Design of Explicit Fuzzy Prediction Controller for Constrained Nonlinear Systems

Abstract

This paper presents an explicit fuzzy predictive control method for a class of nonlinear systems with constrained inputs. The main idea is to construct a terminal invariant set and explicit predictive controller with affine input on the basis of T-S fuzzy model. This method need not compute the complex nonconvex nonlinear programming problem of earlier nonlinear predictive control methods and decreases the number of optimization variables and guarantees stability of the closed-loop system. The simulation results on a numerical example show the validity of the method presented in this paper.

Academic Editor:Kang Li

1, School of Electrical Information and Automation, Qufu Normal University, Rizhao, Shandong 276826, China

Received 31 March 2014; Accepted 26 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Almost all of the industrial processes have strong nonlinearity. Such strongly nonlinear industrial process is difficult to be modeled and controlled. It has attracted much attention in industry and academia by referring to [1, 2]. There are many constraints due to physical conditions to limit the flexible performance of the closed-loop systems, which need high skills in control system design.

Model predictive control (MPC), as an efficient control strategy to handle constraints within an optimal control setting, has received much attention in the past decades (e.g., [3-5]). The nonlinearity of the nonlinear systems makes the optimization problems nonconvex and thus leads to heavy calculation. It makes the parameters difficult to be adjusted online. In literature [6], using norm-bounded linear differential inclusion (LDI) of nonlinear system, a kind of predictive control scheme was put forward based on terminal domain optimization by solving linear matrix inequalities optimization problem. In [7] a robust model predictive control strategy was presented on the basis of polyhedral description systems for discrete-time nonlinear systems with bounded persistent disturbances. Both literatures [6, 7] used linearization of the original nonlinear model approximately. However the methods produce large errors. So these methods can only be applied in weakly nonlinear systems. Literature [8] combined the robust method and hybrid method to design the MPC for constrained piecewise linear (PWL) systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Paper [9] investigated the problem of predictive control for constrained control systems, in which the measurement signal may be multiply missing. However their applications are limited by their linearity.

T-S fuzzy models have become an important tool for researching control problems of the nonlinear system because of their universal approximation capability by referring to [10]. And then fuzzy predictive control scheme is developed. Literature [11] gives a fuzzy multistep linear predictive control strategy taking advantage of T-S model as predictive model and achieves better effect. But it also increases the online optimization computation. In literature [12] a nonlinear predictive control algorithm is proposed based on subsection Lyapunov function and T-S model for the input and output constrained Hammerstein-Wiener nonlinear system. Most of the study results can not obtain explicit expression of controller that is easy to adjust (e.g., [13-15]).

In this paper, we present a model predictive control algorithm for a class of constrained nonlinear system based on T-S fuzzy model. Firstly, the feedback correction is introduced after we establish T-S fuzzy model. For this corrected T-S fuzzy model, the terminal invariant set is designed, and in the terminal set the linear feedback control law is proposed to satisfy constraints and guarantee closed-loop stability. Secondly, we design predictive controller with affine control outside the terminal set which satisfies the constraints as well as making the system states eventually enter the terminal set. Meanwhile, the procedure guarantees closed-loop stability and reduces the amount of computation burden. At last we show the validity of this method by a simulation example.

In literatures [16], filters are given when the systems' states are immeasurable. Inspired by them, we will consider the cases with immeasurable states and stochastic disturbance in the future research. We hope to be given more tips.

Notations introduction: for vector [figure omitted; refer to PDF] and positive definite matrix [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; symbol [figure omitted; refer to PDF] represents symmetric structure [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are symmetric matrices.

2. Problem Statement

Consider the following constrained nonlinear system: [figure omitted; refer to PDF] subject to control constraints [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are state and control vectors, [figure omitted; refer to PDF] is a nonlinear function in [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , so control condition (2) can also be expressed as [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the [figure omitted; refer to PDF] th element of the [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively, [figure omitted; refer to PDF] .

The objective of this paper is to design a predictive controller that can make the following performance index reach to optimization and ensure the stability of the closed-loop system: [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote predicted values of the states and input vectors. Here we choose T-S fuzzy model as the predictive model. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive definite, symmetric weighting matrices.

3. Identification of T-S Model

As T-S fuzzy model can approximate nonlinear systems with arbitrary precision, we use T-S fuzzy model to approximate the nonlinear system. The rules of T-S fuzzy model are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the number of fuzzy rules and [figure omitted; refer to PDF] are fuzzy sets, for all [figure omitted; refer to PDF] . [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Many kinds of the membership functions can be chosen, such as triangular membership function, trapezoidal membership function, and Gaussian membership function. The selection of membership functions depends on the expert experience. In general, the membership functions of fuzzy sets with sharp curve shape have high resolution and high control sensitivity; on the contrary, the membership function with gentle curve has relatively smooth control performance and good stable performance. So, in this paper, we choose the Gaussian membership functions; namely, the membership of [figure omitted; refer to PDF] belonging to the set [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denote the center and the variance of the function.

G-K fuzzy clustering algorithm is chosen to determine the premise parameters combined with the least square method to complete the identification of the consequent parameters of T-S fuzzy model by referring to [12].

The objective function of G-K algorithm is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the data set, [figure omitted; refer to PDF] is the membership matrix, [figure omitted; refer to PDF] is the clustering center and also the center of membership function, the number of clustering is [figure omitted; refer to PDF] (which is also the number of fuzzy rules), [figure omitted; refer to PDF] is the sample size, [figure omitted; refer to PDF] is the fuzzy exponent, [figure omitted; refer to PDF] is the membership of the [figure omitted; refer to PDF] th data relative to the [figure omitted; refer to PDF] th clustering center, and [figure omitted; refer to PDF] denotes the distance norm of the [figure omitted; refer to PDF] th clustering relative to the [figure omitted; refer to PDF] th data: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The necessary conditions obtained by Lagrange multipliers that make the objective function minimum are [figure omitted; refer to PDF]

The variance of the Gaussian membership function is [figure omitted; refer to PDF] .

By the weighted least squares, the amount of the parameters to be identified is [figure omitted; refer to PDF] ; make [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

For identification, define [figure omitted; refer to PDF] and [figure omitted; refer to PDF] which are the [figure omitted; refer to PDF] th line of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is assumed to be the number of data, and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Construct the following matrix: [figure omitted; refer to PDF]

Then by the least squares we get the coefficient matrices [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The proposed T-S fuzzy model parameters identification algorithm can now be specified as follows.

Step 1 . Select the number of fuzzy rule [figure omitted; refer to PDF] , fuzzy exponent [figure omitted; refer to PDF] , and standard termination [figure omitted; refer to PDF] .

Step 2 . Generate fuzzy matrix [figure omitted; refer to PDF] randomly.

Step 3 . Update fuzzy matrix, denote [figure omitted; refer to PDF] , and calculate clustering center by (8).

Step 4 . Calculate distance norm by (6).

Step 5 . If [figure omitted; refer to PDF] , stop; otherwise, repeat Step 3.

Step 6 . Calculate consequent parameters of the T-S model according to (11); that is to say, work out the coefficient matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

The rules of the T-S model we get by G-K clustering algorithm are [figure omitted; refer to PDF]

The expression of T-S fuzzy model by the above algorithm is [figure omitted; refer to PDF]

4. Prediction Control Strategy on the Basis of T-S Fuzzy Models

In this paper, T-S fuzzy model (14) is selected as prediction model. Due to stochastic disturbance, modeling error, and so on, there must be error between predicted values and the actual state values [figure omitted; refer to PDF] . We assume the state error at sample time [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the predictive model state values being gotten by predictive model (14). For the sake of eliminating the error of prediction values caused by several reasons, revising [figure omitted; refer to PDF] by [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the predicted values of state variables and [figure omitted; refer to PDF] is the predicted values achieved from the revised T-S model as follows. Substituting (15) into (14), we can obtain [figure omitted; refer to PDF]

For the revised model, solve the following optimal problem: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a prediction horizon (for simplicity of exposition, the control and prediction horizons are chosen to have identical values in this paper) and [figure omitted; refer to PDF] denotes terminal region.

Now we first design terminal variant set and linear feedback control law which meet the condition of constraints as well as guaranteeing the closed-loop stability of the system in the terminal variant set.

Theorem 1.

Suppose there exist symmetric positive definite matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that the following equations are satisfied: [figure omitted; refer to PDF] Then there exists a terminal variant set [figure omitted; refer to PDF] , such that the controller [figure omitted; refer to PDF] defined in [figure omitted; refer to PDF] meets the constraints and the following inequality holds: [figure omitted; refer to PDF]

Proof.

Give the set [figure omitted; refer to PDF] and candidate Lyapunov function [figure omitted; refer to PDF] .

Then (20) is equivalent to (21) based on (16) and [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Based on (21) and the Schur complement, inequation (18) is obtained.

As [figure omitted; refer to PDF] , it should hold [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Hence we can obtain the following inequality by [figure omitted; refer to PDF] (where [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] th item in standard vector base of the input space [figure omitted; refer to PDF] ): [figure omitted; refer to PDF] Then due to [figure omitted; refer to PDF] , we can obtain a sufficient condition of (22): [figure omitted; refer to PDF] The above inequality can also be expressed as follows: [figure omitted; refer to PDF] Thus constraints (2) can be turned into (19) according to (24) and the Schur complement.

Remark 2.

The stability of the system in the terminal set [figure omitted; refer to PDF] and the invariance of [figure omitted; refer to PDF] are guaranteed by (20).

Remark 3.

[figure omitted; refer to PDF] is designed to deal with the difficulty getting the results brought by considering error.

Remark 4.

In the terminal set [figure omitted; refer to PDF] , the controller is chosen as [figure omitted; refer to PDF] to be convenient to adjust.

Solve the following optimization problem for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

In the terminal variant set [figure omitted; refer to PDF] , feedback control law [figure omitted; refer to PDF] can ensure the stability of the closed-loop system. However with this control law outside the set [figure omitted; refer to PDF] , the system performance will be worse. Therefore define the following affine input structure (referring to [13]): [figure omitted; refer to PDF] The perturbation [figure omitted; refer to PDF] is optimization variables.

Remark 5.

Here we can translate the optimization about [figure omitted; refer to PDF] into the optimization about [figure omitted; refer to PDF] and thus reduce the computational burden. [figure omitted; refer to PDF] denotes augmented state variables, where [figure omitted; refer to PDF] ; then we get the augmented state space model: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

For augmented system (27), the augmented feasible set is denoted by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is symmetric positive definite matrix. Then, from Theorem 1, inequality (30) holds for arbitrary [figure omitted; refer to PDF] if the following conditions are satisfied: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

Adding each side of (30) from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] can be obtained by solving the following convex optimal problem: [figure omitted; refer to PDF]

As a consequence, optimal problem (17) can be transformed as follows: [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] can be obtained by solving optimal problem (32).

Theorem 6.

In the case of the existence of optimal problem (32), the system is Lyapunov stable under input control (26) in the augmented feasible set [figure omitted; refer to PDF] .

Proof.

The optimal solution at time [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] ; meanwhile the feasible sequence at time [figure omitted; refer to PDF] is [figure omitted; refer to PDF] . As a result the cost function at time [figure omitted; refer to PDF] is [figure omitted; refer to PDF] According to (30), we can obtain [figure omitted; refer to PDF] , so [figure omitted; refer to PDF] .

And the performance index with optimal solution must not be larger than that with feasible solution. Namely, [figure omitted; refer to PDF] . Thus we can get [figure omitted; refer to PDF] . In conclusion, [figure omitted; refer to PDF] is a decreasing Lyapunov function of the system. That is to say, the system is closed-loop stable.

Now, we give the nonlinear predictive control strategy on the basis of T-S fuzzy model.

Step 1 . Get T-S model (4) of nonlinear system (1) by T-S fuzzy model parameters identification algorithm in Section 3.

Step 2 . Compute [figure omitted; refer to PDF] , [figure omitted; refer to PDF] by optimal problem (25), and obtain the terminal variant set [figure omitted; refer to PDF] as well as linear feedback control law.

Step 3 . Use affine input structure (26), and solve optimal problem (31) for [figure omitted; refer to PDF] . Consequently, determine the variant set [figure omitted; refer to PDF] of the augmented system and set initial state [figure omitted; refer to PDF] that should be in [figure omitted; refer to PDF] .

Step 4 . Measure the current state [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , adopt the control law [figure omitted; refer to PDF] . Otherwise, solve optimal problem (32) for [figure omitted; refer to PDF] at the current time.

Step 5 . Apply feedback control [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] and go to Step 4.

5. Simulation Example

Consider the problem of balancing and swing-up of an inverted pendulum on a cart from [12]. The equations of motion for the pendulum are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the angle (in radians) of the pendulum from the vertical and [figure omitted; refer to PDF] denotes the angular velocity. [figure omitted; refer to PDF] is the gravity constant, [figure omitted; refer to PDF] is the mass of the pendulum, [figure omitted; refer to PDF] is the mass of the cart, [figure omitted; refer to PDF] is the length of the pendulum, and [figure omitted; refer to PDF] is the force applied to the cart. We choose [figure omitted; refer to PDF] kg, [figure omitted; refer to PDF] kg, and [figure omitted; refer to PDF] m.

Discretize the system by forward difference [figure omitted; refer to PDF] , and here [figure omitted; refer to PDF] s.

Identify the above model using the scheme in Section 2. Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; we obtain the T-S fuzzy model as follows.

: [figure omitted; refer to PDF] : if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] : if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] : if [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Membership functions are as Figure 1.

Figure 1: Membership functions.

[figure omitted; refer to PDF]

The comparison between actual measure output and T-S model output is as Figure 2. The result shows that the simulation acquires good fitting effect.

Figure 2: Comparison between actual output and T-S model output.

[figure omitted; refer to PDF]

Compute the control law by the algorithm in Section 4. Set the algorithm parameters as follows: prediction horizon [figure omitted; refer to PDF] , control constraint [figure omitted; refer to PDF] N, weight matrix of optimal performance index [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Solving the optimal problem (25), we obtain [figure omitted; refer to PDF]

Set the initial state value [figure omitted; refer to PDF] , and for facilitating comparison we adopt the approach in this paper and [12] for state value and control law, respectively. We can obtain the simulation results showed by Figure 3, Figure 4 and Figure 5. From the figures we can see the control law designed by this paper converges to origin faster than that by [12]. The system reaches the stability state at last and the effect is better than that in [12].

Figure 3: State trajectory [figure omitted; refer to PDF] (the solid lines are states obtained by the approach in this paper and the dotted linear are states obtained by approach in the paper [12]).

[figure omitted; refer to PDF]

Figure 4: State trajectory [figure omitted; refer to PDF] (the solid lines are states obtained by the approach in this paper and the dotted linear are states obtained by approach in the paper [12]).

[figure omitted; refer to PDF]

Figure 5: Input profiles (the solid line is the input obtained by the approach in this paper and the dotted line is the input obtained by the approach in [12]).

[figure omitted; refer to PDF]

6. Conclusions

In this paper, we considered the design and stability problem of predictive controller based on T-S fuzzy models for a class of nonlinear system with constrained inputs. We approximated the original nonlinear model by the T-S model on the basis of G-K clustering algorithm and least squares method. And then we designed predictive controller with affine input based on T-S models. Meanwhile the closed-loop stability of the above-mentioned approach was proved by Lyapunov theory and the validity of the approach was showed by a simulation example.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61374004, 61273182, 61473170, and 61104007, the National Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113705120003, and horizontal topic HX201246.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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