Tracking with Prescribed Transient Behavior for Nonlinear Systems of Known Relative Degree

Abstract

Tracking of a reference signal (assumed bounded with essentially bounded derivative) is considered in the context of a class $\Sigma_{\rho}$ of multi‐input, multi‐output dynamical systems, modelled by functional differential equations, affine in the control and satisfying the following structural assumptions: (i) arbitrary—but known—relative degree $\rho \ge 1; (ii) the “high‐frequency gain” is sign definite—but possibly of unknown sign. The class encompasses a wide variety of nonlinear and infinite‐dimensional systems and contains (as a prototype subclass) all finite‐dimensional, linear, m‐input, m‐output, minimum‐phase systems of known strict relative degree. The first control objective is tracking, by the output y, with prescribed accuracy: given $\lambda >0$ (arbitrarily small), determine a feedback strategy which ensures that, for every reference signal r and every system of class $\Sigma_{\rho}$, the tracking error $e=y-r$ is ultimately bounded by λ (that is, $\|e(t)\| < \lambda$ for all t sufficiently large). The second objective is guaranteed output transient performance: the tracking error is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Both objectives are achieved using a filter in conjunction with a feedback function of the tracking error, the filter states, and the funnel parameter φ.