Section: New Results

Homogeneity Theory

Homogeneity is one of the tools we develop for finite-time convergence anasysis. In 2015 this concept has received various improvements:

• The concept of homogeneous evolution equation in a Banach space has been introduced in [67] . It provides the background for the extencion of all homogeneity-based tools for control design and analysis to distributed parameters systems.

• Scalability is a property describing the change of the trajectory of a dynamical system under a scaling of the input stimulus and of the initial conditions. Particular cases of scalability include the scale invariance and fold change detection (when the scaling of the input does not influence the system output). In the paper [19] is shown that homogeneous systems have this scalability property while locally homogeneous systems approximately possess this property.

• In the paper [25] the notion of homogeneity in the bi-limit is extended to local homogeneity and then to homogeneity in the multi-limit. The converse Lyapunov/Chetaev theorems on (homogeneous) system instability are obtained. The problem of oscillation detection for nonlinear systems is addressed. The sufficient conditions of oscillation existence for systems homogeneous in the multi-limit are formulated.

• The notion of weighted homogeneity is extended in [81] to the time-delay systems. It is shown that the stability/instability of homogeneous functional systems on a sphere implies the global stability/instability of the system. The notion of local homogeneity is introduced, a relation between stability/instability of the locally approximating dynamics and the original time-delay system is established using Lyapunov-Razumikhin approach

• In [27] global delay independent stability is analyzed for nonlinear time-delay systems by applying homogeneity theory. It is shown that finite-time stability can be encountered in this class of systems under uniformity of the convergence time with respect to delay. Some additional tools for stability analysis of time-delay systems using homogeneity are also presented: in particular, it is shown that if a time-delay system is homogeneous with nonzero degree and it is globally asymptotically stable for some delay, then this property is preserved for any delay value, which is known as the independent of delay (IOD) stability.

• Theorems on Implicit Lyapunov Functions for finite-time and fixed-time stability analysis of nonlinear systems are presented in [37] . Based on these resutls, new homogenenous nonlinear control laws are designed for robust stabilization of a chain of integrators. The presented results are extended to Multi-Input Multi-Output in [38] . A time-suboptimal control design algorithm based Implicit Lyapunov Function Method is developed in [40] . A robustness-oriented comparison of the optimal and suboptimal solutions in practical implementations of the proposed controller is performed via the numerical example of double integrator. A novel scheme of practical implimentation of the implicit Lypunov function-based control is developed in [79] . It replaces the implicitely defined Lyapunov function (in the feedback law) with the homogenenous norm of the state. Such a modification simplifies the practical application of the finite-time stabiliting feedback control.

• The uniform stability notion for a class of nonlinear time-varying systems is studied in [42] using the homogeneity framework. It is assumed that the system is weighted homogeneous considering the time variable as a constant parameter, then several conditions of uniform stability for such a class of systems are formulated. The results are applied to the problem of adaptive estimation for a linear system. The detailed report on time-varying homogenety is given in [83] .

• In the paper [52] we consider the continuous homogeneous observer defined in the case of the triple integrator. Originally, convergence of the algorithm was only proved when the degree of homogeneity was sufficiently close to 0 without more tractable information. We show here that, in the case of the triple integrator, the observer presents global finite-time stability for any negative degree under constructive conditions on the gains. This is achieved with a homogeneous Lyapunov function design.

• The work [61] addresses the stabilization of dynamical systems in presence of uncertain bounded perturbations using theory. Under some assumptions, the problem is reduced to the stabilization of a chain of integrators subject to a perturbation and is treated in two steps. The evaluation of the disturbance and its compensation. Homogeneous observer and control are the tools utilized to achieve a global asymptotic stability and robustness. The result is formally proven and, to validate the theory, it is applied to the control of the telescopic link of a hydraulic actuated industrial crane used in forestry.

• A geometric homogeneity of evolution equation in a Banach space is introduced in [67] . Scalability property of solutions of homogeneous evolution equations is proven. Some qualitative characteristics of stability of trivial solution are also provided. In particular, finite-time stability of homogeneous evolution equations is studied. Theoretical results are supported by examples from mathematical physics.

• The second order planar nonlinear affine control problem is studied [69] . A homogeneous robust finite-time stabilizing control is developed for the most general case of matched and, the more challenging, mismatched nonlinear perturbations. A homogeneous observer is designed for the planar system. Explicit restrictions on the observer gains and nonlinearities are presented. The main contribution lies in the proposed combination of the explicit and implicit Lyapunov function methods as well as weighted homogeneity while providing finite-time stability analysis.