A Memoryless State Observer For Discrete Time-delay Systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 11, NOVEMBER 1997

M(k) 2 Rn( +1) , and control, N(k) 2

By introducing new state, vectors, as

A Memoryless State Observer for Discrete Time-Delay Systems

Rp(s+1) ,

H. Trinh and M. Aldeen

M(k) = [xT (k); xT (k 0 1); 1 1 1 ; xT (k 0 )]T

(2a)

N(k) = [uT (k); uT (k 0 1); 1 1 1 ; uT (k 0 s)]T

(2b)

and Abstract—This paper presents a reduced-order memoryless state observer with a -stability margin (0 < < 1) for linear discrete-time systems with multiple delays in both the state and control vectors. The dynamics of the observer are derived from a set of unstable and/or poorly damped eigenvalues of the system. A simple and systematic design method is presented. A numerical example is given to illustrate the properties of the new observer and its design method. Index Terms—Delay systems, discrete-time systems, state observers.

I. INTRODUCTION The problem of designing an observer which asymptotically tracks the true states of the system is fundamental for control engineers [1]–[6]. Recently, interest has been focused on the study of observers for continuous time-delay systems. Some results have appeared in this regard (see for example [7]–[9]). A literature search, however, indicates that state estimation of delay-difference systems has not yet been reported. It is well known [10]–[13] that the number of eigenvalues of a delay-difference system is proportional to the number of delays, and this number increases as the number of delays increases. Therefore, a high-order state observer is required in order to estimate the states of such systems. This gives rise to practical and technical difficulties when observer implementation is considered. In this paper, motivated by a recent work by Leyva-Ramos and Pearson [7] on the modal observer for continuous time-delay systems, a reduced-order memoryless state observer for linear discrete systems with multiple delays in both the state and control vectors is introduced. The new state observer has the advantages of being memoryless and having reduced-dynamics. The order of the new state observer is equal only to the number of unstable and/or poorly damped eigenvalues of the system. A simple and systematic design method is presented. A numerical example is given to illustrate the properties of the new observer and its design method. II. PROBLEM FORMULATION AND METHODOLOGY Consider a linear delay difference system described by

x(k + 1) =

 i=0

Ai x(k 0 i) +

y(k) = Cx(k) + Du(k)

s j =0

Bj u(k 0 j)

(1a) (1b)

where x(k+1) 2 Rn and x(k 0i) 2 R n are state vectors, y(k) 2 Rm is an output vector, u(k) 2 Rp and u(k 0 j) 2 R p are control vectors. Matrices Ai (i = 0; 1; 1 1 1 ; ); Bj (j = 0; 1; 1 1 1 ; s); C; and D are constant with appropriate dimensions. Integers   0 and s  0 denote, respectively, the number of time delays in the state and control vectors. The problem to be addressed in this paper is that of designing a memoryless reduced-order state observer to asymptotically track the entire state x(k) of (1). Manuscript received June 5, 1996; revised September 25, 1996. This work was supported by an Australian Research Council Small Grant. The authors are with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria 3052, Australia. Publisher Item Identifier S 0018-9286(97)07633-2.

(1a) and (1b) take the form

M(k + 1) = 2M(k) + 0N(k) y(k) = [C 1 1 1 0]M(k) + [D 1 1 1 0]N(k) where 2 and 0 are defined as A0 A1 1 1 1 A 01 A 0 2 R[n( +1)]x[n( +1)] 2 = I1n 01 11 11 11 00 0 0 0 1 1 1 In 0 and

B0 B1 0 = 01 01 0 0

111 111 111 111

Bs 0 0 0

2Rn

[ ( +1)]x[p(s+1)]

:

(3a) (3b)

(3c)

(3d)

From (3c) it can be easily seen that for a 1 increase in  the order of matrix 2 increases by n1: Thus for large delays in the state, the order of 2 becomes very high. As a result, a high-order state observer for (3) is required. In this paper, the approach taken to addressing this problem consists of two main stages. The first stage involves the determination of a set of unstable and/or poorly damped eigenvalues of the characteristic matrix

(A) = z  +1 In 0

 i=0

Ai z  0i

(4)

as defined by

= z: det z  +1 In 0

 i=0

Ai z  0i = 0; jz j 

(5)

where is a specified positive number in the range 0 < < 1: The set of eigenvalues can be obtained by first computing the eigenvalues of the matrix 2 as defined in (3c). Then for a specified value of ; is obtained. Upon the determination of ; the corresponding right and left eigenvector matrices, denoted as V and W respectively, are derived. The second stage involves the development of a -stable reducedorder observer capable of estimating the state vector, x(k), of system (1). This development is based on the result of the first stage and involves the construction of the matrix quadruplet (Ar ; Br ; Cr ; Dr ); as outlined in Section III. The derivation of V and W and the main result of this paper are given in Section III. Section IV presents a numerical example which serves to illustrate both the design method and the potentials of the new observer. Section V concludes the paper. III. DEVELOPMENT OF OBSERVER DESIGN METHOD For simplicity, but without loss of generality, let us assume that consists of a set of distinct eigenvalues. From (5), the following relationship can be easily derived:

det[(A)] = det z +1 In 0

 i=0



Ai z  0i

= 8(z) det(zIr 0 3r );

8z 2

(6)

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Fig. 1. A diagram representation of the state observer.

where 8(z ) is an entire function on the jz j  plane, i.e., it does not vanish for any z 2 ; and 3r = diag(1 ; 2 ; 1 1 1 ; r ) contains the eigenvalues that belong to : (Note that when contains repeated eigenvalues, the right-hand side of (6), i.e., 3r , will have the Jordan canonical form.) Now [(A)]01 can be expressed in the following form:

[(A)]01 = z +1 In 0

2C

V

In (7),

i=0

Ai z  0i

nxr

[(zq ) Similarly, W

In 0



Ai (zq ) 0 ]vq = 0;  i

i=0

wq

8zq 2 :

T

= [wqT ]

 i=0

Ai (zq ) 0i ]T wq = 0;

q

V~

(8b)

(9a)

(9b)

z

vq ;

p

(12)

p

W = [w1 ; w2 ; 1 1 1 ; 2 Re(wh ); 0 2 Im(wh ); 1 1 1 ; wr ]T : Let us now propose the following rth-order memoryless

(13) state

xr (k + 1) = Ar xr (k) + Br u(k) + L[y(k) 0 CCr xr (k)] (14a) x^(k) = Cr xr (k) + Dr u(k) (14b) rxr rxp nxr where Ar 2 R ; Br 2 R ; and Cr 2 R are defined as

Ar = 3r Br = W B0 Cr = V: (15) The definition of L 2 R rxm and Dr 2 R nxp will be introduced later

in this section when appropriate. A diagram representation for the proposed observer (14) is shown in Fig. (1). ^(k ), and actual Define the error between the observed states, x states, x(k ), as

x~(k) = x(k) 0 x^(k):

is a normalization scalar given by

q = wqT d[(A)] dz

(11)

follows:

8zq 2 ;

q = 1; 2; 1 1 1 ; r and

(8a)

2 C n is a solution to the following equation:

[(zq ) +1 In 0

3r ; V; and W become 3r = block-diag(p 1 ; 2 ; 1 1 1 ; Jh ; 1 1 1 ; r ) p V = [v1 ; v2 ; 1 1 1 ; 2 Re(vh ); 2 Im(vh ); 1 1 1 ; vr ]

observer for (1) as:

2 C rxn is a left-eigenvector matrix defined as follows: W = w1 ; w2 ; 1 1 1 ; wr 1 2 r

where

(7)

2 C n is a solution to the following equation:  +1

(10)

and

is a right-eigenvector matrix defined as follows:

q = 1; 2; 1 1 1 ; r

Jh = 0!h !h : h h Accordingly,

01

= V (zIr 0 3r )01W + V~ (zIn( +1)0r 0 3n( +1)0r )01W~ :

V = [v1 ; v2 ; 1 1 1 ; vr ] = [vq ];

vq

where



follows:

(16)

Using (1b), (14b), and (16), after some manipulations, (14a) becomes

xr (k + 1) = Ar xr (k) + LC x~(k) + (Br + LCDr + LD)u(k):

(17)

q = 1; 2; 1 1 1 ; r:

(9c)

~ in (7) are, respectively, right and left eigenvector matrices and W corresponding to the remaining -stable eigenvalues of (1). They can be similarly derived as shown for the case of V and W; and therefore will not be given here. For the case where matrix 3r contains some complex eigenvalues, then V; W; and 3r can be changed to real matrices [7], by constructing a Jordan sub-block matrix corresponding to zh = h 6 j!h as

Taking the z -transform of (1a) and (17) and after some manipulations, the following equation is obtained:

(A)[In + Cr (zIr 0 Ar )01 LC ]X~ (z ) = (x0 ) s + z Bj z0j 0 (A)Dr 0 (A)Cr (zIr 0 Ar )01 j =0

1 (Br + LCDr + LD) U (z) where

(18)

(x0 ) stands for functions depending on initial conditions.

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In order for the state error, x ~(k), to approach zero with a stability margin, the following two conditions must be satisfied: 1) the ~(k) must be -stable and 2) for steady-state tracking, dynamics of x the term that precedes U (z ) on the right-hand side of (18) must vanish as z ! 1: Now we examine each of these conditions. 1): To ensure -stability of the state error, the zeros of the following function are derived:

(z) = detf(A)[In + Cr (zIr 0 Ar )01 LC ]g = det[(A)] det[In + Cr (zIr 0 Ar )01 LC ]:

(19)

Using (6) and the determinant identity det(In + GH ) = det(Ir and H = (zIr 0 Ar )01 LC; (19) becomes

HG) with G = Cr

(z) = 8(z) det(zIr 0 3r ) det[Ir + (zIr 0 Ar )01 LCCr ]:

+

(20)

In view of definition (15) (i.e., Ar = 3r ) and using the determinant identity det(S ) det(T ) = det(ST ) for any two (r 2 r) matrices S and T; (20) reduces to

(z) = 8(z) det(zIr 0 Ar + LCCr ):

(21)

It is clear from (21) that the dynamics x ~(k) will be -stable if and only if the following condition is satisfied. Condition 1: The observer gain matrix L must be chosen so that matrix (Ar 0 LC C r ) is -stable. Remark 1: In order to be able to choose L to give a -stable (Ar 0 LC C r ); the pair (Ar ; C C r ) must be observable. This condition is always guaranteed under the assumption that (1) is observable 8z 2 : Remark 2: By Condition 1, it follows that the -stability margin of the reduced-order state observer (14) can be arbitrarily assigned. As

can be arbitrarily chosen (0 < < 1) and the observer eigenvalues are chosen within the -circle, the convergence of the observer can be arbitrarily made faster or slower depending on the value of : 2): Now, let us turn our attention to steady-state tracking. For this to happen, the term that precedes U (z ) on the right-hand side of (18) must vanish as z ! 1, i.e., s j =0

Remark 3: It is clear from (23) that steady-state tracking can always be achieved provided that (Ir 0 Ar ) (i.e., (Ir 0 3r )) is invertible. This is always possible provided that 3r does not have eigenvalues equal to one. Accordingly, matrix D3 is also invertible and can be easily shown by showing that det(D3 ) = det[R] det[In + Cr (Ir 0 Ar )01 LC ] 6= 0: In view of the development shown in (19)–(21), it is clear that det(D3 ) 6= 0: Remark 4: For the case where 3r contains eigenvalues equal to ~(k), converges exponentially, with one, then the error state vector, x a prescribed -stability margin, to a residue set in the steady state. Remark 5: The state observer (14a) and (14b) can be extended to provide tracking for other types of inputs such as ramps, parabolas, etc. (see [14] for further details). The above development is summarized in the following Theorem. Theorem: Let S 1 be a linear system described by (1) and S 2 be a reduced-dynamic system described by (14). Assume that S 1 and the pair (Ar ; C C r ) are fully observable. Let L be chosen to satisfy ~(k) = condition one. If S 2 is driven by S 1 ; then as k ! 1; x x(k) 0 x^(k) ! 0 with a -decay rate. Now we summarize the observer design method by the following design algorithm: Design Algorithm Step 1) Compute the eigenvalues of (1) by using (3c). Assign and hence obtain : Step 2) Derive 3r ; V; and W from (11)–(13), respectively, and hence Ar ; Br and Cr from (15). Step 3) Design L so that (Ar 0 LCCr ) is -stable. Step 4) Derive Dr from (23). The rth-order observer for (1) is obtained from (14). IV. NUMERICAL EXAMPLE In this section, a numerical example is given. Through the example, the advantages and potentials of the proposed state observer are illustrated. Let us now consider the following delay-difference system:

x(k + 1) =

where

R = In 0

 i=0

Ai :

(22b)

Dr = (D3 )01 B 3

(23a)

D3 = R[In + Cr (Ir 0 Ar )01 LC ]

(23b)

where

and s j =0

Bj 0 RCr (Ir 0 Ar )01 (Br + LD):

Ai x(k 0 i) + Bu(k)

where

Ai (i = 1; 2; 3); B; C; and D are as given below:

15 ; A0 = 00:3 00::45 A2 = 00:1 00::31 ;

05 00:2 A1 = 00::05 0:05 0 0 : 2 0 2 A3 = 00:1 0::22

B = 12 ; C = [1 0];

Rearranging terms in (22a) gives

B3 =

i=0

y(k) = Cx(k) + Du(k)

Bj 0 RDr 0 RCr (Ir 0 Ar )01 (Br + LCDr + LD) = 0 (22a)

3

(23c)

and

D = 1:

The above system consists of eight eigenvalues. According to Luenberger full-order and reduced-order state observers, an 8th- and a 7th-order dynamic state observers, respectively, are required to estimate the states of the system. In this paper, however, a reducedorder state observer of only first-order is proposed. It will be shown by computer simulations that despite the fact that the reduced-order state observer is only of first-order, it will always lead to asymptotic estimation of the entire states of the system regardless of the initial conditions of the system and of the observer. In the following, the step-by-step design algorithm given in Section III will be used to design a reduced-order state observer for the system.

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(a)

(b) Fig. 2. (a) Response of

x1 (k )

to a step input in

u(k ):

(b) Response of

x2 (k )

Step 1: The eigenvalues of the system, using (3c), are computed by using MATLAB and are found to be

(2) = (0:5161 6 j 0:4623; 00:5854; 00:3172

6 j 0:4693; 0:0056 6 j 0:536; 0:9263):

By choosing = 0:7; is obtained as = f0:9263g: Accordingly, for a prescribed 0.7-stability margin of the observer error dynamics, a first-order reduced observer can be used to construct the states of the system.

to a step input in

u(k ):

3r ; V; and W are derived as follows: 3r = 0:9263; V = 0:6497 1

Step 2: Using (11)–(13),

and

W

= [0:0307 0:5657]:

Accordingly, from (15), Ar ; Br ; and Cr are obtained as

Ar

= 3r = 0:9263;

Br

= 1:1621

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(c)

(d) Fig. 2. Continued. (c) Response of

x1 (k )

to a ramp input in

u(k ):

(d) Response of

x2 (k )

to a ramp input in

u(k ):

Step 4: Using (23a)–(23c), Dr is obtained as

and

Cr

=V =

0:6497 1

Dr

:

Step 3: Since the pair (Ar ; CCr ) is observable, L can be found so that (Ar 0 LCCr ) is -stable. Let us now find L so that the eigenvalue of matrix (Ar 0 LCCr ) is located, say, at 0.5. Accordingly, L is obtained as L = 0:656:

=

01:1433 01:0881

:

Simulation Results The reduced-order state observer was extensively simulated, and it was found that regardless of the initial conditions of the system and observer, asymptotic estimation of the entire states of the system was always achievable. The simulation results are shown in Fig. (2a)

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()

and (b) for the case where u k is a step input. The initial conditions ; x2 k for the system and observer are taken to be x1 k ; k 2 0 ; xr : It is clear from Fig. (2a) and (b) that the reduced-order state observer tracks the true states of the system regardless of its initial condition. As stated in Remark 2, the observer convergence time is dependent on the selection of : To further illustrate Remark 5, simulation results for a ramp input : k k ; ; ; 1 1 1 ; are also considered. In order case, u k to compensate for the steady-state error, an extra term, proportional to the rate of change of the input is added to the right-hand side of 2:5487 (14b). This term is D u k ; where D 3:3779 : The simulation results are shown in Fig. (2c) and (d). The initial conditions are the same as above. The figures show that steady-state tracking of a ramp input is achieved.

0

()=0 ()=

[ 3 0]; (0) = 10

( ) = 01 ( = 0 1 2 ) 1()

=

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On the Existence of Finite-State Supervisors Under Partial Observations Toshimitsu Ushio

Abstract— We consider discrete-event systems under partial observations. We show the equivalence of the (M; 6c ; L(G))-observability of a control specification and the (M; 6c ; 63 )-observability of its corresponding augmented language. We derive necessary and sufficient conditions for the existence of finite-state supervisors under partial observations. Index Terms—Automaton, discrete-event system, partial observations, supervisory control.

I. INTRODUCTION V. CONCLUSION In this paper, a reduced-order memoryless state observer for the estimation of the state vector of linear multivariable discrete-time systems with multiple delays in the state and control vectors has been introduced. The dynamics of the observer are derived from a set of unstable and/or poorly damped eigenvalues of the system. It is shown that arbitrarily chosen dynamics can be assigned to the observer. A step-by-step observer design algorithm has been presented. It is demonstrated that although the transient performance of the reducedorder state observer is dependent on the selection of ; the observer will always track the corresponding states of the driving system. REFERENCES [1] D. G. Luenberger, “An introduction to observers,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 596–602, June 1971. [2] S. D. Cumming, “Design of observers of reduced dynamics,” Electron. Lett., vol. 5, no. 10, pp. 213–214, 1969. [3] T. E. Fortman and D. Williamson, “Design of low-order observers for linear feedback control laws,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 301–308, May 1972. [4] P. Murdoch, “Observer design of a linear functional of the state vector,” IEEE Trans. Automat. Contr., vol. AC-18, pp. 308–310, 1973. [5] K. Ogata, Discrete-Time Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1987. [6] M. Aldeen and H. Trinh, “Observing a subset of the states of linear systems,” Proc. Inst. Elec. Eng.-Control Theory Appl., vol. 141, no. 3, pp. 137–144, 1994. [7] J. Leyva-Ramos and A. E. Pearson, “An asymptotic modal observer for linear autonomous time lag systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1291–1294, July 1995. [8] A. E. Pearson and Y. A. Fiagbedzi, “An observer for time lag systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 775–777, Aug. 1989. [9] K. Watanabe, “Finite spectrum assignment and observer for multivariable systems with commensurate delays,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 543–550, June 1986. [10] E. Gyurkovics, “Robust control design for discrete time linear uncertain systems with delayed control,” Proc. Inst. Elec. Eng.-Control Theory Appl., vol. 140, pp. 423–428, 1993. [11] A. B. Bishop, Introduction to Discrete Linear Controls, Theory and Application. New York: Academic, 1975. [12] C. H. Lee, T. H. S. Li, and F. C. Kung, “D-stability analysis for discrete systems with a time delay,” Syst. Contr. Lett., vol. 19, pp. 213–219, 1992. [13] H. Trinh and M. Aldeen, “D-stability analysis of discrete-delay perturbed systems,” Int. J. Contr., vol. 61, no. 2, pp. 493–505, 1995. , “Comments on ‘An asymptotic modal observer for linear au[14] tonomous time lag systems,’ ” IEEE Trans. Automat. Contr., vol. 42, pp. 742–745, May 1997.

Supervisory control is very useful for logical control problems in discrete-event systems (DES’s) [1]. Ramadge and Wonham [1] show a necessary and sufficient condition for the existence of supervisors under complete observations, and Lin and Wonham [2], [3] and Cieslak et al. [4] show that controllability and observability are the conditions under partial observations. Rudie and Wonham give a fixed-point characterization of observability and propose an algorithm for checking observability for regular languages [5]. A supervisor is implemented by a pair comprised of an automaton and state feedback. A supervisor is called a finite-state supervisor if its automaton is a finite automaton [6]. Obviously, a supervisor is implemented by a finite-state one for any control specification given by a regular controllable and observable language [2]. Finitestate supervisors may exist for nonregular languages. Ushio [7] and Sreenivas [8] show necessary and sufficient conditions for the existence of a finite-state supervisor under complete observations. An algorithm for computing a sublanguage of the control specification for which a finite-state supervisor exists is also proposed [9]. This paper deals with DES’s under partial observations and control specifications given by regular or nonregular languages. We investigate properties of augmented languages of the given languages. Necessary and sufficient conditions for the existence of a finite-state supervisor is shown using augmented languages. II. PRELIMINARIES Let G be a controlled DES modeled by an automaton

G = (Q; 6; ; q0 ; Qm )

(1)

where Q is the state set, 6 is the event set,  : 6 2 Q ! Q is the transition function, q0 2 Q is the initial state, and Qm  Q is the marked set. We write  (; q )! if  (; q ) is defined. Let 63 be the set of all sequences consisting of elements of 6 including the empty sequence : We assume that 6 is partitioned into two disjoint sets 6u and 6c of uncontrollable and controllable events, respectively. Let 0 be the set of control patterns, that is, 0 = f j6u   6g: Let L(G) be a language generated by G: Let M : 6 ! 3 [ fg be a mask, where 3 is a finite set of observed events. Let 6 := f 2 6jM () = g: Intuitively, 6 is interpreted as a set of completely unobserved events in 6: Let T and N be languages over 6: Denoted Manuscript received October 30, 1995; revised April 25, 1996 and September 12, 1996. The author is with the Department of Systems and Human Science, Osaka University, Machikaneyama, Toyonaka, Osaka, 560 Japan. Publisher Item Identifier S 0018-9286(97)07634-4.

0018–9286/97$10.00  1997 IEEE