Switched and hybrid systems with inputs: small-gain theorems, control with limited information, and topological entropy

In this thesis, we study stability and stabilization of switched and hybrid systems with inputs. We consider primarily two topics in this area: small gain theorems for interconnected switched and hybrid systems, and control of switched linear systems with limited information. First, we study input-to-state practical stability (ISpS) of interconnections of two switched nonlinear subsystems with independent switchings and possibly non-ISpS modes. Provided that for each subsystem, the switching is slow in the sense of an average dwell-time (ADT), and the total active time of non-ISpS modes is short in proportion, Lyapunov-based small-gain theorems are established via hybrid system techniques. By augmenting each subsystem with a hybrid auxiliary timer that models the constraints on switching, we enable a construction of hybrid ISpS-Lyapunov functions, and consequently, a convenient formulation of a small-gain condition for ISpS of the interconnection. Based on our small-gain theorem, we demonstrate the stabilization of interconnected switched control-affine systems using gain-assignment techniques. Second, we investigate input-to-state stability (ISS) of networks composed of n ≥ 2 hybrid subsystems with possibly non-ISS dynamics. Lyapunov-based small-gain theorems are established based on the notion of candidate ISS-Lyapunov functions, which unifies and extends several previous results for interconnected hybrid and impulsive systems. In order to apply our small-gain theorem to different combinations of non-ISS dynamics, we adopt the method of modifying candidate exponential ISS-Lyapunov functions using ADT and reverse ADT timers. The effect of such modifications on the Lyapunov feedback gains between two interconnected hybrid systems is discussed in detail through a case-by-case study. Third, we consider the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to compensate for the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization. Last, motivated by the connection between the minimum data rate needed to stabilize a linear time-invariant system and its topological entropy, we examine a notion of topological entropy for switched systems with a known switching signal. This notion is formulated in terms of the number of initial points such that the corresponding trajectories approximate all trajectories within a certain error, and can be equivalently defined using the number of initial points that are separable up to a certain precision. We first calculate the topological entropy of a switched scalar system based on the active rates of its modes. This approach is then generalized to nonscalar switched linear systems with certain Lie structures to establish entropy bounds in terms of the active rate and eigenvalues of each mode.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Yang, Guosong

Contributors dc:contributor

• Liberzon, Daniel
• Belabbas, Mohamed-Ali
• Mitra, Sayan
• Zharnitsky, Vadim

• Switched systems
• Hybrid systems
• Input-to-state stability
• Lyapunov methods
• Small-gain theorems
• Sampling
• Quantization
• Data-rate constraints
• Topological entropy

Statement dc:rights

• Copyright 2017 Guosong Yang

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/98353

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/98353