| This thesis addresses the question of stability and controller synthesis of distributed systems. We will first consider controller synthesis for distributed systems. To represent the distributed systems, we use spatio-temporal graphs to describe the interconnection structure. In Chapter 2, we study systems whose spatio-temporal graph is finite and propose the notion of local dissipativity for performance analysis. The test of local dissipativity is convex and it simplifies the synthesis of distributed controller. In addition, we exploit the structure of local dissipativity and show the distributed computation algorithm which reduce the computation complexity and is useful practically. We also extend local dissipativity to study linear time-invariant systems interconnected with graphs in Chapter 3. Based on similar analysis, we formulate a condition to verify the stability and performance of the distributed system and the test is cast as a semidefinite program.; In Chapter 4, we consider the problem of how to computationally test whether a matrix inequality is positive semidefinite on a semialgebraic set. This problem is deeply related to stability test for systems with parameter uncertainties. We propose a family of sufficient conditions using the theory of matrix Positivstellensatz refutations. When the semialgebraic set is a hypercube, we give bounds on the degree of the required certificate polynomials. We further extend this result in Chapter 5 to consider a more general uncertainty set and we show the certificate is still degree bounded. |
