optimal control problem for descriptor systems

Proceedings of the American Control ConferenceChicago, Illinois June 2000Continuous-time Hz Optimal Control Problemfor Descriptor SystemsJoiio Yoshiyuki Ishihara and Marco Henrique TerraElectrical Engineering Department - EESC - University of Sa0 Paul0 - BrazilE-mails: ishihara, terra@sel.eesc.sc.usp.brAbstractIn this paper we consider the full information and the semistatefeedback H2 control problems for continuous- timedescriptor systems. An explicit characterization of alloptimal controllers has been given for these cases. Wepresent also a parametrization of all optimal static anddynamic controllers for the LQR problem via semi-statefeedback. We show that, for the general semi-statefeedback H2 optimization, the usual gain matrix defined asa linear function of a solution of a generalized Riccatiequation can be not H2-optimal. This contrasts with theclassical results in LQR theory for descriptor systems andwith the results of H2 theory in regular state-spaceframework.1. IntroductionThe H2 control problem and its stochastic interpretation,the linear quadratic Gaussian control problem has beenextensively studied in the literature since the 1970's (seee.g. [7]). The extension of LQR theory for descriptorsystems has been appeared from the early 1980's (see [9],PI, PI, PI, 111, [61, PI, [151). Recently, the H2 outputfeedback control problem for descriptor systems was solvedby Takaba and Katayama [12]. They reduced the Hzcontrol problem to a model matching problemwhere the H2 norm is defined asand solved this problem using inner-outer factorization.However, one can see that for the semi-state feedback plant,the rank condition on 121, assumed by them to perform aco-inner-co-outer factorization, is quite restrictive since itrequires that the number of disturbance signals can not beless than the number of semi-state variables. Moreover, forthe full information system, this rank condition is not0-7803-551 9-9/00 $1 0.00 0 2000 AACC1894satisfied. This shows that although the full information andthe semi-state feedback plants can usually be considered asspecial cases of the output feedback plant, for H2optimization purposes they require a separated study. Thefull information plant has received a great attention in theliterature on control problems for explicit systems once,between all output feedback plants, it requires the weakestassumptions and is also useful to solve the general outputfeedback problem (see [16] and references therein). Thesemi-state feedback Hz control problem deserve interestonce it can be considered as a generalization of the LQRproblem. In this paper, we solve these two problems andgive an explicit characterization of all optimal controllersusing one generalized algebraic Riccati equation (GARE).For the semi-state feedback case, we show that the usualgain matrix defined as a linear function of a solution of theGARE can be not H2-optimal. This contrasts with theclassical results in LQR theory for descriptor systems andwith the results of H2 theory in regular state-spaceframework. We present also a parametrization of alloptimal static and dynamic controllers for the infinityhorizon LQR problem via semi-state feedback. Thepresented static controllers parametrization is an alternativeto the parametrizations given by [ 11 and [ 151.This paper is organized as follows. In Section 2 westate the Hz control problem. In Section 3 we parametrize,for full information and semi-state feedback plants, allclosed loop transfer matrices in RH2 and parametrize alsothe corresponding controllers. In Section 4 we give theparametrization of all full information and semi-statefeedback H2-optimal controllers. We give also a necessaryand sufficient condition under with the usual gain matrixdefined from the solution of the GARE is in fact optimal.Although the GAM can not furnish all optimal gainmatrices for the LQR problem, the specialization of semistatefeedback results to this problem enables us todetermine all optimal static and dynamic controllers.2. Problem FormulationConsider the feedback system in Fig. 1, where thegeneralized plant G(s) is a matrix of rational functions of s

and is described by a descriptor system (1). The H2 controlproblem is to find a rational controller K(s), in descriptorform, such that(i) the feedback system of Fig. 1 is internally stabilizing(ii) the closed loop transfer matrix from w to z, T,, =Gl1 + G&(I - G22K)-lG21, has minimum H2 norm11 Tzw I 12.intemally stabilizing controller to the closed loop system ofFig. 1. Under these conditions there exist matrices F and Lsuch that (E, A + B2F) and (E, A + LC2) are regular,stable and impulse-free. The condition (iv) is assumed,without loss of generality, in order to simplify 'thecomputations.As usual, the controller parametrization can beobtained via doubly coprime factorization technics([3],[13]). For the following two theorems we define thetransfer matrices :UFig. 1. Feedback system.The purpose of this paper is to solve the H2 controlproblem for the full information (FI) and semi-statefeedback (SF) cases.3. Determination of All Closed Loop Transfer Matricesin RH2In this section we consider the systemwherea: E Rn,w E Wd, U E W", t E WPandy E WQ, are thesemi-state, the exogenous disturbance, the control signal,the controlled output and the measured output respectively.We assume also that E E W""", ranlcE = r andi) (E, A) is regular, that is, det(sE - A) # 0 (1.2)ii) (E, A, Bz) is finite dynamics stabilizable and impulsecontrollable (1.3)iii) ,( C2, E, A) is finite dynamics detectable and impulseobservable (1-4)iv) 022 = 0. (1.5)With condition (i), we can define the transfermatrices from (w,u) to (z,y). We denote the transfermatrix of the generalized plant (1) byThese matrices Gi, are rational but not necessarily proper.The conditions (ii) and (iii) guarantee the existence of anTheorem 1. (The full information case). Consider the fullinformation plant1E.I A] B1 B2Dll D12where (E, A) is regular and (E, A, B2) is finite dynamicsstabilizable and impulse controllable. For any F such that(E, A + B2F) is regular, stable and impulse-free and any@ E Wmd, we have that all admissible closed loop transfermatrices can be expressed asand all stabilizing controllers are described asTheorem 2. (The semi-state feedback case). Consider theplantGS, - [yy$+1895

where (E,A) is regular and (E,A, B2) is finite dynamicsstabilizable and impulse controllable. For any F such that(E, A + B2F) is regular, stable and impulse-free, we havethat all stable closed loop transfer matrices are given byand the set of all controllers internally stabilizing the plantis given bywhich there exists Q E RH, such that Tzw(Gs~) E RH2K = F+ (I, + QD~~,(SE-A-B~F)-'B~)-*QD~~,(2) are given byi) Ker Bzl C KerDllwhere Q E RH,,det(sE-A - B2(F - QD21,)) # 0 andCP is an arbitrary matrix of compatible dimension.0We consider now the characterization of the set of allstrictly proper admissible transfer matrices for the fullinformation and semi-state feedback plants, i.e. we searchfor conditions under which T,,(G~F), T,,(GFI) E RH2.For this, consider the SVD coordinate system [ 11 such thatThere exists Q E RH, such that Tzw(Gs~) E RH2 if andonly if there exists 0 = [ 01 0 2 ] satisfyingGF(OO) + UF(OO)OVF(O~) = 0or equivalently,[D12 - (C12 + D12fi)(A22 + B22f~)-~B221.@z(Azz + B22f2m21= [Dll - (C12 + D12f2)(A22 -I- B22f2)-1B211. (4)Lemma 3, The necessary and sufficient conditions underLemma 4. The necessary and sufficient condition for whichthere exists @ such that GF E RH2 is given by thecondition (ii) of the Lemma 3. In this case, choosingCP = @ I, where CP1 is solution ofwe have that all Tzw(G~~)E RH2 are parametrized by :where C E R'"' is diagonal and positive definite andU = [ 21, V E Etnzn are orthogonal. For F E Etmzn wewrite accordingly FV = [ f1 f2]. With D212 = VT weobtainIf the conditions (i) and (ii) of Lemma 3 hold, then we canchooseIn this case, all TZ,(Gs~) E RH2 are parametrized by :v, = [P!g/?zp]E I All + B12fll A12 + &2f2I B111where Q E RH,, &(CO) = [ 01 021 with 01 free and0 2 solution of (4), det(sE-A - B2(F - QDzl,)) # 0.4. The H2 Optimization ProblemIn this section we will solve the H2 optimization problemfor the full information and the semi-state feedback plants.We will consider the following generalized Riccati equation(GARE) :ETX = XTEATX + XTA + CTCi - (CTD12 + XTB2).(7.1)1896

a stabilizing solution. Then the minimal value of norm HZof the closed loop system is given by :We say that the GARE has a stabilizing solution if thereexists X satisfying (7) such that (E, A + B2F2) is regular,stable and impulse free. A sufficient solvability conditionfor the GARE (7) is given by Lemma 4 of [ 121.4. 1. The full information H2 optimization problemThe necessity of studying this particular plantseparately results from the fact that the FI plant studied heredoes not fit the assumptions made to the output feedbackplant studied in [12]. In fact, their assumption (CAO) forDZl can not be satisfied even rewriting the system, since thenumber of measured signals is not lower than the number ofdisturbance signals.Theorem 5. Assume that the FI plant of Theorem 1 satisfy(ii) of Lemma 3 and is such that the GARE (7) has astabilizing solution. Consider F2 given by (8) and Q1solution of (5). Then the minimal value of norm H2 of theclosed loop system is given by :whereFurthermore, the class of all optimal stabilizing c&trollersis given byK = zii + z12(L - Q1Z22)-1Q1Z21whereQ1 E RH,, det(1, - QlZ22) # 0. In particular, forQ1 = 0, we have uOpt = F2x + @lw. 04. 2. The semi-state feedback H2 optimization problemOne can obtain an HZ optimal controller to the SFcase as a particular result of the output feedback plantstudied in [12]. However, their assumption (CA9 is quiterestrictive since it is equivalent to the condition B1 with fullrow rank, that is, it requires more disturbance signals thansemi-state variables. Also, the derivation of all optimalcontrollers is not direct in that context.Theorem 6. Assume that the SF plant of Theorem 2 satisfy(i) and (ii) of Lemma 3 and is such that the GARE (7) haswhere- [E I A+BzF21 IGF? =Cl + D12Fz I Dll B1+B2@2 + D12@2 1 .Furthermore, the class of all optimal stabilizing controllersis given byKopt = Fz + (I + (Q + WII)B2)-'..(Q + WlJ)(sE - A - B2F2) (9)whereFZ is given by (8)II := (I- BIB:), Bf is the Moore-Penrose inverse of B10 2 solution of (4) with F = F2Q := [ 0 02(A22 + BZ2f2)-' ]U, U is the orthogonalmatrix used in the SVD (3)a2 = 02G422 + B22f2PB21W E RH2 and Q are such thatdet(l+ (9 + WII)B2) # 0 0It is known that for the solution of LQR problem,alternative GAREs could be used (see e.g. [8], [4]). Oneinteresting consequence in adopting the GARE (7) is that,in the general semi-state feedback H2 problem, there aresituations where we can not use U = F2x =- ( D:ZD~~)-'(DT~C~ + BFX)~as an optimal controlsignal. In the following lemma we state a necessary andsufficient condition to the plant under which we have F2 asthe usual optimal feedback gain.Lemma 7. A necessary and sufficient condition under whichwe can choose Kopt = F2 is GF?(co) = 0. A sufficientcondition for this is given by Dll = 0 andBTKerET = (0). 0Another consequence in adopting the GARE (7)becomes clear when we specialize the results to the usualLQR problem ([l], [ti]).E&(t) = AX(^) + Bzu(t), Ex(O-) = EXO (10.2)Theorem 8. Consider the LQR problem (10) with thefollowing assumptions(i) (E, A) is regular1897

(ii) (E, A, B2) is finite dynamics stabilizable and impulsecontrollable(iii) the GARE (7) has a stabilizing solutionIn this case, all optimal controllers u(t) can be obtained bythe Laplace inverse of U(.) = K(s)z(s) whereK(s) = F2 - (I + (Z + W(S)UT)U~B~)-'..(E + W(S)U,T)U~(A + B2F2)andW(s) E RH2, E E Rm"(n-') are such that# odet(I + (Z + WU,T)U~B~)F2 is given by (8)U = [ 21 is the orthogonal matrix used in the SVD (3).In particular, all static optimal controllers are parametrizedasK = F2 - (I + EU2&)-'E:V2(A + B2F2), (1 1)z E R"+-'), det(I + EU~B~)# 0.Furthermore, the optimal cost is given byUThe parametrization in Z, (11), of the set of all optimalstatic controllers to the LQR problem is an alternative to thecharacterizations given by [ 13 and [ 151. Also, (11) confirmthe statement of [ 11 p. 686 that: "There is a linear variety ofoperators relating the control ~ (t) to the descriptor variabletrajectory z(t); the dimension of this variety is mz(n - T),the rank defficiency of E times the number of inputs (thedimension of the matrix S)".5. ConclusionsIn this paper we have solved the H2 controlproblem for descriptor systems in two particular cases: theFI and the SF cases. An explicit characterization of alloptimal controllers has been given for the FI, SF and for theLQR problem. We have shown that, in the SF case, theusual gain matrix F2 defined as a function of a solution of aGARE can be not optimal. This contrasts to the classicalresults in LQR theory for descriptor systems ([l], [14],[15]) and the results of H2 theory to regular state-spaceplants ([16]).AcknowledgmentsThis work was supported by FAPESP (Slo Paul0 StateResearch Council) under grant 98/12113-2.ReferencesD. J. Bender and A. J. Laub, "The linear-quadraticoptimal regulator for descriptor systems," IEEE Trans.Automat. Control, 32, pp. 672-688, 1987.D. J. Cobb, "Descriptor variable systems and optimalstate regulation," IEEE Trans. Automat. Control, 28,pp. 601-611, 1983.B. A. Francis, A Course in H, Control Theory.Springer -Verlag, Berlin, 1987.T. Katayama, and K. Minamino, "Linear quadraticregulator problem and spectral factorization forcontinuous-time descriptor systems," Proc. 31st IEEECon$ on Decision and Control, pp. 3304-3305, 1992.V. Kucera, "Stationary LQG control of singularsystems," IEEE Trans. Automat. Control, 31, pp. 31-39, 1986.V. Kucera, "The rational H2 control problem," Proc.31st IEEE Con$ on Decision and Control, pp. 3610-3613, 1992.H. Kwakemaak, and R. Sivan, Linear OptimalControl Theory. Wiley, New York, 1972.F. L. Lewis, "Preliminary notes on optimal control forsingular systems," Proceedings of 24th IEEEConference on Decision and Control, pp. 266-272,1985.L. Pandolfi, "On the regulator problem for lineardegenerate control systems," J. Optim. Theory Appl.,33,241-254, 1981.C. R. Rao and S. K. Mitra, Generalized Inverse ofMatrices and its Applications. John Wiley & Sons,1971.M. A. Rotea, and P. P. Khargonekar, "H2-optimalcontrol with an H,-constraint: the state feedbackcase," Automatica, 27, pp. 307-3016, 1991.[12] K. Takaba and T. Katayama, "H2 output feedbackcontrol for descriptor systems," Automatica, 34, pp.841-850, 1998.[13] M. Vidyasagar, Control System Synthesis: AFactorization Approach. M.I.T. Press, Cambridge,MA, 1985.141 Y.-Y. Wang, P. M. Frank and D. J. Clements"Robustness properties of the linear quadraticregulators for singular systems," IEEE Trans.Automat. Control, 38, pp. 96-100, 1993.151 H. Xu and K. Mizukami, "The linear-quadraticoptimal regulator for continuous-time descriptorsystems: a dynamic programming approach," Int. J.Systems Sci, 25, pp. 1889-1898, 1994.[16] K. Zhou, J. C. Doyle and K. Glover, Robust andOptimal Control. Prentice-Hall, Englewood Cliffs,New Jersey, 1995.1898