Diagonally-Dominant Backstepping Control - IIT Kanpur techniques/ppt/abhay.pdf · ship between response and control variable is known ... • Stability and control (short period,

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


Aircraft Mechanical Degrees of Freedom


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)


Review of Linear Systems

7

�̇�𝑥 = 𝐴𝐴𝑥𝑥 + 𝐵𝐵𝑢𝑢 + 𝐿𝐿𝑤𝑤 𝑧𝑧 = 𝐶𝐶𝑥𝑥 + 𝐷𝐷𝑢𝑢 + 𝐺𝐺𝐹𝐹


Controllable Canonical Form

8

nnnn

nnn

ssssssG

sUsY

αααβββ++++

+++== −−

−−

2

21

1

22

11)(

)()(


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


Full State Feedback

10

�̇�𝑥 = 𝐴𝐴𝑥𝑥 + 𝐵𝐵𝑢𝑢 𝑢𝑢 = −𝐾𝐾𝑥𝑥

-


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


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


Cascade Loops

13

GC2 GC1 GP Σ

Primary controller

Secondary controller Plant

Secondary loop

Primary loop

GFF

Feed Forward


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


Antiwindup


• 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


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


3. Gravity Vector Perturbation

Gravity

Compensate for Gravity Disturbance


Aerodynamic Control Effectors


Control Decoupling & Redundancy

• Explicit Ganging

• Pseudo Control

• Pseudo Inverse

• Daisy Chain 21

𝑢𝑢 = 𝐺𝐺𝛿𝛿

min𝑢𝑢

12𝑢𝑢𝑇𝑇𝑊𝑊𝑢𝑢𝑢𝑢

𝐵𝐵𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑆𝑆𝑠𝑠B𝑆𝑆𝑟𝑟=𝑈𝑈Σ𝑉𝑉𝑇𝑇


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!)


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


Kinematic & Navigation Equations

24

Φ̇ = 𝑓𝑓𝑘𝑘 𝜔𝜔,Φ �̇�𝑋 = 𝑓𝑓𝑛𝑛 𝑉𝑉,Φ


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


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


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


Feedback Control

29

Actuators


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


Feedback Linearization


�̇�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


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


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


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


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



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


38

M = 0.31 and H = 5700m Trim AoA 15deg Longitudinally unstable


Simulation Results (Nonlinear) Responses to Roll Step Input (Multiple Rolls)

40

with and without gravity compensation terms


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


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 transformation matrix

( )βαµ ,,

µ

χ

γ

sT


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


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


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


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


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


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


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


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


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


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


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Documents

Diagonally-Dominant Backstepping Control - IIT Kanpur techniques/ppt/abhay.pdf · ship between response and control variable is known ... • Stability and control (short period,