Tracking Of Constant Reference Signals Research Articles

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.

Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

This paper is concerned with integral control of systems with hysteresis. Using an input-output approach, it is shown that application of integral control to the series interconnection of either (a) a hysteretic input nonlinearity, an L2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L2-stable, time-invariant linear system and a hysteretic output nonlinearity, guarantees, under certain assumptions, tracking of constant reference signals, provided the positive integrator gain is smaller than a certain constant determined by a positivity condition in the frequency domain. The input-output results are applied in a general state-space setting wherein the linear component of the interconnection is a well-posed infinite-dimensional system.

Absolute stability and integral control

Absolute stability results of both circle criterion and Popov type are derived for finite-dimensional linear plants with non-linearity in the feedback loop. The linear plant contains an integrator (and so is not asymptotically stable). The (possibly time-varying) non-linearity satisfies a particular sector condition which allows for cases with zero lower gain (such as saturation and deadzone). The conjunction of stable, but not asymptotically stable, linear plants and non-linearities with possibly zero lower gain is a distinguishing feature of the paper. The absolute stability results are invoked in proving convergence and stability properties of low-gain integral feedback control for tracking of constant reference signals in the context of exponentially stable linear systems subject to input and output non-linearities.

On tracking and disturbance rejection by adaptive control

For linear minimum-phase relative-degree-one systems subject to disturbances and nonlinear perturbations, results pertaining to disturbance rejection and adaptive tracking of constant reference signals are presented and discussed in the context of related results in the literature.

Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities

Continuous-time low-gain integral control strategies are presented for tracking of constant reference signals for finite-dimensional, continuous-time, asymptotically stable, single-input single-output, linear systems subject to a globally Lipschitz and non-decreasing input nonlinearity and a locally Lipschitz, non-decreasing and affinely sector-bounded output nonlinearity. Both non-adaptive (but possibly time varying) and adaptive integrator gains are considered. In particular, it is shown that applying error feedback using an integral controller ensures asymptotic tracking of constant reference signals, provided that (a) the steady-state gain of the linear part of the plant is positive, (b) the positive integrator gain is ultimately sufficiently small and (c) the reference value is feasible in a very natural sense. The classes of actuator and sensor nonlinearities under consideration contain standard nonlinearities important in control engineering such as saturation and deadzone.

Adaptive Low-Gain Integral Control of Multivariable Well-Posed Linear Systems

The principle of low-gain integral control for finite-dimensional systems is well known. More recently, low-gain integral control results have been obtained for classes of infinite-dimensional systems. In this paper we show that integral control with a simple and natural adaptation of the integrator gain achieves tracking of constant reference signals for every exponentially stable, multivariable, well-posed, infinite-dimensional, linear system whose steady-state gain matrix has its spectrum in the open right-half plane. Our results considerably extend, improve, and simplify previous work by the authors [SIAM J. Control Optim. {35} (1997), pp. 78--116].

Discrete-Time and Sampled-Data Low-Gain Control of Infinite-Dimensional Linear Systems in the Presence of Input Hysteresis

We introduce a general class of causal dynamic discrete-time nonlinearities which have certain monotonicity and Lipschitz continuity properties. In particular, the discretizations of a large class of continuous-time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class. It is shown that closing the loop around a power-stable, linear, infinite-dimensional, discrete-time, single-input, single-output system, subject to an input nonlinearity from the class under consideration and compensated by a discrete-time integral controller, guarantees asymptotic tracking of constant reference signals, provided that (a) the positive integrator gain is sufficiently small and (b) the reference value is feasible in a very natural sense. We apply this result in the development of sampled-data low-gain integral control for exponentially stable, regular, linear, infinite-dimensional, continuous-time systems subject to input hysteresis.

Low-Gain Integral Control of Well-Posed Linear Infinite-Dimensional Systems with Input and Output Nonlinearities

Time-varying low-gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable, well-posed, linear, infinite-dimensional, single-input–single-output, systems—subject to globally Lipschitz, nondecreasing input and output nonlinearities. It is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense, provided that (a) the steady-state gain of the linear part of the system is positive, (b) the reference value r is feasible in an entirely natural sense, and (c) the positive gain function t↦k(t) is ultimately sufficiently small and not of class L1. Under a weak restriction on the initial data it is shown that (a), (b), and (c) ensure asymptotic tracking. If, additionally, the impulse response of the linear part of the system is a finite signed Borel measure, the global Lipschitz assumption on the output nonlinearity may be considerably relaxed.

Discrete‐time low‐gain control of linear systems with input/output nonlinearities

AbstractDiscrete‐time low‐gain control strategies are presented for tracking of constant reference signals for finite‐dimensional, discrete‐time, power‐stable, single‐input, single‐output, linear systems subject to a globally Lipschitz, non‐decreasing input nonlinearity and a locally Lipschitz, non‐decreasing, affinely sector‐bounded output nonlinearity (the conditions on the output nonlinearities may be relaxed if the input nonlinearity is bounded). Both non‐adaptive and adaptive gain sequences are considered. In particular, it is shown that applying error feedback using a discrete‐time ‘integral’ controller ensures asymptotic tracking of constant reference signals, provided that (a) the steady‐state gain of the linear part of the plant is positive, (b) the positive gain sequence is ultimately sufficiently small and (c) the reference value is feasible in a very natural sense. The classes of input and output nonlinearities under consideration contain standard nonlinearities important in control engineering such as saturation and deadzone. The discrete‐time results are applied in the development of sampled‐data low‐gain control strategies for finite‐dimensional, continuous‐ time, exponentially stable, linear systems with input and output nonlinearities. Copyright © 2001 John Wiley & Sons, Ltd.

Integral control of linear systems with actuator nonlinearities: lower bounds for the maximal regulating gain

Closing the loop around an exponentially stable single-input/single-output regular linear system, subject to a globally Lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller, is known to ensure asymptotic tracking of constant reference signals, provided that: 1) the steady-state gain of the linear part of the plant is positive; 2) the positive integrator gain is sufficiently small; and 3) the reference value is feasible in a very natural sense. Here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step-response and second-order systems with a time-delay in the input or output. The lower bounds are given in terms of open-loop frequency/step response data and the Lipschitz constant of the nonlinearity, and are hence readily obtainable.

Integral Control of Infinite-Dimensional Linear Systems Subject to Input Saturation

Closing the loop around an exponentially stable single-input single-output regular linear system, subject to a globally Lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller, is shown to ensure asymptotic tracking of constant reference signals, provided that (a) the steady-state gain of the linear part of the plant is positive, (b) the positive integrator gain is sufficiently small, and (c) the reference value is feasible in a very natural sense. The class of actuator nonlinearities under consideration contains standard nonlinearities important in control engineering such as saturation and deadzone.

Stabilizing Predictive Control of Nonlinear ARX Models

In this paper a predictive control algorithm for nonlinear discrete-time systems is presented. Starting from a state-space model, conditions for the asymptotic tracking of constant reference signals in a neighbourhood of a given equilibrium are first derived. Then it is shown that the system under control can be locally described in terms of a suitable NARX (Nonlinear ARX) model, which, in practice, can be identified by means of well-established techniques. For the NARX model, a receding-horizon predictive control algorithm is proposed which guarantees local stability and robust asymptotic tracking in the neighbourhood of the equilibrium. Conditions for local stability and tracking are given in terms of reachability, observability and transmission zeros of the linearization of the original state-space system around the equilibrium. An example is reported to illustrate the effectiveness of the proposed method.