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Stability regions in the parameter space: $D$-decomposition revisited. (English) Zbl 1121.93028
Summary: 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 $M - \Delta$ 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 $A + kB$. Similar technique can be applied to double-input double-output systems with two parameters.

MSC:
 93C05 Linear control systems 93D25 Input-output approaches to stability of control systems 93D99 Stability of control systems
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