Controllers for SISO Discrete Time Systems

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1 vva2-10:20 Mixed fl/?i Controllers for SISO Discrete Tie Systes Mario Sznaier Electrical Engineerin6 The Pennsylvania State University University Park, PA, eail: runaierofrodo.cce.psu.edu Abstract A successful controller design paradig ust take into account both odel uncertainty and design specifications. Model uncertainty can be addressed using either H, or 11 robust control theory, depending upon the uncertainty characterisation. However, these fraeworks cannot accoodate the realistic case where the design specifications include both tie and frequency doain constraints. In this paper we a d b these probles using a ixed h /H, approach. This approach allo for iniising the worst-case peak output due to a persistent disturbance, while, at the nae tie, satisfying an '&-nor constraint upon soe closed-loop transfer function of interest. The ain result of the paper shows that ixed 11/7i optial controllers can be obtained by solving a sequence of probles, each one consisting of a finitediensional convex optiisation and a standard, unconstrained H, proble. I. Introduction A large nuber of control probles involve designing a controller capable of stabilizing a given linear tie invariant syste while iniizing the worst case response to soe exogenous disturbances. This proble is relevant for instance for disturbance rejection, tracking and robustness to odel uncertainty (see [ll] and references therein). When the exogenous disturbances are odeled as bounded energy signals and perforance is easured in ters of the energy of the output, this proble leads to the well known 'H, theory. Since its introduction, the original forulation of Zaes [13] has been substantially siplified, resulting in efficient coputational schees for finding solutions. The 'H, fraework, cobined with p-analysis [4] has been successfully applied to a nuber of hard practical control probles (see for instance [7]). However, in spite of this success, it is clear that plain 'H, control can only address a subset of the coon perforance requireents since, being a frequency doain ethod, it can not address tie doain specifications. Recently, ethodologies incorporating soe classes of tie doain constraints into the 'H, foralis have been developed [&lo]. However, in its present for these techniques allow only for shaping the response to a given, fixed input. The case where the signals involved are persistent bounded signals leads to the I1 optial control theory, forulated and further explored by Vidyasagar [ll-121 and solved by Dahleh and Pearson both in the discrete [2] and continuous tie [3] cases. These ethods are attractive since they allow for an explicit solution to the robust perforance proble [6]. However, they cannot accoodate soe coon classes of frequency doain specifications (such as 'Ha or H bounds). In this paper we propose a ethod for designing ixed f&, controllers. These controllers allow for iniizing the 1, nor of the closed-loop transfer function between an input-ouput pair of signals, while at the sae tie satisfying an H, nor constraint upon the transfer function between a different pair of signals. Our approach resebles that of Boyd et. al. [l] in the sense that we use the Youla paraetrization to cast the proble into a seiinfinite convex optiization for. However, in a significant departure fro [ 11, where several approxiations where used in order to obtain a tractable atheatical proble, we use the special structure of the proble to find a global solution. The ain result of the paper shows that this solution can be found by solving a sequence of odified probles, each one entailing solving of a finate diensional convex, constrained optiization proble and an unconstrained 'H, proble. Moreover, the proposed solution ethod yields, at each stage, a feasible controller (in the sense of satisfying the n, constraint) that provides an upper bound on the optial 11 cost. The paper is organized as follows: In section I1 we introduce the notation to be used and soe preliinary results. Section 111contains the proposed solution ethod. In section IV we present a siple design exaple. Finally, in section V, we suarize our results and we indicate directions for future research. 11. Preliinaries 2.1 Notation f, denotes the space of bounded real sequences q = {qk} equipped with the nor IIqlil,ksup(qkl. f1 denotes the space of real sequences, equipped with the nor llqlll = lqkl < 00. t, denotes the Lebesgue space of coplex valued transfer functions which are essentially bounded on the unit circle with nor ~~!!'(z)~~& sup lt(z)l. 31, (H-) denotes the 1+1 set of stable (antistable) coplex functions G(z) E t,, i.e. analytic in lzl 1 1 ( ). The prefix 72 denotes real rational transfer atrices. Given R E L,, ru(r) denotes its axiu Hankel singular value. Given a sequence q E I, we will denote its a-transfor by Q(z). It is a standard result that q E fl iff Q(z) E 'RH. Throughout the paper we will use packed notation to represent state-space realizations, i.e. k k=o Thia work was supported in part by NSF under grant ECS and by a grant fro Florida Space Grant Consortiu /93/$ IEEE 34

2 Given two transfer atrirrs T = (2; 2')and Q with appropriate diensions, the lower hear fractional transforation is defined as: &(T,Q)$Tii + TiZQ(I - TzzQ)-lTz~ Finally, for a transfer atrix G(z), G-kG((;), where ' denotes transpose. 2.2 Stateent of the Proble Consider the syste represented by the block diagra 1, where S represents the syste to be controlled; the scalar signals w (a bounded energy signal), w1 (a persistent I, signal) and U represent exogenous disturbances and the control action respectively; and (, (l and y represent the regulated outputs and the easureents respectively. Then, the ixed l,/~, control proble can be stated as: Given the noinal syste (S), find an internally stabilizing controller and where F and L are selected such that A t B3F and A + LC, are stable. By using this paraetrization, the scalar closed-loop transfer functions 'Tcww- and T,,,, can be written as: T L (2) ~ = ~ Ti(,) t TF(z)Q(z) T~w>(z) = Ti(z) + T?(z)Q(z) where T,, T,,, Q are stable transfer functions. Moreover (see [lo, 14]), it is possible to select F and L in such a way that Tz(z) is inner (i.e. Ta-Tz = I). By using this paraetrization the ixed ll/?t, proble can be now precisely stated as solving: subject to: IITi(z) + TY(z)Q(z)ll 5 7 (3) where {ti} and {qi} are the coefficients of the ipulse responses of T(,,, and Q respectively. (2) such that worst case peak aplitude of the perforance output llclll due to signals inside the 1,-unity ball is iviized, subject to the constraint llt~=~,il, 5 7. q s Figure 1. The Plant 2.3 Proble Transforation Assue that the syste S has the following state-space rea1izati. (where without loss of generality we assue that all weighting factors have been absorbed into the plant): where D13 has full colun rank, D31 has full row rank, and where the pairs (A, B3) and (C3, A) are stabilizable and detectable respectively. It is well known (see for instance [14)) that the set of all internally stabilizing controllers can be paraetrized in ters of a free paraeter Q E 'H, as: K = &(J, Q) (1) where J has the following state-space realization: Y 111. Proble Solution In this section we show that the ixed 11/%, proble can be solved by solving a sequence of probles, each one requiring the solution of a finite diensional convex optiization proble and an unconstrained H proble. 3.1 A Modified llj?t Proble Since all the solutions to a suboptial Nehari extension proble of the for JIR + Qll- 5 7 can be paraetrized in ters of a free paraeter W(z) E ZH,,I(W~I, 5 7-l proble 11/%, can be thought of as an optiization proble inside the origin centered 7-1-ball. However, even though the space 31, is coplete, it is easily seen that the 7-ball is not copact. Thus a iniizing solution ay not exist. Motivated by this difficulty, we introduce the following odified ixed ll/h proble. Let?t,,6 = {Q(z) EH,: Q(z) analytic inlzl~ 6) and define the 11/~,6 proble as follows: Given Tl(z), TZ(z), Tl(z), T2-(z) E RH,s, find P; = Qj$,s IITLw,lll (11/x,6) subject to: IIT?(z) + 'T?(z)&(z)ll.S 5 7 where 6 < 1 and ~ ~ Q sup ~ \&(z)l. ~,, ~ ~ 1z1=6 Reark 1: Fro the axiu odulus theore, it follows that any solution Q to Z1/H,6 is an adissible solution for ll/?f. It follows that p; is an upper bound for pa. Reark 2: Proble 11/%,6 can be thought as solving proble Z1/%, with the additional constraint that all the poles of the closed-loop syste ust be inside the disk of radius 6. A paraetrization of all achievable closed-loop transfer functions, such that T, T satisfy this additional constraint can be obtained fro (1) by siply changing the stability region fro the unit-disk to the &disk using the transforation z = 62 before perforing the factorization. Furtherore, by cobining this transforation 35

3 with the inner-coinner factorization, the resulting Ta(z) satisfies Ta(6z)Ta(&) = 1. Next we show that a suboptial solution to l1/'h, with cost arbitrarily close to the optiu, can be found by solving a sequence of truncated probles, each one requiring consideration of only a finite nuber of eleents of the ipulse response of TC,~~. To establish this result we will show that: i) ll/'h can be solved by considering a sequence of odified probles 11/~,6. ii) Given e > 0, a suboptial solution to ll/'h,' with cost no greater than pi + e can be found by solving a truncated proble. Lea 1: Consider an increasing sequence 6; Let po and b denote the solution to probles ll/'h and 11/n,64 respectively and assue that rh(taw-tl) < 7. Then the sequence b -+ po. Proof: Fro the axiu odulus theore it follows that the solution Qi to 11/'H,6, is a feasible solution for 11/~,6.+,. Thus, the sequence p; is non-increasing, bounded below by the value of IITclwllll obtained when using the optial I, controller. It follows then that it has a liit p 2 p". We will show next that p = pa. Assue by contradiction that pa < p and select pa < b < p. Let ReTaT1. Since rh(r) < 7, it follows that there exists Q1 E 'Rn, such that IIR + Qlll = rh(r) < 7. Fro the definition of po it follows that, given 9 > 0, there exists Q. E 'R'H, IIR + Qollw 5 7, such that ll~clw,(qo)lll I po +q. Let QAQ. + e(ql - Q.). It follows that: lltcwt(q)lli L. P" + + elita(q1 - Qo)IIi IIR + Qll I eiir + Qlll + (1 - e)iir + Qoll < 7 Since 0 E 'R'H, it follows that there exists 6, < 1 such that + TaQ is analytic in IzI 2 &. Since IITp" + TaQ1l < 7, it follows fro continuity that there exists 61 < 1 such that lltl + TaQll,(, 5 7. Therefore, by taking e and 9 sall enough and 6A ax { 6,, 6a} < 1 We have that IITi + TaQI(,6 I 7 and IITcxwl(Q)lli < b. Hence for 6i 2 6, b 5 ji. However, this contradicts the fact that the sequence p, is non-increasing and that ji < p = lip, 0. 6<-1 Next we show show that, given E > 0, a suboptial solution to l1/?ts6, with cost pi such that pi I p; I p; + e can be found by solving a truncated proble. Lea 2: Let e > 0 be given. Then, there exists N(e,6) such that if Q E 'H,6 satisfies the constraint IIR + Q then it also satisfies Itkl I E, where i=n tk denote the codcients of the ipulse response of Tc,,, = Ti + TaQ. ProoE Since Q E 'H,6, Hence Tt,,, is analytic in IzI 2 6 and: (4) (5) Q"'(qo... qn-1), and where qk, tki denote the kth eleent of the ipulse response of Q(z),Ti(z) respectively. Let Q* and T;lwl denote the optial solution and define pi = ~~T~lw,lll. Then pi I pi I pi + E Proof: pi pi is iediate fro the definition of pi. Denote by T[s,l and T,61,,the solution to probles ll/'h;*6 and 11/?is6 respectively and let tf, tf be the corresponding ipulse responses. Then: N-1 W = ltfl+ lttl i=o i=n N-1 By cobining the results of Leas 1, 2 and 3, the following result is now apparent: Lea 4: Consider an increasing sequence 6; Let pa and pi4 denote the solution to probles ll/'h and respectively. Then the sequence pii has an accuulation point b. such that p* 5 b. I po + E. 3.2 The 31, Perforance Constraint In the last section we showed that ll/'h can be solved by solving a sequence of truncated probles. In principle these probles have the for of a sei-infinite optiization proble, and can be approxiately solved by discretizing the unit-circle and applying outer approxiation ethods (see [5]). In this section we show that each proble ll/~k,6 can be ezactly solved by solving a finite diensional convex optiization proble and an unconstrained 'H, proble. Moreover, since the solution to this 'H, approxiation proble is rational, it follows that the solution to ll/~;,6 is also rational. To establish this result, we recall first a result on constrained Nehari approxiation probles: 36 7

4 N-1. Theore 1: Let R E 'R?i& and Q F = qiz-' be given. Then there exist QR E anh,, z - ~ Q R ~ I, I 7, iff llqlln 5 7 where: x = Lh y = LB,=O such that IIR + QF + and where L, and Lo are the discrete controllability and observability graians of G, i.e they satisfy the discrete tie Lyapunov equations L, = AGL,Ab + bcb" and Lo = A',L,Ac CO. Proof: See Theore 2 in [lo] or the corollary to Theore 3 in 181. Cobining Lea 3 and Theore 1 yields the ain result of this section: Theore 2: A suboptial solution to l1/%,,s, with cost pa 5 pi 5 pa +e is given by &" = &; + rn&g where N- 1 QF =,E qiz-i, 40 = ( q,,... Q ~ )' - solves ~ the following,=0 finite diensional convex optiization proble: (9) and Qk solves the unconstrained approxiation proble Qg(z) = arginiir(z) + Q> + Z-NQR(Z)II,6 QItE'H,,* = argin IIR(z) + Q; + z -~QR(z)I(,,~ Qtl ~n,.r (10) where R = TF-T~D, _tl,r are defined in (7) and N is selected according to Lea 2. to the optiu, can be found using the following iterative algorith. 0) Data: An increasing sequence 6, - 1, e > 0, v > 0. 1) Solve the unconstrained 11 proble (using the standard 11 theory [21)., Copute ltc,w,l. If lltc-w~ll I 7 stop, else set z = 1. 2) For each a, find a suboptial solution to proble ll/?i&, proceeding as follows: 2.1) Let z = 6ii and consider the syste S(i) 2.2) Obtain T,(i), Ti(;) using the Youla paraetrization (1). 2.3) Copute N fro Lea ) Find Q(i) using Theore 2. 3) Let Q = Q(f),K = FdJ,Q). Copute lltc~w~(z)ll. If IIT<y(z)ll v stop, else set a = i + 1 and go to 2. Reark 4: At each stage the algorith produces a feasible solution to l,/?i, with cost p, which is an upper bound of the optial cost po. IV. A Siple Exaple Consider the plant used in [2]: 22 P(2) = 22+2 and assue that the objective is to design a copensator K to iniize IlTlll = ((PK(1 + PK)-llll subject to the constraint llsll = ll(l + PK)-lIl 5 y*. It is shown in [2] that the optial 1, controller is: IC1 = yielding lltlll = 2 and 151, = 3. Table 1 shows a coparison of the optial 1,, optial n, and a ixed l$+ controller (corresponding to 7 = 2.4). The corresponding ipulse and frequency doain response are shown in Figure 2. Reark 3: By using the transforation z = 62 we have that: llr(z) + &;(z) z-n&r(z)ll-,d = llr(6i) + Q;(62) + 6-Ni-NQ~(6i)ll 'llk(i) + &(%) f i-nqr(i)ll = l\in(k(i) + &(i)) + QR(S)(l where we used the fact that in is inner in?i. It follows that the approxiation proble (10) is equivalent to the following unconstrained Nehari approxiation proble: 3.3 Synthesis Algorith Cobining Theore 2 and Lea 4, it follows that a suboptial solution to ll/?i, with cost arbitrarily close The ixed I1 f?i, design requires a 44th order controller, since it can be shown that, for E = 0.001, we need to consider only 20 eleents of the ipulse response. Noteworthy, using odel reduction techniques, we were able to obtain a second order controller, with virtually no perforance loss. The state-space realization of this non-iniu phase second order controller is given by: I ( ' I 1 \ I O J K(z)= 1 0 * The optial?i, controller, K(z) = e, yields 151, = 2. Thus 7 should be

5 [3]. M. A. Dahleh and J. B. Pearson Ll-Optial Copensators for Continuous-Tie Systes, IEEE Trans. Auto. Contr., 32, pp , October No. of Saples [4]. J. Doyle Analysis of Feedback Systes with Structured Uncertainties, ZEE Proceedings, Part D, Vol 129, pp , Saulbviw ,..... I ,.. 1 [5]. C. Gonzaga and E. Polak On Constraint Dropping Schees and Optiality Functions for a Class of Outer Approxiation Algoriths, SIAM J. on Contr. and Opt., Vol 17, 4, pp , Figure 2. Ipulse and Frequency Responses for the I,, I&,, and n, Controllers V. Conclusions In this paper we present a ethod for designing discretetie ixed 11/ controllers. These controllers allow for iniizing the worst case output to persistent bounded excitations, while at the sae tie satisfying a constraint upon the H, nor of the transfer function between a different pair of signals. Thus, they can be thought of as achieving robust stability subject to a noinal perforance specification. Although here we considered only the sipler case of a SISO syste, the proposed design procedure can be easily extended to MIMO systes by using an ebedding procedure to deal with the x, constraint, as proposed in [9]. Perhaps the ost severe liitation of the proposed ethod is that ay result in very large order controllers (roughly 2N), necessitating soe type of odel reduction. Note however that this disadvantage is shared by soe widely used design ethods, such as p-synthesis or 1, optial control theory, that will also produce controllers with no guaranteed coplexity bound. Application of soe well established ethods in order reduction (noteworthy, weighted balanced truncation) usually succeed in producing controllers of anageable order. The exaple of section 4 suggests that substantial order reduction can be accoplished without perforance degradation. Research is currently under way addressing this issue. [6]. M. Khaash and J. B. Pearson Perforance Robustness of Discrete-Tie Systes with Structured Uncertainty, ZEEE Dans. Auto. Contr., 36, pp , April [7]. S. Skogestad, M. Morari and J. Doyle Robust Control of Ill-Conditioned Plants: High-Purity Distillation, IEEE Truns. Auto. Contr., 33, 12, pp , 1988 [8]. A. Sideris and H. Rotstein Single Input-Single Output 3t, Control with Tie Doain Constraints, Autoatica, July 1993, to appear, also in Proceedings of the 2Ph ZEEE CDC, Hawaii, Dec. 5-7, 1990, pp [9]. A. Sideris and H. Rotstein MIM0?t Control with Tie Doain Constraints, Proc. 31. ZEEE CDC, Tucson, Az, Deceber 16-18, 1992, pp [lo]. M. Sznaier A Mixed l,/ & Optiization Approach to Robust Controller Design, PTOC 1992 ACC, Chicago, 11, pp [ll]. M. Vidyasagar Optial Rejection of Persistent Bounded Disturbances, IEEE Trans. Auto. Contr., 31, pp , June [12]. M Vidyasagar Further Results on the Optial Rejection of Persistent Bounded Disturbances, IEEE Trans. Auto. Contr., 36, pp , June [13]. G. Zaes Feedback and Optial Sensitivity: Model Reference Transforations, Multiplicative Seinors and Approxiate Inverses, IEEE Trans. Auto. Contr. Vol26, 4, pp , [14]. K. Zhou and J. Doyle Notes on MIMO Control Theory, Lecture Notes, California Institute of Technology, References [l]. S. Boyd et. al. A New CAD Method and Associated Architectures for Linear Controllers, IEEE Trans. Autoat. Contr., Vol 33, 3, pp , [2]. M. A. Dahleh and J. B. Pearson I1-Optial Feedback Controllers for MIMO Discrete-Tie Systes, IEEE Tw. Auto. Contr., 32, pp , April n