Stabilization for a class of minimum phase hybrid systems under an average dwell-time constraint

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Hybrid internal models for robust spline tracking

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Ball on a beam: stabilization under saturated input control with large basin of attraction

This article is devoted to the stabilization of two underactuated planar systems, the well-known straight beam-and-ball system and an original circular beam-and-ball system. The feedback control for each system is designed, using the Jordan form of its model, linearized near the unstable equilibrium. The limits on the voltage, fed to the motor, are taken into account explicitly. The straight beam-and-ball system has one unstable mode in the motion near the equilibrium point. The proposed control law ensures that the basin of attraction coincides with the controllability domain. The circular beam-and-ball system has two unstable modes near the equilibrium point. Therefore, this device, never considered in the past, is much more difficult to control than the straight beam-and-ball system. The main contribution is to propose a simple new control law, which ensures by adjusting its gain parameters that the basin of attraction arbitrarily can approach the controllability domain for the linear case. For both nonlinear systems, simulation results are presented to illustrate the efficiency of the designed nonlinear control laws and to determine the basin of attraction

Singularly Perturbed Monotone Systems and an Application to Double Phosphorylation Cycles

The theory of monotone dynamical systems has been found very useful in the modeling of some gene, protein, and signaling networks. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. This paper shows that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone systems, convergence to steady states, is still valid. An application is worked out to a double-phosphorylation ``futile cycle'' motif which plays a central role in eukaryotic cell signaling.Comment: 21 pages, 3 figures, corrected typos, references remove

A hybrid algorithm for finite time parameter estimation

A hybrid algorithm inspired by [1] is given that identifies a parameter for a class of nonlinear systems in finite time. We show that this algorithm contains a compact globally asymptotically stable set that is robust to small perturbations. We further give a persistency of excitation condition that ensures convergence of the parameter estimate

Adaptive output regulation for linear systems via discrete-time identifiers

The problem of output regulation for general multivariable linear systems has been solved in the 70s, in the seminal works of Francis, Wonham and Davison, under the assumption that the reference signals and the disturbances acting on the system are generated by a known exogenous linear system (the exosystem). The regulator is designed to embed an internal model of the exosystem, which ensures that asymptotic regulation is maintained under arbitrary plant perturbations that do not destroy linearity and closed-loop stability. This robustness property, however, is inexorably lost whenever the internal model does not match exactly the exosystem. In this paper we endow the linear regulator with a discrete-time adaptive unit that adapts the regulator's internal model on the basis of the closed-loop evolution. Compared to existing approaches, adaptation here is cast as an identification problem, and the corresponding optimal predictor is designed independently from the underlying control system. This permits to separate stabilization and adaptation, thus naturally handling general non-square multivariable non minimum-phase plants. Closed-loop stability is guaranteed and, if the dimension of the internal model is large enough and a persistency of excitation condition is fulfilled, asymptotic regulation is achieved for references and disturbances generated by an unknown exosystem. Robustness to parametric uncertainties is inherited by the linear regulator and robustness to additional unmodeled disturbances is proved to hold

L2-gain analysis for a class of hybrid systems with applications to reset and event-triggered control: A lifting approach

In this paper we study the stability and L2-gain properties of a class of hybrid systems that exhibit linear flow dynamics, periodic time-triggered jumps and arbitrary nonlinear jump maps. This class of hybrid systems is relevant for a broad range of applications including periodic event-triggered control, sampled-data reset control, sampled-data saturated control, and certain networked control systems with scheduling protocols. For this class of continuous-time hybrid systems we provide new stability and L2-gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L2-sense (meaning that the L2-gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in l2-sense (meaning that the l2-gain is smaller than 1) of an appropriate discrete-time nonlinear system. These new characterizations generalize earlier (more conservative) conditions provided in the literature.We show via a reset control example and an event- triggered control application, for which stability and contractivity in L2-sense is the same as stability and contractivity in l2-sense of a discrete-time piecewise linear system, that the new conditions are significantly less conservative than the existing ones in the literature. Moreover, we show that the existing conditions can be reinterpreted as a conservative l2-gain analysis of a discretetime piecewise linear system based on common quadratic storage/ Lyapunov functions. These new insights are obtained by the adopted lifting-based perspective on this problem, which leads to computable l2-gain (and thus L2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied

Small-gain results for discrete-time networks of systems with delay

In this paper input-to-state stability for large-scale discrete-time networks of systems with delay is considered. Firstly, interconnection delays between systems, which can arise due to the propagation of signals over large distances, are treated. It is established that such delays cannot cause the instability of the network if a delay-independent smallgain condition holds. Secondly, local delays in each system, which can arise due to inherent delays in each local dynamical process, are considered. For this set-up, using a small-gain condition, the stability of the network of systems is established via the Razumikhin method and the Krasovskii approach, respectively. By combining the results for networks of systems with communication and local delays, respectively, a framework for ISS analysis for general networks with delay is obtained