Robust stability theory for stochastic dynamical systems

In this work, we focus on developing analysis tools related to stability theory forcertain classes of stochastic dynamical systems that permit non-unique solutions. Thenon-unique nature of solutions arise primarily due to the system dynamics that aremodeled by set-valued mappings. There are two main motivations for studying suchclasses of systems. Firstly, understanding such systems is crucial to developing a robuststability theory. Secondly, such system models allow flexibility in control design problems.We begin by developing analysis tools for a simple class of discrete-time stochasticsystem modeled by set-valued maps and then extend the results to a larger class ofstochastic hybrid systems. Stochastic hybrid systems are a class of dynamical systemsthat combine continuous-time dynamics, discrete-time dynamics and randomness. Theanalysis tools are established for properties like global asymptotic stability in probabilityand global recurrence. We focus on establishing results related to sufficient conditions for stability, weak sufficient conditions for stability, robust stability conditions and converse Lyapunov theorems. In this work a primary assumption is that the stochastic system satisfies some mild regularity properties with respect to the state variable and random input. The regularity properties are needed to establish the existence of random solutions and results on sequential compactness for the solution set of the stochastic system.We now explain briefly the four main types of analysis tools studied in this work.Sufficient conditions for stability establish conditions involving Lyapunov-like functionssatisfying strict decrease properties along solutions that are needed to verify stability properties. Weak sufficient conditions relax the strict decrease nature of the Lyapunov like function along solutions and rely on either knowledge about the behavior of thesolutions on certain level sets of the Lyapunov-like function or use multiple nested non-strict Lyapunov-like functions to conclude stability properties. The invariance principleand Matrosov function theory fall in to this category. Robust stability conditions determinewhen stability properties are robust to sufficiently small perturbations of thenominal system data. Robustness of stability is an important concept in the presenceof measurement errors, disturbances and parametric uncertainty for the nominal system.We study two approaches to verify robustness. The first approach to establish robustnessrelies on the regularity properties of the system data and the second approach isthrough the use of Lyapunov functions. Robustness analysis is an area where the notionof set-valued dynamical systems arise naturally and it emphasizes the reason for ourstudy of such systems. Finally, we focus on developing converse Lyapunov theorems forstochastic systems. Converse Lyapunov theorems are used to illustrate the equivalencebetween asymptotic properties of a system and the existence of a function that satisfiesa decrease condition along the solutions. Strong forms of the converse theorem implythe existence of smooth Lyapunov functions. A fundamental way in which our resultsdiffer from the results in the literature on converse theorems for stochastic systems isthat we exploit robustness of the stability property to establish the existence of a smoothLyapunov function

Approximation, analysis and control of large-scale systems - Theory and Applications

This work presents some contributions to the fields of approximation, analysis and control of large-scale systems. Consequently the Thesis consists of three parts. The first part covers approximation topics and includes several contributions to the area of model reduction. Firstly, model reduction by moment matching for linear and nonlinear time-delay systems, including neutral differential time-delay systems with discrete-delays and distributed delays, is considered. Secondly, a theoretical framework and a collection of techniques to obtain reduced order models by moment matching from input/output data for linear (time-delay) systems and nonlinear (time-delay) systems is presented. The theory developed is then validated with the introduction and use of a low complexity algorithm for the fast estimation of the moments of the NETS-NYPS benchmark interconnected power system. Then, the model reduction problem is solved when the class of input signals generated by a linear exogenous system which does not have an implicit (differential) form is considered. The work regarding the topic of approximation is concluded with a chapter covering the problem of model reduction for linear singular systems. The second part of the Thesis, which concerns the area of analysis, consists of two very different contributions. The first proposes a new "discontinuous phasor transform" which allows to analyze in closed-form the steady-state behavior of discontinuous power electronic devices. The second presents in a unified framework a class of theorems inspired by the Krasovskii-LaSalle invariance principle for the study of "liminf" convergence properties of solutions of dynamical systems. Finally, in the last part of the Thesis the problem of finite-horizon optimal control with input constraints is studied and a methodology to compute approximate solutions of the resulting partial differential equation is proposed.Open Acces

Distributed navigation of multi-robot systems for sensing coverage

A team of coordinating mobile robots equipped with operation specific sensors can perform different coverage tasks. If the required number of robots in the team is very large then a centralized control system becomes a complex strategy. There are also some areas where centralized communication turns into an issue. So, a team of mobile robots for coverage tasks should have the ability of decentralized or distributed decision making. This thesis investigates decentralized control of mobile robots specifically for coverage problems. A decentralized control strategy is ideally based on local information and it can offer flexibility in case there is an increment or decrement in the number of mobile robots. We perform a broad survey of the existing literature for coverage control problems. There are different approaches associated with decentralized control strategy for coverage control problems. We perform a comparative review of these approaches and use the approach based on simple local coordination rules. These locally computed nearest neighbour rules are used to develop decentralized control algorithms for coverage control problems. We investigate this extensively used nearest neighbour rule-based approach for developing coverage control algorithms. In this approach, a mobile robot gives an equal importance to every neighbour robot coming under its communication range. We develop our control approach by making some of the mobile robots playing a more influential role than other members of the team. We develop the control algorithm based on nearest neighbour rules with weighted average functions. The approach based on this control strategy becomes efficient in terms of achieving a consensus on control inputs, say heading angle, velocity, etc. The decentralized control of mobile robots can also exhibit a cyclic behaviour under some physical constraints like a quantized orientation of the mobile robot. We further investigate the cyclic behaviour appearing due to the quantized control of mobile robots under some conditions. Our nearest neighbour rule-based approach offers a biased strategy in case of cyclic behaviour appearing in the team of mobile robots. We consider a clustering technique inside the team of mobile robots. Our decentralized control strategy calculates the similarity measure among the neighbours of a mobile robot. The team of mobile robots with the similarity measure based approach becomes efficient in achieving a fast consensus like on heading angle or velocity. We perform a rigorous mathematical analysis of our developed approach. We also develop a condition based on relaxed criteria for achieving consensus on velocity or heading angle of the mobile robots. Our validation approach is based on mathematical arguments and extensive computer simulations