A study on the applicability of symbolic computation in stabilising control design for switched systems - At-Tasneem, Mohd Amin (2012)
This thesis examines the problem of designing controllers for switched systems that assures stability of the overall system. The switched systems here refer to systems whose dynamic behaviour changes from time to time. Stability concepts for continuous time and discrete event systems cannot be used to assure stability of switched systems as mode switching sequence and dwell time influences the stability of the overall system. One method of ensuring stability of switched systems is by proving the existence of a Common Lyapunov Function for the system. However, finding a Common Lyapunov Function is not trivial. Most methods that have been introduced to solve this problem involve the formulation of the system dynamic model and constraints into a Linear Matrix Inequality (LMI) structure. Then, computational methods are used to solve the LMI problem. Two problems arise from using LMIs to find solutions. Firstly, available LMI solvers use numerical computation which raises the possibility of rounding off errors. Secondly, the computational burden would be quite heavy, especially if the switched system comprises of a large number of subsystems or the subsystems are of a high order. The Haris-Rogers method is an alternative approach that has been previously developed for designing controllers based on the existence of a Common Lyapunov Function. In this approach, the problem is reduced to solving two sets of Linear Inequalities (LI), hence reducing the computational burden, as compared to methods that use LMIs. To overcome rounding off errors, symbolic computation methods should be used. However, this would require more computational power compared to numerical computation methods. In this study, a Switched System Control Design Toolbox employing symbolic computation based on the Haris-Rogers solution method was developed using the Maple software. A switched system with four second order subsystems was used as a test case and the toolbox successfully found a Common Lyapunov Function and subsequently designed the controller. For comparison, an LMI based method was tested with the same switched system using Maple and also three numerical LMI solvers, namely cvx, LMI Solver and Yalmip. Only Yalmip successfully generated a correct solution, while LMI Solver generated an incorrect solution since the controller obtained was clearly unstable. Maple and cvx failed to generate any controller. The Haris-Rogers method was also tested for the same switched system using cvx, LMI Solver and Yalmip, and all three produced correct results. All computational work and testing were performed on a system running Intel Core i5 2.67 GHz, 64-bit operating system with 2 GB RAM.