Vertex results for the steady state analysis of uncertain systems

Determination of the steady state response of a system to an input composed of steps, ramps, and various other signals is considered. If the system and/or the input depend on uncertain parameters then a worst-case philosophy dictates that the analysis goal should be to determine the maximum and minimum possible equilibrium values of the output. It is shown that the system is robustly stable and if the dependence on the uncertain parameters is to an extent multiaffine, then the extreme values of the steady state response over the entire set of parameters is equal to the extreme values over just the vertices of the parameter set. This vertex result is doubly useful because it is valid in both continuous-time and discrete-time. For the transient response of stable systems, the vertices do not provide sufficient information for a complete analysis. Examples are presented which show that the maximum overshoot to the step response of a robustly stable uncertain system does not necessarily occur at a vertex. The systems in these examples have an affine dependence on a single uncertain parameter