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 2014 are as follows:

• The problem of scalability of trajectories in homogeneous and locally homogeneous systems is considered [46] . It is shown that the homogeneous systems have scalability property, and locally homogeneous systems possess this property approximately.

• Constructive conditions for verification of input-to-state stability property for discontinuous systems using geometric homogeneity have been proposed in [48] . The characterization of the asymptotic gain for such systems has been presented in [47] .

• The problem of finite-time output stabilization of the double integrator is addressed in [14] applying the homogeneity approach. Robustness and effects of discretization on the obtained closed loop system are analyzed.

• The paper [24] extends notion of homogeneity to the time-delay nonlinear systems. Generalizations and specifications of the homogeneity approach to time-delay nonlinear systems are given in [57] , where, for instance, the stability independently on delay has been analyzed.

• In [75] the uniform stability notion for a class of non-linear time-varying systems is studied using the homogeneity framework. The results are applied to the problem of adaptive estimation for a linear system.

• The Implicit Lyapunov Function (ILF) method has been applied for homogeneous differentiator design [70] . The procedure for adjustment of differentiator parameters has been resented in the form of semi-definite programming problem. ILF-based algorithms of robust finite-time and fixed-time stabilization of the chain of integrators were developed in [34] . In [69] they were adapted for the second order sliding mode control design.

• The tutorial on homogeneous methods in high sliding mode control has been published [13] . It stresses some recently obtained results of the team about homogeneity for differential inclusions and robustness with respect to perturbations in the context of input-to-state stability.