Abstract: A model predictive controller based on linear matrix inequalities (LMIs) is presented. As ... are subjected to physical constraints, such as an actuator's.
LINEAR MATRIX INEQUALITIES
Linear matrix inequality based model predictive controller E. Granado, W. Colmenares, J. Bernussou and G. Garcı´a Abstract: A model predictive controller based on linear matrix inequalities (LMIs) is presented. As in standard model predictive control (MPC) algorithms, at each (sampling) time, a convex optimisation problem is solved to compute the control law. The optimisation involves constraints written as LMIs, including those normally associated with MPC problems, such as input and output limits. Even though a state-space representation is used, only the measurable output and the extreme values of the unmeasurable states are used to determine the controller, hence, it is an output feedback control design method. Stability of the closed-loop system is demonstrated. Based on this MPC, a Lyapunov matrix is built and the controller computation is set in a more standard MPC framework. The design techniques are illustrated with numerical examples.
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Introduction
Model predictive control (MPC) is the most popular industrial MIMO control strategy [1]. From among the reasons for this popularity we highlight that all real systems are subjected to physical constraints, such as an actuator’s operation limits, and they may be explicitly considered, in the MPC formulation. It is mainly a control technique for systems with slow dynamics but its application to more demanding systems is an area of current interest [2]. In the MPC scheme, the control law is obtained from an optimisation problem whose objective function weights the control efforts and the deviations from the set point. The optimisation problem normally includes constraints on the input (hard constraints), output and state (soft constraints). The optimisation is performed over a (prediction) horizon which is continuously moved forward in time, since only the first control law is applied (out of those calculated over the horizon) to the system [3, 4]. MPC was introduced in the 1970s [5], and considerable research has been performed in the area to ensure, the stability and feasibility of the problem [6 – 10]. MPC is a methodology that, working always in the time domain, lets the operator easily, handle physical performance requirements, such as upper and lower bounds on the process variables, and tuning of the closed-loop. At the same time, very little knowledge of the theory involved is required. An extension to the output feedback case, of the methodology proposed in [11] is now presented.
q IEE, 2003 IEE Proceedings online no. 20030703 doi: 10.1049/ip-cta:20030703 Paper first received 1st October 2001 and in revised form 21st March 2003 E. Granado and W. Colmenares are with the Universidad Simo´n Bolı´var, Dpto. Procesos y Sistemas, Apartado 89000, Caracas 1080, Venezuela J. Bernussou and G. Garcı´a are with the LAAS-CNRS 7, Av. Du Colonel Roche 31077, Toulouse, France 528
The unconstrained case as well as input and output restrictions are considered. To ensure stability, an infinite horizon is included in the objective quadratic function. The optimisation problem is formulated in terms of linear matrix inequalities (LMIs). Dedicated powerful LMI algorithms already exist that allow problem solution in polynomial time, and in many cases in times comparable to those necessary to obtain an analytical solution to a similar problem [12]. An ‘on-line’ and in-time solution is fundamental to MPC. In order to only use the measurable output, a dynamic output feedback controller (not an observer) is calculated at each iteration. Since not all states are available, either the extreme values or some of the statistical properties of the unmeasured states are used. This approach provides, not only a controller but also a Lyapunov matrix. This is used to set the problem in a more standard framework. An observer is required and stability is no longer guaranteed. 2
Problem statement
Consider the discrete linear time invariant system represented by: xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ yðkÞ ¼ CxðkÞ
ð1Þ
where xðkÞ 2
