Elsevier

Automatica

Volume 40, Issue 9, September 2004, Pages 1603-1611
Automatica

Brief paper
General receding horizon control for linear time-delay systems

https://doi.org/10.1016/j.automatica.2004.04.003Get rights and content

Abstract

A general receding horizon control (RHC), or model predictive control (MPC), for time-delay systems is proposed. The proposed RHC is obtained by minimizing a new cost function that includes two terminal weighting terms, which are closely related to the closed-loop stability. The general solution of the proposed RHC is derived using the generalized Riccati method. Furthermore, an explicit solution is obtained for the case where the horizon length is less than or equal to the delay size. A linear matrix inequality (LMI) condition on the terminal weighting matrices is proposed, under which the optimal cost is guaranteed to be monotonically non-increasing. It is shown that the monotonic condition of the optimal cost guarantees closed-loop stability of the RHC. Simulations demonstrate that the proposed RHC effectively stabilizes time-delay systems.

Introduction

In industrial processes, time-delays often occur in the transmission of material or information between different parts of a system. Chemical processing systems, transportation systems, communication systems, and power systems are typical examples of time-delay systems. Because the presence of a time-delay often causes serious deterioration of the stability and performance of the system, considerable research has been devoted to the control of time-delay systems.

For delay-free systems, the receding-horizon control (RHC), or model predictive control (MPC), has received considerable attention (Kwon & Pearson, 1977; Richalet, Rault, Testud, & Papon, 1978; Kwon & Kim, 2000) because of its many advantages, including ease of computation, good tracking performance and I/O constraint handling capability, compared with the popular steady-state infinite horizon linear quadratic (LQ) control. While the steady-state LQ control for linear systems is obtained from the algebraic Riccati equation, the RHC for linear systems is obtained from the differential or difference Riccati equation on a finite interval, which is easier to solve. Therefore, RHC has been widely used, particularly in the chemical process industries (Richalet et al., 1978).

For time-delay systems, there are no general results for the RHC. A simple control method based on the receding horizon concept has appeared in (Kwon, Jin Won Kang, Young Sam Lee, & Young Soo Moon, 2003). However, it does not have a state weighting in the cost function. Furthermore, it does not guarantee closed-loop stability by design, and therefore stability can be checked only after the controller has been designed.

In RHC, the terminal weighting matrices in the cost function are crucial for stability. The cost function taken in this paper differs from the existing forms used in (Krasovskii, 1962; Kushner & Barnea, 1970; Ross, 1971; Uchida, Shimemura, Kubo, & Abe, 1988) which consider the optimal control problem for time-delay systems. The cost function in this paper has an additional terminal weighting matrix on x(t+T+s), −hs⩽0, where T denotes the horizon length. This additional terminal weighting matrix is necessary to guarantee closed-loop stability. For the cost function with this additional weighting matrix, we found no optimal solution in the literature. We first derive the optimal solution for this cost function using the generalized Riccati method taken in (Eller, Aggarwal, & Banks, 1969) and then provide the general solution of the proposed RHC using this solution. The monotonic condition of the optimal cost, under which the optimal cost is guaranteed to be monotonically nonincreasing, is proposed in terms of a linear matrix inequality on the terminal weighting matrices. Using the derived condition, we prove the stability of the proposed RHC in this paper.

Section snippets

Receding horizon control for time-delay systems

Consider a time delay systemẋ(t)=Ax(t)+A1x(t−h)+Bu(t),where the initial condition is x(τ)=φ(τ),τ∈[−h,0], x(t)∈Rn is the state, u(t)∈Rm is the input, and A,A1Rn×n,B∈Rn×m are system matrices. h>0 is the delay, and φ(t) is a continuous function. In order to obtain a receding horizon control, we first consider a finite horizon cost function represented byJ(xt0,u,t0,tf)=t0tf[xT(τ)Qx(τ)+uT(τ)Ru(τ)]dτ+xT(tf)F1x(tf)+−h0xT(tf+s)F2x(tf+s)ds,where xt0=x(t0+s), s∈[−h,0], is a continuous function, t0 is

Monotonic condition of the optimal cost

In this section, we present a condition on the terminal weighting matrices F1 and F2, under which the optimal cost J(xt0,t0,σ) does not increase as the terminal time σ increases. We will call it a monotonic condition of the optimal cost on the terminal weighting matrices.

Theorem 3.1

Assume thatF1andF2in (2) satisfy the following matrix inequality for someK1andK2:P11F1(A1+BK2)+K1TRK2(A1+BK2)TF1+K2TRK1K2TRK2−F2⩽0,whereP11=(A+BK1)TF1+F1(A+BK1)+Q+K1TRK1+F2. The optimal costJ(xt0,t0,σ)then satisfies the

Stability of RHC

This section investigates the stability of the RHC.

Theorem 4.1

GivenQ>0 andR>0, if∂J(xt0,t0,σ)∂σ⩽0forσ>t0, system (1) controlled by the RHC is asymptotically stable.

Proof

If ∂J(xt0,t0,σ)∂σ⩽0 for σ>t0, we haveJ(xt,t,t+T)=tt+θ[xT(τ)Qx(τ)+uT(τ)Ru(τ)]dτ+J(xt+θ,t+θ,t+T),⩾tt+θ[xT(τ)Qx(τ)+uT(τ)Ru(τ)]dτ+J(xt+θ,t+θ,t+T+θ).Rearranging the above inequality and dividing it by θ yieldJ(xt+θ,t+θ,t+T+θ)−J(xt,t,t+T)θ⩽−1θtt+θ[xT(τ)Qx(τ)+uT(τ)Ru(τ)]dτ.If θ→0, we obtaindJ(xt,t,t+T)dt⩽−[xT(t)Qx(t)+uT(t)Ru(t)],

Numerical examples

In this section, a numerical example is presented to illustrate the proposed methods.

Example 5.1

Consider a time-delay system whose system matrices are given byA=−1132,A1=00−1−0.5,B=13.The delay size of the system is h=1. It is noted that this system is open-loop unstable. The weighting matrices Q and R are chosen such that Q=I and R=I. Terminal weighting matrices F1 and F2 guaranteeing the closed-loop stability are obtained by solving the LMI (23)F1=4.78801.97371.97377.1028,F2=1.07990.08310.08310.9533.

We

Conclusions

A stabilizing RHC (or MPC) for linear time-delay systems is proposed. We propose a new receding horizon cost function and have derived the RHC minimizing that cost function. The general solution of the proposed RHC was derived using the generalized Riccati method. Furthermore, the explicit solution was obtained for the case where the horizon length is less than or equal to the delay. A linear matrix inequality condition on the terminal weighting matrix for the RHC, which guarantees that the

Wook Hyun Kwon He received the B.S. and M.S. degrees in electrical engineering from Seoul National University (SNU) in 1966 and 1972, respectively. He received the Ph.D. degree from Brown University in 1975. Since 1977, he has been with the School of Electrical Engineering, SNU. From 1981 to 1982, he was a visiting assistant professor at Stanford University. Since 1991, he has been the founding director of the Engineering Research Center for Advanced Control and Instrumentation of SNU. He is

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Wook Hyun Kwon He received the B.S. and M.S. degrees in electrical engineering from Seoul National University (SNU) in 1966 and 1972, respectively. He received the Ph.D. degree from Brown University in 1975. Since 1977, he has been with the School of Electrical Engineering, SNU. From 1981 to 1982, he was a visiting assistant professor at Stanford University. Since 1991, he has been the founding director of the Engineering Research Center for Advanced Control and Instrumentation of SNU. He is serving the National Academy of Engineering as vice president since 2003 and the International Federation of Automatic Control as President-Elect since 2002. His main research interests are currently multivariable predictive controls, time delay systems, optimal estimation, industrial networks, and industrial applications.

Young Sam Lee Young Sam Lee was born in 1970 in Korea. He received the B.S. and M.S. degree in electrical engineering from Inha University, Inchon, Korea in 1997 and 1999, respectively. He received the Ph.D. in School of electrical engineering and computer science from Seoul National University, Seoul, Korea, in 2003. His research interests include time delay systems, receding horizon control, and signal processing. He has been working for Samsung Electronics Co., Ltd. since September 2003.

Soo Hee Han Soo Hee Han was born in Korea on August 26, 1974. He received the B.S. degree in electrical engineering from Seoul National University, Seoul, Korea in 1998. He received the M.S. and Ph.D. degree in School of electrical engineering and computer science from Seoul National University, Seoul, Korea, in 2000 and 2003, respectively. His main research interests are in the areas of computer aided control system design, distributed control system, time delay system, stochastic signal processing.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Başar. This work was supported by Brain Korea 21.

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He is now with the Digital Media R&D Center, Samsung Electronic Co. Ltd, Suwon City, Gyeonggi-Do, South Korea.

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