On the Sampled-Data H∞ Filtering Problem

Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997

ON THE SAMPLED-DATA H oo FILTERING PROBLEM

Leonid Mirkin*t

Zalman J. Palmor*+

* Faculty of Mechanical Engineering, Technion - lIT, Haifa 32000, Israel t e-mail: mersglm~tec.hunix . technion.ac.il :I:

e-mail: merzpalCOtechunix . technion.ac.il

Abstract: In this paper the Hoc filtering problem of a continuous-time process is considered under the assumption that the measurement is sampled by a generalized sampling device. Both cases of known and unknown initial conditions are considered. The problem is treated via the lifting transformation, which converts the problem to a pure discrete-time setting. It is shown that if a suboptimal filter exists, it can always be presented in the form of a finite dimensional discrete-time (possibly time-varying) filter followed by a stationary generalized hold function . This extends previous results in the literature, where a suboptimal filter of this structure (with an LTI discrete-time part of the filter) was obtained for the case of known initial conditions only. Also, the existence conditions of a suboptimal filter derived in this paper are simpler than those known in the literature. Keywords: Sampled-data systems, Filtering theory, HOC optimization, Discrete-time systems, Holding elements.

vector:

1 INTRODUCTION AND PROBLEM FORMULATION

z(t) = C,x(t),

(2)

The HOC filtering theory has been extensively developed for the last few years, see (Nagpal and Khargonekar, 1991; Shaked and Theodor, 1992) and the references therein. The HOC filtering is a minimax problem where the maximum l2 norm of the estimation error is minimized over all l2 disturbances and uncertain initial conditions. Such an approach is useful if statistic properties of the system disturbances are unknown or known only partially. As argued by Shaked and Theodor (1992) , the HOC filter may also be less sensitive to the uncertainties in the plant parameters than its l2 counterpart.

where Cl E ]Rn: x n is of full row rank. Then the continuous-time HOC filtering problem may be formulated as follows:

Consider a continuous-time LTI process described by the following state space equations:

State-space solutions to such a problem for the cases where n > (i. e., Xo is completely unknown) and n = (L e., Xo is known) were derived by Nagpal and Khargonekar (1991).

x(t)

= Ax(t) + Bw(t),

y(t ) = C2X(t)

+ 02W(t),

x(o) = Xo,

Given the process (la) , the measurements (lb), a matrix n = n' 2:: 0, and a constant y > 0, find a causal filter generating an estimate ~(t) of the signal z(t) in (2) such that

IIz -

~IIIz <

y2(xb ntx o+ IIwllIz)

for all Xo E Im n and w E l2 .

°°

(la) (lb)

In this paper we consider the HOC filtering problem assuming that the measurements and the information processing are performed on the basis of digital hardware. We assume that the measurement equation is of the form

where x( t) E lRn is the state vector, y (t) E ]RT is the measured output, w(t) E ]Rmc is the disturbance vector, which includes both a process disturbance and a measurement noise, and A, B, C2 , and O 2 are matrices of appropriate dimensions. Assume that it is required to estimate by a causal filter the following linear combination of the state

h-

Yk=t

425

c!>s(T)y(kh--T)dT+Odvk,

(3)

where 4>('T) = Dsb('T) + CseAS'TBs for some matrices As, Bs , Cs , and Ds of appropriate dimensions, Vk E IRmd is a discrete-time measurement noise, and h is the sampling period. The ideal sampler corresponds in this setup to the case of As = Bs = Cs = and Ds = I, while the averaging sampler (Shats and Shaked, 1989)-to As = Ds = 0, Bs = ~I, and Cs = I. Since the ideal sampling is an unbounded operator as a mapping l2 H e2 (Chen and Francis, 1995), we make the following assumption

(3) and unknown initial state a solution to FPHoo exists if! there exists a solution in the form of a discrete-time filter followed by a stationary generalized hold function. The necessary and sufficient conditions for the existence of a solution to FPHoo in this case can be given in terms of a discretetime finite-dimensional Riccati recursion. It will also be shown, that the nonsingularity of the matrix MSNK(t) for all t E [0, h] is equivalent to the nonsingularity of n,,(t) , t E [0, h], and the condition p(n" (h)-'n12(h)QO) < y2 , where p(M) denotes the spectral radius of M. The advantage of the latter test is that it simplifies considerably all the calculations involving the solution QO to the Riccati equation (or the Riccati recursion, for the non-stationary case). Furthermore, we consider the problem FPHoo under milder assumption than in (Sun et al. , 1993) : we do not require Dd to be right invertible. Finally, following Nagpal and Khargonekar (1991) we will give sufficient conditions for the existence of a stationary solution to FPHoo , that is a solution of the form of a discrete-time LT! filter followed by a stationary generalized hold function.

°

(AI): DsD2 = 0, which states that the sampler operates over proper signals. By "proper" we mean here that prefiltering by an antialiasing filter is provided if necessary. Hence, (AI) guarantees boundedness of the sampling operations. The sampled-data HOO filtering problem to be considered in this is paper as follows:

FPHoo : Given the process (1), the measurement

(3), a matrix n = n' ~ 0, and a constant y > 0, find a causal filter generating an estimate «(t) of the signal z(t) in (2) such that

IIz- (lit! < y2(XO ntx o + IIwllt2

1.1

The notation throughout the paper is fairly standard. For a real valued matrix M, M' denotes its transpose and p(M) denotes the spectral radius (if M is square). The notation M ' / 2 is adopted for the square root of a matrix M = M' ~ 0, whereas M t - its pseudo-inverse.

+ Ilvllf2)

for all Xo E Im n, w E l2, and v E

Notation

e2 •

The causality of the filter here is understood as the dependence of the estimate «(t) only on the measurements lh, ~ i ~ k, where k is the largest integer such that t ~ kh.

°

Discrete-time signals in the time domain are highlighted by a bar (like ~) , while in the lifted domain - by a breve (like t). Finally, since we will extensively use operator compositions involving operators over infinite dimensional spaces (such as l2[O, Tl), the following notation is aimed to simplify the readability of the formulae: bar above a variable denotes an operator 0 both from and to finite dimensional spaces; grave accent - 6, from a finite dimensional space to an infinite dimensional one; acute accent - 6, from an infinite dimensional space to a finite dimensional one; and breve-a, both from and to infinite dimensional spaces.

The sampled-data HOO filtering problem for the ideal sampler was recently considered by Sun et al. (1993). They gave necessary and sufficient conditions for the existence of ay-suboptimal sampled-data filter in terms of the stabilizing solution to a differential Riccati equation with jumps at t = kh. The suboptimal filter presented in (Sun et al., 1993) is also of the form of a continuoustime filter with jumps. It is however shown in (Sun et al., 1993) that if the initial state Xo is known, then there exists ay-suboptimal sampleddata filter, which has the structure of a discretetime LT! filter followed by a generalized hold junction. Also, in this case the conditions of the existence of a solution to the FPHoo is given in terms of the stabilizing solution QO to a discrete-time algebraic Riccati equation and the nonsingularity of the matrix MSNK(t) == n" (t) + n12(t)Qo, 'it E [0, h], where the matrices nl l (t) and n12 (t) are subblocks of a matrix exponential of the form e Mt .

2 PRELIMINARY: LIFTING TECHNIQUE In this section some preliminary materials about the so-called lifting technique are collected. For more details on this subject the reader is referred to (Yamamoto, 1994; Bamieh et al., 1991; Bamieh and Pearson, 1992). The new representation of the lifted system is due to Mirkin and Palmor (1995) .

The purpose of this paper is to show that even in the general case of the measurement equation

The notion of lifting is based on a conversion of real valued signals in continuous time into Junc426

Next , define the impulse and the sampling operators. The impulse operator Je transforms a vector " E IRn into a modulated 5-impulse as follows :

tional space valued sequences, that is sequences that take values not from IR but rather from some general Banach space (e[O, hJ in this paper). Formally, let t l 2[O,n) be the space of sequences, each element of which is a function from L2[0, h], that is

t l 2[O,n) =

{t:t[kJ E L2 [O, hJ

Vk E

e

The sampling operator J transforms a function {, E Cn[O, hJ into a vector from Rn as follows:

Z+} .

Then given any h> 0, the lifting operator Wn : l~ H el 2[O,n) is defined as an operator so that:

t = WnE.

{=::}

(t[kl) (T)

= E.(kh + Tl.

e

It is worth stressing that J is not the adjoint of :Je. We, however, will proceed with this abuse of notation for the reasons discussed in (Mirkin and Palmor, 1995).

T E [0, hJ.

It is easy to see that the lifting operator is a linear bijection between l~ and el 2[O,n). Moreover, if we restrict the domain of Wn to the Hilbert space L2 , then by appropriate choice of a norm on el 2[O,n) the lifting operator can be made an isometry (Bamieh and Pearson, 1992). Hence, treating a system {, = 9w not as a mappin~ from w to {, but rather as a mapping from wto (, gives essentially the same system. Indeed, lifting preserves system stability and induced system norms. This allows one to conclude that 9 and

9 == Wn 9w h

1

:

t L 2[O,n)

H

Now consider the representation of lifted systems in state-space. Let 9 be an LTI finite dimensional

9, is also LTI and

(Mirkin

where

el 2[O,nl>

[~

D

3 LIFTED DOMAIN SOLUTION The treatment of the FPHoo is complicated owing to its hybrid continuous/discrete nature and inherent periodicity. To circumvent these problems let us apply the lifting technique described in the previous section. We get the following discretetime LTI system:

First, define systems with two-point boundary conditions (STPBC), which are linear continuoustime operators 9 : L2 [0, hJ H L2 [0, h], described by the following state equations:

+ Ix(h)

0] (ACOD I 0B) [:J0o 10]. 10

B ] == [:Jh. ~ 01

Note, that the dimension of the state vector is preserved under lifting.

Before discussing the representation of LTI systems in the lifted domain, some definitions are in order.

o.x(O)

~ I ~ ].

Then the lifting of 9 , and Palmor, 1995)

which is called lifting of 9, are equivalent. The advantage of treating systems in the lifting domain is due to the fact that 9 is time-invariant in discrete time even if 9 is h-periodic in continuous time. Hence, any periodic problem in continuous time can be reduced to a time-invariant one in discrete time.

x(t) = Ax(t) + Bw(tl, y(t) = Cx(t) + Dw(t),

system with the state-space realization [

+ 13Wk , C2Xk + D2 wk ,

(4a)

Xk+l = AXk

h

= 0,

=

(4b) (4c)

= x (kh), E.k = Yk+l , Zk = WnZ(T) , W~W(T)] (for T E [kh, kh + h)) , and

where Xk

where the square matrices Q and I shape the boundary conditions of the state vector x. The boundary conditions are said to be well-posed if and in this case the map det(o. + leAn) i= (, = 9w is well defined "Iw E L2[O, h). Throughout the paper STPBC will be denoted by the following compact block notation:

Wk = [

°

13

== [Bc 0] : L2 [0, hJ EEllRmcl

C== [g~]

where

and the term STPBC will be used for systems with well-posed boundary conditions only. In the case where 0. = I and I = the boundary condition "window" will be omitted:

°

427

Vk+l

: Rn

H

H

IRn ,

L2 [0, h) EEl IRT ,

and

One can see now that the measured output tk of the lifted system is just 'Yk+ 1. Therefore, the causality of an admissible filter in time domain is equivalent to strict causality of an admissible filter in the lifted domain. So, the Hoo filtering problem can now be reformulated in the lifted domain as follows:

L(M) == -(AMC* = [ Ll (M)

+ BO*)H(M)-l l2(M)].

(note, that lz(M) is finite dimensional). A solution P k = PL k E Z+, to (5) is said to be stabilizing if it is bounded, det(H(P k )) =F 0, and the system

FP eq : Given the process (4a), the measurements (4b), a matrix n = n' ;::: 0, and a constant y > 0, find a strictly causal filter generating an estimate t of the signal z in (4c) such that

is asymptotically stable. Then the formal solution to the FP eq is as follows:

for all Xo E Im nand W E

°

Theorem 1. Given ay> and n > 0, then an estimator which solves FP eq exists ifJ the Riccati recursion (5) has stabilizing solutions P k such that 'Vk E Z+

e{Z[O,h]elR'

Remark 3.1. It is worth stressing that in FPHoo (and, hence, in FP eq ) the computational delays are assumed to be negligible. Nevertheless, the HOO filtering problem in the lifted domain is always the so-called a priory filtering problem (Shaked and Theodor, 1992). This is a direct consequence of the choice of the upper integral limit in (3) as h - rather than h. Such a choice is due to the fact that computational delays, even negligible ones, do not permit instantaneous information processing.

(a) P k > 0;

If these conditions hold, then one possible ysuboptimal filter is

tk = Cliik, iik+l = Aiik + lZ(Pk)(tk - CZfikl.

The problem FP eq is almost a standard discretetime Hoo a priori filtering problem (see e. g. (Shaked and Theodor, 1992; Hassibi et al., 1996)). It differs from the problems considered in the literature in that the signal to be estimated and the disturbance operate not over lR, but rather over infinite dimensional spaces lZ [0, hJ and lZ [0, hJ $lR, respectively. Nevertheless, the two spaces above are Hilbelt spaces and the dimension of the state vector of the process (4a) is finite. Hence, the formal solution to FP eq can be obtained using precisely the same reasoning as in the finite dimensional case.

The solution to FP eq given in Theorem 1 is in general time varying. In some situations, however, one is interested in a stationary solution. Below we present sufficient conditions for the existence of such a solution. To this end we need the following discrete-time algebraic Riccati equation (DTARE): Y = AYA' + BB* - l(Y)H(Y)l(Y)*.

Theorem 2. Given ay> 0, which solves FP eq exists only has the stabilizing solution Y 00 C1 Yoo C; < yZ. If in addition LTIobserver

(A2): The pair (Cz,A) is detectable;

B]

A_e i8 C2 02

(6)

Then we have:

To this end we need the following assumptions:

(A3): The operator [

fio = 0.

is right invert-

then an estimator if the DTARE (6) such that 0 1 0; + Yoo ;::: n, then the

tk = Clfik. fik+l = Aiik + l2(Y)(tk - C2iik),

ible 'Vs E [0,2n).

fio

= 0,

solves FP eq .

Also consider the following discrete Riccati recursion:

4

P k+ 1 = APkA' + BB* - L(Pk)H(Pk)L(P k)* po=n, (5)

MAIN RESULTS

The solutions to FPHoo presented in the previous section are given in the lifted domain. Hence for implementation purposes the results of Theorems 1 and 2 should be "translated" to the time domain. The latter problem is complicated due to

where

428

(b) Pk > 0;

the fact that the designed lifted filters are inherently infinite-dimensional. Hence, the reduction of the problems to equivalent finite-dimensional ones as done in (Bamieh and Pearson, 1992) for the sampled-data H OO problem appears to be impossible. In this respect the new representation of the lifted systems and the machinery, proposed by Mirkin and Palmor (1995), will be used. Because of the space limitations, all the proofs will be omitted.

(c) p(ri31r32Pk) < y2.

If these conditions hold, then one possible ysuboptimal filter is l,(kh+'T) = C1eATiik, iik+1 = eAhiik - 812(Pk)822(P k )-1 €k+1, iie = 0,

In order to represent the lifted solutions in Theor-

where €k+1 == Yk+1 - C2eA~k.

ems 1 and 2 in implementable forms we need the matrix exponential r12 r22 r32

Remark 4.1. The requirement on y to be larger then Ye is clearly necessary for the solvability of the FPHoo. Otherwise, 6 16; -/. y2I and hence condition (b) of Theorem 1 is violated for each P k. See (Chen and Francis, 1995, §13.5) and (Gu et al., 1996) where different approaches to computing the l2[O, h)-induced norm of LTI systems are presented.

r13 r23 r33

° °

where

BsD2B' BSD2DZBSj BB' BDzB -A' -CzB

o

-As

s s .

The formulae in Theorem 3 are presented for the general case. They can be simplified if additional assumptions are imposed. For example, assume that DdD~ > and define the matrices

°

Also, define the following matrix function: 8(M) =

[00 DdDd 0 ] + Ms [r23 + r22M r24] r13 + r12 M r14

where Ms

==

1 [Ds C2

Then the Riccati recursion (7) can be written as

~s].

P k+ 1 = [ r 23 + r22P k

Note, that 8(M) = 8(M)' whenever M = M'. Finally, denote

x ['1'2

r 1 [ ~ ].

+ C2(DdD~)-1 C2(r23 + r 22 Pkl) -1,

°

(7')

(7")

°

while if Ds = and Cs = I (this case includes the average sampling as well as the H2 and HOO optimal samplers), then the Riccati recursion becomes:

(Pk) - 812(P k )822(Pk)-le21 (P k), Pe=n,

'1'3

P k+1 = (r23 + r2 2P k)((r3 3 + r32 P kl

Theorem 3. Given ay> Ye and n > 0, then det(r33 ) =f:. and an estimator which solves FPHoo exists ifJ the Riccati recursion 11

+ '1'1 Pk

Further, if h is the ideal sampler (As = Bs = Cs = 0, Ds = I), then (7') simplifies to:

that is Ye is the l2[0, h)-induced norm of the operator from w(t) to z(t). Then the time domain equivalent of Theorem 1 is:

P k+ 1 = 8

r 24 ]

(7)

is such that \fk E Z+ (a) The system Xk+1 = (r22 - P k+ 1r 32 - 812(Pkl822(Pk)-1

Now consider a stationary solution of FPHoo. The following theorem gives sufficient conditions for the existence of such a solution and also the formulae for one possible stationary filter:

x (D SC2 r 22 + CSrd)Xk

is asymptotically stable; 429

Theorem 4. Given ay> Yo, then det(f33) f. 0 and an estimator which solves FPHoo exists only if the DTARE

Chen, T. and B. Francis (1995). Optimal SampledData Control Systems. Springer-Verlag. London.

Y = 8" (Y ) - 812(Y)822(Y)-'S2' (Y)

(8)

Gu, G., J. Chen and O . Toker (1996) . Computation of l 2[0, hJ induced norms. In: Pro-

has the (unique) stabilizing solution Y00 such that p(fi3'f32Yoo ) < y2. If in addition Yoo 2: n, then the LTI observer

ceedings of the 35th IEEE Conference on Decision and Control. Kobe, Japan, Dec. 1996. pp. 4046-4051. Hassibi, B., A. H. Sayed and T. Kailath (1996) . Linear estimation in Krein spaces-Part II: Applications. IEEE Transactions on Automatic Control 41(1) , 34-49.

l.(kh + 'T) = C, eA-cTik,

Tik+, = eAhTik - 812 (Y00)822 (Y00)-' €k+', Tio = 0,

Mirkin, L. and Z. J. Palmor (1995). A new representation of lifted systems with applications. Technical Report TME-439. Faculty of Mechanical Engineering, Technion - Israel Institute of Technology.

solves FP eq. Remark 4.2. It can be shown that equation (8) can be solved by finding the stable deflating subspace of the extended simplectic matrix pair (Al , A T ), where

Nagpal, K. M. and P. P. Khargonekar (1991). Filtering and smoothing in an Hoo-setting.

IEEE Transactions on Automatic Control 36(2) , 151-166. Shaked, U. and Y. Theodor (1992). Hoo-optimal estimation: A tutorial. In: Proceedings of the

31st IEEE Conference on Decision and Control. Vo!. 2. Thcson, AZ, Dec. 1992. pp. 22782286.

and

Shats, S. and U. Shaked (1989) . Exact discretetime modelling of linear analogue systems. International Journal of Control 49(1) , 145160.

In this case Yoo and [2 ~ -812(Yoo )822(Yoo )-' satisfy the equation

Sun, W. , K. M. Nagpal and P. P. Khargonekar (1993). H OO control and filtering for sampleddata systems. IEEE Transactions on A utomatic Control 38(8) , 1162-1174.

for some Schur matrix Acl.

Yamamoto, Y. (1994). A function space approach to sampled data control systems and tracking problems. IEEE Transactions on Automatic Control 39(4), 703- 713.

ACKNOWLEDGMENT This research has been supported by the fund for the promotion of research at the Technion. REFERENCES Bamieh, B. and J. B. Pearson (1992). A general framework for linear periodic systems with applications to HOO sampled-data control. IEEE Transactions on Automatic Control 37(4) , 418-435. Bamieh, B., J. B. Pearson, B. A. Francis and A. Tannenbaum (1991) . A lifting technique for linear periodic systems with applications to sampled-data control. Systems fj Control Letters 17, 79-88. 430