Robust stabilization by linear output delay feedback

The main result establishes that if a controller $C$ (comprising of a linear feedback of the output and its \emph{derivatives}) globally stabilizes a (nonlinear) plant $P$, then global stabilization of $P$ can also be achieved by an output feedback controller $C[h]$ where the output derivatives in $C$ are replaced by an Euler approximation with sufficiently small delay h>0. This is proved within the conceptual framework of the nonlinear gap metric approach to robust stability. The main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain. Results are also given for systems with non-zero initial conditions

Robustness of funnel control in the gap metric

Copyright © 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.49th IEEE Conference on Decision and Control, Atlanta, USA, 15-17 December 2010For m-input, m-output, finite-dimensional, linear systems satisfying the assumptions (i) minimum phase, (ii) relative degree one and (iii) positive high-frequency gain), the funnel controller achieves output regulation in the following sense: all states of the closed-loop system are bounded and, most importantly, transient behaviour of the tracking error is ensured such that its evolution remains in a performance funnel with prespecified boundary. As opposed to classical adaptive high-gain output feedback, system identification or internal model is not invoked and the gain is not monotone. Invoking the conceptual framework of the nonlinear gap metric we show that the funnel controller is robust in the following sense: the funnel controller copes with bounded input and output disturbances and, more importantly, it may even be applied to a system not satisfying any of the classical conditions (i)–(iii) as long as the initial conditions and the disturbances are “small” and the system is “close” (in terms of a “small” gap) to a system satisfying (i)–(iii)

Robustness of λ-tracking and funnel control in the gap metric

PublishedCopyright © 2009 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on 15-18 December 2009For m-input, m-output, finite-dimensional, linear systems satisfying the classical assumptions of adaptive control (i.e., (i) minimum phase, (ii) relative degree one and (iii) positive definite high-frequency gain matrix), two control strategies are considered: the well-known λ-tracking and funnel control. An application of the λ-tracker to systems satisfying (i)–(iii) yields that all states of the closed-loop system are bounded and |e| is ultimately bounded by some prespecified λ > 0. An application of the funnel controller achieves tracking of the error e within a prescribed performance funnel if applied to linear systems satisfying (i)–(iii). Moreover, all states of the closed-loop system are bounded. The funnel boundary can be chosen from a large set of functions. Invoking the conceptual framework of the nonlinear gap metric, we show that the λ-tracker and the funnel controller are robust. In the present setup this means in particular that λ-tracking and funnel control copes with bounded input and output disturbances and, more importantly, may be applied to any system which is “close” (in terms of a “small” gap) to a system satisfying (i)–(iii), and which may not satisfy any of the classical conditions (i)–(iii), as long as the initial conditions and the disturbances are “small”

Differential-algebraic systems are generically controllable and stabilizable

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1-61. https://doi.org/10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats