Quasi-Min-Max MPC algorithms for LPV systems

doi:10.1016/S0005-1098(99)00176-4

Automatica

Volume 36, Issue 4, April 2000, Pages 527-540

https://doi.org/10.1016/S0005-1098(99)00176-4 Get rights and content

Abstract

In this paper a new model predictive controller (MPC) is developed for polytopic linear parameter varying (LPV) systems. We adopt the paradigm used in gain scheduling and assume that the time-varying parameters are measured on-line, but their future behavior is uncertain and contained in a given polytope. At each sampling time optimal control action is computed by minimizing the upper bound on the “quasi-worst-case” value of an infinite horizon quadratic objective function subject to constraints on inputs and outputs. The MPC algorithm is called “quasi” because the first stage cost can be computed without any uncertainty. This allows the inclusion of the first move u(k|k) separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints. Proposed optimization problems are solved by semi-definite programming involving linear matrix inequalities. It is shown that closed-loop stability is guaranteed by the feasibility of the linear matrix inequalities. A numerical example demonstrates the unique features of the MPC design.

Introduction

Model Predictive Control (MPC), also known as moving or receding horizon control, has originated in industry as a real-time computer control algorithm to solve linear multivariable problems that have constraints and time delays (Cutler & Ramaker, 1980; Richalet, 1980). Today most process industries use MPC in one form or another as their advanced control technology. Basically MPC solves on-line an open-loop constrained optimization problem at each time instant and implements only the first element of the optimal control profile. The optimization is repeated at the next sampling time by updating the initial condition with the new state. It is well known that this receding horizon implementation of the open-loop optimal control profile gives rise to a stationary feedback control law (Bitmead, Gevers & Wertz, 1990). In the past most industrial applications have used the finite horizon implementation of MPC. However, despite many reported successful applications, the finite horizon MPC algorithms are difficult to analyze theoretically since closed-loop asymptotic stability depends on many tuning parameters in an unnecessarily complicated way and no guarantees are provided (Bitmead et al., 1990). Realizing this drawback several researchers have recently revisited MPC and studied it as a constrained infinite horizon LQR problem, which led to useful stability results. For example for linear plants with input and output constraints reference Rawlings and Muske (1993) is able to perform the optimization over an infinite prediction horizon by using only the first N control moves and setting the remaining (infinitely many) moves to zero. It is shown that feasibility of the resulting quadratic program guarantees stability. Instead of setting the control inputs to zero after a certain horizon, Scokaert and Rawlings (1998) and Chmielewski and Manousiouthakis (1996) use a fixed feedback control law to obtain a finite parameterization of the inputs over an infinite prediction horizon.

For nonlinear plants similar infinite horizon MPC algorithms have been developed to guarantee closed-loop stability. For example Michalska and Mayne (1993) suggests a switching dual-mode horizon controller in which a local linear feedback control law is applied once the states enter an invariant terminal region and a finite horizon predictive controller is applied outside the terminal region. In Chen and Allgöwer (1998) the infinite horizon MPC cost function is bounded by a finite horizon stage cost with a terminal penalty term so that the resulting nonlinear optimization problem can be solved numerically. A local state feedback law is used to compute the terminal penalty term off-line. Local stability is proven by the existence of a feasible solution to the optimization problem.

In this paper we consider the class of linear parameter-varying or LPV systems whose state-space matrices are affine functions of some time-varying parameter vector p(k). In recent years there has been a renewed interest in LPV systems, especially to provide useful theoretical foundations for gain scheduling control (see Shamma & Athans, 1991; Apkarian, Gahinet & Becker 1995; Wu & Packard, 1995). The common theme in these works is to make the controller parameter dependent so that when time-varying parameters are measured in real-time, the controller becomes self-scheduling and offers potential performance improvements over a fixed robust controller. The purpose of this paper is to develop infinite horizon scheduling MPC algorithms for LPV plants. In doing so MPC is extended to apply to an important class of systems. Both input and output constraints are addressed and stability guarantees are provided.

The rest of the paper is structured as follows. Section 2 defines the basic system of interest as a polytopic Linear Parameter Varying (LPV) model and introduces the necessary notation. In Section 3, Model Predictive Control (MPC) algorithm “Quasi-Min-Max” is formulated for the unconstrained case first, and then modified to include both input and output constraints. Stability proofs and comparison with some of the existing MPC algorithms are also given. In Section 4, simulations on a numerical example illustrate the application of the developed algorithms. Finally, Section 5 concludes the paper.

Section snippets

Problem statement

We consider discrete polytopic LPV systems whose system matrices are affine functions of a parameter vector p(k): $x(k+1)=A(p(k))x(k)+B(p(k))u(k)$ where $A(p(k))= ∑ j=1 l p_{j} (k)A_{j}, B(p(k))= ∑ j=1 l p_{j} (k)B_{j}$ with $x∈ R^{n_{x}}$ denoting the state, $u∈ R^{n_{u}}$ the control and $p=[p_{1},p_{2},…,p_{l}]∈ R^{l}$ the parameter vector. The time-varying parameter vector p(k) belongs to a convex polytope $P$ , i.e. $∑ j=1 l p_{j} (k)=1, 0≤p_{j} (k)≤1.$ Therefore, when p_i=1 and p_j=0 for $j=1, 2,…,l$ and j≠i, the LPV model , reduces to its ith linear time-invariant “local”

MPC problem

Consider the following infinite horizon objective function which is split into two parts: $J_{0}^{∞} (k)= ∑ i=0 N_{0} x(k+i|k)^{T} Qx(k+i|k)+u(k+i|k)^{T} Ru(k+i|k)+ ∑ i=N_{0} +1 ∞ x(k+i|k)^{T} Qx(k+i|k)+u(k+i|k)^{T} Ru(k+i|k)=J_{0}^{N_{0}} (k)+J_{N_{0}+1}^{∞} (k)$ where Q and R are positive definite.

Example

We consider an LPV system which belongs to a polytope formed by two local discrete models: $A_{1} = 0.2730 0.0660 0.3021 −0.5012 0.2717 0.4416 0.5602 −0.7123 0.3051 −0.7865 0.7651 0.3121 0.7962 −0.1452 0.5231 −0.9345,$ $B_{1} = 0.2300 0.2601 0.1213 1.3452,$ $A_{2} = 0.2093 −0.1981 −0.2394 0.5671 0.2717 0.4598 0.5602 1.3782 −0.4700 0.6700 −0.8600 −1.2400 0.3456 −0.6312 −1.4594 1.8936, B_{2} =B_{1} .$ A₁ is stable, with eigenvalues of −0.5982; −0.1160 and 0.6297±0.4254i, and A₂ is unstable with eigenvalues of −1.5901; 1.7151; 1.4980; 0.0798. The parameter vector

Conclusions

In this paper, infinite horizon Quasi-Min-Max MPC algorithms have been developed for discrete, polytopic linear parameter varying systems. The on-line optimization problems include input and output constraints and can be solved by semi-definite programming. It is shown that receding horizon implementation of the feasible solutions guarantees closed-loop stability. The implemented control moves are dependent on parameters which are assumed to be available (measured) in real-time; therefore, the

Acknowledgements

The authors gratefully acknowledge the financial support of E. I. duPont de Nemours and Co., Inc. and the National Science Foundation through Grant CTS-9522564.

References (16)

P. Apkarian et al.
Self-scheduled H_∞ control of linear parameter-varying systems
Automatica
(1995)
H. Chen et al.
Quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability
Automatica
(1998)
D. Chmielewski et al.
On constrained infinite-time linear quadratic optimal control
Systmes & Control Letters
(1996)
M.V. Kothare et al.
Robust constrained model predictive control using linear matrix inequalities
Automatica
(1996)
Arkun, Y., Banerjee, A., & Lakshmanan, N. M. (1998). Self scheduling MPC using LPV models. In R. Berber, Nonlinear...
A. Banerjee et al.
Estimation of nonlinear systems using linear multiple models
AICHE Journal
(1997)
Bitmead, R. R., Gevers, M., & Wertz, V. (1990). Adaptive optimal control: The thinking man's GPC. Englewood Cliffs, NJ:...
Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory....

There are more references available in the full text version of this article.

Cited by (338)

Multi-model predictive control strategy for path-following of unmanned surface vehicles in wide-range speed variations
2024, Ocean Engineering
This paper proposes a robust multi-model predictive control strategy to solve the path-following control problem of unmanned surface vehicles (USVs) in wide-range speed variations. The dynamic characteristics of USVs tend to vary greatly in wide-range speed variations. A discrete piecewise affine system with polytopic uncertainty is established to describe the speed and steering dynamics characteristics of USVs in wide-range speed variations. Then, an improved adaptive line-of-sight (IALOS) guidance law is designed to adapt the USV maneuvering characteristics in different speed ranges, which features a variable look-ahead distance subject to speed and lateral error. Finally, the speed and heading controllers are designed by combining the multi-model switching control theory and the robust model predictive control algorithm based on linear matrix inequalities (LMIs), and the stability proofs are given. Comparative numerical simulations are performed to validate the effectiveness and superiority of the proposed control strategy.
SG-PBFS: Shortest Gap-Priority Based Fair Scheduling technique for job scheduling in cloud environment
2024, Future Generation Computer Systems
Job scheduling in cloud computing plays a crucial role in optimizing resource utilization and ensuring efficient job allocation. But cloud resources may be wasted, or service performance may suffer because of under-utilization or over-utilization because of poor scheduling. Existing approaches often overlook the dynamic nature of cloud environments, resulting in issues like missed deadlines, prolonged flowtime, extended makespan, and unacceptable total tardiness. To address this issue, the main objective of this research is to improve the existing Priority Rules (PR) cloud schedulers by developing a new dynamic scheduling algorithm by manipulating the gaps in the cloud job schedule. Firstly, a Priority-Based Fair Scheduling (PBFS) algorithm has been introduced to schedule jobs so that jobs can access the required resources at optimal times. Then, a backfilling strategy called Shortest Gap - Priority-Based Fair Scheduling (SG-PBFS) is developed that attempts to manipulate the gaps in the schedule of cloud jobs. Finally, the performance evaluation demonstrates that the proposed SG-PBFS algorithm outperforms SG-SJF, SG-LJF, SG-FCFS, SG-EDF, and SG-(MAX-MIN) regarding flow time, makespan time, and total tardiness, which conclusively demonstrates its effectiveness. To conduct this experiment, we employed the CloudSim simulator, which is implemented using the Java programming language.
Robust control and flexible operation for commercial-scale coal-fired power plant with solvent-based post-combustion carbon capture
2023, International Journal of Greenhouse Gas Control
Coal-fired power plant integrated with post-combustion carbon capture (CFPP-PCC) process exhibits strong nonlinearity and multi-variable connections. These dynamics along with unknown disturbances make it a challenging task to realize robust control and flexible operation in CFPP-PCC system. For this reason, this paper develops an extended state observer based stable model predictive control (ESOSMPC) in order to reject unknown disturbances and to achieve flexible operation with closed-loop stability. In the proposed control structure, an extended state observer is utilized to give accurate estimation for plant behavior change and unknown disturbances. This information will be used in active disturbance rejection, which aims to alleviate negative effects of disturbances and to enhance control performance. Stable model predictive control with infinite input-output based objective function is developed in order to fulfill Lyapunov stability condition, guaranteeing a stable operation for the integrated process. The proposed controller is compared with conventional MPC/PID controller under varying working conditions. Simulation results show that the presented ESOSMPC is able to achieve robust control in a multi-disturbance situation and to attain flexible operation in a broad-range load variation scenario.
Offline output feedback robust anti-windup MPC-LPV using relaxed LMI optimization
2023, European Journal of Control
This paper addresses a new technique of constrained output feedback robust model predictive control (RMPC) with anti-windup (AW) synthesis adopting linear parameter varying state-space (LPV-SS) systems via relaxed linear matrix inequalities (LMI) procedure. We proposed an output feedback control design with a conservatism reduction, including a robust AW-LPV synthesis to solve the robust stability under saturated signal control mode. Lyapunov stability functions through the LPV emphasis are considered. To evaluate the efficiency of the proposed method, we presented a time response analysis, closed-loop z-plane, worst-case upper bound, and we compared the performance of the proposed method with some MPC benchmarks using a numerical example.
Robustness analysis of gain-scheduled Model Predictive Control for Linear Parameter Varying systems: An Integral Quadratic Constraints approach
2023, International Journal of Robust and Nonlinear Control
Adaptive event-triggering mechanism-based N-step predictive load frequency control for power systems with cyber attack
2023, Optimal Control Applications and Methods

View all citing articles on Scopus

Yaman Arkun is the Dean of the College of Engineering at KOC University in Istanbul, Turkey. He received his B.S. from University of Bosphorous, Istanbul, Turkey, and his M.S. and Ph.D. from University of Minnesota. He was on the faculty at RPI from 1979 to 1986. Before he moved to Turkey he was at Georgia Institute of Technology from 1986 to 1999 as Professor of Chemical Engineering. Dr. Arkun has held industrial visiting positions at Tennessee Eastman, Du Pont and Weyerhaeuser companies. He is the North American Editor of the Journal of Process Control. He has served as Editor of Automatica, editorial board member of International Journal of Control, Trustee and Secretary of CACHE, and CAST Director. He is the 1986 recipient of the Donald P. Eckman Award given by the American Automatic Control Council. He has chaired the Systems and Control Area 10b of CAST (1988–1990) and served as the AIChE Director to American Automatic Control Council (1989–1991). His research interest is in process dynamics, modeling and control. In particular he is interested in robust, nonlinear and predictive control, and synthesis of control systems for large-scale plants.

Yaohui Lu was born in Tangshan, China, in 1970. He received the B.Sc. and M.Sc. degree in Chemical Engineering from Tianjin University, China in 1992 and 1995 respectively. He is now a Ph.D. student in School of Chemical Engineering of Georgia Institute of Technology. His current research interests include nonlinear process and model predictive control, system dynamics, and robust stability theory.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Tits under the direction of Editor T. Basar.

View full text

Quasi-Min-Max MPC algorithms for LPV systems☆

Abstract

Introduction

Section snippets

Problem statement

MPC problem

Example

Conclusions

Acknowledgements

Automatica

Automatica

Systmes & Control Letters

Automatica

Estimation of nonlinear systems using linear multiple models

AICHE Journal