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This paper deals with the problem of robust model predictive control (RMPC) for a class of linear time-varying systems with constraints and data losses. We take the polytopic uncertainties into account to describe the uncertain systems. First, we design a robust state observer by using the linear matrix inequality (LMI) constraints so that the original system state can be tracked. Second, the MPC gain is calculated by minimizing the upper bound of infinite horizon robust performance objective in terms of linear matrix inequality conditions. The method of robust MPC and state observer design is illustrated by a numerical example.

Model predictive control (MPC) [

From such a viewpoint, it is a significant problem to develop the MPC algorithms, which are robust against model uncertainties, and guarantee a certain control performance objective [

With the increasing requirement of reliability, environmental sustainability, and profitability, we begin to apply the communication networks to the practical industrial process [

In this paper, we describe an industrial process system as the linear time-varying system with packet dropouts from the MPC controller and dynamic output controller to the plant. Furthermore, a Bernoulli random binary distribution with known probabilities is used to describe the packet dropouts phenomena. Thus, the MPC controller can be analyzed and designed to guarantee that the closed-loop system is stochastically stable. Additionally, the Lyapunov function is adopted to handle the problem of designing the controller and state observer, which make the result less conservative. A numerical example is proposed to prove the effectiveness of the design method. The main merit of this paper is the following one: a robust MPC controller is developed for a control system with uncertainties, saturations, and packet dropouts under time-varying probabilities.

The organization of the paper is given as follows. The considered problem is denoted in Section

First, we consider the linear time-varying system as follows:

In this paper, the system matrices

In this paper, we consider a state observer to estimate the state of the plant (

The MPC controller is to be determined as follows:

The form of dynamic output controller is as follows:

Combining (

In practical industrial processes, we have to not only consider the uncertainties in (

Here, we consider a stochastic variable

In a similar situation, we consider stochastic variables

The following Lemma is used in the process of proof.

Let

In this paper, the aim of the robust MPC is to determine the model predictive control law

We consider transferring the minimization of

Firstly, we construct a Lyapunov function

In this section, we consider the design of state observer. The form of state observer is (

Next we use LMI to determine the state observer gain

If there exist

Inequality (

Now our goal is to design a robust MPC to generate an optimal control law so that the performance objective can be achieved. In the following, Theorem

Assume that the uncertainty

subject to

At each future time instant, there exists the optimal control law to minimize the performance objective. Finally, the closed-loop system will be asymptotically stable.

In order to achieve the robust MPC performance objective under input constraint and packet dropouts, there exist three LMIs that need to be feasible. The first LMI is obtained by (

Let

Assume that the uncertainty

Let

Consider the following uncertain polytope system:

Another weighting matrix

The state observer gain

In Figure

The state response of closed-loop system.

Estimation of original system state

Estimation of original system state

The control input signal

The output of closed-loop system.

Consider the following uncertain polytope system:

The dynamic output controller gains,

In Figure

The state response of open-loop system.

The state response of closed-loop system.

The state response of dynamic output controller.

The control input signal

The control output signal

The output of closed-loop system.

In this paper, we have designed an output-feedback MPC to solve the problem of the robust MPC with input constraints and successive packet dropouts. The method makes use of infinite horizon min-max algorithm with LMI constraints. First, we have constructed a state observer. Then, the optimization problem can be solved by dealing with some LMI constraints. We can obtain the control input sequence by dealing with the infinite horizon robust MPC and input constraint based on the estimated state of observer. From the simulation, the design method of robust MPC with input constraint has been verified feasiblely. As a future work, we will develop the output-feedback MPC algorithm combined with nonlinear MPC and its optimization.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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