This thesis presents solutions to robust and networked control problems using the semidefinite relaxation of quadratic optimization problems.

In the first part, we solve the problem of exact computation of the phase and gain margins of multivariable control systems. These stability margins are studied using the concept of the Davis-Wielandt shell of complex matrices. Calculation of the phase and gain margins requires solving a quadratically constrained quadratic program (QCQP) and a parametrized quadratically constrained problem, respectively, which are known to be difficult in general. The semidefinite relaxation (SDR) technique is often used as a computationally efficient approximation technique to solve QCQPs. It turns out that the QCQPs formulated from the phase and gain margin pro...[ Read more ]

This thesis presents solutions to robust and networked control problems using the semidefinite relaxation of quadratic optimization problems.

In the first part, we solve the problem of exact computation of the phase and gain margins of multivariable control systems. These stability margins are studied using the concept of the Davis-Wielandt shell of complex matrices. Calculation of the phase and gain margins requires solving a quadratically constrained quadratic program (QCQP) and a parametrized quadratically constrained problem, respectively, which are known to be difficult in general. The semidefinite relaxation (SDR) technique is often used as a computationally efficient approximation technique to solve QCQPs. It turns out that the QCQPs formulated from the phase and gain margin problems in this thesis fall under the class of quadratic optimization problem, for which the SDR is exact. This is so in the sense that optimal values of the QCQP and its SDR are equal, and an optimal solution for the original QCQP problem can be obtained from an optimal solution of its SDR. Thus, we are able to propose computationally efficient algorithms to compute the phase and gain margins with an arbitrarily high precision. In addition, we define the gain-phase margin of MIMO control systems and propose an algorithm to compute its lower bound.

In the second part, we study the networked stabilizability of discrete-time MIMO system over MIMO communication. The communication channel is modelled by a MIMO transceiver, which consists of a transmitter, a receiver and an AWGN MIMO channel. We assume that the MIMO transceiver is subject to transmission power constraints in transmitted signals. The aim is to find a fundamental limitation on the information constraints so as to stabilize the networked control system over MIMO communication. We obtain necessary and sufficient conditions on the predetermined admissible power level for networked stabilizability.