Abstract

This paper investigates the problem of model predictive control for a class of nonlinear systems subject to state delays and input constraints. The time-varying delay is considered with both upper and lower bounds. A new model is proposed to approximate the delay. And the uncertainty is polytopic type. For the state-feedback MPC design objective, we formulate an optimization problem. Under model transformation, a new model predictive controller is designed such that the robust asymptotical stability of the closed-loop system can be guaranteed. Finally, the applicability of the presented results are demonstrated by a practical example.

1. Introduction

The ideas of model predictive control and receding horizon control have been developed since 1960s. It has been shown from [1] that model predictive control (MPC) is an effective way to handle multivariable constrained control problems, which appear in the chemical process control, the petrochemical industries, gas pipeline, and so on. In [2, 3], the authors gave us an overview of the origins of model predictive control and the recent results. The original MPC technique is aimed at solving an open-loop optimization problem with constraints at every sampling instant, implementing only the first control step of solutions.

In practical control systems, parameter uncertainties cannot be avoided. In the literatures, two kinds of parameter uncertainties are often included in the uncertain systems. They are norm-bounded parameter uncertainty and polytopic parameter uncertainty. In addition, time-delay often appears in industrial processes, which results in degradation and instability in such systems [4]. References [5, 6] studied the networked control with time-delay; [7] investigated linear switched systems with time-varying delay. The authors in [8] discussed the problem of dissipativity analysis of stochastic neural networks systems of discrete-time form with time-varying and finite-distributed delays. The authors in [9] designed a novel output-feedback controller for the suspension systems with input delay. Moreover, there exist some physical limits, for instance power limitations and value saturation, in many industrial processes, which result in constraints on input and output. Therefore, considerable researchers have been attracted to study the robust control problem of constrained uncertain systems with state delays [10, 11].

Many results about MPC technique for time-delay systems have been addressed. To mention a few, in [12], the authors proposed that the control strategy for uncertain systems could be developed into a delay system via the MPC. However, it is proper only if the delay indices are known. Recently, the authors in [10] presented an improved delay-dependent robust MPC to reduce the conservatism, still with a known delay. The work in [4] put forward an MPC method for time-varying state-delay systems with uncertainty and constrained control input. However, since the stability is guaranteed under the fixed constant weighting matrix at all time, the method is very limited and the conservatism may be generated.

Motivated by the above observation, the problem of MPC for time-varying delay systems with parameter uncertainties and input constraints is studied in this paper. We summarize the main contributions of this paper as follows. (1) The uncertainty is supposed to be polytopic uncertainty type, and the state with unknown delay with both specified upper and lower bounds is handled by an approximated model. (2) In the controller design process, for the state-feedback MPC design objective, we formulate an optimization problem over an infinite time horizon. A new model predictive controller is designed under the model transformation by approximating the state delay, such that the robust asymptotical stability of the closed-loop system is guaranteed. The existence of the controller can be expressed by the convex optimization algorithm. (3) It is shown that the approach proposed in this paper is effective and performs better with the faster response, smaller overshoot, stronger robustness and so on by a practical example.

The rest of this paper is organized as follows. Section 2 formulates the problem to be solved. Section 3 proposes an MPC method for delay systems with uncertainties and constraints. Section 4 illustrates the effectiveness of the method proposed in this paper with a practical example. This paper is concluded in Section 5.

Notation. stands for the-dimensional Euclidean space, denotes the set of real matrices, denotes the identity matrix, and denotes a block-diagonal matrix.   () denotes that the matrix is a positive definite (resp. a positive semi-definite) matrix. denotes transposition of , and denotes the inverse matrix of . , when is symmetric matrix. denotes 2-norm, and , where The symbol induces a symmetric structure in a matrix.

2. Problem Formulation

Consider the following system: where denotes the state variable with the initial condition ,   means the unknown value of delay units, being supposed with known integers and , and stands for the control input variable and satisfies is unknown but belongs to a polytope at each time , that is in which indicates the convex hull and are vertices of the convex hull. The nonnegative coefficients for each time satisfies the following: In this paper we aims at designing the following controller for system in (1): with the performance index as follows at every time : where where and are known positive definite symmetric weighting matrices, denotes the predicted state at time and denotes the control signal at time , when   . Based on the concept of MPC, before the next sampling time comes, we just implement the first compute input signal . Then we repeat the aforementioned optimization problem after updating it with the actual state.

Remark 1. The input-output technique is one of the most effective ways to handle the time delay, which was presented in the robust control theory [13, 14]. Before using the approach to dispose time-varying delay, a proper approximation with small error for should be found. In [15, 16], the authors used different variants of the state variable as the approximation of . In this paper, we utilize as the approximation of .

Next, the lower and upper bounds and are used to estimate . The two-term approximation leads to the following error: where ,  , and Then, system (1) is replaced by the following one: Now let Then we gain the following: As in [17], (8) can be converted into the following equivalent descriptor form:

We introduce the following lemma for our main result.

Lemma 2. According to [18], suppose that , , and; then for any matrices , , and , the following inequality holds: where .

3. Main Results

The following function is introduced in order to obtain the main results For every , and satsifies the following: we can obtain the upper bound of . Since should be limited, one can get . Then, one can get Therefore, It follows that , by calculating the sum of inequality (17) from to . Hence where Therefore, it can be seen from (18) that we transform the organical min-max optimization problem in (6)–(9)into the following optimization problem which can minimize the upper bound of as follows: Equations (9), (14), and (17) are the constraint conditions.

Theorem 3. Considering the uncertain system in (1) with time-varying delay and input constrains (2), if there exist matrices , , , , , , , , , , , and with appropriate dimensions and the scalar , the following problem is solvable: subject to where and indicates the th element of and represents the th diagonal element of ; then the upper bound of the desired performance index is minimized by the MPC law, .

Proof. Set , , , and . Then, the problem of minimizing can be regarded as follows: subject to We can easily deduce (21) and (22) by using the Schur complement. According to the definition of in (16), one can obtain the following: By Lemma 2 and the descriptor system (14), we have where and matrices and satisfy the following conditions: After substituting (31) into (30), we obtain the following inequality: where , Replacing (35) with (36), it can be found that Replacing (17) with (37), we have For obtaining LMI, we give the definition as follows: Pre- and postmultiplying (38), (33), and (34) by and , and , and and , respectively, then (38) is equivalent to the following:where Equation (33) is equivalent to the following: Equation (34) is equivalent to the following: If and only if inequalities in (23)–(25) hold, respectively, inequalities in (40)–(43) are fulfilled for any , for that (40)–(43) are affine on the basis of matrices .
Now we consider the input constraints (9) and discuss how to transform it into LMI. First we introduce the following invariant ellipsoid: where As discussed in [12], it is shown that where .
Therefore, there exists a symmetric matrix and the following inequality is satisfied: It is easily shown that (47) is equivalent to (26) with the definitions of and . This completes the proof.