A gradient flow approach to decentralised output feedback optimal control

This paper visits the quadratic optimal control problem of decentralised control systems via static output feedback. A gradicrrt flow approach is introduced as a tool to compute the optimal output feedback gain. Several nice properties are revealed concerning the convergence of the gain matrix along the trajectory of an ordinary differential equation obtained from the gradient of objective cost, i.e. the objective cost is decreasing along this trajectory. If the equilibrium points are IsoIated, the convergence can be guaranteed. A simulation example is given to illustrate the effectiveness of this approach, K(,j}tords; Output feedback; Optimal control; Decentralised controi; Gradient flow; Metric 1. hrtroduction and preliminary results The problem of the linear quadratic optimal control problem is well understood for centralised control. For example, see [1]. However, theories are not complete for application to large scale systems using decentralised control strategy. In such cases, the full state of the system is not available at each control station, and the controller of necessity can feedback only locally available outputs to locally available controls. Such controllers can be structured as static decentralised output feedback controllers. Even for a centralised time-invariant linear system, the computation of the quadratic optimal static output feedback gain is not a simple problem, see [2,4,6, 7’]. The approaches in these references are iterative in nature :Ind require intricate adjustment of step size, except the gradient flow approach in [7]. In this paper, we will apply the gradient approach to the decentralised control systems. For simplicity the approach is developed on two station decentralised control systems. It is straightforward to extend all results to the decentralised control systems with more control stations. Consider the following two station decentralised control system: i = Ax + Blul + &u2;, (la) -n = Clx, (lb) Y2 = C2X, (It) “correspondingauthor. 224 Danchi Jiang, J. B. Moore 1Systems & Control Letters 27 (1996) 223-231 where x E Rn, ul E Rml, u2 E IW, yl E R’{, yz E R’z, and A, BI, B2, Cl, C2 are constant matrices with appropriate dimensions. Assume also that the system ( 1) has no fixed unstable modes, see [5]. Our target is to find a static decentralised output feedback controller ul =Flyl, (2a) U2= F2 y2 (2b) such that the following performance index, J(UI, U2)=E / ‘[XTQX + u; f?IUI + u$&u21d;, (3) o is minimised over all decentralised controllers of the form (2a), (2 b), where Q, RI, R2 m positive definite matrices. Let .I(FI yl~ FI y2 ) be denoted as J(FI, Fz ). The following results are properties of this index: Lemma 1. For the decentra[ised control system (1),(2) with the performance index (3): (l) The index is jinite f and only if the closed-loop system is stable. (2) For any real number q >0, the set s(q) := {(FI, F2) e Rm’x” x Rm’x” :~(Fl, F2) < q} is a compact set. Proof. Note that m 1“,le[A+B,~,CI+B~~~CIlf Ildt < .7(FI,F2) o / co ~(Fi, F2) q+ do for some dO >0 or the closed-loop system corresponding to (FI, ~2 ) is unstable. In both cases, there is a time T > 0 such that e[A+B, ~ICI+BZ~@E(xoX:) d > q + 60, which implies that ~(F~, F; ) > q for all n > N where N is a proper integer by the continuous dependence of the index of finite time on (Fl, F2 ). Hence (2) is established. ❑ By the continuous that the set s:= IJ s(q) q20 Danchi Jiang, J.B. Moore lSysiems& Control Letters27 (1996) 223-231 225 dependence of the closed-loop system pole on output feedback gain matrices we know is an open set, which contains all stabilizing output feedback gains. If the closed-loop system is stable, the performance index of the closed-loop system is easily computed by standard technique as where K(FI, F2 ) satisfies the following matrix Lyapunov equation: K (FI, F2)[A + BIFIC1 +%2 F2C2] + [A + BIFICI + BZFZCZ]7K(FI,FZ) 2. Gradient flow of deeentralised systems In this section, the gradient of ~(FI, Fz ) will be computed and some properties of the corresponding gradient flow obtained. Let a Hamiltonian function be defined as follows: H(K, Fl, F2, ~) = tr[KE(x@~)] + tr(~’ {K[A + BIFI CI + B2F2C2] +[A + BIFI CI + B2F2C21’K + C:F:RtFi CI + C; F;R2F2C2 + Q}). (6) For any K satisfying the Lyapunov equation (5) and any r, it is obvious that H(K, FI, F2, r) = .7(F1,F2) Given initial stabilizing static output feedback control gain (F!, F$’), let So denote the largest connected subset of,$ that contains (F!, F;). Let 0:= {(K, FI, F2) : K E R“X”, (FI, F2) c S’O.K(F1, Fz) and (Fl, Fz) subject to the Lyapunov equation (5)}. (7) Then, f2 is an (m I x 11 + m2 x 12) dimensional sub-manifold of IWx”+m’x ‘I‘m’ x”. Now the Hamiltonian function (6) is a map from f2 x 0?”‘n to R. Define the following maps (see Fig. 1): L : (F,,F2) e so w L(F,, F2) =(F,, F2, K(F,, F2)) e Q, (8) ld : (K(Fl, F2), F1, F2, r) E ~ X k!nx”I-+I~(K(Fl, F2), Fl, F2, r) = (K(FI, F2), FI, F2) ● Q. (9) Since L is a smooth isomorphism, it induces another function ~ on Q by J= JO L-l. Then, the map in Fig. I is commutative. For any (Xl ,X2) c T(F,F,)(SO),DL(XI ,X2) = (X3, XI, X2 ) E T and for any (x3,~],x2,x4)~ T(K,Fl,Fj, r)(~ x R“x”), DId(x3,xI,x2,x4) = (X3, XI, X2) ~ 7’(K,F,,F2,r)(f2 X k!nxn). 226 Danchi Jian,q, J. B, Moore I Sysiems & Control Letters 27 (1996) 223-231