Input-to-state stability of infinite-dimensional control systems

We define the notion of local ISS-Lyapunov function and prove, that existence of a local ISS-Lyapunov function implies local ISS (LISS) of the system. Then we consider infinite-dimensional systems generated by differential equations in Banach spaces. We prove, that an interconnection of such systems is ISS if all the subsystems are ISS and the small-gain condition holds. Next we show that a system is LISS provided its linearization is ISS. In the second part of the thesis we deal with infinite-dimensional impulsive systems. We prove, that existence of an ISS Lyapunov function (not necessarily exponential) for an impulsive system implies ISS of the system over impulsive sequences satisfying nonlinear fixed dwell-time condition. Also we prove, that an impulsive system, which possesses an exponential ISS Lyapunov function is uniform ISS over impulse time sequences, satisfying the generalized average dwell-time condition. Then we generalize small-gain theorems to the case of impulsive systems

Hybrid dynamics in large-scale logistics networks

We study stability properties of interconnected hybrid systems with application to large-scale logistics networks. Hybrid systems are dynamical systems that combine two types of dynamics: continuous and discrete. Such behaviour occurs in wide range of applications. Logistics networks are one of such applications, where the continuous dynamics occurs in the production and processing of material and the discrete one in the picking up and delivering of material. Stability of logistics networks characterizes their robustness to the changes occurring in the network. However, the hybrid dynamics and the large size of the network lead to complexity of the stability analysis. In this thesis we show how the behaviour of a logistics networks can be described by interconnected hybrid systems. Then we recall the small gain conditions used in the stability analysis of continuous and discrete systems and extend them to establish input-to-state stability (ISS) of interconnected hybrid systems. We give the mixed small gain condition in a matrix form, where one matrix describes the interconnection structure of the system and the other diagonal matrix takes into account whether ISS condition for a subsystem is formulated in the maximization or the summation sense. The small gain condition is sufficient for ISS of an interconnected hybrid system and can be applied to an interconnection of an arbitrary finite number of ISS subsystems. We also show an application of this condition to particular subclasses of hybrid systems: impulsive systems, comparison systems and the systems with stability of only a part of the state. Furthermore, we introduce an approach for structure-preserving model reduction for large-scale logistics networks. This approach supposes to aggregate typical interconnection patterns (motifs) of the network graph. Such reduction allows to decrease the number of computations needed to verify the small gain condition

On Stability and Stabilization of Hybrid Systems

The thesis addresses the stability, input-to-state stability (ISS), and stabilization problems for deterministic and stochastic hybrid systems with and without time delay. The stabilization problem is achieved by reliable, state feedback controllers, i.e., controllers experience possible faulty in actuators and/or sensors. The contribution of this thesis is presented in three main parts. Firstly, a class of switched systems with time-varying norm-bounded parametric uncertainties in the system states and an external time-varying, bounded input is addressed. The problems of ISS and stabilization by a robust reliable $H_{\infty}$ control are established by using multiple Lyapunov function technique along with the average dwell-time approach. Then, these results are further extended to include time delay in the system states, and delay systems subject to impulsive effects. In the latter two results, Razumikhin technique in which Lyapunov function, but not functional, is used to investigate the qualitative properties. Secondly, the problem of designing a decentralized, robust reliable control for deterministic impulsive large-scale systems with admissible uncertainties in the system states to guarantee exponential stability is investigated. Then, reliable observers are also considered to estimate the states of the same system. Furthermore, a time-delayed large-scale impulsive system undergoing stochastic noise is addressed and the problems of stability and stabilization are investigated. The stabilization is achieved by two approaches, namely a set of decentralized reliable controllers, and impulses. Thirdly, a class of switched singularly perturbed systems (or systems with different time scales) is also considered. Due to the dominant behaviour of the slow subsystem, the stabilization of the full system is achieved through the slow subsystem. This approach results in reducing some unnecessary sufficient conditions on the fast subsystem. In fact, the singular system is viewed as a large-scale system that is decomposed into isolated, low order subsystems, slow and fast, and the rest is treated as interconnection. Multiple Lyapunov functions and average dwell-time switching signal approach are used to establish the stability and stabilization. Moreover, switched singularly perturbed systems with time-delay in the slow system are considered

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

Qualitative Properties of Stochastic Hybrid Systems and Applications

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems

Sampled-data control of linear systems subject to input saturation : a hybrid system approach

In this work, a new method for the stability analysis and synthesis of sampled-data control systems subject to variable sampling intervals and input saturation is proposed. From a hybrid systems representation, stability conditions based on quadratic clockdependent Lyapunov functions and the generalized sector condition to handle saturation are developed. These conditions are cast in semidefinite and sum-of-squares optimization problems to provide maximized estimates of the region of attraction, to estimate the maximum intersampling interval for which a region of stability is ensured, or to produce a stabilizing controller that results in a large implicit region of attraction, through the maximization of an estimate of it.Neste trabalho é proposto um novo método para a análise da estabilidade de sistemas de controle amostrados aperiodicamente e com saturação na entrada, e também para a síntese de controladores estabilizantes. A partir de uma representação por sistemas híbridos, condições de estabilidade baseadas em funções quadráticas de Lyapunov dependentes do clock e na condição de setor generalizada para o tratamento de saturação são desenvolvidas para o sistema amostrado em questão. Essas condições são incorporadas como restrições em problemas de otimização. Os problemas de otimização são baseados em programação semidefinida e em programação sum-of-squares, e têm o objetivo de obter estimativas maximizadas da região de atração do sistema, estimativas do intervalo de amostragem máximo para o qual uma dada região de estados iniciais seja uma região de estabilidade, ou para produzir controladores (dados por ganhos estáticos estabilizantes) que resultem em uma região de atração implicitamente grande, através da maximização da estimativa dessa região de atração