Singular Value Analysis of Linear-Quadratic Systems

Robert Stengel Optimal Control and Estimation MAE 546

Princeton University, 2018

Copyright 2018 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

Δy s( ) = A s( ) ΔyC s( )− Δy s( )⎡⎣ ⎤⎦1+ A s( )⎡⎣ ⎤⎦Δy s( ) = A s( )ΔyC s( )Δy s( )ΔyC 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( )ΔOL s( )

ΔCL s( ) = ΔOL s( ) 1+ A s( )⎡⎣ ⎤⎦!  Closed-loop polynomial is

open-loop polynomial multiplied by return difference function

Δy s( )ΔyC s( ) =

kn s( ) ΔOL s( )1+ kn s( ) ΔOL s( )⎡⎣ ⎤⎦

=kn s( )

ΔOL s( ) 1+ kn s( ) ΔOL s( )⎡⎣ ⎤⎦

=kn s( )

ΔOL s( ) + kn s( )⎡⎣ ⎤⎦=

kn s( )ΔCL s( )

3

Return Difference Function Matrix for the Multivariable LQ Regulator

sΔx(s) = FΔx(s)+GΔu(s)sI− F( )Δx(s) =GΔu(s)

Δx(s) = sI− F( )−1GΔu(s)

Δu s( ) = −R−1GTPΔx s( ) ! −CΔx s( )= −C sI− F( )−1GΔu(s) ! −A s( )Δu(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

Δδ(s) = ΔuC (s)−A s( )Δδ(s)⎡⎣ ⎤⎦Im +A s( )⎡⎣ ⎤⎦Δδ(s) = ΔuC (s)

Δδ(s) = Im +A s( )⎡⎣ ⎤⎦−1ΔuC (s)

Analogy to SISO closed-loop transfer function

−Δu(s) = A s( )Δδ(s) = A s( ) Im +A s( )⎡⎣ ⎤⎦−1ΔuC (s)

Analyze signal flow from δ s( ) to δ 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−Δu(s) = A s( )Δδ(s) = A s( ) ΔuC (s)+ Δu(s)[ ]

− Im +A s( )⎡⎣ ⎤⎦Δu(s) = A s( )ΔuC (s)−Δu(s) = Im +A s( )⎡⎣ ⎤⎦

−1A s( )ΔuC (s)7

Closed-Loop Transfer Function Matrix is Commutative

−Δu(s) = A s( ) Im +A s( )⎡⎣ ⎤⎦−1ΔuC (s)

2nd-order example

A I+A[ ]−1 = I+A[ ]−1A

−Δu(s) = Im +A s( )⎡⎣ ⎤⎦−1A s( )ΔuC (s)

8

a1 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( )

Im +A s( ) = Im +C sIn − F( )−1 G

= Im +CAdj sIn − F( )G

ΔOL s( )

ΔOL s( ) Im +CAdj sIn − F( )G

ΔOL s( )= ΔOL s( ) Im +A s( ) = ΔCL 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

ΔCL s( )ΔOL s( )

= Im +A s( ) = Im +CAdj sIn − F( )G

ΔOL s( )= a s( )+ jb s( ) Scalar

Scalar Control

ΔCL s( )ΔOL s( )

= 1+A s( )⎡⎣ ⎤⎦= 1+CAdj sIn − F( )G

ΔOL s( )

⎡

⎣⎢

⎤

⎦⎥

= a s( )+ jb s( ) Scalar

Multivariate Control

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

Nyquist Stability Criterion

11Determinant is scalar in either case

ΔCL s( )ΔOL 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

12Counter-clockwise encirclements of RHP (unstable) poles

ΔCL s( )ΔOL 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

σ i D( ) = λi DTD( ), i = 1,n

Maximum singular value of D

Minimum singular value of D

σmax D( ) σ D( ) D = max

x =1Dx

σmin D( ) σ 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 : σ A1( ) = 2; σ A1( ) = 1A2 : σ A2( ) = 100.025; σ A2( ) = 0.02A3 : σ A3( ) = 100.025; σ A3( ) = 0

e.g.,x2

σ 2 +y2

σ 2 = 1

20

Stability Margins of Multivariable LQ Regulators

21

Bode Gain Criterion and the Closed-Loop Transfer Function

Δy jω( )ΔyC jω( ) =

A jω( )1+ A jω( )

A jω( )→0⎯ →⎯⎯⎯ A jω( )A jω( )→∞⎯ →⎯⎯⎯ 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 jω( )

22

Additive Variations in A(s)

Ao s( ) = Co sIn − Fo( )−1GoA s( ) = Ao s( ) + ΔA s( )

Connections to LQ open-loop transfer matrix

ΔAC s( ) = ΔC sIn − Fo( )−1Go

ΔAG s( ) = Co sIn − Fo( )−1 ΔG

ΔAF s( ) = Co sIn − Fo + ΔF( )⎡⎣ ⎤⎦−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

σ ΔA jω( )⎡⎣ ⎤⎦ < σ Im + Ao jω( )⎡⎣ ⎤⎦, 0 <ω < ∞

Sandell, 197924

“Bode Plot” of Singular Values

σ ΔA jω( )⎡⎣ ⎤⎦

σ Im + Ao jω( )⎡⎣ ⎤⎦

20 logσ

logω

Singular values have magnitude but not phase

Stability guaranteed for changing σ ΔA jω( )⎡⎣ ⎤⎦

up to the point that it touches σ Im + Ao jω( )⎡⎣ ⎤⎦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 + ΔL s( )

ΔL s( ) affects first row of Ao s( ) for pre-multiplicationΔL 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

σ ΔL jω( )⎡⎣ ⎤⎦ < σ Im + Ao−1 jω( )⎡⎣ ⎤⎦, 0 <ω < ∞

σ ΔL jω( )⎡⎣ ⎤⎦

σ Im + Ao−1 jω( )⎡⎣ ⎤⎦

20 logσ

logω27

Desirable “Bode Gain Criterion” Attributes

At low frequencyσ Im + A jω( )⎡⎣ ⎤⎦ > σmin ω( ) > 1

At high frequency

σ Im + A−1 jω( )⎡⎣ ⎤⎦−1{ } = 1

σ Im + A−1 jω( )⎡⎣ ⎤⎦< σmax ω( )

28

Desirable “Bode Gain Criterion” Attributes

σ A jω( )⎡⎣ ⎤⎦

σ A jω( )⎡⎣ ⎤⎦

ωC1 ωC2 logω

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 jω( ) Im + A jω( )⎡⎣ ⎤⎦−1

T jω( ) A jω( ) Im + A jω( )⎡⎣ ⎤⎦−1

31

Small S(jω) implies low sensitivity to parameter variations as a function of input frequency

Small T(jω) implies low response to noise as a function of input frequency

Sensitivity and Complementary Sensitivity Matrices of A(s)

•  But

S jω( ) + T jω( ) ! Im +A jω( )⎡⎣ ⎤⎦−1+A jω( ) Im +A jω( )⎡⎣ ⎤⎦

−1

= Im +A jω( )⎡⎣ ⎤⎦ Im +A jω( )⎡⎣ ⎤⎦−1

= Im

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

T jω( )S jω( )

logω32

Next Time: Probability and Statistics

33

Supplemental Material

34

Alternative Criteria for Multiplicative Variations in A(s)

ΔOL s( ) : Open-loop characteristic polynomial of original systemΔOL s( ) : Perturbed characteristic polynomial of original systemΔCL s( ) : Stable closed-loop characteristic polynomial of original system

•  Definitions

ΔOL jω( ) = 0{ } implies that ΔOL jω( ) = 0

for any ω on ΩR (i.e., vertical component of "D contour")α = σ Im + Ao jω( )⎡⎣ ⎤⎦ for any ω on ΩR

Lehtomaki, Sandell, Athans, 1981

35

Alternative Criteria for Multiplicative Variations in A(s)

σ L−1 jω( ) − Im⎡⎣ ⎤⎦ <α = σ Im + Ao jω( )⎡⎣ ⎤⎦And at least one of the following is satisfied:• α < 1• LH jω( ) + L jω( ) ≥ 0

• 4 α 2 −1( )σ 2 L jω( ) − Im⎡⎣ ⎤⎦ >α2σ 2 L jω( ) + LH jω( ) − 2Im⎡⎣ ⎤⎦

Perturbed closed-loop system is stable if

36

Guaranteed Gain and Phase Margins

K =1

1±αo

•  Guaranteed Gain Margin

•  Guaranteed Phase Margin

ϕ = ± cos 1− αo2

2⎛⎝⎜

⎞⎠⎟

In each of m control loops

Ifσ Im +Ao jω( )⎡⎣ ⎤⎦ >α o ≤1

37

Guaranteed Gain and Phase Margins

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

ϕ = ±90°

In each of m control loops

If LH jω( ) +L jω( ) ≥ 0 and AoH jω( ) +Ao jω( ) ≥ 0

38