Abstract--This note contains recurrence state estimation and control algorithms for general time-delay, discrete linear systems with the noises correlated on time ...
Automatica, Vol. 24• No. 6, pp. 841-843. 1988
0005-1098188 $3.00 + 0.00 Pergamon Press pie © 1988 International Federation of A u t o m a t i c Control
Printed in G r eat Britain.
Technical C o m m u n i q u e
On State Estimation and Control in Discretetime Systems of Linear Type* A L E K S A N D E R KOWALSKU" and DOMINIK SZYNALt Key Words--Linear filtering; separation principle; discrete-time linear system•
Abstract--This note contains recurrence state estimation and control algorithms for general time-delay, discrete linear systems with the noises correlated on time intervals whose lengths change in time. 1)C(k) I] then the system (1) reduces to the standard Kalman system [cf. Anderson and Moore (1979) and Zabczyk ~19762)]. For a ( k ) = b ( k ) = c ( k ) = O, d ( k ) = - 1, qt = rk = s k = s t = 0 we get the Kalman system with coloured observation noises (Anderson and Moore, 1979). Under the assumptions of white noises we obtain a class of time-delayed systems [cf. Medrano-Cerda (1983), Zabello and Lebedev (1982)]• From (1), one can also get time-delayed systems with coloured measurement noises, which are correlated with white state noises [cf. Biswas and Mahalanabis (1972), Mishra and Ramajani (1975) and Liang and Christensen (1975), Tarn and Zaborsky (1970)]. The aim of this note is to give a recurrence algorithm for the state estimation and control in (1). This algorithm reduces to the analogous algorithms considered in early papers. Let W denote the Hiibert space of square integrable r-dimensional random vectors with zero mean• Suppose that {[x~ y~¢]' k = 0, 1. . . . } ~-W +:. Then the linear least square estimate -~klt of the vector x k based on the process {y/ j = 0 . . . . . l} is the orthogonal projection of the vector xk on the e-dimensional linear space Y'~ spanned by this process. We write Skit = P~tXk. Moreover, by the projection lemma (Anderson and Moore, 1979) and (Ansley and Kohn, 1983)
1. Introduction
THE STATE ESTIMATION algorithm for linear time-delayed systems with white noise processes was derived by Kwakernaak (1967) and others. Biswas and Mahalanabis (1972) and Mishra and Ramajani (1975) extended the problem to systems with coloured measurement noise• Liang and Christensen (1975) extended the results to systems with correlated state and measurement noise• Medrano-Cerda (1983) considered systems with delays in the noise process. Kowalski and Szynal (1986) introduced linear systems with noises correlated on a finite time interval and derived the filtering equations. We now consider linear time-delayed systems with the noise processes correlated on time intervals whose lengths change in time. We give the state estimation and general cost control algorithms. The solutions are presented in a recurrence form. However, Kucera (1979) and Nguen and Hoang (1982) considered more general systems but they did not obtain any recurrence algorithms. Let the linear, time-delayed system with non-white noises be given by [x'k+t, y'k+t]'=[F',(k)
r~(k)l'x,,
k=0,1...
(1)
where Xk = ix'k" " • X'k-a(k)
I
Y'k " " " Y'k--b(k) w'~. . . w'k-¢(k)
v ' k ÷ t ' ' " v'~-~¢k~]'
1=0
and xk, Yk, Wk and Ok are random vectors of e,f, g and h dimensions, respectively. ([x~, y~ w~ v~] = 0 for k < 0), F/(k) j = 1,2 are.,~al matrices and a ( k ) , b ( k ) , c ( k ) and d ( k ) are integer-valued non-increasing functions• Forl-k k,l=0,1.--supposethat E[x'k
y~
w;, o~]=0,
E([x~y~l'[w'~
whereei=(I-P~/_0yi, W_t=0 j=0...1, are the innovation vectors, and A - denotes a generalized inverse of A (Ansley and Kohn, 1983). 2. Filtering equations
v;,])---0,
Rewrite the equations (1) in the form
E ( w k w ; ) = O ( k , 1), E ( v k v ; ) = R ( k , l),
xk+ 1 = A ( k ) x k + C ( k ) w k,
E(WkV;) = S t ( k , l), E ( v , w ; ) = SZ(k, l),
(3)
P~tx k = ~ , E ( x k e ; ) ( E ( e / e ; ) ) - e 1,
Yk = B ( k ) x k , Xk = Ix(k)xk
(2)
where wk = [w~+l, v~+2]' , Ix(k ) = [I
where Q ( k , l) = 0 for k - 1 > q k , R ( k , l) = 0 for k - 1 >rkz, S l ( k , l) = 0 for k - 1 >s~, S2(k, l) = 0 for l - k >s~, and qk, rk, S~, S2, are non-negative numbers may be depending on k. We see that the system contains as particular cases discrete-time systems of the references. If a ( k ) = c ( k ) = O, b ( k ) = - 1 (no vector), d ( k ) = - 1 , q k = r k = s t ~ = s 2 = O , r , ( k ) = [A(k) C ( k ) 01, Fz(k ) = [ B ( k + 1)A(k) n(k +
rt(k) I
0---
0...01 -o-
o 0-1
I
0
0
0 • • .
r2(k) A(k)=
0
I
0-..
(r)
k=0,1...
0
B(k) =
C(k) =
0
01
I
01
0
O" -•
0
*Received 12 August 1987; revised 19 April 1988. Recommended for publication by Editor W. S. Levine. t Instytut Matematyki UMCS, PI. Marii CurieSklodowskiej 1, 20-031 Lublin, Poland.
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0••
0 841
0
•
I
0
0-..
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o
842
Fechnical C o m m u n i q u e
We see that the system (1) is equivalent to the standard Kalman type system (1') without measurement noises and with correlated state noises wk. Note that the assumptions (2) for the system (1) are equivalent to the following assumptions for (1')
where u~ is an n-dimensional r;mdom vector, 1 p k ) ~ [Fl(k ) F"(k)] is a real matrix, and m(k) is an integer-valucd non-increasing function. Let the process [x~', .v~', v,,[,, ,,', w', t"l . . . . ; be ()au,',svan and (2) holds. Then -¢k1/= E(.t~ ! ;~.). where :Y'~, k -: !), 1 is the a-field generated by the observation process
E(w,x;,7 = [~ O ( k + l , O ) S ' ( k + 2, 0)
{y~,, y~. . . . . y~}. 0 0
St(k+l,1) R ( k + 2, 1)
S'(k+l.0) R ( k + 2, 0)
,, [ Q ( k + l , l + l ) E(w~w;)=Q(k,t)=LS2(k+2`1+l)
R] ,
(2')
k,l=O, 1.'.
~ where Q ( k , l ) = O for k - l > q k = m a x { q , + ~ , r,+,, S ~+1, -~ • , 7g + h s~.~, s ~ q } . The last assumptmns ~mp~ that wg ± Yt and Wk~LXt[ +~ for l ~ k - q ~ , where Xtg h denote the g + h dimensional linear space spanned by the process {x k k =
0.1..-/}. Define
ikl I = P~Xk,
itlt = Xk -- iklt,
ff klt = P~ Wk'
ff kl! = Wk -- WkF
F(k I1) = E(iktti'~lt), [D(k, l)Dt(k, l)O2(k, t, j)] = E(wk[e;i'tlt_ , ~';IJ])' L(k, 1) = E(x,e;),
of
admissible
control
strategies
,.m
and a cost functional J N--[
v(,., xo) = E( E
/si
+
where .r, = [(x~)' -" ' (X~ko~{k))'] ', U4. = [U'k, .. U'k ,,,(k~]', and qk, k = 0, 1. . . . N - 1, q,'%. are real. non-negative, measurable functions. As in the previous section we introduce an augement state vector x~ and rewrite the equation (1") as in (1'). Then Theorem 1 implies that the estimate x~l~: is given by i,~+q,~+l--A(k)i~l,t + C(k) ~ G(k, i)ek_ , i=O
N(I) = E(ete;).
+ K(k + 1)ek+t +
Using (3) we get the following result.
Theorem 1. The linear least square filtering and one-step prediction estimates for the system (1') under the assumptions (2') satisfy the following recurrence equations: iglk = ikl~-t + K(k)ek,
class
'~//= {u = (u o . . . . . UN_ O, Uk is ~ - m e a s u r a b l e , u, e D, DisaclosedsubsetofgJ'.k=0. 1...N-I} i4)
S'(k+l.l+2)] R ( k + 2 , 1 + 2 ) J' l
