Interval observers for continuous-time linear systems

International audienceWe consider continuous-time linear systems with additive disturbances and discrete-time measurements. First, we construct an observer, which converges to the state trajectory of the linear system when the maximum time interval between two consecutive measurements is sufficiently small and there are no disturbances. Second, we construct interval observers allowing to determine, for any solution, a set that is guaranteed to contain the actual state of the system when bounded disturbances are present

Boundary observers for a reaction–diffusion system under time-delayed and sampled-data measurements

We construct finite-dimensional observers for a one-dimensional reaction-diffusion system with boundary measurements subject to time-delays and data sampling. The system has a finite number of unstable modes approximated by a Luenberger-type observer. The remaining modes vanish exponentially. For a given reaction coefficient, we show how many modes one should use to achieve a desired rate of convergence. The finite-dimensional part is analyzed using appropriate Lyapunov-Krasovskii functionals that lead to linear matrix inequalitie (LMI)-based convergence conditions feasible for small enough time-delay and sampling period. The LMIs can be used to find appropriate injection gains

Controllability, Observability, and Stability of Mathematical Models

International audienceThis article presents an overview of three fundamental concepts in Mathematical System Theory: controllability, stability and observability. These properties play a prominent role in the study of mathematical models and in the understanding of their behavior. They constitute the main research subject in Control Theory. Historically the tools and techniques of Automatic Control have been developed for artificial engineering systems but nowadays they are more and more applied to "natural systems". The main objective of this article is to show how these tools can be helpful to model and to control a wide variety of natural systems

Dynamic observers for unknown populations

Dynamic observers are considered in the context of structured population modeling and management. Roughly, observers combine a known measured variable of some process with a model of that process to asymptotically reconstruct the unknown state variable of the model. We investigate the potential use of observers for reconstructing population distributions described by density-independent (linear) models and a class of density-dependent (nonlinear) models. In both the density-dependent and -independent cases, we show, in several ecologically reasonable circumstances, that there is a natural, optimal construction of these observers. Further, we describe the robustness these observers exhibit with respect to disturbances and uncertainty in measurement

Automated Model Generation and Observer Design for Interconnected Systems : A Port-Hamiltonian Approach

This work addresses the automated generation of physical-based models and model-based observers. We develop port-Hamiltonian methods, which for the first time allow a complete and consistent automation of these two processes for a large class of interconnected systems

Thermal Neural Networks: Lumped-Parameter Thermal Modeling With State-Space Machine Learning

With electric power systems becoming more compact and increasingly powerful, the relevance of thermal stress especially during overload operation is expected to increase ceaselessly. Whenever critical temperatures cannot be measured economically on a sensor base, a thermal model lends itself to estimate those unknown quantities. Thermal models for electric power systems are usually required to be both, real-time capable and of high estimation accuracy. Moreover, ease of implementation and time to production play an increasingly important role. In this work, the thermal neural network (TNN) is introduced, which unifies both, consolidated knowledge in the form of heat-transfer-based lumped-parameter models, and data-driven nonlinear function approximation with supervised machine learning. A quasi-linear parameter-varying system is identified solely from empirical data, where relationships between scheduling variables and system matrices are inferred statistically and automatically. At the same time, a TNN has physically interpretable states through its state-space representation, is end-to-end trainable -- similar to deep learning models -- with automatic differentiation, and requires no material, geometry, nor expert knowledge for its design. Experiments on an electric motor data set show that a TNN achieves higher temperature estimation accuracies than previous white-/grey- or black-box models with a mean squared error of $3.18~\text{K}^2$ and a worst-case error of $5.84~\text{K}$ at 64 model parameters.Comment: Preprint; Fix typos, streamline math. notation; 10 page

Observability and observer design for switched linear systems

Hybrid vehicles, HVAC systems in new/old buildings, power networks, and the like require safe, robust control that includes switching the mode of operation to meet environmental and performance objectives. Such switched systems consist of a set of continuous-time dynamical behaviors whose sequence of operational modes is driven by an underlying decision process. This thesis investigates feasibility conditions and a methodology for state and mode reconstruction given input-output measurements (not including mode sequence). An application herein considers insulation failures in permanent magnet synchronous machines (PMSMs) used in heavy hybrid vehicles. Leveraging the feasibility literature for switched linear time-invariant systems, this thesis introduces two additional feasibility results: 1) detecting switches from safe modes into failure modes and 2) state and mode estimation for switched linear time-varying systems. This thesis also addresses the robust observability problem of computing the smallest structured perturbations to system matrices that causes observer infeasibility (with respect to the Frobenius norm). This robustness framework is sufficiently general to solve related robustness problems including controllability, stabilizability, and detectability. Having established feasibility, real-time observer reconstruction of the state and mode sequence becomes possible. We propose the embedded moving horizon observer (EMHO), which re-poses the reconstruction as an optimization using an embedded state model which relaxes the range of the mode sequence estimates into a continuous space. Optimal state and mode estimates minimize an L2-norm between the measured output and estimated output of the associated embedded state model. Necessary conditions for observer convergence are developed. The EMHO is adapted to solve the surface PMSM fault detection problem

Lectures on algebraic system theory: Linear systems over rings

The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are: (1) discrete-time systems over a ring of scalars such as the integers; (2) continuous-time systems containing time delays; (3) large-scale discrete-time systems; and (4) time-varying discrete-time systems

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