Convex Optimization Formulation of Controller Design Problems.

In this thesis we derive convex optimization formulations for four controller design problems. In the first and second problems, we consider a discrete-time time-varying linear dynamical system, perturbed by process noise, with linear noise corrupted measurements, over a finite horizon. In the first problem we address the problem of designing a general affine causal controller, in which the control input is an affine function of all previous measurements, in order to minimize a convex objective, in either a stochastic or worst-case setting. This controller design problem is not convex in its natural form, but can be transformed to an equivalent convex optimization problem by a nonlinear change of variables. Our method is related to the classical Q-design procedure for time-invariant, infinite-horizon linear controller design. In the second problem we consider the design of nonlinear dynamic controllers, in the same setting. We describe an extension of the Q-design method that affinely parametrizes all nonlinear causal controllers, and reduces the design problem to a convex stochastic optimization problem, which can be solved by standard methods such as sampling. In principle (for a large enough basis, and enough sampling) this method can solve the controller design problem to any degree of accuracy; in any case it can be used to find a suboptimal controller, using convex optimization methods. The third problem concerns a continuous-time linear system with sampled constant linear state-feedback control and a convex quadratic performance measure. The sample times, however, are subject to variation within some known interval. We use linear matrix inequality (LMI) methods to derive a Lyapunov function that establishes an upper bound on performance degradation due to timing jitter. We also show how to synthesize a constant state-feedback controller that minimizes the performance bound, for a given level of timing jitter. Our last problem concerns linear controller coefficient truncation, while guaranteeing that a given set of relaxed performance constraints is met. We describe a method that sequentially and greedily truncates individual coefficients, using a Lyapunov certificate, typically in LMI form, to guarantee performance. We then show how to extend the basic method to handle nonlinear plants and controllers.