Self-Excited Dynamics of Discrete-Time Lur'e Systems With Affinely Constrained, Piecewise-C$^{1}$ Feedback Nonlinearities

Abstract

Self-excited systems (SES) arise in numerous applications, such as fluid-structure interaction, combustion, and biochemical systems. In support of system identification and digital control of SES, this paper analyzes discrete-time Lur'e systems with affinely constrained, piecewise-C $^{1}$ feedback nonlinearities. In particular, a novel feature of the discrete-time Lur'e system considered in this paper is the structural assumption that the linear dynamics possess a zero at 1. This assumption ensures that the Lur'e system have a unique equilibrium for each constant, exogenous input and prevents the system from having an additional equilibrium with a nontrivial domain of attraction. The main result provides sufficient conditions under which a discrete-time Lur'e system is self-excited in the sense that its response is 1) nonconvergent for almost all initial conditions, and 2) bounded for all initial conditions. Sufficient conditions for 1) include the instability and nonsingularity of the linearized, closed-loop dynamics at the unique equilibrium and their nonsingularity almost everywhere. Sufficient conditions for 2) include asymptotic stability of the linear dynamics of the Lur'e system and their feedback interconnection with linear mappings that correspond to the affine constraints that bound the nonlinearity, as well as the feasibility of a linear matrix inequality.