Topological Feedback Entropy and Nonlinear Stabilization

SIAM Journal on Control and Optimization, 2012

Minimal bit rates and entropy are studied for exponential stabilization of control systems in continuous time. Upper and lower bounds for the stabilization entropy are derived. In particular, for linear systems, a formula is given in terms of the real parts of eigenvalues. Then the minimal bit rate is related to the stabilization entropy.

Journal of Dynamics and Differential Equations

The invariance pressure of continuous-time control systems with initial states in a set K which are to be kept in a set Q is introduced and a number of results are derived, mainly for the case where Q is a control set.

Journal of Dynamics and Differential Equations

For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.

2020

We introduce discrete-time linear control system on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs. In the case of solvable Lie groups the upper bound coincides with the outer invariance entropy.

2021

For deterministic continuous-time nonlinear control systems, epsilon-practical stabilization entropy and practical stabilization entropy are introduced. Here the rate of attraction is specified by a KL-function. Upper and lower bounds for the diverse entropies are proved, with special attention to exponential KL-functions. The relation to feedbacks is discussed and, in particular, it is shown that the existence of a feedback making the system Input-to-State Dynamically Stable (ISDS) implies that the practical stabilization entropy vanishes. Two scalar examples are analyzed in detail.

Mathematics of Control, Signals, and Systems, 2011

For continuous-time control systems with outputs, this paper analyzes invariance entropy as a measure for the information rate necessary to achieve invariance of compact subsets of the output space. For linear control systems with compact control range, relations to controllability and observability properties are studied. Furthermore, the notion of asymptotic invariance entropy is introduced and characterized for these systems.

Ergodic Theory and Dynamical Systems, 2016

Two notions of metric invariance entropy are constructed with respect to conditionally invariant measures for control systems in discrete time and it is shown that they are invariant under conjugacies.

Journal of Systems Science and Complexity, 2011

This paper addresses a robust stabilization problem of a class of uncertain nonlinear systems using output measurements via a finite data-rate communication channel. The authors assumes that there exist an observer and a control law for the systems in the absence of any finite data-rate communication channel. Based on the observer and the control law, the authors constructs an encoder/decoder pair and provides a sufficient condition, including suitable sampling period and data rate, which will guarantee the stability of the closed-loop systems when a finite data-rate communication channel is introduced.

Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, 2019

This paper studies topological entropy and stability properties of switched linear systems. First, we show that the exponential growth rates of solutions of a switched linear system are essentially upper bounded by its topological entropy. Second, we estimate the topological entropy of a switched linear system by decomposing it into a part that is generated by scalar multiples of the identity matrix and a part that has zero entropy, and proving that the overall topological entropy is upper bounded by that of the former. Third, we prove that a switched linear system is globally exponentially stable if its topological entropy remains zero under a destabilizing perturbation. Finally, the entropy estimation via decomposition and the entropy-based stability condition are applied to three classes of switched linear systems to construct novel upper bounds for topological entropy and novel sufficient conditions for global exponential stability.

Journal of Dynamical and Control Systems, 2015

In this paper we study the chain control sets of right-invariant control systems on the flag manifolds of a non-compact semisimple Lie group. We prove that each chain control set is partially (skew-) hyperbolic over the associated control flow. Moreover, we characterize those chain control sets that are uniformly (skew-) hyperbolic in terms of the flag type of the control flow.

International Journal of Robust and Nonlinear Control, 2009

Control systems over networks with a finite data rate can be conveniently modeled as hybrid (impulsive) systems. For the class of nonlinear systems in feedfoward form, we design a hybrid controller which guarantees stability, in spite of the measurement noise due to the quantization, and of an arbitrarily large delay which affects the communication channel. The rate at which feedback packets are transmitted from the sensors to the actuators is shown to be arbitrarily close to the infimal one.

Mathematics of Control, Signals, and Systems, 2020

This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states. We construct an upper bound for the topological entropy in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times, and a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct upper bounds for the topological entropy that only depend on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well.

Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control, 2021

This paper studies topological entropy of switched nonlinear systems. We construct a general upper bound for the topological entropy in terms of an average of the asymptotic suprema of the measures of Jacobian matrices of individual modes, weighted by the corresponding active rates. A general lower bound is constructed in terms of an active-rate-weighted average of the asymptotic infima of the traces of these Jacobian matrices. For switched systems with diagonal modes, we construct upper and lower bounds that only depend on the eigenvalues of Jacobian matrices, their relative order among individual modes, and the active rates. For both cases, we also construct more conservative upper bounds that require less information on the switching, with their relations illustrated by numerical examples of a switched Lotka-Volterra ecosystem model.