Stability regions in the parameter space: D-decomposition revisited

The challenging problem in linear control theory is to describe the total set of parameters (controller coefficients or plant characteristics) which provide stability of a system. For the case of one complex or two real parameters and SISO system (with a characteristic polynomial depending linearly on these parameters) the problem can be solved graphically by use of the so-called D  -decomposition. Our goal is to extend the technique and to link it with general M-ΔM-Δ framework. In this way we investigate the geometry of D-decomposition for polynomials and estimate the number of root invariant regions. Several examples verify that these estimates are tight. We also extend D-decomposition for the matrix case, i.e. for MIMO systems. For instance, we partition real axis or complex plane of the parameter k   into regions with invariant number of stable eigenvalues of the matrix A+kBA+kB. Similar technique can be applied to double-input double-output systems with two parameters.