Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 154158, 10 pages

http://dx.doi.org/10.1155/2015/154158

## Robust Output Feedback Model Predictive Control for a Class of Networked Control Systems with Nonlinear Perturbation

^{1}Designing Institute, Hangzhou Hangyang Co., Ltd., Hangzhou 310014, China^{2}Department of Information Engineering, Research Center for Smart Agriculture and Forestry, Zhejiang A&F University, Lin’an 311300, China

Received 20 June 2014; Revised 27 August 2014; Accepted 29 August 2014

Academic Editor: Yun-Bo Zhao

Copyright © 2015 Qiuxia Chen 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.

#### Abstract

This paper is concerned with the design problem of robust dynamic output feedback model predictive controllers for a class of discrete-time systems with time-varying network-induced delays and nonlinear perturbation. The designed controllers achieve on-line suboptimal receding horizon guaranteed cost such that the system can be stabilized for all admissible uncertainties. A novel delay compensation strategy is proposed to eliminate the effects of the time-varying network-induced delays. By using multistep prediction and the receding optimization, the delay-dependent sufficient condition is derived for the existence of delay compensation controllers. By employing the cone complementarity linearization (CCL) idea, a nonlinear minimization problem with linear matrix inequality (LMI) constraints is formulated to design the desired output feedback controllers, and an iterative algorithm involving convex optimization is presented to solve the nonlinear minimization problem. Finally, an example is given to illustrate the feasibility and effectiveness of the proposed results.

#### 1. Introduction

With the rapid development of digital systems and communication networks, more and more control engineers would like to use a real-time communication channel interfaced to a digital system to exchange information and to complete the control task. Such networked control systems (NCSs) have received increasing attention in recent years because of their many advantages, such as lower cost and more convenience for installation and maintenance. Industrial applications of NCSs include automobiles, vehicle systems, robotic systems, jacking systems for trains, and process control systems [1–3]. However, the streams of data exchange between NCS components are prone to delay, losses, and missynchronization, which degrade the performance and even cause the instability of the systems [4, 5]. Network-induced delays typically have negative effects on the NCS’s stability and performance. So far, different techniques have been presented to deal with the problem of network-induced delays, such as the stochastic system approach [6, 7], the hybrid system approach [8, 9], and the time-delay system approach [10, 11]. It is noted that, in the aforementioned results, all the proposed controllers are operated in an off-line fashion, and the controllers are designed such that the overall closed-loop NCSs can tolerate certain amount of network-induced delays. Another constructive scheme is to use future control sequences to directly eliminate the negative effects of the network-induced delays, and model predictive control (MPC) which can provide future control sequences and operate in an on-line fashion is such a desired algorithm.

Model predictive control is probably one of the most successful modern control technologies during recent years. This is due to several thousand applications in process control because of its many advantages, including ease of computation, good tracking performance, and I/O constraint handling capability. Recently, MPC has received increasing attention in NCSs for its ability to on-line compensate time delays as well as its good tracking performance. Reference [12] proposed a new approach of predictive compensation for simultaneous network-induced delays and packet losses and addressed the codesign of both network and controller. In [13], a model-based networked predictive output tracking control scheme is proposed to actively compensate for the random round-trip time delay. Reference [14] proposed a data-driven networked predictive control scheme which consisted of the control prediction generator and network delay compensator for MIMO NCSs with random network delays. However, all the above designed model predictive controllers are state feedback controllers which are quite difficult for implementation when the system states cannot be directly measured for availability. Therefore, it is necessary to design a feedback controller using the measured output. To the best of the authors’ knowledge, few results have been reported on the dynamic output feedback MPC for NCSs [15]. Besides, the above approaches are not applicable to the case that there are parameter uncertainties in the model. Since the uncertainties are frequently the sources of instability and performance deterioration, the stability analysis and controller synthesis for NCSs with model uncertainty and external disturbances have been some of the most challenging issues. Up to now, a few robust MPC algorithms have been presented to improve the robustness of the networked control systems [16, 17].

Taking the network-induced delay into consideration, the NCSs can be modeled as an uncertain discrete-time system with time-varying delays, and this motivates us to apply the theory of time-delay systems and robust MPC strategy to design the feedback controllers for such NCSs. The main results of this paper will contribute to the development of the delay-dependent dynamic output feedback robust MPC for a class of NCSs with norm-bounded nonlinear perturbation and time-varying communication delays. A sufficient delay-dependent condition that guarantees the robust stability of the closed-loop NCS is derived. An optimization problem is also formulated to construct the dynamic output feedback MPC controllers subject to a set of nonlinear matrix inequalities. Based on the linearization idea [18], an iterative algorithm involving convex optimization is proposed to solve the nonlinear matrix inequality system, and the iterative optimization algorithm is guaranteed to be feasible at each time step if it is feasible at the first step. The control inputs applied to the system from the solutions of the MPC optimization problems guarantee an on-line suboptimal receding horizon guaranteed cost. Finally, an example is given to illustrate the effectiveness of the proposed results.

*Notation*. Throughout the paper, stands for the set of all real -dimensional vectors and is the set of all -dimensional matrices. denotes identity matrix of appropriate dimensions; denotes the block diagonal matrix. (, , ) means that is a real symmetric positive-definite matrix (positive-semidefinite, negative-definite, and negative-semidefinite). denotes the symmetric part.

#### 2. Problem Formulation and Preliminaries

The detailed assumptions about the NCS studied in this paper are described as follows.(1)The sensor is clock-driven and has the sampling period , and the controller and the actuator are event-driven.(2)The sensor-to-controller delay and the controller-to-actuator delay are both uncertain but bounded and can be obtained by best case analysis and worst case analysis. The total time delay satisfies , where and are known positive integers corresponding to maximum and minimum of .(3)Controller computational delay can be absorbed into either or [19].

The uncertain discrete-time system with nonlinear perturbation is described by the following state space model: where is the state, is the control input, is the measured output, and , , and are known real constant matrices of appropriate dimensions. and are unknown matrices representing time-varying parameter uncertainties in the system model. It is assumed that the uncertainties are norm-bounded and can be described as , where is an unknown matrix satisfying and is a known positive scalar representing the upper bound of the unknown matrix , and , , and are known constant matrices of appropriate dimensions. The nonlinear function satisfies the following: where and are known constant matrices and is the bounding parameter on the uncertain function .

The physical system consisting of sensor and actuator nodes is connected to the controller through a communication medium. For the convenience of system analysis and controller design, the sensor-to-controller delay and the controller-to-actuator delay are lumped up as the feed-forward delay . Considering the delay effect, the input of the controller can be given as , and then the full-order dynamic output feedback controller to be determined has the following form: where is the controller state.

Applying controller () to the system () results in the following networked closed-loop system: where

In this way, the NCS () and () with nonlinear perturbation and communication delays is modeled as the uncertain discrete-time system () with time-varying state delay, which enables us to apply the theory of time-delay systems and the receding optimization of the MPC to deal with the analysis and design problem of such NCS.

*Remark 1. *Equation () is used to express the mathematical model of the networked control systems when the transmitted data is single packet. For the multiple-packet transmission case, since the arrival time of the sensor messages at the controller or the arrival time of the controller messages at the actuator may be different, especially for the case when the sampling times of the sensors are different, a buffer before the controller and actuator is needed. By employing the buffer technology on the network, model () can also be used to express the NCS with multiple-packet transmission.

The objective of this paper is to find a stabilizing dynamic output feedback controller of the form () for the uncertain system () with time-varying delays by MPC strategy. To this end, we define the following min-max optimization problem, which is considered at each sampling time :
where is the control horizon, is the prediction horizon, and are given weighting matrices, and and denote the predicted variables of the state and the input, respectively, with , , and for . Besides, we have the terminal constraints for .

Associated with the closed-loop system (), the min-max optimization problem () becomes of the following form:
where , , and .

The future control sequence can be obtained by solving the above optimization problem. In order to eliminate the delay effects, the control input should be but not . By using the multistep prediction and the linear interpolation method, the control sequence can be obtained. In the receding horizon framework, only the first control variable actuates the system in time.

Lemma 2 (see [20]). *Let and , be given matrices with appropriate dimensions. Then, it follows that
**
holds for any matrix satisfying , if and only if there exists a scalar such that
*

*3. Main Results*

*In this section, we first derive an upper bound on the worst value of the cost .*

*Theorem 3. Consider the uncertain system () and the cost function . If there exist a controller of the form (), a scalar , , , , and matrices , , such that the following matrix inequality holds
where
then the closed-loop system () is asymptotically stable and the cost function satisfies the following:
where .*

*Proof. *Choose the following candidate Lyapunov functional:
where
Taking the forward difference for the Lyapunov functional , one obtains
where
Direct computation gives
Note that
Hence, we have
is computed as follows:
By the well-known inequality , we obtain
where , .

Combining ()–() yields
where
and implies that ; that is, is degenerated. Therefore, the closed-loop system () is asymptotically stable. Besides, for any integer , we have
To guarantee the existence of the upper bound on the robust performance index , we must have , ; hence, , . Let , and then ; we get ().

By Schur complement, () is equivalent to the following matrix inequality:
where .

By -procedure, is equivalent to the existence of matrices , , , , and and a scalar such that
Define matrix
and then inequality () can be rewritten as
By Lemma 2, inequality () holds for any matrix satisfying , if and only if there exists a positive scalar such that
By Schur complement, inequality () is equivalent to (). This completes the proof.

*It is noted that in the matrix inequality (), the controller parameters , , and are unknown and occur in nonlinear fashion; therefore, () is not an LMI problem. In the sequel, we will use a method of changing variables [21] to obtain an equivalent matrix inequality representation of the nonlinear matrix inequality (), which enables us to use the CCL technique to design the output feedback controllers.*

*Now, we present a sufficient condition for the existence of the output feedback delay compensation controller of the form () for the NCS ().*

*Theorem 4. Consider the NCS () and the cost function . Suppose that for some prescribed matrices , ; there exist scalars , , matrices , , , and , and matrices , , , , and such that the following optimization problem is feasible at the initial time step :
and then the networked predictive control derived from the solutions to the above optimization problem robustly asymptotically stabilizes the system (), and the cost function satisfies the bound
where
*

*Proof. *First, partition and its inverse as
where and . Note the identity gives
Define
Then,
Define the new controller variables as
Therefore, given , , and invertible matrices and , the controller matrices , , and can be uniquely determined by , , and .

Pre- and postmultiply () by and by , respectively, set , , , , , , , and , and consider the change of controller variables () and Schur complement, and then () can be obtained.

By Theorem 3, the original min-max problem () can be redefined as the following optimization problem that minimizes an upper bound on the worst value of the original cost function :
By Schur complement, is equivalent to the following matrix inequality:
where
Define and pre- and postmultiply () by and , respectively, and it follows from () and that inequality () is equivalent to inequality (). This completes the proof.

*Remark 5. *Theorem 4 presents an optimization problem to construct the desired output feedback model predictive controllers. Note that the conditions in Theorem 4 are no more LMI conditions due to the terms , , , , , and . As a result, we cannot find a minimum guaranteed cost by using convex optimization algorithms. However, by using a complementarity idea [18], we can cast the original nonconvex optimization problem to a nonlinear minimization problem involving LMI constraints and, by applying a related iterative algorithm, some suboptimal guaranteed costs can be obtained.

Replace the terms , , , and in , by , , , and , respectively, and denote the obtained matrices by , , respectively. Let the cost bound be lower than some specific value , and then the nonlinear minimization problem involving LMI constraints can be formulated as follows:
If the minimum of the above nonlinear minimization problem is 3, that is, , we can say from Theorem 4 that the closed-loop system () is asymptotically stable with guaranteed cost . We propose an iterative algorithm shown in the following paragraph to solve the above nonlinear minimization problem. Since it is numerically very difficult in practice to obtain the optimal solution such that is exactly equal to 3, we use () and () as a stopping criterion in the iterative algorithm, and, thus, only some suboptimal guaranteed costs can be obtained within a specified number of iterations.

*Now, we summarize the proposed robust networked predictive control algorithm as follows (Figure 1).*