Stabilization of invariant sets for nonlinear non-affine systems

Shiriaev, Anton S.; Fradkov, Alexander L.

doi:10.1016/s0005-1098(00)00077-7

Cited by 56 publications

(42 citation statements)

References 20 publications

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“…Almost all these techniques have been devoted to nonlinear systems stabilization with respect to an equilibrium or a reference trajectory. Another promising topic deals with the problem of nonlinear systems stabilization with respect to a set [12], [21], [23], [24], [25], [28] (part of variables or output). Such problem arises in oscillations or synchronization control, energy level stabilization in mechanical systems, maneuvering problem or in robotic applications.…”

Section: Introductionmentioning

confidence: 99%

Oscillating system design applying universal formula for control

Efimov

Perruquetti

2011

IEEE Conference on Decision and Control and European Control Conference

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Abstract-The problem of oscillating system design applying the homogeneity approach is studied. The Anti-control Lyapunov Function (ALF) is introduced as a counterpart of Control Lyapunov Function (CLF) for the control design that destabilizes a nonlinear system. A universal anti-control formula is proposed. Next, the universal control formulas based on ALF and CLF are used to design an oscillating system. Efficiency of the proposed approach is demonstrated on example.

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Section: Introductionmentioning

confidence: 99%

Oscillating system design applying universal formula for control

Efimov

Perruquetti

2011

IEEE Conference on Decision and Control and European Control Conference

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“…There exist different versions of analytic conditions guaranteeing that the control goal Q(z(t ), t ) can be achieved in the system (7.1) with adaptation law (7.4); see [Fradkov, 1979;Shiriaev and Fradkov, 2000]. The main condition is that a constant value of the parameter u * has to exist ensuring attainability of the goal in the systeṁ…”

Section: Speed-gradient Methodsmentioning

confidence: 99%

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

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In this thesis, I consider the control of synchronization in delay-coupled complex networks. As one main focus, several applications to neural networks will be discussed. In the first part, I focus on the stability of synchronization in complex networks meaning that the control is realized by considering the stability of synchrony in dependence on the parameters. In the second part, adaptive control of synchronization is studied. To this end, adaptive control algorithms are developed that tune the system parameters such that the desired control goal is reached.Besides zero-lag synchrony -a state where all nodes follow the same dynamics without a phase lag -groups and cluster states are considered, i.e., states where the network consists of several groups where the nodes within one group are in zero-lag synchrony and, in the case of cluster synchrony, with a constant phase lag between the clusters.The stability of synchronization can be accessed via the master stability function [Pecora and Carroll, 1998]. This convenient tool allows for treating the node dynamics and the network topology in two separate steps, and, thus, allows for a quite general treatment of different network topologies. In this thesis, I will discuss the generalization of the master stability function to group and cluster states and to non-smooth systems.The master stability function can be used to investigate synchrony in neural networks. Neurons are excitable systems where type-I and type-II excitability can be distinguished. Here, the stability of synchrony for both types in complex networks with excitatory and inhibitory links is investigated on two generic models, namely the normal form of the saddle-node infinite period bifurcation and the FitzHugh-Nagumo system. Furthermore, synchronization in systems with heterogeneous delays or node parameters is studied.In situations where parameters are unknown or drift, adaptive control methods are useful since they allow for an automatically realized adaption of the control parameters. A convenient adaptive method is the speed-gradient method that minimizes a predefined goal function [Fradkov, 1979[Fradkov, , 2007. I first apply this method in the control of an unstable focus and an unstable periodic orbit embedded in the Rössler attractor. Furthermore, I show that clusters states in networks of delayed coupled Stuart-Landau oscillators can be controlled by adaptively tuning the phase of the complex coupling strength or by adapting the topology. The first method is particularly simple because only one parameter has to be adapted, while the second method is more reliable in the sense that its success is widely independent of the choice of the control parameters and the initial conditions. ZUSAMMENFASSUNGIn dieser Arbeit wird die Kontrolle von Synchronisation in komplexen Netzwerken mit retardierten Kopplungen untersucht. Dabei werden verschiedene Anwendungen auf neuronale Netzwerke diskutiert. Der erste Teil der Arbeit beschäftigt sich mit der Stabilität von Synchronisation in komplexen Ne...

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“…Moreover, since Steps 2 and 3 are the stabilization of A for the input-affine polynomial system (17), various stabilization methods (e.g. [3,[33][34][35][36]) can be employed.…”

Section: Behavior Outside the Algebraic Setmentioning

confidence: 99%

Lie Derivative Inclusion for a Class of Polynomial State Feedback Controls

Yuno

Ohtsuka

2014

Transactions of ISCIE

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In this paper, we deal with two problems of input-affine polynomial dynamical systems. One aims to obtain a state feedback controller such that a prescribed algebraic set is invariant for the resulting closed-loop system. The other aims to obtain a state feedback controller such that the resulting closed-loop system has a prescribed vector field on a given algebraic set. It is shown that the two problems can be represented by a particular inclusion of polynomials, and the inclusion can be solved. As a result, all the state feedback controllers required in the problems can be exactly represented by using free polynomial parameters.

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Stabilization of invariant sets for nonlinear non-affine systems

Cited by 56 publications

References 20 publications

Oscillating system design applying universal formula for control

Oscillating system design applying universal formula for control

Controlling Synchronization Patterns in Complex Networks

Lie Derivative Inclusion for a Class of Polynomial State Feedback Controls

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