Switching between different controllers to achieve a required behavior is rather common in practical applications. A common way of implementing switching is for example by the use of PID controllers with selectors [1]. This can be used to guarantee that certain process variables are kept within specified bounds. Simple simulation studies and results from optimal control theory such as bang-bang control show that performance can be enhanced by switching between a number of state-feedback laws. A sufficient condition to be able to design control laws based on switching between different controllers, is that we can choose a controller that makes the time derivative of a positive definite function negative along all trajectories in a neighborhood of the desired set-point [14]. Here we give a computational test of this requirement, formulated as a quantifier elimination problem.
For nonlinear systems, switched controllers can be used for stabilization. Brockett has given a necessary condition for when a system with continuously differentiable right hand side can be stabilized by a continuously differentiable state feedback [3, 15]. However, there are many examples of systems that do not satisfy this condition but can be stabilized by a discontinuous state feedback [11].
In output regulation and tracking problems, the zero dynamics [7, 4] plays an important role. Stable zero dynamics is necessary for proper regulation or tracking, since otherwise there are internal variables that exhibit unbounded behavior. There are nonlinear systems whose zero dynamics cannot be stabilized by continuous state feedback but can be stabilized by switched controllers [11, 10]. In this report we approach the problem of analysis and design of discontinuous control laws in a constructive way. Quantifier elimination is used to answer a number of questions. Is it possible to stabilize a system by switching between a number of so called basic controllers? Can we obtain stable zero dynamics for the output regulation or tracking problem using a switched controller? Can we obtain an invariant set of states using a switched controller?
The report is organized as follows. In Section 2 we review some basic concepts and definitions related to stability. Since we focus on switched systems we also need to define what we mean by a solution to a system of differential equations with discontinuous right hand sides. Section 3 deals with the problem of stabilizing a polynomial system by switching between a number of continuous controllers. These results are further applied in Section 4 where we show how zero dynamics of a nonlinear system which is not affine in the control can be stabilized by switching between different output zeroing controllers. A number of examples, where we use quantifier elimination software, is presented in Section 5, and Section 6 gives a summary of the report. Throughout the report we utilize quantifier elimination to deal constructively with the computational questions that arise.