Qing-Chang Zhongmypages.iit.edu/~qzhong2/NUS_HinfDelay_slides.pdf · Qing-Chang Zhong...

ON THE STANDARD H∞ CONTROL OF

TIME-DELAY SYSTEMS

Qing-Chang [email protected]

Dept. of Electrical Eng. & Electronics

The University of Liverpool

United Kingdom

Outline of the talk

Problem statement

Notations and preliminaries

Some simpler problems

Solving the problem

Solutions to the problems

Summary

Q.-C. ZHONG: ON THE STANDARD H∞ CONTROL OF TIME-DELAY SYSTEMS – p. 2/53

Problem statement

Standard H∞ control problem (SPh): For a givenγ > 0, find a proper controllerK(s) such that theclosed-loop system is internally stable and

∥

∥Fl(P, Ke−sh)∥

∥

∞< γ.

Pe−shI

K

�

� �

�

y

z

u

w

u1

-


NotationsGiven a matrixA, AT andA∗ denote its transpose andcomplex conjugate transpose respectively.A−∗ standsfor (A−1)∗ when the inverseA−1 exists.

G(s) = D + C(sI − A)−1B.=

[

A B

C D

]

G∼(s) = [G(−s∗)]∗ =

[

−A∗ −C∗

B∗ D∗

]

Fl andFu are the commonly-used lower/upper linearfractional transformation (LFT).


Preliminaries

Two operators

Chain-scattering representation

Homographic transformation


Completion operatorA completionoperatorπh is defined forh ≥ 0 as:

πh{G}.=

A B

Ce−Ah 0

−e−sh

A B

C D

.= G(s)−e−shG(s).

0 0.2 0.4 0.6 0.8 h=1 1.2 1.40

1

2

3

4

Time (sec)

Impu

lse

resp

onse

G

πh(G)

e−shG

shift


Truncation operatorA truncationoperatorτh is defined forh ≥ 0 as:

τh{G}.=

A B

C D

− e−sh

A eAhB

C 0

.= G(s) − e−shG(s)

0 0.2 0.4 0.6 0.8 h=1 1.2 1.40

1

2

3

4

Time (sec)

Impu

lse

resp

onse

G

τh(G)


Representations of systems

Input-output rep.

Cl(M)

(a) the left CSR

Cr(M)

(b) the right CSRChain-scattering rep.


Chain-scattering representation

For M =

[

M11 M12

M21 M22

]

, the right and left chain-

scattering representations are defined as:

Cr(M).=

[

M12 − M11M−121 M22 M11M

−121

−M−121 M22 M−1

21

]

Cl(M).=

[

M−112 −M−1

12 M11

M22M−112 M12 − M22M

−112 M11

]

provided thatM21 andM12, respectively, are invertible.If both M21 andM12 are invertible, then

Cr(M) · Cl(M) = I.Q.-C. ZHONG: ON THE STANDARD H

∞ CONTROL OF TIME-DELAY SYSTEMS – p. 9/53

Homographic transformations (HMT)

Theright HMT:

Hr (M,Q) = (M11Q + M12) (M21Q + M22)−1

Theleft HMT:

Hl (N,Q) = −(N11 − QN21)−1(N12 − QN22)

M

@@

Q

-

� �

6

N

��

Q

-

� �

6

Hr (M, Q) Hl (N, Q)


Properties of the HMTLemma 1Hr satisfies the following properties:(i) Hr(Cr(M), S) = Fl(M,S);(ii) Hr(I, S) = S;(iii) Hr(M1,Hr(M2, S) = Hr(M1M2, S);(iv) If Hr(G,S) = R andG−1 exists, then

S = Hr(G−1, R).

Lemma 2 LetΛ be any unimodular matrix, then theH∞ control problem‖Hr(G, K0)‖∞ < γ is solvableiff ‖Hr(GΛ, K)‖∞ < γ is solvable. Furthermore,K0 = Hr(Λ, K) or K = Hr(Λ

−1, K0).


Problem statement



Solving the problem


Summary



One-block problemOPh

Extended Nehari problemENPh


One-block problem

OPh: Given a not necessarily stableGβ with

Gβ(∞) =

[

0 I

I 0

]

, find a proper (but not necessar-

ily stable)K(s) such that∥

∥Fl(Gβ, Ke−sh)∥

∥

∞< γ,

for a givenγ > 0.Actually, the(1, 2) and(2, 1) blocks ofGβ have stableinverses as well here.


Extended Nehari problem

ENPh: Given Gβ11, which is not necessarily stable,find a proper (but not necessarily stable)Qβ(s) suchthatGβ11 + e−shQβ is stable and

∥

∥Gβ11 + e−shQβ

∥

∥

∞< γ

for a givenγ > 0.


Problem statement



Solving the problem


Summary


Solving the standard problem

Via solvingsimpleproblemsusingsimpleideas andelementarytools.

Key steps:

Step 1: reducingSPh to SP0 + OPhStep 2: reducingOPh to ENPhStep 3: solvingENPhStep 4: recovering the controllers


The delay-free problem (I)

P (s) =

[

P11(s) P12(s)

P21(s) P22(s)

]

=

Ap Bp1 Bp2

Cp1 0 Dp12

Cp2 Dp21 0

Standard assumptions:(A1) (Ap, Bp2) is stabilisable and(Cp2, Ap) is de-tectable;(A2)

[

Ap − jωI Bp2

Cp1 Dp12

]

and[

Ap − jωI Bp1

Cp2 Dp21

]

have full column

rank and full row rank, respectively,∀ω ∈ R;(A3) D∗

p12Dp12 = I andDp21D∗p21 = I.

Assumption (A3) is made to simplify the exposition al-though, in fact, only the nonsingularity of the matricesD∗

p12Dp12 andDp21D∗p21 is required.


The delay-free problem (II)

H0 =

[

Ap γ−2Bp1B∗p1

−C∗p1Cp1 −A∗

p

]

−

[

Bp2

−C∗p1Dp12

]

[

D∗p12Cp1 B∗

p2

]

,

J0 =

[

A∗p γ−2C∗

p1Cp1

−Bp1B∗p1 −Ap

]

−

[

C∗p2

−Bp1D∗p21

]

[

Dp21B∗p1 Cp2

]

.

Solvability conditions:(i) H0 ∈ dom(Ric) andX = Ric(H0) ≥ 0;(ii) J0 ∈ dom(Ric) andY = Ric(J0) ≥ 0;(iii) ρ(XY ) < γ2.


The delay-free problem (III)Controller parameterisation:

K0 = Hr(Gα, Qα)

where

Gα =

Aα − B1C2 B2 B1

C1 I 0

−C2 0 I

andQα(s) ∈ H∞ is a free parameter satisfying

‖Qα(s)‖∞ < γ.


The delay-free problem (IV)

Aα = Ap+LCp2+γ−2Y C∗p1Cp1+

(

Bp2 + γ−2Y C∗p1Dp12

)

FΨ,

B1 = −L, B2 = Bp2 + γ−2Y C∗p1Dp12,

C1 = FΨ, C2 = −(

Cp2 + γ−2Dp21B∗p1X

)

Ψ,

F = −(B∗p2X+D∗

p12Cp1), L = −(Y C∗p2+Bp1D

∗p21),

Ψ = (I − γ−2Y X)−1.


Reducing SPh to SP0 + OPh

Cr(P )

@@ e−shI

K

-

� � �

w

z u

y

u1

6

Cr(P )

@@

Gα

@@

Cr(Gβ)

@@ e−shI

K

Delay-free problem 1-block delay problem

-

�

-

�

-

� � �

w

z u

y

u1

6

w1

z1

y

u1

Gα is the controller generator of the delay-free prob-lem. Gβ is defined such thatCr(Gβ)

.= G−1

α . Gα andCr(Gβ) are all bistable.


Gβ

Gβ =

[

Gβ11 Gβ12

Gβ21 Gβ22

]

=

A B1 B2

−C1 0 I

−C2 I 0

with A.= Aα − B1C2 − B2C1. AlthoughAα and/or

A may be unstable butA + B2C1 = Aα − B1C2 andA + B1C2 = Aα − B2C1 are always stable sinceGα

andCr(Gβ) are all bistable.


Reducing OPh to ENPh (I)

Qα(s) = Hr(Cr(Gβ), e−shK) = Fl(Gβ, e

−shK)

= Gβ11 + Gβ12Ke−sh(I − Gβ22Ke−sh)−1Gβ21

h

Gβ11

Gβ12

Gβ22

Gβ21h

∆2

e−shI

h

Kh(s)

�

-

+

-

K(s)�

- -

��

6

6

6

?

?

?

?

z1

w1

u1

y

u

Introducing a Smith predictor∆2(s) = −πh{Gβ22}

.= −Gβ22 + e−shGβ22


Reducing OPh to ENPh (II)

h

Gβ11

Gβ12e−shI

Gβ21

Gβ22

h

Kh(s)

�

-

+Qβ(s)�

-

��

6

6

6

?

?

z1

w1

u2

z2

Qβ.= Hr(

[

Gβ12 0

−G−1β21Gβ22(s) G−1

β21

]

,Kh)

The OPh is now reduced to ENPh:‖Qα(s)‖∞ =

∥


∥

∥

∞< γ.


An intermediate result

TheSPh is solvable iff its delay-free counterpartSP0

and the corresponding extended Nehari problem witha delayENPh are all solvable.

SPh ⇐⇒ SP0 + ENPh


ENPhin an different formDefine a stable FIR block

Q∆ = Gβ11 − e−shGβ11 = τh{Gβ11},

then theENPh is converted to a conventional Nehariproblem

∥

∥eshQ∆ + Q0

∥

∥

∞< γ,

whereQ0 = Gβ11 + Qβ is stable.


Solvability condition of ENPhIt is well-known that this problem is solvable iff

γ > γh.=

∥

∥ΓeshQ∆

∥

∥

whereΓ denotes the Hankel operator.

In fact,γh is the L2[0, h]-induced norm ofGβ11:

γh = ‖Gβ11‖L2[0,h] = sup‖u‖[0,h] 6=0

‖y‖[0,h]

‖u‖[0,h]

.

A simple representation ofγh:

γh = max{γ : det Σ22 = 0}.


Σ-matrix

Σ =

[

Σ11 Σ12

Σ21 Σ22

]

.= eHh,

where

H =

[

A γ−2B1B∗1

−C∗1C1 −A∗

]

.


What is the real problem left?

To find the parameterisation ofQβ under the conditionγ > γh.


Equivalent conditions for γ > γh

There existsQ0 ∈ H∞ such that∥


∥

∥

∞< γ, where

Qβ = Q0 − Gβ11;

Σ22 (which is a function ofγ) is nonsingular not only forγ but also for any number

larger thanγ;

There exists a unique functionX(t) satisfying the differential Riccati equation

X(t) =[

I −X(t)]

H

X(t)

I

,

with X(0) = 0 for t ∈ [0, h]. In particular,X(h) = Σ12Σ−1

22;

There exists a unique solution pairU(t), V (t) satisfying the differential Riccati equation

U(t)

V (t)

= H

U(t)

V (t)

with U(0) = 0, V (0) = I for t ∈ [0, h]. In particular,U(h) = Σ12, V (h) = Σ22.


Solving the ENPhENPh can be written in the chain-scattering represen-tation:

‖Hr(G(s), Qβ(s))‖∞ < γ.

with

G(s).=

[

e−shI Gβ11

0 I

]

.

A H∞ control problem is closely related to the matrix

G∼(s)JG(s) with J =

[

I 0

0 −γ2I

]

. If G(s) is com-

pleted to beJ-lossless, then theH∞ problem is solved.


Rationalise the problem

h

Gβ11 ∆1(s)

e−shI

W−1(s)

h

Q(s)

Qβ(s)@@�

--

z1

w1

u2

z2

�

-

��

6

6

6

?

?

∆1(s) = −πh{Fu(

[

Gβ11 I

I 0

]

, γ−2G∼β11)}.

The following matrix is rationalised:

Θ.=

[

I ∆1(s)∼

0 I

]

G∼JG

[

I 0

∆1(s) I

]


Realization ofΘ(s)

A B1B∗1 0 0 0 0 0 0 0 0

−γ−2C∗1C1 −A∗ 0 0 0 0 0 0 −γ−2C∗

1 0

0 0 A 0 0 0 B1B∗1Σ

∗21 −B1B

∗1Σ

∗11 0 B1

0 0 −C∗1C1 −A∗ 0 0 0 0 0 0

0 0 0 Σ21B1B∗1 −A∗ C∗

1C1 −γ2Σ21B1B∗1Σ

∗21 γ2Σ21B1B

∗1Σ

∗11 0 −γ2Σ21B1

0 0 0 −Σ11B1B∗1 −γ−2B1B

∗1 A γ2Σ11B1B

∗1Σ

∗21 −γ2Σ11B1B

∗1Σ

∗11 0 γ2Σ11B1

0 0 0 0 0 0 A γ−2B1B∗1 0 0

0 0 0 0 0 0 −C∗1C1 −A∗ γ−2C∗

1 0

C1 0 0 0 0 −γ−2C1 0 0 I 0

0 0 0 B∗1 0 0 −γ2B∗

1Σ∗21 γ2B∗

1Σ∗11 0 −γ2I


Simplified Θ−1(s)

A γ−2B1B∗1 0 γ−2Σ11B1

0 −A∗ −C∗1 γ−2Σ21B1

−C1 0 I 0

γ−2B∗1Σ

∗21 −γ−2B∗

1Σ∗11 0 −γ−2I


Factorisation of Θ−1

Amazingly,γ−2B1B

∗1 + Σ12Σ

−122 A∗ + AΣ12Σ

−122 ≡

−Σ12Σ−122 C∗

1C1Σ−∗22 Σ∗

12 + γ−2Σ−∗22 B1B

∗1Σ

−122 .

Hence,

Θ−1 = W−1(s) · J−1 · W−∼(s),

where

W−1(s) =

A Σ12Σ−122 C∗

1 −Σ−∗22 B1

−C1 I 0

γ−2B∗1Σ

∗21 0 I

.


J-losslessness of the system

h

Gβ11 ∆1(s)

e−shI

W−1(s)

h

Q(s)

Qβ(s)@@�

--

z1

w1

u2

z2

�

-

��

6

6

6

?

?

The completed matrix:

M(s) =

[

M11 M12

M21 M22

]

.= G(s)

[

I 0

∆1(s) I

]

W−1(s)

is stable andJ-unitary. Furthermore, it isJ-losslessbecauseM22(s) can be shown bistable.


Realization ofM(s)

M(s) =

[

e−shI 0

0 I

]

+

τh{

A γ−2B1B∗1 0 −B1

−C∗1C1 −A∗ Σ−1

22 C∗1 Σ−1

22 Σ21B1

C1 0 0 0

0 −γ−2B∗1 0 0

}.


Near the end (sufficiency)We have now obtained that

Gβ11 + e−shQβ = Hr(M(s), Q(s))

is stable and∥


∥

∥

∞< γ for

Qβ(s) = Hr(

[

I 0

∆1(s) I

]

W−1(s), Q(s))

with any stableQ(s) having‖Q(s)‖∞ < γ.


NecessityBy construction,

∥


∥

∥

L∞

< γ iff

‖Q(s)‖L∞

< γ for

Q(s) = Hr(W (s)

[

I 0

−∆1(s) I

]

, Qβ(s)).

Q(s) given above is proper iffQβ(s) is proper.

The set of proper operators inL∞ is in factH∞. So ifQβ(s) solvesENPh, thennecessarily‖Q(s)‖∞ < γ.


Solution to ENPhSolvability⇐⇒ γ > γh

Parameterisation:

Qβ(s) = Hr(

I 0

∆1(s) I

W−1(s), Q(s)),

where

∆1(s) = −πh{

A γ−2B1B∗

10

−C∗

1C1 −A∗ C∗

1

0 γ−2B∗

10

},

W−1(s) =

A Σ12Σ−1

22C∗

1−Σ−∗

22B1

−C1 I 0

γ−2B∗

1Σ∗

210 I

,

and‖Q(s)‖∞

< γ is a free parameter.


Moving back to OPh and SPh


Recovering the controller

hh

Gβ22∆2(s) ∆1(s)

G−1β12

G−1β21

W−1(s)

h

Q(s)

@@

- - - --- -

u

y

�

-

��

6

?

?

?

?

?

?

K = Hr(

[

I 0

∆2 I

]

[

G−1β12 0

Gβ22G−1β12 Gβ21

]

[

I 0

∆1 I

]

W−1, Q)


The simplified controller

∆(s) V −1(s)

h

Q(s)

@@

--

u

y-

��

6

?

?

K(s) = Hr(

[

I 0

∆ I

]

V −1, Q),

where

∆(s) = −πh{Fu(Gβ, γ−2G∼

β11)}

V −1(s).=

[

G−1β12 0

Gβ21F0 − F2G−1β12 Gβ21

]

W−1.


Problem statement



Solving the problem


Summary


Solution to OPhSolvability⇐⇒ γ > γh

Parameterisation:

K(s) = Hr(

[

I 0

∆(s) I

]

V −1(s), Q(s))

where

∆(s) = −πh{Fu(Gβ, γ−2G∼

β11)}

V −1(s) =

A + B2C1 B2 − Σ12Σ−122 C∗

1 Σ−∗22 B1

C1 I 0

−γ−2B∗1Σ

∗21 − C2Σ

∗22 0 I

,

and‖Q(s)‖∞ < γ is a free parameter.


Solution to SPhSolvability⇐⇒ :

H0 ∈ dom(Ric) andX = Ric(H0) ≥ 0;

J0 ∈ dom(Ric) andY = Ric(J0) ≥ 0;

ρ(XY ) < γ2;

γ > γh.

Parameterisation:

K(s) = Hr(

[

I 0

∆(s) I

]

V −1(s), Q(s))

where∆(s) andV −1(s) are given in in the last slideand‖Q(s)‖∞ < γ is a free parameter.


Some commentsV (s) is bistable:Σ−∗

22 (A + B1C2)Σ∗22 ∼ A + B1C2

The lower-right block ofCr(P )

[

I 0

∆ I

]

V −1 is

bistable.

The solvability conditions ofSPh are those ofSP0 plusγ > γh.

ForSP0, Σ ≡ I. SoΣ22 ≡ I is always nonsingularfor anyγ > 0. The last condition disappears andV −1(s) becomesGα(s). Hence, the last conditionis the additional cost of the delay.


Hints for multi-delay problemsTwo steps to obtain the FIR block: in the first step,a common modified Smith predictor∆2(s) is ob-tained; in the second step another FIR block∆1(s)is obtained;

It can also be solved in a clear and simple way: re-ducing the original problem to a delay-free prob-lem and a one-block problem with delay(s). Oncethe one-block problem with delay(s) is solved, theoriginal problem is solved.


Interpretation of H∞ fixed-lagsmoothing problemGivenG1(s), G2(s), γ > 0 andh ≥ 0, find a properK(s) such that

∥

∥e−shG1(s) − K(s)G2(s)∥

∥

∞< γ.

h

e−shG1 ∆s

G2

h

Ks

�

-

- +K(s)� �

6

6

6 6

6

Nothing but afeedforwardFIR block∆s is needed.


Summary

Standard problem with a single delay (SPh)

One-block problem (OPh)

Extended Nehari problem (ENPh)

Recovering the controller

SPh = SP0 + OPh

SPh ⇐⇒ SP0 + ENPhQ.-C. ZHONG: ON THE STANDARD H

∞ CONTROL OF TIME-DELAY SYSTEMS – p. 51/53

Implementation of the controllerAs seen above, the control laws associated with delay systems

normally include a distributed delay like

v(t) =

∫ h

0

eAζBu(t − ζ)dζ,

or in thes-domain, Z(s) = (I − e−(sI−A)h) · (sI − A)−1.The implementation ofZ is not trivial becauseA

may be unstable. This problem had confused the

delay community for several years and was pro-

posed as an open problem inAutomatica in 2003.

It was reported that the quadrature implementation

might cause instability however accurate the imple-

mentation is.

My investigation shows that:

The quadrature approximation error converges to0

in the sense ofH∞-norm.10

−210

−110

010

110

210

310

−4

10−3

10−2

10−1

100

101

Frequency (rad/sec)

N=1

N=5

N=20 A

ppro

xim

atio

n er

ror


Rational implementation

1x2xΠ

Nx 1−Nx

B1−Φbu

u

rv

…

ΦΦ+−=Π −1)( AsI

Π Π

…

Π = (sI − A + Φ)−1Φ,

Φ = (∫ h

N

0 e−Aζdζ)−1.


AB 3

1

› springer.com

isbn 1-84628-264-0

Qing-Chang Zhong

Robust Control of Time-delay Systems

ZhongRobust Control of Tim

e-delay Systems

Systems with delays appear frequently in engineering; typical examples include communication networks, chemical processes and tele-operation systems. The presence of delays makes system analysis and control design much more complicated. During the last decade, we have witnessed significant develop-ments in robust control of time-delay systems. This volume presents a system-atic and comprehensive treatment for robust (H∞) control of such systems in the frequency domain. The emphasis is on systems with a single input or out-put delay, although the delay-free part of the plant can be multi-input-multi-output, in which case the delays in different channels should be the same.

This synthesis of the author’s recent work covers the whole range of robust control of time-delay systems: from controller parameterization and design to controller implementation; from the Nehari and one-block problems to the four-block problem; from theoretical developments to practical issues. The major tools used in this book are similarity transformation, the chain-scattering approach and J-spectral factorization. The idea is, in the words of Albert Einstein, to “make everything as simple as possible, but not simpler”. A website associated with the book, is a source of MATLAB® and Simulink® material which will assist in the simulation of the material in the text.

Robust Control of Time-delay Systems is self-contained and will interest control theorists, researchers and mathematicians working with time-delay systems and engineers looking to design commercial controllers or to use them in plants, biosystems or communication systems with time delays. Its methodical approach will also be of value to graduates studying either general (robust) control theory or its particular applications in time-delay systems.

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