Robust Kalman filtering for discrete state-delay systems

MSMahmoud, L.Xie and Y.C.Soh

Abstract: A robust estimator design methodology has been developed for a class of linear uncertain discrete-time systems. It extends the Kalman filter to the case in which the underlying system is subject to norm-bounded uncertainties and constant state delay. A linear state estimator is constructed via a systematic procedure such that the estimation error covariance is guaranteed to lie within a certain bound for all admissible uncertainties. The solution is given in terms of two Riccati equations involving scaling parameters. A numerical example is provided to illustrate the theory.

1 Introduction

The problem of optimal filtering has been well studied for more than three decades in various branches of science and engineering. The focus has been on dynamical systems subject to stationary gaussian input and measurement noise processes [ 11. The celebrated Kalman filtering provides a solution to this problem. When the available plant model contains uncertain parameters, the robust state estimation problem comes into the scene for which several techniques have been proposed [2-51. On another front of research, uncertain systems with state delay have received increasing interest in recent years [6-81. Most of the research effort has been concentrated on robust stability and stabilisation [9, 101. The problem of estimating the state of uncertain systems with state delay has been overlooked despite its importance for control and signal processing.

This paper considers the state estimation problem for linear discrete-time systems with norm-bounded parameter uncertainties and constant state delay. This delay factor arises naturally in different engineering fileds [lo]. It could result from constant processing delays as in digital systems, inherent gestation lags as in production systems or finite transit time as in industrial mills. Indeed, this delay is among the main sources of instability in control systems. A related problem is the design of deterministic observers with unknown inputs [ 111 using algebraic methods. Here, we address the state estimator design problem such that the estimation error covariance has a guaranteed bound for all admissible uncertainties and state delay.

The main tool for solving the foregoing problem is the Riccati equation approach. It is proven that the stabilising solution of robust Kalman filtering is given in terms of two algebraic Riccati equations. The existence of the solutions hinges on the quadratic stability of the uncertain system. In principle, all the developed results can be cast into the framework of linear matrix inequalities to yield solution (not necessarily stabilising) [ 101. Finally, a numerical example is provided to illustrate the theory.

Notations and facts. In the sequel we denote by W', W-' and L( W) the transpose, the inverse and the eigenvalues of any square matrix W We use W > O( W < 0) to denote a positive- (negative-) definite matrix W and I to denote the n x n identity matrix. Sometimes the arguments of a function will be omitted in the analysis when no confusion can arise.

Fact 1: For any real matrices C,, C, and C3 with appropriate dimensions and XiC3 5 I, it follows that

~ 1 ~ 3 ~ 2 + x ; x ~ c ~ , 5 c ' x ~ c ~ , + MC;C,, VM > 0

Fact 2: Let C , , C,, C, and 0 < R =R' be real constant matrices of compatible dimensions and H(t) be a real matrix function satisfying H'(t)H(t) 5 1. Then for any p > 0 satisfying pCiX, < R, the following matrix inequal- ity holds:

(Z3 + C,H(t)C,)R-'(C: + C:H'(t)C:)

- < p-'C,C{ + C,(R - pCiX,)-'C:

2 Class of uncertain time-delay systems

0 IEE, 2000 IEE Proceedings online no. 20000749 Dol: I O . 1049/ip-cta:20000749 Paper first received 12th October 1999 and in revised form 28th April 2000 M.S. Mahmoud was with the Department of Electrical and Computer Engineering, Kuwait University, Kuwait and is now with the Department of Engineering, MSA University, Amer St., Mesaha Square, Dokki, Egypt E-mail: magdim@yahoo.com L. Xie and Y.C. Soh are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

IEE P~oc.-Contvol Theory AppL, Vol. 147. No. 6, November 2000

Consider a class of uncertain time-delay systems repre- sented by

(2)

(3)

613

where x, E !)Ift is the state, yk E 91" is the measured output, z, E gip is a linear combination of the state variables to be estimated and w, E 91' and vk E 3'" are, respectively, the process and measurement noise sequences. The matrices A , E 91" ", Dk E 9?" " and Ck E W' " are real-valued matrices representing the nominal plant with A, being invertible for all k. Here, 5 is a known constant scalar representing the amount of delay in the state. The matrices AA, and AC, represent time-varying parametric uncertain- ties given by

(4)

where H I , k E 9i" ', H2, E 3"' matrices and A, E 3' B is an unknown matrix satisfying

and Ek E 9$' " are known

ALA, 5 I k = 0 , 1 , 2 .... (5) The initial condition is specified as (x,,$(s)), where q 5 ( . ) ~ l , [ - z , O ] . The vector x, is assumed to be a zero- mean gaussian random vector. The following standard assumptions on x, and the noise sequences {wx} and {v,}, are assumed:

(a) €[wk] = 0; €[w,w;] = W,6(k - j ) ; W, > 0; V k , j (6 )

(b) E [ v ~ ] = 0; &[V~V;] = Vk',6(k - j ) ; Vk > 0; V k , j (7)

(c) E[w,vJ] = 0; E[x,w;] = 0 , V k , j (8)

(4 &l.,x:l = R, (9)

where E [ . ] stands for the mathematical expectation and 6(.) is the Dirac function.

3 Robust Kalman filter design

Our objective is to design a stable state-estimator of the form

(10) .E,+, = Go,,?, + K,,ky, .E, = 0

where Go,k E lli" " and KO., E %" "' are real matrices to be determined such that there exists a matrix Y 2 0 satisfying

E[{xk - i k ] { x k - .?k} t ] = E[e,e:] 5 y

V A : ALA, 5 I (1 1)

Note that exprs. 11 implies

E[{xk - &]'{xk -&I] = E[e;ek]

- < t r ( Y ) V A : AiA, 5 I (12)

In this case the estimator (eqn. 10) is said to provide a guaranteed cost (GC) matrix Y . The proposed estimator is now analysed by defining

Go,k = (Ak + 6 A k ) - &,kCk (13)

where 6A, and KO,, are unknown matrices to be determined later on. Using eqns. 1, 2 and 13 to express the dynamics of the state estimator in the form

Introduce the augmented state vector

It follows from eqns. 1 and 14 that - - U

t k + l + H k A k E l - l t k + D k S k - 7 + ;k l?k - - - = A k , A t ! i + Dkth--7 + Bk Y k (16)

where q, is a stationary zero-mean noise signal with identity covariance matrix and

(17)

Dejinition 2: Estimator of eqn. 10 is said to be a quadratic estimator (QE) associated with a sequence of matrices {R,} > 0 for the system of eqns 1 and 2 if there exist a sequence of scalars {i,} 0 and a sequence of matrices {Q,} such that

satisfying the algebraic matrix inequality

( I + Ak)Ak,AQkAk,A - ! 4 + l

- -t

-I ,-.I

+(1 + A;i)&fi,-7D, + B,B, 5 0, k 2 0 (21)

for all admissible uncertainties satisfying eqns. 4 and 5. The following result shows that if eqn. 10 is a QE for the

system of eqns. 1-3 with cost matrix Q k , then 0, defines an upper bound for the filtering error covariance, that is 1;[e,e;] 5 Q2,, ,Vk> 0. Theorem I : Consider the time delay system ofeqns. 1-3 satisfying exprs. 4 and 5 and with known initial state. Suppose there exists a solution Rk = Ri 2 0 to expr. 2 1 for some A, > 0 and for all admissible uncertainties. Then the estimator (eqn. 10) provides an upper bound for the filtering error covariance, that is

& [ e k e i ] 5 [0 I ] Q ~ [ O I ] ' v ~ > o (22)

Proofi Suppose that the estimator (eqn. 10) is a QE with cost matrix Q k . By evaluating the one-step ahead covar- iance matrix = ( ; + , I , we get

1 - C<,k+l = E[A,,AI", + w-, + Bkvlkl - - -

[ A k , A t k f D k c k - 7 + B k q k l t 1 A t -I

= E I A k , A t k t i A k , A l + EIAk.A(ktL-rDkl ,_I

+ E [ & l k - 7 t i 1 4 k , ~ l - ,t ,I

+ w k ' ~ k - r S L - ? D k l + E[&7,y;B,I (23)

Note that

614 IEE Proc.-Control Theory Appl., Vol. 147, No. 6, November 2000

Using exprs. 24 in eqn. 23 and arranging terms, - ,I

C ( , k + l 5 ( l + ' L k ) A k , A C { , k A k , A A t -1 + (1 + &')6Jt,,+TDk + BkB, (25)

Letting E, = E(., - R, with e, =xk - .fk and considering exprs. 21 and 25, gives

A A t , ,t

5 (1 + Ak)Ak,AEkAk,A + (1 + Ail)DkEk-TDk (26)

By considering that the state is known over the period [ -7, 01, it justifies letting E(,, = OVk E [ - z, 01. Then it follows from exprs. 26 that Z, 5 0 for k > 0; that is Ct,k 5 Rk for k > 0. Hence &[e&] 5 [0 I]R,[O T]'Vk > 0.

4 Riccati equation approach

Motivated by the recent results of robust filtering theory [3-51, we employ hereafter a Riccati equation approach to solve the robust Kalman filtering for time-delay systems. To this end define matrices 0 < P, = PL E %" "; 0 < S, =Si E 91'' '* as the solutions of the Riccati differ- ence equations (RDEs)

Pk+l = (l + A!,){Ak(l + p!iPkYk)PkA:}

+ (1 + lLi l )DkPk-~Di + wkk;

Pkpr = 0 V k E [o, z ] (27)

sk+l = (1 f A k ) i k ( l f pkskyk)skiL + ( 1 + l.k)6Ak(I 4- ,UkPkYk)PkdA6

+ (1 + Ak)p&P, Y,S&

+ (1 + AiI)D,Pk-rD; + w, + (1 + Ak)ik/l$kYkPk6Ai - h L ( ? k + F,)-'hk

S,-r = 0 Vk E [ 0 , z ] (28)

where pk > O,A, > 0 are scaling parameters ,such th%t P k ' - p(,'EL.Ek > 0 and the matrices A , , 6A , , vk, W,, rk and M, are given by

Note that the assumption that A , being invertible for all k is needed for the existence of l k and 6 A k . Let the (A#)- parametrised estimator be expressed as

where the Kalman gain matrix E 92" x m is to be determined. The following theorem summarises the main result: Theorem 2: Consider the system of eqns. 1 and 2 satisfying the uncertainty structure of exprs. 4 and 5 with zero initial condition and A , being invertible Vk. Suppose the process and measurement noises satisfy eqns. 6 and 8. For some p k > o , & > O , let o < P k = P k and O<S,=Si be the solutions of RDEs 27 and 28, respectively. Then the (L, ,u)-parametrised estimator (eqn. 37) is a QE estimator with GC

Moreover, the gain matrixK is given by

Proo) Let

where Pk and sk are the positive-definite solutions to eqns. 27 and 28, respectively. By using facts 1 and 2 and combining eqns. 27-36, it is a simple task to show that

wJhere-rI, E D I ~ ~ ' ~ , r 1 2 ~ X n x f 1 , r I 3 ~ 9 i n x " and T,, gk, H,, D , are given by eqns. 17-19. One way to verify this is to expand eqn. 41 use eqns. 17-19 and 40 to yield

n, = (1 + ~L,){A& + iUkPkYk)PkAbl

r I 2 = (1 + /&(I + ,u,S,Y,)S&

+ (1 + A~~)D,P~-~D: + wk - P , + ~ (42)

+ (1 + R,)6Ak(I + @kYk)Pk6AL

+ (1 + ~ . , J ~ , ~ , P , E ; ( I - ~,E,P,E~)-~E,S,~;

+ ( l + A k ) A k ~ k s k y k p k 6 A i

- hi(?, + F , ) - l h k - & + I

+ (1 + L~~)D,P~-~D; + w, (43)

n3 = -(1 + A,)A,P,GAi

+ (1 + Ak)A$k [ Ai + 6Ai - CiK;,, ] - pk(1 + )Lk)A$kEi(z - pkE&Ei)-'E,P&A;

+ (1 + A;l)D,Pk-rD; - s,+, + ( l + A k ) & ' H l , k [ H ; . k - " 2 , k K ; , k ] (44)

By setting nl - 0 in eqn. 42 and using eqns. 29-36 we immediately obtain eqn. 27. Next, we enforce 112 0 in eqn. 43. By using eqns. 3 1-36 with some standard matrix manipulations define KO,, as in eqn. 39 to yield (28). Finally, by using eqns. 28-34 in eqn. 44 and setting 6A, as in eqn. 35, we find that r13 E O .

IEE Proc.-Control Theory Appl., Vol. 147, No. 6, November 2000 615

Now using lemma A of [ 1 2 ] , it is easy to see that using fact 2 with some algebraic manipulation eqn. 41 implies

V A : AkA, 5 I Vk

It follows from theorem 1 that eqn. 37 is a quadratic estimator and

&[eke ; ] = E [ O Z]X,[O 11'5 S,

which implies that &[e; ek] 5 tr(Sk). 0

Remark I : From the foregoing analysis it is seen that our results are independent of the size of the delay. This can be considered to yield a conservative design method [ 1 2 ] . As shown in the simulation example, our method works well for a wide range of the delay factor z. Remark 2: In the case of systems without uncertainties and delay factors, that is H I , , = 0, H2 , = 0, E, = 0, D, = 0, it can be easily shown that

Yk = 0; X , = 0; w k = w k ; 7, = (1 + A!,)(P, -&)A:

2, = ( 1 + /2,)2A,skcL(( 1 + A,)cks,cL + v,)-' C,S,Ai

Now in terms of L, = P, - S, and

yk = S,cL((l + %,)c,S,cL + v,)-'Cks,; @k = A,yklf i

R, = Akf(Pk - S,)-I ; A k̂ = (1 + A,)R,@,

we manipulate eqns. 27 and 28 to arrive at

Lk+l = (1 + A,)(A,L,AI + A,); L,-, = 0 v k E [O, Z] A, = @,

- (1 + Rk){Ak(Pk - L,)@:R: + R,@,(P, - L,)A: + ( 1 f lLk)Rk@)k(2Pk - Lk)@iR;] (46)

By iterating eqns. 46 and 27 it follows that L, = Pk - S, > 0Vk > 0. It can be shown in the general case that manipulation of eqns. 27-38 yields

&+I = ( 1 + A,)[A,(I + P&Yk)LkAi + n,]; LkP7 = 0 Vk E [O, Z]

In this case n, depends on A,, HI, , , H2,,, D,, C,, P,. The derivation of n, requires tedious mathematical manipula- tion and it is therefore omitted.

Remark 3: Note that P, does not depend on the filter matrices and the structure of X, is identical to that of the joint covariance matrix of the state of a certain system and its standard H,-optimal estimator. By similarity to the standard H2-optimal filter, an estimate of z , in eqn. 3 will be given by 2, = Cl,,&.

Remark 4: In the delay-free case (D, = 0) we supress the parameter A, and observe that eqn. 37 reduces to the recursive Kalman filter for the system

where Gk and C, are zero-meaq white noise sequences with covariance matrices W, and rk, respectively, and having cross-covariance matrix M, . Hence, our approach to robust filtering in theorem 2 generalises [4 , 51 and corresponds to designing a standard Kalman filter for a related discrete- time system which captures all admissible uncertainties and time delay, but does not involve parameter uncertain- ties. In this regard the matrix 6A, reflects the effect of uncertainties (AA,, AC,) and time delay factor Dk on the structure of the filter.

5 Steady-State robust Kalman filter

This section investigates the asymptotic properties of the recursive Kalman filter of Section 4. Consider the uncer- tain time delay system

Xk+l = [A + HlA&]Xk + D Xk-, + W k

= AAXk + D X k - , + W k (48)

= CAX, + vk (49 ) Yk = [c + H2 A@] Xk + vk

where A, satisfies eqn. 5. In the sequel, we assume that A is a Schur matrix; that is 12(A)I < 1 . The matrices A E 91" n, C E Sm are constant matrices representing the nominal plant. The uncertain parameter matrix A, is, however, time-varying. In this regard the objective is to design a shift-invariant a priori estimator of the form

= 22, + Ko[y, - Ci,] (50)

that achieves the following asymptotic performance bound

lim E{(;, - xk)(2, - xk)'] 5 s (51) k+ cc

Theorem 3: Consider the uncertain time delay system eqns. 48 and 49 with A being invertible. If for some scalars p > 0, A > 0, there exist stabilising solutions P > 0, S 2 0 for the ARES

P = (1 + A){A(I + p ~ ~ ) ~ ~ ' ~ + (1 + I,-')DPD' + I+; (52)

S = (1 + A) i ( I + pSY)Si ' + (1 + A)hA(Z + ptpY)PSA'

+ ( 1 + A)/LAIPYSAI' + (1 + A)AIp,SYPGA'

- 2(i. + P ) - ' h (53 )

Y = E'(1 - pEPE')-'E; W = W + (1 + A)pu'Nl H,' (54)

= V + ( 1 + 2)pL1H2H:; = (1 + 2)CSC' (55)

(56) 16 = (1 + IL)[CSA' + pSYPSA' + pH2Hi]

Then the estimator (eqn. 50) is a stable quadratic (SQ) estimator and achieves expr. 51 with

2 = A + 6A; 6A = T ' ( X + Z )

2 = ( 1 + n>fiyi. + ?)-I

(57)

KO = ${? + p ] - ' ; 7 = (1 + A)(P - S)(I + pYP)A' (58 )

(CS(I+ pYP)At + / L - ' H ~ H ~ ) (59)

X = (1 + A)pASYPA'

+ (1 + i )p - lH ,Ht + ( I + 3,-')DSD' (60)

616 IEE Proc.-Contvol Theoy Appl., Vol. 147. No. 6. November 2000

Proofi To examine the stability of the closed-loop system augment eqns. 48 and 49 with (wk = 0, vk = 0) to obtain

the ?&,-nom of C is less than (p+ ) - ' / * . It then follows, given a 1, that the system of eqns. 48 and 49 is quad- ratically stable for some ,U 5 ,U+.

Introduce a discrete Lyapunov-Krasovskii functional [ 131

for some A>0. By evaluating the first-order difference AVk = vk+l - Vk along the trajectories of eqn. 61 and arranging terms, we get

with P and S being the stabilising solutions of eqns. 52 and 53, respectively. Following a similar procedure as in the proof of theorem 2 and in view of definition 2, it follows in the steady-state as k + 00 that the augmented system eqn. 6 1 is asymptotically stable. The guaranteed perfor- mance &[eke;] SS follows from similar lines of argument as in the proof of theorem 2. U

Remark 5: Note that the invertibility of A is needed for the existence of RT and 6A. In the delglesscase ( D f O , it follows from eqns. 49 and 51 with W= BB' that

P = (1 + 2){APAf +AP[(p- ' I + EPEf)-lPAf) + 6' (66)

which is a bounded real lemma equation for the system C = (AJ(1 + A), B, E, 0). Suppose that for p = p+, the ARE (eqn. 66) admits a solution P = P+. This implies that

6 Example

Consider the following discrete-time delay system:

0.2 -0.1

xh+l = ([ 0 . 7 0:; ~ I

which is of the type of eqns. 1 and 2 with three-units of delay. Further assume that W=I, V=0.21. To determine the Kalman gains, solve eqns. 52 and 53 with the aid of eqns. 54 and 60 for selected values of A, p . The numerical

Table 1: Guaranteed cost (GCI against scaling para- meters (F, A)

2 = 0.2

P 0.2 0.45 0.7 1.1 1.5 2.3 GCx 219.241 141.706 37.029 27.583 65.742 120.333 i. = 0.4 P 0.2 0.45 0.7 1.1 1.5 2.3 GCx 173.601 121.016 31.125 25.113 60.142 101.471 i, = 0.8 Ll 0.2 0.45 0.7 1.1 1.5 2.3 GCx IOF4 96.711 101.315 29.451 24.881 51.371 89.116 1= 1.4 / L 0.2 0.45 0.7 1.1 1.5 2.3 GCx 109.345 117.996 38.222 24.003 61.332 110.541 i. = 2.7 P 0.2 0.45 0.7 1.1 1.5 2.3 GCx 165.124 122.236 46.113 35.723 72.119 121.171 i. = 3.4 P 0.2 0.45 0.7 1.1 1.5 2.3 GCx 259.943 165.176 51.152 41.907 77.139 151.454

Table 2: Comparison with nominal Kalman filter

Actual cost Filter A k = -0.8 A k = O A, = 0.8 Nominal Kalman filter 72.034 x IO-' 45.147 x 235.654 x IO-' Robust Kalman filter 66.802 x 61.147 x IOp4 67.113 x

IEE Proc-Control Theory Appl., 1/01, 147, No. 6, November 2000 617

6

computation is basically of the form of iterative schemes and the results for a typical case of p = 0.7, 1. = 0.3 are given by

0.141 0.005 0.003

0.003 0.175 0.501 0.284 -1.17 -2.966

-2.966 12.208 30.962 -0.841 -1.309

KO = IOp6 [ 3.463 5.388 1, 8.782 13.665

0.331 -0.019 -0.034 -0.277 0.254 0.175 -1.641 -0.897 0.961

The developed estimator is indeed asymptotically stable since

A(A1) = (0.302,0.48,0.765) E (0, 1)

The guaranteed cost is 36.059 x Several simulation studies have been carried out to examine the performance of the steady-state Kalman filter. In Table 1, the guaranteed cost is presented for selected values of the scaling para- meters (p, A).

It is readily evident that the scaling parameters (,U, A) have a crucial impact on the optimality of the guaranteed cost. This is eequally true for specified ,U while changing A or given A and varying p.

For comparison a standard Kalman filter was designed for the nominal delayless system of eqns. 67 and 68 by setting Ak = 0, x k - 3 = 0. Then we applied the developed robust Kalman filter and the standard Kalman filter with Ak = 0, A k = 0.2, Ak = - 0.2 and retained the delayed state. The resulting filtering costs for both filtering schemes are provided in Table 2. Again, it is clearly shown that the developed robust Kalman filter outperforms the standard nominal Kalman filter in the presence of uncertainty and delay factor.

Together, Tables 1 and 2 demonstrate the superior performance of the developed robust Kalman filter.

7 Conclusions

A robust Kalman filter has been developed for a class of uncertain systems with constant state delay. Both time- varying and steady-state Kalman filtering algorithms have been treated. It has been proven in both cases that the filter design amounts to solving two Riccati equations which involve scaling parameters. Important properties of the proposed filter have been disclosed. A numerical example has been provided to illustrate the theory.

8 References

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5

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I 1

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