H∞ controller and observer design for linear systems with point and distributed time-delays

IFAC

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Copyright (:' IFAC Linear Time Delay Systems, Ancona, ital y, 2000

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Hoc CONTROLLER AND OBSERVER DESIGN FOR LINEAR SYSTEMS WITH POINT AND DISTRIBUTED TIME-DELAYS Anas Fattouh, Olivier Se name and Jean-Michel Dion

Laboratoire d'Automatique de Grenoble, ENSIEG - BP 46, 38402 Saint Martin d'Heres Cedex, FRANCE

Abstract: A memoryless state feedback control law is developed such that the closedloop system containing point and distributed time-delays is asymptotically stable and the Hoc-norm of the transfer function from the disturbance to the controlled output is reduced to some predefined level. The feedback is obtained by solving a parameterdependent linear matrix inequality, When the state variables are not available , an observer is built using a similar technique which generates an estimation of the state variables in order to implement the above state feedback. Copyright © 2000 IFAC Keywords: Time-delay, Memoryless, Hoc-Control, Observers, Linear Matrix Inequality (LMI) .

1. INTRODUCTION

bined with the algebraic Riccati equation technique (Ivanescu et al. 1999).

Many engineering processes can be modeled using point and/or distributed time-delay in the dynamics equations.

This paper deals with Hoc controllers and observers design for linear systems with point and distributed time-delays using the Lyapunov Krasovskii approach combined with the LMI technique. The proposed method not only guarantees the asymptotic stability of the closed-loop system (resp. the observer) but also reduces the Hoc-norm of the transfer function from the disturbance to the controlled output (resp. estimated error) to a predefined level.

Stability, state feed backs and observers for linear systems with only point time-delays have been throughout studied in the literature (see for example (Watanabe 1986, Lee et al. 1994, Fattouh et al. 2000)) . Controllability, observability and stability of linear systems with distributed time-delays are studied by several authors, see for example (Olbrot 1978, Verriest 1995, Kharitonov 1998). Distributed time-delays systems have been stabilized by a distributed control law using different approaches: polynomial approach (Manitius and Olbrot 1979), reducing approach (Fiagbedzi and Pearson 1987), infinite dimensional algebraic Riccati equations (Germani et al. 1994) , algebraic approach (Brethe 1997) and comparison principal approach (Tchangani et al. 1998). Recently, those systems have been stabilized by a memoryless state feedback using the Popov theory com-

The paper is organized as follows . Section 2 is devoted to develop a criteria for the Hoc stability of autonomous linear systems with point and distributed time-delays. This criteria serves to construct Hoo memory less state feedback controller in Section 3 and Hoc asymptotic observer in Section 4. Section 5 shows that the separation principle holds for the asymptotic stability, however, the Hoc-norm of the transfer function from the disturbance to the controlled output is not bounded by the pre-designed value but by some bigger value. The paper concludes with Section 6.

2. Hoo STABILITY CRITERIA

where «t,O)

Consider the following system: «t) =

~(t) + Al({t -

z(t) = V(t);

(p) = cp(p);

h)

+j

2: to

t

[

Q(O) =

o A2(O)(t

+ O)dO + £w(t) (1)

-h

pE [-h,OJ

where (t) E IRn is the state vector, z(t) E IRm is the controlled output vector, w(t) E IRq is the square-integrable disturbance vector, cp(p) is the continuous initial-value function vector, hE IR+ is the fixed known delay duration, ~, AI, £ and V are real matrices of appropriate dimensions. Finally, A2(O) is a (n X n) continuous matrix realvalued function . It should be noted that the controlled output z(t) contains point and distributed time-delay terms via the state vector (t). Definition 1. System (1) is Hoo asymptotically stable if - the trivial solution of (1) with w(t) == 0 is asymptotically stable, and - the IITzw(s)lloo :S "I for zero initial condition and some positive scalar "I.

The following Theorem provides a criteria for testing the Hoo asymptotic stability of system (1).

:=

[('(t) ('(t - h) ('(t

+ O)J' and

A~P+P~+R+S PAl hPA2(O)]

AP

0

-R

hA;(O)P

(5)

0-5

1- The asymptotic stability of system (1) under the assumption w(t) == 0: using the Schur complement (2) is equivalent to ~P+ P~

+ R+ S+ PAIR-IA'IP

+h 2PA2(0)S-IA;(0)P

+ V'v + ~pu' P < 0 "I

I;f

0 E [-h,OJ.

This implies

~P+P~

+ R+ S+PA1R-1A'lP

+h2 PA2(0)S-1 A;(O)p

<0

(6)

0 E [-h,OJ since V'v + ;frpu' P 2: o. Once again, usin~ the Schur complement one can conclude from (6) that Q(O) < 0 for all 0 E [-h,O] and so dV J:.9) < 0 for all t 2: 0 and for all 0 E [-h,O] which implies the asymptotic stability of system (1) according to Lyapunov-Krasovskii theory. 2- The H00 norm bound under the zero functional initial condition: let us introduce

I;f

T

JT

:=

j [z' (t)z(t) - 'Y2w' (t)w(t)]dt

o

then, for any nonzero square-integrable function w(t) T

Theorem 2. Consider system (1). Given a positive scalar "I, if there exist symmetric positive definite matrices P, R and S such that

~~ hP~2(O) ~£

;':P J(O):=

hA;(O)P

0

-s

0

£'p

0

0

-"1 2 Iq

J T = j[Z'(t)Z(t) - 'Y2w'(t)w(t)

1

T

:S

< 0 (2)

I;f 0 E [-h,OJ , where M := A~P + P~ + v'v + R+ s, then system (1) is Hoo asymptotically stable.

Proof. Assume there exist symmetric positive definite matrices P, R and S such that (2) is satisfied for some positive scalar "I and for all 0 E [-h,OJ. Let us introduce the following LyapunovKrasovskii functional t

+

j

x'(o)Rx(o)da

t-h

o

+~ j

t

j

x'(f3)Sx(f3)df3dO

- V(T)

o

(3)

-h t+6

Note that V(t) > 0 for all t 2: 0 and for all 0 E [-h,OJ. The time-derivative of V(t,O) along the state trajectory of the system (7) equals to

j[z'(t)Z(t) - 'Y2w'(t)w(t)

+ V(t)]dt

o

[

V(t, 0) = x'(t)Px(t)

+ V(t)Jdt

since V(O) = o. From (1) and (4), one can find that: 0

T

JT

:S

~j o

j

(' (t, O)J(O)(t, O)dOdt

-h

where (t,O):=[('(t) ('(t-h) ('(t+O) w'(t)J' . Finally, from (2) one can conclude that J < 0, i.e. z'(t)z(t)dt < 'Y2w'(t)w(t)dt for all T > o. Therefore, z(t) is square-integrable function for any nonzero square-integrable function w(t) and IIz(t)1I2 :S 'Yllw(tllI2 i.e. IITzw(s)lloo :S "I. 0

f;

f;

T

Remark 3. Under the assumption that A2(0) is a matrix of affine functions on 0 E [-h,OJ, the LMI (2) has a solution (P, R, S) for a given positive scalar "I if and only if (P, R, S) is a solution to J(O) < 0 and J(-h) < 0 for the same "I (Gahinet et al. 1994).

o

dV~~,O)

=

~j

('(t, O)Q(O)«t,O)dO

+ ('(t)P£w(t)

-h

+w'(t)£' P(t)

(4)

248

Remark 4. For A2(0) == 0, LMI condition (2) of Theorem 1 reduces to the condition proposed by (Lee et al. 1994) to test the Hoo stability of linear systems with point delayed-state.

Consider now the linear time-delay system:

all 8 E [-h,O}. The closed-loop system (7) with control law (9) is given by

o

+ AIX(t -

x(t) = Aox(t)

+

h)

!

A2(8)x(t

+ 8)d8

x(t) = (Ao - BoYPe- 1 )x(t)

-h

o

+Bou(t)

+ Bl U(t -

h)

+

!

B2(8)u(t

+ 8)d8 + Ew(t)

+

(7)

h)

+

!

C2(8)x(t

+ 8)d8 + Fw(t)

(10)

Inequality (8) means that there exists a vector t- 0 such that

t ~ to

Ze(t) = Dx(t) ;

+ 8)d8 + Ew(t)

v(t,8)

-h

x(p) = 4>(p);

(A2(8) - B2(8)Y Pe-l)x(t

Ze(t) = Dx(t)

o

+ CIX(t -

!

-h

-h

y(t) = Cox(t)

- Bl Y p e- l )x(t - h)

+ (AI

o

v'(t , 8)Jc(8)v(t, 8)

<0

(11 )

pE [-h , O}

Let v(t) then

where x(t) E IRn is the state vector, u(t) E IRr is the control input vector, y(t) E IRI> is the measured output vector, ze(t) E IRm is the controlled output vector, w(t) E IRq is the square-integrable disturbance vector, 4>(p) is the continuous initialvalue function vector, h E IR+ is the fixed known delay duration, Ao, AI, Bo, Bl, Co, Cl, E and D are real matrices with appropriate dimensions. Finally, A2(8), B2(8) and C2(8) are continuous matrix real-valued functions of appropriate dimensions. Note that the matrices E and F may also include parameter uncertainties or modeling errors.

:= [x'(t)Pe-

l , x'(t-h)P - l , x' (t+8)P e e

v' (t, 8)Je(8)v(t, 8) = v' (t, 8)Je(8)v(t , 8)

l

,

w'(t)J',

<0

(12)

where x'(t - h),

M

p e- l Al -pe- l ~Pe-l

1 hPe- A2(8)

o o

-Pe-1SePe-l

0

o

-'Y; Iq

A~Pe-1

[

+ 8) ,

v(t) = [x'(t),

hA;(8)Pe- 1

E' p e-

l

x'(t

w'(t)]'

o

p e-

l

0

E1 (13)

and M=

3. Hoc CONTROLLER PROBLEM

A~Pe-l

,.10 = Ao

+ p e- l ,.10 + D'D + Pe-l(~ + Se)pe- l ,

+ BoK,

+ BIK, = A2(8) + B2(8)K.

Al = Al

The objective of this section is to design a me moryless state feedback control law u(t) = Kx(t) , K E IRrxn, for system (7) such that the closed-loop system is asymptotically stable for w(t) :: 0 and IIzc(t)112 ::; 'Yellw(t)112 for zero initial condition and some positive scalar 'Ye .

A2(8)

The matrix (13) corresponds to LMI (2) applied on system (10) with P = pc-I, R = Pe-l~Pe-l and S = Pe-1SePe-l. Since (13) is negative definite then, by Theorem 2, system (10) is asymptotically stable for w(t) :: 0 and Ilze(t)112 ::; 'Yellw(t)112 for zero initial condition.

Theorem 5. Consider system (7) . Given a positive scalar 'Ye, if there exist symmetric positive definite matrices Pc, ~, Se and Y E IRr X n satisfying the following linear matrix inequality (LMI):

Remark 6. Under the assumption that A2(8) and are matrices of affine functions on 8 E [-h , O} , the LMI (8) has a solution Pc and Y for a given positive scalar 'Ye and some symmetric positive definite matrices ~ and Se if and only if Pc and Y are a solution to Jc(O) < 0 and J e( -h) < 0 for the same 'Ye, ~ and Se. B2(8)

[

:~ ~~:2 M~

0

-Se

E'

0

0

:1

<0

(8)

0

-'Y~Iq

Example 1. Consider the time-delay system (7) with

'V 8 E [-h,O}, where Mc := AoPe + PeA~ - BoY Y'B~ + PeD'DPe + ~ + Se , Ml = AIPe - BlY and M2 = h(A2(8)Pe - B2(8)Y), then the system (7)

closed by the control law u(t) = -YPe-

l

x(t)

is asymptotically stable for w(t) :: 'Yellw(t)1I2

Ao

= [ -~ ~ ] , AI = [~ ~]

Bo

= [ 1~ ],

Co

= [0

(9) 0

and

IIze(t ll l2 ::;

for zero initial condition.

Proof. Assume there exist symmetric positive definite matrices Pc, ~, Se and Y E IRrxn such that (8) is satisfied for some positive scalar 'Ye and for

= [ -0.5~ ],

10] , Cl

E=[n, 249

Bl

, A2(8)

= [0

0],

= [~

B2(8) =

=:] , [~] ,

C2(8) = [0 0] ,

D=[12], F=1

In order to design Hoo feedback controller for = 0.2, Theorem 5 is applied. Solving LMI (8) for 0 = 0 and 0 = -1 simultaneously gives, for example,:

4. Hoo OBSERVER DESIGN

'Ye

2.0500 -1.0520] [ ] Pc = [ -1.0520 1.3705 , Y = 1.2200 2.4000

Consider the system (7). A Luenberger-like observer for system (7) is given by the following dynamical system: a ±(t) = Aax(t) + AIX(t - h) + /

The corresponding feedback gain equals to: K = [ -2.4646 -3.6430]

o

The state variables of the closed-loop system are shown in Fig. 1. The transfer function ft¥1ction from the disturbance w(t) to zc(t) equals to Tzcw(s) = D{sI n - (Aa

+ BaK) -

(AI

+ BIK)e-

+BIU(t - h)

+/

B2(0)U(t

+ O)dO -

Fig. 2 shows the singular values of Tzcw(jw) versus the frequency. It shows that llzc(t)llz :S 0.21Iw(t)llz.

(14)

a y(t) = Cax(t) + CIX(t - h) + / C2(0)X(t + O)dO

+ Bz(0)K)e- 8s dO}-IE

a

L(Y(t) - y(t))

-h

sh

h

- /(Az(O)

Az(O)x(t + O)dO + Bou(t)

-h

-h

where x(t) E IRn is the estimated state of x(t), y(t) E IRP is the estimated output of y(t) and L E IRnxp is the observer gain matrix. The estimated error, defined as e(t) := x(t) - x(t), obeys the following dynamical behavior, obtained from equations (7) and (14) :

+ (AI - LCt}e(t - h)

e(t) = (Aa - LCa)e(t)

ooe

a

006

+ / (Az(O) - LC2(0)) e(t + O)dO + (E - LF)w(t) (15) -h

The objective of this section is to design an observer gain matrix L such that system (15) is asymptotically stable for w(t) == 0 and lle(t)llz :S 'Yollw(t)llz for zero initial condition and some positive scalar 'Yo . -0.06

Theorem 7. Consider systems (7), (14) and (15). Given a positive scalar 'Yo, if there exist symmetric positive definite matrix Po and X E IRnxp satisfying the following linear matrix inequality (LMI):

-000

-O'0~~--~--7---~-75--~--7-~~~~,0 time l..c !

Fig. 1. State variables of system (10).

Mo Jo(O) := [ 'V 0 E (-h,O],

Por(O) XA(O)

r'(o)Po -I2n+q A'(O)X'

1

<0

0

(16)

-I2p+q

0

where

I " 1 Mo = AoPo + PoAa - XCa - CoX + (2+ "2)In

'Yo

r(lJ)

= v2

A(O) =

[AI, hA2(0),

v2 (Cl,

El.

(17)

hC2(0), F] .

Then system (15) is asymptotically stable for w(t) == 0 and Ile(t)112 :S 'Yollw(t)llz for zero initial condition. Moreover, the observer gain matrix is given by: 002

(18)

Proof. Suppose that there exist symmetric positive definite matrix Po and X E IRn xp such that LMI (16) is satisfied. Using the Schur complement, LMI (16) is equivalent to

Fig. 2. Singular values of Tzcw(jw). When the state variables are not available to implement the designed state feedback control law, an observer has to be construct in order to estimate the state variables. This will be done in the next section.

I

I

I

1

AaPo + PoAa - XCo - CoX + (2 + "2 )In 'Yo +porr'po + XAA'X' < 0

Using (17)-(18), inequality (19) equals to

250

(19)

rnpulu(l)

+2Po L[C I , hC2(8), FJ[CI , hC2(8), Fl'L'Po

<0

(20)

Using the following inequality:

-uu' - vv'

~

.1

uv' + vu'

10

(21)

where U and V are any two matrices with suitable dimensions, one can write:

,

1

.:'V'(--------------j

.

time

-LFF'L' - EE' ~ LFE'

+ EF'L' Fig. 3. Estimated errors e(t)

By (22) equation (20) leads to the following inequality: (Ao - LCo)' Po

+ Po(Ao

- LCo)

,

10

I.-: !

= x(t) -

x(t) .

Yo'" 0.5

OM r---~----~----~----~-----r----~

+ (2 + -;. )In 70

+Po(AI - LCI )(AI - LCd Po +h 2Po(A2(8) - LC2(8»(A2(8) - LC2(8»' Po +Po(E - LF)(E - LF)' Po

02'

< 0 (23) 02

Using the Schur complement, inequality (23) corresponds to LMI (2) applied on system (15) with R = S = In. Since (23) is satisfied then, by Theorem 2, system (15) is asymptotically stable for w(t) == 0 and Ile(t)112 ~ 70 1Iw(t)112 for zero initial condition. 0

'" 0 1

005

Remark 8. Under the assumption that A2(8) and C2(8) are matrices of affine functions on 8, the LMI (16) has a solution Po and X for a given 70 E IR+ if and only if Po and X are solution to Jo(O) < 0 and J o ( -h) < 0 for the same 70 '

O L---~----~----~--~~----~----~ ~

~

m

~

~

~

~

w lraa'MC ]

Fig. 4. Singular values of TewUw). 5. SEPARATION PRINCIPAL

Example 2. Consider again the Example 2. In order to design Hoo-observer of system (7) for "fa = 0.5 , Theorem 7 is applied. Solving LMI (16) for 8 = 0 and 8 = -1 simultaneously gives, for example,: p. o

=[

2.0634 -1.5123] - 1.5123 2.0371 '

In this section, it is shown that if the control law u(t) = Kx(t) is implemented using the estimated state variables in the place of the real state variables then the separation principal holds.

Consider the system (7) and the dynamics output controller:

X = [0.0 197]

3.5008

The corresponding observer gain equals to:

u(t) = Kx(t) = K(x(t) - e(t»

L = [2.7836]

(24)

where K is given by (9), x(t) and e(t) obey the dynamical equations (14) and (15) respectively. The closed-loop system of system (7) and controller (24) is given by:

3.7850

The estimated errors are shown in Fig. 3. The curve between -1 and 0 sec. represents the initialvalue function. The transfer function function from the disturbance w(t) to e(t) equals to

o £ (t)

= Aox(t) + Alx(t -

h)

+j

A2(8)x(t

+ 8)d8 + Ew (t)

-h

Te",(s) = {sIn - (Ao - LCo) - (AI - LClle-· h

Zc(t) = Dx(t)

(25 )

h

where

- j(A2(8) - LC2(8))e-· s d8}-I(E - LF )

o

-(t)_[X(t)] Ao=[AO+BOK -BOK] , x e(t) , 0 Ao - LCo

Fig. 4 shows the singular values of Te",(jw) versus the frequency. It shows that Ile(t)1 12 ~ 0.51Iw(t)1I 2.

251

~LF ],

E= [ E

D = [ DO]

o1 ~

.

I

The spectrum of system (25) is given by o det(sI2n -

.10 -

A2(O)e-· 8 dO) =

Ale-oh - / -h

det(sI n - ;':e)det(sIn -

;':0) -00.

where ;':e

+ BoK) + {AI + BIK)e-· h

= (Aa

-o10~--~~~~~--~~'~~~~~~~--~~'0

o

+/

tirMl..c·J

(A2{1I)

+ B2(II)K)e-· 8 dO

Fig. 5. State variables of system (25).

-h ;':0

= (Aa - LCo)

+ (AI

7. REFERENCES

- LCde-· h

o

+/

(A2{O) - LC2{1I))e- Oe dll

-h

and the separation principal holds. As an illustration, Fig. 5 shows the state variables of system (25) for Examples 2 & 3. The transfer function from w(s) to ze{S) is given by: ze{t)

= {D{sIn -

;':e)-I

E - D{sIn -

;':e)-I

o .(Bo

+ Ble-· h + /

B2{II)e- 38 dO)

-h

.K{sIn - ;':o)-I{E - LF)}w(s)

(26)

From (26), one can find //Zc(S)//2 :S (-ye

+ ho)//w(s)//2

(27)

From (27) one can conclude that when applying the control law (24) the Hoc-norm of the transfer function from w(s) to Ze(S) is not bounded by 'Ye but by some maximal bound 'Ye + ho. This means that constructing a controller and an observer separately cannot guarantee the performance obtained by constructing the two systems simultaneously. Hence, a possible extension of this work is to solve the Hoc observer-based controller which leads to a condition involving two coupled LMls.

6. CONCLUSION

In this paper, a method for designing Hoc controller and observer for linear systems with point and distributed time-delays has been developed. A sufficient condition for the existence of such a controller and an observer is given in term of parameter-dependent linear matrix inequality to be solved. The proposed method can be extended to the case of linear systems with multiple commensurate point and distributed time-delays.

Brethe, D. (1997) . Contribution a I 'etude de la stabilisation des systemes lineaires a retards. PhD thesis in french. Universite de Nantes, France. Fattouh, A., O . Sename and J.-M. Dion (2000) . Robust observer design for linear uncertain time-delay systems: A factorization approach. In: 14th Int. Symp. on Mathematical Theory of Networks and Systems. Perpignan, France, June, 19-23. Fiagbedzi, Y. A . and A. E . Pearson (1987). A multistage reduction technique for feedback stabilizing distributed time-lag systems. Automatica 23, 311-326. Gahinet, P., P . Apkarian and M. Chilali (1994) . Parameterdependent lyapunov functions for real parametric uncertainty. Internal note, CERT-ONERA. Germani, A., L. Jetto, C. Manes and P. Pepe (1994) . The LQG control problem for a class of hereditary systems: A method for computing its approximate solution. In: Proc. 33th Confer. on Decision £; Control. pp. 13621367. Ivanescu, D ., J.-M. Dion, L. Dugard and S.-I. Niculescu (1999). Delay effects and dynamical compensation for time - delay systems. In: Proc. 38th IEEE Confer. on Decision £; Control. Phoenix, Arizona, USA. pp. 1999-2004. Kharitonov, V. L. (1998) . Robust stability analysis of time delay systems: A survey. In: IFAC Conj. System Structure and Control. Nantes, France. pp. 1-12. Lee, J. H., S. W . Kim and W. H. Kwon (1994). Memoryless Hoo controllers for state delayed systems. IEEE Trans. on Automatic Control 39(1), 159- 162. Manitius, A. Z. and A. W. Olbrot (1979) . Finite spectrum assignment problem for systems with delays . IEEE Trans . on Automatic Control 24(4), 541-553. Olbrot , A. W. (1978). Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans. on Automatic Control 23(5), 887-890. Tchangani, A . P., M . Dambrine and J.-P. Richard (1998). Robust stabilization of delay systems with discrete or distributed delayed control. In: Proc. 37th Confer. on Decision £; Control. Tampa, Florida, USA. pp. 40514056. Verriest, E . I. (1995). Stability of systems with distributed delays. In: IFAC Conj. System Structure and Control. Nantes, France. pp. 294-299. Watanabe, K. (1986). Finite spectrum assignment and observer for multi variable systems with commensurate delays. IEEE Trans. on Automatic Control 31(6), 543- 550.

ACKNOWLEDGEMENT Mr. Anas FATTOUH is supported by the University of Aleppo, SYRIA.

252