This dissertation considers the decentralized control of switched linear systems with parameter dependent cost and system matrices. This problem class is investigated under a number of different formulations of player information structure, performance criteria and switching architecture. Such decentralized switched systems can be encountered in various applications like network control, control in a changing environment, economic theory, power systems, decision making in organizations, resource allocation. The thesis is roughly divided into three parts.
The first part of the thesis focuses on the static quadratic team problem, where players observe partial observations of an underlying random state and generate actions with the objective of minimizing the expected value of a common quadratic cost function in the player actions. One of the motivations behind studying this problem is to solve a static stochastic-parameter problem useful in solving dynamic switched control problems encountered later. The problem however is studied in full generality and an operator theoretic framework is presented to analyze the same. We prove that a scheme where strategies are updated by sequentially applying the best responses of players, converges to the team optimal strategy. Such an update scheme provides a mechanism to numerically compute arbitrarily close approximations of the team optimal strategy. It also acts as a tool for validating structure of the team optimal strategy which can be beneficial in some cases for analytical computation of these strategies.
The second part of the thesis considers dynamic switched optimal control problems with quadratic cost and players having local parameter knowledge. One of these problems is studied under full state feedback and i.i.d. parameter; the remaining two problems are output feedback, distinguished by the type of information structure: partially nested and one-step delayed sharing. For the former output feedback problem, parameters and measurements follow a partially nested structure with the parameters possibly being correlated across all stages. For the latter case, parameters are assumed to be Markov processes, with their values along with measurements available instantaneously to local controllers, but with a one time step delay to others. The solution to all these problems rely on the optimal solution to a static (one-stage) stochastic-parameter problem with local parameter dependent Gaussian measurements, and for this purpose the static quadratic team problem, examined in first part is used. The strategies obtained in all these dynamic problems are affine in the measurements with the parameter dependent coefficients obtained by solving a set of linear equations. These equations are immediately solvable when the total number of parameter values is finite. However, for the case of infinite parameter values, the update scheme examined in the first section also provides a mechanism to determine an approximation to the team optimal strategy.
In the final part of the thesis, we consider a setup with switched linear nested plant whose system matrices switch between a finite number of values, with transitions in time governed by a finite state automaton. A linear nested controller is sought with corresponding system matrices dependent on a finite path history of the plant’s system matrices in order to stabilize the plant and achieve a desired level of l2-induced norm performance. The nested structures of both plant and controller are characterized by block lower-triangular system matrices with compatible dimensions. For this setup, exact conditions are provided for the existence of a finite path dependent synthesis. These include conditions for the completion of scaling matrices obtained through an extended matrix completion lemma. When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to a set of linear matrix inequalities. The completion lemma also provides an algorithm to complete the closed loop scaling matrices leading to inequalities for controller synthesis
