Structural decomposition of multiple time scale Markov processes

Caption title. "October 1987."Includes bibliographical references.Supported in part by a grant from the Air Force Office of Scientific Research. AFOSR-82-0258 Supported in part by a grant from the Army Research Office. DAAG-29-84-K005J.R. Rohlicek, A.S. Willsky

Singular perturbations of linear systems with multiparameters and multiple time scales

AbstractIn this paper, an alternate approach to the method of asymptotic expansions for the study of a singularly perturbed, linear system with multiparameters and multiple time scales is developed. The method consists of developing a linear, non-singular transformation that enables one to transform the original system into an upper triangular form. This process of upper triangularization will enable us to investigate (i) stability and (ii) approximation of solutions of the original system in terms of the overall reduced system and the corresponding boundary layer systems

Stability Analysis of Droop-Controlled Inverter-Based Power Grids via Timescale Separation

We consider the problem of stability analysis for distribution grids with droop-controlled inverters and dynamic distribution power lines. The inverters are modeled as voltage sources with controllable frequency and amplitude. This problem is very challenging for large networks as numerical simulations and detailed eigenvalue analysis are impactical. Motivated by the above limitations, we present in this paper a systematic and computationally efficient framework for stability analysis of inverter-based distribution grids. To design our framework, we use tools from singular perturbation and Lyapunov theories. Interestingly, we show that stability of the fast dynamics of the power grid depends only on the voltage droop gains of the inverters while, stability of the slow dynamics, depends on both voltage and frequency droop gains. Finally, by leveraging these timescale separation properties, we derive sufficient conditions on the frequency and voltage droop gains of the inverters that warrant stability of the full system. We illustrate our theoretical results through a numerical example on the IEEE 13-bus distribution grid

Асимптотика приближения нулевого порядка решения трехтемповой линейно-квадратичной задачи оптимального управления

This paper is devoted to the construction of a zero-order approximation of the solution of a three-time scale singular perturbed linear-quadratic optimal control problem with the help of the direct scheme method. The algorithm of the method consists in immediate substituting a postulated asymptotic expansion of solution into the problem condition and constructing a family of control problems to define the terms of the asymptotic expansion. Asymptotic approximation of the solution contains regular functions and four boundary ones of exponential type which are determined from the five linearquadratic optimal control problems. It is shown, that the system of equations for a zero-order approximation appeared from control optimality conditions of the initial perturbed problem corresponds to control optimality conditions appeared in respective five optimal control problems constructed for finding zero-order asymptotic approximation with the help of the direct scheme method. An illustrative example is given.Данная работа посвящена построению приближения нулевого порядка решения сингулярно возмущенной линейно-квадратичной трехтемповой задачи оптимального управления методом прямой схемы. Алгоритм метода заключается в непосредственной подстановке постулируемого асимптотического разложения решения в условие задачи и построении серии задач для нахождения членов асимптотики. Асимптотическое разложение решения в данном случае содержит регулярные функции и четыре пограничные функции экспоненциального типа, которые определяются из решения пяти линейно-квадратичных задач оптимального управления. Показано, что система уравнений для членов приближения нулевого порядка асимптотического решения задачи, вытекающей из условий оптимальности управления исходной возмущенной задачи, соответствует системе уравнений, получаемой из условий оптимальности управления в построенных пяти задачах оптимального управления для нахождения асимптотического приближения решения нулевого порядка методом прямой схемы. Приведен иллюстративный пример

외란 관측기의 이론적 해석 : 안정성 및 성능

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2014. 8. 심형보.This dissertation provides the stability and performance analysis of the disturbance observer and proposes several design methods for guaranteeing the robust stability and for enhancing the disturbance rejection performance. Compared to many success stories in industry, theoretic analysis on the disturbance observer itself has attracted relatively little attention. In order to enlarge the horizon of its applications, we provide some rigorous analysis both in the frequency and time domain. In the frequency domain, we focus on two main issues: disturbance rejection performance and robust stability. In spite of its powerful ability for disturbance rejection, the conventional disturbance observer rejects the disturbance approximately rather than asymptotically. To enhance the disturbance rejection performance, based on the well-known internal model principle, we propose a design method to embed an internal model into the disturbance observer structure for achieving the asymptotic disturbance rejection and derive a condition for robust stability. Thus, the proposed disturbance observer can reject not only approximately the unmodeled disturbances but also asymptotically the disturbances of sinusoidal or polynomial-in-time type. In addition, a constructive design procedure to satisfy the proposed stability condition is presented. The other issue is to design of the disturbance observer based control system for guaranteeing robust stability under plant uncertainties. We study the robust stability for the case that the relative degree of the plant is not exactly known and so it happens to be different from that of nominal model. Based on the above results, we propose a universal design method for the disturbance observer when the relative degree of the plant is less than or equal to 4. Moreover, from the observation about the role of each block, we generalize the design of disturbance observer and propose a reduced order type-k disturbance observer to improve the disturbance rejection performance and to reduce the design complexity simultaneously. As a counterpart of the frequency domain analysis, we analyze the disturbance observer in the state space for the purpose of extending the horizon of the disturbance observer applications and obtaining the deeper understanding of the role of each block. Based on the singular perturbation theory, it reveals not only well-known properties but also interesting facts such as the peaking in the transient response. Moreover, we investigate robust stability of the disturbance observer based control systems with and without unmodeled dynamics and derive an explicit relation between the nominal performance recovery and the time constant of Q-filter. Since the classical linear disturbance observer does not ensure the recovery of transient response, a nonlinear disturbance observer, in which all the benefits of the classical one are still preserved, is presented for guaranteeing the recovery of transient as well as steady-state response.Abstract List of Figures Symbols and Acronyms 1. Introduction 1.1 Motivation 1.2 Contributions and Outline of the Dissertation 2. Robust Stability for Closed-loop System with Disturbance Observer 2.1 Structure of Disturbance Observer 2.2 Robust Stability Condition for Closed-loop System with Disturbance Observer 2.3 Illustrative Example 3. Embedding Internal Model in Disturbance Observer with Robust Stability 3.1 Design Method for Embedding Internal Model of Disturbance 3.2 Design of Q-filter for Guranteeing Robust Stability 3.2.1 Robust Stability Condition of Closed-loop System 3.2.2 Selecting a_i's for Robust Stability 3.3 Illustrative Example 3.4 Discussions on Robustness 3.4.1 Pros and Cons of Proposed Design Procedure 3.4.2 Bode Diagram Approach 4. Disturbance Observer with Unknown Relative Degree of the Plant 4.1 Robust Stability 4.2 A Guideline for Selecting Q and P_n 4.2.1 A Universal Robust Controller 4.3 Technical Proofs 4.4 Illustrative Examples 5. Reduced Order Type-k Disturbance Observer under Generalized Q-filter 5.1 Concept of Disturbance Observer with Generalized Q-filter Structure 5.2 Robust Stability 5.3 Reduced Order Type-k Disturbance Observer 5.4 Illustrative Examples 6. State Space Analysis of Disturbance Observer 6.1 State Space Realization of Disturbance Observer 6.2 Analysis of Disturbance Observer based on Singular Perturbation Theory 6.3 Discussion on Disturbance Observer Approach 6.3.1 Relation of Robust Stability Condition between State Space and Frequency Domain Approach 6.3.2 Effect of Zero Dynamics 6.3.3 Stability of Nominal Closed-loop System 6.3.4 Infinite Gain Property with p-dynamics 6.3.5 Peaking in Fast Transient 6.4 Nominal Performance Recovery with respect to Time Constant of Q-filter 7. Nominal Performance Recovery and Stability Analysis of Disturbance Observer under Unmodeled Dynamics 7.1 Problem Formulation 7.2 Stability and Performance Analysis based on Singular Perturbation Theory 7.2.1 Nominal Performance Recovery 7.2.2 Multi-time-scale Singular Perturbation Analysis 7.3 Nominal Performance Recovery by Disturbance Observer under Unmodeled Dynamics 8. Extensions of Disturbance Observer for Guaranteeing Robust Transient Performance 8.1 Extensions to MIMO Nonlinear Systems 8.1.1 SISO Nonlinear Disturbance Observer with Nonlinear Nominal Model 8.1.2 MIMO Nonlinear Disturbance Observer with Linear Nominal Model 9. Conclusions Appendix Bibliography 국문초록Docto

Analysis and Control of Non-Affine, Non-Standard, Singularly Perturbed Systems

This dissertation addresses the control problem for the general class of control non-affine, non-standard singularly perturbed continuous-time systems. The problem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for non-affine systems. This is done by generalizing the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization is used to identify conditions under which non-affine systems can be rendered passive. Asymptotic stabilization for non-affine systems is guaranteed by using these conditions along with well-known passivity-based control methods. Unlike previous non-affine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow manifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and simultaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closed-loop system stability is guaranteed. Performance of these methods is validated in simulation for several problems from science and engineering including the continuously stirred tank reactor, magnetic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase helicopter and conventional take-off and landing aircraft models. Results show that the proposed technique applies both to standard and non-standard forms of singularly perturbed systems and provides asymptotic tracking irrespective of the reference trajectory. This dissertation also shows that some benchmark non-minimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking