# Approximate Decoupling of Multivariable Control Systems A Time-domain Approach.

код для вставкиСкачатьDev. Chem. Eng. Mineral Process. 13(3/4). pp. 361-3 70, 2005. Approximate Decoupling of Multivariable Control Systems: A Time-domain Approach Wei-Jie Mao* and Shao Liu National Lab of Industrial Control Technology, Institute of Advanced Process Control, Zhejiang University, Hangzhou 310027, P.R.China A new approximate decoupling method for linear time-invariant systems based on the concept of energetic diagonal dominance is proposed in this paper. It is a timedomain approach and the design procedure is simpler than that of j-equencydomain approach. The conditions for the energetic decoupling with simultaneous stabilizability are established in terms of linear matrix inequalities (LMIs). It is very convenient to solve this problem using the powerfil LMI toolbox. The results are derived first for proper systems with nonsingular D, and then extended to strictly proper systems. Illustrative examples demonstrate that this method has satisfactory decoupling pe~ormance. Introduction Decoupling technique for the design of multivariable control systems is of considerable theoretical and practical importance. A great deal of interest has been brought to this problem over recent decades. The aim of decoupling is that each output of a multi-input multi-output system can be independently controlled by a corresponding input. On the view of the depth of decoupling, it can be divided into complete decoupling and approximate decoupling. A number of state-space methods aimed at Morgan’s problem [e.g. 1-41 all belong to the complete decoupling approach. Based on complete cancellation, the decoupling performance of these methods will be impossibly achleved in the case of perturbation. Furthermore, if an original system has zeros, pole-zero cancellation will take place in the decoupling procedure. For instance, we consider the following linear timeinvariant system: . According to [2], L it is found that J this system can be decoupl d by the following control law: * Authorfor correspondence (wjmao@iipc.zju.edu.cn). 361 Wei-Jie Ma0 and Shao Liu u ( t ) = Fx(t)+ Gv(t) where F = [-2 -1 -4 0 0],G=[ 1 -1 -1 1. 0 1 {::I The closed-loop transfer matrix is: H,,(s) = diag -,- Indeed, a 3rd-order system has been decoupled to a 2nd-order system. Only two poles of the decoupled system can be determined. The lost pole (-1), which is fixed, is merely the zero of the original system. As a result, if the original system has zeros in right half plane, it can be impossibly decoupled with closed-loop stability. However, the frequency-domain method [6, 71 belongs to the approximate decoupling approach. The aim of this method is the diagonal dominance of the controlled plant, not the diagonalization. Thus, avoiding the drawback of the complete decoupling approach is possible. The key step of this method is how to achieve the diagonal dominance by a certain pre-compensator. Until now practical and efficient methods do not exist to acheve the diagonal dominance, except for the pseudo-diagonalization method which is based on a single frequency and unlikely to be extended to the multi-frequency case. In this paper, we are looking for a new approximate decoupling method. Similar to the concept of diagonal dominance [6,7], energetic diagonal dominance in timedomain framework is introduced. Thus, the proposed method is a time-domain approach and the design procedure is simpler than that of Rosenbrock [6, 71 . The problem to achieve energetic diagonal dominance is called as an energetic decoupling problem (EDP) here. Our contribution consists in establishing the conditions for the EDP with simultaneous stabilizability. All the results are derived using the LMI approach, for which there are efficient algorithms such as LMI toolbox in MATLAB. Problem Formulation Consider the linear time-invariant (LTI) system described by state-space model of the following form: X(t) = Ax(t) + Bu(t) y ( t ) = Cx(t)+ Du(t) ...(1) where x ( t ) E R" is the state vector; u(t) E R" is the input vector; y(z) E R" is the output vector of the above system satisfying m I n ; A, B,C, D are constant real matrices of suitable dimensions satisfying detD # 0 . Employing a control law of the form: u(t) = Fx(t)+ Gv(t) with v ( t )E R" , the resultant system is: 362 ...(2) Approximate Decoupling of Multivariable Control Systems x ( t ) = ( A + BF)x(t)+ BGv(t) y ( t ) = (C + DF)x(t)+ DGv(t) ...(3) Definition 1: The LTI system of Equation (3) with zero-initial condition is said to be energetically diagonal row dominant 8 I,'j:(t)dt I I,'yi(t)dt n ...(4) V r 2 0,vj = vi,i = 1,2,--.,rn where yii represents the part in thoutput caused by iIh input vi , jii represents the part in f h output caused by the other inputs but thinput (denoted as Gi). Energetic diagonal column dominance can be defined similarly by: j,'j,'(t)ji(t)dt I Iiyf(t)dt vz 2 0,vj = ~ (# i),i j = 1,2,..-,m ... ( 5 ) where j i E R"-' represents the vector composed of all outputs in y but thoutput yi. In contrast to diagonal dominance in the frequency-domain framework, the concept of energetic diagonal dominance by Definition 1 is in time-domain framework, and thus easy to understand. However, the inequalities of Equations (4) and ( 5 ) are not easy to solve. They need to be transformed into more solvable forms. For instance, Equation (4) can be transformed into: jiyii(t)vi(t)dt2 aijiv,?(t)dt ...(6) Iiji(t)dt I fliIiGr(t)Ci(t)dt,Vr 2 0,vj = vi,i = 1,2,...,m ...(7) It is obvious that Equation (4) can be guaranteed by Equations (6) and (7) with properly chosen positive constants ai,pi. Then, the EDP of LTI system (1) can be summarized as: to find F E R""",G E R""" , such that, for the closed-loop system of Equation (3), then Equations (6) and (7) are satisfied. Remark 1. The inequalities in Equations (6) and (7) may be meaningless with r + 00 . However, we validate them in this paper if ~im~;yri(r)vr(t)dr and r+- Iiii(tvf !! I;v,?(tWt exist. It is the same for inequalities of Equations (4) and (5). j~iy(1)qI)dl Remark 2. The decoupling perjormance of closed-loop system is dependent on the prescribed positive constants a = [a, a2 am ,fl = p, . p,,, , and on the controller F,G . Thus, we denote the EDP as EDP(a,fl I F,G) in detail. However, a,P may not be regarded as the final decoupling measure because of the inequality feature of (6), (7). Final decoupling per$ormance will be determined by the input-output responses of closed-loop system with designed controller. p b, p 363 Wei-Jie Ma0 and Shao Liu Note that the stability is the most important goal in the system synthesis, we shall consider the stable version of the above problem (namely EDPS) in the following. Definition 2: The EDPS(a,P I F,G) of LTI system (1) is said to be solvable $ (1) there exist a,P and F,G satisfiing inequalities (6) and (7) with zeroinitial condition; (2) the closed-loop system (3) is asymptotically stable. Conditions for EDPS If the LTI system (1) is open-loop stable, the control law (2) may be simplified as: ...(8) u(t) = Gv(t) Then the closed-loop system (3) becomes: i ( t ) = Ax(t) + BGv(t) y ( t ) = &(t) + DGv(t) ...(9) Theorem 1. The EDPS (a,P I0,G) of LTI system (1) is solvable if there exist matrices X i , q > 0 , i =1,2,.-.,m, such that: X,A' + AX, g:Br - C,X, 2a,I - (D,g,+ gfD:) KA'+AK Bg, KC: -pi+I iTD: < O gfBT c,< where D,i, -dI 1 iiE Rmx("'-')represents the matrix composed of all columns in ...(11) G but the ih column g, , Ci E RIXnand D, E RIXmrepresent the rows of C and D respectively. Proof. Consider the i" output of Equation (9),which can be divided into two subsystems described as: X(t) = Ax@)+ Bg,v,(t) yii(t) = Cix(t)+ Digivi(t) and: I m x(t) = Ax(t)+ BCg,v,(t) = Ax@)+BiiCi(t) 1 364 j=l jti I" Approximate Decoupling of Multivariable Control Systems Define quadratic h c t i o n s K ( t , x ) = x'(t)@(t) q ( t , x ) = x'(t)Q.x(t) , e,Qi> 0 , for above subsystems respectively. It is easy to show that: d dr M , ( t ) = -~ , ( t , x-) 2yii(t)vi(t) + 2aiv,?(t) = x T ( t ) ( A T+4@ ) x ( t ) + [ 2x'(t)eBg,vi(t)- 2yii(t)v,(t)+2a,vf(t) A T 4+ 4 A eBgi - Cr vi(t) g,?BT4-Ci 2ai1-(Digi = x(t)]'[ d dt + pl:+j;(t) N i ( t ) = - (t,x) - pif;,?(t)i(t) $11 $1 = xT(t)(ATQi +QiA)x(r)+2~T(~)QiB~i~i(t)-pi~~,?(t)ti(f)+~~~ji:(r) =[ A'Qi +QiA QiBii if13'Qi ci -Ddi iIi C: i,?Dr][ pi ' y , Pi 'Yii c-', Set X i = = Ql:' . Then it follows immediately from inequalities of Equations (10) and (11) that M i ( t ) < O , N i ( t ) < O . Integrating both sides of the above inequalities from 0 to 5 with initial condition x(0) = 0 , we obtain: j,'Mi(t)dt= y(r,x(r))-jl(2yiivi- 2aiv,2)dt 5 o j,'Ni(t)dt = q ( r , x ( s ) ) + j , ' ( ~ ; -/$?,?~~)dt +jj Io Thus, for all r 20: j,'yiividt 2 aijOrv,?dt, j,'jtdt 5 pij,';:Cidt In addition, with v(t) = 0 , selecting the quadratic function V;,(t,x) or q . ( t , x ) as Lyapunov hnctional candidate with nonzero-initial condition, we have: <(t,x) <o , q ( t , x ) <o Therefore the condition of Equations (10) or (1 1) can guarantee the asymptotic stability of system (9). This completes the proof of the theorem. For the LTI system (1) with unstable system matrix A, a simple way to achieve energetic decoupling may be described as: fKst, to stabilize LTI system (1) with state feedback; then, to design input transformation with the above method. Due to insufficient utilization of state feedback, it may result in conservative design. In the following we consider the EDPS with simultaneous state feedback and input transformation. Theorem 2. The EDPS(a,PI F,G)of LTI system (1) is solvable if there exist matrices Z and X > 0, such that: 365 Wei-Jie Ma0 and Shao Liu X.4' + A X + Z T B T+ BZ gTBT -C,X-D,Z Bg, -XCT -ZTDT 2aiI-(Dig,+g,?DT) M T+ AX + Z TBT + BZ g;BT CiX + DiZ Bi, XCT + 2'0: -p/Z $DT - p,!I [ where Diii ...(12) 1 <O ...(13) iiE Rmx(m-"represents the matrix composed of all columns in G but the ilh column gi , CiE RIXnand D, E RIXmrepresent the rows of C and D respectively. Furthermore, the state feedback matrix is: ...(14) F=m-' Proof. Applying Theorem 1 to the closed-loop system (3), this theorem follows immediately from the definition of Z = FX and Xi= $ = X , i = 1,2,...,m Extension to Strictly Proper Systems Usually, D in the LTI system ( 1 ) is zero. In th~scase, the LTI system ( ) can be described by state-space model of the following form: X(t) = Ax(t)+ Bu(t) Y ( t )= CxO) ..(15) Let H(s) be the transfer matrix of (15). There are unique integers di, i = 1, 2, ..., m, such that: lims"'"H,(s) = E~ ...(16) I +m where H,(s)is rh row of H ( s ) , and Ei is both finite and nonzero. Obviously, d , , i = 42;. .,m , are nonnegative and there results in the following lemma. Lemma 1 [2]:Let d,, i=1,2;--,m begiven by Equation (16). Then: di= { min{j : C,A'B f 0 , j = 0,1,...,n - 1) n-1 if C,A'B=O forallj=O,l,...,n-l ...(17) Based on Lemma 1 , the following LTI system: H ( s ) = diagb'' +I, ..,sd- )H(s) can be described by state-space model of the following form: 366 ...(18) Approximate Decoupling of Multivariable Control Systems If d e 6 # 0 , applying the results above to Equation (19), we find the control law (2) such that, for the resulting closed-loop system HJs) from (19) with zero-initial condition, energetic diagonal row dominance is achieved. Let H,,(s) be the resulting closed-loop system from (15) with the same control law. Then: H,,(S) = diag Sd"""-1 { l Sd, + I } H , ( S ) Therefore, H,,(s) is of the same decoupling performance as ...(20) pc,(s). However, despite the stability of gc/(s), H,,(s) is an integrator system. What is still required is a procedure for specifying the poles of H,,(s) while reserving the decoupling performance. In fact, we can incorporate these two procedures. Let: , .(21) where y,(j = 1,2,...,d, + 1,i = 1,2,...,m ) are selected by the requirement of closedloop system poles. fi(s) can be described by state-space model of the form: ( X ( t ) = Ax(t) + Bu(t) If d e G # 0,applying the results of the above section to Equation (22), we will find the control law (2) such that, for the resulting closed-loop system Ec,(s)from (22) with zero-initial condition, energetic diagonal row dominance is achieved. Then, with the same control law: ..(23) Therefore, H,(s) is of the same decoupling performance as gc,(s). The stability of H,(s) can be guaranteed by the appropriate selection of y, and the application of the results of the above section. 367 Wei-Jie Ma0 and Shao Liu Illustrative Examples Example 1. Consider the LTI system as in (l), with: In this example, A is unstable. By Theorem 2, choose a,= a2= Dl = p2 = 1 , then we get the state feedback and input transformation matrices: ’=[ -4.6898 0.1132 [ 8.1118 3.53211 -5.7065 -1.02781 ,G= - 0.373 1 7.0004 -2.9086 -3.0030 The step response of the resulting closed-loop system H,,(s) is shown in Figure 1. Example 2. Consider the LTI system as in (15), with This example is a variant of Example 1 with D = 0 . Note that C,B = [l 01f 0 , C2B= [l 1]+ 0 , thus d , = d, = 0 . Selecting y,, = y2, = 2 , we get i ( s ) described by state-space model of the form (22) with: Apply Theorem 2 to E(s) and choose a,= a, = p, = p2 = 1 , then we get the state feedback and input transformation matrices: - 3.4457 0.1528 ] [ 1.2026 - 5.0742 - 0.2349 ,G = -0.7155 -0.6899 -2.0727 0.28561 1.0993 The step response of the resulting closed-loop system H,,(s) is shown in Figure 2. The poles of this closed-loop system are: eig(A + BF) = -1.4056, -0.8070 f 0.7334i. It is obvious that none of the poles is fixed to -1, zero of the original system. Therefore, this method avoids the pole-zero cancellation in the decoupling procedure as stated for the example in the Introduction. Conclusions This paper has presented an energetic decoupling method, which regards the reduction of energetic relationship as the main goal. In contrast to the complete decoupling approach, this method avoids the sensitivity to the model perturbation 368 Approximate Decoupling of Multivariable Control Systems Figure 1. Step response of Example1 with designed controller. Time (secs) 0.2 Tim a (secs) Time (ssca) Figure 2. Step response of Example2 with designed controller. and the pole-zero cancellation in the decoupling procedure. Conditions for the proper linear time-invariant systems, as well as strictly proper linear time-invariant systems, are obtained in terms of LMIs. A memory-less state feedback and input transformation controller can be constructed by solving LMIs to guarantee both stability and decoupling performance. 369 Wei-Jie Ma0 and Shao Liu Acknowledgments This work was supported by the National Natural Science Foundation of People's Republic of China under grant 60004002, and by the Zhejiang Provincial Natural Science Foundation of China under grant 602055. References 1. 2. 3. 4. 5. 6. Morgan, B.S. 1964. The synthesis of linear rnultivariable systems by state-variable feedback. IEEE Trans. Automatic Control, 9,405-41 1. Falb, P.L., and Wolovich, W.A. 1967. Decoupling in the design and synthesis of multivariable control systems. IEEE Trans. Automatic Control, 12,651-659. Gilbert, E.G.1969. The decoupling of rnultivariable systems by state feedback. SIAM J. Control Optimization, 16,83-105. Morse, A.S., and Wonham, W.M. 1971. Status of noninteracting control. IEEE Trans. Automatic Control, 16,568-581. 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