ELSEVIER Systems & Control Letters 25 (1995) 113-119

S JITUlS k CONTIIOL UTTIIW

On the evaluation of the least achievable tolerance in the disturbance feedforward problem

Vlad Ionescu*

Department of Control Engineering, University Politechnica Bucharest, Bucharest, Romania

Received 1 May 1994

Abstract

Based on the so-called generalized Popov-Yakubovich theory the well-known formula of the least achievable tolerance for the "two block problem" is easily recovered in a self-contained way.

Keywords." General distance problem; Disturbance attenuation; Nehari problem; Popov-Yakubovich theory

1. Introduction

As is well known the (suboptimal) H~-control problem of prescribed tolerance gamma is equivalent to the y-disturbance attenuation (y-DA) problem which consists in designing a controller such that the resultant closed-loop system exhibits simultaneously two properties: internal stability and y-attenuation i.e. the quotient "output"/"disturbance" is less than V. If the Youla parametrization is used (see [21]) then one can prove that the y-DA problem reduces to the (suboptimal) generalized distance (GD) problem (see [2-4, 6, 8, 9]). A problem which has been intensively investigated is that of the evaluation of the least achievable tolerance 7 (= Vmi,), i.e. Ymin := inf{y: the 7-DA problem has a solution}. Clearly Ymi, coincides with the optimal norm for the GD problem. Technical reasons lead to consider the following terminology. The most general case of the GD problem is called the "four-block problem" while as special eases we refer to the "two-block problem" and "one-block problem". The last problem is usually known as the "best (or Hankel) approximation" problem and Nehari's theorem (see [19]) provides an elegant solution to it (see for instance [9, 10]). Special attention has been paid to the "two-block problem" or extended Nehari problem (see [14] for the discrete case) since it is closely related to the so-called mixed sensitivity problem which plays an important role in the H~-control system design. In this case the evaluation of Ymin is achieved in terms of the spectral radius of a Hankel + Toeplitz operator (see [2, 20]). Conceptually (see [2]), it is easy to generalize the above-cited result to the "four block problem". A complete solution to this case is given in [6]. An elegant treatment of the same topics may be also found in [7].

A special ease of 7-DA problem is the so called y-disturbance feedforward (v-DF) problem which reduces to the "two-block problem". Hence, in this case, Ymi, can be rapidly evaluated in accordance with the above mentioned results.

Recently the y-DA problem received a new treatment by imbedding it in the framework of the so called generalized Popov-Yakubovich theory (see [5, 12-14, 16, 17]).

* Correspondence address: 3 Emile Zola, 71272 Bucharest, Romania. E-mail: cris@karla.indinf.pub.ro.

0167-6911/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 01 67-69 11 (94)00069-7

114 ~ lonesculSystems & Control Letters 25 (1995) 113-119

Based on the above-mentioned theory the present paper gives a simple, self contained proof for the Ymin-formula in the case of the ?-DF problem, formulated for discrete-time systems. Connections with the classical "two block problem" are also presented. As a relevant application of the developed theory a simple and new evaluation of the maximum stability margin in the normalized left coprime factorization (LCF) robust stabilization problem is given.

The following notations will be used. By D we denote the open unit disk of the complex plane C. By ~ and we denote the ring of integers and the field of real numbers, respectively, l 2,q will stand for the square norm

summable ~q-valued sequences v = (vk)k~e. The norm and the inner product in l 2,q will be denoted by H ll2 and (,), respectively. By l~ q (12"_ q ) we denote the closed subspace of 12,q consisting of those v for which vk = 0 if

k < 0 (k>~0). Orthogonal projection of l 2'q onto l~_ q (12'_ q) wil l be denoted by P+'q (P - ' q ) . The adjoint of an f-operator T will be denoted T*. If T = T* and (u, Tu) ~>611u112, 6 > 0, we call T coercive and we shall write T>>O ( - r ~ o ) .

IfA is any square matrix, A(A) stands for its spectrum.

2. Preliminaries and problem statement

Consider the generalized (discrete-time) linear system

ax = Ax + Blul + Bzu2, yj = Clx + Dllul + D12u2, Y2 = C2x + D21ul (2.1)

together with the controller

aXc= Acxc + BcY2, u2 = Ccxc + DcY2, (2.2)

and a given tolerance y > 0. Here x E ~", ui E ~m,, Yi E ~P~, i = 1,2, xc C ~nc and the matrix coefficients in (2.1) and (2.2) are constant matrices of appropriate dimensions.

The ?-DA problem (see for instance [ 13]) consists in finding a controller (2.2) such that internal stability and input-output y-attenuation are simultaneously achieved by the closed-loop resultant system, To be more specific the above two requirements mean

A(AR ) C D f°r AR = [ A + B2DcC2BcC2 B2Cc

and

Ilry,~,ll < Y,

(2.3)

(2.4)

where Ty,., : l 2,m' --+ l 2,p' is the linear bounded input-output operator defined by the stable resulting system. A particular y-DA problem is the y-DF problem for which the following hypothesies are assumed to be valid.

(DF1) D21 is nonsingular. (DF2) A - BID~C2 is stable. (DF3) (A, B2) is stabilizable. (DF4) The matrix

[ z l - A - B 2 ] c 1 D12 (2.5)

has full column rank for z = e j°, V0 E [0, 2~). As (DF3) and (DF4) both hold it follows (see for instance [ 16]) that the discrete-time algebraic Riccati equation

(DTARE)

ATXA - X - (ATXB2 + CTD,2)(DT2D,2 + BTXB2)-'(DT2C, + BTXA) + cTc , = 0 (2.6)

has a positive semidefinite stabilizing solution X>~O, i.e. AF := A + BF is stable for F := --(DT2DI2 + B~XB2)-'(DT2 C' + B~XA) and, in addition, D~zD,2 + B~XB2 > O.

V. lonescu I Systems & Control Letters 25 (1995) 113-119 115

Let C1F := CI +D21F. According to Proposition 5 [13, p. 163] both (2.1) and (2.2) provide the same resultant system as

6x = AFX + Bi Ul -]- B2U2,

to which the controller

6X¢ = Acx c + [Bc 0]Y2e, u2 = Ccxc + [De - F]y2e (2.8)

is connected. As AF is stable in (2.7) Tll : ul ~ Yl and TI2 : u2 --~ Yl are well defined and bounded operators from l 2,m' into l 2,p' and l 2"m2 into l 2"p' , respectively.

One can easily prove (see also (2.11), (2.12) [13, p. 164]) that

rl*2T12 : oY2Ol2 + BTXB2 : v T v > 0 (2.9)

with V nonsingular. Thus TI* 2 Tl2 is an 12-coereive operator. Moreover,

/~12 := TIzV -1 (2.10)

is inner as directly follows from (2.9). Introduce now the following Toeplitz like operator:

0 p+,m2 ] (rT;l ['~0/ ~LT;2] [Tll T12] - : ] ) [Po m, 0 p+,mz ]

= [ "~11 ~2t

Notice that

~12] (2.11) "~22 J "

~22 : p+,m2 T72 Tl2P+, m2 = p+,m2 vT V (2.12)

as directly follows from (2.9). Hence "~22 is also coercive and has a bounded inverse. Introduce the Schur complement of ,-~22, that is,

~11 : : ~11 -- ~ 1 2 ~ 1 ~ ; 2 • (2.13)

Notice now that due to the above-cited Proposition 5 [13] the 7-DF problem formulated for (2.1) is equivalent to a special 7-DA problem formulated for the modified system (2.7) which, in addition, is stable. Based on this fact, by combining Theorem 4, p. 162 with Theorem 5, p. 182 [13], one obtains that the following result is true.

Theorem 1. The 7-DF problem has a solution if and only if

~11 40. (2.14)

AsCii clearly depends on 7 it is now quite transparent that the evaluation of least achievable tolerance reduces to the problem of finding the infimum of the family of T for which the inequality (2.14) still holds.

3. Main result

First we shall compute~ll . To this end from (2.11) with (2.12) one obtains

[ P+'ml T~x TIIP+'m' - 72p+'m' P+'m' TT1T12P+'m2 ]

~l= L P+'m2T~zTIlp+'m' P+'mzvTvp+'m2

116 V. lonescu I Systems & Control Letters 25 (1995) 113-119

from where according to (2.13) we get

~11 -~- p+,m, T~ 1 T11P+'m' -- 72P +'m' -- p+.,nt TI* I T12P+,m2 V- ! V-TT~2TI iP +'mr

_p+,m~T~lTllP+,m, o + , m l T * '1% o+,m2'~'* T D+,ml 72p+,mL ( 3 . 1 ) - - - - 1 *111 12 z l 1 2 1 1 1 r -- ,

where (2.10) and the commutativity of V with p+.m2 have been used.

As is well known (see [11, 13]) there exists an orthogonal completion 1~2 of the inner operator/~12 such that

[1~127~1 is allpass. Let ql := PJ - ml. Then (3.1) becomes

= e+,mt TII P+,rnl ,°~11 TI*I[TI2 ~lA_2] p+m, q_p--,ml 0 [ T,2 0 -}- P - ' q ' l (TI±2) *

_p+,m, Tll T12 P + ' m : ]g~2TI tP +'m' - 22p+,rn,

=p+,m,T~lf. lzp-.m,÷* T o+.m, ~± ^± * +m _72p+,m, t t z * l l * + P+'m'T~IT12(TI2) T I IP" '

= HT. t f,, H;,~ ;~,2 + T(T,.~/~ )(T,~ f~ )* -- 72P+'m' (3.2)

where by H p and TQ the causal Hankel and Toeplitz operators associated to the/2-operators P and Q have been denoted, respectively.

From (3.2) we rapidly deduce that (2.14) holds if and only if y > Ymin where

_l/2z 71 7i* "Ymin := P ~.nT;,t,zt*rat,2 + T(r?, r,2"~ )(r u• r,2*~ )* )' (3.3)

where p stands for the spectral radius. Thus we have the following theorem.

Theorem 2. The least achievable tolerance in y-DF problern is given by (3.3). [~

Remark 1. As is well known (see [8]) by using Youla's parametrization, the 7-DA problem is equivalent to the (suboptimal) model rnatchiny problem which consists in finding an 12-operator Q such that

lIT, - TzQT3][ < 7, (3.4)

where T1, /'2, 7"3 and Q are all causal/2-operators. Remember that an/2-operator T : l 2,m ---+ 12,p is said causal if P+,PTP +''n = TP +,". Using the algorithm given in [8], (3.4) reduces, in the case of the y-DF problem, to the (suboptimal) two block Nehari's problem

r,, ;?~z - Q < 7. (3 .5 )

The computations have been omitted since they are trivial. Following the theory developed in [2, 6, 20], the least 7 (= 7min) for which there exists a causal Q such that (3.5) holds for all y > Ymi,, is given by the same formula (3.3). []

4. Application to the normalized LCF robust stabilization problem

Let

ax = Ax + Bu, y = Cx (4.1)

be a stabilizable and detectable system. Here A E R "×n, B E ~nxm and C E ~pXn. Hence the following two (standard) Riccati equations:

ATXA - X - ATXB(I + BTXB)- lBXXA + c T c = 0 (4.2)

II.. lonescu / Systems & Control Letters 25 (1995) 113-119 117

and

AYA T - Y - ` 4 y c T ( I + C Y C T ) - I C Y A T + BB T = 0 (4.3)

have positive semidefinite stabilizing solutions X and Y, respectively. Therefore, ̀ 4F : : .4 + B F and AK := A + K C are both stable for F := - ( I + B T X B ) - I B T X A and K := - - A Y C T ( I + c Y C T ) -1 .

Let I + B T X B = V T V and I + C Y C T : /7-/7 -T. Let the following/Z-operators

M =[`4F, BV-1,F,V-I], N :[AF,BV-1,C,O], (4.4)

G = [ A K , B , - V F , V], H = [`4K,K, VF, O]

and

^ - I ~t = [AK,K, ~-1C, ~-~], ~ = [AK,8, V C, 0],

: [ ` 4 F , - - K ~ ' , C, re], /-~ : [`4F,KV, F,O]

be defined. Then the following doubly coprime normalized factorization holds, i.e.

T = N M -1 = I Q - I ~ ,

(4.5)

(4.6)

/Q N M = ' (4.7)

M * M + N * N = I, (4.8)

~¢~Q* + NN* = I. (4.9)

For proof see for instance in [13, Theorem 14, p. 106]. Here T : l +,m ~ l +,p is the operator defined by (4.1) and l +'q stands for the space of Rq-valued sequences

v = (vk)ke~ for which there exists an integer v (depending on v) such that vk = 0 for k < v. For more details see [1]. Notice that T is not necessarily an 12-operator.

As is well known (see [ 18]) the normalized LCF robust stabilization problem is equivalent to the ~-DA problem formulated for the fictitious system

ax = Ax - K~'Ul + Bu2, (4.10)

For (4.10) the associated modified system (2.7) is, in this case,

fix = AFX -- KI)Ul + Bu2, (4.11)

0 Yl = [Flx-~- [~] Ul q- [10] u2, Y2e= [Ci]x-~- IVo] Ul.

According to (4.4) and (4.5) the corresponding operators 7"11 and/~lz are given by

as can be checked immediately.

118

As

V. lonescu / Systems & Control Letters 25 (1995) 113-119

is all-pass (see (4.8) and (4.9)) one obtains that

/ ~ = [ ~ / ~ * ] . (4.13)

Thus from (4.12) and (4.13) we have

(4.14)

r;,r~=[-,O* d*] 4" =(~?~+~d)=t, (4.15)

where in (4.15), (4.7) has been used. With (4.14) and (4.15) formula (3.3) becomes

?rain = (1 + p(HsH*s)) 1/2. (4.16)

With a little computations (omitted here) it can be proved (see [17]) that the controllability and observability Gramians associated to the stable part of S are Ps = Y(I +X Y)-1 and Qs = (I + XY)X. As p( HsH* s) = P(Ps Qs ) (see [18] for the continuous case and [13] for the discrete case) the well-known Glover-McFarlane formula

~min = (1 -~" p ( S Y ) ) I/2

is recovered.

5. Conclusions

Based on the generalized Popov-Yakubovich theory a very simple proof of the 7min-formula has been presented. Such proof is self-contained and powerful tools as commutant lifting theorem [6, 7] or Helton's lemma [20] is avoided. It can be easily remarked that the whole development runs similarly for the continuous-time case.

References

[1] J.A. Ball, 1. Gohberg and M.A. Kaashoek, Nevanlinna-Pick interpolation for time-varying input-output maps: the discrete-case, in: Operator Theory: Advances and Applications, Vol. 56 (Birkhauser, Basel, 1992) 1-51.

[2] Cheng Cin Chu, J.C. Doyle and E. Bruce Lee, The general distance problem in H °° optimal control theory, Internat. J. Control 40 (1986) 565-596.

[3] J.C. Doyle, Lecture Notes in Advances in Multivariable Control, ONR/Honeywell Workshop, Minneapolis, MN, 1984. [4] J.C. Doyle and B. Francis, Linear control theory with an H °° optimality criterion, S l A M J. Control Optim 2.5 (1987) 815-844. [5] V. Dragan, A. Halanay and V. lonescu, Infinite horizon disturbance attenuation for discrete-time systems: a Popov-Yakubovich

approach, Integral Equations Operator Theory 19 (1994) 153-215. [6] A.Feintuch and B.A. Francis, Uniformly optimal control of linear systems, Automatica 21 (1985) 563-574. [7] C. Foias, Commutant lifting techniques for computing optimal HC~-controllers, in: H°°-Control Theory, Como 1990, Lecture Notes

in Mathematics, Vol. 1496 (Springer, Berlin, 1991 ). [8] B.A. Francis, A guide to H°°-control theory, in: Modeling, Robustness and Sensitivity Reduction in Control Systems (Springer in

cooperation with NATO Scientific Affairs Division, 1987) 1-30. [9] B. Francis, A course in H °° control theory, Lecture Notes in Control and Information Science, Vol. 88 (Springer, Berlin, 1987).

[10] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L°°-error bounds, lnternat. J. Control 39 (1984) 1115-1193.

V. Ionescu I Systems & Control Letters 25 (1995) 113-119 119

[I 1] D.W. Gu, M.C. Tsai, S.D. O' Young and I. Postlethwaite, State-space formulae for discrete-time H°°-optimization, Internat. ,L Control 49 (1989) 1683-1724.

[12] A. Halanay and V. lonescu, Generalized discrete-time Popov-Yakubovich theory, Systems Control Lett. 20 (1993) 1~. [13] A. Halanay and V. Ionescu, Time-varying discrete linear systems. Input-output operators. Riccati equations. Disturbance attenuation,

in: Operator Theory: Advances and Applications, Vol. 68 (Birkhauser, Basel, 1994). [14] A. Halanay and V. Ionescu, Extended Nehari problem for time-variant discrete systems, Systems Control Lett. 22 (1994) 195-201. [15] V. lonescu, Maximum stability margin evaluation with respect to coprime factor perturbations: a generalized Popov-Yakubovich type

approach, submitted to: Automatica. [16] V. lonescu and M. Weiss, Two Riccati formulae for the discrete-time H°°-control problem, lnternat. J. Control 57 (1993) 141-195. [17] V. lonescu and M. Weiss, Continuous and discrete-time Riccati theory: a Popov function approach, Linear Algebra Appl. 193 (1993)

173-209. [18] D.C. McFarlane and K. Glover, Robust controller design using normalized coprime factor plant descriptions, Lecture Notes in Control

and Information Science, Vol. 138 (Springer, Berlin, 1990). [19] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957) 153-162. [20] M.S. Verma and E. Jonckheere, L °° compensation with mixed sensitivity as a broadband matching problem, Systems Control Lett.

4 (1984) pp 125-129. [21] D.C. Youla, H.A. Jabr and J.J. Bongiorno, Modern Wiener-Hopf design of optimal controllers: Part II, IEEE Trans. Automat.

Control AC21 (1976) 319-338.