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.
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 and positive definite matrix , ; symbol represents symmetric structure , where and are symmetric matrices.
2. Problem Statement
Consider the following constrained nonlinear system:subject to control constraintswhere and are state and control vectors, is a nonlinear function in , and , , and , so control condition (2) can also be expressed as ; , , and are the th element of the , , and , respectively, .
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: and denote predicted values of the states and input vectors. Here we choose T-S fuzzy model as the predictive model. and 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 arewhere , is the number of fuzzy rules and are fuzzy sets, for all . and .
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 belonging to the set is , where , 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 iswhere is the data set, is the membership matrix, is the clustering center and also the center of membership function, the number of clustering is (which is also the number of fuzzy rules), is the sample size, is the fuzzy exponent, is the membership of the th data relative to the th clustering center, and denotes the distance norm of the th clustering relative to the th data:where
The necessary conditions obtained by Lagrange multipliers that make the objective function minimum are
The variance of the Gaussian membership function is .
By the weighted least squares, the amount of the parameters to be identified is ; make , where
For identification, define and which are the th line of and , is assumed to be the number of data, and , . Construct the following matrix:
Then by the least squares we get the coefficient matriceswhere
The proposed T-S fuzzy model parameters identification algorithm can now be specified as follows.
Step 1. Select the number of fuzzy rule , fuzzy exponent , and standard termination .
Step 2. Generate fuzzy matrix randomly.
Step 3. Update fuzzy matrix, denote , and calculate clustering center by (8).
Step 4. Calculate distance norm by (6).
Step 5. If , 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 , .
The rules of the T-S model we get by G-K clustering algorithm are
The expression of T-S fuzzy model by the above algorithm is
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 . We assume the state error at sample time is , where 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 by , we obtainwhere is the predicted values of state variables and is the predicted values achieved from the revised T-S model as follows. Substituting (15) into (14), we can obtain
For the revised model, solve the following optimal problem:where is a prediction horizon (for simplicity of exposition, the control and prediction horizons are chosen to have identical values in this paper) and 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 , , such that the following equations are satisfied:Then there exists a terminal variant set , such that the controller defined in meets the constraints and the following inequality holds:
Proof. Give the set and candidate Lyapunov function .
Then (20) is equivalent to (21) based on (16) and in . Considerwhere , .
Based on (21) and the Schur complement, inequation (18) is obtained.
As , it should hold , , where . Hence we can obtain the following inequality by (where is the th item in standard vector base of the input space ):Then due to , we can obtain a sufficient condition of (22):The above inequality can also be expressed as follows: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 and the invariance of are guaranteed by (20).
Remark 3. is designed to deal with the difficulty getting the results brought by considering error.
Remark 4. In the terminal set , the controller is chosen as to be convenient to adjust.
Solve the following optimization problem for , :
In the terminal variant set , feedback control law can ensure the stability of the closed-loop system. However with this control law outside the set , the system performance will be worse. Therefore define the following affine input structure (referring to [13]):The perturbation is optimization variables.
Remark 5. Here we can translate the optimization about into the optimization about and thus reduce the computational burden. denotes augmented state variables, where ; then we get the augmented state space model:where
For augmented system (27), the augmented feasible set is denoted by , where is symmetric positive definite matrix. Then, from Theorem 1, inequality (30) holds for arbitrary if the following conditions are satisfied:where , , and
Adding each side of (30) from to , we get , where can be obtained by solving the following convex optimal problem:
As a consequence, optimal problem (17) can be transformed as follows:Then 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 .
Proof. The optimal solution at time is denoted by ; meanwhile the feasible sequence at time is . As a result the cost function at time isAccording to (30), we can obtain , so .
And the performance index with optimal solution must not be larger than that with feasible solution. Namely, . Thus we can get . In conclusion, 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 , by optimal problem (25), and obtain the terminal variant set as well as linear feedback control law.
Step 3. Use affine input structure (26), and solve optimal problem (31) for . Consequently, determine the variant set of the augmented system and set initial state that should be in .
Step 4. Measure the current state . If , adopt the control law . Otherwise, solve optimal problem (32) for at the current time.
Step 5. Apply feedback control . Let 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 arewhere denotes the angle (in radians) of the pendulum from the vertical and denotes the angular velocity. is the gravity constant, is the mass of the pendulum, is the mass of the cart, is the length of the pendulum, and is the force applied to the cart. We choose kg, kg, and m.
Discretize the system by forward difference , and here s.
Identify the above model using the scheme in Section 2. Let and ; we obtain the T-S fuzzy model as follows. : if is , is , and is , then : if is , is , and is , then : if is , is , and is , then
Membership functions are as Figure 1.
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.
Compute the control law by the algorithm in Section 4. Set the algorithm parameters as follows: prediction horizon , control constraint N, weight matrix of optimal performance index , and .
Solving the optimal problem (25), we obtain
Set the initial state value , 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].
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.