Low‐gain integral control for a class of discrete‐time Lur'e systems with applications to sampled‐data control

Abstract

We study low-gain (P)roportional (I)ntegral control of multivariate discretetime, forced Lur'e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur'e system in feedback with sampled-data low-gain integral control. The theory is illustrated by two examples.

Identification of Self-Excited Systems Using Discrete-Time, Time-Delayed Lur'e Models

“…CT [20], [21], [22], [23], [24] [25], [26], [27], [28] [29], [30], [31] DT [32], [33], [34], [35], [36], [37] [38], [39], [40], [41], [42] [43], [44], [45] In order to address the special features of self-excited discrete-time Lur'e models, the main contribution of the present paper is to prove that a class of discrete-time Lur'e models with affinely constrained feedback are self-excited in the sense that 1) all trajectories are bounded and 2) the set of initial conditions for which the state trajectory is convergent has measure zero. Although an affinely constrained function need not be bounded or even sector-bounded, it must have linear growth, thus ruling out superlinear nonlinearities, as necessitated by the fact that discrete-time strictly proper linear systems are not high-gain stable.…”

“…The stability of Lur'e models is a classical problem, expressed by the Aizerman conjecture for sectorbounded nonlinearities [17], [18], [19]. Although the Aizerman conjecture is false, the stability of Lur'e models has been widely studied in both continuous time [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and discrete time [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45].…”