Observability Of Linear Systems Research Articles

Optimal Quantizer Scheduling and Controller Synthesis for Partially Observable Linear Systems

Optimal Quantizer Scheduling and Controller Synthesis for Partially Observable Linear Systems

An Enhanced Feedback Adaptive Observer for Nonlinear Systems With Lack of Persistency of Excitation

The design of exponentially convergent adaptive observers is addressed for linear observable systems which are perturbed by linearly parameterized nonlinearities depending on measured signals (inputs and outputs). When there is a lack of persistency of excitation a new robust adaptive observer is presented which performs an additional feedback depending on the kernel of the Gramian of the regressor matrix, which is computed online, and generates state variables estimates whose estimation errors are exponentially convergent to zero, provided that a design parameter is chosen to be sufficiently small. The boundedness of the parameter and observer estimation errors is always guaranteed. Parameter estimates do not converge to their true values unless the regressor matrix is persistently exciting (i.e. the Gramian of the regressor matrix is nonsingular): in this case, a well-known exponentially convergent adaptive observer is reobtained, since the additional feedback is zero.

Synthesis of Adaptive Observer of State Variables for a Linear Stationary Object in the Presence of Measurement Noise

The paper is devoted to the problem of state variables observers synthesis for linear stationary system operating under condition of noise or disturbances in the measurement channel. The paper considers a completely observable linear stationary system with known parameters. It is assumed that the state variables are not measured, and the measured output variable contains a small amplitude (in general, modulo less than one) additive noise or disturbance. It is also assumed that there is no a priori information about the disturbance or noise in the measurement channel (for example, frequency spectrum, covariance, etc.). It is well known that many observer synthesis methods have been obtained for this type of systems, including the Kalman filter, which has proven itself in practice. Under the condition of complete observability and the presence of some a priori information about a random process (which is typical for the case when a disturbance in the measurement channel can be represented as white noise), approaches based on Kalman filtering demonstrate the highest quality estimates of state variables convergence to true values. Without disputing the numerous results obtained using the application of the Kalman filter, an alternative idea of the state variables observer constructing is considered in this paper. The alternative of the new approach is primarily due to the fact that there is no need to use the usual approaches based on the Luenberger observer. The paper proposes an approach based on the estimation of unknown parameters (in this case, an unknown vector of initial conditions of the plant state variables) of a linear regression model. Within the framework of the proposed method, after a simple transformation, a transition is made from a dynamic system to a linear regression model with unknown constant parameters containing noise or disturbing effects. After that, a new nonlinear parametrization of the original regression model and an algorithm for identifying unknown constant parameters using the procedure of dynamic expansion of the regressor and mixing are proposed which ensure reduction the influence of noise. The article presents the results of computer simulations verifying the stated theoretical results.

Stochastic Linear-Quadratic Optimal Control with Partial Observation

This paper studies a class of quadratic optimal control problems for partially observable linear dynamical systems. In contrast to the full information case, the control is required to be adapted to the filtration generated by the observation system, which in turn is influenced by the control. The variation method fails in this case due to the fact that the filtration is not fixed. To overcome the difficulty, we use the orthogonal decomposition of the state process to write the cost functional as the sum of two parts: one is a functional of the control and the filtering process and the other part is independent of the choice of the control. The first part possesses a mathematical structure similar to the full information problem. By completing the square, it is shown that the open-loop optimal control is given by a feedback representation via the filtering process. The optimal value is also obtained explicitly.

Online Learning of the Kalman Filter With Logarithmic Regret

In this paper, we consider the problem of predicting observations generated online by an unknown, partially observable linear system, which is driven by Gaussian noise. In the linear Gaussian setting, the optimal predictor in the mean square error sense is the celebrated Kalman filter, which can be explicitly computed when the system model is known. When the system model is unknown, we have to learn how to predict observations online based on finite data, suffering possibly a non-zero regret with respect to the Kalman filter's prediction. We show that it is possible to achieve a regret of the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{poly}\log (\mathsf {N})$</tex-math></inline-formula> with high probability, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the number of observations collected. This is achieved using an online least-squares algorithm, which exploits the approximately linear relation between future observations and past observations. The regret analysis is based on the stability properties of the Kalman filter, recent statistical tools for finite sample analysis of system identification, and classical results for the analysis of least-squares algorithms for time series. Our regret analysis can also be applied to other predictors, e.g. multiple step-ahead prediction, or prediction under exogenous inputs including closed-loop prediction. A fundamental technical contribution is that our bounds hold even for the class of non-explosive systems (including marginally stable systems), which was not addressed before in the case of online prediction.

Distributed Observers for State Omniscience with Stochastic Communication Noises

The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each local observer measures only part of the system output, and the state estimation can not be accomplished within a single observer. Then, all observers communicate through a pre-specified graph to make up information in the remaining system output. By solving a parametric algebraic Riccati equation (ARE), a simple method to calculate parameters in the observers is proposed. Furthermore, using the stability theory of stochastic differential equations, state omniscience is discussed in almost sure sense and in the mean square sense for the cases of state-dependent noises and non-state-dependent noises, respectively. It is shown that, for observable linear systems, the resulting observers work in a coordinated mode to reach state omniscience under the connected graph. Illustrative examples are provided to show the effectiveness of the distributed observers.

An efficient approach to stabilization for linear systems subject to output saturation

This paper studies the stabilization problem for controllable and observable linear systems subject to output saturation. An efficient approach to global asymptotical stabilization is proposed, where multiple output components can be collectively steered to escape from saturation. This differs from previous approaches in which each output component is driven one by one. Sufficient conditions for collectively driving all output elements are presented in single-batch driven strategy. In the case of the above sufficient conditions not being satisfied, a multi-batch driven strategy is constructed to guarantee that each component is steered away from saturation at least once. Moreover, an algorithm is put forward for selecting controller parameters in the multi-batch driven case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.

Unknown input observer design for linear time-invariant multivariable systems based on a new observer normal form

In various applications in the field of control engineering, the estimation of the state variables of dynamic systems in the presence of unknown inputs plays an important role. Existing methods require the so-called observer matching condition to be satisfied, rely on the boundedness e variables or exhibit an increased observer order of at least twice the plant order. In this article, a novel observer normal form for strongly observable linear time-invariant multivariable systems is proposed. In contrast to classical normal forms, the proposed approach also takes the unknown inputs into account. The proposed observer normal form allows for the straightforward construction of a higher-order sliding mode observer, which ensures global convergence of the estimation error within finite time even in the presence of unknown bounded inputs. Its application is not restricted to systems which satisfy the aforementioned limitations of already existing unknown input observers. The proposed approach can be exploited for the reconstruction of unknown inputs with bounded derivative and robust state-feedback control, which is shown by means of a tutorial example. Numerical simulations confirm the effectiveness of the presented work.

State Observer for Linear Systems with Explicit Constraints: Orthogonal Decomposition Method.

In this paper, an orthogonal decomposition-based state observer for systems with explicit constraints is proposed. State observers have been an integral part of robotic systems, reflecting the practicality and effectiveness of the dynamic state feedback control, but the same methods are lacking for the systems with explicit mechanical constraints, where observer designs have been proposed only for special cases of such systems, with relatively restrictive assumptions. This work aims to provide an observer design framework for a general case linear time-invariant system with explicit constraints, by finding lower-dimensional subspaces in the state space, where the system is observable while giving sufficient information for both feedback and feed-forward control. We show that the proposed formulation recovers minimal coordinate representation when it is sufficient for the control law generation and retains non-minimal coordinates when those are required for the feed-forward control law. The proposed observer is tested on a flywheel inverted pendulum and on a quadruped robot Unitree A1.

Decentralised navigation systems for bearing-based position and velocity estimation in tiered formations

This paper presents a novel decentralised navigation system based on bearing measurements for tiered vehicle formations. In the proposed framework, some vehicles have access to measurements of their own position, whereas others have access to bearing measurements to one or more neighbouring vehicles. Depth measurement may also be available. Local observers for the position and fluid velocity are designed based on the derivation of an equivalent observable linear time-varying system, thus yielding globally exponentially stable error dynamics. The local observers rely on local measurements, as well as limited communications between the vehicles. The stability of the system as a whole is analysed by studying the robustness of the local observers to exponentially decaying perturbations. Thorough Monte Carlo simulations are presented and discussed to compare the performance of the proposed solution with the extended Kalman filter, the unscented Kalman filter, and the Bayesian Cramér-Rao bound.

Attitude, body-fixed Earth rotation rate, and sensor bias estimation using single observations of direction of gravitational field

This paper addresses the problem of estimating the attitude of a robotic platform using biased measurements of: (i) the direction of the gravitational field and (ii) angular velocity obtained from a set of high-grade gyroscopes sensitive to the Earth’s rotation. A cascade solution is proposed that features a Kalman filter (KF) tied to a rotation matrix observer built on the special orthogonal group of order 3. The KF, whose model stems from a uniformly observable linear time-varying system, yields estimates of: (i) the Earth’s total rotational rate; (ii) two sensor biases associated with the aforementioned measurements; and, (iii) noise-filtered and bias-corrected accelerometer data. All estimates are expressed in the platform’s body-fixed frame. In turn, the attitude observer put forward is shown to be almost globally asymptotically stable, in particular locally input-to-state stable with respect to the KF errors. Experimental results are showcased that successfully demonstrate the efficiency of the proposed attitude estimation solution.

THE CONTROLLABILITY OF RLCM NETWORKS OVER F(Z) AND THEIR APPLICATIONS

The subject of controllability and observability in dynamical systems is one of the important matters in linear system theory. The controllability and observability in linear systems over the field R of real numbers were heavily studied. If the values of resistors, capacitors, and self and mutual inductors in RLCM network are regard as independently variable parameters, the network is defined over the field F(z) of all rational functions in its physical parameters. Some concepts and results on the reducibility and separability for this network are presented; the controllability problem and application are discussed. In this paper the author will discuss the relationship between separability and reducibility of RLCM networks over F(z), the relationship between the physical structure of RLCM networks and structural controllability. The results in this paper can be used to study different kinds of linear or nonlinear electrical networks (over F(z)).

Minimal Control of Constrained, Partially Controllable & Observable Linear Systems

We aim at developing control strategies for constrained linear systems without requiring full system controllability and observability. Given a fixed, potentially small set of actuators and sensors, we first design a static affine-linear output feedback controller that guarantees both the asymptotic stability of the closed-loop system and the adherence of the steady state to a set of linear inequality constraints in the presence of interval-bounded, constant inputs. Subsequently, the derived method is used to find the minimum number of actuators and sensors with which one can fulfill such partial controllability and observability requirements. The approach is applied to electrical power networks modeled as a set of linearly coupled oscillators.

Cooperative Navigation Based on Bearing and Range Measurements to Different Vehicles

This paper presents a navigation solution for a vehicle operating in cooperation with two other. The vehicle is assumed to measure bearing to one of the aiding vehicles and range to the other. An observer with globally exponentially stable error dynamics is designed by obtaining an equivalent observable linear time-varying system using an artificial output and state augmentation. The observer relies on local measurements, as well as limited communication between the vehicles. Simulations are performed to assess the proposed solution.

Algebraic approaches for the design of simultaneous observers for linear systems

In this study, algebraic techniques are proposed to design observers capable of estimating the state of multiple linear continuous-time systems. In order to pursue this objective, first an algebraic technique is given to compute the set of all the linear inverses of the observability map of a single plant. Such a result is then used to characterise, through algebraic geometry tools, the simultaneous observability of multiple linear systems both in the forced and in the autonomous case. Such a characterisation is finally employed to design a single observer that is capable of estimating the state of multiple linear systems.

Observers for Linear Systems by the Time Integrals and Moving Average of the Output

In this paper, it is shown that, under some mild assumptions, it is possible to design observers for linear time-invariant continuous-time and discrete-time systems by feeding classical linear observers (e.g., the Kalman filters and the Luenberger observer) with the successive integrals and the moving average of the measured output, respectively. The main interest in these observers relies on the fact that both the integral and the moving average exhibit low-pass behaviors, thus allowing the design of observers that are less sensitive to high-frequency noise. Examples are reported all throughout this paper to corroborate the theoretical results and to highlight the improved filtering properties of the proposed observers.

Complete and Finite-Time Stabilization of a Delay Differential System by Incomplete Output Feedback

For a spectrally controllable and simultaneously spectrally observable linear time-invariant system with commensurable delays, we construct one-dimensional output feedback controls that ensure the complete stabilization of the closed-loop system (the complete damping of the original system and the asymptotic stability of the closed-loop system) by an integro-difference controller and the finite-time stabilization (the complete damping of the original system) by a differential-difference controller. The results are illustrated with an example.

Attitude estimation using high-grade gyroscopes

This paper addresses the problem of attitude estimation using the angular velocity of Earth as a reference vector. A nonlinear observer is proposed that evolves on the special orthogonal group and is aided by angular velocity readings containing implicit measurements of the Earth’s spin. Additionally, the observer resorts to body-fixed measurements of one constant inertial reference vector. The observer’s sole tuning parameter, shaped as a matrix gain, is computed from a time-varying Kalman filter strategically applied to a uniformly observable linear time-invariant system, which is obtained from the linearized rotation matrix error dynamics. The nonlinear observer is proved to be locally exponentially stable but, most noticeably, in spite of this local-based inception, a Monte Carlo simulation analysis demonstrates the good properties of the observer in terms of convergence rate, tuning capability, and large basin of attraction. Furthermore, extensive experimental results confirm the properties of the proposed technique and validate its usage in real world applications.

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

Robust $H_{\infty}$ Interval Observer for Linear Systems With a Controllable Convergence Rate: A Parametric Method

In this paper, the design method of a robust interval observer for linear systems with time-varying disturbances is proposed. First, the $H_{\infty }$ -gain performance is established by constructing the transfer function from disturbances to error dynamic systems of the interval observers. Second, the design problem about a robust interval observer, equivalent to the eigenstructure assignment of the observer error systems under the above form and the idea of the eigenstructure decomposition, is solved. Finally, in view of this situation where there does not exist an observation gain ensuring the cooperativity of the error systems, a novel parametric approach to design an interval observer with a controlled convergence rate and the robustness with respect to disturbances is proposed by a linear transformation and the solutions to a type of generalized Sylvester equations. Besides, the correctness and efficiency of the obtained results are illustrated by numerical examples and an actual physical system about the longitudinal motion of a Charlie Aircraft.