ICIC Express Letters ICIC International c 2008 ISSN 1881-803X Volume 2, Number 4, December 2008 pp. 395 399 AN OPTIMIZATION-BASED APPROACH FOR QUASI-NONINTERACTING CONTROL Jose M. Araujo, Alexandre C. Castro and Eduardo T. F. Santos Research Group in Signals and Systems Federal Center in Technological Education of Bahia Rua Emidio dos Santos, S/N, Barbalho, Salvador-BA, Brazil 40301-015 { jomario; castro; eduardot }@cefetba.br Received June 2008; accepted September 2008 Abstract. Decoupling or noninteracting control (NIC) on MIMO linear time-invariant systems is a result of great importance on systems and control. Algebraic and geometric techniques are very interesting in presenting the solvability conditions for a perfect decoupling problem, but quite complex in nature. On this paper, a very simple approach for quasi-noninteracting (quasi-nic) control for MIMO square systems is presented. Static state feedback, output feedback and the forward gains are determined by an unconstrained minimization problem where the cost function is based on the distance between the matrix of polynomial norms of numerators and the identity matrix. The proper choice of initial guess for numerical minimization algorithm yields stable closed-loop poles simultaneously with minimal interacting. The merit of the proposed approach is also shown from its application to examples found in literature. Keywords: Linear multivariable systems, Noninteracting control, State feedback, Output feedback, Norm minimization 1. Introduction. Noninteracting or decoupling control (NIC) for MIMO linear timeinvariant systems is a modern result, with consolidated results on algebraic and geometric approaches. The concept of multivariable linear control of a plant without interactions between crossed inputs and outputs is very interesting and useful, and some practical applications can be found in literature [5,13-14]. The state feedback and output feedback are two techniques widely used in order to achieve the noninteracting control, and some of the first works on this subject can be seen in [9,11]. More on the algebraic approach is presented in [7]. The geometric control theory arose during the seventies and new results were brought on the scene in several important contributions [3,10]. Many of these results were compiled on important books about linear systems theory [4,12]. Although algebraic and geometric methods are well-known, numerical techniques for noninteracting control are still under development. A few works present numerical alternatives for implementation of noninteracting control [6]. On this paper, a simple numerical optimization-based approach is proposed to solve the quasi-noninteracting control problem. The proposed approach uses a properly chosen cost function based on the resulting diagonal form of the system transfer matrix and in its polynomial norms. It yields very good results for model order reduction and disturbance attenuation as shown in [1,2]. The results obtained from numerical examples show that the proposed approach is very effective despite of its simplicity, yielding good solutions even when the method presented in [8] increases the system order. 2. Preliminaries. Let the system under study be MIMO, linear time-invariant, n-order with m x m inputs-outputs. It can be represented by the following state-space form: 395
396 J.M.ARAUJO,A.C.CASTROANDE.T.F.SANTOS ẋ(t) =Ax(t)+Bu(t) (1) y(t) =Cx(t) (2) The NIC problem consists of obtaining a static gain for full state or output feedback plus forward gain for control laws v(t) = Kx(t)+Fu(t) (3) v(t) = K o y(t)+fu(t) (4) in the way that the closed-loop system transfer matrix: G(s) =C(sI A + BK) 1 BF (5) G(s) =C(sI A + BK o C) 1 BF (6) is a diagonal matrix. The required conditions on system matrices for the solvability of this problem are given by algebraic or geometric approaches. A rational transfer function for the entries of G(s) given by Eq.(5) or Eq.(6) has a very useful norm called polynomial norm. Let an entry for the aforementioned transfer matrix be: G ij (s) = N ij(s) d(s) = b ms m + b m 1 s m 1 +... + b 1 s + b 0 s n + a n 1 s n 1 +... + a 1 s + a 0 (7) The polynomial norm is then given by: coeff(n ij ) = b m b m 1... b 1 b 0 (8) That norm was successfully applied for model order reduction and disturbance attenuation as respectively shown in [1,2]. 3. The Proposal. In this paper, a numerical methodology for achieving quasi-nic in linear systems as described in Section 2 is proposed on a very intuitive reasoning. Let the square matrix in which the entries are polynomial norms of entries in G(s) be: S =[s ij ],s ij = coeff(n ij ) (9) A minimization problem is formulated for the determination of the NIC solution, if any exists. The cost function is given by the square of the distance between the S matrix and theidentitymatrixi: f(k, F) =[distance(s, I)] 2 = X (S I) 2 (10) which is minimized for K and F. The unconstrained minimization can be carried out on this problem and a proper choice on initial guess can guarantee convergence and stable solution, if any exists. For the particular case of fixed poles, it can guarantee the stability of the other poles. The proposed approach is applied to examples found on literature. These results are presented in the following sections and its features and limitations are discussed. 4. Numerical Examples. 4.1. Fourth order 2 2 system. This example is a fourth order 2 2 system, presented in [7]. The proposed approach is applied in order to confirm its effectiveness. The system matrices are: A = 0 1 1 0 0 1 B = 1 0 0 1 CT = 1 0 0 1 (11)
ICIC EXPRESS LETTERS, VOL.2, NO.4, 2008 397 The numerical minimization is performed using the initial guess given by the LQR solution for state-feedback and the suitable forward matrix gain calculated as in [7]. The obtained gains and final transfer function for this example are: G(s) = K = 0.9294 2.2437 2.2437 0 0 2.2537 0.6225s+0.7826 F = 0 s 4 +3.5010s 3 +5.0647s 2 +3.7504s+1.1585 0 0.6225 0.2895 0.2895s 3 +0.6495s 2 +0.6495s+0.2690 s 4 +3.5010s 3 +5.0647s 2 +3.7504s+1.1585 Figure 1 shows the step response for the system excited by inputs 1 and 2 in statefeedback design. (12) (13) Figure 1. Step response of the system in Example 4.1 4.2. Third order 2 2 system with output feedback. This example was presented in [8], which states that the decoupling is not possible by output feedback: A = 1 1 0 0 4 1 B = 1 0 1 1 C T = 1 0 1 1 (14) 0 2 1 0 1 0 1 The proposed approach was applied to this case using output feedback and scaling the forward gain by a factor of one hundred, yielding: 1.7542 1.0630 K o = F = 100F = 0.8713 1.5434 5.25 2.18 2.62 5.25 (15)
398 J.M.ARAUJO,A.C.CASTROANDE.T.F.SANTOS G(s) = Ã 7.8793s2 +52.7349s+84.5966 10 1 (8.8659s 2 3.5649s+1.3941) s 3 +10.6457s 2 +36.6343s+40.1997 s 3 +10.6457s 2 +36.6343s+40.1997 10 3 (4.6271s 2 +10.3402s+7.9004) 8.3172s2 +55.061s+83.0503 s 3 +10.6457s 2 +36.6343s+40.1997 s 3 +10.6457s 2 +36.6343s+40.1997 Figure 2 shows the step response for this example. It can be seen that quasi-nic was achieved for this case, and that the drawback on methodology shown in [8] was overcame. Furthermore, there is no need of dynamic forward compensation, avoiding this way the system order increase.! (16) Figure 2. Step response of the system in Example 4.2 5. Conclusions. A simple and new approach to quasi-nic in linear time-invariant systems was presented. It aims to minimize the cost function of the feedback and feed-forward gains on state and output feedbacks. This cost function computes the distance between the resulting matrix obtained from the polynomial norms of closed-loop transfer matrix and the identity matrix. Numerical issues due to ill-conditioning or instability of the final solution can be overcome with constraints on the poles of the closed-loop systems. The proposed approach was successfully applied to numerical optimization examples, yielding good results. Acknowledgment. This work was partially supported by Federal Center in Technological Education of Bahia. The authors also would like to acknowledge the helpful comments and suggestions from reviewers.
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