Synthesis of robust disturbance-feedback strategies by using semi-definite programming

Abstract

Systems in real-life have to deal with uncertainty in such a manner that a high level of performance is guaranteed under all conditions. The objective in this thesis is to obtain robust strategies that provide an upper bound (worst-case) on the performance of an uncertain system against all allowable disturbances and uncertainties. Semi-definite programming has proven to be an effective tool to design robust control strategies. Traditionally, robust strategies are designed in open-loop form leading to conservative strategies and, therefore, poor performances. This is not surprising as the open-loop control, contrary to feedback control, does not take into account the fact that in the future new measurements will be available. Directly designing state-feedback or output-feedback strategies leads to non-convexity and, hence, computational intractable problems. The synthesis of robust disturbance-feedback strategies can, on the other hand, easily be reformulated as a convex problem. This approach hence bypasses the non-convexity issue while maintaining the advantages of feedback strategies implying that robust disturbance-feedback problems can effectively be solved by semi-definite programming which is the main topic of this thesis. As a key result it is shown that for some systems both sources of conservatism attributed to this approach, namely the relaxation method and the affine parametrization, can be removed at the expense of an increase in computational effort. In addition, the framework for disturbance-feedback strategy synthesis needs to be extended such that it can effectively deal with a large class of model mismatches and uncertainties e.g. system uncertainties and non-linearities. The major obstacle of the implementation of these strategies as receding horizon control strategies is the computational load. Therefore, a gain-scheduling approach is suggested to transfer the computational effort to offline computations. As a result an automatic scheduling of the robust strategies is provided that does not require any online optimization. This work hopefully contributes to the usage of robust disturbance-feedback strategies, not only as a nice academic exercise, but as a valuable engineering tool to design robust strategies and reduce the conservatism so often encountered.