Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 897853, 13 pages

http://dx.doi.org/10.1155/2015/897853

## Optimal Control Algorithm of Constrained Fuzzy System Integrating Sliding Mode Control and Model Predictive Control

Department of Information Science and Engineering, Northeastern University, Liaoning 110004, China

Received 26 September 2014; Accepted 25 December 2014

Academic Editor: Bo Shen

Copyright © 2015 Chonghui Song. 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.

#### Abstract

The sliding mode control and the model predictive control are connected by the value function of the optimal control problem for constrained fuzzy system. New conditions for the existence and stability of a sliding mode are proposed. Those conditions are more general conditions for the existence and stability of a sliding mode. When it is applied to the controller design, the design procedures are different from other sliding mode control (SMC) methods in that only the decay rate of the sliding mode motion is specified. The obtained controllers are state-feedback model predictive control (MPC) and also SMC. From the viewpoint of SMC, sliding mode surface does not need to be specified previously and the sliding mode reaching conditions are not necessary in the controller design. From the viewpoint of MPC, the finite time horizon is extended to the infinite time horizon. The difference with other MPC schemes is that the dependence on the feasibility of the initial point is canceled and the control schemes can be implemented in real time. Pseudosliding mode model predictive controllers are also provided. Closed loop systems are proven to be asymptotically stable. Simulation examples are provided to demonstrate proposed methods.

#### 1. Introduction

The modeling and control of fuzzy systems is a very active research area [1–9]. In the optimal control of fuzzy systems, the value functions for deterministic and uncertain systems satisfy the first order Hamilton-Jacobi-Bellman (HJB) equations and the first order Hamilton-Jacobi-Isaac (HJI) equations, respectively. References [10, 11] tried to solve the optimal control problem of fuzzy systems by linear system methods. References [12, 13] solved HJB equations in the optimal control of constrained fuzzy systems by dynamic programming. References [14, 15] solved HJI equations in the optimal control of uncertain constrained fuzzy systems by differential game. Usually, such equations are difficult to be solved in closed form due to their high nonlinearity. Some numerical methods with convergence proofs [13–21] are proposed to solve the value function. For the development of the above mentioned numerical methods, refer to [13, 14] and the references therein. References [22–25] use the adaptive dynamic programming to approximate the value function by neural networks. The finite difference approximation with sigmoidal transformation (FDAST) algorithms [13–16] is proposed to solve HJB equations arising in receding horizon control (MPC) schemes and HJI equations arising in robust receding horizon control (MPC) schemes, respectively.

Model predictive control (MPC), also known as receding horizon control, is a powerful tool to integrate the control and optimization of constrained nonlinear systems. Many MPC schemes have been developed [13, 16, 26–38]. The development and limitations of traditional MPC schemes are summarized in [13, 16]. Variable structure systems (VSS) first appeared in the late 1950s. Variable structure control (VSC) is an important control method for nonlinear systems [39]. Since VSC has been proposed, it has undergone great development [40]. The dominant role in VSS theory is played by sliding modes, and the core idea of designing VSC algorithms consists of enforcing this type of motion in some manifolds of the system state space into the sliding mode surface. The design of the sliding mode surface generally takes the artificially selected linear equation form [41, 42]. This form facilitates the design of variable structure controllers and the discussion of the stability region. However, artificially selected sliding mode surface due to the restriction of the sliding mode reaching condition will limit the stable region of the closed-loop system. Furthermore, the design of sliding mode surface and control is still a difficult problem for constrained uncertain nonlinear systems and often uses the local linearization method. The results obtained are locally stable, and there are no discussions on global stability and semiglobal stability of constrained uncertain nonlinear systems under SMC. The relationship between the optimal control and the sliding mode was not discussed in the literatures.

In this paper, we present the connection between the optimal control and the sliding mode via the value function. We briefly present the FDAST algorithm to solve HJB equations for constrained fuzzy systems. Then, new conditions for the existence and the stability of the sliding mode are proposed. Those conditions are constructed by the optimal value function and the system equation. They integrate the sliding mode existence criteria and the stability criteria for constrained fuzzy systems. This leads to some big variations in the SMC design. Then, sliding mode model predictive control (SMMPC) and pseudosliding mode model predictive control (PSMMPC) schemes are proposed for some kind of constrained fuzzy systems. Those controllers are MPC and also SMC. From the viewpoint of SMC, the sliding mode surface does not need to be specified previously in the SMC design. Sliding mode reaching conditions are not necessary in the controller design, since the closed-loop stability of the trajectory, which cannot reach the sliding mode surface, is guaranteed by MPC schemes. Therefore, it can cancel limitations on the stable zone due to the selection of the sliding mode surface and the sliding mode reaching condition. The closed-loop system is globally stable. From the viewpoint of MPC, those controllers are state-feedback (SF) which is very easy to be adjusted. The finite time horizon is extended to the infinite time horizon which guaranteed the global stability without added terminal penalties and constraints. The value function is used to design a controller instead of using the optimal control as the current control action. The dependence on the feasibility of the initial point is canceled and the online repeated optimization in MPC can be avoided. Therefore, the control schemes can be implemented in real time.

This paper is organized as follows. Section 2 begins with a description of HJB equations which includes MPC schemes for fuzzy systems. A brief review of the FDAST algorithm is given. Section 3 gives new conditions for the existence and the stability of a sliding mode. Section 4 gives the SMMPC design for some kind of constrained fuzzy systems by using the value function as a design parameter. Those controllers integrate MPC and SMC. Section 5 provides simulations to demonstrate the proposed method. Section 6 concludes the paper with some further remarks.

#### 2. Problem Formulation and Preliminary Results

We consider following constrained fuzzy systems: where is the state, is the input, is a nonempty compact convex subset including the original point , is the premise variable and is some function of and , denotes the th rule of the fuzzy model, is the number of fuzzy rules, and are input fuzzy terms in the th rule.

Assume that is continuous. The origin is assumed to be the balance point of the global model of system (1); that is, if system (1) is assembled into the global expression by using singleton fuzzifier, product inference, and center-average defuzzifier, then . In the following, when the context is clear, the time label will be omitted.

Define the cost functional where if and only if .

The value functional in the optimal control of (1) is

The optimal control problem (1) and (4) satisfies the following HJB equation:

*Remark 1. *For the detailed derivation of (5), please refer to [14, 15]. Such HJB equation (5) covers MPC of constrained fuzzy systems.

*Remark 2. *Throughout the rest of this paper, we use the following notations. For a scalar , sign function is
and the saturation function is
where and .

For a vector and , the vector sign function is defined as , the vector saturation function is defined as , and is defined as For the function and the function vector and , is defined as (Isidori [43])

#### 3. New Conditions for the Existence and Stability of a Sliding Mode

Since we vary for the closed-loop system (27) and (36) in simulations, the closed-loop trajectories produce the sliding mode motion. This directly motivates us to consider the reason why the sliding mode motion exists. But we find it impossible to apply the commonly used conditions to judge the sliding mode motion. We first give the two commonly used conditions for the sliding mode existence and then discuss the reason why they cannot work. First, define the sliding mode plane and some notations by Then, the two most commonly used conditions for the existence of a sliding mode are or Generally, is chosen by the designer as

The reasons why the commonly used conditions cannot work are as follows.(i)Since the sliding mode surface is not selected by the designer previously, this means that the sliding mode surface equation is unknown. But in conventional SMC, is a known function selected by the designer. This means the commonly used condition (12) for the existence of a sliding mode does not work.(ii)Since is unknown, the gradient is unknown. This means the commonly used condition (13) does not work.(iii)The stability of the sliding mode motion in the conventional SMC depends on the analysis of the equivalent control and the mean motion on the sliding mode surface . Since is unknown, the analysis of the equivalent control and the mean motion can not proceed.(iv)The controller design and the attraction region in the conventional SMC depend on the reaching condition. Since is unknown, the reaching condition cannot be obtained.

To deal with this encountered situation, the new conditions of the existence and the stability of the sliding mode for general constrained uncertain fuzzy systems are constructed by system (1) and the value function (4).

Theorem 3 (new conditions for the existence and stability of a sliding mode). *Suppose that the system is (1), is (4), and is a hyperplane. For , is chosen as
**
Define
**
For , define
**
The conditions for the existence and the stability of a sliding mode are
*

*Proof. *First, we need to prove . Since
we get
That is, .

From (17) and (20) and referring to Figure 1, the vector is a vector starting at and pointing to the space . For and , if (18) is satisfied, the projection of the vector onto the vector has the same direction as the vector . This means that moves from the inside of the space toward the sliding mode plane . For and , if (18) is satisfied, the projection of the vector onto the vector has the opposite direction to the vector . This means that moves from the inside of the space toward the sliding model plane . From the basic sliding mode principle (i.e., the trajectory of the system in the vicinity of the sliding mode plane moves from the inside toward the sliding mode plane), we know the hyperplane is the sliding mode plane.

From (15) and referring to Figure 1, the projections of and onto have the opposite direction to the vector . This means that moves from the higher level surface of the value function toward the lower level surface, so the sliding mode motion is stable. That completes the proof.