Funnel control for systems with relative degree two

PublishedJournal ArticleTracking of reference signals yref (·) by the output y(·) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e = y - yref and its derivative e within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k0(t)2e(t)-k 1(t)e(t), where the simple proportional error feedback has gain functions k0 and k1 designed in such a way to preclude contact of e and e with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. © 2013 Society for Industrial and Applied Mathematics

Funnel control for systems with relative degree two

Tracking of reference signals yref(·) by the output y(·) of linear (as well as a considerably large class of nonlinear) single-input, single-output system is considered. The system is assumed to have strict relative degree two with ("weak") stable zero dynamics. The control objective is tracking of the error e = y − yref and its derivative e˙ within two prespecified performance funnels, resp. This is achieved by the so called 'funnel controller': u(t) = −k0(t)2e(t) − k1(t)e˙(t), where the simple proportional error feedback has gain functions k0 and k1 designed in such a way to preclude contactof e and e˙ with the funnel boundaries, resp. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller is (i) applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control

Feedback control: systems with higher unknown relative degree, input constraints and positivity

The thesis deals with the control of m-input, m-output systems with unknown but bounded relative degree and Volterra-Stieltjes systems. The two control strategies are: adaptive high-gain output derivative feedback and funnel control. The aim is the development of feedback controllers which achieves output regulation without system identification. Particularly systems with higher relative degree are considered. Firstly, a universal adaptive lambda-tracking controller is designed which achieves a prespecified control objective and tracks any system with known relative degree. This controller is extend to systems with unknown relative degree but an upper bound is known. Secondly, the well known concept of funnel control for relative degree one systems achieves in presence of input saturation the control objectives of funnel control. The presence of explicit input constraints is a distinguished feature of this thesis. The results for funnel control will be generalized to systems with relative degree two and input constraints. This new funnel controller is robust for systems of unknown relative degree one or two. Finally, Volterra-Stieltjes systems are considered with regard to positivity, various stability concepts, zero dynamics and funnel control. These results are exploited to generalize funnel controller to Volterra-Stieltjes systems. Positivity of the closed-loop system is guaranteed, input constraints are possible and non-negativity of the input can be guaranteed

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”

Normal forms, high-gain and funnel control for linear differential-algebraic systems

We consider linear differential-algebraic m-input m-output systems with positive strict relative degree or proper inverse transfer function; in the single-input single-output case these two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties - such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the system classes and secondly, to solve the output regulation problem by funnel control. The funnel controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in particular, the transient behaviour of the output error can be specified and the funnel controller does neither incorporate any internal model for the reference signals nor any identification mechanism, it is simple in its design. The results are illuminated by position and velocity control of a mechanical system encompassing springs, masses, and dampers

Funnel control for a moving water tank

We study tracking control for a moving water tank system, which is modelled using the Saint-Venant equations. The output is given by the position of the tank and the control input is the force acting on it. For a given reference signal, the objective is to achieve that the tracking error evolves within a prespecified performance funnel. Exploiting recent results in funnel control we show that it suffices to show that the operator associated with the internal dynamics of the system is causal, locally Lipschitz continuous and maps bounded functions to bounded functions. To show these properties we consider the linearized Saint-Venant equations in an abstract framework and show that it corresponds to a regular well-posed linear system, where the inverse Laplace transform of the transfer function defines a measure with bounded total variation.Comment: 11 page

Global distribution of modern shallow marine shorelines. Implications for exploration and reservoir analogue studies

Acknowledgments Support for this work came from the SAFARI consortium which was funded by Bayern Gas, ConocoPhillips, Dana Petroleum, Dong Energy, Eni Norge, GDF Suez, Idemitsu, Lundin, Noreco, OMV, Repsol, Rocksource, RWE, Statoil, Suncor, Total, PDO, VNG and the Norwegian Petroleum Directorate (NPD). This manuscript has benefited from discussion with Bruce Ainsworth, Rachel Nanson and Christian Haug Eide. Boyan Vakarelov and Richard Davis Jr. are thanked for their constructive reviews and valuable comments that helped to improve the manuscript.Peer reviewedPostprin

Exploring the Free Energy Landscape: From Dynamics to Networks and Back

The knowledge of the Free Energy Landscape topology is the essential key to understand many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers are, how the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times or rate constants, and the hierarchical relationship among basins, complete the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, the dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.Comment: PLoS Computational Biology (in press