Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems

Abstract

Incremental stability and convergence properties for forced, infinite-dimensional, discretetime Lur'e systems are addressed. Lur'e systems have a linear and nonlinear component and arise as the feedback interconnection of a linear control system and a static nonlinearity. Discrete-time Lur'e systems arise in, for example, sampled-data control and integro-difference models. We provide conditions, reminiscent of classical absolute stability criteria, which are sufficient for a range of incremental stability properties and input-to-state stability (ISS). Consequences of our results include sufficient conditions for the converging-input convergingstate (CICS) property, and convergence to periodic solutions under periodic forcing.

Semi-global incremental input-to-state stability of discrete-time Lur’e systems

“…Sufficient conditions for the CICS property are obtained as a corollary. The relation between the present paper and our earlier work [5,13,14] is discussed in Remarks 3.3, 3.4 and 4.7.…”

“…Although this stability notion is semi-global, it is suitable for almost all practical applications, as all relevant initial conditions and inputs are likely to have their norm bounded by some R > 0. We refer the reader to papers such as [2,13,14] for varying notions of global incremental stability.…”

“…Related ideas, which have been explored in the contexts of contraction methods and convergent systems, can be found in [1,22,25] and the references therein. Recently, in [13,14], sufficient conditions have been determined which guarantee that certain infinitedimensional Lur'e systems are exponentially incrementally ISS. Moreover, in [13,14], exponential incremental ISS is used to infer convergence properties such as the converging-input converging-state (CICS) property (see, for example, [5]) and the asymptotic periodicity of the response to asymptotically periodic inputs.…”

Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities

“…Hence unique solutions of (2.3) exist, proving statement (1). To prove statement (2), we realize that the estimate (2.6) follows from an application of [13,Statement (ii) of Theorem 3.2] to (2. 19) and (2.20).…”

“…19) and (2.20). We proceed to verify that the hypotheses of [13,Theorem 3.2] hold. To this end, we note that inequality (2.5) implies the existence of an ε > 0 such that…”