Section: New Results

Homogeneity theory and analysis of nonlinear systems

Homogeneity is a kind of symmetry, if it is presented in a system model, then it may simplify analysis of stability and performance properties of the system. The new results obtained in 2013 are as follows:

• The notion of geometric homogeneity has been extended for differential inclusions in [44] . This kind of homogeneity provides the most advanced coordinate-free framework for analysis and synthesis of nonlinear discontinuous systems. Theorem of L. Rosier on a homogeneous Lyapunov function existence and an equivalent notion of global asymptotic stability for differential inclusions have been presented. Robustness properties (ISS) of sliding mode systems applying the homogeneity concept have been considered in [46] .

• Retraction obstruction for time-varying stabilization on compact manifolds has been revisited in [13] .

• Several conditions have been proposed to check different robustness properties (ISS, iISS, IOSS and OSS) for generic nonlinear systems applying the weighted homogeneity concept (global or local) in [14] , [45] . The advantages of this result are that, under some mild conditions, the system robustness can be established as a function of the degree of homogeneity.

• A new algorithm for the analysis of strange attractors has been presented in [51] . An application of that results for observability-singularity manifolds in the context of chaos based cryptography has been given in [52] .

• Exciting multi-DOF systems by feedback resonance has been considered in [20] .

• Some conditions on existence of oscillations in hybrid systems have been established in [23] , [57] . An application to a humanoid robot locomotion has been considered.

• Considering two chaotic Rossler systems, the paper [83] presents a study on the forced synchronization of two systems, bidirectionally coupled by transmitting unidirectional signals which explicitly depend on a single state variable (from the emitter) and only affect directly the dynamics corresponding to the transmitted state variable (of the receiver).

• The paper [33] is concerned with the construction of local observers for nonlinear systems without inputs, satisfying an observability rank condition. The aim of this study is, first, to define a homogeneous approximation that keeps the observability property unchanged. This approximation is further used in the synthesis of local observer which is proven to be locally convergent for Lyapunov-stable systems.

• The paper [74] addresses the problem of exact average-consensus reaching in a prescribed time. The communication topology is assumed to be defined by a weighted undirected graph and the agents are represented by integrators. A nonlinear control protocol, which ensures a finite-time convergence, is proposed. With the designed protocol, any prescribed convergence time can be guaranteed regardless of the initial conditions.

• The Implicit Lyapunov Function (ILF) method for finite-time stability analysis has been introduced in [75] . The control algorithm for finite-time stabilization of a chain of integrators has been developed. The scheme of control parameters selection has been presented by LMIs. The robustness of the finite-time control algorithm with respect to system uncertainties and disturbances has been studied. The new high order sliding mode control has been derived as a particular case of the developed finite-time control algorithm. The settling time estimate has been obtained using ILF method. The algorithm of practical implementation of the ILF control scheme has been discussed.