7& Control with Time Domain Constraints

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1 Proceedings of the Aerican Control Conference Philadelphia, Pennsylvania June & Control with Tie Doain Constraints Mario Sznaier and Takeshi Aishia Departent of Electrical Engineering The Pennsylvania State University University Park, PA eail (sznaier, Abstract In this paper we study the proble of iniizing the 'Ha nor of a given transfer function subject to tie-doain constraints on the tie response of a different transfer function to a given test signal. The ain result of the paper shows that this proble adits a iniizing solution in Moreover, rational solutions with perforance arbitrarily close to optial can be found by constructing a faily of approxiating probles. Each one of these probles entails solving a finite-diensional quadratic prograing proble whose diension can be deterined before hand. 1 Introduction In any cases the objective of a control syste design can be stated siply as synthesizing an internally stabilizing controller that iniizes the response to soe exogenous inputs. When these exogenous inputs are assued arbitrary but with bounded energy and the outputs are also easured in ters of energy, this proble leads to the iniization of an 3t-nor of the closed loop syste. The case where the exogenous inputs are bounded persistent signals and the outputs are easured in ters of the peak tie-doain agnitude, leads to the iniization of an C1/tl-nor. X,-optial control can now be solved by elegant state space forulae [5] while L1/.tl-optial control can be (approxiately) solved by finite linear prograing [4]. Finally, the case where the input is a bounded energy signal and perforance is easured in ters of the t- nor leads to the generalized 212 proble [9], also solvable via finitediensional convex optiization. closed-loop syste to a given, fixed test input (such as bounds on the rise tie, settling tie or axiu error to a step). In this case, if the output is easured in ters of its energy the probles leads to the iniization of the closed-loop '&-nor, extensively studied in the 1960's and 1970's. On the other hand, if the outputs are easured in ters of the peak tiedoain agnitude, it leads to the iniization of C/t-nor [3, 10, 11, 12, 7, 6, 13. In general, a realistic control proble is likely to involve specifications on both the energy and peak values of the output. It is well known that, for discrete-tie stable systes, the 7-12 nor is an upper bound of the? f nor. Thus, in principle one can try to enforce restrictions on the peak value of a (weighted) tie-doain response through the iniization of a weighted 212 nor. However, this approach can be arbitrarily conservative. In this paper we propose a solution to 212 probles subject to tie doain constraints given in ters of the response to a fixed, given signal. The ain result of the paper shows that these probles adit a solution in 212. Moreover, we show that coputing a rational (and thus ipleentable) controller yielding perforance -away fro the optial can be accoplished by solving a sequence of finitediensional quadratic progras. 2.1 Notation 2 Preliinaries t1 denotes the space of absolutely suable sequences h = {hi} equipped with the nor - In any cases, following a coon practice in llhlll1 lhkl < denotes the Ba- engineering, soe of the perforance require- ents are stated in ters of the response of the Ihk I < 00. k=-w IThis work was supported in part by NSF under e denotes the space of bounded sequences grant ECS h = {hi} equipped with the nor llhlll & /98 $ AACC 3229 k=o nach space of sequences {hn} :

2 sup lh;l k/o < o. We denote by lp, the space of bounded vector sequences (h(k) E RP). In this space we define the nor 1lhllt I supilhi(k)lj~~. Given a sequence h E.!I, its i z-transfor is defined as H(z) = biz-'. In the sequel, by a slight abuse of notation we will soeties use the notation llhllt to denote Ilhllt- By 'Hz(X2') we denote the Banach space of coplex valued atrix functions G(z) with analytic continuation outside (inside) the unit disk, and square integrable there, equipped with the usual 3tz nor 11G11: G s u ~ & ~ > ~ l~(z)i$$, where ~ denotes the Frobenious nor. 72x2 denotes the subspace of 'Hz consisting of real rational transfer atrices. The projection operator P,, : 'HZ 4 R'Hz is defined by 00 i=o 7%-1 P, [G(z)] G G;z-; (2-1) i=o 2.2 The 7 l2 with tie doain constraints proble spectively, obtained when connecting a stabilizing controller fro y to U. Using the Youla Paraeterization, the set of all such transfer atrices can be paraeterized by [13] "(2) = Tii(z) + Tiz(z)Q(z)Tzi(z) S(z) = Sii(z) + ~12(z)Q(~)~zi(z) where T;,, Si, are stable transfer atrices, and Q(z) E Xz is the "free paraeter" in the paraeterization. In order to stress the dependence on Q, the notation T(Q), S(Q) is soeties used in the sequel. The paraeterization allows for precisely stating the 'H2 with tie-doain constraints proble as: Proble 1 Given sequences of upper and lower bounds (ubi) and {Ibi), find the optial value of the perforance easure: subject to lb; 5 (S * d); < ubi, i = 0,1,... and the corresponding optial controller &*. Without loss of generality (by using weighting functions and absorbing these weights in the generalized plant (see [ll] for details)) this proble can be recast into the following for: Proble 2 Find the optial value of the perforance easure: Figure 1: The 'H2 with tie doain constraints setup Lea 1 Let T12, Tzl have generically full colun and row rank respectively, and assue that a solation to Proble 2 exists. Then this solution is unique. Consider the syste shown in Figure 1, where the signal wz E RPy (white noise) and d E R represent exogenous disturbances and a known, given test signal respectively, U E RP- represents the control action, CZ E R"' and Ct E R represent regulated outputs, and where y E R". represents the easureents available to the controller. Assue that the generalized discrete-tie plant P is finite-diensional, linear tie invariant. Let T(z) and S(z) denote the closed-loop trans- fer atrices fro wz to & and fro d to &, re In the sequel we solve Proble 2 by constructing sequences of super and sub-optial controllers, {Q} and {a} respectively such that the corresponding S(Qi) satisfies IIS(Qi)lll.. < 1 and such that IIT(Qi)llz -+ p. 3 Proble Solution 3.1 Proble Transforation It is a standard result that the paraeterization of all stabilizing controllers can be selected so that Tlz, Tal are inner and co-inner respectively and such that R G T12"T11T21- E R'H;

3 [13]. Since the 7-12 nor is invariant under pre(post)-ultiplication by inner (outer) atrices, we have that subject to where G,, denotes the strictly proper part of G G R" and DO its feed-through ter. It follows that Proble 1 ay be reforulated as follows. Proble 3 Proble 3 is a convex infinite-diensional proble, for which no closed-for solution is known to exist. In this paper, a solution will be coputed by taking the liit of the solution to soe finite-diensional iniization probles. For notational siplicity, in the sequel we consider SISO systes, but the proofs generalize to the MIMO case at the price of a ore involved notation. Additionally, by redefining S11 as S11 - SlzD~Sal if necessary, we can assue without loss of generality that DG = Coputation of super-optial solutions In this section, a sequence of finite diensional convex optiization probles is introduced. The n-th proble has U(n) variables, and its optial cost pn satisfies pn 5 p. The sequence of probles approxiates Proble 1 in the sense that pn --t p and the partial solutions converge to the optial solution (in the 7-12 nor) as Using the projection operator defined in (2-1), consider the optiization proble Theore 1 Assue that there exists Q E 7-12 such that llsl+ SzQllt 5 1. Then E' t p and 119" - Q*l , where Q* E 7-12 is a solution to Proble Coputation of sub-optial solutions Theore 1 shows that a solution to Proble 2 can be obtained by solving a sequence of quadratic prograing probles. However, it does not furnish inforation on how to select n to achieve soe desired error bound. To solve this difficulty, in this section we introduce a sequence of suboptial solutions converging to the optial fro above. Consider the following finitely any variables approxiation to Proble 2: Proble 5 n-1 Theore 2 jp 1 p and I/&" - &*I , where Q* E 7-12 is the solution to Proble 2. Proble 4 subject to IIPn(S1 + S2Q)IIt Proble 4 can be thought of as a finitely-any constraints approxiation to the original proble, where the constraints are enforced only over a finite horizon n. In the sequel we show that this proble is equivalent to a finite di- In principle, Proble 5 is a sei-infinite diensional quadratic prograing proble, since it has an infinite nuber of constraints. However, as we show in the sequel, under ild conditions only finitely any of these constraints are active. Theore 3 Assue that S2 does not have any zeros on IzI = 1. Then Proble 5 is equivalent to: -n c1= in ensional quadratic prograing proble. [Qg Q;... Q2-ll i=d 3231 n-1 ~IIQTlI$

4 subject to 5 MQ, i = 0,l..., n and yield the expansion where = xiel 4ili(s) where MQ and N are constants that depend only on the proble data. 4 The continuous-tie case In this section we consider the continuous-tie counterpart of Proble 2, naely: Proble 6 Given upper and lower boundfinctions ub(t) and Zb(t), find the optial value of the perforance easure: subject to 2b(t) 5 (S * d)(t) 5 ub(t),t 2 0, and the corresponding optial controller Q*. The ain result of this section shows that this proble is equivalent to a discrete-tie proble siilar to Proble 2 that can be solved proceeding as in section 3. To establish this result, begin by introducing the Laguerre functions, defined as VQij s-a <-l, i = 1,2... Zi(S) = -- s+a L+a) where a is a positive real. It is a standard fact (see for instance 181, Chap. 18) that the faily {Zi} is an orthonoral basis in 312. Therefore, any function G(s) E 312 can be expanded as: 00 G(s) = I ili. Since these functions are or thonoral it follows that llgll: = IlI ill$. In the sequel, for siplicity, we will assue that Sl(s) is strictly proper. Since Q E 3-12 this iplies that C(&) = SI + S2Q is also strictly proper. Let SI, S2, Q, ub and Zb have the following Laguerre expansions: a=o 00 and Zb(t) 5 (S * d)(t) 5 ub(t),t 2 0. lbi 5 q5i 5 ubi,i = 1,... Hence Proble 6 is equivalent to the following discretetie proble: Proble 7 pt = in f(q) QEUa sabject to lbi 5 l&(z) +.!&(z)q(z)li l,..., where &(z) = M nl,iz-ij Clearly this proble is siilar to Proble 2 (the only difference being in the objective quadratic function) and thus can be solved using the techniques proposed in section 3. 5 Illustrative Exaple Consider the proble of iniizing the 7-l~ nor of the sensitivity function for the unstable noniniu phase syste shown in Figure 2, subject to a constraint on the peak of the control due to a unit-ipulse disturbance w. Assue that the transfer function of the Figure 2: Block diagra for the exaple %(s) = u~,di(s), &(s) = ~ Z, O uz,ik(s> plant is given by P(z) = =&. In this case the optial (unconstrained) TLz controller achieves 00 W Ub(8) = Ubili(S), Zb(s) = Ibili(5) ~/T w~~~2 = with IITvwIIl_ = Suppose that it is required that the agnitude ~($1 = e,c(s) of the control action ust reain below 6.8, i.e. [ITvw IIl l. An inner-outer factorization Straightforward but tedious coputations using Using the techniques in [3] it can be shown that the fact that I;(.) * lj(s) = (k+j-l - k +j) infq l l ~ ~ ~ = 6.75 l l t ~ 3232

5 of the plant is given by tolerance. Additional results show that the sequence of controllers thus obtained converges strongly to the optial solution. Kqt= [ loulse resonse of T w stape Figure 3: Ipulse responses of Tu, 6 Conclusions : In this paper we consider the proble of optiizing the 3tz nor of a given syste subject to additional specifications given in ters of the response to a given test signal. The ain result shows that both in the discrete and continuous tie cases this proble adits a so- lution in 7&. Moreover, suboptial solutions can be obtained by solving sequences of finitediensional quadratic prograing probles until the gap _ - between upper -_ and lower bounds of the solution is saller than a pre-specified 3233 References [I] F. Blanchini, S. Miani and M. Sznaier, Step Response Bounds for Tie-Varying Uncertain Systes with Bounded Disturbances, Autoatica, to appear. [2] Mischa Cotlar and Roberto Cignoli, An Introduction to Functional Analysis, North- Holland Publishing Copany, [3] M. A. Dahleh and J. B. Pearson, Miniization of a Regulated Response to a Fixed Input, IEEE Trans. Ado. Contr., vol. AC- 33, pp , October [4] I. J. Diaz-Bobillo and M. A. Dahleh, Miniization of the Maxiu Peak-to-Peak Gain: The General Multiblock Proble, IEEE Trans, Auto. Contr., vol. AC-38, pp , October, [5] J. Doyle, K. Glover, P. Khargonekar and B. Francis, State-Space Solutions to Standard 3t2 and 3t, Control Probles, IEEE Trans. Auto. Contr., vol. AC-34, pp , August, [6] N. Elia,, P. M. Young and M. A. Dahleh, Robust Perforance for Fixed Inputs, PTOceedings of the 3Yd IEEE CDC, Lake Buena Vista, Florida, Dec. 1994, pp [7] M. Khaash, Robust Steady-State Tracking, Proceedings of the 1994 Aerican Control Conference, Baltiore, Md., June 29- July 1, 1994, pp [8] Y. W. Lee. Statistical Theory of Counications. John Wiley, New York, [9] Rotea M., The generalized 3t2 control proble, Autoatica, Vol 29, No 2, pp , [lo] M. Sznaier and F. Blanchini, Mixed L /X, Suboptial Controllers for Continuous-Tie Systes, IEEE Trans. Autoat. Contr., 40, 11, pp , Noveber [ll] Z. Q. Wang and M. Sznaier, URational L-Suboptial Controllers for SISO Continuous Tie Systes, IEEE Trans. Auto. Contr., 41, 9, pp , Septeber [12] Z. Q. Wang and M. Sznaier, C--Optial Control of SISO Continuous Tie Systes, Autoatica, 33, 1, pp , [13] K. Zhou, J. Doyle, and K. Glover. Robust and Optial Control. Prentice-Hall, New Jersey, 1995.