Elsevier

Information Sciences

Volume 178, Issue 3, 1 February 2008, Pages 920-930
Information Sciences

An efficient model predictive controller with pole placement

https://doi.org/10.1016/j.ins.2007.09.001Get rights and content

Abstract

For model predictive control (MPC) of constrained systems, enlarging the feasible region is usually in conflict with improving the dynamic performance. To resolve the conflict, we proposed an efficient model predictive controller with pole placement for a class of discrete-time linear systems. By specifying a group of circular regions that contain the desired closed-loop poles, appropriate terminal weighting matrices and local controllers are calculated to construct a time-varying terminal convex set, which is a significant constraint for the online optimization problem. During the online optimization, the size of the terminal convex set can adjust itself according to the actual state at each sampling time. In this way, a large initial feasible region can be achieved while maintaining the good dynamic performance. An illustrative example is used to show the effectiveness of the proposed approach.

Introduction

Model predictive control (MPC) is an effective control algorithm for handling constrained control problems encountered in the chemical process industries. At each sampling instant, with the current state of the plant as the initial condition, a control sequence is obtained by solving online a finite horizon optimal control problem and the first control in this sequence is applied to the plant. At the next sampling instant, when the new state is available, the same procedure is repeated over a shifted horizon [7]. In the past decades, most of the studies on MPC focused on the closed-loop stability [8], [11]. Ref. [13] summarized three key ingredients for establishing the closed-loop stability of an MPC system: terminal performance cost, terminal constraint set and local controller.

In the design of predictive controllers with guaranteed closed-loop stability, a terminal constraint set is usually imposed artificially on the online optimization problem. For a discrete-time linear system subject to input and state constraints, Ref. [10] presented a sufficient condition for the exponential stability of the constrained system based on a terminal ellipsoid constraint. For a class of nonlinear systems, Ref. [12] approximated the nonlinear behavior by combining a linear model with a linear time-varying model, and developed a scheduling quasi-min–max MPC algorithm.

For constrained systems, the terminal constraint set has influence not only on the closed-loop stability, but also on the initial feasible region and the dynamic performance. The usual way to enlarge the initial feasible region is to optimize the terminal weighting matrix. In terms of LMIs, Ref. [9] provided an approach to enlarging the initial feasible region by optimizing the terminal weighting matrix offline. However, the initial feasible region was enlarged at the cost of poor dynamic performance. To enlarge the initial feasible region without sacrificing local optimality of the controller, Ref. [1] introduced a new approach to designing MPC controllers by optimizing online the end-point state-weighting matrix. Although effective in most cases, this approach incurs heavy computational burden. For high-dimensional systems, it is difficult to complete the tremendous computation within one sampling period.

In addition, it is well known that the dynamic performance of a system depends on the locations of the closed-loop poles. Refs. [5], [6] indicated that placing all closed-loop poles within a specified disk could achieve the desired dynamic performance. In practical applications, we hope to place all closed-loop poles into a disk near the origin in order to obtain good dynamic performance. Doing so, however, will cause the feasible region to shrink. To solve this problem, this paper presents an approach to designing efficient model predictive controllers with pole placement. The key idea is as follows: Design offline a group of terminal weighting matrices by placing closed-loop poles into different circular regions within the unit circle, then combine them into a terminal constraint convex set. In the online optimization, coefficients of the terminal convex set can be adjusted flexibly according to the actual state so as to achieve a large feasible region as well as good dynamic performance.

Section snippets

Problem statement

Consider the following discrete-time linear system:x(k+1)=Ax(k)+Bu(k)where u(k)  Rm is the control input, x(k)  Rn is the state of the plant, A and B are matrices of appropriate dimensions.

The input constraints are given by|u(k)|iuimax(i=1,,m)where ∣·∣i denotes the absolute value of the ith element of a vector, and uimax represents the maximal value of the corresponding element.

For the above system, consider the following infinite horizon objective function:J(k)=i=0[xT(k+i|k)Qx(k+i|k)+uT(k+i|k

MPC with pole placement

It is well known that the dynamic performance of a controlled system is to a large degree determined by the locations of its closed-loop poles. To obtain good dynamic performance, it is necessary to place all of the poles at certain prescribed locations. For linear systems without constraints, Refs. [5], [6] studied the pole placement problems within circular regions by using state feedback and output feedback, respectively. In the presence of constraints, however, placing the closed-loop poles

Feasibility and stability

For the feasibility and stability of the proposed algorithm with pole placement, we have the following conclusion.

Theorem 2

Given linear system (1) with input constraint, if optimization problem (31) is initially feasible, the implementation of the first control in u˜(k) stabilizes system (1) asymptotically.

Proof

Suppose that optimization problem (31) has a feasible solution at sampling instant kη˜1(k)ηˆ2(k)u˜(k)α1(k)αb-1(k)The corresponding state sequence is x(k|k)x(k+N-1|k), and the terminal state

An example

Consider the following model for a single, nonisothermal continuous CSTR[16]:x˙=Ax+Buwhere x is a vector of the reactor concentration and temperature, and u is the coolant flow (which is constrained). The matrices A and Bdepend on operating conditions as follows:A=-FV-k0e-E/RTs-ERTs2k0e-E/RTsCAs-ΔHrxnk0e-E/RTsρCp-FV-UAVρCp-ΔHrxnρCpE/RTs2k0e-E/RTsCAsB=0210000Ts-365VρCpwhere F = 1 m3/min, V = 1 m3, k0 = 1010/min−1, −ΔHrxn = 1.3 × 108 cal/kmole, E/R = 8330.1K, ρ = 106g/m3, UA = 5.34 × 106 cal/K,Cp = 1 cal/(gK), And Ts and

Conclusions

This paper presents an efficient MPC algorithm with pole placement. By pre-specifying several different disk regions, a terminal convex set with free coefficients is designed. During the online optimization, the maximal domain of attraction can be achieved by adjusting free coefficients of the terminal convex set. As the state converges to the zero, the constraint on input is relaxed gradually, which makes it possible to select the controller with good dynamic performance. The LMI-based online

Acknowledgement

This work was supported in part by the National Science Foundation of China under Grant Nos. 60474002 and 60674041, and in part by the National High Tech. Project of China under Grant No.2006AA04Z173.

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