A NEW OPTIMAL ROOT LOCUS TECHNIQUE FOR LQR DESIGN

A.A. Mohammad

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  40. [40] P.K. Aghaee et al., Principle of frequency-domain balancedstructure in linear systems and model reduction, Computersand Electrical Engineering, 29(3), 2003, 463–477.AppendixProof of Theorem 1: The weighting matrix Q issymmetric positive semi-definite as it is the sum of thesymmetric positive semi-definite matrices αCTC andα2WOBR−1BTWO. This means that Q can be factoredout as Q = QT/2Q1/2, where (A, Q1/2) is guaranteed to beobservable as will be shown. If (A, B) is a controllablepair, Lyapunov stability theory guarantees closed loopsystem stability and the uniqueness of the positive definitesolution of the ARE. In this case, the ARE solution ispredetermined and is given by P = αWO. To see this,consider the performance index given by (2). Substitutingfor Q from (15), the resulting performance index becomes:J =∞0(XT[αCTC + α2WOBR−1BTWO]X + UTRU) dt.Substituting U = KX, we get:J =∞0(XT[αCTC + α2WOBR−1BTWO + KTRK]X) dt(A.1)Choose a Lyapunov function of the form V (X, t) = XTPX,so that:˙V (X, t) =ddt(XTPX)= −XT(αCTC + α2WOBR−1BTWO+ KTRK)X (A.2)According to the second theory of Lyapunov [9], if thereexists a positive definite P satisfying (A.2), the closed loopsystem (A + BK) will be stable. Substituting (A.2) into(A.1) results in:J = −∞0ddt(XTPX) dt = −XTPX|∞0= XT(0)PX(0) − XT(∞)PX(∞) (A.3)If (A + BK) is stable, X(∞) → 0 and the cost reduces toJ = XT(0)PX(0). The control law that minimizes J whilestabilizing the closed loop system is found by using systemequation (1) and a control law as given by (3) into (A.2),one gets:˙V (X, t) = ˙XTPX + XTP ˙X= (AX + BKX)TPX + XTP(A + BK)X= XT[(A + BK)TP + P(A + BK)]X= −XT(αCTC + α2WOBR−1BTWO+ KTRK)XXT[(A + BK)TP + P(A + BK)]X= − XT(αCTC + α2WOBR−1BTWO+ KTRK)XSince this must hold for all X, we have:(A + BK)TP + P(A + BK)= −(αCTC + α2WOBR−1BTWO + KTRK)ATP + PA + (BK)TP + PBK + αCTC+ α2WOBR−1BTWO + KTRK = 0 (A.4)Assuming R to be symmetric positive definite, R canbe factored as R = RT/2R1/2. Using this, (A.4) can berewritten as:ATP + PA + [R1/2K + R−1/2BTP]T[R1/2K+ R−1/2BTP] − PBR−1BTP= −αCTC − α2WOBR−1BTWO (A.5)To minimize J, it is necessary to make P minimum. Thisis achieved by making the first quadratic term in (A.5)equal to zero. Thus: [R1/2K + R−1/2BTP]T[R1/2K +R−T/2BTP] = 0. This results in the control matrix:K = −R−1BTP. Thus, (A.5) reduces to:22ATP + PA − PBR−1BTP= −αCTC − α2WOBR−1BTWO = −Q (A.6)By the second theory of Lyapunov, we know that fora given Q, there exists a unique positive definite solutionfor (A.6) if and only if (A, Q1/2) is observable. Thus, anypositive definite solution will be unique. By inspection,therefore:P = αWO (A.7)is the unique positive definite solution for (A.6). To showthis, substituting (A.7) into (A.6), results in:αATWO + αWOA − α2WOBR−1BTWO= −αCTC − α2WOBR−1BTWOATWO + WOA = −CTC.This is guaranteed to have a unique positive definite solution because we assumed A to be a stable system matrix. doi:10.1016/S0045-7906(01)00045-3

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