A Polynomial Approach to ℓ1 Optimal Control Problems

Copyright @ IF AC Robust Control Design, Prague, Czech Republic, 2000

A POLYNOMIAL APPROACH TO RI OPTIMAL CONTROL PROBLEMS Alessandro Casavola' Domenico Famularo'

• Dipartimento di Elettronica, Informatica e Sistemistica Universita della Calabria Via P. Bucci, 41C-42C - Rende (CS), 87036 Italy email: {casavola.famularo}~deis.unical.it

Abstract: In this paper , the scalar multi-block RI-optimal control problem is considered. It is shown that it can be converted via polynomial equation techniques to an infinite dimensional linear programming (LP) problem. Finite dimensional sub/super approximations can be determined by considering two sequences of modified finite dimensional linear programming problems derived directly from the YJBK parameterization by exploiting the underlying algebraic structure. This approach induces the application of a consistent truncation strategy that leads to a redundancyfree constraint formulation and , as a consequence, to linear programming problems less affected by degeneracy. Further, more insight on the algebraic structure of the problem and on the achievement of exact rational solutions is provided , allowing the development of a simple and conceptually attractive theory. Copyright @2000 IFAC Keywords: RI Optimal Control , Linear Programming, Polynomials equation , Discrete-time Systems.

Definitions and notations Z(·): The Z-transform operator. Given a matrix H {H(k)}r'=o of causal real sequences: Z(H) = 1i(d) := L:;"=o i H(k) 1 d k , in the complex variable d. ll,pxm: The real normed linear space of all p x m matrices H of absolutely summable real causal sequences H;j = {H;j(k)}r'=o with norm IIH lb := max;E~ L:::':o i 1

H(k)

1< 00 ,

L:::

where p

:=

{I , 2, ... ,p} .

Ap x m: The real normed linear space of all p x m matrices

1i(d) which are Z-transform of some matrix sequence HE ll,pxm. Note that A is isomorphic to ll ,p x m ' 1UI ,pxm: The subspace of ll,p x m of all real causal matrix sequences each of whose entries has a rational Z transform . nA: The subspace of Apx m consisting of elements each of whose entries is a stable stable real rational functions . 8 (X) : Denotes the degree of the polynomial X .

In the sequel, with the hat we denote (matrix) sequences, whereas the same unhatted variable denotes its correspond-

ing Z-transform ; the space of polynomials will be denoted by n[d] whereas rational transfer functions by n(d). With nt [d] will be denoted all polynomial with degree lower than or equal to t .

1. INTRODUCTION

The discrete-time model matching RI-optimization problem (Dahleh and Pearson, 1987; Dahleh and Pearson, 1988) amounts to the minimization of the A-norm of the closed-loop error transfer matrix E(d)

= H(d) -

U(d)Q(d)V(d)

(1)

where H , U , and V are given stable rational matrices (E RA) of appropriate dimensions and Q E A is the free parameter. It is now well understood that one-block problems (MacDonald

and Pearson, 1991; Staffans, 1991) admit rational solutions, that is there exists a minimizer Qo E RA provided that U and V have not transmission zeros on the unit circle.

mial approach and extends directly to the general MIMO case, that will be presented elsewhere.

In bad-rank cases such a strong result of existence is not so far available and minimizers were shown to exist only in A. In this case, finite dimensional approximations are of interest, along with an estimate of the gap existing between the corresponding sub-optima and the true optimum. Various strategies have been proposed to that purpose: Q-design (Boyd and Barratt, 1991) , FMV-FME (Staffans, 1993), Delay Augmentation Method (Diaz-Bobillo and Dahleh, 1993) , Semidefinite (Quadratic) programming methods (Elia and Dahleh, 1998) . All of them, except the last method, consist of adding additional constraints to the original optimization problem in order to achieve sub-optimal finite dimensional solutions. Further, in order to have an estimate of the quality of the approximated solution, in (Staffans, 1993) was firstly introduced the idea of building a dual sequence of linear programming problems whose solutions form a not-decreasing sequence of super-optima converging from the below to the optimum. The latter was obtained by dropping some of the structural constraints existing between [ and Q in (1). The combined use of such a sub/super-optimization scheme allows one to obtain sequences of lower and upper bounds both converging to the optimum. The Semidefinite programming approach (Elia and Dahleh, 1998) instead, embeds the original £1 problem into two finite dimensionalH2 quadratic programming problems (sub- and super-optimal) and uses the LMI paradigm to numerically approach the problem.

2. PROBLEM FORMULATION

Consider the following system

the YJBK parameterization of all admissible closed-loop error maps can be characterized by a matrix transfer function H and polynomial matrices U and V as

for some Q E A. In (3), we can assume w.l.o.g. that Hij(d) ERA, Ui E R[d] and Vi E R[d], i = 1, 2. Let Hij = Nij/D, Q = P/(DS) be defined as polynomial ratios for some given polynomials N ij and D, with D strictly-Schur, viz. all of their possible zeros are outside of the unit disk, and S and P free with S strictly-Schur and PEA of possibly infinite degree. Further, let U(d) := gcd(U1 ,U2 ) and V(d):= gcd(Vi , V2) denote the respective greatest common polynomial divisors so that Ui = UUi and Vi = vl% . Throughout the paper we assume that (A .I) {

• U(d) . V(d) have no roots over the unit circle .

The problem we want to solve is the following £1 four-block model-matching problem jJ-opt:=

The main goal of this paper is to present a different characterization of the closed-loop error transfer function (1). This is done by resorting the polynomial equation approach of Kucera (Kucera, 1979) . This allows a more direct achievement of unconstrained suboptimal and superoptimal linear programming problems that are free of most of the above defects. The key idea consists of parameterizing both the closed-loop error [ and the free parameter Q in terms of a polynomial matrix, that really represents the available degrees of freedom existing as long as the closed-loop stability and feedback structural constraints underlying (1) have been satisfied. As a consequence, the original optimization problem can be expressed in terms of this new polynomial matrix, resulting in an unconstrained linear programming problem.

inf

QEA

II[II A

(4)

2.1 Structural conditions In order to solve the problem we first characterize the class of all admissible closed-loop maps [ = ([ij) , i, j = 1,2, viz. compatible with the feedback structure (3) . To this end, for the sake of clarity, we consider first the case in which the terms Hi1 V2 - Hi2 VI, i = 1, 2, and H 1jU2 - H 2jU1 , j = 1, 2 are polynomials. Specifically, denote as T i , i = 1, ... ,4

H

[!~J = [~~]

[U2 -ud

H

= [T3

T4]

(5) (6)

Note that the following identity holds true

Such a scheme was considered in (Casavola, 1996) for scalar mixed-sensitivity problems. Here the 2 x 2 multi-block case is considered. The case treated exhausts the ideas involved in the present polyno-

T1 U2

+ T4 VI

= T2 U1

+ T3 V2 •

Under the above assumption we have. 2

(7)

Lemma 1 - Let Ti in (5)-(6) be polynomials. Then, the all and the only £ij(d) that jointly satisfy (3) can be parameterized in terms of a possibly infinite degree free polynomial X (d) E A as

where the last equality follows from (13). Then, in order to ensure A-stability is necessary and sufficient that

(8)

for a possibly infinite degree polynomial PI E A . In (16), we have factorized (j = (j- (j+, (j+ strictly-Schur and (j- monic anti-Schur and the same for V = V-V+. Now, all polynomials PI and X that satisfy (15) must satisfy equivalently the following Diophantine equation

P

In (8), Et E R[d] are particular solutions of the following pair of uncoupled polynomial Diophantine equations

= (j-V- PI,

(16)

(j-V-P1 +DX=W .

The above equation is always solvable because ((j-V-:,D) are coprime. Let (XO,PlO ) the minimum degree solution of (17) w.r.t. X, that is 8(X) < 8((j-V-). Then, the general solution has the following expression

Remark 1 - Notice that the above parameterization characterizes the class of all admissible responses with finite support (deadbeat responses) whenever X (d) is restricted to be a polynomial of finite degree. A convenient choice in order to avoid degree inflation in the solution is to select for Et as the minimal degree solutions of (9) and (10) . However, it is worth pointing out that the minimal degree solutions of (9) and (10) with respect to the first or the second of their arguments may differ in general. This does not happen, e.g. for (9), when 8(U1 V2) +8(VIU2) > 8(T1 U2 -T3 V2), where 8(T) denotes the degree of the polynomial T. A similar condition holds for (10) 0

X { PI

Eij = EPj

(18)

+ UiVjXO + Uiv,

((j-V-T) . (19)

If each E?j is the minimal degree solution, we know that 8(Et) < 8(Ui V,). Moreover 8(Xo) < - o' • • • - 8(U-V-) and 8(Eij +Ui VjXO) < 8(Ui VjU-V-). Then, if T is a polynomial of degree t we have that 8(£ij)

The assumptions (5)-(6) imply also that

< 8(Ui V, (j-V-) + 8(T) < mo + t .(20)

where mo := maxi,j 8(Ui V, (j-V-).

(11)

(Nlj - DEiJ U2 = (N2j - DE~j) UI , (12)

The parameterization (19) hinges upon the limitative assumptions (5)-(6) . When they don't hold true, one can adopt a truncating strategy. Several equivalent alternatives are possible. A simple idea, exemplified for (5), consists of finding a polynomial pair (TI(N) , T1(N», with TiN) of degree lower or equal to N, such that

where i = 1, 2,j = 1,2. From the above relations it follows that there exists a single polynomial W (d) such that N ij - DE?j = UiV,W,

= Xo - (j-V-T , = PlO + DT .

with T(d) is a free polynomial. As a consequence, all admissible and A-stable closed-loop error maps £ij can be parameterized in terms of a possibly infinite degree free polynomial TEA as

2.2 Stability conditions

(Nil - DEPd V2 = (Ni2 - DEP2) VI"

(17)

i,j = {1,2}, (13)

!!Jf

on the roots which implies that EPj interpolate of Ui v,. This consideration is not sufficient for ensuring closed-loop A-stability. In fact, rewrite the error sequences £ij as

and similarly for (6). Such a polynomial pair is unique for any N and can be computed as the minimal degree solution w.r.t. Ti(N) of the following set of Diophantine equations

(14) P

where Q = DS and P and S polynomials to be determined. From (14) one obtains

DT1(N) Dr,2(N)

+ dN+ 1T-(N) 1 N 1 + d + r,-(N) 2

(N)

N+l -(N)

+d

(N)

+d

DT3 DQ = P = W - DX S UV'

(15)

DT4 3

T3

N+l -(N)

T4

-- N 11 TT Y2

-

-- N 21 TT Y2

-

N 12 v,'1, (22) N 22 v,'1, (23)

•

•

•

•

= N 11 U2 - N 21 U1 , (24) = N 12 U2 - N 22 U1 • (25)

whose value is ilt. A convenient choice for N, in (N) •• order to have all coefficients of Eij + Ui V; Xo influenced by T, is

Notice that (22)-(25) are always solvable with 8(Ti(N» < N + 1 because (D,d N + 1 ) are coprime for all N. Then, the parameterization (8) of Lemma 1 becomes

N

= N(t) = .,)=1,2 ,min 8(Oi~ [J-V-) + 8(T).(31)

Then, by denoting with where E~) and i;~) are the minimal degree solutions (see Remark 1 for details) of the Diophantine equations (9) and (10) with Ti replaced by T}N) and, respectively, Ti(N), i = 1,2,3,4.

itt :=

.) + N ij = W,

ilt

P

=S=

W-DX UV'

+ DX = W.

(N)

••

+ UiV;XO

(28)

• • - _ - _ dN + 1 - (N) - UiV;U V T+ ---v-Eij . (30)

IIE(N) + [ql] (Xo + (j-V-T) [VI U2

V2]

~ J.Lopt, V t ~ 0

lim itt

t-+oo

=0

In order to derive linear programming problems whose solutions provide a sequence of lowerbounds to J.Lopt, it is necessary to rule out some constraints from (4). Because one cannot eliminate constraints related to stability, the only possibility is to relax some structural constraints. This can be done easily by considering four free polynomials (T + ~+!Tij) instead of a single T in (19). It is evident that this choice remove some structural conditions depending on the degrees of T and Tij . Once such substitutions have taken place, (19) becomes

(29)

t ij

= EPj

+ (l;i'jxo -

UiV;(T

+ dt+ITll ) ,

(33)

with £ij any longer admissible for OPT. A further simplification can be accomplished by considering that the closure of the space all polynomials U(d)T(d) E RA (of possibly infinite degree) generated by an arbitrary polynomial T(d) ERA, with U(d) polynomial, is given by U(d)*T(d), where U(d)* denotes the anti-Schur factor of U(d) with all repeated zeros on the unit circle replaced by simple zeros (Vidyasagar, 1991). This means that U(d)*T(d) E RA if and only if T(d) ERA. In particular, U(d)*T(d) polynomial if and only if T(d) is a polynomial. As a consequence, one can usefully consider the following substitutions in (33)

The parameterizations (19) and (30) allow one to directly construct suboptimization schemes by imposing that the closed-loop error maps are polynomials. In fact, the above conditions impose additional constraints to (4) and the corresponding solutions are of course sub-optimal. Then, any arbitrarily tight approximating solution to (4) can be obtained by solving the following finite-dimensional linear programming problem (SUP-OPT t ) for a sufficient large value for t:= 8(T): min

+ itt+!

= J.Lopt,

4. SUPEROPTIMIZATION

3. SUB OPTIMIZATION

TER.' Cd]

(32)

Further, the sequence of solutions T(t) admits a subsequence T(t,) that converges in the A-norm (component-wise) to an op~imal solution of the OPT problem as t -+ 00. IT such a solution is unique, the whole sequence converges to it.

Specifically, let (PlO, Xo) be the minimal degree solution w.r.t. X of (29), viz. 8(Xo) < 8([J-V-). Then, the general solution of (29) is given by X = Xo - [J-V-T and PI = PlO + DT in the possibly infinite degree free polynomial TEA. Finally, for any integer N, the parameterization of all admissible and A stable closed-loop error maps £ij (19) modify as Eij

~ ilt+!

t-+oo

is A-stable, viz. by requiring that P = [J-v- PI, with PI E A solution of the following Diophantine equation [J-v- PI

+ itt

lim ilt

(27)

Again, the stability issue is resolved by requiring that DQ

1

Lemma 2 - Let (A .l) be fulfilled and T(t) denote a solution of SUB-OPT t . Then, the sequence ilt is non-increasing and

-DEN. - d N + 1 i;(N)

Oi~

1

the part of the cost due to truncated amounts a link with the OPT problem is established by the following Lemma.

Next, by observing that eqs. (11)-(12) modify coherently for all integer N i = 1,2, j = 1,2, and similarly for one finally concludes that there exists a single polynomial W, independent of N, such that

.)

//D- dN + E(N) t,

IIA' 4

without loss of generality. The above discussion leads to the following finite dimensional, unconstrained, superoptimization sequence of linear programming problems in the unknown polynomials (T, T ij ), i , j = 1, 2, indexed by t = 8(T) ~ 0, for given fixed tij = 8(Tij) ~ 0 and N selected as in (31) . SUP-OPTt : ~=

min

TER' [dj ;TijER'ij [dj

Hm l!-t = J1.opt, and lim Pt = J1.opt

t~oo

t~oo

and the sequence of polynomial solutions T(t) admits subsequences T(t · ) that converges in the A-norm (component-wise) to an optimal solution of OPT as t ~ 00, while T;j) converge strongly to zero. If such a solution is unique, the whole sequence T(t) converge to it.

IltllA '

Based on Lemma 3, it is possible to use SUP-OPT t to achieve both lower and upper bounds on the optimum along with an admissible approximating minimizer. In special cases, however, the exact minimizer can be determined. In particular, this is the case when the problem has a block that is dominant, in the sense that the structural constraints corresponding to the inactive blocks can be removed without changing the solution. In such a case, SUP-OPT t is able to capture the exact finite support (rational) solution.

Remark 2 - The key feature of this approach is that the number of equations retained with respect to the number of variables considered can be modified by varying the degrees of T and Tij . In particular, if tij are too small, the required superoptimal behavior for the sequence of l!-t cannot be ensured because the quantity of constraints relaxed is not still significant if compared with the cost improvement achieved by increasing t . On the other hand, tij should be chosen as small as possible because the more the constraints are relaxed, the worse the lower bounds are achieved. Values for t ij which suffice for priming the superoptimal behavior in SUP-OPT t can be obtained by solving the latter with the extra condition T == O. In fact, in such a case SUP-OPT t reduces to four independent one-block problems in the unknowns Tij for which finite finite degree polynomial solutions Tij) always exist (Dahleh

Lelllma 4 - Let (A .l) be fulfilled and assume that the dominant block is given by £11 . Further, let [TO,t. , Ti~,t.], i = 1,2, j = 1,2 be a subsequence of solutions to SUP-OPT t . Then, limt-too T~it. = 0, Topt cancels all the stable factors in U I VI , that is ppt = TO /(Ut VIt) with TO a finite degree polynomial and v;.t denotes the Schur factor of UI VI ; each pair [To ,t., T~/'], satisfies

ut

(ut VIt) TO ,t. + dt.+IT~it. = TO ,

and P~arson, 1987), with tij = 8(T;j») . Therefore, when tij are used in SUP-OPT t , the increasing of t doesn't improve any longer the cost because the finite dimension of the admissible set is just spanned by the unknowns Tij of degree f ij and the only effect is that of adding more and more constraints. 0

(35)

(ut V/).

If the optimum for any ts > 8 (TO) - 8 is unique, the whole sequence satisfies (35) .

5. AN EXAMPLE: 2 x 2 FOUR-BLOCK MIXED SENSITIVITY PROBLEM

A further characteristic of SUP-OPT t is that a suboptimal admissible solution, in general worse than the one achieved by SUB-OPT t , is directly available. In fact, denoting as Tt + ~+ITij the optimal solution of SUP-OPT t at each t, one has that the sequence Pt of sub-optimal problems obtained by putting Tlj = 0 and computing the A-norm of t. The sequence Pt provides an upper bound to [Lt. Finally it can be shown that SUP-OPT t is well-posed for each t ~ 0 and a link with the OPT problem is established by the following result.

Consider the plant

p

=

d(d - 0.5) (d - 0.1)(1 - 0.5d) ,

(36)

and the filtering weighting functions

W d -

0.02

W _

1 _ 0.2d '

n -

0.004 1 - 0.6d

(37)

The task is to find a SISO controller C that minimize the A norm of

Lemma 3 - Let (A.I) be fulfilled. Then, provided that tij are sufficiently large, the sequence l!-t is non-decreasing and bounded from the above by J1.opt, that is

where S denotes the sentisivity, T the complementary sensitivity and M is the control effort. The term p is a scalar whose role is to properly weight the second row in the matrix (38) and in this example has been chosen equal to 0.1. We have that

Further, regardless of values used for tij , the sequences l!-t and Pt converge to J1.opt, that is

Xo(d)

5

= 6;:0030,

FlO(d)

= - 565960390 + ~~~~ d.

In table 1 the convergence of the norms is showed when the degree of T increases. Note that the optimum turns out to be j.lopt = 0.986267 as in the two block case (Casavola, 1996) because the first row is dominant. In table 2 the convergence SUP-OPT!

SUB-OPTt t

ilt

0 1 2 3

1.377954 1.245985 1.153142 1.095022

... 26 27 28 29 30

the rational solution determined. A generalization to the present approach to the general multivariable four-block problem is in progress and first results seem promising.

v 0.956980 0.968695 0 .975724 0 .979941

...

...

0 .986268 0 .986267 0.986267 0 .986267 0.986267

0 .986266 0 .986266 0.986266 0.986266 0.986267

v! 1.754364 1.713944 1.545537 1.383160

7. REFERENCES Barrodale, I. and F.D.K. Roberts (1973). An improved algorithm for discrete f1 linear approximation. SIAM J. Num . Anal. 10, 839848. Boyd, S.P. and C.H. Barratt (1991). Linear Controller Design: Limits of Performance. Prentice-Hall. Englewood Cliffs, NJ. Casavola, A. (1996) . A polynomial approach to the f 1-mixed sensitivity optimal control problem. IEEE Trans. Automatic Control 41, 751-756. Dahleh, M.A. and J.B . Pearson (1987). £l-optimal feedback controllers for mimo discrete-time systems. IEEE Trans . Automatic Contr. 32, 314-322. Dahleh, M.A. and J.B. Pearson (1988). Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization. IEEE Trans. Automatic Contr. 33,924-930. Diaz-Bobillo, I.J. and M.A. Dahleh (1993). Minimization of the maximum peak-to-peak gain: The general multiblock problem. IEEE Trans. Automatic Control 38, 1459-1482. Dugundji, J. (1966). Topology. Allyn and Bacon. Boston, MA. Elia, N. and M.A. Dahleh (1998) . A quadratic programming approach for solving the £1 multiblock problem. IEEE Trans. Automatic Control 43, 1242-1252. Kucera, V. (1979). Discrete Linear Control. Wiley. New York, NY. MacDonald, J .S. and J.B . Pearson (1991) . £1optimal control of multivariable systems with output norm constraints. Automatica 37, 317-329. Staffans, O.J. (1991). Mixed sensitivity minimization problems with rational f l-optimal solutions. J. Optimiz. Theory Appl. 70, 173-189. Staffans, O.J. (1993). The four-block model matching problem in £1 and infinite-dimensional linear programming. SIAM J. Contr. Optimiz. 31, 747-779. Vidyasagar, M. (1991) . Further results on the optimal rejection of persistent bounded disturbances. IEEE Trans. A utom. Control 36,642-652.

0.986272 0.986269 0.986268 0.986267 0 .986267

Table 1. Convergence of the Norms of the error sequences for SUP-OPT for the component (1,1) in the dominant row is shown. It's evident that the super-optimal component remains identical when t varies. The optimal Q SUP-OPT

o 2 5 8

0 .02 0 .02 0 .02 0.02

-

0.25111 0.25111 0.25111 0.25111

d + 0 .51111 d + 0.51111 d + 0.51111 d + 0.51111

d d2 d2 d2

Table 2. Convergence of the first component of the error sequences in the dominant row and C can be retrieved from the expression of [P1 in table 2 and are given by

t

0.164445 Q(d)opt = 0.02(1 - 0.5 d) , C(d)opt

(d - 2)(d - 4.991304) 5) .

= 0.5 (d _ 0.391304)(d -

Notice that the pole in d = 5 and the zero in d = 4.991304 in the controller transfer function cannot be cancelled without affecting remarkably the optimality.

6. CONCLUSIONS New sub/superoptimization schemes have been presented for the scalar £1 general four-block control problem which result less affected by unnecessary redundant constraints and hence more efficiently solvable. This has been obtained by exploiting a polynomial equation approach and the Y JBK parameterization of the admissible maps. In this way all structural and stability constraints have been taken care by the parameterization and the resulting optimization problems are unconstrained. An example has been fully analyzed and 6