Solving and visualizing nonlinear parametric constraints in control based on quantifier elimination

Abstract

We present a new method with a software tool for parametric robust control synthesis by symbolic-numeric computation. The method is a parameter space approach and it is especially effective for analysis and design of fixed-structure controllers of rational type, which encompass PI and PID controllers. The real quantifier elimination (QE), which is one of the recent progresses in the symbolic computation, plays a key role in our development. The QE-based approach can uniformly deal with a lot of important design specifications for robust control such as frequency restricted H ∞ norm constraints, stability (gain/phase) margin and stability radius specifications, and pole location requirement by reducing such specifications to a particular type of formulae called a “sign definite condition (SDC)”. This is also useful for improving the efficiency of QE computations since we can utilize an efficient QE algorithm specialized to the SDC using the Sturm-Habicht sequence. We have developed a MATLAB toolbox for robust parametric control based on a parameter space approach accomplished by QE. The QE-based parameter space approach and numerical simulation of performances for specific controller parameter values taken from a controller parameter space are integrated conveniently in our toolbox with the assistance of a graphical user interface (GUI). With our toolbox the feasible regions of controller parameters are visualized in a parameter space for the controllers with three or two parameters. This enables control engineers to achieve multi-objective robust controller synthesis smoothly. We also discuss how to merge the numerical computation and the symbolic operation to make our new design methods more efficient in practical control design.