IQC arguments for analysis of the 3-state Moore-Greitzer compressor system

Abstract

The Integral Quadratic Constraint (IQC) framework developed by Professor Yakubovich and his co-workers, see Yakubovich et.al. (2004), is one of few available constructive tools for establishing robust stability of nonlinear systems. An explicit format of stability conditions, procedures for computing a Lyapunov function and developed libraries of IQCs for common nonlinearities in dynamics, all together have made the approach unique and at the same time so to say automatic for recovering stability conditions for many applications: in the process of analyzing a dynamical system, an engineer is just required to search for a suficiently rich set of IQCs describing nonlinearities in the dynamics so that such nonlinearities can be substituted in analysis by quadratic constraints, which they satisfy. The power of the methodology becomes also its weak part in an analysis of concrete systems. Searching IQCs is the dificult task in new examples, where a lack of a rich set of IQCs for concrete nonlinearities makes the method inconclusive or too rough to detect (in)-stability. The paper is aimed at a discussion of such an example of a nonlinear dynamical system (the classical 3-state Moore-Greitzer compressor model) augmented with the dynamical feedback controller, whose parameters should be adjusted to meet a stability condition. The closed-loop system has several nonlinearities and searching the corresponding IQCs to meet the stability conditions for this example is rather involved. To overcome the problem, we have previously described by different methods a set of parameters for which any solution of the closed loop system, if bounded, will converge to the origin and that the origin is locally asymptotically stable. However, the proof is incomplete without showing a boundedness of all solutions. To solve the task we have re-used some of the IQC framework ideas, where the method has been utilized and the corresponding IQCs have been found only for unbounded trajectories, if they would be present in closed loop system. The arguments have allowed completing the proof of stability and illustrating deliberate use of the IQC framework aimed at analysis of behavior of specific trajectories.