In this paper we are interested in the problem of state observation of state-affine nonlinear systems. Our main contribution is to propose a globally exponentially convergent observer that requires only the necessary assumption of observability of the system. To the best of the authors' knowledge this is the first time such a result is reported in the literature.
Introductory state-space linear control courses focus on linear, time-invariant systems and spend intense efforts by introducing system realizations that allow the student to grasp fundamental concepts, among which controllability, observability, and controller and observer design. This note describes a graphical mechanism to transform a system expressed in observability form into its observer form based on elementary block diagram manipulations, a technique whose pedagogical usefulness tends to be overlooked. Given the well-known duality principle in linear systems with respect to controllabi…
We propose globally exponentially convergent continuous observers for invariant kinematic systems on finite-dimensional matrix Lie groups. Such an observer estimates, from measurements of landmarks, vectors and biased velocity, both the system state and the unknown constant bias in velocity measurement, where the state belongs to the state-space Lie group and the velocity to the Lie algebra of the Lie group. The main technique is to embed a given system defined on a matrix Lie group into Euclidean space and build observers in the Euclidean space. The theory is illustrated with the special Eucl…
Interval Observer Design Using Observability Decomposition for Detectable Linear Systems
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Conservatoire National des Arts et Métiers (CNAM)（国立工艺学院）；Harbin Institute of Technology（哈尔滨工业大学）
This paper examines the global convergence problem of SLAM algorithms, an issue that faces topological obstructions. This is because the state-space of attitude dynamics is defined on a non-contractible manifold: the special orthogonal group of order three SO(3). Therefore, this paper presents a novel, gradient-based hybrid observer to overcome these topological obstacles. The Lyapunov stability theorem is used to prove the globally asymptotic convergence of the proposed algorithm. Finally, comparative analyses of two simulations were conducted to evaluate the performance of the proposed schem…
The paper addresses the problem of attitude estimation for rigid bodies using (possibly time-varying) vector measurements, for which we provide a necessary and sufficient condition of distinguishability. Such a condition is shown to be strictly weaker than those previously used for attitude observer design. Thereafter, we show that even for the single vector case the resulting condition is sufficient to design almost globally convergent attitude observers, and two explicit designs are obtained. To overcome the weak excitation issue, the first design employs to make full use of historical infor…
This work attempts to discuss the observability of linear time-invariant systems with event-triggered measurements. A new notion of observability, namely, $ε$-observability is defined with parameter $ε$, which relates to the worst-case performance of inferring the initial state based on not only the received measurement but also the implicit information in the event-triggering conditions at no-event instants. A criterion is developed to test the proposed $ε$-observability of discrete-time linear systems, based on which an iterative event-triggered set-membership observer is designed to evaluat…
The robust distributed state estimation for a class of continuous-time linear time-invariant systems is achieved by a novel kernel-based distributed observer, which, for the first time, ensures fixed-time convergence properties. The communication network between the agents is prescribed by a directed graph in which each node involves a fixed-time convergent estimator. The local observer estimates and broadcasts the observable states among neighbours so that the full state vector can be recovered at each node and the estimation error reaches zero after a predefined fixed time in the absence of …
We present the first solution to the problem of estimation of the state of multimachine power systems with lossy transmission lines. We consider the classical three-dimensional \fluxdecay" model of the power system and assume that the active and reactive power as well as the rotor angle and excitation voltage at each generator is available for measurement|a scenario that is feasible with current technology. The design of the observer relies on two recent developments proposed by the authors: a parameter estimation based approach to the problem of state estimation and the use of the dynamic reg…
Australian Centre for Field Robotics & Sydney Institute for Robotics and Intelligent Systems, The University of Sydney（悉尼大学澳大利亚野外机器人中心与悉尼机器人与智能系统研究所）；DJI Innovation Inc.（大疆创新科技有限公司）
In this paper we propose a novel observer to solve the problem of visual simultaneous localization and mapping (SLAM), only using the information from a single monocular camera and an inertial measurement unit (IMU). The system state evolves on the manifold $SE(3)\times \mathbb{R}^{3n}$, on which we design dynamic extensions carefully in order to generate an invariant foliation, such that the problem is reformulated into online \emph{constant parameter} identification. Then, following the recently introduced parameter estimation-based observer (PEBO) and the dynamic regressor extension and mix…
The control approaches generally resort to the tools from the mathematics, but whether and how the mathematics can benefit from the control approaches is unclear. This paper aims to bring the "control design" idea into the mathematics by providing an observer-based iterative method that focuses on solving linear algebraic equations (LAEs). An inherent relationship is revealed between the problem-solving of LAEs and the design of observer-based control systems, with which the iterative method for solving LAEs is exploited based on the design of the basic state observers. It is shown that all (l…
This paper proposes a novel discrete-time multi-virus SIR (susceptible-infected-recovered) model that captures the spread of competing SIR epidemics over a population network. First, we provide a sufficient condition for the infection level of all the viruses over the networked model to converge to zero in exponential time. Second, we propose an observation model which captures the summation of all the viruses' infection levels in each node, which represents the individuals who are infected by different viruses but share similar symptoms. We present a sufficient condition for the model to be l…
The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high-dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that…
A General Nonlinear Observer Design for Inertial Navigation Systems with Almost Global Stability Guarantees
Capgemini Engineering（凯捷工程）；University of Quebec in Outaouais（渥太华大学）；I3S-UniCA-CNRS, University Cote d’Azur（尼斯大学）；Institut Universitaire de France（法国高等研究院）
The University of Sydney（悉尼大学）；DJI Innovation Inc.（大疆创新有限公司）
The design of a globally convergent position observer for feature points from visual information is a challenging problem, especially for the case with only inertial measurements and without assumptions of uniform observability, which remained open for a long time. We give a solution to the problem in this paper assuming that only the bearing of a feature point, and biased linear acceleration and rotational velocity of a robot -- all in the body-fixed frame -- are available. Further, in contrast to existing related results, we do not need the value of the gravitational constant either. The pro…
Convergent, contractive or incremental stability properties of nonlinear systems have attracted interest for control tasks such as observer design, output regulation and synchronization. The convergence property plays a central role in the neuromorphic (brain-inspired) computing of reservoir computing, which seeks to harness the information processing capability of nonlinear systems. This paper presents a small-gain theorem for discrete-time output-feedback interconnected systems to be uniformly input-to-output convergent (UIOC) with outputs converging to a bounded reference output uniquely de…
Conditions for detectability in distributed consensus-based observer networks
School of Engineering and IT, University of NSW at the Australian Defence Force Academy（新南威尔士大学澳大利亚国防军学院工程与信息技术学院）；Australian National University（澳大利亚国立大学）
Western University（西安大略大学）；University of Quebec in Outaouais（魁北克大学乌塔韦校区）；Lakehead University（湖首大学）
This paper deals with the simultaneous estimation of the attitude, position and linear velocity for vision-aided inertial navigation systems. We propose a nonlinear observer on $SO(3)\times \mathbb{R}^{15}$ relying on body-frame acceleration, angular velocity and (stereo or monocular) bearing measurements of some landmarks that are constant and known in the inertial frame. Unlike the existing local Kalman-type observers, our proposed nonlinear observer guarantees almost global asymptotic stability and local exponential stability. A detailed uniform observability analysis has been conducted and…
This paper presents a new observer design approach for linear time invariant multivariable systems subject to unknown inputs. The design is based on a transformation to the so-called special coordinate basis. This form reveals important system properties like invertability or the finite and infinite zero structure. Depending on the system's strong observability properties, the special coordinate basis allows for a straightforward unknown input observer design utilizing linear or nonlinear observers design techniques. The chosen observer design technique does not only depend on the system prope…
The goal of this survey paper is to provide an introduction to chaos synchronization using nonlinear observers and its applications in cryptography. I start with an overview of cryptography. Then, I recall the basics of chaos theory and how to use chaotic systems for cryptography, with an introduction to the problem of chaos synchronization. Then, I present the theory of non-linear observers, which is used for the synchronization of chaotic systems. I start with an explanation of the observability problem. Then, I introduce some of the classical observers: Kalman filter, Luenberger observer, E…
Observer design for position and velocity bias estimation from a single direction output
French Direction Générale de l’Armement (Technical Directorate)（法国国家军备总局）；I3S UNS-CNRS（尼斯-索菲亚-安蒂波利斯大学与法国国家科学研究中心联合研究所）；School of Engineering, Australian National University（澳大利亚国立大学工程学院）；INRIA（法国国家信息与自动化研究所）
This paper deals with the problem of full state estimation for vehicles navigating in a three dimensional space. We assume that the vehicle is equipped with an Inertial Measurement Unit (IMU) providing body-frame measurements of the angular velocity, the specific force, and the Earth's magnetic field. Moreover, we consider available sensors that provide partial or full information about the position of the vehicle. Examples of such sensors are those which provide full position measurements (e.g., GPS), range measurements (e.g., Ultra-Wide Band (UWB) sensors), inertial-frame bearing measurement…
Although an appropriate choice of measured state variables may ensure observability, designing state observers for the state estimation of epidemic models remains a challenging task. Epidemic spread is a nonlinear process, often modeled as the law of mass action, which is of a quadratic form; thus, on a compact domain, its Lipschitz constant turns out to be local and relatively large, which renders the Lipschitz-based design criteria of existing observer architectures infeasible. In this paper, a novel observer architecture is proposed for the state estimation of a class of nonlinear systems t…
In this paper, we propose a new approach to design globally convergent reduced-order observers for nonlinear control systems via contraction analysis and convex optimization. Despite the fact that contraction is a concept naturally suitable for state estimation, the existing solutions are either local or relatively conservative when applying to physical systems. To address this, we show that this problem can be translated into an off-line search for a coordinate transformation after which the dynamics is (transversely) contracting. The obtained sufficient condition consists of some easily veri…
In this paper we are interested in the problem of adaptive state observation of linear time-varying (LTV) systems where the system and the input matrices depend on unknown time-varying parameters. It is assumed that these parameters satisfy some known LTV dynamics, but with unknown initial conditions. Moreover, the state equation is perturbed by an additive signal generated from an exosystem with uncertain constant parameters. Our main contribution is to propose a globally convergent state observer that requires only a weak excitation assumption on the system.
A globally exponentially stable position observer for interior permanent magnet synchronous motors
ITMO University（圣彼得堡信息、光学和机械大学）；The University of Sydney（悉尼大学）；University of Belgrade（贝尔格莱德大学）；Pohang University of Science and Technology (POSTECH)（韩国科学技术院）
In this work, we present a mixed sensorless strategy for Permanent Magnets Synchronous Machines, combining a torque/current controller and an observer for position, speed, flux and stator resistance. The proposed co-design is motivated by the need of an appropriate signal injection technique, in order to guarantee full state observability. Neither the typical constant or slowly-varying speed assumptions, nor a priori mechanical model information are used in the observer design. Instead, the rotor speed is modeled as an unknown input disturbance with constant (unknown) sign and uniformly non-ze…
Synchronous Observer Design for Landmark-Inertial SLAM with Magnetometer and Intermittent GNSS Measurements
Centre for Systems and Control, Indian Institute of Technology Bombay（印度理工学院孟买分校系统与控制中心）；School of Aerospace, Mechanical and Mechatronic Engineering (AMME), Faculty of Engineering, University of Sydney（悉尼大学工程学院航空航天、机械与机电工程系）
