A new perspective on the robustness of Markov jump linear systems

This paper is concerned with the robust analysis and control of continuous-time Markov jump linear systems. The centerpiece of the paper is an alternative to the small-gain theorem, which hinges on the robustification of a certain adjoint Lyapunov operator. By means of this technique, it is proven that the small-gain theorem of Markov jump linear systems may sometimes be arbitrarily conservative, even when nonlinear Lipschitz disturbances are taken into account. The adjoint approach, on the other hand, provides the maximal degree of robustness for this particular setup. In addition, we prove that the adjoint design of controllers is solved more efficiently than the design based on small-gain analysis. Bearing these new facts in mind, we derive an iterative algorithm for the design of robust controllers, based on linear matrix inequalities. By means of numerical examples, regarding the robust control of an underactuated manipulator arm and of a simplified power systems model, it is shown that the adjoint design methodology can be much more advantageous than its small-gain counterpart.