Workshop] Robust and Adaptive Part 1

Robust and Adaptive Control Workshop

Kevin A. Wise, Ph.D.email: [email protected] Phone: (314) 232-4549

Eugene Lavretsky, Ph.D.email: [email protected] Phone: (714) 235-7736

Prof. Naira Hovakimyanemail: [email protected] Phone: (217) 244-1672

All rights reserved. No part of this publication may be reproduced, distributed, or transmitted, unless for course participation, in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the Publisher and/or Author. Contact the American Institute of Aeronautics and Astronautics, Professional Development Programs, Suite 500, 1801 Alexander Bell Drive, Reston, VA 20191-4344



Kevin A. Wise, Ph.D.email: [email protected] Phone: (314) 232-4549

Eugene Lavretsky, Ph.D. Prof. Naira Hovakimyanemail: [email protected] Phone: (714) 235-7736 email: [email protected] Phone: ((540) 231 7989


Introduction Required Background Senior/Master Level Understanding of Control System Design, Frequency Domain Analysis, State Space Methods Design Goal: Augment Robust Baseline Architecture With Adaptive Control Baseline Architecture Required For Many Reasons Do Not Sacrifice Performance in Order To Be Robust Part 1: Robust Control Methods and Lessons Learned Part 2: MRAC Methods and Lessons Learned Part 3: L1 Adaptive Control

2K. Wise E. Lavretsky, N. Hovakimyan


Course Outline Part 1Review of BasicsState space models, Linear vector spaces, Operators, Similarity transformations Norms, Eigenvalues, Eigenvectors, Matrix norms, Singular values, Singular vectors, State transition matrices, Controllability, Observability, Stability, Power signals, Norms for systems, Function spaces, Wellposedness and stability

Frequency Domain AnalysisReview of transfer functions and transfer function matrices, Classical frequency response methods, Nyquist theory, Multivariable Nyquist Theorem, Stability margins, Singular value stability margins, Performance specifications in the frequency domain, Robust stability analysis, Small Gain Theorem, Frequency dependent weights, Robust stability tests to specify hardware requirements, Analysis methods for real parameter uncertainties, Singular value robustness tests

Robust Control System DesignPole placement with state feedback, Observer feedback, Robust servomechanism, Optimal control theory, Linear Quadratic Regulator (LQR), Projective control theory. Motivation for optimal control, optimal state feedback control, optimal control with synthesis, H optimal control



Course Outline Part 2 Motivating Example Review of Lyapunov Stability Theory Model Reference Adaptive Control Basic concepts 1st order systems nth order systems Robustness to Parametric / Non-Parametric Uncertainties Architectures Using sigmoids Using Radial Basis Functions, (RBF)

Neural Networks, (NN)

Adaptive NeuroControl Adaptive Backstepping F-16 Control Design Example Adaptive Reconfigurable Flight Control using RBF NN-s mod nonlinear-in-control design

Adaptive Control Modifications Adaptive Control Augmentation of a Baseline Flight ControllerDynamic Inversion Linear Gain-Scheduled Regulator 4

Open Problem: Validation Metrics for Adaptive Control

K. Wise E. Lavretsky, N. Hovakimyan


Course Outline Part 3Limitations of MRAC L1Adaptive Control Theory Systems with Matched Uncertainties Performance Bounds Robustness/Stability Margins Verification and Validation Examples Wing Rock UCAV Aerial Refueling Rohrs Example Hypersonic Vehicle Output Feedback Extension Autopilot Augmentation Architecture Systems with Unmatched Uncertainties L1Adaptive Backstepping Topics of Current Research5K. Wise E. Lavretsky, N. Hovakimyan

AFOSR Technology Transitioned Into Advanced Weapon Systems PW-STL Using AFOSR Funded Technology In Flight Control Systems For Advanced Weapon Systems Projective Control Theory Used To Project Optimal Control Optimal Design Into Output Feedback X-45A Projection Into Eliminated State Feedback Sensors Output Feedback Reduced Weapon Cost/Weight Improved Performance/Accuracy89 90 91 92 93 Technology Transition Timeline 94 95 96 97 98 99 00 JDAM and MMT Flight Test At Eglin AFB 01 02 03

MA-31 04 05 06 07

AFOSR Funds Basic R&D @ Univ of ILL MDC Develops Flight Control Design Tool Called AUTOGAIN

Tomahawk Validates Improved Performance in 6DOF

AUTOGAIN JDAM Family

SDB 4GENMk-84 Mk-83, BLU-110 BLU-109 Mk-82 (Growth)

SLAM-ER and JDAM Use AUTOGAIN

BQM-74 MMT SLAM-ER

JDAM

6

AFOSR Adaptive Control Transitioned To Advanced Weapon Systems Adaptive Control Based upon Earlier Aircraft Application Extended to Munitions (00-02) with GST Boeing IRAD Improvements Focus on System ID, Implementation, and Actuator Saturation Issues Design Retrofits Onto Existing Flight Control Laws Flight Proven on X-36, Mk-84, MK-82, and L-JDAM Transitioned To Production JDAMReference Model

+

-

Adaptive Control Baseline Autopilot

Guidance

+

+

Airframe

X-45CBoeing Collaborates with Prof. N. Hovakimyan (VaTech) and Dr. Annaswamy (MIT) on V&V methods for Adaptive Systems

X-45A

Robust Adaptive Control Technology Transition Timeline93 94 95 96 97 98 99

J-UCAS03 04 05 06 07 10

00

01

02

Intelligent Flight Control System (NASA/Boeing) F-15 ACTIVE

Adaptive Control For Munitions (AFRL-MN/GST//Boeing) MK-84

Boeing IRAD/CRAD

MK-82 L-JDAM

Gen I, flown 1999, 2003 Gen II, 2002 2006 flight test 4th Q 2005 Gen III, 2006

Reconfigurable Control For Tailless Fighters (AFRL-VA/Boeing) X-36

MK-84 JDAM

MK-82 JDAM

Theoretically justified, numerically efficient, Theoretically justified, numerically efficient, and flight proven technology and flight proven technology POC: K. Wise, E. Lavretsky


Robust and Adaptive Control Challenges4th Generation Escape System90 80 70 60 50 40 30 20 10 0

Air Superiority Missile TechnologyAngle of Attack (deg)

Maneuver

Launch

Fly-Out/End-Game

0

0.5

1.0

1.5

2.0

2.5

3.0

X-45A J-UCAS

Nonlinear Aero Large Uncertainties Nonlinear Control Effectors Limited Actuation

Mach Number

X-36 Tailless Agility Research Aircraft

Unstable In Multiple Axes, Non-minimum PhaseK. Wise E. Lavretsky, N. Hovakimyan

8


Adaptive Control Proven In FlightX-36 Tailless Agility Research Aircraft Joint Direct Attack Munition

JDAM

Mk-84

Mk-83, BLU-110 BLU-109 Mk-82 (Growth)

9



X-45A, X-45C at Edwards AFB



Video



Adaptive Control Transitions X-45A and X-45C J-UCAS

evaluated in flight simulation environmentLaser guided MK82 scores direct hit against a moving target during tests at Eglin AFB.

JDAM MK-82 and MK-82L in production currently in flight testingAffordable hit-to-kill accuracy minimizes collateral damage

JDAM MK-84 IDP 2000

scheduled to go into production FY 08 evaluated in fuel lab ASDRE evaluated in flight simulation environment evaluated in flight simulation environment

HyFly J-DRADM EAPS (Army)Technology Transitioned into Air Force Advanced Aircraft and Weapon Systems 12



Adaptive Augmentation of Pitch Autopilot

( azAZcmd + Cperc

azcmd

q)

Adaptive Control

eada r eFin Mixing 600 Hz

Incremental Elevator Command

Turn Rate qcmd KAZ + KI Inner Loop + 1/s KP q

Vehicle 3 Actuators

AZ

3rd Order Elliptic Filter

Lever Arm * s 1st Order Lag Noise Filter 100 HzK. Wise E. Lavretsky, N. Hovakimyan

IMU3rd Order Elliptic Filter

+

Mean AZ Filter

13


Adaptive Augmentation of Roll-Yaw Autopilota cmd y ps rs )

(Error Aycmd=0 + AY -

err

ay

Adaptive Control Inner Loop + KI + KP p r

(a

ad a

rad )

Incremental Ail/Rud Commands

Turn Rate rcmd KPHI KAY ps + + rs 1/s 1/s

Vehicle

lead-lag filter

r

Fin Mixing

3

Actuators

e600 Hz 4th Order Elliptic Filter 4th Order Elliptic Filter 4th Order Elliptic Filter14

Cperc

Transform to Stability Axes Lever Arm * s 1st Order Lag Noise Filter 100 HzK. Wise E. Lavretsky, N. Hovakimyan

IMU

+

Mean AY Filter


LJDAM on Target



Review of Basics


Modern ControlControl Design Problem: Design control inputs to produce satisfactory output response in the presence of disturbances and plant uncertainties.Disturbances

wOutput Variables

Control Inputs

u

P

yPlant ModelDone

Steps to a good design

Implementation Signal Constraints Robustness Sensitivity Disturbance Rejection Transient Behavior Steady State Accuracy Stability

17



What is control system design about?Plant model for describing/modeling control system dynamicsExogenous Variables Regulated Variables

wP

zxKFeedback Variables

uControl

Objective: Make transfer function from w z small with internal stability



Formulating the Control Design ProblemLook at the transfer function from w z

z Pzw y = P yw

Pzu w Pyu u

u = KyControl Law

1 z = Pzw + Pzu K I Pyu K Pyw = Tzw w

(

)

Plant Model

Linear Fractional Transformation

Measure size of Tzw using different norms. Most common are 2-norm and -norm

2 2 z 2 = z ( ) d z = sup z ( t )tK. Wise E. Lavretsky, N. Hovakimyan

1

2 2 Tzw 2 = 1 Tzw ( j ) d 2

1

Tzw = sup Tzw ( j )19


Tzw Tzw ( j )Tzw 0 dB

Peak versus frequency of the frequency response



State Space Models Math model for system dynamics, controls, sensors& x = Ax + Bu y = Cx + Du

xR

nx

u R nu

yR

ny

Assume x(0)=0, Compute transfer function matrixsX ( s ) x ( 0 ) = AX ( s ) + BU ( s ) X ( s ) = ( sI A ) BU ( s ) C ( sI A )1 B + D U ( s ) = G ( s )U ( s ) Y (s) = 1444 2444 3G( s )K. Wise E. Lavretsky, N. Hovakimyan

1

G (s)C

n y nu21


Linear Vector Spaces Important axioms for linear vector space X Set of vectors xi ; i = 1,L, nx are independent IFF

{

}

1x1 + 2 x 2 + L + nx 1x nx 1 + nx 1x nx = 0

If all

i ' s = 0

A set of linearly independent vectors is a BASIS SET for X if every vector in X can be expressed as a linear combination of the basis vectors

x = 1e + 2e + L + nx e

1

2

nx

is the representation of x wrt basis Change of basis:K. Wise E. Lavretsky, N. Hovakimyan

e1 L enx M = E = 4 14 244 3 nx { basis

1

x = E = E

= E 1E 22


Operators Two vector spaces X , Y operator a codomain

Y = aXdomain

Apply a to every x X The resulting y Y is called the range or image of a

R ( a ) Range space of a& x = Ax + BuRange space of B is very important Null space (kernel) of a : N

( a ) = { x X : ax = 0}

Matrix multiplies are operators23



Example0 1 1 2 A = 1 2 3 4 2 0 2 0 We see that

0 1 R ( A ) = sp 1 , 2 2 0 Compute the Null space for

1 0 1 3 = 1 + 2 2 2 0

dim (R ( A ) ) = 2

Range space is spanned by these two vectors.

x2

0 1 A= 0 0 x1

0 1 x1 0 x2 0 Ax = = = 0 0 x2 0 0 0

Any vector on x1 -axis maps to Null Space24



Similarity Transformations& x = Ax + BuLet z = Tx

T 1 exists

& 1 3 z = TAT 1 z + TB u { 2 B A Similarity transformations preserve eigenvalues Transform dynamics to diagonal/modal form & x = Pz P = v1 L vn x = Ax + Bu Let x eigenvectors & P 3 z = 11 AP z + {u = diag L P 1 B 2 nx 1 Spectral radius of the matrix A : ( A ) = max i& & z = Txi

Eigenvectors form a basis for the state space and describe the coupling of the modes ei t25



Eigenvalue Facts Hermitian matrix Unitary matrix Orthogonal matrix

M = M* U * = U 1 RT = R 1

1. s of a Hermitian matrix are real 2. s of a Unitary matrix have unit magnitude 3. If A is Hermitian, then P = [ vectors 5. Anm , 4. The matrix A is singular IFF i ( A ) = 0 for some i

]

is Unitary

Bmn ,

6. ( A ) = 1 A1 7. ( A ) = ( A )

( )

ABnn is singular if n > m



Eigenvalues and Eigenvectorsn n Eigenvalues and Eigenvectors of the matrix A R x x

Avi = i vi-vector -value

( A I ) vi = 0

( I A) vi = 0right eigenvector

vi N

( A i I ) v1 v 2 vi = v3 M vn x

Elements of the eigenvector describe t coupling of the mode e i into the state space Very important to understand the & eigenstructure associated with x = Ax + Bu

27



Aerodynamics

Integrate pressure distribution over body and compute force/moment. If aero center of pressure occurs in front of cg unstable If aero center of pressure occurs aft of cg stable If aero center of pressure occurs at cg neutrally stable Too stable or unstable causes increased trim drag, reduced maneuverabilityK. Wise E. Lavretsky, N. Hovakimyan

Unstable

Stable

28


Aircraft/Missile DynamicsLinear momentum: P=mV Angular momentum: H=I Differentiate to get 6-DOF Equations of motion

& mV = V + G + A + T & = I 1 ( I ) + I 1MAero Gravity Thrust

& u = rv qw + X + Gx + Tx & v = pw ru + Y + Gy + Ty & w = qu pv + Z + Gz + Tz & p = Lpq pq Lqr qr + L + LT

q - Body Pitch Rate v - Body Velocity ybody w - Body Velocity

u - Body Velocity

p - Body Roll Rate xbody

- Angle of Attackr - Body Yaw Rate

Sideslipxstability

& q = M pr pr Mr 2 p 2 r p + M + MT2 2

c

h

zbody

Vxwind

& r = N pq pq Nqr qr + N + NT

u = V cos ( ) cos v = V sin ( )


w = V sin ( ) cos ( )

29


Aircraft/Missile Dynamics = + & q = + &IMU,

CG

V

xb

xI

zI

zb

b g & = a1 / Vc f s b X + G + T g + c b Z + G + T g + q p tana f & = a1 / V f cs b X + G + T g + c cY + G + T h ss b Z + G + T g r& V = cc X + Gx + Tx + s Y + Gy + Ty + sc Z + Gz + Tzx x z z s

b

g cx

h

LM u OP LMc 0 s OPLMc s 0OPLMV OP v 0 MMwPP = MMs0 1 c0 PPMMs0 c0 1PPMM 0 PP N Q N 0 QN QN 0 Q LM u& OP LMcc sc cs OPLM V& OP & & v s MMwPP = MMs c0c scs PPMMV PP & N &Q N c QNV Q LM V& OP 1 Lcc sc sc OL u& O P& & c P M v P 0 MMV PP = c MM s & MN1444442 ssc PQMMNwPPQ & V Q s cc c N 44444 3 a f LM V& OP F L pO L u O L X O LG O LT OI & MMV PP = W a , fGGG MMq PP MM v PP + MMY PP + MMG PP + MMT PPJJJ & H NM r QP NMwQP NM Z QP NMG QP NMT QPK NV QBody Wind2 2 2 W ,

x

x

y z

y z

x

y

y

z

z

s


rs = r cos ( ) p sin ( )

ps = p cos ( ) + r sin ( )

30


Aircraft/Missile DynamicsLM LM X OP qS LMC OP LM L OP MM I MMY PP = m MMC PP MMM PP = MM N Z Q NC Q N N Q M MN I Aero Forcesx y z

qSl Cl I zz + Cn I xz 2 Izz I xz xx qSl Cm I yy qSl Cn I xx + Cl I xz 2 I zz I xz xx

b

b

OP g PP PP gPPQ

& =&

1 V

dZ + q + Z

e

e sin 0 Tx + cos 0 Tz

b g

b g iz

& q = M + Mq q + M e e + MTZ = Z e = M = = cos = 0x x

Aero Moments

& e

= 0

LM a fFG Z G T XIJ sina fFG X + G + T + ZIJ OP K H KQ N H L Z cosa f X sina fOP =M N Qz e e

=

= 0

Linear Model of Pitch Dynamics

M = 0

Mq =

M q = 0

M e =

M e

= 0

Z & q = V & M K. Wise E. Lavretsky, N. Hovakimyan

Z e 1 + q V e Mq M e 31


Zero Shaping To Eliminate Non-Minimum Phase Accelerometer ZerosAccelerometers Located in TailNormal Acceleration (g)

Large Non-Minimum Phase Effect Impacts Stability Margin Affects AOA Estimation

Step Response1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 0 0.5 1 1.5 2

Sensor LocationIMU CG CPerc

Solution: Lever-Arm Correction on Acceleration Signals Virtual IMU Slightly Ahead of Center of Percussion Eliminates Non-Minimum Phase Effect Transfer Function Zeros Better Located Tail Force IMU CG CPerc vIMU

Time (sec)

ImvIMU Nose

ReAccel Due to Tail Force vs Longitudinal PositionApproved for Public Release 9 Oct 1998 K. Wise E. Lavretsky, N. Hovakimyan

Zeros

IMU

CG32


Pitch Axis Linear ModelMissile Dynamics Nonlinear Pitch Dynamics:& = 1 + Z cos ( )( G 4+ Z 4 T3) sin ( )( G442 4 T 3) + q Z X + X + X V 1z-axis Forces 42 4 1x-axis Forces 4

& q = M + MT 1 24 4 3

- angle-of-attackq - pitch rate

Moments about y-axis

2 2 && + 2 n& + n = n c

2nd order actuator modelZ & & Az = Az + Z q + Z e V M M Z & q = Az + M q q + M Z Z && e = 2&e + 2 ( c e )

Linear Model:1 Z & Z q M M q M & = 0 0 & 0 && 2 0 n 0 Az = VZ + VZ 0 0 q 0 + 1 0 c 2 & 2n n 0

e

Design autopilot to track Azc commandsK. Wise E. Lavretsky, N. Hovakimyan

& x = Ap x + B p u y = Cpx

33


HAVE SLICK Missile DataMach 0.8 (V = 886.78 ft/s) and an altitude of 4000 ftZ VForce coefs divided by V

(deg) 6 16 (deg) 6 16 (deg) 6 16 (deg) 6 16K. Wise E. Lavretsky, N. Hovakimyan

PITCH Z Z (1/s) (1/s) -0.8757 -0.1531 -1.2100 -0.1987 ROLL-YAW Y Ya (1/s) (1/s) -0.0251 0.1228 -0.0052 0.1338 L La (1/s2) (1/s2) 574.7 195.5 556.8 70.2 N Na (1/s2) (1/s2) 16.20 -53.61 5.679 -57.03

M (1/s2) -68.0209 44.2506 Yr (1/s) -0.2763 -0.1004 Lr (1/s2) -529.4 -1879.0 Nr (1/s2) 33.250 5.7471

M (1/s2) -74.9210 -97.2313

34

Always Check Units


Eig and EigV of Missile ModelZ & q = V & M D= -7.2846e+000 0 0 0 S= -1.6243e-001 9.8672e-001 0 0 -1.3600e-001 -9.9071e-001 0 0 -2.8100e-004 +6.1484e-005i 6.0275e-003 -2.0058e-002i -8.8206e-003 -1.1761e-002i 9.9967e-001 -2.8100e-004 -6.1484e-005i 6.0275e-003 +2.0058e-002i -8.8206e-003 +1.1761e-002i 9.9967e-001 35K. Wise E. Lavretsky, N. Hovakimyan

Z 1 + V 0 q M q M

Added second order actuator dynamicsAp = -1.2100e+000 0 -1.9870e-001 0 4.4251e+001 0 -9.7231e+001 0 0 0 0 1.0000e+000 0 0 -4.6240e+003 -8.1600e+001

>> [S,D]=eig(Ap)

0 6.0746e+000 0 0

0 0 0 0 -4.0800e+001 +5.4400e+001i 0 0 -4.0800e+001 -5.4400e+001i


Autopilot Design Model: Mach 5.0(M6, ~65K ft) (~M3.5, ~40K ft) (M6, ~90K ft) (M0.9, 40K ft)

Cruise Initiate Mach Control

Pre-Boost Release Pre-Launch

Boost

(M1.5, 10K ft) Submunition Release

& x = Ap x + B p u y = Cpx Sensors located at virtual IMU36K. Wise E. Lavretsky, N. Hovakimyan


Missile Model With Zero Shaping%************************************************************************* % Plant Model %************************************************************************* % State Names % ----------% AZ fps2 % q rps % Dele deg % Dele dot dps % % % Input Names ----------Dele cmd deg Ap = [ -0.576007 -0.0410072 0 0 Bp = [ 0 ; 0 Cp = [ 1 0 0 0 1 Dp = 0.*Cp*Bp;

-3255.07 4.88557 -0.488642 -2.03681 0 0 0 -8882.64 ; 0 ; 8882.64]; 0; 0 0];

9.25796; 0 ; 1 -133.266

; ];



Definiteness of Matrices The matrix A is POSITIVE DEFINITE if Re ( ( A ) ) > 0 The matrix A is POSITIVE SEMIDEFINITE if Re ( ( A ) ) 0 The matrix A is NEGATIVE DEFINITE if Re ( ( A ) ) > 0 The matrix A is NEGATIVE SEMIDEFINITE if Re ( ( A ) ) 0 Notation

( A) > 0

( A) 0

( A) < 0

( A) 0



Norms Vector norm x x R nx , R Axioms 1. x = 0 IFF x = 0 2. x = x 3. x + y x + y Triangle Inequality

Norms we will use on vectors and matrices: p n p=1,2,

x R nx

x 1 = xii =1

n 2 2 x 2 = xi i =1 x = max xiiK. Wise E. Lavretsky, N. Hovakimyan

1

39


Matrix NormsA C mn Induced Normm

p =1 p= p=22

A 1 = max aijj i =1 m i

Ax p A p = sup = sup Ax p x0 x p x p =1

A = max aijj =1

A 2 = max A* A

(

( ))

1

2

= max AA*

(

( ))2

1

2

Let x be a -vector of A* A . A* A 0 is Hermitian.


Ax 2 = x* A* Ax = max ( A ) x* x = max ( A ) x 2 Ax 2 sup = max A* A x0 x 2

( )

40


Matrix Norms (cont)Frobenius Matrix Norm Properties of norms: 1. 2. 3. 4.

n m 2 2 A F = Tr A* A = a 2 j =1 i =1 ij

( )

1

1

x* y x 2 y 2 ( A) A pAB A B

5. U is Unitary U *U = I

AB F A 2 B F AF B2

Ux 2 = x 2 UA 2 = A 2 UA F = A F

(

)

AU 2 = A 2 AU F = A F

Unitary Matrices Are Norm Preserving41K. Wise E. Lavretsky, N. Hovakimyan


Singular Values, Singular Vectors Singular values used to measure size of matrices Like -values, there is a singular value decomposition (SVD) for each matrix.

A C m n

A = U V *

U , V Unitary (norm preserving)

U = [u1 L um ] C m m V = [ v1 L vn ] C nn 1 m>n m 0, If A is nonsingular A 1 = 1

n n If A C

U C n n V C n n

i

( )

( A)

A 1 = 1 ( A)

( )

2 2 A F = 1 + 2 + L + 2 min ( m, n ) W is Unitary, then i (WA ) = i ( A )

A 2 = ( A)

(2-norm of A is the max singular value)

Let l = rank ( A ) , A = i ui v* ii =1

l

43



Singular Values, Singular Vectors(cont)

Transfer Function Matrix G ( s )

G (s) V** v1 M * vn

0 1 % 2 O n 0 %

U

[u1

L un ]

v1 - Highest gain input direction,1 - Highest gain vn - Lowest gain input direction, n - Lowest gain u1 - Highest gain output direction un - Lowest gain output directionK. Wise E. Lavretsky, N. Hovakimyan

44


Useful Properties of Singular Values1. A C mm

, E Cmn

m

and det ( A +E) > 0, then ( E) < ( A ).

2. C, A C m 3. A ,B C 4. A C 5. A C 7. A Cm m m k m k

, i ( A ) = i ( A ).n

,(A +B) (A ) + (B). , (AB ) (A ) (B). , (AB ) (A ) (B). 17. A ,B C n 18. A C m 19. A C m 20. A Cn k n

, B Ck ,B C

6. A ,B C

m n

, (A )- (B) (A - B).

n n n n

, (A )-1 (I +A ) (A )+1.

n n

, i (AB ) i (BA ), in general,k

i.

, (A ) i (A ) (A ). 8. A C 9. Rank ( A ) = Number of nonzero i ' s . 10. A C mn

, B Cn , B Cn ,B C

, n k only, , n m only, , no restrictions,

(A ) (B) (AB ). (A ) (B) (AB ). (AB ) (A ) (B). (AB ) (A ) (B).

k

, i (A *) =i (A ).n

m n m n

n k n k

11. A ,B C m 12. A ,B C m 13. A ,B C m14. A C mn

, (A )- (B) (A +B) (A )+(B). , (A )- (B) (A +B) . , (A )- (B) (A - B) (A )+(B).Trace ( A *A ) n ( A ).k i =1 k

n n

,B C , no restrictions, 21. A C 22. (A )- (B) (A +B) (A )+(B). 23. (A )- (B) (A - B) (A )+(B).

, ( A )

15. A C mn , Tr A* A = i2 ( A ), k = min ( n, m ) 16. A C mn , det A* A = i2 ( A ), k = min ( n, m )i =1

( )

24. If B is square or has more columns than rows (A ) (B) (AB ) (A ) (B) (AB ) (A ) (B). 25. If A is square or has more rows than columns (A ) (B) (AB ) (A ) (B) (AB ) (A ) (B).

( )

45



How Do We Compute Singular Values i = i A* A = i AA*

( )

( )

Form A* A or AA*. Compute eigenvalues and eigenvectors for this matrix. Singular values are the square root of the eigenvalues. The singular vectors are the eigenvectors associated with the i2Lets look at the SVD.

Given A C mn , Unitary U , V A = U V * = i ui v* i

l

i =1 Right Singular Vectors

Left Singular Vectors

A* Avi = i2 ( A ) vi* * ui A* A = i2 ( A ) uiK. Wise E. Lavretsky, N. Hovakimyan

( Right ) ( Left )

46


FactA in Unitary if A* = A1. AA* = AA1 = I . i AA* = 1 iFor Unitary A,A 2 = ( A ) = ( A ) = 1.

( )

Therefore, the norm of a unitary matrix is always unity. Thus unitary matrices are norm invariant, and when multiplied against another matrix will not change the norm of that matrix.

For Unitary U ,

A 2 = UA 2



Solution to the State Equation Linear Time Invariant System Model

x ( t ) = ( t , t0 ) x ( t0 ) + 124 4 3

( t , ) Bu ( ) dt0

t

& x = Ax + Bu

State Transition Matrix

t k Ak 1 (t, 0) = = e At = L -1 ( sI A ) k! k =0 Easiest to form ( t , t0 ) using eigenvectors/eigenvalues

P 1 = v1 L vnx eigenvectors

= diag 1 L nx eigenvaluesK. Wise E. Lavretsky, N. Hovakimyan

(t, 0) = e

At

=e

P 1Pt

= P 1et P

48


Modal Expansion

& Homogeneous solution for x = Ax

x ( t ) = ( t , t0 ) x ( t0 ) = e At x ( t0 ) = P 1et Px ( t0 ) = ei t vi xi {i =1 nx

1 1 A= det ( I A ) = ( + 2 ) + 1 2 12 3 2 1 2 1 Avi = i vi 1 = 2 v1 = 2 = 2 v2 = 1 1 1 2t 2 1 2 t 2 1 2 t 2t 1 2 e + e e 0 3 1 2 e 3 3 3 3 At e = = 1 t 1 1 1 1 1 4 3 0 e 2 1 e 2 t 1 e 2t 2 e 2t + 124 14 244 14 3 3 4 3 3 24 3 3 3 1

Example

i-th component of Px ( t0 )

(

)

2

P

e t

e 3 1 t 1 e 2 3 2 t

P

49



Modal Expansion (cont)1 1 = 2 v1 = 1 2 1 2 = 2 v2 = 1Initial conditions elsewhere excite both modes1 t e 2

x2Initial conditions on this line excite only this mode

x1

e 2t

1 2t 2 12t x ( t ) = e x ( t0 ) = e x1 + e x2 1 1At

State Response Is Always a Linear Sum of ModesK. Wise E. Lavretsky, N. Hovakimyan

50


Controllability0.9w 0.1w

& x = Ax + Bu

Controllable Controllability Matrix Pc = B Rows of B Test

Controllable

A2 B L Anx 1B rank ( Pc ) = nx Rows of B not zero & & x = Ax + Bu z = z + Bu AB B = nx s = i ( A )51

Eigenvectors not contained in Nullspace of BT Hautus Test rank ( sI A )



Observability Test Summary C CA Observability Matrix: Q = M n 1 CA x rk = n x Columns of C Test

& x = Ax + Bu , y = Cx + Du & z = z + Bu , y = Cz + Du

Sensor Placement

Columns are not zero Eigenvectors contained in the Null space of C Hautus Test

sI A s = i ( A ) C rk = nx52



Stabilizability/Detectability Pair ( A, B ) is Controllable Pair ( A, B ) is Stabilizable Pair ( C , A ) is Observable

rank ( Pc ) = nx u = Kx,

( A BK )

Stable

C rank M = nx nx 1 CA

Pair ( C , A ) is Detectable

( A LC )

Stable

These tests are all Yes/No answers. How can we tell how controllable/observable a system is?53K. Wise E. Lavretsky, N. Hovakimyan


Degree of ControllabilityLTI System

c 1 B = b1 L bnu C = M c ny

& x = Ax + Bu y = Cx

= P 1 AP

P - Right Eigenvectors P 1 - Left Eigenvectors 1 v1 L vn P 1 = M P= x nx

Degree of Controllability Between Mode i and Control j i b j cos ( c ) = 1 - Controllable cos ( c ) = i 2 b j cos ( c ) = 0 - Not Controllable 2 Degree of Observability Between Mode i and Measurement j vi c j cos ( o ) = 1 - Observable cos ( o ) = vi 2 c j cos ( o ) = 0 - Not Observable 2


cos() Near Zero Will Cause Large Gains in Controller or Kalman Filter

54


Kalman Decomposition TheoremCO uCO y C - Controllable

C - Not ControllableO - Observable

CO CO G ( s ) = C ( sI A )1

O - Not ObservableOnly Controllable + Observable Modes Appear In Transfer Function

B+D

( A, B, C , D )

Is A Minimal Realization IFF

( A, B ) - Controllable ( C , A) - Observable55



Controllability DecompositionAc

u

Bc

s

1

xc

Cc

A12

y

s 1Ac& xc Ac & = xc 0 A12 xc Bc x + u Ac c 0

Cc

y = Cc

xc Cc x c56



Transmission ZerosZeros of a SISO Transfer Function

y = G (s)u

G (s) =

G ( s ) = c ( sI A )Zeros of a MIMO Transfer Function Matrix

1

d (s)

n(s)

b+d

y = G (s)u

G ( s ) = C ( sI A )

1

B+D

& x = Ax + Bu y = Cx + Du

sX = AX + BU Y = CX + DU

( sI A) X BU = 0CX + DU = Y

( sI A ) B X 0 = D U Y CIs There An s To Make Y = 0 for X , U 0 ?57K. Wise E. Lavretsky, N. Hovakimyan


Stability& x = Ax + Bu y = Cx + DuLyapunov Equation

( A ) Are the poles of G ( s ) Re ( A ) < 0 For StabilityAT P + PA = Q

Q>0

If P > 0 Exists That Solves The Lyapunov Eq., Then A is stable



Frequency Domain Analysis


Review of Transfer Functions and Transfer Function Matrices Math model for system dynamics, controls, sensors& x = Ax + Bu y = Cx + Du

xR

nx

u R nu

yR

ny

Assume x(0)=0, Compute transfer function matrix0 sX ( s ) x ( 0 ) = AX ( s ) + BU ( s )X ( s ) = ( sI A ) BU ( s ) C ( sI A )1 B + D U ( s ) = G ( s )U ( s ) Y (s) = 1444 2444 3G( s )K. Wise E. Lavretsky, N. Hovakimyan

1

G (s)C

n y nu60


Important Transfer FunctionsSISOReturned Signal

r +-

e

K(s)Controller

uo

Injected Signal

ui

G(s)Plant

y

U o ( s ) = K ( s ) G ( s )U i ( s ) 14 3 24 L (s) U i ( s ) U o ( s ) = U i ( s ) + K ( s ) G ( s )U i ( s ) = (1 + K ( s ) G ( s ) ) U i ( s )1 + L ( s ) - The Return Difference Dynamics E (s) 1 = = S (s) R (s) 1+ L (s)SensitivityK. Wise E. Lavretsky, N. Hovakimyan

Inject signal ui , examine returned signal uo

Scalar Variables K(s),G(s) Transfer Functions

Y (s)

R (s)

=

1+ L ( s)

L (s)

= T (s)

Complementary Sensitivity

Zeros of the return difference are the poles of the closed loop 61 system


Control Design DilemmaSISO Error Transfer Function

S (s) + T (s) = 1 1 + =1 1+ L ( s) 1+ L (s) L (s)Closed Loop Transfer Function

Want the errors to be small for command tracking. Want the plant to roll-off for robustness to high frequency unmodeled dynamics and sensor noise. When S ( s ) is small, T ( s ) is not small. When T ( s ) is small, S ( s ) is not small. What happens when neither are small? - This is where stability margins are determined. Same for MIMO Systems62K. Wise E. Lavretsky, N. Hovakimyan


MIMO System Transfer Function MatricesReturned Signal

r+-

e

K (s)

uo u i

Injected Signal

Returned Injected Signal Signal

G (s)

y

r+-

e

K (s)

G (s)

uo ui

y

Input Loop Break Point

Output Loop Break Point

U o ( s ) = K ( s ) G ( s )U i ( s ) 14 3 24 L1 ( s )n y n y nu nu L1 ( s ) C , L2 ( s ) C

U o ( s ) = G ( s ) K ( s )U i ( s ) 14 3 24 L2 ( s )L1 ( s ) L2 ( s )

n n n n n u R nu , y R y , K ( s ) C u y , G ( s ) C y u

For scalar systems loop gain is identical at plant input and plant output. Not true for MIMO systems. Must specify where the loop break point is.63K. Wise E. Lavretsky, N. Hovakimyan


SISO vs. MIMO Bode Nyquist Nichols

Loop Gain TF

Loop Gain Matrix

L = KG KG = GKReturn Difference

L1 = KG, L2 = GK KG GKReturn Difference Matrix

Singular Values Multivariable Nyquist

1+ LClosed Loop TF

I nu + L1, I n y + L2Closed Loop TF Matrix

T=

L 1+ L

T = [ I + L]

1

L

Inverse Closed Loop TF 1 1

T

= 1+ L

Stability Robustness Matrix 1

I+L L1 must exist

64



Example: Missile Pitch AutopilotAzc +-

K Az ( s )

+ -

Kq ( s )

q

(s)

q

Az (s) q

AzSIMO

Inner Rate Loop Outer Accel Loop

& Z q = M &

1 Z q + M 0 1

Az VZ q = 0

VZ q + 0 1

G ( s ) = C ( sI A )At plant input:

Az B + D = K ( s ) = K Az ( s ) K q ( s ) K q ( s ) 12 q 21

At plant output:K. Wise E. Lavretsky, N. Hovakimyan

L ( s ) = K ( s ) G ( s ) = Az K Az ( s ) K q ( s ) + q K q ( s ) Az K ( s ) K ( s ) Az K ( s ) q Az q L (s) = G (s) K (s) = q q K (s) K (s) K q ( s ) 65 Az q 22


Classical Frequency Response MethodsG (s) = s ( s + 2 )( s + 5 )10 ( s + 10 )

Well understood for scalar system. What if loop gain is a matrix?K. Wise E. Lavretsky, N. Hovakimyan

66


Nyquist TheoryIm

F (s)

s1

s2 s3Re

F ( s1 )

Im

Principle of the Argument

F ( s2 ) F ( s3 )

Re

Clockwise

s F ( s ) is analytic on s F ( s ) has Z zeros inside s F ( s ) has P poles inside s

F

F ( s ) Locus will encircle the origin Z P times as s traverses s N =Z P67K. Wise E. Lavretsky, N. Hovakimyan


Nyquist Theory (cont)For Stability Test:Im

n ( s ) Polynomial with Z zeros inside DR F (s) = d ( s ) Polynomial with P zeros inside DRDetermine if closed loop control system is stable by examining the loop gain transfer function

RRe

DR

F ( s) = 1+ L (s) 1 F ( s ) 1 1 L (s)Check for encirclements about origin



Nyquist Theory (cont)Look at closed loop TF

Y (s)

R (s)

=

1+ L (s)

L (s)

Return Difference

1+ L ( s) = 1+

d (s)

n(s)

=

d (s) + n(s) d (s)

cl ( s ) ol ( s )

Y (s)

d = n = R (s) 1+ n d d + n

n

For closed loop stability do not want any zeros of d + n in RHP

What if use F ( s ) = 1 + L ( s ) and check for encirclements about the origin (instead of (-1, j0))? Ans: Multivariable Nyquist Theorem69K. Wise E. Lavretsky, N. Hovakimyan


Nyquist CriterionIm

RRe

L (s)-1

Im

=Enclosed Region

Re

DR

=0

L ( j )Im

If plant is open loop stable, need only check j axis

=-1

Re

Enclosed Region

L ( j ) =070K. Wise E. Lavretsky, N. Hovakimyan


Multivariable SystemsReturn difference is a matrix: I + L ( s ) MFD: L ( s ) = N ( s ) D 1 ( s )

(s) I + L ( s ) = cl det ol ( s )det D ( s ) + N ( s ) det D ( s )

det I + L ( s ) = det ( D ( s ) + N ( s ) ) D 1 ( s ) = State space:

(s) = cl

ol ( s )

-

B

( sI A)K

1

x

cl ( s ) = det [ sI A + BK ]1

& x = ( A BK ) x

& x = Ax + Bu

u = Kx

cl ( s ) = det [ sI A] det I + ( sI A ) BK = ol ( s ) det I + ( sI A ) BK

1

cl ( s ) = ol ( s ) det I + K ( sI A ) B = ol ( s ) det [ I + L ( s )]

1

71



Multivariable Nyquist Theorem (s) I + L ( s ) = cl det ol ( s )The control system will be closed loop stable ( cl ( s ) has no RHP zeros) IFF,

R sufficiently large, N 0, det I + L ( s ) , DR = PolOrigin # of encirclements

(

)

Unstable open loop poles

Nyquist D-contour

N = Z P = Pol

No closed loop poles in RHP Z = 0

Plot det I + L ( s ) and count encirclements. If I + L ( s ) is a scalar,K. Wise E. Lavretsky, N. Hovakimyan

det I + L ( s ) = 1 + L ( s )

72


Gain and Phase Margins-GM-1

+GM

Im Re -1

Im

+GM

Re

L ( j1 )

+PM

L ( j 2 ) L ( j 3 )

Gain Margins:

GM = L ( j1 ) +GM = L ( j3 )

Phase Margins: = L ( j2 )

Close to (-1,j0). Small simultaneous gain and phase uncertainty will destabilize L ( j )

+PM

Good stability margins do not imply robustness. Must look at distance to (-1,j0).73K. Wise E. Lavretsky, N. Hovakimyan


Gain and Phase Margins (cont)Gain and phase uncertaintyIm

r+-

K (s)

ki e

ji

G (s)

y

1-1

Re

1+ L

L

To compute gain margin: L ( j ) ji 1) Set i = 0 ( e = 1 ). 2) Increase gain ki until system is unstable. +GM 3) Decrease gain ki until system is unstable. GM To compute phase margin: 1) Set ki = 1 . 2) Increase i until system is unstable. +PM 3) Decrease i until system is unstable. -PM

Distance to (-1,j0) measured by size of return difference.

What happens if ki and i vary at same time? Distance to (-1,j0) measures stability robustness.K. Wise E. Lavretsky, N. Hovakimyan

74


Gain and Phase Margins (cont)Multivariable Systems Distance to (-1,j0) is measured by the size of the Return Difference. For multivariable systems Return Difference is now a matrix. Need to measure size of a matrix - use singular values.

A + B Argument Assume matrix A is nonsingular. Assume A + B is singular. If A + B is singular, then A + B is rank deficient. Must have nullspace. ( A + B ) x = 0 Ax = Bx Ax 2 = Bx 2 ( A ) Ax 2 = Bx 2 ( B )To be singular, ( A ) ( B ) To be nonsingular, Return DifferenceK. Wise E. Lavretsky, N. Hovakimyan

( A) > ( B )Simultaneous gain and phase uncertainties75


Gain and Phase Margins For Multivariable Systems(Applies to SISO systems) & u = Kx Consider the following feedback control system: x = Ax + Bu Assume feedback stabilizes the system. Insert gain and phase uncertainties. Simultaneous gain and phase uncertainties x 1 diag ki e ji sI A ) B (

% Let = diag ki e ji

-

KReturn difference for nominal system 1

( ki = 1, i = 0 ) I + K ( sI A ) B = I + L ( s ) Is Nonsingular% det I + L ( s ) = cl ( s ) ol ( s )

Add gain/phase uncertainties to make system unstable. Return difference matrix becomes singular. This will change the number of encirclements of:76



Gain and Phase Margins For Multivariable Systems (cont)Nominal system + Uncertainties Return difference matrix: I + K ( sI A )1

% % B = I + L ( s )

1 % % % I + L = 1 I ( I + L ) + I ( I + L ) 1444 24444 4 3

(

)

Singular

This matrix must be singular

Nonsingular

% Want to find the smallest that destabilizes the nominal system.



Gain and Phase Margins For Multivariable Systems (cont)Use A + B argument:

(

1 % 1 I ( I + L ) + { I 144 2444 A 4 3 B

)

To be nonsingular:

( A) > ( B ) (I ) >

(( % 1 > (( % 1 > (( ( (

% 1 I1 1

( I + L ) 1 ) ( I + L ) 1 ) ( I + L ) 1 )

I I

) ) % ) (

1

I

) (

( I + L )1

)

Sufficient Test For StabilityK. Wise E. Lavretsky, N. Hovakimyan

(( I + L ) )1

1

% 1 I

) )

% ( I + L ) 1 I

Min singular value of the return difference must be larger than the uncertainty

78


Gain and Phase Margins For Multivariable Systems (cont) Let min ( I + L ( j ) ) = ( I + L)

( % i ( 1 I ) =

% 1 I1

)

ki e

ji

1

1

What if GM:

% Classical Gain Margin: 1 = 1 ki 1 1 1 + ki1 ki 1 1 + 1 ji % Classical Phase Margin: 1 = e 1 PM = 2 sin 2

= 1 ? (Best it can be) ki + dB -6 dB2 1

( )

o PM: 60

Min singular value of the return difference measures positive GM. What about negative GM?K. Wise E. Lavretsky, N. Hovakimyan

79


Gain and Phase Margins For Multivariable Systems (cont)Let

% % % = I +

% % % % % I + L ( s ) = I + L ( s ) + L ( s ) = L ( s ) I + L1 ( s ) + 1 24 4 3 144 244 3Nonsingular Return Difference Nonsingular Must be singular

Use A + B argument: 1

% % B= To be nonsingular: ( A ) > ( B ) % % I + L1 > A=I +L

(s)

Sufficient Test For StabilityK. Wise E. Lavretsky, N. Hovakimyan

( ) () % ( I + L1 ) > ( I )

Min singular value of the stability robustness matrix must be larger than the uncertainty80


Gain and Phase Margins For Multivariable Systems (cont)Let

min I + L

(

1

) =

I + L1

(

) = 1 ? (Best it can be)0 ki 2 dB +6 dB

1

Classical Gain Margin:

1 ki 1 + Classical Phase Margin: PM = 2sin1

What if GM:

2 I+L1

o PM: 60

( s ) - Stability robustness matrix81



Singular Value Stability Margins Also called multivariable stability margins Always more conservative than SISO classical marginsSISO Nyquist

-1

Im Re

T ( j )20 log10 0 dB

SISO Comp Sensitivity

L ( j )

min I + L1 = PM I + L = 2sin 1 2

MIMO SV Margins

min ( I + L ) =

1 1 , GM I + L = 1 + 1

(

)

GM

I + L1

= [1 ,1 + I + L1

]

PM

= 2sin 1 2 I + L1

GM = GM I + L U GMK. Wise E. Lavretsky, N. Hovakimyan

PM = PM I + L U PM

I + L1

82


Control Design DilemmaWould like

( I + L ) and I + L1

I + L = S 1Inverse of Sensitivity Small min singular value indicates poor stability robustness Make sensitivity small at low frequencies for command following

(

)

to both be large.

I + L1 = T 1Inverse of Complementary Sensitivity Small min singular value indicates large peak resonance Make complementary sensitivity small at high frequencies for robustness to unmodeled dynamics and sensor noise

S +T = I

Cant make both small at same time. Must trade off. When neither are small at loop gain crossover stability margins.K. Wise E. Lavretsky, N. Hovakimyan

83


Have Slick Example

Azc+-

K a ( s + az ) + s

K q K + aq s s q( ) s

(

)

2nd Order Actuator

(s) c

q

(s)

q Azq

(s)

AzSIMO

Inner Rate Loop Outer Accel Loop

Classical proportional plus integral control autopilot design N.nd order K. Wise E. Lavretsky, 2 Hovakimyan actuator dynamics included

84


Frequency Domain Plots

Inverse of Peak Resonance

Min Distance to (-1,j0)

Peak Resonance



Performance Specifications In The Frequency DomainDoes not matter if SISO or MIMO

r+-

e

K (s)

u

G (s)

y

e 1 ( s ) = ( I + K ( s ) G ( s )) r

= ( I + L ( s ))

1

= S (s)

Sensitivity

Important Transfer Functions (Transfer Function Matrices)Loop Gain

i ( L)

( L)

L = KG

Loop gain crossover frequency

0 dB

( L)Roll off plant for robustness to noise, high frequency unmodeled dynamics86

Need DC gain for command trackingK. Wise E. Lavretsky, N. Hovakimyan


Performance Specifications In The Frequency Domain (cont) (S )0 dB

Sensitivity

Stability

Roll off plant

Complementary Peak Resonance Sensitivity

(T )0 dB

e ( s ) = ( I + L )1 = S ( s ) r

Roll off plant

Want errors small at low freq for command tracking + disturbance rejection

y ( s) = ( I + L)1 L = T ( s) r Roll off plant for robustness to noise, high freq unmodeled dynamics

Shapes of these frequency responses are specified in the design of H optimal controllers.87K. Wise E. Lavretsky, N. Hovakimyan


Useful Analysis Method Vazsony DiagramVazsony Diagram Relative stability analysis Define new Nyquist contour x = sin 2 s = x + j 1 xIm

Second order CE:2 s 2 + 2 n s + n

x n

n 1 2

0

Workshop] Robust and Adaptive Part 1

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Transcript of Workshop] Robust and Adaptive Part 1