Robust Constrained Model Predictive Control for T-S Fuzzy Uncertain System with Data Loss and Data Quantization

Qu, Hongchun; Li, Yu; Liu, Wei

doi:https://doi.org/10.1155/2021/8865701

Complexity

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Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 8865701 | https://doi.org/10.1155/2021/8865701

Robust Constrained Model Predictive Control for T-S Fuzzy Uncertain System with Data Loss and Data Quantization

Hongchun Qu,¹Yu Li,²and Wei Liu²

Academic Editor: Alex Alexandridis

Received08 Sept 2020

Revised19 Apr 2021

Accepted21 Jun 2021

Published07 Jul 2021

Abstract

This paper addresses the robust constrained model predictive control (MPC) for Takagi-Sugeno (T-S) fuzzy uncertain quantized system with random data loss. To deal with the quantization error and the data loss over the networks, the sector bound approach and the Bernoulli process are introduced, respectively. The fuzzy controller and new conditions for stability, which are written as the form of linear matrix inequality (LMI), are presented based on nonparallel distributed compensation (non-PDC) control law and an extended nonquadratic Lyapunov function, respectively. In addition, slack and collection matrices are provided for reducing the conservativeness. Based on the obtained stability results, a model predictive controller which explicitly considers the input and state constraints is synthesized by minimizing an upper bound of the worst-case infinite horizon quadratic cost function. The developed MPC algorithm can guarantee the recursive feasibility of the optimization problem and the stability of closed-loop system simultaneously. Finally, the simulation example is given to illustrate the effectiveness of the proposed technique.

1. Introduction

Model predictive control (MPC) is an effective advanced control method that has aroused extensive attention among the academic and industrial communities in the past decades [1–3]. The main idea of MPC is online solution of an optimization problem to obtain a sequence of optimal control inputs; then, only the first one is implemented and it repeats this procedure with new measurements at the next sampling time [4]. The defining feature of MPC comes from its ability to handle the multivariable systems with hard constraints. A large amount of results on MPC of ideal systems without model uncertainties were reported in [5–7]. However, model uncertainty exists in practical industrials, and hence robust MPC is of theoretical and practical significance and need to be considered. To solve this problem, the authors in [8] firstly proposed a robust MPC synthesis approach for the uncertain system by solving a linear matrix inequalities (LMIs) optimization problem which explicitly considered the input and state constraints. The key point of [8] was to utilize the concept of robust invariant set to guarantee the recursive feasibility of the LMI optimization problem. Since then, much progress has been made in the research of MPC synthesis approach. There are mainly two aspects: one is to reduce the computational burden (see [9, 10]) and the other is to improve the control performance (see [11]).

Meanwhile, networked control systems (NCSs) have been a hot research topic in the last decade which combines sensors, controllers, and actuators via a shared communication network. Compared with traditional point-to-point control systems, NCSs bring incomparable advantages, such as high reliability, easy installation and maintenance, and excellent flexibility [12, 13]. However, some phenomenons, such as data loss and quantization, inevitably exist in the communication channels which would deteriorate the performance or even destroy the stability of the systems. There have emerged many works about the controller design and stability analysis for the NCSs with packet loss and/or quantization (see, e.g., [14–16]). Works on the stabilization problem for NCSs with packet loss can be found in [17–20]. Some nice results about the quantized feedback control problem for NCSs were reported in [21, 22]. Further, most recently, some researchers considered more complex network environment where packet loss and quantization were coexisted in NCSs [23–27].

Some interesting results are presented in the aforementioned documents; however, all of the results are under the assumption that the controlled plant is a linear system without system uncertainty. Unfortunately, almost all the systems in industry are nonlinear with uncertainty. The research studies on the uncertain system have been made by many researchers, and several common descriptions of uncertain systems can be found in [8, 28, 29], such as polytopic paradigm and structured uncertainty. In addition, Takagi–Sugeno (T-S) fuzzy model is employed in this paper to describe the nonlinear system. As we all know that the T-S fuzzy model can transform the nonlinear system into a set of linear submodels by IF-THEN rules instead of handling it directly. In the analysis of T-S fuzzy systems, the controller is designed based on the linear submodels, while the implementation of the controller clearly depends on the membership functions which explicitly consider the nonlinearity of the system. The local linearities of the T-S fuzzy system provide a bridge between the analysis of nonlinear systems and the fruitful linear control theory results [30–35]. For the T-S fuzzy system, some interesting technologies are introduced for the purpose of reducing conservatism, such as nonquadratic Lyapunov function [36, 37], extended nonquadratic Lyapunov function [38, 39] with nonparallel distributed compensation (non-PDC) law, and slack matrices [40]. Recently, some researchers begin applying the T-S fuzzy model to describe complex nonlinear plant in NCSs, and more investigations have been carried out. The controller design and stability analysis for T-S fuzzy model-based NCSs with data losses were considered in [41, 42], and for NCSs with quantization, the same problems were addressed in [44, 45].

There also have been some nice works that introduce MPC to deal with the problem for NCSs represented by the T-S fuzzy system (see, e.g., [43, 46, 47]). The authors in [43] used Bernoulli random binary distribution to model data loss occurring intermittently between the controller and the physical plant. Meanwhile, piecewise approach is more effective for fuzzy systems with trapezoidal membership functions. Consequently, the fuzzy predictive control problem of nonlinear NCSs subject to parameter uncertainties and data loss was considered in [46]. The difference between [43, 46] is that the authors in [46] employed slack matrices to develop less conservative conditions for stability analysis of the closed-loop system. Yu et al. [47] considered more intricate network where data loss and quantization were existed simultaneously. In [48], a fuzzy predictive controller which guaranteed the stability of the closed-loop system was designed in terms of sector bound approach. Tang et al. [49] investigated the output feedback model predictive control for networked control systems with packet loss and data quantization. Although some remarkable results are presented in the foregoing documents, most of the T-S fuzzy systems are described without uncertainty. In addition, the same problem for NCSs, packet loss, data quantization, and the model predictive control strategy, were considered in both of this paper and [49]. However, this paper is different from [49]. First, the interval-type 2 T-S fuzzy model is introduced to describe the nonlinear NCSs, while the type 1 T-S fuzzy model is applied in this paper and the system uncertainty is also considered which is not involved in [49]; second, the result for [49] was obtained based on the assumption that the system states were not measurable and the output feedback MPC strategy was addressed, while this paper assumed that the system states can be sampled and state feedback MPC approach was investigated. It is noted that different assumptions resulted in completely different solutions. Indeed, few works are focused on the problem of reducing the conservatism for fuzzy MPC for NCSs which motivates our research. To illustrate this research more clearly, we provide Table 1 to compare published works with this paper.

This paper focuses on the synthesis approach of the robust constrained MPC with quantization and data loss for the uncertain fuzzy system. The quantization error is regarded as sector bound uncertainties by using the sector bound approach, and the data loss process is described by Bernoulli distributed sequence. A new stability condition for NCSs is achieved based on extended nonquadratic Lyapunov function and additional slack and collection matrices. By optimizing an objective function in the infinite time horizon at each sampling time, a robust MPC optimization problem which explicitly considers the data loss and quantization is presented. Furthermore, the optimization problem is proved to be recursive feasible, and the networked control system turns out to be asymptotically stable.

The remainder of this paper is summarized as follows. The problem formulation is given in Section 2. Section 3 provides the new stability results for fuzzy NCSs. Online synthesis of MPC for fuzzy NCSs is presented in Section 4. Section 5 gives the analysis of feasibility and stability. Section 6 presents a simulation example, and Section 7 draws the conclusion.

Notations. For any vector and positive-definite matrix , . and denote n-dimensional Euclidean space and the set of real matrices, respectively. is the identity matrix with appropriate dimension. For any vector, is the value of at sampling instant . The superscript denotes the transpose of matrix or vector. The symbol stands for a symmetric element or submatrix. For simplicity, and are used in place of , and , , respectively. , , , and , with .

2. Problem Formulation

Consider a discrete-time T-S fuzzy model with rules and describe its th rule aswhere stands for the fuzzy set of rule and is the corresponding premise variable which depends on the states of the system; and denote the system state and control input, respectively; and are known matrices; and and are introduced to describe the system uncertainty, which satisfywhere is an uncertain matrix and , , and are known matrices.

According to the above discussion, the definition of the inferred T-S fuzzy model can be represented as follows:wherein which represents the weight of the th rule and is the membership grade of in and satisfies

In this paper, we assume that the communication network exists in sensor-to-controller (S/C) and controller-to-actuator (C/A) links, and the configuration of quantized NCSs with data loss is shown in Figure 1. At time , the output of controller is quantized aswhere denotes a logarithmic quantizer which is defined asin which and is the quantization density. The set of quantization levels is presented as follows: , , . Each quantization level corresponds to a segment such that the quantizer maps the whole segment. Based on the sector bound approach in [48], the control input can be expressed as follows:where .

When and are transmitted in the communication channel, it may be lost due to the poor network traffic. Hence, two Bernoulli processes are introduced to model the data loss in NCSs.where is input of controller. and are two mutually independent Bernoulli processes. (1 means that the data are transmitted successfully and 0 indicates that data packet is lost).

Then, we assume that and satisfy the following conditions:

In the following, the fuzzy controller is presented.where and are the controller gains. Then, the state feedback fuzzy controller is defined by

Based on (9) and (12), the following state feedback fuzzy controller is achieved:

Let . It is obvious that if and , ; otherwise, . Thus, we have

According to (3), (8), and (13), the overall uncertain closed-loop fuzzy system is described as follows:where , , , , , and . , , and . Obviously, and .

3. New Stability Results for Fuzzy NCSs

Compared with the conditions of stability for traditional control fuzzy systems, a new stability condition which can reduce the conservatism is presented in this section by introducing the notation of slack matrices. This new stability condition is the foundation of the MPC for fuzzy NCSs which will be provided in the next section.

First, the following two lemmas that play a significant role in the proof of Theorem 1 are introduced.

Lemma 1 (see [50]). For vectors and and matrices , , , and of any appropriate dimensions, if , then the following inequality holds:

This inequality is generally called Moon inequality. It can be seen that if and , then . For any and any positive-definite matrix , the following inequality holds:

Lemma 2 (see [51]). Symmetric matrix and matrices satisfy the following inequality:for , if there exists a scalar such that

Next, the following theorem is given to obtain the conditions of the stability for NCSs.

Theorem 1. Considering the T-S fuzzy system (3), the asymptotic stability is guaranteed if there exist scalars , matrices , , and , symmetric matrices and , and such thatwhere

Proof. See Appendix A for details.

4. Online Synthesis of MPC for Fuzzy NCSs

Based on the stability condition achieved in the above section, the online synthesis of MPC is derived for fuzzy NCSs with the quantization and data loss. Furthermore, the model uncertainty and physical constraints are considered.

The following non-PDC fuzzy predictive controller is used for the T-S fuzzy system:

In view of (25), we can get the predictive model of uncertain T-S fuzzy NCSs:where , and .

The following is the optimization problem for the online constrained robust fuzzy MPC, which is optimized in the infinite time horizon at each time:where , with and the weighting matrices. It is obvious that this objective function is related to the model uncertainty, quantization density, and expected values of stochastic variables.

The following extended nonquadratic Lyapunov function is considered:

Then, imposing the following stability constraint at each time,

Summing up both sides of (29) from to , we have

As we all know, when , . Then, we can obtain . Thus, , which is equal towhere .

Hence, the problem of minimizing is transformed into minimizing a scalar satisfying

Moreover, input and state constraints arewhere and are the component of the vector and , respectively.

Can be solved by the following theorem.

Theorem 2. Consider the uncertain fuzzy NCS (26) and assume that the communication link parameter and quantization density are known. The non-PDC state feedback control law (25) can guarantee the robust asymptotically stability of the closed-loop fuzzy system if there exist scalars , , , , and , positive-definite matrices , , and , and any matrices , , , and such that the following minimization problem is feasible:subject towhere

Proof. See Appendix B for details.

Then, we handle the input constraint (33). For any , we have

If there exists a symmetric matrix X that satisfies

By applying the Schur complement and [52], (45) becomeswhich equals to

Using Lemma 2 and Schur complement, it yieldswhere is a positive-definite matrix. In consequence, (33) is guaranteed by (49) through the convex feature of and .

In addition, we deal with the state constraint (34). For any , we can achieve

If there exists a symmetric matrix that satisfies

Using Lemmas 1 and 2 and Schur complement, (50) holds iffor all satisfying . Equation (52) can be transformed into

According to Lemma 1 and utilizing Schur complement, the following inequality is obtained:where and is a positive-definite matrix. We can conclude that (34) is guaranteed by (53). The proof is completed.

Based on the above discussions, the solution of (35) can be obtained by solving the following optimization problem:where . At time instant , optimization problem (54) is solved online and we can get optimal control sequence . Then, only the first control input is implemented.

The implementation steps of designing RMPC are summarized in Algorithm 1.

(1)	Set the communication link parameter and quantization density
(2)	At time , solve the optimization problem (54) and get ,
(3)	Using (25), calculate the optimal control sequence
(4)	Quantize and transmit the first control input into the communication network
(5)	Update time to and then return to step (2)

The complexity of solving the LMI optimization problem is polynomial time, which (regarding the fastest interior-point algorithms) is proportional to , where is the total number of scalar variables and is the total row size of the LMI system. For the optimization problem which is described by (54), the total number of scalar variables is and the total row size of the LMI system is , where is the number of inference rules. It can be seen that the increase in , and will increase the computational complexity exponentially. Let and take the most influential parts of and , respectively. Then, the computational complexity is .

5. The Analysis of Feasibility and Stability

5.1. The Analysis of Feasibility

Theorem 3. Consider the uncertain fuzzy NCS (26), if any feasible solution is achieved from the optimization problem (55) at time , then the optimization problem (55) is feasible for all time .

Proof. See Appendix C for details.

5.2. The Analysis of Stability

Theorem 4. Consider the uncertain fuzzy NCS (26). The control input , which is applied in a receding horizon way by solving optimization problem (55), robustly asymptotically stabilizes the closed-loop fuzzy NCSs.

Proof. See Appendix D for details.

6. Simulation Example

In this section, the following example of a continuous stirred tank reactor (CSTR) is given to illustrate the effectiveness of the proposed method.

Example 1. As shown in Figure 2, consider the nonlinear model of CSTR which has been described in [11].
Assuming constant liquid volume, the CSTR liquid volume, the CSTR for an exothermic, irreversible reaction, i.e., , is presented by the following dynamic model based on a component for reactant A and an energy balance:where , , and denote the concentration of A in the reactor, the reactor temperature, and temperature of the coolant stream, respectively.
The aim of using CSTR is to regulate by manipulating , satisfying the constraint . The system state and input variables are set as , . In addition, let . In this example, we choose as the premise variable of the fuzzy system and , where and . Then, the following equations are defined:The membership functions are described asThus, (55) can be transformed into the following T-S fuzzy model:Choose sampling period . The values and meaning of other parameters are given in Table 2.In this section, for simplicity, we adopt the following T-S fuzzy model:withAccording to the above discussion, the T-S fuzzy model can be summarized aswithConsidering the uncertainty of the T-S fuzzy system, the following conditions must be satisfied:with , , . For more rational analysis, we assume that is norm-bounded and satisfies , .
The control input and state constraints must satisfy . We choose the initial conditions as and . Let and . Assume . The comparison in this example is made in the view of Theorem 2 with quantization density and the method in [43], which provided a designing approach of MPC for fuzzy NCSs with data loss. The state feedback gains at , and the performance costs for different methods are shown in Table 3. As we can see from Figures 3–5 and Figures 6–8, the state, upper bound of the objective function and control input all converge to zero as . However, some differences can be found from these figures. The convergence speed of Theorem 2 in this paper is faster than that of the method in [43]. Further, the control performance of the system in this paper is better than that in [43]. Figures 9 and 10 show the state trajectories and invariant sets of these two methods which show the superiority of the presented approach. It is obvious that the recursive feasibility and asymptotic stability of the closed-loop T-S fuzzy system can be guaranteed, which illustrates the effectiveness of our method.

7. Conclusion

In this paper, the robust constrained MPC scheme with quantization and data loss for the uncertain fuzzy system has been studied. Compared with other documents, the key idea of our approach is that the non-PDC control law and nonquadratic Lyapunov function are employed in the case of taking the problem of NCSs into consideration. Based on the nonquadratic Lyapunov function, the new conditions for stability related to slack and collection matrices are obtained. Further, the fuzzy controller is designed, and online synthesis of MPC for fuzzy NCSs is presented by solving the optimization problem in view of non-PDC control law and an extended nonquadratic Lyapunov function, respectively. In future work, we will focus on reducing the impact of network-induced delay through the MPC method or study the problem of reducing computational burden of the networked MPC method.

Appendix

A. Proof of Theorem 1

Consider the nonquadratic Lyapunov functionwhere and are in a convex sum property for all . Then, the following condition must be satisfied:

The following condition is put forward, so the asymptotic stability of (3) is guaranteed:where . Premultiplying and postmultiplying (A.3) by and and utilizing the inequality , inequality (A.3) is turned into

Using Schur complement, we can see that (A.4) becomes

Using (2), we have

Applying Lemma 2, inequality (A.6) holds for all admissible uncertainties satisfying if and only if there exist some constants and such that

Applying Schur complement, we havewhich is equivalent towhere .

Utilizing Lemma 1, one can get

Condition (A.9) holds if

By applying Schur complement, the following is achieved:

Equation (A.12) guarantees (24) on account of the convex sum property of membership functions. Resorting to (21) and (22), the above inequality yields

Therefore, there always existsor equivalently

It is obvious that the uncertain fuzzy system in (1) is asymptotically stable via the fuzzy controller (13), and the proof is completed.

B. Proof of Theorem 2

Set . According to (32) and Schur complement, one can get

Due to , (37) is achieved.

Considering (26), (28), and (29), we have

Premultiplying and postmultiplying both sides of (B.2) by and and using the inequality , we havewhere . Because of the uncertainty of the fuzzy system and , the following inequality is obtained:

Using Lemma 2 and Schur complement, if and only if there are and satisfyingwhere

Employing the similar method to deal with quantization density in Theorem 1, the following inequality is established:where

By utilizing Schur complement, one can getwherewhere . Owing to , (42) is guaranteed. Using the same method which is described in [53, 54], based on (B.9), one can getwhere , , and .

Introducing matrices , , and , it yields

From (B.11) and (B.12), we can get

Under the above discussion, (B.9) is established and the stability of the T-S fuzzy system (15) is guaranteed. Obviously, (B.12) equals to

Therefore, we can get (40) and (41). The proof is completed.

C. Proof of Theorem 3

It is obvious that only (37) is related to state ; in other words, we only need to prove that this LMI is still feasible for all future measurable system states .

At time , we assume that and are the optimal solution which are derived from the optimization problem (55). Since (37) is satisfied, we havewhere . Then, a feasible solution is established at time . Referring to Theorem 3, we can conclude that and

We infer that (37) is feasible at time . Thus, it can be summarized that the optimization problem (55) is also feasible for all times , and the proof is completed.

D. Proof of Theorem 4

In this paper, in order to guarantee the asymptotic stability of the fuzzy system, we choose the extended nonquadratic Lyapunov function. Thus, if we can guarantee that the extended nonquadratic Lyapunov function is strictly decreasing, the asymptotic stability of the fuzzy system is guaranteed.

According to Theorem 3, the feasibility of optimization is guaranteed. At time , if we obtain the optimal solution , it means that the solution at time must satisfywhere and . Then, considering (29) and setting , we havewith and . Since , we can get

In summary, the strictly decreasing of the extended nonquadratic Lyapunov function is proofed except . Therefore, the closed-loop fuzzy system is robust asymptotically stable, and the proof is completed.

Data Availability

Data availability and data simulation program used to support the findings of this study have been deposited in the figshare repository https://doi.org/10.6084/m9.figshare.12947621.v1.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2021 Hongchun Qu et al. 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.

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