ISSN 0005-1179, Automation and Remote Control, 2008, Vol. 69, No. 12, pp. 1991–2026. c  Pleiades Publishing, Ltd., 2008. Original Russian Text c  E.N. Gryazina, B.T. Polyak, A.A. Tremba, 2008, published in Avtomatika i Telemekhanika, 2008, No. 12, pp. 3–40. REVIEWS D-decomposition Technique State-of-the-art 1 E. N. Gryazina, B. T. Polyak, and A. A. Tremba Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia Received December 24, 2007 Abstract—It is a survey of recent extensions and new applications for the classical D-decompo- sition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H ∞ -criterion. It is shown that the D-decomposition technique can be naturally linked with M - Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique. PACS number: 02.30.Yy DOI: 10.1134/S0005117908120011 1. INTRODUCTION Consider a linear system depending on a vector parameter k with a characteristic polyno- mial a(s, k). The boundary of a stability domain (in the space k) is given by the equation a(jω,k)=0, ω ∈ (-∞, +∞), (1) that is the imaginary axis (the boundary of instability in the root plane) is mapped into the parameter space. If k ∈ R 2 (or k ∈ C) then we have two equations (real and imaginary part of (1)) in two variables and (in general) can define the parametric curve k(ω), -∞ <ω< ∞ defining a boundary of the stability domain. Moreover, the curve k(ω) divides the plane into root invariant regions (i.e., regions with a fixed number of stable and unstable roots of a(s, k)). This is the basic idea of D-decomposition approach. The idea can be traced to Vishnegradsky [85] who reduced a cubic polynomial to the form a(s, k)= s 3 + k 1 s 2 + k 2 s + 1 and treated the coefficients k 1 ,k 2 as parameters. Then Eq. (1) yields k 1 ω 2 =1,ω(k 2 - ω 2 ) = 0. Eliminating ω we get that D-decompo- sition is given by the hyperbola k 1 k 2 = 1. The stability domain is the set k 1 k 2 > 1. For the general case, similar ideas were exploited in [1, 27, 51] (the latter two papers deal with time-delay systems). Moreover, Nyquist plot can be considered as the realization of the same idea. But it was Yu. Neimark [13,14] who developed the rigorous algorithm (and coined the name “D-decomposition”). In the Western literature the technique is described first in [68]; the mapping of contours other that imaginary axis was also proposed. This line of research was significantly developed by Siljak [80–82]. He extended the approach for nonlinear systems and for the case of nonlinear parameter dependence. In his works D-decomposition (which he calls the parameter plane method) was broadened to become a useful tool for design purposes. Neimark’s method is also 1 This work was supported by the Russian Foundation for Basic Research, project nos. 08-08-00371, 05-08-01177 and Presidium RAS Programm no. 22. The preliminary version was presented at IX Chetaev Conference in the 12–16 of July 2007. 1991