Andrey V. Savkin

Stability results for switched controller systems

There are many practical control problems where the control action is determined by switching among a given set of control laws. This paper presents necessary and sufficient conditions to test for quadratic stabilizability and for robust stabilizability with a quadratic storage function for switched controller systems. Algorithms which can be used to construct appropriately stabilizing control laws are also presented.

Qualitative Theory of Hybrid Dynamical Systems

1 Introduction.- 1.1 Hybrid Dynamical Systems.- 1.2 Two Contrasting Examples of Discretely Controlled Continuous Variable Systems.- 1.3 The Main Goal of This Book.- 1.4 Organization of the Book.- 1.5 List of Notations.- 2 Qualitative Analysis of Some Simple Hybrid Dynamical Systems.- 2.1 Introduction.- 2.2 Differential Automata and Their Trajectories.- 2.3 Cyclic Linear Differential Automata.- 2.4 Qualitative Analysis of Cyclic Linear Differential Automata.- 2.5 Switched Server Systems with a Cyclic Switching Policy.- 2.6 Switched Server Systems with Several Limit Cycles.- 2.7 Qualitative Analysis of Closed Switched Server Systems.- 2.8 Essentially Non-Periodic Dynamics of Switched Arrival Systems.- 3 General Theory of Multivalued Differential Automata.- 3.1 Introduction.- 3.2 Multivalued Differential Automata.- 3.2.1 Basic assumptions and definitions.- 3.2.2 Illustrative examples.- 3.2.3 Invariant sets.- 3.2.4 A partial classification of points in the phase space.- 3.2.5 Deterministic and well-posed systems.- 3.2.6 The skeleton and the backstepping mapping.- 3.2.7 Asymptotically stable limit cycles.- 3.3 Decomposition of Well-Posed Differential Automata.- 3.4 Existence of Periodic Trajectories.- 3.5 Proofs of the Theorems and Lemmas from Section 3.2.- 3.6 Proof of Theorem 3.2.26.- 3.7 Proofs of the Theorems from Sections 3.3 and 3.4.- 3.7.1 Proof of Theorem 3.4.3.- 4 Two-Dimensional Hybrid Dynamical Systems.- 4.1 Introduction.- 4.2 An Analog of the Poincare-Bendixon Theorem.- 4.2.1 Basic assumptions.- 4.2.2 A simple periodic dynamics.- 4.2.3 A criterion for a simple periodic dynamics.- 4.3 A Switched Arrival System with Three Buffers.- 4.4 A Switched Server System with Three Buffers.- 4.5 Proofs of the Statements from Section 4.2.- 4.5.1 Proofs of the lemmas from Section 4.2.- 4.5.2 Proof of Theorem 4.2.10 and the remarks following it.- 5 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: General Theory.- 5.1 Introduction.- 5.2 Basic Assumptions and Definitions.- 5.2.1 Multivalued differential automata with constant derivatives.- 5.2.2 Key assumptions.- 5.3 Criteria for Existence and Stability of Limit Cycles.- 5.3.1 A complement concerning deterministic systems.- 5.4 Proofs of the Lemmas from Section 5.2.- 5.5 Proofs of the Theorems and Lemmas from Section 5.3..- 5.6 Proofs of the Theorem and Lemmas from Subsection 5.3.1.- 6 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: Examples.- 6.1 Introduction.- 6.2 Qualitative Analysis of a Switched Server System.- 6.2.1 Description of a switched server system.- 6.2.2 A cyclic control policy.- 6.2.3 The Clear-the-Largest-Buffer-Level Policy.- 6.2.4 Structural stability of a switched server system.- 6.3 A Switched Arrival System with Three Buffers.- 6.4 Qualitative Analysis of Switched Single Server Flow Networks.- 6.4.1 Single server flow networks.- 6.4.2 A cyclic control policy.- 6.4.3 A composed cyclic control policy.- 6.4.4 A combined control policy.- 7 Globally Periodic Behavior of Switched Single Server Flow Networks.- 7.1 Introduction.- 7.2 Description of Switched Single Server Flow Networks.- 7.3 Analysis of Switched Single Server Flow Networks.- 8 Regularizability of Switched Multiple Server Flow Networks.- 8.1 Introduction 315 8.2 Description of Switched Multiple Server Flow Networks.- 8.3 Regularizable Switched Multiple Server Flow Networks.- 8.4 Illustrative Example.- 9 Open Problems.- 9.1 Introduction.- 9.2 Switched Server Systems.- 9.3 Essentially Nonperiodic Multidimensional Switched Arrival Systems.- 9.4 Switched Server/Arrival Systems with Several Servers.- 9.5 A Generalized Processor Sharing Model.- 9.6 Stabilizability of Switched Multiple Server Flow Networks.- 9.7 Chaotic Switched Flow Networks.- 9.8 Existence and Global Stability of Limit Cycles in Nonlinear Differential Automata.- References.

Technical communique: Robust state estimation and model validation for discrete-time uncertain systems with a deterministic description of noise and uncertainty

The paper presents a new approach to robust state estimation for a class of uncertain discrete-time systems with a deterministic description of noise and uncertainty. The main result is a recursive scheme for constructing an ellipsoidal state estimation set of all states consistent with the measured output and the given noise and uncertainty description. The paper also includes a result on model validation whereby it can be determined if the assumed model is consistent with measured data.

Brief Paper: Robust output feedback stabilizability via controller switching

The paper considers the output feedback robust stabilizability problem for uncertain dynamical systems. The uncertain system under consideration is a composite of a continuous-time plant and a switched controller. A necessary and sufficient condition of absolute stabilizability is given in terms of the existense of suitable solutions to a dynamic programming equation and a Riccati algebraic equation of the H^~ filtering type. A real time implementable method for robust stabilization is also presented.

Multirate Stabilization of Linear Multiple Sensor Systems via Limited Capacity Communication Channels

The paper addresses a feedback stabilization problem involving bit-rate communication capacity constraints. A discrete-time partially observed linear system is studied. Unlike classic theory, the signals from multiple sensors are transmitted to the controller over separate finite capacity communication channels. The sensors do not have constant access to the channels, and the channels are not perfect: the messages incur time-varying transmission delays and may be corrupted or lost. However, we suppose that the time-average number of bits per sample period that can be successfully transmitted over the channel during a time interval converges to a certain limit as the length of the interval becomes large. Necessary and sufficient conditions for stabilizability are established. They give the tightest lower bounds on the channel capacities for which stabilization is possible. An algorithm for stabilization is also presented.

An Analogue of Shannon Information Theory for Detection and Stabilization via Noisy Discrete Communication Channels

The paper addresses both detection and stabilization problems involving communication errors and capacity constraints. Discrete-time partially observed linear systems are studied. Unlike the classic theory, the sensor signals are transmitted to the estimator/controller over a noisy digital communication link modeled as a stochastic stationary discrete memoryless channel. It is shown that for noise-free plants, the Shannon capacity of the channel constitutes the border separating the cases where stabilization and reliable detection (asymptotic state estimation) with arbitrarily large probability are and are not possible, respectively.

Model validation and state estimation for uncertain continuous-time systems with missing discrete-continuous data

This paper considers two related problems of state estimation and model validation for a class of uncertain linear systems. The main contribution of the paper is that it considers a general information structure which allows for discrete and continuous measurements as well as missing data. The results are given in terms of a recursive state estimator involving a jump Riccati differential equation and jump state equations. These equations can be solved on-line.

Cyclic linear differential automata: a simple class of hybrid dynamical systems☆

Abstract We introduce a special class of hybrid dynamical systems: cyclic linear differential automata (CLDA). We show that any CLDA can be reduced to a linear discrete-time system with periodic coefficients. Any CLDA has no equilibrium points. Therefore, the simplest attractor in such system is a periodic trajectory. We call a CLDA globally stable if it has a periodic trajectory which attracts all other trajectories of the system. A necessary and sufficient condition for global stability of CLDA is given. We apply our result to prove global stability of a flexible manufacturing system modelled as a switched server system.

The problem of optimal robust sensor scheduling

Abstract This paper considers the sensor scheduling problem which consists of estimating the state of an uncertain process based on measurements obtained by switching a given set of noisy sensors. The noise and uncertainty models considered in this paper are assumed to be unknown deterministic functions which satisfy an energy type constraint known as an integral quadratic constraint. The problem of optimal robust sensor scheduling is formulated and solution to this problem is given in terms of the existence of suitable solutions to a Riccati differential equation of the game type and a dynamic programming equation. Furthermore, a real time implementable method for sensor scheduling is also presented.

Decentralized state-feedback stabilization and robust control of uncertain large-scale systems with integrally constrained interconnections☆

Abstract This paper is concerned with a problem of stabilization and robust control design for interconnected uncertain systems. A new class of uncertain large-scale systems is considered in which interconnections between subsystems as well as uncertainties in each subsystem are described by integral quadratic constraints. The problem is to design a set of local (decentralized) controllers which stabilize the overall system and guarantee robust disturbance attenuation in the presence of the uncertainty in interconnections between subsystems as well as in each subsystem. The paper presents necessary and sufficient conditions for the existence of such a controller. The proposed design is based on recent absolute stabilization and minimax optimal control results and employs solutions of a set of game-type Riccati algebraic equations arising in H ∞ control.