𝑟𝑟 ℎ + 𝐹𝐹
𝐹𝐹(𝑟𝑟) = 𝐹𝐹
Nonlinear Dynamic Inversion Inherent Nonlinearities : Kinematic coupling, Inertia Coupling, Gravity Correction NDI enables direct design of nonlinear controller by cancelling out the aircraft dynamics
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 32
Feedback Linearization (NDI)
• Feedback linearization can be accomplished with systems that have relative degree less than n
• However, the normal form of the system will have states that are not observable from the output of the system (zero dynamics)
• The unobservable states must be stable by themselves or need to be stabilized by feedback
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 33
34
Backstepping Affine system with two states:
(2) xbxfx (1) bxxfx
12222
12111
)(),(+=+=
δ
Desired trajectory obtained as:
( )( ) (4) xfxbx
(3) xxfxbdd
d
)(
),(
2221
21
21111
1
−=
−=−
−
δ
An innovation:
( )( ))(ˆˆ
),(ˆˆ
222221
21
2111111
1
xfekxbx
xxfekxbdd
d
−−=
−−=−
−
δ
),( 222111dd xxexxe −=−=
22
21 2
121 eeVlyap +=
Backstepping: used as a pseudo-control obtained by inverting Eq.(2) then used in Eq.(3) after differentiating to get Stability in sense of Lyapunov can be proved
dx1
δ
34 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Simplified Nonlinear Dynamic Inversion (SNDI)
35
SNDI enables direct design of nonlinear controller by cancelling out the aircraft dynamics
Inner Loop: Fast states in stability axis (q, ps, rs) Outer Loop: Slow states in wind axis α and β
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 35
SNDI Controller for Fighter Aircraft
36
Cascaded Structure of NDI controller
AircraftControl
AllocationInnerloop
controlOuterloop
control
LOE
LIE
RIE
ROE
R
δδδδδ
pitch
roll
yaw
δ
δδ
scmd
cmd
scmd
pqr
s
s
pqr
cmd
cmd
αβ
αβ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 36
SNDI Controller for Fighter Aircraft
37
Stability Axis equations:
Wind Axis equations: Linear Aircraft Model:
Mixed Axis Formulation for NDI controller design 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 37
SNDI Controller for Fighter Aircraft
38
M = 0.31 and H = 5700m Trim AoA 15deg Longitudinally unstable
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 38
Simulation Results (Nonlinear) Responses to Roll Step Input (Multiple Rolls)
40
with and without gravity compensation terms
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 40
41
V
yh
rqp
thr
s
s
yaw
pitch
roll
→
→
→
→
→
δ
γχ
βαµ
δ
δδ• Back-Stepping:
• Equations of motion grouped to get a cascaded controller:
PositionControl
ref
ref
hy Trajectory
ControlWind Axis
ControlInner Loop
ControlControl
AllocationAircraft
c m d
c m d
χγ
c m d
c m d
αµ
re fV
s c m d
c m d
s c m d
rqp
y a w
ro ll
p itc h
δδδ
refβ
−
−
−
−
rud
ailright
aille ft
elright
elle ft
δδδδδ
th rδ
41
Trajectory Controller Based on Backstepping
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Aircraft Equations of Motion in Mixed-axis System
1. Rotational Equations: 2. Wind-axis Equations:
+
+
−−
−−=
−
T
T
T
a
a
a
nml
nml
rqp
Ipq
prqr
Irqp
0 0 0
1
=
−
=
rp
Trp
rp
ss
sαααα
cossinsincos
( ) γχβαβµ sinsincos +−+= qps
( )β
γµβαcos
cos cos tanmV
mgLpq s−
−−=
mVYs
Vg
rs −+−= γµβ cos in
where, are “fast” states,
I is moment of inertia matrix
aerodynamic contribs.
thrust contributions
) , ,( rqp
,
( )aaa nml ,,
( )TTT nml ,,
where, are “slow” states,
is velocity roll angle,
is velocity yaw angle (heading),
is flight-path angle, and
is body to stability axis trans- formation matrix
( )βαµ ,,
µ
χ
γ
sT
42 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Aircraft Equations of Motion (contd)
3. Velocity Vector Equations: where, are “very slow states”, is thrust angle of attack, and
is thrust side-slip angle
( ) γνε sincoscos gm
DTV −−
=
( )
( )mV
mgLmV
YmV
sT
γµ
µµνεµεγ
coscos
sinsin in coscos sin
−+
+−
=
( )γµ
γµεµνε
γµχ
coscos
cossin sincos sin cos
cossin
mVY
mVT
mVL
−+
+=
4. Navigational Equations:
χγ coscosVx =
χγ sincosVy =
γsinVh =
Equations of motion are grouped into fast-, slow-, and very slow-states to facilitate the design of a backstepping control law with cascaded structure.
),,( χγV
εν
43 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Linear Aircraft Model • A linear model of aircraft is required for designing the inner-most control loop dealing with the fast rotational states. • The rest of the control law design uses the nonlinear formulation of the aircraft dynamics.
,BuAyx +=
[ ]Trpqx =
[ ]Trpqy βα=
=
−
−
−
−
r
righta
lefta
righte
lefte
u
δ
δ
δ
δ
δ
where Plant and control matrices: (vel=82.6 m/s, alt=600m)
−−−−−
−−=
0169.0 0007.0 0007.0 0074.0 0074.0 0340.0 0842.0 0842.0 0549.0 0549.0
0 0005.0 0005.0 0299.0 0299.0B
−−−−
−−−=
2424.0 0482.0 8847.1 0025.0 0 8792.0 7533.1 1548.14 0003.0 0 0029.0 0 0066.0 6491.0 8145.0
A
44
−−
−−
−−
=
rrudrelerele
prudpailpailpelepele
qeleqele
b b b
b b bb b
b b
B
00
000
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
DDBS Controller Design A. Inner loop control design: 1. Three pseudo-controls decouple control of rotational axes. 2. Transform state and output vectors 3. Transformation matrices
[ ]Tyawrollpitchu δδδ=
[ ] Tss rpqx =
[ ]Tss rpqy βα=
0
0111
12
21 xTxT
xs
=
= −
×
× 0
021
32
2333 yTyT
Iy
s=
= −
×
××
Differential elevators used to generate additional roll and yaw where, helps accommodate failure of both the ailerons. extends rudder fault tolerance limit, and suppresses side-slip during coordinated turns
uSuu =
=
1K001001-0
KK1K-K-1
ari
reiaei
reiaei
aeiK
ariK
reiK
45 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
DDBS Controller Design (contd) A. Inner loop control design: (contd) 4. Linearized equations for rotational dynamics: matrix pseudo-inverse of BS
5. Choose , , and S to make diagonal, and decouple equations:
6. Decoupled controls Proposed pitch control law: commanded pitch rate input. Substituting into eqn with results in first order pitch rate resp with Similarly:
uBSyTxT 21 +=
( ) ( ) uyTBSxTBS += ++ 21
( ) +BS
reiK ariK( ) 1TBS +
pitchqq δα ++−=− 9.106.137.16
rollsss rpp δβ +−+=− 8.57.79.675
yawsss rpr δβ ++−−=− 1.223.102228.46
cmdq
sradKq /deg/105−=
s16.01057.16 =−−=τaeiK
46
( )qqKq cmdqcmdpitch −+−≅ 7.16δ
( ) ( )( )qqKqq cmdqcmd −≅− 7.16
( )sscmdpscmdroll ppKps
−+−≅ 5δ
( )sscmdrscmdyaw rrKrs
−+−≅ 46δ
pitchδ q
ssradK psps 2.0/deg/25 =→−= τ
ssradK rsrs 26.0/deg/180 =→−= τ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
47
B. Wind axis loop control design: 1. First order approx. of equations: 2. Control laws based on above approx. 3. Selection of gains, resulting time const Outer: Inner loop time constants = 2.5:1
C. Velocity vector loop control: 1. First order approx of equations where 2. Control law for tracking loop:
DDBS Controller Design (contd)
µβµαVgrpq ss +−≅≅≅ ,,
( )ααα −+= cmdalphacmdcmd Kq
( )µµµ −+= cmdmucmdscmd Kp
( )
−−+−= µβββ
VgKr refbetarefscmd
sradsradK alphaalpha 5.0//0.2 =→= τ
sradsradK mumu 33.0//0.3 =→= τ
sradsradK mubeta 0.1//0.1 =→= τ
( ) ( )γδγ gbgm
DTV thrvthr −≈−−
≅
( ) ( )trimL
mVSCqbar
mVmgL ααγ α −
−≈
−≅
µµχVg
mVL
≈≅
thr
vthrVbδ∂∂
=
( )[ ]γδ gVVKVb refvelref
vthrthr +−+= 1
( )[ ] trimcmdgamcmdL
cmd KCSqbar
mV αγγγαα
+−+=
( )[ ]χχχµ −+= refcmdchicmdcmd KgV
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
48
DDBS Controller Design (contd) Velocity vector loop Control (Contd) : 3. Gains and Time Constants: The time constants of flight path and Heading angle loops are at least 2.5 times those of angle of attack and bank angle loops, ensure dynamic separation. D. Navigational Equations: 1. First order approximations
Navigational equations (contd): 2. Control law for the position loop 3. Chosen gains, resulting time const. The time constants of the cross track and altitude loops are at least 1.9 times those of bank angle and flight path angle loops.
15.0 −= sKvel
rad/s/radK gam 2.1=
15.0 −= sKchi
svel 0.2=τ
sgam 8.0=τ
schi 0.2=τ
)( refcmdVy χχ −⋅≅ γ⋅≅ Vh
[ ])(1)( yyKyV refyrefrefcmd −+=− χχ
[ ])(1 hhKhV refhrefcmd −+= γ
11.0 −= sK y 165.0 −= sKh
sy 0.10=τ
sh 5.1=τ
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
DDBS Controller - Longitudinal
Airc
raft
Actu
ator
s
Con
trol A
lloca
tion
δe-left
δe-right
δa-left
δa-right
δr
δthr
δpitch
δyaw
δroll
qαγh
Failure(s)
href
Saturation status of actuators
AoA Saturation status
Vref
+ + + +
+
_ _ _
__
_
++
114.014.1
ssKV
αK
g−
V
++
+ hKs
s10
+
sKK h
h05.0
αLqbarSCMV ++
−−
7.16/107.16
qKss
++
+ αKs
s10
trimα
γK ++
+ γKss
10
++ V1
qK
vthrb1++
+ VKs
s10
49 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
DDBS Controller – Lateral-Directional
Airc
raft
Actu
ator
s
Con
trol A
lloca
tion
δe-left
δe-right
δa-left
δa-right
δr
δpitch
δyaw
δroll
ps
Failure(s)
_
µχ/ψχref
+ +_ _ +
yref-y
Actuator saturation status
rs
+++
ββref
_+ PIDψ
+_
++
ψ
0
≤2mh
Vg−
µKgV
βK
+
sK
KV
yy
05.01
p sK ++
−−
5/105
psKss
++
+ µKss
10
+
+ χKss
10
+χK++
_+ r sK ++
−−
8.46/108.46
rsKss
+ βKss
10
-1
50 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
State Limiting and Antiwindup
• Each of the successive outer loops of the controller is treated as a PID
• State or control surface saturation of an inner loop of the cascaded controller results in the integrators in the outer loops to be held for the duration of the time the variable is in saturation
23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 51
Conclusions
• Control design is an inverse problem for dynamic systems • If mathematical model is available it’s a good idea to develop a controller based
on that based on integrator backstepping • Even if mathematical model is not available or complex it’s a good idea to think
in terms of models for control design • Attempt to cancel only the most significant nonlinearities. If required use
alternate formulation of the EoM • Pay attention to time scale separation for gain selection • Inner loop must have as high gain as possible for robustness to failure while
avoiding actuator position and rate saturation 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur 52
References 1. Shaik Ismail, Abhay A. Pashilkar, Ramakalyan Ayyagari, Narasimhan Sundararajan,
“Diagonally Dominant Backstepping Autopilot for Aircraft with Unknown Actuator Failures and Severe Winds”, The Aeronautical Journal, Vol. 118, No. 1207, September 2014, pp. 1009-1038
2. P. Lathasree, Shaik Ismail and A A Pashilkar, “Design of Nonlinear Flight Controller for Fighter Aircraft”, Published in the Third International Conference on Advances in Control and Optimization of Dynamical Systems, ACODS 2014, IIT Kanpur, March 2014
3. P Lathasree, Abhay A Pashilkar, N Sundararajan, “Fast Nonlinear Flight Controller Design for a High Performance Fighter Aircraft”, Indian Control Conference 2015 held at IIT Madras, Chennai during 5-7 January 2015
53 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur
Thank You
54 23 September 2016 Dr. Abhay Pashilkar, IIT Kanpur