Performance funnels and tracking control

Abstract

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).

Keywords:

• output feedback
• non-linear systems
• functional differential equations
• transient behaviour
• tracking

Acknowledgement

This research was supported by the UK Engineering & Physical Sciences Research Council (Grant Ref. GR/S94582/01) and DFG (Grant IL 25/4).

Notes

¹Let 𝒟 be a domain in ℝ₊ × ℝ (that is, a non-empty, connected, relatively open subset of ℝ₊ × ℝ). A function F : 𝒟 × ℝ ^q → ℝ, is deemed to be a Carathéodory function if, for every ‘rectangle’ [a, b] × [c, d] ⊂ 𝒟 and every compact set K ⊂ ℝ ^q , the following hold: (i) F(t, ·, ·) : [c, d] × K → ℝ is continuous for all t ∈ [a, b]; (ii) F(·, x, w) : [a, b] → ℝ is measurable for each fixed (x, w) ∈ [c, d] × K; (iii) there exists an integrable function γ : [a, b] → ℝ₊ such that |F(t, x, w)| ≤ γ(t) for almost all t ∈ ℝ₊ and all (x, w) ∈ [c, d] × K.