Robust N, filtering for polytopic uncertain systems via convex optimisation

S.H.Jin and J.B. Park

Abstract: A robust ‘l-t, filtering technique is proposed for convex polytopic uncertain systems. This class of uncertainty can describe the parametric uncertainty more precisely, without conservatism, than the norm-bounded uncertainty. By applying a bounded real lemma to the error dynamics and .using the Schur complement with the appropriate change of variables, a nonlinear matrix inequality is obtained. It is then shown that the congruence transformation, with some newly defined variables, converts this nonlinear matrix inequality into the convex optimisation problem for the design of robust X, filters, which is expressed by linear matrix inequality and can be solved very efficiently by so called interior point algorithms. The optimal tolerance level can be directly computed without the aid of the conventional bisection method, and the proposed algorithm does not require the additional search procedures needed for dealing with the norm-bounded uncertainty. Numerical examples are given to show that the proposed filter is more robust than the robust X2 filter against the parameter variation, as well as the noise in the worst-case frequency range and to illustrate the advantage of describing the uncertainty as polytopic rather than norm bounded.

1 Introduction

Over the past four decades, state estimation has been widely studied and has found many practical applications. The most famous filtering scheme is the celebrated Kalman filter [ 11, which requires two fundamental assumptions: the availability of an exact internal model of the system and the a priori information on the external noises. However, it is very well known that these assumptions are not usually satisfied in real situations [2, 31, hence considerable inter- est has been devoted to the X w filtering technique, in which external noise signals are assumed to be unknown but energy bounded rather than Gaussian. An Xm filter is designed such that the X, norm of the system, which reflects the worst-case energy gain of the system, is minimised. As pointed out in [3] and [4], the advantage of an X, filter in comparison with an N 2 (Kalman) filter is twofold. First, no statistical assumptions on the noises are needed. Secondly, the filter tends to be more robust when there exists additional parameter uncertainty in the system. These features make the X, filtering technique useful in certain applications. There have been several approaches to X w filter design: the algebraic Riccati equation (ARE) [2,4-61, interpolation [7], polynomial equations [8], game- theoretic [9] and, more recently, linear matrix inequality (LMI) approaches [3, 101. However, most of the works mentioned above require that the system model is precisely known, apart from the exogenous noises. Therefore the robustness of X, filters against the parameter uncertainty

0 IEE, 2001 IEE Proceedings online no. 20010237 DOI: 10.1049/ip-cta: 200 10237 Paper first received 3rd May and in revised form 9th October 2000 The authors are with the Department of Electrical and Computer Engineering, Yonsei University, Seoul 120-749, Korea E-mail: shjin@control.yonsei.ac.!a E-mail: jbpark@control.yonsei.ac.!a

IEE Proc.-Control Theory Appl., Vol. 148, No. I , January 2001

has to be taken into account. Several results have been obtained on robust X, filtering [2, 3, 51. Some of these results deal with the so called norm-bounded uncertainty acting in the state and/or in the output matrices and are obtained using ARE. The ARE approach for both X, [2, 51 and X2 [l 13 filters involves the conversion of a robust filtering problem into a scaled filtering problem, which seems not to include uncertainty but introduces another scaling parameter. However, the introduction of the scaling parameter requires an additional unidimensional search procedure with respect to this parameter. Further, the norm-bounded uncertainty assumption is somewhat conservative in many application [3, 121. Recently, the robust X w filtering technique for a wider class of para- meter uncertainty described by the integral quadratic constraints (IQCs) has been addressed [3], which uses LMI and an S-procedure, but unfortunately yields only sufficient conditions for the existence of a solution. In this paper, a design methodology of robust X, filters for polytopic uncertain systems [ 12-14] is proposed. All matrices contained in the system are assumed to be dependent on uncertain parameters and, in addition, struc- tured uncertainties are also considered, By applying a bounded real lemma [ 151 to the error dynamics, a quadratic matrix inequality (QMI) is obtained and, using a simple change of variables and the Schur complement formula [ 131, we have a nonlinear block matrix inequality. Defining an appropriate nonsingular matrix and then using the congruence transformation, together with the introduction of new matrix variables, LMI constraints constituting the convex optimisation problem can be obtained, which immediately solves the robust X, filtering problem. The proposed method is significant in that an extremely effi- cient and powerful numerical algorithm can be used and moreover, a larger class of parameter uncertainty can be handled. Although the robust 7t2 filter has been developed for the same class of uncertainty [12], it is shown that the proposed robust ‘Hm filter enjoys the benefit of fi-equency- domain properties and robustness against parameter uncer-

55

tainty in comparison with that filter, which is a similar argument as for the nominal case [4].

2 Problem statement

In many practical situations, physical models of a system often lead to a state-space description of its dynamical behaviour. The resulting state-space equations typically involve physical parameters whose values are not exactly known, as well as the approximation of complex and possibly nonlinear phenomena and hence, result in the following uncertain state-space model called a polytopic system [13, 141

(1) : i = A(A)x + B(A)w

y = C(A)X + D(A)w (2)

z = L x (3) where A denotes an independent uncertain parameter. It is assumed that x E R" is the state, w E R4 is an exogenous noise input in C2[0, m), y E R" is the measured output and Z E Rp is the signal to be estimated. The state-space matrices A , B, C, D and L depend on uncertain and/or time-varying parameters or vary in some bounded sets of the space of matrices. To make this point clear, let us define a matrix:

The matrix S(1) varies within a fixed polytope of matrices, i.e.

k

i= 1 i= 1

where SI , . . . , s k are given vertex systems:

In other words, S(A) is a convex combination of the vertex matrices S1,. . , , s k . The non-negative numbers a , , . , . , a k are called the poiytopic coordinates of S [ 141. The matrix L and the matrix

which defines the nominal system, are also assumed to be known. Consider the filter with minimum state-space realisation of the form

Zf (t> = Cf X f (t) (8)

where the matrices As E R"f nf, Bf E R"f ", Cf E Rp "f

and the dimension nf > 0 of the filter are constant matrices and a scalar to be chosen, respectively.

Now, the problem to be considered can be stated as follows: Check whether or not, for any S E V ~ , it is possible to find a filter of the form given by eqns. 7-8 that guarantees the following two conditions: (i) Transfer function 7 ( s ) is BIB0 stable, (ii) Given y > 0, 11 7 ( s ) 1 1 oo < y, where the matrix S is arbitrary but has constant values for its elements and T(s ) denotes the transfer function from an

56

exogenous noise w to the estimation error e(t):= z (0 - Zf (0.

3 Robust 7-& filter design

In this section, the stationary case is only addressed where all matrices defining the uncertain system, as well as the filter, are time invariant. It is assumed that matrix S E Vp is arbitrary but has constant values, i.e. S can be represented by

(9)

Connecting the filter given by eqns. 7-8 to the system given by eqns. 1-3 corresponding to matrix S yields

: k = A ( + B w (10)

e = Cg (1 1) where ( = [xTxTIT and the coefficient matrices are

C = [ L -cf] (13) By applying the bounded real lemma [15], the condition 11 7 1 1 o3 < y in the previous section, just as in the nominal case, can be shown to be equivalent to the following QMI

3 P > O s . t A T P + P A + y - 2 P B B T P + C T C < 0 (14)

Defining P := yp-' and multiplying it by both sides of eqn. 14 yields

AP + PAIT + y-'PCTCP + y-'BBT < 0 (15) and, by using the Schur complement formula [13], the following block matrix inequality can be obtained

AP+PAT PCT B CP -yl 0 ] < 0 (16) [ BT 0 - y l

Now, partition P and its inverse P-' as

where P l l , PI, E R" ", and PZ2, P22 E RS 'f are all symmetric and positive definite matrices. By substituting the partitioned matrix P, together with eqns. 12-13, for eqn. 16, the nonlinear matrix inequality can be described as follows:

APl l + P l l A T

BfCP, 1 + AfPT2 + &AT

BT

LPIl - qpT2

-yI * BfcP12 + AfP22 + PT2CTBfT + P22AfT * *

0 * - y l * I *

LP12 - CfP22

D'B;

I < 0 (18)

IEE Proc.-Control Theory Appl., Vol. 148, No. I, January 2001

By permuting the third and fourth columns and rows, an equivalent inequality can be obtained as

AP, , + P1,AT

BfCPl + AfPT2 + PT2AT

BT I L LPll - CfPT2

* * * BfCPI2 + AfP2, + PT2CTBfT + P,,AfT * *

D'B; -yI * LP,, - CfP22 0 -YI1

=: A(P11,P12,P22,AfrBf,CfrY) < O (19)

where * denotes the blocks obtained easily by symmetry. However, the above inequality is not jointly linear in each matrix variable. The main object in this paper is to convert the nonlinear matrix inequality, eqn. 19 into a LMI. If this is achieved, the filter design turns out to be a convex optimisation problem that can be solved very efficiently by the so called interior point algorithm [14]. Let us define a non-singular matrix

rpll' 0 0 0 1 c := i"' PI, I 0 0 o ]

0 o r By applying the congruence transformation [ 161, with the matrix C above, to the block matrix A(.) in eqn. 19, The following transformed block matrix can be obtained

X A + ATX X A +A?,, + c T y T + K~

A ~ x + P , , A + Y c + K P , , A + A ~ P , , + y c + c T y T

BTX BTP, + DT YT

L - Z L

X B L T - Z T ]

P , , B + Y D LT < 0 (21)

-YI 0

0 -YI _I where X ,

X := P;;, Y := P12Bf , Z := CfPT2X, K := P,,AfPT,X

Z and K are new matrix variables defined as

(22)

The above matrix inequality is now expcessed as an LMI, not only over the matrix variables X , P , , , E: K and Z but also over the scalar y. This implies that the scalar y > 0 can be included as one of the optimisation variables for the LMI eqn. 21, which readily yields the following conclusion: by using linear programming in conyex opti- misation, one can easily find feasible matrices X , P , ,, E: K and 2 that yield the following relation:

Yinf 5 Y 5 Yinf + E (23)

where E is an extremely small positive number that depends on the accuracy of the linear programming algo- rithms and the precision of a computer. Furthermore, from

IEE Proc.-Control Theory AppL. Vol. 148, No. I , January 2001

the definition given by eqns. 22, the filter matrices can be calculated as A f -P-' - 12 K(PT,X)-', Bf = hhl Y , Cf = Z(Pr2X)-'

(24) However, there seem to be no systematic ways to determine the matrices P12 and P , , needed for the filter matrices. To deal with such a problem, first of all, let us denote the filter transfer function by

If = Z P , ~ ( S P ~ , P ~ ~ - KP,,)- 'Y (25)

which depends not on each of the matrices P12 and PT,, but only on the prodyt of those matrices. This means that one of the two matrices P , , a;d P I , can be arbitrarily assigned by the designer such that PI2PT2 = I - P I , P I , holds. Therefore one can now solve the '&-optimal' Em filtering problem [lo], where the 'e-optimality' implies that the am norm of the resulting error dynamics is greater than or equal to the infimum, say y in of the X, norms of all possible error dynamics, but is r',ss than or equal to yinf + E for an extremely small positive number E , which depends on the accuracy of the algorithms and the precision of a computer. The follow- ing theorem summarises the above discussion and will play the role of a basis for handling the robust filtering problem.

Theorem 1 (LMI formulation of &-optimal Xm filtering): Assume that S E Dp is arbitrary but has a constant value and that n = nf . Then the &-optimal Xm filter can be found by solving the following convex optimisation problem:

Minimise y > O over X = X T , fill =@T,, I: K , Z and y subject to

X A + ATX A ~ x + @ , , A + Y C + K

BTX L - z

< 0 (27) -yI *

0 * -yI :I * P , , A + A ~ P , , + YC + cTyT *

BTPl l + DTYT L

The optimal solution to the above problem can be used to construct the filter of the form giyen by eqns. 7-8 with the matrices in eqns. 24. Matrices P, , and P, , can be arbi- trarily assignet to be nonsingular such that P12PT2 = I - PI ,PI1 holds.

One can easily extend theorem 1 for the rpbust Xm filtering problem, thanks to the inherent properties of polytopic systems [12, 141, to yield the main results of this paper as follows:

Theorem 2 (LMI formulation of robust &-optimal Flm filtering): Assume that S E Dp represents a time-varying uncertain system and that n = n f . Then the robust E-

optimal 'Hm filter can be found by solving the following convex optimisation problem:

Minimise y > O over X = X T , 8,, =@TI, I: K , Z and y subject to

5 1

XA, + A,TX

ATX + kl lA , + YC, + K

BTX

L - z *

i l l A i +Arkl1 + YC, + CfYT * * BTill + DTYT -?I * *

L 0 - y z

V i = l , . . . , k (29)

where A , , B , , C, and D, are the sub-matrices of the vertex matrices S,, i = 1, . . . , k. Then the desired filters are given by eqns. 7-8 along with the matrices in eqns.&24. Any nonsingular matrices such that = Z - P I ,PI holds can be freely assigned to matrices P I , and Plz.

4 Illustrative examples

In this section, numerical examples are given to demon- strate the properties of the proposed filter. Consider the following system:

y( t ) = [ 0 - I .2 h(t) + [ 0 1 ]w(t) (31)

z(t) = [O lIx(t) (32)

where w(t) is an exogenous noise in L,[O,oo) and ((t) is an uncertain parameter satisfying I ((t) I 5 0.4. ( ( t ) = 0 corre- sponds to the nominal system and two vertex systems SI and S, , which define a fixed polytopic uncertain domain 'Dp, are easily determined such that Vp can equally be considered as a norm-bounded uncertain domain. Theorem 2 provides the optimal l-t, performance of yopt = 0.7624, which is a tolerance level and can be regarded as an indication of the quality of the filter.

Fig. I shows the singular value comparison between the robust XFt, filter and the robust l-t2 filter [12] for different values of t;(t). 30 points between the vertex systems SI and

10-1 100 IO' 102 1 03 frequency, rad/s

Fig. 1 - robust H , filter

_ _ _ robust X 2 filter

58

Maximum singular value comparison

S, are used, and the effect of ((t) is observed. As expected, in the frequency range from approximately 3 rad/s to 5 rad/s, which exhibits the worst case, 1 1 CT 11 , for the robust l-t, filter is lower than that for the robust l-t, filter. Also, it is clear that the robust l-tFt, filter is less sensitive to the variations in ((t), but the trade-off in the other frequency ranges is apparent. Note that the peak of the maximum singular value plot for the robust X, filter changes its position from left to right according to the values of ((t) .

Next, consider the uncertain system of the form

-4 -0.6 X(t) =

y( t ) = [ 0 -1.2 + 0.3y(t) ]x(t) + [ 0 1 ]w(t) (34)

z(t) = [ O 1 lx(t> (35) where ((t) and y(t) are uncertain parameters such that (,(t) + q2(t) p 1 and ( ( t ) = y( t ) = 0 defines the nominal system. In this case, the norm-bounded uncertain domain cannot be exactly represented by Dp with a finite number of vertices. Fortunately, however, as the number of vertices k increases, it is approximated by Dp with vertices given by

2ni [-0.6 4+cos-

k si= I -4 -0.6 1.5 0 I , i = l , . . . , k

(36) Table 1 shows the achievable optimal l-tm performance yopt provided by theorem 2 for each value of k. The case of k = 2 yields the same uncertain domain as the first example with the bound of I ((t) I replaced by 1. As k increases, the value of yppt also increases and converges to the constant value, which can be regarded as being obtained from the conventional ARE-based approach [2] for a class of norm- bounded uncertainty. From this second example, it is clearly shown that the polytopic uncertainty description can represent an uncertain domain in more detail than the norm-bounded uncertainty one and, hence, does not intro- duce the conservatism.

Finally, consider the uncertain system

-4 -0.6 X(t ) =

y(t) = [ 0.3y(t) -1.2]x(t) + [ 0 1 ]w(t) (38)

z(t> = [ O 1 ]x(t) (39) The aim of this last example is to show the prefered ability of the proposed algorithm to incorporate a particular structure of the uncertainty and additional a priori infor- mation into the design procedure. To this end, we consider two cases of ( ( t ) # y( t ) and ( ( t ) = y(t), respectively. Assume, first, that ( ( t ) and y( t ) are uncertain parameters such that I ((t) I p 1 and I y(t) I 5 1 and (((t), y(t)) = (0,O) also defines the nominal system. Recalling that the norm-

Table 1: Optimal 8, performance

k 2 3 4 6 8

yopt 0.8283 0.9779 1.0078 1.0081 1.0081

IEE Proc-Control Theory Appl., bl. 148, No. I , January 2001

bounded uncertainties are represented as the following form [2]:

the appropriately assignable matrices H , , H2 and E are given by

H,=[O 0.31, E = [ ’ ‘1 1 0 ’

Even although the uncertain matrix F(t) is known to have the diagonal structure, the norm-bounded uncertainty description cannot take this particular structure into account because design equations do not contain the uncertain matrix term at all but only the matrices H I , H2 and E. In other words, the ARE approach for a class of norm-bounded uncertainty may allow the non-diagonal structures of F(t) and, as a result, introduce a degree of conservatism. On the other hand, for the uncertain system given by eqns. 37-39 the polytopic uncertainty description can exactly define the uncertain domain by four vertices. Using theorem 2, the optimal XFt, performance is obtained by

yopt = 2.0518 (42)

Assume now that the uncertain parameters ((t) and y ( t ) in the previous example are the same. The norm-bounded uncertainty description, eqns. 4 1, cannot make use of this a priori information either and, hence, will not result in a better performance than the case of [ ( t ) # y( t ) for the same reason mentioned above. However, the proposed approach can incorporate this a priori information into the design procedure and reduce the number of vertices for the uncertain domain. Specifically, only two vertices are needed to define the uncertain domain and, thanks to the reduction of the uncertain domain, the optimal 7-1, perfor- mance can be obtained such that

yap+ = 1.2505 (43)

which turns out to be more improved than in eqn. 42.

5 Conclusion

This paper has proposed a robust NFt, filter design proce- dure for convex polytopic uncertain systems. This approach can successfully solve the robust filtering problem for systems with uncertainty in all the matrices. The motivation for choosing this type of uncertainty is that the uncertainty varying in the fixed convex polytope may

represent an uncertainty domain more precisely than the norm-bounded uncertainty and, consequently, causes no conservatism for a particular structure. The convex opti- misation has been used to design the robust 7-1, filter and as a result, the optimal tolerance level can be directly computed as one of the decision variables, which implies that the bisection method for the optimal value of y need not to be employed. Moreover, the proposed algorithm does not require the cumbersome unidimensional search procedure indispensable for dealing with the norm- bounded uncertainty. Several numerical examples have shown that the robust 7-1, filter proposed here is more robust than the robust 7-1, filter against the parameter variation, as well as the noise in the worst-case frequency range and have illustrated the advantages of describing the uncertainty as polytopic rather than norm bounded. It is expected that the proposed techniques can also be extended to the design of robust mixed 7-1,/I-l, filters.

6 Acknowledgment

This work was supported by Brain Korea 2 1.

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