Constrained predictive control based on T-S fuzzy model for nonlinear ...

Journal of Systems Engineering and Electronics, Vol. 18, No. 1,2007, pp. 95-100

Constrained predictive control based on T-S fuzzy model for nonlinearsystems* Su Baili"2, Chen Zengqiang' & Yuan Zhuzhi' 1. Dept. of Automation, Nankai UNv. ,Tianjin 300071, P. R. China; 2. Qufu Normal Univ., Qufu 273165, P. R. China

(Received July 7,2005)

Abstract: A constrained generalized predictive control (GPC) algorithm based on the T-S fuzzy model is presented for the nonlinear system. First, a Takagi-Sugeno (l-S) fuzzy model based on the fuzzy cluster algorithm and the orthogonal least square method is constructed to approach the nonlinear system. Since its consequence is linear, it can divide the nonlinear system into a number of linear or nearly linear subsystems. For this T-Sfuzzy model, a GPC algorithm with input constraints is presented. This strategy takes into account all the constraints of the control signal and its increment, and does not require the calculation of the Diophantine equations. So it needs only a small computer memory and the computational speed is high. The simulation results show a good performance for the nonlinear systems.

Keywords: Generalized predictive control (GPC), Nonlinear system, T-Sfuzzy model, Input constraint, Fuzzy cluster.

1. Introduction The generalized predictive control (GPC), which was presented by Clarke et al"', has been successfully applied in industry and has become an attractive research field in automatic control for its advantages over conventional techniques. Because the variables of the system are often limited in the real system, especially the control variables being limited by the physical conditions, it is advisable to study GPC algorithms with constrained variables. In general, the nonlinear search algorithm is applied to solve this kind of constrained optimization problems[21.The nonlinear search algorithm is mature in theory, but its computational load is too great, so it is difficult to be used in industry. A GPC algorithm was presented in Ref.[3]. where only the increment of the input was limited and there were two constrained conditions and two variables- Au(t) and Au(t+l) . But it still required the nonlinear search algorithm. The soften-gene /3 was applied to make the control variable smooth in Ref.[4]. That GPC algorithm presented took into account all kinds of constraints of the input and its increment. Furthermore, it didn't require the solving of Diophantine equations and the inverts of matrixes, so its computational load was not too great and its computational speed was high. The predictive control algorithms were originally developed for linear processes, but the basic idea can be transferred to nonlinear system^[^-^]. In these literatures the conventional GPC was applied based on * This

a T-S model, but the constraints of the variables were not taken into account. In this paper, a constrained GPC algorithm based on the T-S fuzzy model is presented for nonlinear systems. First, a T-S fuzzy model is constructed to approach the nonlinear system by applying a fuzzy cluster algorithm"'] and an orthogonal least square method["]. Then the constrained GPC strategy is adopted based on this T-S model. It takes all the constraints of the input into account and has good dynamic tracking performance. Finally, the results of simulation show the validity of this strategy for nonlinear systems.

2. The T-S fuzzy model and its identification 2.1 The structure of T-S fuzzy model A nonlinear discrete system can be expressed by a T-S fuzzymodelwith n rules: R' : i f ? is Af and...and x, is 4, (1) then y' = p ~ + p f x , + p ~ x , + . . . + p ~ x , , , where X, = y ( k - l ) , . . . , x , = y ( k - v ) , xv+, = u ( k - l ) , . . . , x,,, = u ( k - I ) , m = v + l , A' is the fuzzy set, pj is constant, where i = l , . . . , n , j = l , . - . , m . Assuming that the number of the fuzzy sets for y is s and the number of the fuzzy sets for u is t, then the number of rules is s" .t' . If any xi is A; does not appear in the premise of

project was supported by the National Natural Science Foundation of China (60374037 and 60574036). and the Opening Project Foundation of National Lab of Industrial Control Technology (0708008).

Su Baili, Chen Zengqiang & Yuan Zhuzhi

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1)

the rules, it denotes that x j is not constrained and it d , = Ilxk - ci = 01 (the Set of centers), can vary freely in its universe of discourse. ?! I - I, (the set of noncenters), so we have Given a generalized input vector ( ~ o , x m , - - ~ , x , , , o ) , the output of the T-S fuzzy system at any time instant ('k = 4) k is computed as the weighted average of the y i (i = 1,2; ,n) in formula (I), that is,

and

i=l

where the weights pi are computed as

n

where (b is empty set, and

m

pi =

n

A; ( x j o )

aik: x u i k = 1, Vi E I, # 4

(3)

where, is a fuzzy operator, usually applied as the "min" of the '>roducf' operator. Identification of the T-S fuzzy model involves two steps: the identification of the rules' premise using the fuzzy cluster algorithm, and the identification of the rules' consequence in which the orthogonal least square method is adopted.

2.2 The identification of the rules' premise using fuzzy cluster algorithm Given a group of input vectors xk E O,l< k d N , and divided them into c fuzzy clusters A,7...,Ac , such that the self-contained condition is satisfied: VXkE 3 1< i d c, s.t. 4 ( x k )> 0 (4)