Robust Observer Design for Hybrid Dynamical Systems with Linear Maps and Approximately Known Jump Times

This paper proposes a general framework for the state estimation of plants given by hybrid systems with linear flow and jump maps, in the favorable case where their jump events can be detected (almost) instantaneously. A candidate observer consists of a copy of the plant's hybrid dynamics with continuous-time and/or discrete-time correction terms multiplied by two constant gains, and with jumps triggered by those of the plant. Assuming that the time between successive jumps is known to belong to a given closed set allows us to formulate an augmented system with a timer which keeps track of the time elapsed between successive jumps and facilitates the analysis. Then, since the jumps of the plant and of the observer are synchronized, the error system has time-invariant linear flow and jump maps, and a Lyapunov analysis leads to sufficient conditions for the design of the observer gains for uniform asymptotic stability in three different settings: continuous and discrete updates, only discrete updates, and only continuous updates. These conditions take the form of matrix inequalities, which we solve in examples including cases where the time between successive jumps is unbounded or tends to zero (Zeno behavior), and cases where either both the continuous and discrete dynamics, only one of them, or neither of them are detectable. Finally, we study the robustness of this approach when the jumps of the observer are delayed with respect to those of the plant. We show that if the plant's trajectories are bounded and the time between successive jumps is lower-bounded away from zero, the estimation error is bounded, and arbitrarily small outside the delay intervals between the plant's and the observer's jumps

상태변수 영역 접합을 통한 하이브리드 시스템의 상태변수 추정 및 추종 제어

학위논문 (박사)-- 서울대학교 대학원 공과대학 전기·컴퓨터공학부, 2017. 8. 서진헌.In this dissertation, we propose a new observer and tracking controller design approach for a class of hybrid dynamical systems with state jumps. The hybrid dynamical system exhibits characteristics typical of both continuous-time dynamical system and discrete-time dynamical system. Therefore, it can be modeled as differential equation of the continuous-time dynamics, difference equation of the discrete-time dynamics, the interaction between them. Since the interaction of continuous-time and discrete-time dynamics in a hybrid system leads to rich dynamical behavior and unfamiliar phenomena, several challenges are encountered when we deal with this system. The observer design considered in this dissertation is to construct a dynamical system called an observer that estimates the state of a given hybrid dynamical system (without any input), from an output of the given system. In addition, the tracking controller design is to construct a dynamical system called a tracking controller that makes an input for a given hybrid dynamical system (with an input) such that the state of the given system tracks a given reference. There many results of the observer and tracking controller designs for the continuous-time and discrete-time dynamical systems, but the results for the hybrid dynamical systems are insufficient. Moreover, the results are applied to some classes of hybrid systems (switched systems, hormone systems, powertrain systems, and so on) rather than general hybrid dynamical systems. The proposed idea dealing with the hybrid dynamical system is to "glue" the jump set (a part of the domain where the jumps take place) onto its image. Then, on the "glued" domain, the hybrid dynamical system becomes a continuous-time dynamical system without any jumps. Especially, for some class of the system, the continuous-time dynamical system has a smooth vector field via some notion, "smoothing". Furthermore, we specify this concept of gluing as a map and investigate the essential conditions of the map. By this map, we obtain the "glued" hybrid dynamical system (which is a continuous-time dynamical system) and it may be possible to construct an observer and/or a tracking controller through conventional methods for continuous-time dynamical systems. From these constructions, we obtain the observer and tracking controller for the hybrid system. Especially, the proposed observer does not require any detection of the state jumps while many previous results does. Furthermore, the proposed tracking controller does not need to make the state jump whenever the jumps of the reference happen. Simulation results for examples including mechanical system with impacts and ripple generator in AC/DC converter illustrate the effectiveness of the proposed approach.1 Introduction 1 1.1 Research Background 1 1.2 Organization and Contributions of the Dissertation 4 2 Mathematical Preliminaries 9 2.1 Calculus in Rn 9 2.2 Differential Geometry 11 2.3 Viability Theorems for Ordinary Differential Equations 23 3 Reviews of Related Previous Works 27 3.1 Gluing Manifolds and Vector Fields 27 3.2 Viability Condition 36 3.3 State Estimation 38 3.4 Tracking Control 42 4 Gluing Domain of Hybrid System 45 4.1 Frameworks 45 4.2 Gluing and Smoothing 48 4.3 Frameworks in Rn and Gluing Function 53 5 State Estimation Strategy 71 5.1 Standing Assumptions 71 5.2 State Estimation 75 5.3 Observer with Linearized Error Dynamics 83 5.4 Observer for Lipschitz Continuous Systems 88 6 Tracking Control Strategy 99 6.1 Standing Assumptions 99 6.2 Tracking Control 101 6.3 Using Discontinuous Feedback to Counteract Dynamics Jumps 108 6.4 Output Tracking Controller for Normal Form 119 7 Conclusions 129 BIBLIOGRAPHY 133 국문초록 139Docto