Some remarks on adaptive stabilization of infinite-dimensional systems

It is the purpose of this note to show that a first-order adaptive controller stabilizes a large class of infinite-dimensional systems described by strongly continous semigroups. It is assumed that the plant is minimum-phase and has invertible high-frequency gain. Knowledge of the sign of the high-frequency gain is not required

PID control of second-order systems with hysteresis

The efficacy of proportional, integral and derivative (PID) control for set point regulation and disturbance rejection is investigated in a context of second-order systems with hysteretic components. Two basic structures are studied: in the first, the hysteretic component resides (internally) in the restoring force action of the system ('hysteretic spring' effects); in the second, the hysteretic component resides (externally) in the input channel (e.g. piezo-electric actuators). In each case, robust conditions on the PID gains, explicitly formulated in terms of the system data, are determined under which asymptotic tracking of constant reference signals and rejection of constant disturbance signals is guaranteed. Keywords: hysteresis; non-linear systems; PID control; tuning regulator

Semi-Global Persistence and Stability for a Class of Forced Discrete-Time Population Models

We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur’e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur’e system and, in this context, the multiplicative forcing may correspond toharvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control

The converging-input converging-state property for Lur'e systems

Using methods from classical absolute stability theory, combined with recent results on input-to-state stability (ISS) of Lur’e systems, we derive necessary and sufficient conditions for a class of Lur’e systems to have the converging-input converging-state (CICS) property. In particular, we provide sufficient conditions for CICS which are reminiscent of the complex Aizerman conjecture and the circle criterion and connections are also made with small gain ISS theorems. The penultimate section of the paper is devoted to non-negative Lur’e systems which arise naturally in, for example, ecological and biochemical applications: the main result in this context is a sufficient criterion for a so-called “quasi CICS” property for Lur’e systems which, when uncontrolled, admit two equilibria. The theory is illustrated with numerous examples

An adaptive servomechanism for a class of infinite-dimensional systems

A universal adaptive controller is constructed that achieves asymptotic tracking of a given class of reference signals and asymptotic rejection of a prescribed set of disturbance signals for a class of multivariable infinite-dimensional systems that are stabilizable by high-gain output feedback. The controller does not require an explicit identification of the system parameters or the injection of a probing signal. In contrast to most of the work in universal adaptive control, this paper is based on an input-output approach and the results do not require a state-space representation of the plant. The abstract input-output results are applied to retarded systems and integrodifferential systems

Input-to-state stability of discrete-time Lur'e systems

An input-to-state stability theory, which subsumes results of circle criterion type, isdeveloped in the context of discrete-time Lur’e systems. The approach developed is inspired by the complexified Aizerman conjecture

A circle criterion for strong integral input-to-state stability

We present sufficient conditions for integral input-to-state stability (iISS) and strong iISS of the zero equilibrium pair of continuous-time forced Lur'e systems, where by strong iISS we mean the conjunction of iISS and small-signal ISS. Our main results are reminiscent of the complex Aizerman conjecture and the well-known circle criterion. We derive a number of corollaries, including a result on stabilisation by static feedback in the presence of input saturation. In particular, we identify classes of forced Lur'e systems with saturating nonlinearities which are strongly iISS, but not ISS