Eugene P. Ryan

The initial-value problem for a class of Volterra functional differential equations—of sufficient generality to encompass, as special cases, ordinary d...

Tracking of an absolutely continuous reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class of non-linear, single-input, single-output, dynamical systems modelled by functional differential equations satisfying certain structural hypotheses (which, interpreted in the highly specialised case of linear systems, translate into assumptions of (i) relative degree one, (ii) positive high-frequency gain and (iii) stable zero dynamics). The control objective is evolution of the tracking error within a prespecified funnel, thereby guaranteeing prescribed transient performance and prescribed asymptotic tracking accuracy. This objective is achieved by a control which takes the form of linear error feedback with time-varying gain. The gain is generated by a non-linear feedback law in which the reciprocal of the distance of the tracking error to the funnel boundary plays a central role. In common with many established adaptive control methodologies, the overall feedback structure exploits an intrinsic high-gain property of the system, but differs from these methodologies in two fundamental respects: the funnel control gain is not dynamically generated and is not necessarily monotone. The main distinguishing feature of the present article vis à vis its various precursors is twofold: (a) non-linearities of a general nature can be tolerated in the input channel; (b) a more general formulation of prescribed transient behaviour is encompassed (including, for example, practical (M, μ)-stability wherein, for prescribed parameter values M > 1, μ > 0 and λ > 0, the tracking error e(·) is required to satisfy |e(t)| < max {Me ^−μt |e(0)|, λ} for all t ≥ 0).

A tracking problem is considered in the context of a class of multi-input, multi-output, nonlinear systems modelled by controlled functional differential equations. The class contains, as a prototype, all finite-dimensional, linear, m-input, m-output, minimum-phase systems with sign-definite “high-frequency gain". The first control objective is tracking of reference signals r by the output y of any system in : given , construct a feedback strategy which ensures that, for every r (assumed bounded with essentially bounded derivative) and every system of class , the tracking error is such that, in the case , or, in the case , . The second objective is guaranteed output transient performance: the error is required to evolve within a prescribed performance funnel (determined by a function φ). For suitably chosen functions α, ν and θ, both objectives are achieved via a control structure of the form with , whilst maintaining boundedness of the control and gain functions u and k. In the case , the feedback strategy may be discontinuous: to accommodate this feature, a unifying framework of differential inclusions is adopted in the analysis of the general case .

Three related but distinct scenarios for tracking control of uncertain systems are reviewed: asymptotic tracking, approximate tracking with prescribed asymptotic error bound, tracking with prescribed transient behaviour. A variety of system classes are considered, ranging from finite‐dimensional linear minimum‐phase systems to nonlinear, infinite‐dimensional systems described by functional differential equations. These classes are determined only by structural assumptions, such as stable zero dynamics and known relative degree. The objective is a single (and simple) control structure which is effective for every member of the underlying system class: no attempt is made to identify the particular system being controlled.

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 L 2-stable, time-invariant linear system and a non-decreasing globally Lipschitz static output nonlinearity, or (b) an L 2-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 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.

Tracking of reference signals (assumed bounded with essentially bounded derivative) is considered for a class of single-input, single-output, nonlinear systems, described by a functional differential equation with a hysteresis nonlinearity in the input channel. The first control objective is tracking, by the output, with prescribed accuracy: determine a feedback strategy which ensures that, for every reference signal and every system of the underlying class, the tracking error ultimately satisfies the prescribed accuracy requirements. The second objective is guaranteed output transient performance: the evolution of the tracking error should be contained in a prescribed performance funnel. Under a weak sector boundedness assumption on the class of hysteresis operators, both objectives are achieved by a memoryless feedback which is universal for the underlying class of systems

This paper is concerned with stability properties of second-order and third-order systems with hysteresis subject to delay, which naturally arise in control and mechanical engineering. Conditions ensuring boundedness of solutions and related asymptotic and integrability properties are expressed in terms of parameters associated with the linear component of the overall system and a Lipschitz constant associated with the hysteretic component

(2004). Asymptotic Behaviour of Nonlinear Systems. The American Mathematical Monthly: Vol. 111, No. 10, pp. 864-889.