We have investigated the application of quantifier elimination to the analysis of stabilizability of polynomial systems by switching between a number of controllers. We have shown how to check if a polynomial system is stabilizable by switching but also how to estimate the region of attraction in such cases.
In exact output tracking, the zero dynamics plays an important role. For control-affine systems the zero dynamics is uniquely determined and stability of the zero dynamics or, equivalently, the concept of minimum phase, is usually used as a feasibility condition for exact output tracking problems. For more general nonlinear systems the zero dynamics or the output zeroing controller are no longer unique. We argue that stabilizability of the zero dynamics is a more appropriate condition than just stability. In a number of examples we show how to stabilize the zero dynamics of different systems by switching between different output zeroing controllers, where no continuous stabilizing controller exists. Quantifier elimination is used to carry out the necessary computations, where we also can take control and state constraints into account in a direct manner.