Download - 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
Robust and Adaptive Control Workshop
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
3K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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 Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
X-45A, X-45C at Edwards AFB
10K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
Video
11K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
LJDAM on Target
15K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
Review of Basics
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
18K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
Tzw Tzw ( j )Tzw 0 dB
Peak versus frequency of the frequency response
20K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
26K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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 ( )
K. Wise E. Lavretsky, N. Hovakimyan
w = V sin ( ) cos ( )
29
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
rs = r cos ( ) p sin ( )
ps = p cos ( ) + r sin ( )
30
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
; ];
37K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
38K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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.
K. Wise E. Lavretsky, N. Hovakimyan
Ax 2 = x* A* Ax = max ( A ) x* x = max ( A ) x 2 Ax 2 sup = max A* A x0 x 2
( )
40
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
47K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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 )
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
cos() Near Zero Will Cause Large Gains in Controller or Kalman Filter
54
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
58K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
Frequency Domain Analysis
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
68K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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.
77K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
Frequency Domain Plots
Inverse of Peak Resonance
Min Distance to (-1,j0)
Peak Resonance
85K. Wise E. Lavretsky, N. Hovakimyan
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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
Robust and Adaptive Control Workshop
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