Veselý, Vojtech

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Institute of Mathematics AS CR, v. v. i.

KybernetikaVojtech VeselýDecentralized control of linear dynamical systems with partial aggregationKybernetika, Vol. 25 (1989), No. 5, 408--418Persistent URL: http://dml.cz/dmlcz/124327Terms of use:© Institute of Information Theory and Automation AS CR, 1989Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - VOLUME 25 (1989), NUMBER 5 DECENTRALIZED CONTROL OF LINEAR DYNAMICAL SYSTEMS WITH PARTIAL AGGREGATION VOJTECH VESELY A new method for designing the decentralized control of linear dynamical systems with conti-nuous or numerically controlled stations is presented. The proposed method is intended for the use in designing the suboptimal decentralized control via solving the L-Q problem with output feedback for one subsystem under the assumption that the remaining part of the system is stable. Since the problem can be very complex a special partial aggregation of the mathematical model minimizing the interaction between the considered subsystem and the other part of the system is proposed. The design procedure is demonstrated on an example. 1. INTRODUCTION One of the problems occurring in connection with large scale systems is the design of a decentralized control station using the output feedback only from one sub-system, supposing that the other part of the system being stable. This problem can be solved by optimizing a given objective function while stability of the overall system has to be ensured. As one of many examples the design of an excitation controller of a synchronous generator integrated into a system of N generators can be mentioned. Two approaches have been adopted so far: 1. Design of suboptimal output feedback control taking the whole high order mathematical model of the system into the consideration. 2. Design of the control for the isolated subsystem and checking the stability of the global system [1, 6]. In this paper a new approach is developed. It is based on the use of decentralized control design procedures for constructing an optimal output feedback control scheme using a partial aggregated mathematical model of the system [5]. An essential advantage of the proposed method is that it enables to design suboptimal decentralized control for the chosen subsystem with the remaining part of the system being aggre-gated. Although the properties of the basic system are retained, the aggregation 408 of a part of the system considerably decreases the computation burden. This method is also effective in the special case when the whole system to be controlled is sequential-ly built by adding subsystems to the first one and the suboptimal behaviour of in-dividual portions of the technological process have to be assured. The proposed solution guarantees the stability and suboptimal control of the whole dynamical system. At first, the approach is presented for linear discrete-time systems and then the main results are summarized for linear continuous systems in the concluding remarks. 2. PROBLEM STATEMENT Let us suppose that a large-scale system can be divided into two subsystems. The first is to be controlled, while the second is supposed to be stable. Let us consider a linear time-invariant system S?\ (2.1) Sf\ xx(t + 1) = Axx xx(t) + AX2 x2(t) + Bxx ux(t), x2(t + 1) = A2X xx(t) + A22 x2(t) + B21 ux(t) , y(t) = Cx xx(t) , where xx(t) e 1R"1 and x2(t) e R"2 are the state vectors of the subsystems, ux(t) e Um is the control vector of the first subsystem, y(t) e IR1 is the output vector of the first subsystem, t = 0 , 1 , . . . is the time sample, Atp Bix, Cx are constant matrices of appropriate dimensions. Further, we assume that nx <̂  n2, the matrix A22 is stable and the triple (Axx, Bxx, Cx) is controllable and observable. Suppose the objective function in the form oo (2.2) J = £ x[(t) Qx xx(t) + u[(t) Rx ux(t) t = t0 where Qx and Rx must be positive definite symmetric matrices. The problem is that of determining a feedback matrix K\ (2.3) ' ux(t) = KCX xx(t) with a prescribed information structure and ensuring i) the minimization of the objective function, ii) the best possible realization of stability conditions of the global system. Assuming this conditions, the first subsystems' feedback matrix K must be chosen so that the contribution of the first subsystem to the stability of the whole dynamical system will be optimal. The criterion function taking into account these requirements will be formulated later. 409 3. MAIN RESULTS FOR THE LINEAR DISCRETE-TIME SYSTEM Let us define a new state vector Vl(t) as follows (3.1) v,(t) = L x2(t) . where vx(t) e Up. Matrix L will be referred to as the aggregation matrix [3]. The dimension of matrix L and the magnitude of its elements are determined by the requirements following from the aggregation objective. In our case, this objective is to retain those properties of the aggregated system portion which are essential when con-sidering the stability of the whole system. In addition, the minimal strength of inter-connections between the first subsystem and the rest of the system must be achieved. Hence the biggest eigenvalues of the matrix A22 have to be retained in the aggre-gated part of the system. The way of findingL to solve this problem is introduced in [7]. The choice of p depends on the number of eigenvalues, which are kept in the aggregated matrix. LettingL = A12, the most simple but not optimal (in the sense of the aggregation objective) results are obtained. Now, suppose A12m Eq. (2.1) in the form (3.2) A12 = M12L + E1 Minimizing the norm of the matrix Ex w.r.t. M12 (3.3) m i n i m i = \\A12 - M12L | | , we obtain (3.4) M12 = AX2L+ , where L+ =L T (LL T ) - 1 Similarly let us suppose in Eq. (2.1) (3.5) LA22 = M22L + E2. Minimizing the norm of the matrix E2 w.r.t. M22 we obtain min j |L 2 | = min |LA 2 2 — M22L\\ . M22 M22 It results in ' (3.6) M22 =LA22L+ Applying (3.1)-(3.5) to (2.1) the system y can be rewritten in the following form: (3.7) STX: xx(t + 1) = An xx(t) + M12 vx(t) + Ex x2(t) + Bn ux(t) v,(t + 1) = L A 2 1 x,(t) + M22 Vl(t) + E2 x2(t) +LB21 ux(t) x2(t + 1) = A21 xx(t) + A22 x2(t) + B21 ux(t) y(t) = Cx xx(t) 410 Let zx(t) be the state vector of the extended subsystem "*i(0" :>(') = and г2(i) = x2(t), Jl(t)_ then we obtain for the global system (3.8) <7>2: Zl(t + 1) = D1X zt(t) + D12 z2(t) + Bx ut(t) , z2(t + 1) = D21 Zi(t) + D22 z2(t) + B21 Ul(t) y(t) = Czx(t) where DX1 M, _, *,a-[j]. *г Bц LB21 All - r - 1 2 _LA2 1 M22_ D21 = [ A 2 1 0 ] , D22=A22, C = [ C 1 0 ] . Then the interactions between the two subsystems in Eq. (3.8) are much weaker than those between the original subsystems. Minimizing the norms of Ex and E2, the norm of D12 is minimized too. If its effect could be even neglected, local control scheme could be applied for the first subsystem using any design procedure. In order to check whether D12 is sufficiently small, we construct a Lyapunov function and determine the sufficient conditions of stability for the interactions term D12. Applying the feedback gain K, the closed loop system has the form (3.9) 9>3\ Zl(t + 1) = ( D n + BXKC) zx(t) + D12 z2(t) , Z2(t + 1) = (D 2 1 + B2XKC) zx(t) + D22 z2(t) In order to check the sufficient conditions for the stability of an interconnected system, Lyapunov functions of the subsystems are found. Using the scalar approach, we obtain the Lyapunov function for the global system V(t): Let (3.10) V(t) = ax z\(t) Px zx(t) + a2 zT2(t) P2 z2(t), where ax, a2 > 0 are real constants, Px, P2 are symmetric positive definite Lyapunov matrices. Using some properties of the matrix norms for the first difference of (3.10) we obtain an inequality constraint which ensures stability of the global system. We obtain (see Appendix A) (3.11) AV(t) ^ -zyGz = 0 , where *T = [k(.)|. MOU. [|*,(<)I = (zl(t) zfr))"2. ' = 1.2-The norm of the matrix D is calculated as follows \\D\\=X^X(D^D) 411 and Mmin (Q) - a2Xmax(D'2\P2D'21) « 1 2 aí2 a2 - ax ІЫ-̂ m a x ^ l ) Ä 1 2 - -м l> 1 2i iiлDuii + a2 Џl P2D2\ II) ->il = Dгi + BXKC D'21 — A21 + B21KC ß / , is the identity matrix . 2max('), Amin(') a r e the maximum and minimum eigenvalues of the matrix (•) re-spectively. If the feedback gain matrix K has been chosen to stabilize the first subsystem, then Pt can be found as a solution of the following equations: (3.12) DWP.D^ -PX = -(Q + C^FR.KC), D22P2D22 — P2 = — I„2 ' 0 i 0 ' .0 Q2 The matrix Q2 is a positive definite matrix associated with the extra state vector v^t). The state vector vx(t) is regarded as the interaction between the original first subsystem and the other part of the system. An appropriate choice of Q2 leads to a minimization of this interaction. The stability conditions for the global system (3.11) yield: (3.13) (i) Amax(D21P2Z)21)->min or Tr (D'2\D'2i) -> mm 00 ^max(^i) -> min or Tr (Pt) -> min . (hi) the matrix G must be positive semidefinite or definite one. (Tr(-) denotes trace function of the matrix (•).) The third condition can be used to determine the range of values for the inter-action term J  Z>121| => which ensures the satisfaction of the sufficient stability conditions. Conditions (3.13i) and (3.13ii) ensure the best possible satisfaction of stability conditions from the viewpoint of the first subsystem. For any positive numbers alta2 > 0 a maximum value of | |D1 2 | can be found by solving Eq. (3.11). A maximiza-tion routine can then be used to find the constants at and a2 which give the maximum possible range of ||D*2||- The equation (3.11) yields the requirement on the matrix G (3.14) G(alta2t\D12\) = 0 — max ||D12|| . It is possible to calculate ||Df2| for instance using a gradient method. If the inequality (3-15) Org \D12\ g | Z ) * 2 | holds, the feedback gain matrix K can be computed taking into account simply the first extended subsystem in Eq. (3.8). The augmented subsystem consists of the 412 original first subsystem and the aggregated mathematical model of the remaining part of the system. Thus, it is possible to design the feedback gain matrix K taking into account all the properties of the original subsystem and the approximation of the rest of the system. The extra part of the original subsystem is determined by the choice of the matrix L in (3.1). 4. CONTROL SYSTEM DESIGN The aim of this part is to determine the output feedback matrix K which minimizes (3.11) and the extended objective function (2.2) subjected to the constraints (3.12). Let us define the Lagrange function Lx: (4.1) Lx = min max Tr {Px + (D2X + B2XKC)T (D2X + B2XKC) + PlyK W + W[(DXX + BXKC)T PX(DXX + BXKC) - Px + Q + CTKTRXKC)} where W is a symmetric matrix of Lagrange multipliers. If the last term in Eq. (4.1) is equal to zero, it can be stated using the fact [8] that the minimization of Tr (Px) is in the average the same as the minimization of the cost function (2.2). Under the assumption that B2X is very "small" it is possible to neglect the second part of the cost function (4.1). The differentiation of (4.1) yields three necessary optimality conditions (4.2) a) VLlfi - I2lH + (DXX+ BXKC)W(DXX + BXKC)T -W=0 b) VLlw = (DXX + BXKC)T PX(DXX + BXKC) - Px + Q + CTKTRXKC = 0 and VLlK = 0 implies c) K = -(BTXPXBX + R,)-1 [BTXB21KC + BT2XD2X + BTXPXDXXW] CT . .(CWC1)'1 If C is the identity matrix and B2X = 0 Eq. (4.2c) reduces to the well known equation for the linear quadratic problem K= -(BjP^+Rj'B^D,,. The three nonlinear matrix equations (4.2) are in the same form used in [2]. There-fore we may suppose the same iterative algorithm. The steps of the solution are as follows: 1. Choose K1 such that (Dxx + BXKXC) is stable. 2. Substitute K1 into (4.2a) and (4.2b) and calculate P[ and W1. 3. Substitute K1, P[ and W1 into (4.2c) to calculate the next approximation K2. 4. If \\K2 - K11| > £ > 0 then K1 = K2 and go to Step 2. 5. End. The convergence of the solution represents another interesting problem, and it is not solved in this paper. According to [2] the choice of K1 corresponding to the 413 conditions introduced in the first step represents the necessary convergence condition. If only maintaining of the stability of the system is required without minimization of the objective function, then the right-hand side of Eq. (3.12a) reduces to the identity matrix, Amin(Q) = 1 and RA = 0. 5. MAIN RESULT FOR LINEAR CONTINUOUS-TIME SYSTEMS Consider a linear time-invariant dynamical system in the form (5.2) ^ 4 . Xx = An*! + A12X2 + #11«1 , x2 = A21x1 + A22x2 , y = c1x1 with the objective function (5.2) J = J£ (xlQlXl + •I.K1»1) dt -> min . Applying the procedure analogous to Eqs. (3.1) —(3.6) and (3.7), we obtain Ses\ i j = DllZl + D12z2 + Bux , z 2 = "21^1 + "22^2 5 y = Czt where -И Since D12 is sufficiently small it is possible to find the condition for disconnecting the two subsystems while maintaining the stability of the global system. Substituting (2.3) into (5.3) the closed loop system can be described as follows (5.4) «?V z. = D'llZl + D12z2, Z 2 ~ -"'21^1 + D22Z2 , where D1X = Du + BKC. In order to check the sufficient stability conditions of the interconnected system, the Lyapunov function for the subsystems are constructed which are followed by the ensuring that some linear combination of them is the Lyapunov function for the global system which is analogous to Eq. (3.10). The main result for the sufficient stability condition is given by the following lemma. Lemma. If D'1X is stable in the system (5.4) under the assumption of the stability of D22, then the condition KUQ) (5-5) Í-9.JI = 4-°i \\\\DJ2P2\ 414 satisfying the Lyapunov matrix equations #i i-°i + -°i->ii = ~[Q + CTKTR,KC^ D22P2 + PzD22 = —/„_ni ensures that the global system is stable. It makes possible the disconnection of the two subsystems (5.4). The proof of this Lemma is given in Appendix B. Now, suppose that for Eq. (5.4) the condition (5.5) holds. Then it is obvious from Eq. (5.5) that Px is to be minimized To minimize (5.2) and Pu we solve the following problem (5.7) L2 = min max Tr {Pt + W[(Dn + BKC)J Pt + P1(Dil + BKC) + K,J»! W + Q + CTKTR1KC]} . The necessary optimality conditions are: (5.8) a) VL2pi = I2ni + W(Dn + BKC)T + (Dn + BKC)W = 0 . b) VL2w = Px(Dn + BKC) + (Dn + BKC)T Pt + Q + CTKTR1KC = 0 c) L2 = 0 implies (5.8) K = -R1-1BTP1WCT(CWC)i When Cis the identity matrix, Eq. (5.8c) reduces to the well known equation (5.9) K = -R1BTPl . The nonlinear matrix equations (5.8) can be solved using the two level iterative procedure in the same way as Eqs. (4.2). 6. EXAMPLE FOR THE LINEAR DISCRETE-TIME SYSTEM Let us consider the linear discrete-time system An = Aí2 — Aт -л2\ ~ 0-5 0-05 004 001 0 - 0 0 1 -0-2 - 0 0 1 - 1 1 002 - 0 0 1 0 0 0 0 0 1 " - 0 0 2 0 0 1 0 0 0 0-02 0 0 0 0 0 0 001 0 0 0 0 1 0 0 0-05" 0 0 0 012 0 0 1 0 0 0-2 0 0 0 - 0 1 004 415 ~ 0-2 - 0 0 1 0-1 0 0 0-15 0 " 0 - 0 1 5 0 0-05 -0-02 0 0 0 0 0 0 0 -0-05 002 ^ 2 2 = 01 0 0 0 015 -0-02 0 0-02 0-04 0 0-08 0 0 0-12 - 0 0 8 0 0-05 - 0 0 2 0 0 0-12 _ - 0 0 2 0 1 0 001 0-2 0 0 Bïг = ;o oi i ] . BJ_ = [0- 0 0.001 00005 0 0 0-001] , C. = [ 0 1 1] . Let L = A12. The performance index is given 00 t = t0 where Q = diag {0-5 1 1 5 5 5}, r_ = l . Choosing K1 — 0-3 after the fourth iteration it is obtained from (4.2) |K 2 - Kx_ = [0-3416 - 0-36037[| = 0-02 . K = 0-3416 The matrix G is given for \\D12\\ = 0-025 as follows: o_ . 0-5 - a2 . 005739 -(«_ . 047 + a2 . 0034959)1 -(fl_ . 0-47 + a2 . 0-034959) a2 - a_ . 0-03501 J Without minimizing the relation (3.13) it is obvious that for a1 = a2 = 1 the matrix G is a positive definite one and therefore, the system is stable and the output feedback gain matrix K minimizes the given objective function. G = 7. CONCLUSIONS A method of decentralized control design based on an aggregated model is present­ed. This design method is particularly applicable for a global system consisting of many recognizable subsystems. In such case, the design method is used to calculate local control schemes for each subsystem in turn incorporating the effects of the previous local control schemes at each stage. This simplifies the control calculation when compared to the use of a centralized control design with a full system model. The method yields a robust design which optimizes the objective function of the subsystems while ensuring the best stability conditions of the global system. 416 APPENDIX A We get from Eqs. (3.8) and (3.9) AV(t) = V(t + 1) - V(t) = z\(t)[(-Q - CrKrR1KC)a1 + + a2(D21 + B21KC)r P2(D21 + B21KC)~] z,(t) + z\(t). .[-a2I+.axDT2PxDX2] z2(t) + 2z](t) . . [ax(Dxl + B.KCfP.D^ + a2(D21 + B21KC)T P2D22] z2(t) Using some properties of the matrix norms we obtain AV(t) = ^ ZT r«iAmi„(G + CrKrRxKC) - a2^max(Z>2TP2Z)21) - ^ "I L ~ a 1 2 a2 - a 1 A m a x ( D [ 2 P 1 D i a ) J 012 = «i4/ax[(-^ii-0i^i2)T(l>iii°i^i2)] + + a2^[(D'2\P2D22)T (D21P2D22)] or more strongly AV(ř) = z ' - l U f i ) - «2Amax(D2TiP2D21) «12 1 a12 a2 - ax(\Dx2\2 X^Di)] APPENDIX B Proof of Lemma. In order to check the sufficient stability conditions of the interconnected system (5.4) Lyapunov functions of the subsystems are found. Let (B.l) F = axVx + a2V2, a1>0, a2>0 and (B.2) ~ = axz\(D'x\Px + PXD'XX) zx + a2z\(Dr22P2 + P2D22) z2 + at + 2z[[a1P1D12 + a2Dr21P2] z2 If Px and P 2 are given by Eqs. (5.6) using some properties of the matrix norms for the first derivative of (B.2) we get Mm.n(-2) -£12" C12 a2 (вз) fi-Є dř zû 0 where Q12 = flJPJ \\D12\\ + a2\\Dr21P2\ 417 We get from the stability conditions of the overall system for |D 1 2 | | (B.4) ||Z)12I| ^ ~fl2llZ>^/>2ll +V( f l- f l2W6)) «ilp i l Let us find ax > 0 and a2 > 0 such that | |#12f -* max. We get from Eq. (B.4) (B.5) gjY lo12|l2 \\Ptf + (^)(-l-»„| |M |J>IA| - Amln(o)) + loI.P.l2 S 0 Since we require at and a2 to be real constants, Eq. (B.5) yields the following solution II 1>12II = M ^ 4[1 \ | | JDJ.P-1| (Received June 25, 1987.) REFERENCES [1] M. G. Singh: Decentralized Control. North Holland, Amsterdam 1981. [2] T. C Xinogalis: Hierarchical Optimisation Structures. PhD Thesis, UMIST Manchester 1982. [3] M. Aoki: Control of large scale dynamic systems by aggregation. IEEE Trans. Automat. Control AC-13 (1968), 3, 246-253. [4] F. R. Gantmacher: The Theory of Matrices. Chelsea Publishing Company, New York 1959. [5] V. Vesely and J. M. Edmunds: Decentralized Control with Partially Aggregated Systems. Control Systems Centre Report No. 664, UMIST Manchester 1984. [6] V. Vesely, J. Murgas and P. Hudzovic: Contribution to decentralized control of linear systems. In: Fourth Formator Symposium on Mathematical Methods for the Analysis of Large-Scale Systems (L. Bakule, J. Benes, eds.), Academia, Prague 1983, pp. 541—548. [7] M. S. Mahmoud and M. G. Singh: Large Scale Systems Modelling. (International Series on Systems and Control, vol. 3.) Pergamon Press, Oxford 1981. [8] V. M. Kuncevic and M. M. Lycak: Design of Automatic Control via Lyapunov Functions (in Russian). Nauka, Moskva 1977. Prof. Ing. Vojtech Vesely, DrSc, Katedra automatizovanych systemov riadenia technologickych procesov, Elektrotechnickd fakulta SVST (Department of Automated Technological Process Control Systems, Faculty of Electrical Engineering— Slovak Technical University), Mlynskd dolina, 812 19 Bratislava. Czechoslovakia. 418 

Decentralized control of linear dynamical systems with partial aggregation

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Decentralized control of linear dynamical systems with partial aggregation

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Institute of Mathematics AS CR, v. v. i.