Singular Value Analysis of Linear-Quadratic Systems !

Robert Stengel! Optimal Control and Estimation MAE 546 !

Princeton University, 2017

Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE546.html

http://www.princeton.edu/~stengel/OptConEst.html

!! Multivariable Nyquist Stability Criterion!! Matrix Norms and Singular Value Analysis!! Frequency domain measures of robustness

!! Stability Margins of Multivariable Linear-Quadratic Regulators

1

Scalar Transfer Function and Return Difference Function

A s( ) : Transfer Function

1+ A s( )!" #$ : Return Difference Function

•! Block diagram algebra

!y s( ) = A s( ) !yC s( )" !y s( )#$ %&1+ A s( )#$ %&!y s( ) = A s( )!yC s( )!y s( )!yC s( ) =

A s( )1+ A s( ) = A s( ) 1+ A s( )#$ %&

"1

•! Unit feedback control law

2

Relationship Between SISO

Open- and Closed-Loop Characteristic

PolynomialsA s( ) = kn s( )!OL s( )

!CL s( ) = !OL s( ) 1+ A s( )"# $%!! Closed-loop polynomial is

open-loop polynomial multiplied by return difference function

!y s( )!yC s( ) =

kn s( ) !OL s( )1+ kn s( ) !OL s( )"# $%

=kn s( )

!OL s( ) 1+ kn s( ) !OL s( )"# $%

=kn s( )

!OL s( ) + kn s( )"# $%=

kn s( )!CL s( )

3

Return Difference Function Matrix for the Multivariable LQ Regulator

s!x(s) = F!x(s)+G!u(s)sI" F( )!x(s) =G!u(s)

!x(s) = sI" F( )"1G!u(s)

!u s( ) = "R"1GTP!x s( ) ! "C!x s( )= "C sI" F( )"1G!u(s) ! "A s( )!u(s)

Open-loop system

Linear-quadratic feedback control law

4

Multivariable LQ Regulator Portrayed as a Unit-Feedback System

5

Broken-Loop Analysis of Unit-Feedback Representation of LQ Regulator

!""(s) = !uC (s)#A s( )!""(s)$% &'Im +A s( )$% &'!""(s) = !uC (s)

!""(s) = Im +A s( )$% &'#1!uC (s)

Analogy to SISO closed-loop transfer function

!"u(s) = A s( )"##(s) = A s( ) Im +A s( )$% &'!1"uC (s)

Analyze signal flow from ! s( ) to ! s( )Cut the loop as shown

6

Broken-Loop Analysis of Unit-Feedback Representation of LQ Regulator

Analyze signal flow from ! u s( ) to ! u s( )Cut the loop as shown

A s( ) = C sIn ! F( )!1G!"u(s) = A s( )"##(s) = A s( ) "uC (s)+ "u(s)[ ]

! Im +A s( )"# $%&u(s) = A s( )&uC (s)!&u(s) = Im +A s( )"# $%

!1A s( )&uC (s)7

Closed-Loop Transfer Function Matrix is Commutative

!"u(s) = A s( ) Im +A s( )#$ %&!1"uC (s)

2nd-order example

A I+A[ ]!1 = I+A[ ]!1Aa1 a2a3 a4

"

#$$

%

&''

a1 +1 a2a3 a4 +1

"

#$$

%

&''

!1

=a1 +1 a2a3 a4 +1

"

#$$

%

&''

!1a1 a2a3 a4

"

#$$

%

&''

=a1a4 ! a2a3 + a1( ) 1

a3 a1a4 ! a2a3 + a4( )"

#

$$

%

&

''det I2 +A( )

!"u(s) = Im +A s( )#$ %&!1A s( )"uC (s)

8

Im +A s( ) = Im +C sIn ! F( )!1G

= Im +CAdj sIn ! F( )G

"OL s( )

!OL s( ) Im +CAdj sIn " F( )G

!OL s( ) = !OL s( ) Im + A s( ) = !CL s( ) = 0

Relationship Between Multi-Input/Multi-Output (MIMO) Open- and Closed-Loop

Characteristic Polynomials

Closed-loop polynomial is open-loop polynomial multiplied by determinant of return difference function matrix

9

Multivariable Nyquist Stability Criterion!

10

!CL s( )!OL s( ) = Im +A s( ) = Im +

CAdj sIn " F( )G!OL s( )

= a s( ) + jb s( ) Scalar

Scalar Control

!CL s( )!OL s( ) = 1+ A s( )"# $% = 1+

CAdj sIn & F( )G!OL s( )

"

#'

$

%(

= a s( ) + jb s( ) Scalar

Multivariate Control

Ratio of Closed- to Open-Loop Characteristic Polynomials Tested in

Nyquist Stability Criterion

11

!CL s( )!OL s( ) = Im + A s( ) ! a s( ) + jb s( ) Scalar

Multivariable Nyquist Stability Criterion

Same stability criteria for encirclements of –1 point apply for scalar and vector control

Eigenvalue Plot, “D Contour” Nyquist Plot Nyquist Plot, shifted origin

12

!CL s( )!OL s( ) = Im + A s( ) ! a s( ) + jb s( ) Scalar

Limits of Multivariable Nyquist Stability Criterion

•! Multivariable Nyquist Stability Criterion –! Indicates stability of the nominal system–! In the |I + A(s)| plane, Nyquist plot depicts

the ratio of closed-to-open-loop characteristic polynomials

•! However, determinant is not a good indicator for the size of a matrix–! Little can be said about robustness–! Therefore, analogies to gain and phase margins

are not readily identified13

Determinant is Not a Reliable Measure of Matrix Size

A1 =1 00 2

!

"#

$

%&; A1 = 2

A2 =1 1000 2

!

"#

$

%&; A2 = 2

A3 =1 1000.02 2

!

"#

$

%&; A3 = 0

•! Qualitatively,–! A1 and A2 have the same determinant–! A2 and A3 are about the same size

14

Matrix Norms and Singular Value Analysis!

15

Vector NormsEuclidean norm

x = xTx( )1/2Weighted Euclidean norm

Dx = xTDTDx( )1/2

For fixed value of ||x||, ||Dx|| provides a measure of the size of D

16

Spectral Norm (or Matrix Norm)Spectral norm has more than one size

Also called Induced Euclidean normIf D and x are complex

x = xHx( )1/2

Dx = xHDHDx( )1/2

D = maxx =1

Dx is real-valued

dim x( ) = dim Dx( ) = n !1; dim D( ) = n ! n

17

xH ! complex conjugate transpose of x= Hermitian transpose of x

Spectral Norm (or Matrix Norm)Spectral norm of D

D = maxx =1

Dx

DTD or DHD has n eigenvaluesEigenvalues are all real, as DTD is symmetric and DHD is

HermitianSquare roots of eigenvalues are called singular values

18

Singular Values of DSingular values of D

! i D( ) = "i DTD( ), i = 1,n

Maximum singular value of D

Minimum singular value of D

!max D( ) ! ! D( ) ! D = max

x =1Dx

!min D( ) ! ! D( ) = 1 D"1 = min

x =1Dx

19

Comparison of Determinants and

Singular Values

A1 =1 00 2

!

"#

$

%&; A1 = 2

A2 =1 1000 2

!

"#

$

%&; A2 = 2

A3 =1 1000.02 2

!

"#

$

%&; A3 = 0

•! Singular values provide a better portrayal of matrix size , but ...

•! Size is multi-dimensional•! Singular values describe

magnitude along axes of a multi-dimensional ellipsoid

A1 : ! A1( ) = 2; ! A1( ) = 1A2 : ! A2( ) = 100.025; ! A2( ) = 0.02A3 : ! A3( ) = 100.025; ! A3( ) = 0

e.g.,x2

! 2 +y2

! 2 = 1

20

Stability Margins of Multivariable LQ Regulators!

21

Bode Gain Criterion and the Closed-Loop Transfer Function

!y j"( )!yC j"( ) =

A j"( )1+ A j"( )

A j"( )#0$ #$$$ A j"( )A j"( )#%$ #$$$ 1

!! Bode magnitude criterion for scalar open-loop transfer function!! High gain at low input frequency!! Low gain at high input frequency

!! Behavior of unit-gain closed-loop transfer function with high and low open-loop amplitude ratio

A j!( )

22

Additive Variations in A(s)

Ao s( ) = Co sIn ! Fo( )!1GoA s( ) = Ao s( ) + !A s( )

Connections to LQ open-loop transfer matrix

!AC s( ) = !C sIn " Fo( )"1Go

!AG s( ) = Co sIn " Fo( )"1 !G

!AF s( ) = Co sIn " Fo + !F( )#$ %&"1" sIn " Fo( )#$ %&

"1{ }Go

Gain Change

Control Effect Change

Stability Matrix Change

23

Conservative Bounds for Additive

Variations in A(s)Assume original system is stable

Ao s( ) Im + Ao s( )!" #$%1

Worst-case additive variation does not de-stabilize if

! "A j#( )$% &' < ! Im + Ao j#( )$% &', 0 <# < (

Sandell, 197924

Bode Plot” of Singular Values

! "A j#( )$% &'

! Im + Ao j"( )#$ %&20 log!

log!

Singular values have magnitude but not phase

Stability guaranteed for changing ! "A j#( )$% &'

up to the point that it touches ! Im + Ao j#( )$% &'25

Multiplicative Variations in A(s)

Ao s( ) = Co sIn ! Fo( )!1Go

A s( ) = LPRE s( )Ao s( ) or A s( ) = Ao s( )LPOST s( )

Very complex relationship to system equations; suppose

L s( ) = I3 +l11 s( ) 0 00 0 00 0 0

!

"

###

$

%

&&&= I3 + 'L s( )

!L s( ) affects first row of Ao s( ) for pre-multiplication!L s( ) affects first column of Ao s( ) for post-multiplication

Sandell, 1979 26

Bounds on Multiplicative Variations in A(s)Ao s( ) = Co sIn ! Fo( )!1Go

! "L j#( )$% &' < ! Im + Ao(1 j#( )$% &', 0 <# < )

! "L j#( )$% &'

! Im + Ao"1 j#( )$% &'

20 log!

log!27

Desirable Bode Gain Criterion Attributes

At low frequency! Im + A j"( )#$ %& > !min "( ) > 1

At high frequency

! Im + A"1 j#( )$% &'"1{ } = 1

! Im + A"1 j#( )$% &'< !max #( )

28

Desirable Bode Gain Criterion Attributes

! A j"( )#$ %&

! A j"( )#$ %&

!C1 !C2 log!

Undesirable Region

Undesirable Region

Crossover Frequencies29

Sensitivity and Complementary Sensitivity Matrices of A(s)

S s( ) ! Im + A s( )!" #$%1

Sensitivity matrix

Inverse return difference matrix

Im + A!1 s( )"# $%

T s( ) ! A s( ) Im + A s( )!" #$%1

Complementary sensitivity matrix

30

Sensitivity and Complementary Sensitivity Matrices of A(s)

S j!( ) ! Im + A j!( )"# $%&1

T j!( ) ! A j!( ) Im + A j!( )"# $%&1

Small T j!( ) implies low noise response as a function of frequency

Small S j!( ) implies low sensitivity to parameter variations as a function of frequency

31

Sensitivity and Complementary Sensitivity Matrices of A(s)

•! But

S j!( ) + T j!( ) ! Im + A j!( )"# $%&1 + A j!( ) Im + A j!( )"# $%

&1

= Im + A j!( )"# $% Im + A j!( )"# $%&1 = Im

•! Therefore, there is a tradeoff between robustness and noise suppression

T j!( )S j!( )

log!32

Next Time: !Probability and Statistics!

33

Supplemental Material

34

Alternative Criteria for Multiplicative Variations in A(s)

!OL s( ) : Open-loop characteristic polynomial of original system!!OL s( ) : Perturbed characteristic polynomial of original system!CL s( ) : Stable closed-loop characteristic polynomial of original system

•! Definitions

!!OL j"( ) = 0{ } implies that !OL j"( ) = 0

for any " on #R (i.e., vertical component of "D contour")$ = % Im + Ao j"( )&' () for any " on #R

Lehtomaki, Sandell, Athans, 1981

35

Alternative Criteria for Multiplicative Variations in A(s)

! L"1 j#( ) " Im$% &' <( = ! Im + Ao j#( )$% &'And at least one of the following is satisfied:• ( < 1• LH j#( ) + L j#( ) ) 0

• 4 ( 2 "1( )! 2 L j#( ) " Im$% &' >(2! 2 L j#( ) + LH j#( ) " 2Im$% &'

Perturbed closed-loop system is stable if

36

Guaranteed Gain and Phase Margins

K =1

1±!o

•! Guaranteed Gain Margin

•! Guaranteed Phase Margin

! = ± cos 1" #o2

2$%&

'()

In each of m control loops

If

! Im + Ao j"( )#$ %& >'o ( 1

37

Guaranteed Gain and Phase Margins

K = 0,!( )•! Guaranteed Gain Margin •! Guaranteed Phase Margin

! = ±90°

In each of m control loops

If LH j!( ) +L j!( ) " 0 and AoH j!( ) +Ao j!( ) " 0

38