**1. Introduction**

More than 25 years after model predictive control (MPC) or receding horizon control (RHC) appeared in industry as an effective tool to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge, see [1-3] for reviews of results in this area.

The focus of this chapter is on MPC of constrained dynamic systems, both linear and nonlin‐ ear, to illuminate the ability of MPC to handle constraints that makes it so attractive to in‐ dustry. We first give an overview of the origin of MPC and introduce the definitions, characteristics, mathematical formulation and properties underlying the MPC. Furthermore, MPC methods for linear or nonlinear systems are developed by assuming that the plant un‐ der control is described by a discrete-time one. Although continuous-time representation would be more natural, since the plant model is usually derived by resorting to first princi‐ ples equations, it results in a more difficult development of the MPC control law, since it in principle calls for the solution of a functional optimization problem. As a matter of fact, the performance index to be minimized is defined in a continuous-time setting and the overall optimization procedure is assumed to be continuously repeated after any vanishingly small sampling time, which often turns out to be a computationally intractable task. On the con‐ trary, MPC algorithms based on discrete-time system representation are computationally simpler. The system to be controlled which usually described, or approximated by an ordi‐ nary differential equation is usually modeled by a difference equation in the MPC literature since the control is normally piecewise constant. Hence, we concentrate our attention from now onwards on results related to discrete-time systems.

By and large, the main disadvantage of the MPC is that it cannot be able of explicitly dealing with plant model uncertainties. For confronting such problems, several robust model pre‐

© 2012 Dai et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Dai et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

dictive control (RMPC) techniques have been developed in recent decades. We review dif‐ ferent RMPC methods which are employed widely and mention the advantages and disadvantages of these methods. The basic idea of each method and some method applica‐ tions are stated as well.

more, MPC can handle control problems where off-line computation of a control law is diffi‐

Discrete-Time Model Predictive Control http://dx.doi.org/10.5772/51122 79

Specifically, an important characteristic of this type of control is its ability to cope with hard constraints on controls and states. Nearly every application imposes constraints. For in‐ stance, actuators are naturally limited in the force (or equivalent) they can apply, safety lim‐ its states such as temperature, pressure and velocity, and efficiency, which often dictates steady-state operation close to the boundary of the set of permissible states. In this regard, MPC is one of few methods having applied utility, and this fact makes it an important tool for the control engineer, particularly in the process industries where plants being controlled

In addition, another important characteristic of MPC is its ability to handle control problems where off-line computation of a control law is difficult or impossible. Examples where MPC may be advantageously employed include unconstrained nonlinear plants, for which online computation of a control law usually requires the plant dynamics to possess a special

A fairly complete discussion of several design techniques based on MPC and their relative

As mentioned in the excellent review article [2], MPC is not a new method of control design. Rather, it essentially solves standard optimal control problems which is required to have a finite horizon in contrast to the infinite horizon usually employed in *H*2and *H∞*linear opti‐ mal control. Where it differs from other controllers is that it solves the optimal control prob‐ lem on-line for the current state of the plant, rather than providing the optimal control for all

The on-line solution is to solve an open-loop optimal control problem where the initial state is the current state of the system being controlled which is just a mathematical programming problem. However, determining the feedback solution, requires solution of Hamilton-Jaco‐ bi-Bellman (Dynamic Programming) differential or difference equation, a vastly more diffi‐ cult task (except in those cases, such as *H*2and *H*∞linear optimal control, where the value function can be finitely parameterized). From this point of view, MPC differs from other control methods merely in its implementation. The requirement that the open-loop optimal control problem be solvable in a reasonable time (compared with plant dynamics) necessi‐

MPC has become an attractive feedback strategy, especially for linear processes. By now, lin‐ ear MPC theory is quite mature. The issues of feasibility of the on-line optimization, stability

tates, however, the use of a finite horizon and this raises interesting problems.

and performance are largely understood for systems described by linear models.

cult or impossible.

are sufficiently slow to permit its implementation.

merits and demerits can be found in the review article by [4].

states, that is determining a feedback policy on-line.

**3. Linear Model Predictive Control**

structure, and time-varying plants.

**2.2. Essence of MPC**

Most MPC strategies consider hard constraints, and a number of RMPC techniques exist to handle uncertainty. However model and measurement uncertainties are often stochastic, and therefore RMPC can be conservative since it ignores information on the probabilistic distribution of the uncertainty. It is possible to adopt a stochastic uncertainty description (in‐ stead of a set-based description) and develop a stochastic MPC (SMPC) algorithm. Some of the recent advances in this area are reviewed.

In recent years, there has been much interest in networked control systems (NCSs), that is, control systems close via possibly shared communication links with delay/bandwidth con‐ straints. The main advantages of NCSs are low cost, simple installation and maintenance, and potentially high reliability. However, the use of the network will lead to intermittent losses or delays of the communicated information. These losses will tend to deteriorate the performance and may even cause the system to become unstable. MPC framework is partic‐ ularly appropriate for controlling systems subject to data losses because the actuator can profit from the predicted evolution of the system. In section 7, results from our recent re‐ search are summarized. We propose a new networked control scheme, which can overcome the effects caused by the network delay.

At the beginning of research on NCSs, more attention was paid on single plant through net‐ work. Recently, fruitful research results on multi-plant, especially, on multi-agent net‐ worked control systems have been obtained. MPC lends itself as a natural control framework to deal with the design of coordinated, distributed control systems because it can account for the action of other actuators while computing the control action of a given set of actuators in real-time. In section 8, a number of distributed control architectures for inter‐ connected systems are reviewed. Attention is focused on the design approaches based on model predictive control.
