Optimal control of systems with constraints

Abstract

In this thesis, we present new developments resulting from our work on constrained LQG control. Our work can be divided into two broad areas: Generalizations of well known results associated with unconstrained LQG control such as the Separation Theorem, and development of new computational algorithms for solving these problems. A summary of the topics we are presenting is as follows: Linearly constrained LQG control In this chapter, we study the LQG control problem with finitely many and infinitely many linear inequality constraints. We derive the optimal control for these problems, and prove the Separation Theorem. We show (using duality theory) that when there are finitely many constraints, the optimal control can be calculated by solving a finite dimensional optimization problem. When there are infinitely many constraints, the optimal control is. determined by solving an infinite programming problem. LQG control with IQ constraints We consider the LQG control problem with finitely many integral quadratic constraints. Using duality theory, we derive the optimal control, and show that it can be calculated by solving a finite dimensional optimization problem. Relevant gradient formulae pertaining to this finite dimensional problem are derived. We prove that the Separation Theorem does not hold. Rather, a result we call a Quasi-separation Theorem is proven. Indefinite LQG control with IQ constraints We extend recently discovered results for full observation LQG control with an indefinite control weight to the constrained case. We derive conditions under which the optimal control can be explicitly derived, and calculated by solving a finite dimensional optimization problem. We also derive relevant gradient formulae so that algorithms for nonlinear optimization problems can be used to solve this problem. Infinite quadratic programming In this chapter, we derive an alternative method for solving the linearly constrained LQG problem. Drawing inspiration from the field of interior point methods, we .derive a path following interior point method for linearly constrained quadratic programming to infinite dimensions. In this way, an interior point method for linearly constrained LQG problems is derived. We also prove global convergence of this algorithm. Infinite linear programming We generalize the potential reduction interior point method for finite dimensional linear programming to the infinite linear programming case. We show how this algorithm can be used to solve linear optimal control problems with continuous state constraints, as well as continuous linear programming problems. In this way, we derive new methods for solving these problems. We also examine some convergence issues. Show more