^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

This paper focuses on the problem of asymptotic stabilization for a class of discrete-time multiple time-delayed uncertain linear systems with input constraints. Then, based on the predictive control principle of receding horizon optimization, a delayed state dependent quadratic function is considered for incorporating MPC problem formulation. By developing a memory state feedback controller, the information of the delayed plant states can be taken into full consideration. The MPC problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions. Then, based on the Lyapunov-Krasovskii function, a delay-dependent sufficient condition in terms of linear matrix inequality (LMI) can be derived to design a robust MPC algorithm. Finally, the digital simulation results prove availability of the proposed method.

Model predictive control (MPC), also known as receding horizon control, has received much attention in control societies. The reason for its success is its ability to handle both constraints and time-varying behaviors [

All physical systems (biological, economical, industrial, etc.) inherently possess time delay, which is often a source of poor performance and even instability. Since the parameters of uncertainties and time-delays are frequently the main cause of performance degradation and instability, there has been an increasing interest in the robust control of uncertain time-delay systems in the control literature [

In [

In this section, we propose a new RMPC algorithm for multiple time-delayed uncertain linear systems with input constraints, in which the gain matrices of the memory state feedback controller consisting of only current state and all the time-delayed states are determined from the sufficient condition for cost monotonicity. The sufficient condition is derived using the delayed state dependent quadratic function. Compared with previous results [

The paper is organized as follows. In Section

We consider a class of discrete-time uncertain linear systems with multiple time-delayed states described as follows:

The structured uncertainties

Given any vectors

Given matrices

In the section, we will focus on the controller synthesis for the linear system with multiple time-delayed states and the objective is to construct a memory state feedback controller as follows:

Firstly, we consider the following min-max optimization problem, which minimizes the worst case infinite horizon quadratic objective function:

In order to design such a controller (

Then, the cost monotonicity is guaranteed for the uncertain linear system (

Therefore, the min-max problem is relaxed to the following minimization problem which minimizes the upper bound of the worst cost as

In order to convert the minimization problem (

First, we derive an LMI condition for the cost monotonicity (

The system (

Consider the following:

Firstly, we derive the LMI condition for the inequality (

From (

Substituting (

From (

Therefore,

Substituting

Multiplying both sides of the inequality (

Secondly, we convert the input constraint (

Now, the minimization problem (

If the optimization problem (

Let

Since

In this section, a numerical example is presented to illustrate the effectiveness of the proposed memory state feedback robust MPC algorithm. Let us consider the following discrete time multiple time-delayed uncertain linear system:

The initial value state is given as

The following terminology is used for the controllers designed:

State responses for

Control input.

Cost index.

In this paper, a new robust memory state feedback MPC technique was proposed for a class of multiple time-delayed uncertain linear systems with input constraints. The minimization problem for infinite horizon cost was derived using the time-delayed state dependent Lyapunov function. The optimization problem was reformulated in the form of a finite number of LMIs. Numerical examples demonstrated the effectiveness of the proposed method.

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

This work is supported in part by the National Natural Science Foundation (NNSF) of China under Grants 61304001 and 61203007. The authors would also like to thank He Bin and Hu Xiao-xiang for Matlab simulations.