Model predictive control for stochastic systems by randomized algorithms

The main topic of this thesis is control of dynamic systems that are subject to stochastic disturbances and constraints on the input and the state. The main motivation for dealing with control of such systems is that there is no method available that adequately deals with this problem, despite the fact that stochastic, constrained systems are often encountered in real world problems. For example, in process industry the margins of physical quantities such as temperature, pressure, concentration, velocity and position can be expressed as amplitude constraints in a natural way. Such constraints are usually persistent in that suitable control actions need to be implemented that respect these constraints irrespective of the presence of uncontrolled disturbances that effect the system. Goals of the thesis are to 1. Formulate a mathematical problem for the synthesis of a controller that will achieve desired performance of the controlled system. More precisely, to minimize a performance measure that captures desired performance while respecting constraints in the face of stochastic disturbances. 2. Deduce verifiable conditions under which the problem formulated in 1. is solvable. 3. Formulate a solution concept for the problem in 1. that is based on the model predictive control technique. 4. Create feasible computational algorithms for the synthesis of controllers that solve control problems from 1. within the solution setup from 3. 5. Investigate convergence properties of the approximate solutions obtained by computational algorithms from 4. The main tool that is used in the thesis to solve the problem formulated in 1. is the model predictive control technique. Model predictive control has had a significant and widespread impact on industrial process control. When dealing with stochastic systems, however, application of the standard model predictive control algorithms results in a significant loss in the controlled system performance. Therefore, to deal with the problem 1. within the model predictive control framework, it was necessary to develop alternative model predictive control techniques. Contributions of the thesis are twofold. The first set of contributions is made with regard to the model predictive control of constrained, stochastic systems. In this thesis, we develop a novel approach to the model predictive control of such systems, that is based on the optimization in closed loop over the control horizon and stochastic sampling of the disturbance i.e. a randomized algorithm. The second set of contributions has been made in more general framework of the optimal control of stochastic systems that are subject to input and state constraints. We present a novel problem setup for control of such systems and give initial results that are concerned with solvability conditions for the posed optimization problem and the characterization of the optimal solution.