Backstepping Design for the Predefined-Time Stabilization of Second-Order Systems

The backstepping design of a controller which stabilizes a class of second-order systems in predefined-time is studied in this paper. The origin of a dynamical system is said to be predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through an appropriate selection of the system parameters. The proposed backstepping construction is based on recently proposed Lyapunov-like sufficient conditions for predefined-time stability. Different from other approaches, the proposed backstepping design allows the simultaneous construction of a Lyapunov function which meets the conditions for guaranteeing predefined-time stability. A simulation example is presented to show the behavior of a developed controller, and to show its advantages against similar schemes.ITESO, A.C

Discrete Nonlinear Planar Systems and Applications to Biological Population Models

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates

Polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities

This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions

Stability and bifurcations in neural fields with axonal delay and general connectivity

A stability analysis is presented for neural field equations in the presence of axonal delays and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the delays play a crucial role in non-stationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the kernel. Bounds are determined for the frequencies of bifurcating periodic solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillatory solutions, and traveling waves. For high transmission speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the connectivity kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible second order systems with combinations of Gaussian excitatory and inhibitory connections

Асимптотические свойства и стабилизация одной системы нейтрального типа с постоянным запаздыванием

The problem of obtaining sufficient conditions for the asymptotic stability for a certain class of linear systems of a neutral type with constant delay is analyzed in the article. Some coefficients of these systems in the right side have an exponential factor. As a consequence, the study of the stability of such systems with the help of the Lyapunov-Krasovskii functionals is not possible; methods of receiving asymptotic appreciations lead to extremely rough results. By applying the apparatus of difference systems and the properties of simpler systems, which the author examined previous, sufficient conditions for the exponential stability of such systems are obtained. As an example, a second-order system is considered. The graphs of the solutions of the corresponding system, both without neutral members and with the original system where the right-hand side contains neutral terms, are provided. On the basis of theory difference systems, the author proposes an algorithm of stabilization for some systems of a similar type. © 2021 Saint Petersburg State University. All rights reserved

Control of Piecewise Smooth Systems: Generalized Absolute Stability and Applications to Supercavitating Vehicles

Many systems in engineering applications are modeled as piecewise smooth systems. The piecewise smoothness presents great challenges for stability analysis and control synthesis for these systems. Over the years, the theory of absolute stability has been one of the few tools developed by control theory researchers to meet these challenges. For systems in which the nonlinearity is known to be bounded within certain sectors, many stability and control problems can be addressed using results from absolute stability theory. During the last few decades, many important advances have been made in the study of the absolute stability. In these studies, it is commonly assumed that the sector bound for the system nonlinearity is \textit{symmetric} with respect to the origin in state space. However, in many practical engineering systems, the nonlinearity does not satisfy such a symmetry assumption. To study stability and control problems for these systems, in this work the author studies generalized absolute stability problems involving \textit{asymmetric} sector bounds. Nonlinear systems with Lure' structure are considered. For second-order systems, conditions that are both necessary and sufficient for generalized absolute stability are obtained. These conditions can be easily tested in engineering applications. For general finite-order systems, sufficient conditions are provided for generalized absolute stability. The derived conditions may be easily tested by using numerical tools for linear programming. With the generalizations in this work, absolute stability theory becomes a more powerful tool in the sense that it applies to an extended class of piecewise smooth systems in which the nonlinearities can be asymmetric with respect to the state variables. This work includes general theoretical questions as well as detailed investigations of an application to models of supercavitating vehicles. For these high-speed underwater vehicles, the dive-plane motion is naturally modeled as a piecewise smooth system with a dead zone. The strong nonlinear planing force plays an important role in determining the dive-plane dynamics. To design control laws that stabilize the dive-plane motion, the necessary and sufficient condition for generalized absolute stability of second-order systems is applied to a reduced-order model obtained through the backstepping control approach. The obtained sufficient conditions for generalized absolute stability of finite-order systems can also be successfully applied for stabilizing the dive-plane motion. In comparison with alternative control approaches, control designs with the aid of theoretical findings in generalized absolute stability lead to stability that is robust to the modeling errors in the nonlinearity such as the magnitude, local slope and the dead zone location. The dissertation also includes basic results on bifurcation and bifurcation control of supercavitating vehicles. The presence of bifurcations in the dive-plane dynamics is demonstrated, and control techniques for modifying the bifurcation behavior to improve the vehicle dynamic performance are developed. These results complement the absolute stability results to give a more complete picture of the dynamics and control of supercavitating vehicles