Linear systems with polynomial uncertainty structure: Stability margins and control

This paper is concerned with the stability and control of linear systems with uncertain physical parameters. The case where the characteristic equations of the systems are polynomially dependent on uncertain parameters is studied. A new algorithm is presented for the calculation of stability margins in the parameter space in general / ^p-norms. The stability is defined with respect to a desired region as the location of poles in the complex plane (so-called D-stability). The procedure is straight forward and computationally tractable. It contains the sweeping of the contour of the region (frequency sweeping) and at each point on the contour, the numerical solution of a system of equations. The Minimum Distance approach used in several previous works for polynomials with independent and linearly-dependent coefficients is generalized here for the case of polynomials with polynomially-dependent coefficients. Based on the above approach for the calculation of D-stability margins, a method is also presented for the design of robust controllers. Robust pole placement inside the regions of interest is addressed. The method provides a simple and computationally tractable solution for the synthesis of robust controllers for linear systems with polynomial uncertainty structure.