This paper provides an overview of commercially available model predictive control (MPC) technology, both linear and nonlinear, based primarily on data provided by MPC vendors. A brief history of industrial MPC technology is presented first, followed by results of our vendor survey of MPC control and identification technology. A general MPC control algorithm is presented, and approaches taken by each vendor for the different aspects of the calculation are described. Identification technology is reviewed to determine similarities and differences between the various approaches. MPC applications performed by each vendor are summarized by application area. The final section presents a vision of the next generation of MPC technology, with an emphasis on potential business and research opportunities.

More than 15 years after Model Predictive Control (MPC) appeared in industry as an effective means to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge. The issues of feasibility of the on-line optimization, stability and performance are largely understood for systems described by linear models. Much progress has been made on these issues for nonlinear systems but for practical applications many questions remain, including the reliability and efficiency of the on-line computation scheme. To deal with model uncertainty "rigorously" an involved dynamic programming problem must be solved. The approximation techniques proposed for this purpose are largely at a conceptual stage. Among the broader research needs the following areas are identified: multivariable system identification, performance monitoring and diagnostics, nonlinear state estimation, and batch system control. Many practical problems like control objective prior...

This paper is concerned with the optimal control of linear discrete-time systems, which are subject to unknown but bounded state disturbances and mixed constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with memory of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This result implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods without having to introduce any conservatism in the problem formulation. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and for all allowable disturbance sequences. The cost that is chosen to be minimized in the associated finite horizon optimal control problem is a quadratic function in the disturbance-free state and input sequences. It is shown that the value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic or given by a 1-norm or ∞-norm bound, or a second-order cone program (SOCP) if the disturbance set is ellipsoidal or given by a 2-norm bound.

receding horizon control, model predictive control, discrete-time systems. The first part of this paper studies a specific class of uncertain quadratic and linear programs, where the uncertainty enters the constraints in an affine manner and the uncertainty set is a polytope. It is shown that one can convert the resulting semi-infinite optimization problem into a standard QP or LP with a finite number of decision variables and a finite number of constraints. This transformation is achieved in a computationally tractable way by solving as many LPs as there are constraints in the optimization problem without uncertainty. It is also shown that if the uncertainty set is given by upper and lower bounds only,then one need not solve any LPs in order to do this transformation; computing the 1-norms of the rows of the matrix by which the uncertainty enters the constraints is sufficient. The second part of the paper reviews and extends some definitions and results on input-to-state stability for nonlinear discrete-time systems. The third part of the paper shows how one can translate a class of robust finite-horizon optimal control problems (RFHOCPs) into the class of robust convex optimization problems that was studied in the first part of the paper. It is assumed that the system under consideration is linear, that there is a persistent, but bounded state disturbance that assumes values in a polytope and that there are mixed affine constraints on the input and state. By using the results from the previous sections, it is shown that one can set up a receding horizon controller (RHC) that is input-tostate stable (ISS) and guarantees robust constraint satisfaction for all time. The number of decision variables and constraints in the RFHOCP that define the RHC law increases linearly with the horizon length. 1

In this paper a method for nonlinear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the state-space known as clusters the plant is assumed to be an element in a polytope which vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to being linear systems. The clusters cover the region of interest in the state-space. An affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single nonconvex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed-loop. The feasibility problem is attacked by a branch-andbound based global approach. If the feasibility check is successful, the Liapunov matrix and the piecewise ...

Abstract — In order to effectively control nonlinear and multivariable models, and to incorporate constraints on system states, inputs and outputs (bounds, rate of change), a suitable (sometimes necessary) controller is Model Predictive Control (MPC). MPC is an optimization-based control scheme that requires abundant matrix operations for the calculation of the optimal control moves. In this work we propose a mixed software and hardware embedded MPC implementation. Using a codesign step and based on profiling results, we decompose the optimization algorithm into two parts: one that fits into a host processor and one that fits into a custom made unit, that performs the computationally demanding arithmetic operations. The profiling results and information on the coprocessor design are provided. I.

Abstract — This paper presents a decentralized robust Model Predictive Control algorithm for multi-vehicle trajectory optimization. The algorithm is an extension of a previous robust safe but knowledgeable (RSBK) algorithm that uses the constraint tightening technique to achieve robustness, an invariant set to ensure safety, and a cost-to-go function to generate an intelligent trajectory around obstacles in the environment. Although the RSBK algorithm was shown to solve faster than the previous robust MPC algorithms, the approach was based on a centralized calculation that is impractical for a large group of vehicles. This paper decentralizes the algorithm by ensuring that each vehicle always has a feasible solution under the action of disturbances. The key advantage of this algorithm is that it only requires local knowledge of the environment and the other vehicles while guaranteeing robust feasibility of the entire fleet. The new approach also facilitates a significantly more general implementation architecture for the decentralized trajectory optimization, which further decreases the delay due to computation time.

Abstract — Model Predictive Control (MPC) has recently been applied to several relevant classes of hybrid systems with promising results. These developments generated an increasing interest towards issues such as stability and computational problems that arise in hybrid MPC. Stability aspects have been addressed only marginally. In this paper we present an extension of the terminal cost and constraint set method for guaranteeing stability in MPC to the class of constrained piecewise affine systems. Semidefinite programming is used to calculate the employed terminal weight matrix that ensures stability for quadratic cost based MPC. A procedure for computing a robust positively invariant set for piecewise linear systems is also developed. The implementation of the proposed method is illustrated by an example. Index Terms — Piecewise affine systems, Model predictive

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Hammersteinsy5 ems are a class ofsy] ems represented by a static nonlinearity at the input followed by a linear dy1 mic block. In this paper the static input nonlinearity is transformed into a poly opic description. The remaining uncertain linear model is used in a MPC algorithm of which the optimization problem involves minimization of a linear objective function subject to Linear Matrix Inequalities (LMIs), which is a convex problem. A procedure is presented to remove a number of LMIs from the optimization problem, prior to solving it.By means of an iterative procedure the conservatism of the poly13#R description can be reduced. Nominal closed loop stability of this Hammerstein MPC algorithm is guaranteed. A comparison is presented between the proposed algorithm and an algorithm which removes the nonlinearity from the control problem via an inversion. Key words: Predictive control, Hammersteinsynj160 Linear matrix inequalities, Stability , Actuator nonlinearities 1 Introduction In...