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We study low‐gain (P)roportional (I)ntegral control of multivariate discrete‐time, 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. [ABSTRACT FROM AUTHOR]

We suppose that a continuous-time feedback is input--output stabilizing for an infinite-dimensional system. We address the question of whether the sampled-data controller obtained by applying idealized sample-and-hold to this continuous-time feedback is also input--output stabilizing if the sampling time is small enough. This question has been previously addressed for fairly general systems under various conditions. In this note, we restrict our attention to Riesz spectral systems, for which we generalize the existing results. Specifically, we give two relatively simple conditions which, combined, are sufficient for the sampled-data controller to be stabilizing. The first condition is a spectrum decomposition for the open-loop system generator, which by itself is necessary, but not sufficient, for the system to be stabilizable by sampled-data control. The second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators. Index Terms--Infinite-dimensional systems, sampled-data control, stabilization.

An adaptive switching feedback control scheme is proposed for classes of discrete-time, positive difference equations, or systems of equations. In overview, the objective is to choose a control strategy which ensures persistence of the state, consequently avoiding zero which corresponds to absence or extinction. A robust feedback control solution is proposed as the effects of different management actions are assumed to be uncertain. Our motivating application is to the conservation of dynamic resources, such as populations, which are naturally positive quantities and where discrete and distinct courses of management actions, or control strategies, are available. The theory is illustrated with examples from population ecology. [ABSTRACT FROM AUTHOR]

It is a well-known principle, for finite-dimensional systems, that applying sampled-and-hold in the feedback loop around a stabilizing state feedback (or dynamic) controller results in a stable sampled-data feedback control system if the sampling period is small enough. The principle extends to infinite-dimensional systems with compact state feedback if either the input operator is bounded or the given state-space system is analytic. In this note, we give an example for which this principle fails but which nevertheless satisfies certain necessary conditions arising in sampled-data control of infinite-dimensional systems. Index Terms--Infinite-dimensional systems, nonrobustness, sampled-data control.

It is well known that proportional output feedback control can stabilize any relative-degree one, minimum-phase system if the sign of the feedback is correct and the proportional gain is high enough. Moreover, there exist simple adaptation laws for tuning the proportional gain (so-called high-gain adaptive controllers) which do not need to know the system and do not attempt to identify system parameters. In this paper the authors consider sampled versions of the high-gain adaptive controller. The motivation for sampling arises from the possibility that the output of a system may not be available continuously, but only at sampled times. The main point of interest is the need to develop techniques for adapting the sampling rate, since the stiffness of the system increases as the proportional gain is increased. Our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1/log k. Index Terms: Adaptive stabilization, minimum-phase systems, proportional control, sampled-data control.

A class of recurrent neural networks is shown to possess a stable limit cycle. A gradient type algorithm is used to modify the parameters of the network so that it learns and replicates autonomously a time varying periodic signal. The results are applied to controlling the repetitive motion of a two-link robot manipulator. Index Terms - Hopf bifurcation, learning systems, neural networks, nonlinear dynamics.

Using a frequency-domain analysis, it is shown that the application of a feedback controller of the form k/(z - 1) or kz/(z - 1), where k [element of] R, to a power-stable infinite-dimensional discrete-time system with square transfer-function matrix G(z) will result in a power-stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that i) all the eigenvalues of G(1) have positive real parts, and ii) the gain parameter k is positive and sufficiently small. Moreover, if G(1) is positive definite, we show how the gain parameter gain k can be tuned adaptively. The resulting adaptive tracking controllers are universal in the sense that they apply to any power-stable system with G(1) [greater than] 0; in particular, they are not based on system identification or plant parameter estimation algorithms, nor is the injection of probing signals required. Finally, we apply these discrete-time results to obtain adaptive sampled-data low-gain controllers for the class of regular systems, a rather general class of infinite-dimensional continuous-time systems for which convenient representations are known to exist, both in state space and in frequency domain. We emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval. Index Terms - Adaptive tracking, discrete-time systems, frequency-domain methods, infinite-dimensional systems, sampled-data control, state-space methods.