5/13/2003

Robust Control 8 Robust Stability

Harry G. KwatnyDepartment of Mechanical Engineering & MechanicsDrexel University

Outline• Modeling Uncertainty• Robust Stability/Robust Performance• Robust Stability of the M-∆ Structure• Uncertainty using Coprime Factors• The Structured Singular Value

Modeling Uncertainty( )Let be a stable transfer matrix with

may be , i.e., full may be , e.g., block diagonal or some elements

unstructuredstructured

Additive uncertainty

may be real.

:

Multiplicative uncertain

p

s

G G

γ∞

∆ ∆ ≤

∆∆

= + ∆

( )( )

: inpu

tyt

outputp

p

G G I

G I G

= + ∆

= + ∆

Actual plant

Nominal plant

Uncertainty

Example

( ) ( )2

2 2

A spacecraft attitude control model that includes the rigid body dynamics plus one flexible mode is given by the transfer function

2 21

A nominal model that includes only the rigid body d

ps sG s

s s s+ +=

+ +

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

2

2

2 22 2

2

2

ynamics is2

Additive uncertainty: 2 2 2 1

11

Multiplicative uncertainty:

12 1

p

pp

G ss

s sG s G ss s ss s s

G G sG s G sG s s

=

+ + −= + ∆ ⇒ ∆ = − =+ ++ +

− −= + ∆ ⇒ ∆ = =+ +

M-∆ Structurey∆u∆

y∆u∆

( ) 111 12 22 21N F P P I P K P−= + −

y∆u∆

11M N=

Obtaining P, N, M

K

G W

κ∆

z

y−u

y∆ u∆

w

( ) ( )( ) ( )

11

1 1

To generate , noteinputs: , ,outputs: , ,

0 0

Pu w u

y z y

IK I GK

P WG WKG I K

WG NWG I KG W I GKG I G

G −

∆

∆

−

− −

− + = ⇒ = + + −

− +

− −

Singular ValuesRecall a singular value and the corrsponding singular vectorsof a rectangular matrix are, respectively, the scalar andthe two vectors , that satisfy

The singular value decomposition of

T

Au v

Av uA u v

σ

σσ

==

isT

AA U V= Σ

Robust Stability/Robust Performance

( ) 122 21 11 12

22

The perturbed system closed loop transfer function is

Nominal Stability (NS) is stableNominal Performance (NP) 1; & NS

Robust Stability (RS)

w z

F N N I N NNN

−∆

∞

→

= + ∆ − ∆⇔⇔ <

⇔ is stable , 1; & NS

Robust Performance (RP) 1, , 1; & NS

F

F

∆ ∞

∆ ∞ ∞

∀∆ ∆ ≤

⇔ < ∀∆ ∆ ≤

Small Gain Theorem( ) ( )

( )( )

Recall spectral radius: ( ) max ( )

Consider a systemwith stable open loop transfer function Theorem (Spectral radius stability theore

. Then the closed loop

is

m).

stTheo

able if ( ) 1e

, .r

iiL j L j

L s

L j

ρ ω λ ω

ρ ω ω

=

< ∀

( ) Consider a system

with stable open loop transfer function . Then the closed loop

is stable if ( ) 1, , where deno

m (Small Gain T

tes any norm sa

heo

tis

rem)

fying

.

.L s

L j L

AB A B

ω ω< ∀

≤ ⋅

Determinant Stability Condition ( ) ( ) Assume , are stable and belongs to

a convex set of perturbations, such that if ' is a memberthen so is ' for any real scalar with 1. Then the

system is stable for all admissib

Theorem: M s s

c c cM

∆ ∆∆

∆ ≤− ∆

( )

( )( )( )( ) ( )

le perturbations if and only if

The Nyquist plot of det does not encirclethe origin for all

det 0, ,

1, , , 1, ,dimi

I M

I M j

M j i M

ω ω

λ ω ω

− ∆∆

⇔ − ∆ ≠ ∀ ∀∆

⇔ ∆ ≠ ∀ ∀∆ = ∆…

Spectral Radius Stability Condition( ) ( ) Assume , are stable and belongs to

a set of perturbations, such that if ' is a member then so is ' for any complex scalar with 1. Then the

system is stabl

The

e for all admissibl

ore

e

m: M s s

c c c M

∆ ∆∆

∆ ≤ − ∆

( )( )

( )( )

perturbations if and only if

1, ,

max 1,

M j

M j

ρ ω ω

ρ ω ω∆

∆ < ∀ ∀∆

∆ < ∀

Robust Stability of M-∆ Structure

( )( )

. Assume that the nominal system is stable and that the

perturbations ar

Theorem (Robust stability

e stable. Then the system is

stable for all perturbations

for unstructured perturbations

sa

)

M s

s M∆ − ∆

( )( )

( ) ( ) ( ) ( ) ( )Remarks on Proof: If satisfies 1, then

max max max .

This gives necessity. The small gain theorem with gives

sufficiency.

tisfying 1 if and only if

( ) 1, 1

M M M M

L M

M j M

σ

ρ σ σ σ σ

σ ω ω

∆ ∆ ∆

∞

∞

∆ ∆ ≤

∆ = ∆ = ∆ =

= ∆

∆ ≤

< ∀ ⇔ <

Coprime Factorization

1 1

A transfer matrix can be written in (left or right) coprime factored form

where , are stable, i.e., contains all of the RHP-zeros of and

contains all of the RHP-poles of

r rG D N N D

N D N G D

− −= =

• as RHP-zeros.

, are coprime, i.e., they have no common RHP-zeros which result in cancelation. Formally, they satisfy the Bezout identity: there exist stable , that satisfy

orl l

GN D

U VN U DV I

•

+ =

* * * *

a coprime factorization is called normalized if

or

r r

l l l l r r r r

UN VD I

N N D D I N N D D I

+ =•

+ = + =

Example

( ) ( )( )( )( )

( ) ( )

( ) ( )( ) ( ) ( )( )1 02 2

1 0 1 0

1 23 4

An obvious factorization is1 3,4 2

Another one is1 2 3 4

, , , 0

s sG s

s s

s sN s D ss s

s s s sN s D s a a

s a s a s a s a

− +=

− +

− −= =+ +

− + − += = >

+ + + +

Computing Coprime Factors( )

( ) ( )

( )

1/ 2 1/ 2 1/ 2

1

, a minimal realization

A normalized left coprime factorization is given by

where

,

and is the unique positive definite soluti

l l

T T T

A BG s C D

A HC B HD HN s D s R C R D R

H BD ZC R R I DD

Z

− − −

−

⇔

+ + ⇔

= − + = +

( ) ( )1 1 1 1

on to the ARE

0

with

TT T T T

T

A BS D C Z Z A BS D C ZC R CZ BS B

S I D D

− − − −− + − − + =

= +

Uncertainty Using Coprime Factors

( ) ( )

1 1

1

Recall that a transfer matrix can be written in coprime factored form

Suppose we allow separate perturbations in the numeratorand denominator so that the actual plant is

,

r r

p D N

G D N N D

G D N

− −

−

= =

= + ∆ + ∆ ∆[ ]

[ ] ( )

[ ]

1 1

This is equivalent to a system in structure with

= ,

RS 1/

N D

N D

N D

MK

M I GK DI

M

ε

ε ε

∞

− −

∞∞

∆ ≤

− ∆

∆ ∆ ∆ = − +

⇒ ∀ ∆ ∆ ≤ ⇔ <

Uncertainty Using Coprime Factors, Contunued

y∆u∆

( ) 1 1KI GK D

I− −

− +

[ ]N D∆ ∆D∆N∆

lN 1lD−

K−

M-∆ Structure for CoprimeUncertainty

[ ] ( )

( ) [ ]

( ) [ ]

1 1 1 1

1 1

1 1

1 1

,

, ,

This is equivalent to a system in structure with

= ,

D N D N

N D

N D

N D

y Gu D y D u u Ky y GKy D y D Ky

Ky I GK D y

I

Ky I GK D u u y y y

I

MK

M I GK DI

− − − −

− −

− −

∆ ∆ ∆ ∆

− −

= − ∆ + ∆ = − ⇒ = − − ∆ − ∆

= − + ∆ ∆

= − + = ∆ ∆ =

− ∆

∆ ∆ ∆ = − +

Controller Design with CoprimeUncertainty

( ) ( ) [ ]{ }

( )

1

1 1

Consider the family of perturbed plants

with some stability margin 0.The system is robustly stabilized by the controller ifand only if

1

Robust Stabili

p D N N DG D N

u Ky

KI GK D

I

ε

ε

γε

−

∞

− −

∞

= + ∆ + ∆ ∆ ∆ <

>=

− ≤

Find the lowest achievable

and the corresponding maximum stability margin and thecorresp

zer Design

onding cont

Problem:

roller .Kγ ε

Solution to RSDP( )( )

( ) ( )( ) ( )

( ) ( )

1/ 21min max

1 1 1 1

1 1 1 1

1

where , are the unique positive definite sol'ns of the ARE's

0

0

and minimal realization , , ,

TT T T T

TT T T T

XZ

X Z

A BS D C Z Z A BS D C ZC R CZ BS B

A BS D C X X A BS D C XBS B X C R C

G s A B C D

R I

γ ε ρ−

− − − −

− − − −

= = +

− + − − + =

− + − − + =

⇔

= + ,T TDD S I D D= +

Solution to RSDP, Cont’d

( )

( ) ( ) ( )

( )( )

1 1min

1 12 2

1

2

a solution that guarantees

for some

is given by

1

T T T T

T T

T T

KI GK D

I

A BF L ZC C DF L ZCK

B X D

F S D C B X

L I XZ

γ γ γ

γ γ

γ

− −

∞

− −

−

− ≤ >

+ + + = −

= − +

= − +

Example: Combustion Stabilizationmicrophone

speaker

controller

fuel

air

air

( ) ( )( )( )

2 2

2 2 2 21 1 1 2 2 2

flame acoustics

1 1 2 2

2

2 2

1500, 1000, 0.045, 4500

0.5, 1000, 0.3, 3500

z z zf

f

f f z z

s ss zG s K

s p s s s s

z p

ρ ω ωρ ω ω ρ ω ω

ρ ωρ ω ρ ω

+ + += + − + + +

= = = − =

= = = = Poles & zeros in RHP

A. M. Annaswamy and A. F. Ghonien, "Active Control in Combustion Systems," IEEE Control Systems, vol. December, pp. 49-63, 1995

Control Design• We will consider 3 designs

Classical approach – lead plus notchRobust stabilizationRobust performance (loop shaped stabilization)

• In each case look atRoot locusSensitivity functionError response to command stepGain and phase margins

Classical Lead + Notch( ) ( )

( )( )( )

2

2

0.70.0525

5 36c

s ssG s

s s s

+ ++=

+ + +

Classical Sensitivity Function

Classical Error Response to Step

Classical MarginsGMFrequency: [0.9109 4.0482 Inf]GainMargin: [0.7418 1.7252 Inf]PMFrequency: [0.6182 3.6938 5.2243 7.3372]PhaseMargin: [-55.1366 25.4364 179.2394 31.6964]

Classical Nyquist

Robust Stabilization (CoprimeUncertainty)

>> cd C:\Projects\Examples>> combustiongammin =

5.1270>> zpk(K)Zero/pole/gain:-32.5492 (s+0.9894) (s-0.3953) (s^2 + 2.101s + 12.25)---------------------------------------------------------(s+1.511) (s^2 + 9.959s + 32.55) (s^2 + 1.209s + 14.59)

Robust Stabilization Root Locus

Robust Stabilization Sensitivity

Robust Stabilization MarginsGMFrequency: [0 1.0251 2.6936 7.0934]GainMargin: [1.8545 0.6793 1.7478 19.1030]PMFrequency: [0.5216 1.6291]PhaseMargin: [-29.1102 19.5459]

Robust Stabilization Nyquist

Robust Stabilization Error Response to Step

Loop Shaping for Performance: Closed Loop Transfer Functions

( )( )( ) ( )

( )( )

Assume the closed loop is stable. Then:1. disturbance rejection 1

2. noise attenuation 1

3. reference tracking 1

4. control energy 1

5. robust stability with additive uncertainty 1

S

T

T T

KS

KS

σσσ σ

σσ

⇔ <<

⇔ <<

⇔ ≈ ≈

⇔ <<

⇔ <<

( )6. robust stability with mult output uncertainty 1Tσ⇔ <<

Loop Shaping for Performance: Open Loop Transfer Function

( )( )( )

( )( )

Assume the closed loop is stable. Then:1. disturbance rejection 1

2. noise attenuation 1

3. reference tracking 1

4. control energy 1

5. robust stability with additive uncertainty 1,

if

L

L

L

K

K

σσσ

σσ

⇔ >>

⇔ <<

⇔ >>

⇔ <<

⇔ <<

( )( )

1

6. robust stability with mult output uncertainty 1

L

L

σσ

<<

⇔ <<

Effect of Weighting Function( )1

1sW ss+= Shaped plant

Original plant

Robust Performance (Shaped Coprime Uncertainty)

>> combustion2gammin =

6.7266>> zpk(K)Zero/pole/gain:-54.29 (s^2 - 0.102s + 0.1931) (s^2 + 2.101s + 12.25)---------------------------------------------------------(s+1.517) (s^2 + 11.74s + 42.92) (s^2 + 1.122s + 15.35)

Robust Stabilization Root Locus

Robust Stabilization Sensitivity

Robust Stabilization MarginsGMFrequency: [0.4591 1.1951 2.7718 7.5896]GainMargin: [5.0727 0.7377 1.5541 14.4468]PMFrequency: [0.2363 0.7681 1.7705]PhaseMargin: [89.2292 -32.9393 14.9418]

Robust Stabilization Step Response

Robust Stabilization Nyquist

Combustion Example Summary• Classical design does stabilize, but margins and

performance are poor.• Robust stabilization dramatically improves

margins, but performance is still poor.• Robust performance dramatically improves

performance, but margins are reduced.• Can additional loop shaping improve margins?• Can γ-iteration improve design?

Structured Singular Value (µ)The goal is to find a generalization of , that allows generalizationof the above results to structured . Here is one approach.Find the smallest (infinity-norm) in a structured class that makes

d

σ ρ∆

∆ D

( ) ( ) ( )

( ) ( ) ( ){ }

( ) ( ) ( ) ( ) ( )

1

et 0; then : 1That is

Notice that for an unstructured class, the smallest

min d

withdet 0 has 1/

et 0M

I M

M

I

I

M

M

M M M

µ

µ σ

σ

σ σ µ σ

−

∆∈

− ∆ = = ∆

∆

∆ − ∆ =

− ∆ = ∆ = ⇒ =

D

Robust Stability: Structured Uncertainty

( ) ( )

( )Recall

. Assume that , are stable. Then the system

is stable for all admissible per

Theorem (Robust stabi

turbations satisfying

lity for structured perturbat

1 if

and only if ( ) 1, .

ions)M s s M

M jµ ω ω∞

∆ − ∆

∆ ≤

< ∀( )( )

( ) ( )

( )

: RS det 0, , , 1

If 1 at all frequencies, the required perturbation to make det 0

is larger than 1, so the system is stable.

If 1 at some frequency, there exists a perturbation wi

I M j

M I M

M

ω ω

µ

µ

∞⇔ − ∆ ≠ ∀ ∀∆ ∆ ≤

< ∆ − ∆ =

= ( )( )

th 1 such that

det 0 at this frequency and the system is unstable.I M

σ ∆ =

− ∆ =