## Abstract

This dissertation concerns numerical procedures for the problems arising in moving horizon estimation and control. Moving horizon estimation and control is also referred to as model predictive control as well as receding horizon estimation and control. Model predictive control is the most successful and applied methodology beyond PID-control for control of industrial processes. The main contribution of this thesis is introduction and definition of the extended linear quadratic optimal control problem for solution of numerical problems arising in moving horizon estimation and control. An efficient structure-employing methodology for solution of the extended linear quadratic optimal control problem is provided and it is discussed how this solution is employed in solution of constrained model predictive control problems as well as in the solution of nonlinear optimal control and estimation problems.

Chapter 1 motivates moving horizon estimation and control as a paradigm for control of industrial processes. It introduces the extended linear quadratic control problem and discusses its central role in moving horizon estimation and control. Introduction, application and efficient solution of the extended linear quadratic control problem is the key contribution of this thesis. In addition chapter 1 provides a comprehensive survey of existing methods for model predictive control.

Chapter 2 discusses computational methods and inherent approximations in model predictive control. By considering the stochastic optimal control problem, the approximations and assumptions of model predictive control are pin-pointed. In an ad hoc fashion the separation principle and certainty-equivalence are assumed to prevail, such that the stochastic optimal control problem may be separated into an estimation problem and a deterministic optimal control problem. Both the estimation problem and the obtained deterministic optimal control problem are demonstrated to be instances of a constrained non-linear optimal control problem. In the sequential quadratic programming algorithm for solution of constrained nonlinear optimal control problems, the quadratic subproblem generated at each iteration is shown to be a constrained linear-quadratic optimal control problem. Procedures for generation of the constrained linear-quadratic optimal control problem and its data from the nonlinear estimation problem, the nonlinear control problem, the linear moving horizon estimator, and the linear moving horizon controller are provided. The significance of these conversions to constrained linear-quadratic optimal control problems is that the entire model predictive control problem can be solved efficiently by having efficient algorithms tailored for solution of the con-strained linear-quadratic optimal control problem. The major intention in this chapter is to emphasize the central role of the constrained linear-quadratic optimal control problem in model predictive control such that tailored algorithms for the constrained linear-quadratic optimal control problem is motivated and justified.

Chapter 3. A primal active set, a dual active set, and an interior point algorithm for solution of the constrained linear quadratic optimal control problem are outlined. The major computational effort in all these algorithms reduces to solution of certain unconstrained linear quadratic optimal control problems, i.e. the extended linear quadratic control problem. A Riccati recursion procedure for effective solution of such unconstrained problems is stated.

Chapter 4. Based on dynamic programming, Riccati recursion procedures for the linear-quadratic optimal control problem as well as the extended linear-quadratic optimal control problem are developed. Compared to alternative solution procedures such as control vector parameterization by elimination of the states, the Riccati based procedure is highly efficient for long prediction horizons. The extended linear-quadratic optimal control problem may also be regarded as an equality constrained quadratic program with special structure. The computation of the optimal solution-Lagrange multiplier pair for a convex equality constrained quadratic program is specialized to the extended linear-quadratic optimal control problem treated as a quadratic program. Efficient solution of the highly structured KKT-system corresponding to the extended linear-quadratic optimal control problem is facilitated by the Riccati recursion developed by dynamic programming.

Chapter 5 presents the principles for efficient solution of unconstrained non-linear optimal control problems described by ordinary differential equations. These principles are presented through numerical solution of a continuous-time nonlinear optimal control problem of the Bolza form. To focus on the basic principles involved and for illustrative purposes, the continuous-time Bolza problem is discretized by the explicit Euler method. The discrete-time nonlinear optimal control problem of the Bolza form is solved by different SQP methods and an algorithm based on the discrete maximum principle. The SQP algorithms presented are implementations based on open- and closed-loop feasible path control vector parameterizations as well as an infeasible path simultaneous procedure. Two procedures for solution of the quadratic programs are presented. In the first procedure, the structure of the quadratic programs arising in the solution of the nonlinear optimal control problem is utilized by a Riccati iteration based factorization of the resulting KKT-system. In the second procedure, an efficient procedure for elimination of the states and solution of a dense reduced space quadratic program is presented. These methods are compared for a simple process example operated around an unstable equilibrium. The infeasible path and the closed-loop feasible path algorithms converge for this example. The implemented open-loop feasible path algorithms are not able to converge to an unstable equilibrium. The Riccati based solution procedure enables implementation of the stabilized infeasible path SQP algorithm as well as the closed-loop feasible path SQP algorithm. The methods are presented in a framework that is easily extended to constrained nonlinear optimal control problems. Such extensions and methodologies for efficient integration of the ordinary differential equations as well as the corresponding sensitivity equations are discussed.

Chapter 6 summarizes the main contribution of this thesis. It briefly discusses the pros and cons of using the extended linear quadratic control framework for solution of deterministic optimal control problems.

Appendices. Appendix A demonstrates how quadratic programs arise in sequential quadratic programming algorithms. Appendix B uses a control vector parameterization approach to express various extended constrained linear quadratic optimal control problems as standard quadratic programs. Appendix C discuss construction of maximal output admissible sets. It provides an algorithm for computation of the maximal output admissible set for linear model predictive control. Appendix D provides results concerning linear regression. Appendix E discuss prediction error methods for identification of linear models tailored for model predictive control.

Chapter 1 motivates moving horizon estimation and control as a paradigm for control of industrial processes. It introduces the extended linear quadratic control problem and discusses its central role in moving horizon estimation and control. Introduction, application and efficient solution of the extended linear quadratic control problem is the key contribution of this thesis. In addition chapter 1 provides a comprehensive survey of existing methods for model predictive control.

Chapter 2 discusses computational methods and inherent approximations in model predictive control. By considering the stochastic optimal control problem, the approximations and assumptions of model predictive control are pin-pointed. In an ad hoc fashion the separation principle and certainty-equivalence are assumed to prevail, such that the stochastic optimal control problem may be separated into an estimation problem and a deterministic optimal control problem. Both the estimation problem and the obtained deterministic optimal control problem are demonstrated to be instances of a constrained non-linear optimal control problem. In the sequential quadratic programming algorithm for solution of constrained nonlinear optimal control problems, the quadratic subproblem generated at each iteration is shown to be a constrained linear-quadratic optimal control problem. Procedures for generation of the constrained linear-quadratic optimal control problem and its data from the nonlinear estimation problem, the nonlinear control problem, the linear moving horizon estimator, and the linear moving horizon controller are provided. The significance of these conversions to constrained linear-quadratic optimal control problems is that the entire model predictive control problem can be solved efficiently by having efficient algorithms tailored for solution of the con-strained linear-quadratic optimal control problem. The major intention in this chapter is to emphasize the central role of the constrained linear-quadratic optimal control problem in model predictive control such that tailored algorithms for the constrained linear-quadratic optimal control problem is motivated and justified.

Chapter 3. A primal active set, a dual active set, and an interior point algorithm for solution of the constrained linear quadratic optimal control problem are outlined. The major computational effort in all these algorithms reduces to solution of certain unconstrained linear quadratic optimal control problems, i.e. the extended linear quadratic control problem. A Riccati recursion procedure for effective solution of such unconstrained problems is stated.

Chapter 4. Based on dynamic programming, Riccati recursion procedures for the linear-quadratic optimal control problem as well as the extended linear-quadratic optimal control problem are developed. Compared to alternative solution procedures such as control vector parameterization by elimination of the states, the Riccati based procedure is highly efficient for long prediction horizons. The extended linear-quadratic optimal control problem may also be regarded as an equality constrained quadratic program with special structure. The computation of the optimal solution-Lagrange multiplier pair for a convex equality constrained quadratic program is specialized to the extended linear-quadratic optimal control problem treated as a quadratic program. Efficient solution of the highly structured KKT-system corresponding to the extended linear-quadratic optimal control problem is facilitated by the Riccati recursion developed by dynamic programming.

Chapter 5 presents the principles for efficient solution of unconstrained non-linear optimal control problems described by ordinary differential equations. These principles are presented through numerical solution of a continuous-time nonlinear optimal control problem of the Bolza form. To focus on the basic principles involved and for illustrative purposes, the continuous-time Bolza problem is discretized by the explicit Euler method. The discrete-time nonlinear optimal control problem of the Bolza form is solved by different SQP methods and an algorithm based on the discrete maximum principle. The SQP algorithms presented are implementations based on open- and closed-loop feasible path control vector parameterizations as well as an infeasible path simultaneous procedure. Two procedures for solution of the quadratic programs are presented. In the first procedure, the structure of the quadratic programs arising in the solution of the nonlinear optimal control problem is utilized by a Riccati iteration based factorization of the resulting KKT-system. In the second procedure, an efficient procedure for elimination of the states and solution of a dense reduced space quadratic program is presented. These methods are compared for a simple process example operated around an unstable equilibrium. The infeasible path and the closed-loop feasible path algorithms converge for this example. The implemented open-loop feasible path algorithms are not able to converge to an unstable equilibrium. The Riccati based solution procedure enables implementation of the stabilized infeasible path SQP algorithm as well as the closed-loop feasible path SQP algorithm. The methods are presented in a framework that is easily extended to constrained nonlinear optimal control problems. Such extensions and methodologies for efficient integration of the ordinary differential equations as well as the corresponding sensitivity equations are discussed.

Chapter 6 summarizes the main contribution of this thesis. It briefly discusses the pros and cons of using the extended linear quadratic control framework for solution of deterministic optimal control problems.

Appendices. Appendix A demonstrates how quadratic programs arise in sequential quadratic programming algorithms. Appendix B uses a control vector parameterization approach to express various extended constrained linear quadratic optimal control problems as standard quadratic programs. Appendix C discuss construction of maximal output admissible sets. It provides an algorithm for computation of the maximal output admissible set for linear model predictive control. Appendix D provides results concerning linear regression. Appendix E discuss prediction error methods for identification of linear models tailored for model predictive control.

Original language | English |
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Place of Publication | Kgs. Lyngby |
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Publisher | Technical University of Denmark |

Number of pages | 462 |

Publication status | Published - 2004 |