Lim, Andrew E.B.

### Description

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...[Show more] 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.

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