Abstract

A design method is established for the mixed output-feedback control of stochastic nonlinear systems with multiplicative noise. Firstly, using T-S fuzzy rules, we obtain a fuzzy model to approximate the original nonlinear system. Then, by Schur’s complement, the suboptimal output-feedback control design is transformed into a two-step convex optimization problem. A numerical example is given to show the effectiveness of the proposed method.

1. Introduction

One of the objectives of system control is to design a controller for the object model so that the closed-loop system achieves good performance while ensuring internal stability [14]. control and control have been attractive subjects since they are of great practical significance in the field of engineering [59]. control has a high request of model accuracy and generally does not consider the influence of model error. However, in practical control systems, the system could not exclude the implication of uncertain factors. Being put forward by Zames in 1981, nowadays the design idea has grown into an important robust control theory to eliminate external interference [10]. Accordingly, the combination of and control design methods will ensure the robustness and optimality of the controlled system at the same time (we refer the readers to [1114]).

It is noticeable that randomness is ubiquitous in the real world [15]. For example, there is a great deal of randomness in financial risk managements. Correspondingly, stochastic control has been an attractive subject in recent decades. Chen et al. [16] implemented a detailed study on stochastic control problems for linear systems with state-dependent noise. Subsequently, [1720] reported research progress on control of Markov jump systems.

On the other hand, nonlinearity is a universal phenomenon existing in engineering systems [21, 22]. Giving an example, a buck-boost circuit is rich in nonlinear dynamics. Generally speaking, control problems of nonlinear systems are more complicated than those of linear systems [2325]. Linearizing the nonlinear system has become mature technology to treat the nonlinear problems. [26] introduced a suitable linear model gained by the T-S fuzzy rule to approximate a nonlinear system. [27] designed a mixed controller for nonlinear systems based on fuzzy observer.

According to all above, robust control for stochastic nonlinear systems is definitely worthy both from the theoretical and practical application views. Compared with [27], in which the considered system model does not contain multiplicative noise, it is clear that control for nonlinear systems with multiplicative noise has broader application prospects. The other contribution of this paper is that the suboptimal output-feedback control design is transformed into a two-step convex optimization problem, which is convenient for solving by MATLAB efficiently.

This article is organized as follows. Section 2 builds up an approximate model of the original nonlinear system by the T-S fuzzy rule. Section 3 designs a fuzzy observer-based output-feedback controller by solving a two-step convex optimization problem. A numerical example is given to illustrate the efficiency of the proposed design method in Section 4. Section 5 gives the summary of this paper.

For convenience, we adopt the following notations:: the trace of matrix : the transpose of matrix : a positive semidefinite (positive definite) matrix : the identity matrix: the Euclidean 2-norm of the -dimensional real vector

2. Problem Description

Consider the following nonlinear random perturbation system:where is the state vector, is the input of the system, and is the measured output. We assume that is a one-dimensional standard Wiener process. , , and are supposed to be smooth functions. System (1) is influenced by which is a bounded measurement noise, that is, .

Using T-S fuzzy rule, we establish a linear fuzzy model for the stochastic model (1). Specifically, by the fuzzy rule, : if is , …, is , , then we havewhere represents the th rule, demotes the rule number, are the measurable prerequisite variables, is the fuzzy set, and , , are matrices with right dimensions.

By using the single point blur method, the product of reasoning, and the average weighted fuzzification, the following form of fuzzy model is obtained:where , , , and is the grade of membership of in . We suppose that , . It is easy to see

Thus, system (1) is equivalent to the following system:whererepresent the approximate error between system (3) and nonlinear model (1).

Select the finite-dimension compensator shown below:where is the fuzzy controller and is the control parameter.

Setting and , we get the following closed-loop system:where

Next we consider control performance. Given and weighting matrix , if ,then we call the performance is satisfied.

control aims to eliminate the influence of external interference, but the performance of the closed-loop system may not be ideal. Therefore, a mixed control design based on fuzzy observer will be implemented. performance is defined as follows:where and .

3. Output-Feedback Control Design Based on Fuzzy Observer

In the previous work, using the T-S fuzzy rule, we got a fuzzy model (3) and the approximate error between nonlinear system (1) and the fuzzy model. This section attempts to design an output-feedback control satisfying performance and performance for fuzzy model (3).

Let the following inequalities be true:

By computation and the above inequalities, we havewhere , , , .

For the smooth progress of subsequent work, let us choose a Lyapunov function for system (8):where is a weighted matrix with appropriate dimensions.

By integrating (14), we have

Based on (15), we can derive the following theorem.

Theorem 1. If there exists a satisfying the following inequalities:where , then(a) control performance (10) is fulfilled.(b) performance (11) has a upper bound, that is,

Proof. For given , by Schur’s complement and (15), one can see thatwhere . Therefore, is directly derived, i.e., conclusion (a) is valid.
Now let us prove (b). Under the constraint of (8), with the help of the method of completing square, we assert thatwhich shows that (b) holds. The proof of this theorem is concluded.
According to Theorem 1, suboptimal control design has been transformed into solving the optimization problem under the constraint of (16) and (17). However, because , , and are coupled in some components, the optimization problem is not convex. So, we need to convert it into convex optimization problems.
Express , , and as follows:Plugging these representations into (16) and (17), one getswhereNext, let with . Multiplying both sides of (22) and (23) by and setting , , we havewhereBy Schur’s complement, (25) and (26) can be rewritten aswhereNoticing that if (16) and (17) are true, then and , the following two inequalities hold:which can be written as the following LMIs:whereTherefore, the observer-based suboptimal stochastic control design can be transformed into solving a two-step convex optimization problem.
The first step: under the constraint of (31) and (32), solve the convex optimization problem:It can be obtained that , and .
The second step: under the constraint of (25) and (26), solve the following convex optimization problem:We can get and the feedback gain . A suboptimal solution and compensator (7) are achieved.
To sum up, we state the following main result.

Theorem 2. If the above convex optimization problems (34) and (35) have solutions, then and . Moreover, we have , and .

4. A Numerical Example

For system (5), we define the fuzzy number as “big and small” and assume its coefficient matrices are

Give

We have , and , .

Choose . Using LMI toolbox in MATLAB, we get

According to , , the parameters of observer and controller are

Taking the controller into account, the simulation results are shown in Figure 1. It is shown that the system can achieve the desired control effects under the fuzzy controller.

5. Conclusions

In this paper, the mixed output-feedback control problem for stochastic nonlinear systems in a finite horizon has been studied. Firstly, the nonlinear system is transformed into a linear fuzzy model by T-S rules, and the error between the original system and the fuzzy one has been considered. A fuzzy observer-based two-step convex optimization method has been proposed to treat the suboptimal problem. The method is simple and effective. The closed-loop system can guarantee the robustness and minimize the energy output. Since time delays exist widely in practical systems, how to generalize the obtained output-feedback controller design method to stochastic nonlinear systems with delays is one of the directions of future research.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China under grant no. 62073204, Key Research and Development Plan of Shandong Province under grant no. 2019GGX101052, and Natural Science Foundation of Shandong Province under grant no. ZR2020MF071.