Admissible MIMO singular observation LQG designs

Automatica, Vol, 24, No. l, pp. 43-51, 1988 Printed in Great Britain.

0005-1098/88 $3.00 + 0.00 Pergamon Journals Ltd. (~) 1988 International Federation of Automatic Control

Admissible MIMO Singular Observation LQG Designs* Y. H A L E V D and Z. J. PALMOR~:

Singular observation optimal LQG systems may have improper sensitivity matrices which implies lack of stability robustness. Necessary and sufficient conditions for admissible designs avoiding this phenomenon are presented for the multiple-input~multiple-output case. Key Words--Optimal control; linear optimal regulator; optimal systems; robustness; stability; optimal filtering; suboptimal control.

Abstract--The widely known LQG design method for linear time-invariant MIMO systems may lead under certain circumstances to closed loop systems with improper sensitivity matrices. Such systems have no stability robustness, and we refer to them as inadmissible ones. This phenomenon is analyzed in the paper. Necessary and sufficient conditions guaranteeing admissibility of LQG designs, both in the frequency and the time domains, are presented and proven. These conditions are based on the available information in the statement of the problem, thus enabling a priori detection of inadmissibility in optimal systems. The key tool in the analysis is the extension of results from polynomial matrices to rational matrices. A suboptimal design, aimed directly at avoiding inadmissibility and which involves the minimum necessary changes in the LQG cost function, is suggested, lllustrative examples are used to demonstrate the main results.

In some cases some of the measurements may have negligibly small errors and may be considered free of noise. These cases are commonly referred to as singular observation LQG designs. Singular observation LQG designs are naturally handled in the frequency domain within the framework of the Wiener-Hopf technique and its extensions, e.g. Grimble (1978) and Shaked (1976). Introduction of colored noises in state space formulations is accomplished via augmentation of the state but this leads to a singular measurement equation. Optimal reconstruction of the state cannot then be done via the standard Kalman filter since its formulation requires a nonsingular covariance matrix of the measurement noise. As was shown in Bryson and Johansen (1965) and Halevi and Palmor (1986a), optimal estimation in singular measurement systems may require differentiation of the outputs and the inputs. An extremely limited number of reports refer to "difficulties" witl, singular observations LQG design. Simakov (1974), perhaps for the first time, demonstrated that LQG systems may loose stability due to infinitesimal changes in the plant parameter. Recently, Halevi and Palmor (1986b) have discussed the phenomenon of inadmissible SISO optimal LQG systems and presented conditions for its occurrence. It should be noted that the H ~ design method (e.g. Francis et al., 1984; Kwakernaak, 1985) avoids these difficulties by requiring certain structural relations involving the plant and the disturbances transfer matrices. In the SISO case the term "inadmissible" was used to denote systems with no stability margins at all. In the context of MIMO systems inadmissibility means systems with improper sensitivity matrices. In MIMO systems there are several different sensitivity matrices and for

1. INTRODUCTION

ONE OFTHE MAJORAPPROACHESto feedback design is the L Q G procedure (Anderson and Moore, 1971; Kwakernaak and Sivan, 1972). Except for the selection of weighting matrices for the mean square error criterion and stochastic models for the disturbances and measurement noise, the design of LQG controllers is automatic. This and the stabilization property (under mild conditions) are probably the main reasons why the LQG designs attracted both practitioners and theoreticians. Most of the literature on LQG designs focuses on cases where the measurement noises (MN) are white. In many practical systems, however, MN are found to be colored. * Received 15 October 1986; revised 19 May 1987. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. M. Guimaraes Ferreira under the direction of Editor H. Kwakernaak. t Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802, U.S.A. Currently with the Faculty of Mechanical Engineering, Technion-lsrael Institute of Technology, Haifa, Israel. :t:Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa, Israel. 43

44

Y. HALEVl and Z. J. PALMOR

admissibility all of them must be proper otherwise the system has no stability robustness. This paper is concerned with singular observation LQG designs for MIMO systems. It shows that the automatic design via linear timeinvariant LQG may lead to inadmissible closed loop systems. These cases are analyzed and conditions for a priori detection of such possible "traps" are stated and proven in the form of necessary and sufficient conditions for the admissibility of LQG systems. These conditions are given in both the frequency domain and the state space. In the SISO case the admissibility conditions involved only the degree differences between the denominator and numerator polynomials of the transfer function of the process and the noises. Application of these conditions can be carried out merely by inspection. The extension to MIMO systems requires different tools and results in more involved conditions due to the increased complexity and structural richness of these cases. These complexities give room to hidden mechanisms that cause inadmissibility even in cases that otherwise seem to be quite simple and "innocent". The key tool in our analysis is the extension of the notion of row-reduced polynomial matrix to rational matrices (Krishnarao and Chen, 1984). Some known as well as new relevant definitions and results concerning row-reduced rational matrices are given in Section 2. These results are used in the following sections. The statement of the LQG optimization problem and its solutions for linear time-invariant systems in both the frequency domain and the state space are given briefly in Section 3. In Section 4 we discuss inadmissible systems, not necessarily optimal, and give admissibility conditions for two basic control schemes. Section 5 deals with inadmissible optimal LQG systems and presents the main results, namely conditions for a priori detection of inadmissibility in optimal systems. These conditions are given in both the frequency domain and the state space. In Section 6 a method for achieving admissibility is presented. We discuss and summarize the results of this paper in Section 7.

give herein some definitions and lemmas, some new as well as known, to be employed in the following sections where optimal MIMO closed loop systems will be treated. If A(s) is a rational transfer matrix then each of its entries, aq(s), may be expanded in a descending power series in s in the following form

2. PRELIMINARIES The notion of row-reduced (or columnreduced) polynomial matrices is well known and was found to be a useful tool in analyzing MIMO systems given by matrix fraction description (MFD) (Kailath, 1980). Recently it was shown (Krishnarao and Chen, 1984) that this notion can be extended to rational transfer matrices. Most of the available results for polynomial matrices hold for rational transfer matrices as well and we

C(s)CT(--S) = A(s)AT(-s)

aij(s) = ~ aii,ts'

(2.1)

l=ki/

where kq is the degree difference between the numerator and the denominator polynomials. This notation is similar to the one used in the context of polynomial matrices except for the infiniteness of the series. In the sequel k 0 denotes the degree of ao(s) and aij.k,, is the leading coefficient. The degree of the i-th row, ki, is defined as the maximum degree of all the entries in this row. In complete analogy to polynomial matrices, A(s) may be written as (Kailath, 1980)

where

A(s) = A(s)Ah~ + A(s)

(2.2)

A(s) = diag {s k'}

(2.3)

and the highest degree in the i-th row of A(s) is strictly less than k~. With the representation (2.2) we have the following definitions and results. The proofs of the lemmas are given in Halevi (1985) and are omitted here for brevity.

Definition 2.1. If Ant has full row rank then A(s) is called row reduced.

Lemma 2.2. For any A(s) with normal full row rank there exists a unimodular matrix U(s) such that Al(s) = U(s)A(s) is row reduced. Lemma 2.3 (Krishnarao and Chen, 1984). Given H(s)=A-t(s)B(s) where A(s) and B(s) are rational transfer matrices, then H(s) will be strictly proper (proper) only if every row of B(s) has degree less than (less than or equal to) that of the corresponding row of A(s). If A(s) is row reduced then "only if" is replaced by "if and only if".

Lemma 2.4. Let A ( s ) ( r x m ) be given by equation (2.2). If C(s)(r x q) satisfies (2.4)

then C(s) = A(s)Chr + C(s), where Ch,C~r= Ah~A~.

Corollary 2.5. If A(s) is row reduced, so is C(s). Definition 2.6. A rational transfer matrix H(s) has a degree k if s-(k-t)H(s) is improper and s-kH(s) is proper.

Admissible singular observation LQG designs

Lemma 2.7. If degree satisfies

H(s)=A-l(s)B(s)

then

its

lw

[Ocs~ I

k - max (bi - ai) i

(2.5) I'=0 *Q,,

where bi and ai are the row degrees of B(s) and A(s), respectively. If A(s) is row reduced then (2.5) becomes an equality.

Lemma

2.8. Let A(s) be a square and row-reduced rational transfer matrix given by equation (2.2). Then

A-'(s) = A L ' A - ' ( s ) + A l ( S )

(2.6)

where the degree of the j-th column of fi, l(s) is strictly less than -kj. We refer now to a state space realization of a MIMO system, and define a row-reduced system in terms of its realization. Let the system be described by 2 = Ax + Bu (2.7)

x~R",

45

y = Cx ueR",

(2.8)

y~RL

If ci denotes the i-th row of C, then to every such a row there corresponds a positive integer li such that

ciAi-lB = 0 1 <-j < l~ ciAti-tB ¢ 0.

(2.9) (2.10)

In the following we refer to li as the index of the i-th row.

Definition 2.9. A system is called row reduced if

ClAtl-lB rank

]

= r.

(2.11)

_c,A t'-' B _1 The significance of equation (2.11) is that the transfer matrix which corresponds to (2.7)-(2.8), i.e. G(s)= C ( s I - A ) - ~ B , is row reduced and the matrix in (2.11) is the leading coefficients matrix of G(s). Consequently any system can be made row reduced by means of a dynamical output transformation which consists of linear combinations of the various outputs and their derivatives. 3, THE OPTIMAL REGULATOR

3.1. Frequency domain We consider the system in Fig. 1 where u ~ R ' , wl e R q', w2 e R q2, y e R r, r e R r', and C(s), D(s), P(s), H(s) and M(s) have the appropriate dimensions, wl and wz are mutually independent, zero mean, unit intensity, Gaussian white noise vectors. The scheme is completely general since any other existing noise in the loop may be grouped to either wx or w2. To avoid the possibility of a noise which may be

t *iv2

lw2 FIG. 1. The regulator system.

introduced to the loop via an unstable system we assume that D(s) and M(s) may be factored as follows:

D(s) = P(s)Dt(s) + Dz(s) M(s) = H(s)ML(s) + M2(s)

(3.1) (3.2)

where Dx(s), D2(s), Mr(s) and Mz(s) are asymptotically stable or constant matrices. We further assume that there is no cancellation of unstable poles in H(s)P(s). The optimization criterion is J = r--,2 T l i1m E(f;(yT(t)Qy(t)+uT(t)Ru(t) ) _

dt}. (3.3)

The optimal compensator is given by (Grimble, 1978):

Co(s) = Go(s)[I- H(s)P(s)Go(s)]-'

(3.4)

where Go(s) is the optimal transfer matrix from the reference r to the input u and is determined from

Go(s) = Z-'(s){(zT(--S))-'PT(--s)QD(s)DT x (-s)(YT(-s))-t}+Y-'(s). (3.5) { }+ denotes expansion over the poles of the causal part of the expression in brackets, namely those of D(s). Z(s) and Y(s) are square matrices that are calculated via the following spectral factorizations

zT(--s)Z(s) = PT(-s)QP(s) + R

(3.6)

Y(s) YT(-s) = H(s)D(s)DT(-s)HT(-s) + M(s)MT(-s).

(3.7)

The poles of Z(s) are those of P(s) and z - l ( s ) is stable. Similarly the poles of Y(s) are those of H(s), D(s) and M(s) and Y-l(s) is stable. 3.2. State space We consider the case of totally singular measurement. This case includes the case of colored measurement noise. The state space

46

Y. HALEVI and Z. J. PALMOR

realization is 2 = A x + Bu + Fw

(3.8)

y = Cx

(3.9)

x ~ R ~,

u ~ R m,

y ~ R ~,

-,tI 0

J

-- I i =

C.(s)

wER q

=

(3.17)

"

lr--

1

. "

E Crml,-i-Is i

and

E{W(t)wT(Td)}= lqt~(g -- ~).

(3.10)

i=1)

The optimal control law is

The optimization criterion is 1

_

u = -/72

(3.18)

r

J= lim-~E(( (xT(t)Qx(t)+uT(t)Ru(t))dt}. T~ /.1 L J - - T

where again the exact value of F is of no interest to us.

(3.11) Assumption 3.1. The system (C, A, F) is row reduced in the sense of Definition 2.9. With this assumption, 2, the optimal estimation of x is given by (Halevi and Palmor, 1986) - a f[H

4. INADMISSIBLE OUTPUT FEEDBACK SCHEMES

The closed loop shown in Fig. 1 may be broken at three points, x, xx and xxx thus giving rise to three different sensitivity matrices

]

(s)

l[L(s I - L2)-I(L3S(s) + L4HI(S))_] y(s) + .(SI - L2)-x[Z5

- -

t3Ca(S ) + L4n2(s)]

x Bu(s)}

]

(3.12)

where Ll(n × n), Lz(nl × hi), L3(nl × r), La(nl x 7) and Ls(n~ x n) are constant matrices• As will be shown later the exact values of these matrices are of no importance to the current development• The dimension i is given by i = 2 l,

(3.13)

i=1

where the l~s are the row indices of (C, A, F) and n~ = n -7. The nonconstant matrices in equation (3.12) are S(s) = diag {s"} (3.14) rls..

"S11-1

/-/ds)= L 0 . 0 - 0 .

0"" "0 ls'"s

0

''-~.

$1 = (I + CHP) -1

(4.1a)

$2 = (I + PCH) -~

(4. lb)

$3 = (I + H P C ) - ' .

(4. lc)

Definition 4.1. A system is admissible if and only if the sensitivity matrices Sk, k = 1, 2, 3 are proper• This requirement is expressed by the following condition lim sup s

0...0

"0 1 • • • s I'-1 0 Cl

12= Clah-i-2s i

i=O

(4.2)

The equivalent therefore

$1 = I - G H P

(4.3a)

Sz = I - P G H

(4.3b)

$3 = I - HPG.

(4.3c)

admissibility

lim sup IIG ( s ) n ( s ) P ( s ) I t

c~s + ClA

2, 3.

For reasons that will become clear in Section 5 we wish to express Condition 4.2 in terms of G(s), the transfer matrix from the reference r to the input u. Substitution of equation (3.4) [with general C(s) and G(s) rather than the optimal Co(s) and Go(s)] into equations (4.1) yields after some algebraic manipulation

1T

(3•15)

II&(s)ll < ~ k = l,

~

conditions < o~

are (4.4a)

lira sup IIP(s )G(s )H (s )ll <

(4.4b)

lim sup [IH(s)P(s)G(s)ll < o~.

(4.4c)

S~Oe

(3.16)

/-/2(s) = -

0 ¢r

lr--1 E Crml,-i-2S i i~o

We turn now to the system in Fig. 2 which describes the case of an observer-state feedback type controller• Tl(s)(n × r) and T2(s)(n x m ) are the transfer matrices from the output y and the input u, respectively, to the estimated state 2. F is a constant gain matrix• No optimality is assumed in both the observer and the state

Admissible singular observation LQG designs r--o

+

I

Y

FIG. 2. A system with an o b s e r v e r - s t a t e feedback type controller.

feedback. The overall control law is u(s) = - ( I + FT2(s))-lFT~(s)y(s).

(4.5)

Substituting equation (4.5) into equations (4.1), (H(s) = I), the corresponding sensitivity matrices are given by S~(s) = [I + (I + FTz(s))-~FT~(s)P(s)] -: = I - [I + F(T2(s) + T:(s)P(s))]-: × FT~(s)P(s)

(4.6a)

S2(s) = [I + P(s)(I + FTz(s))-1FT:(s)] -1 = I - P(s)[l + F(Tz(s) + Tl(s)P(s))] -1 x FTa(s). (4.6b) A common feature of any asymptotic observer, whether optimal or not, is that the transfer function from u to the estimated state ~" is identical to the one from u to the actual state x. Hence Tz(s) + T:(s)P(s) = ( s I - A ) - I B . As the right-hand side strictly proper, so must even when some or all of proper or even improper.

(4.7)

of equation (4.7) is be the left-hand side, its components are just Therefore

lira [T2(s) + T~(s)P(s)] = 0.

(4.8)

s.-..~ zc

This in turn implies that the admissibility conditions for the state space representation, (4.6), are equivalent to lim sup IIFT~(s)P(s)II < ~

(4.9a)

tle(s)fT~(s)ll < ~.

(4.9b)

lira sup

Equation (4.9a) leads to the following. L e m m a 4.2. A closed loop system with an observer-state feedback type controller will be inadmissable for almost every state feedback F if derivatives of the input signal u are used by the observer to reconstruct the state. Proof. When derivatives of u are used for the estimation then T2(s) is improper and from equation (4.8) so is T:(s)P(s). Hence condition 4.9a is violated. Remark 4.3. Absence of derivatives of the input

47

in the observer is not a sufficient condition for admissibility since then only condition (4.9a) is satisfied while condition (4.9b) may not hold. Although observers involving output or input derivatives are not a common practice and normally they will not be encountered in deterministic designs, optimal estimation when the measurement is singular may lead to such observers. 5. CONDITIONS FOR ADMISSIBLE LOG DESIGNS Conditions (4.4) or (4.9) are completely general and apply to any feedback system including, certainly, optimal systems. The point is that one does not have Co(s) or Go(s) unless the procedure in equations (3.4)-(3.7) is carried out. Thus the admissibility of the design can be checked only after the optimization problem is solved. We present in this section conditions that enable a priori detections of the inadmissibility phenomenon. These conditions are based on the available information in the statement of the problem. We consider the system in Fig. 1 and the optimization criterion (3.3) and assume that P(s) is a proper transfer matrix. A matrix ~(s)(r~ × (ql + q2)) is defined as • (s) = [ n ( s ) D ( s ) i M(s)].

(5.1)

Ul(S) is a unimodular matrix such that qJ~(s) = U:(s)W(s)

(5.2)

is row reduced. We also define the following matrices A1 = U1HP;

m2=

U1H;

~1 = I;

~2 =

m 3 = Ol'~

P;

~k3 =

HP.

(5.3)

With these definitions we have the following theorem. Theorem 5.1. The optimal system minimizing criterion (3.3) will be admissible if 3k+max(6ki--~Oi)<--I

k = 1, 2, 3; i=1 ..... m

(5.4)

where 6~ and ~Pi are the row degrees of Ak(S) and ~t(s), respectively, and ~k is the degree of the matrix/~k. Proof. Define E(s) by E(s) = { ( z T ( - - s ) ) - I P T ( - - s ) Q D ( s )

x OT(-s)I4T(-s)(U(-s))-'}+.

(5.5)

Then the admissibility conditions (4.4) may be

48

Y. HALEV! and Z. J. PALMOR and Ap, Ah and Am are defined in a similar fashion. For M(s) = O, Am = oo.

written as lim sup II~,k(s)Z-~(s)E(s)Y-~(s)Ak(s)ll <

(5.7)

Remark 5.4. When the measurement noise is white with nonsingular covariance, state reconstruction is accomplished via the Kalman filter. Under these circumstances the system is always admissible. Here nonsingular white measurement noise is represented by a constant, full row rank M(s). Hence U2(s) is row reduced a n d ]]Yi= 0Vi. Substituting this into conditions (5.4) shows that admissibility is always guaranteed in these cases.

(5.8)

Remark 5.5. Properness of $2 and $3 implies properness of S~. To see this note that

k = 1, 2, 3.

(5.6)

Next we find the degrees of the matrices in (5.6). As was mentioned earlier, the { }÷ denotes expansion over the poles of D(s). Each entry of E(s) is a sum of terms of the form at/(s + be) and therefore if the (i, j) entry of E(s) has the form of Olij(S)/~ij(S), then almost always degree (o~,i(s)) - degree (fl~i(s)) = 1 Vi, j.

E(s) is therefore given by E(s) = s-t/• Ehr + Consequently the degree of E(s) is - 1 . From equation (3.6) we see that

31max+ max (6~ -- V/i) = max (deg (U1HP)i - v/i) i i

Z(s) =/~t/2 + Z(s)

(5.9)

-< max deg (tiP) + max (deg (UOi - ~Pi) i = 63ax + max (63 - ~p,) (5.16)

where Z(s) is strictly proper and

= R + PV(oo)Qe(oo).

(5.10)

/~ is nonsingular. From Lemma 2.8 we get

z-t(s) = + 2(s) (5.11) where 2](s) is strictly proper. Hence, the maximum degree in Z-~(s) is 0. From equation (3.7) and the definition of tF(s): =

(5.12)

Premultiplication of equation (5.12) by Ut(s) and postmultiplication by UT(-s) yields UI(S ) r(s) YT(--S) u T ( - - S ) = UxIt(S)I-I-/1T(--S).

(5.13) Now, according to Lemma 2.4, if W~(s) is row reduced so is Ut(s)Y(s) and the degree of the i-th row in the latter matrix is V/~. From Lemma 2.7 the degree of Y-t(s)Ak(s) is max (6~--~p~). It is therefore concluded that condition (5.6) holds if (5.4) is satisfied.

Remark 5.2. Only in rare cases is the degree of ;Xk(S)Z-I(s)E(s)Y-I(s)Ak(s) less than the sum of the degrees of its individual components. Therefore the above sufficient condition is almost always a necessary one as well.

Remark 5.3. Condition (5.4) reduces to the results obtained in Halevi and Palmor (1986a) for SISO systems and takes the form min(Ad+Ah,

Am)-Ap-Ah<-i

(5.14)

where Ad--a no. of poles of D(s) - no. of zeros of D(s)

(5.15)

where deg (A)i is the maximum degree in the i-th row of the matrix A(s). Similarly it can be shown that properness of $2 implies that of Sx. On the other hand, properness of either $2 or $3 does not guarantee the properness of the other.

Remark 5.6. In certain cases inadmissibility may be determined without the need to make tF(s) row reduced. If the difference between the maximum row-degree in H(s)P(s) and the minimum row-degree in W(s) is larger than 1, then the design will be inadmissible. On the other hand, if the above difference is 1 or less the conditions of the theorem must be applied. In the state space the corresponding condition for admissibility is as follows. Theorem 5.7. Given the system in equations (3.8)-(3.9) and the optimization criterion (3.11), assume that (C, A, F) is row reduced and its row indices are li. Similarly, let mj be the row indices of (C, A, B). Then the optimal system will be admissible if max i

(li)

-

min (mj) < 1 i, j = 1, 2 . . . . . r. i

(5.17) Proof. For condition (4.9) to hold it is sufficient that the sum of the maximum degrees of P(s) and T~(s) is nonpositive. The maximum degrees of P(s) and Tl(s) are - m i n (mj) and max (li - 1), respectively. The former degree comes from the definition of the index of the j-th row [see (2.9) and (2.10)] of P(s) or equivalently of (C, A, B). The latter degree is the highest degree in T~(s) as can be seen from (3.12). Hence (5.17).

Admissible singular observation LQG designs

Remark 5.8. The admissibility condition (5.16) was derived based on the sensitivity matrix $2 (4.6b). It can be shown that the corresponding condition for $1 is max(li-mi)<-I i

i=l, 2,...,r.

(5.18)

It is clear however that (5.17) implies (5.18). Hence, properness of $2 implies properness of $1 but not vice versa.

Remark 5.9. Similarly to Remark 5.2 condition (5.17) will be almost always necessary too. Theorems 5.1 and 5.7 may be used to analyze the sensitivity properties of the LQG/LTR (loop transfer function recovery) design method (Doyle and Stein, 1979). In the limit the measurement equation may be regarded as singular and the process noise input matrix F approaches B. Under these circumstances the systems (C, A, B) and (C, A, F) become identical. Therefore li = mi and from equation (5.18) it is seen that the sensitivity matrix Sl(s) is always proper. However, the sensitivity matrix S2(s) will be improper if max (li) - min (li) > 1. This means that in such cases the LQG/LTR method drives asymptotically to inadmissible designs. Two sensitivity matrices may differ considerably if the condition number of P(s) is large (Freudenberg and Looze, 1986). This is consistent with our result regarding the LQG/LTR since whenever the row-degrees in P(s) are not the same its condition number goes to infinity at high frequencies. In Examples 5.10-5.11 we present two cases of inadmissible optimal systems. While in Example 5.11 both S~(s) and S2(s) turn out to be improper, in Example 5.10 just S2(s) is improper. Such conclusions regarding optimal designs are certainly not easy to reach without the admissibility conditions in Theorems 5.1 and 5.7.

Ilj

Example 5.10. Let Q=0.512, M(s) = O. P(s) =

R=H(s)=I2,

(s+l

D(s)=

49

s+2)

(s+l)(s

2)(s+3)

To apply condition (5.4) we first check if W(s) is row reduced.

W(s) = O(s) = L[SoZ

0

1

Wh, is singular, which means that W(s) is not row reduced. Multiplying ~ ( s ) by

U~(s)D(s) 2 (s + 1)(s + 2)

we get fft/l($)

=

2 (s + 1)(s + 2)(s + 3)

0 s-Z

0][2 s -3 0

0

0

+ tP,(s)

Al(s) = VxH(s)e(s) I

2 s+l

=

2

(s + 1)(s + 2) S01

I

0

2

= 62 - tF2 = 1. Hence Sl(s) is proper. However,

so that 32 + max (62 - Wi) = - 1 + (0 - ( - 3 ) ) = 2. From Theorem 5.1 we see that S2(s) is improper and the system is inadmissible. A direct solution of the optimization problem leads to the following optimal controller,

Co(s) = [-0.010Is2 + 0.1645 0.0101s 2 + 0.1322s + 0.4309] and the corresponding Sl(S) is given by

s +2 s+3

and

Sl(s) =

(s + 1)(s + 2)

(s + 1)(s + 2) 1.1524(s z + 3.4641s + 3)

which is proper, but 1

S2(s) = 1.1524(s 2 + 3.4641s + 3)

×

0.0101s + 1.1624s 2 + 3.4309s + 3.2928 0.0101s 3 + 0.0303s 2 - 0.1645s - 0.4935 &0TO 24:1-D

.

-(0.010Is 3 + 0.1423s 2 + 0.5632s + 0.4309)] -0.0101s 3 + 0.9899s 2 + 3.1645s + 2.1645 _1

50

Y. HALEVI and Z. J. PALMOR

is improper. Consequently the system is stable for the perturbation C(s)=KCo(s), - 1 . 3 7 < K<~, but becomes unstable for P ' ( s ) =

KtP(s), Kt

Example

=

[l+e 0

~]

_..> + e

0

.

[277-49

5.11. Let the system be given by

2=

8 -1 18 4

-16 -33

10

-19

2

11

design of Youla et al. (1976) the admissibility problem is avoided by assumptions made on the system and the cost criterion. These assumptions can always be satisfied by an appropriate modification of the criterion. We give herein a new method that also consists of a modified criterion. The modification is the minimal one required to achieve admissibility. Consider the optimization criterion

J= lim --~TE{f (yT(t)Qy(t) +

x

+ ~'~ u(i)'(t)Riu(i)(t)) dt i=1

+

Y= Now ctF =

~ u+

-3

c~AF = 0

4 w

0 _1]

-1

5

}

(6.1)

where u(~)(t) is the i-th derivative of u(t). For simplicity we assume that all the R~ are diagonal matrices. We define pi as the maximal derivative of uj(t) on which weight is put in (6.1), so that

0 x.

(R,)jj

but

>

i = p,

(6.2)

i >pj.

The system is row reduced l~ = 3, 12 = 1. For mt we have c~B = [1 0 ] ~ 0

Clearly p. is the largest p# Now the question is what are the minimal pj's that lead to an admissible optimal system. We define 6~ as the column degree of Ak(s). Then we have the following theorem.

so that m~ = 1 and since l~ - m~ = 2 the optimal L Q G system will be inadmissible with all sensitivity matrices improper.

Theorem 6.1. The optimal system for the optimization criterion (6.1) will be admissible if and almost always only if

c2F J

6. SUBOPTIMAL ADMISSIBLE LQG DESIGNS So far the phenomenon of admissible optimal L Q G system was presented and conditions for its occurrance were derived. Suppose now that the application of Theorems 5.1 or 5.7 leads to the conclusion that the optimal system will be inadmissible. If some sort of optimality is required then a deviation from the original optimization problem must be made. When such a deviation is relatively small the design may be regarded as a suboptimal one. It should be emphasized that inadmissible systems lack stability robustness. Thus small deviations from the original problem lead to low robustness. The design should therefore be a compromise between the two conflicting demands, that of being as close as possible to the original optimal cost and at the same time having a sufficient robustness. A straightforward method to overcome the admissibility problem is by simply adding a fictitious nonsingular white measurement noise. This was shown in Remark 5.4 to guarantee admissibility. In the modern Wiener-Hopf

pj > 6~ + max (6~ - Wi) - 1 j=l .....

m;

k=1,2,3.

(6.3)

Proof. The solution of the problem in the frequency domain is the same as in Section 3 with the distinction that R is replaced by R(s) given by P.

R(s) = ~ (-s2)iRi.

(6.4)

i =0

Z(s) is now given by Z(s) = Zhc diag {s °,} + Z(s). (6.5) degree of Ak(s)Z-I(s) is max (6~--pj).

Consequently

The Condition (6.3) guarantees that

~,k(s)Z-l(s)E(s)y-l(s)Ak(s) and so is S~(s) ( k = 1, 2,

is proper 3). The "almost always only if" applies in the sense of Remark 5.2.

Remark

6.2. A nonpositive pj resulting from condition (6.3) means that no modification is required on the weight on the input uj(t).

Admissible singular observation LQG designs

Remark 6.3. The method suggested by Youla requires that weight will be put on higher derivatives of u in the optimization criterion than the degree required by Theorem 6.1. The corresponding condition in the state space is as follows.

Theorem 6.4. The optimal system which minimizes criterion 6.1 [with x(t) replacing y(t)] will be admissible if and almost always only if pj -> max (li) - mj - 1

(6.6)

where mj is the index of the j-th column of (C, A, B) which is dual to the row index defined in equations (2.15) and (2.16). The proof can be found in Halevi (1985). It is omitted for the sake of brevity.

Example 6.5. Consider the system in Example 5.10. Applying condition (6.3) with k = 2 yields p1_>-1+3-1=1. Therefore weighting the first derivative of u is sufficient to make the optimal system admissible.

Example 6.6. Consider the system in Example 5.11. The column indices of ( C , A , B ) are m I = 1, m2 = 2. Max (li) = 3. Hence it is clear from Theorem 6.4 that only Ux should be weighted in the PI for admissibility.

7. CONCLUSIONS

The well-known optimal LQC design method for linear time-invariant MIMO plants is shown to lead, under certain circumstances, to inadmissible closed loop systems. Such situations may arise when colored noises are considered or when MN is white but with singular covariance matrix. This rather surprising phenomenon is analyzed in this paper. It shows that there exists a class of problems that the ordinary LQG design method cannot cope with. In MIMO systems there are several sensitivity matrices and for admissibility all of them must be proper otherwise the system has no stability robustness. Necessary and sufficient conditions guaranteeing admissible optimal closed loop systems were presented in Section 5 both in the frequency and time domains. These conditions are based on the known plant and noise transfer matrices or equivalently on the known realization. They

51

enable an a priori detection of inadmissible designs. While the conditions in this paper apply to all sensitivities it was shown that the critical condition for admissibility is the properness of $2 and $3 which implies the properness of S~. The L Q G / L T R was shown to guarantee in the limit the properness of S~ but not necessarily that of $2 or $3. In fact if the difference between the degrees of two rows of the plant transfer matrix is greater than 1 the resulting system will be inadmissible in the limit. A suboptimal method, aimed directly at avoiding the admissibility problem, was derived in Section 6. The advantage of this method is that it indicates the minimum required changes in the original cost which guarantee an admissible design. REFERENCES Anderson, B. D. O. and J. B. Moore (1971). Linear Optimal Control. Prentice-Hall, Englewood Cliffs, NJ. Bryson, A. E. and D. E. Johansen (1965). Linear filtering for time varying systems using measurements containing colored noise. IEEE Trans. Aut. Control, AC-10, 4-10. Doyle, J. C. and G. Stein (1979). Robustness with observers. IEEE Trans. Aut. Control, AC-24, 607-611. Francis, B. A., J. W. Helton and G. Zames (1984). H®-Optimal feedback controllers for linear multivariable systems. IEEE Trans. Aut. Control, AC-29, 888-900. Freudenberg, J. S. and D. P. Looze (1986). Relation between properties of multivariable feedback systems at different loop-breaking points. Proc. ACC. Grimble, M. J. (1978). Design of stochastic optimal feedback control systems. Proc. IEE, 125, 1275-1284. Halevi, Y. (1985). Inadmissible optimal LQG systems. Ph.D. thesis, Faculty of Mechanical Engineering, Technion, Haifa, Israel. Halevi, Y. and Z. J. Palmor (1986a). Extended limiting forms of optimum observers and LQG regulators. Int. J. Control, 43, 193-212. Halevi, Y. and Z. J. Palmor (1986b). Inadmissibility of SISO singular observation LQG design. Opt. Control Applic. Meth., 7, 219-241. Kailath, T. (1980). Linear Systems. Prentice-Hall, Englewood Cliffs, NJ. Krishnarao, I. S. and C. T. Chen (1984). Properness of feedback transfer matrices. Int. J. Control., 39, 57-61. Kwakernaak, H. (1985). Minimax frequency domain performance and robustness optimization of linear feedback systems. IEEE Trans. Aut. Control, AC-80, 994-1004. Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems. Wiley Interscience, NY. Shaked, U. (1976). A general transfer function approach to the steady state linear quadratic Gaussian stochastic control problem. Int. J. Control, 24, 771-800. Simakov, I. P. (1974). Incorrectness of certain methods of synthesis of optimal systems of automatic control subjected to the action of random disturbances. Automatica Telemekh., 3, 186-189. Youla, D. C., J. J. Bongiorno and H. A. Jabr (1976). Modern Wierner-Hopf design of optimal controllers, Part II: the multivariable case. IEEE Trans. Aut. Control, AC-21, 319-338.