Stability of Equilibria of Discrete-Time Systems and Localization of Invariant Compact Sets

The stability (or asymptotic stability) of equilibria of an time-invariant discrete-time system can be verified with the use of stability and asymptotic stability criteria stated in terms of invariant sets. An earlier proposed method reduces the verification of these criteria to some set operations. However, the method is analytical and hard to implement. We propose another approach to the verification of these criteria based on the functional method for localizing invariant compact sets.     

Localization of invariant compact sets in uncertain discrete systems

A functional method for the localization of invariant compact sets in discrete autonomous systems is generalized to discrete systems with uncertainty. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for positively invariant compact sets of the discrete Henon system with uncertainty.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

Global asymptotic stability analysis by the localization method of invariant compact sets

We study the asymptotic stability and the global asymptotic stability of equilibria of autonomous systems of differential equations. We prove necessary and sufficient conditions for the global asymptotic stability of an equilibrium in terms of invariant compact sets and positively invariant sets. To verify these conditions, we use some results of the localization method for invariant compact sets of autonomous systems. These results are related to finding sets that contain all invariant compact sets of the system (localizing sets) and to the behavior of trajectories of the system with respect to localizing sets. We consider an example of a system whose equilibrium belongs to the critical case.     

Functional method for the localization of invariant compact sets in discrete systems

We suggest a method for the localization of invariant compact sets in discrete autonomous systems. We describe the properties of the corresponding localizing sets. By using this method, we construct localizing sets for invariant compact sets of the discrete Henon system.     

Asymptotic stability analysis of autonomous systems by applying the method of localization of compact invariant sets

The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

Stability of equilibria of discrete-time systems in terms of invariant sets

We suggest a new approach to the verification of the stability (or asymptotic stability) of the equilibria of time-invariant discrete-time systems based on stability and asymptotic stability criteria stated in terms of invariant sets. A set-theoretic method for the verification of the conditions in these criteria is presented.     

Invariant sets and Knaster-Tarski principle

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.     

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.     

Some topological dynamics properties of discrete-time control systems

This paper is concerned with the study of certain stabilityproperties of discrete-time systems by means of topologicalmethods. It turns out that suitably defined sets guarantee thestability of the reachable sets in the vicinity of compact sets.Furthermore, special properties about affine control systemsare given.     

Invariant polytopes of linear systems

Stable linear systems possess invariant sets which have hyperellipsoidalregions associated with their Lyapunov function. In real systems,however, state and control variables are often confined in boundedpolyhedral regions(polytopes) so that a set of linear inequalitieshas to be satisfied. In this paper, necessary and sufficientconditions for the existence of positively invariant polytopesfor both discrete-time and continuous-time linear systems aregiven in terms of their spectral properties.     

Construction of Lyapunov functions by the method of localization of invariant compact sets

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered.     

Contractive Markov Systems

Certain discrete-time Markov processes on locally compact metricspaces which arise from graph-directed constructions of fractalsets with place-dependent probabilities are studied. Such systemsnaturally extend finite Markov chains and inherit some of theirproperties. It is shown that the Markov operator defined bysuch a system has a unique invariant probability measure inthe irreducible case and an attractive probability measure inthe aperiodic case if the vertex sets form an open partitionof the state space, the restrictions of the probability functionson their vertex sets are Dini-continuous and bounded away fromzero, and the system satisfies a condition of contractivenesson average.     

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.     

Negatively Invariant Sets and Entire Trajectories of Set-Valued Dynamical Systems

Strongly negatively invariant compact sets of set-valued autonomous and nonautonomous dynamical systems on a complete metric space, the latter formulated in terms of processes, are shown to contain a weakly positively invariant family and hence entire solutions. For completeness the strongly positively invariant case is also considered, where the obtained invariant family is strongly invariant. Both discrete and continuous time systems are treated. In the nonautonomous case, the various types of invariant families are in fact composed of subsets of the state space that are mapped onto each other by the set-valued process. A simple example shows the usefulness of the result for showing the occurrence of a bifurcation in a set-valued dynamical system.     

Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator

This paper is concerned with the localization problem of compact invariant sets of the system describing dynamics of the nuclear spin generator. We establish that all compact invariant sets of this system are located in the intersection of a ball with two frusta and compute its parameters. In addition, localization by using the two-parameter set of parabolic cylinders is described. Our results are obtained with help of the iteration theorem concerning a localization of compact invariant sets. One numerical example illustrating a localization of a chaotic attractor is presented as well.     

Numerical Solutions to the Bellman Equation of Optimal Control

In this paper, we present a numerical algorithm to compute high-order approximate solutions to Bellman’s dynamic programming equation that arises in the optimal stabilization of discrete-time nonlinear control systems. The method uses a patchy technique to build local Taylor polynomial approximations defined on small domains, which are then patched together to create a piecewise smooth approximation. The numerical domain is dynamically computed as the level sets of the value function are propagated in reverse time under the closed-loop dynamics. The patch domains are constructed such that their radial boundaries are contained in the level sets of the value function and their lateral boundaries are constructed as invariant sets of the closed-loop dynamics. To minimize the computational effort, an adaptive subdivision algorithm is used to determine the number of patches on each level set depending on the relative error in the dynamic programming equation. Numerical tests in 2D and 3D are given to illustrate the accuracy of the method.     

Bounding a domain containing all compact invariant sets of the permanent-magnet motor system

In this paper we characterize a locus of compact invariant sets of the system describing dynamics of the permanent-magnet synchronous motor (PMSM). We establish that all compact invariant sets of this system are contained in the intersection of one-parameter set of ellipsoids and compute its parameters. In addition, localizations by using a parabolic cylinder, an elliptic paraboloid and a hyperbolic cylinder are obtained. Simple polytopic bounds are derived with help of these localizations. Most of localizations mentioned above remain valid for more specific motor systems; namely, for the interior magnet PMSM and for the surface magnet PMSM. Yet another localization set for the interior magnet PMSM is described. Examples of localization of chaotic attractors existing for some values of parameters of PMSMs are presented as well.