Uniform Complete Observability Research Articles

Distributed Exponential State Estimation for Discrete-Time Linear Systems Over Jointly Connected Switching Networks

The existing results on distributed state estimation problems for discrete-time linear systems were obtained over connected static networks, or every-time connected switching networks by a two-time-scale design. This paper further studies the same problem over jointly connected switching networks which can be disconnected at every time instant by a single-time-scale distributed design. Since the existing distributed designs critically rely on the connectedness assumption on the networks and thus may not apply to jointly connected switching networks, we have to modify the local observers and develop a novel approach which makes use of the uniform complete observability property for a discrete-time time-varying system. Additionally, we establish two exponential stability results for two classes of discrete-time switched systems, which may be of independent interest.

H‐representation to stochastic observability conservation by output feedback and Gramian bounds

AbstractIn this paper, we are concerned with uniform complete observability of linear stochastic time‐varying systems conserved by output feedback and Gramian bounds via so called ‐representation method. We obtain the lower and upper bounds for the Gramian of output feedback stochastic system from the original system and vice versa. A numerical example is constructed to verify the effectiveness of our main result. The uniform complete observability is useful for the analysis and synthesis of the Kalman filter.

Observability is sufficient for the design of globally exponentially stable state observers for state-affine nonlinear systems

In this paper we are interested in the problem of state observation of state-affine nonlinear systems. It is well-known that observability of the system is a necessary assumption for the state reconstruction. Our main contribution is to prove that it is also sufficient to design a globally exponentially stable observer This should be contrasted with existing results that require the strictly stronger assumption of uniform complete observability of the system. To the best of the authors’ knowledge this is the first time such a result is reported in the literature.

An Active Perception Framework for Autonomous Underwater Vehicle Navigation Under Sensor Constraints

Inertial navigation for autonomous underwater vehicles (AUVs) is challenging because of the drift error caused by the noise and measurement errors of inertial sensors, typically packaged as an inertial measurement unit (IMU), integrated over time. To mitigate the drift error, recent AUV state estimation approaches incorporate external references or environmental information obtained from exteroceptive sensors, with increased costs and limited operational domains. For improved navigation under sensor constraints, this article proposes an active perception framework that exploits vehicle motion to estimate the flow state together with the vehicle state using IMU and depth sensors only. The proposed framework uses the estimated flow state as external information to improve vehicle state estimation. We construct a linear time-varying system for the flow state, separated from a nonlinear system for the vehicle state. This formulation allows us to analyze uniform complete observability for the flow state, which is found to depend on vehicle motion. Then, along with vehicle and flow state estimators, we design a vehicle controller to enable vehicle motion to maximize an information metric pertaining to estimation performance based on either observability or constructability Gramian for the flow state. The proposed framework is validated through simulations for a case study with a vehicle descending through the water column in a time-varying flow field. The effectiveness of the framework is demonstrated by comparing results obtained from its four implementations with those from baseline approaches without active perception.

Globally convergent visual-feature range estimation with biased inertial measurements

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 proposed approach builds upon the parameter estimation-based observer recently developed in Ortega et al. (2015) and its extension to matrix Lie groups in our previous work. Conditions on the robot trajectory under which the observer converges are given, and these are strictly weaker than the standard persistency of excitation and uniform complete observability conditions. Finally, as an illustration, we apply the proposed design to the visual inertial navigation problem.

Regularized adaptive Kalman filter for non-persistently excited systems

Most available methods for joint state-parameter estimation in discrete-time stochastic time-varying systems suffer from severe performance degradation if the system is not persistently excited. We revisit and extend a recently proposed adaptive Kalman filter in order to enhance its robustness against periods of poor excitation. A Levenberg–Marquardt-like regularization algorithm is integrated in the filter to preclude the estimator from becoming unreliable due to wind-up effects. It is proven that, under uniform complete observability–controllability conditions and a persistent excitation condition, the expectations of the state and parameter estimation errors tend to zero exponentially fast. Furthermore, their stability (in the sense of Lyapunov) is guaranteed even if the persistent excitation condition is not satisfied. The effectiveness of the algorithm is demonstrated using real sensor data from a vehicle motion estimation application. Unlike the original algorithm, the proposed regularized adaptive Kalman filter provides accurate and reliable estimates of the states and parameters despite periods of poor excitation.

On Necessary and Sufficient Conditions for Exponential Consensus in Dynamic Networks via Uniform Complete Observability Theory

In this article, we deal with the consensus problem of multiple partial-state coupled linear systems, which are neutrally stable. These systems communicate over dynamic undirected networks, which change continuously and can be disconnected at any time. We develop an analysis framework from uniform complete observability theory to work out <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">the necessary and sufficient conditions</i> for exponential consensus. It turns out that, with a suitably designed feedback matrix, exponential consensus can be realized globally and uniformly <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">if and only if</i> a jointly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\delta,T)$</tex-math></inline-formula> -connected condition and an observability condition relying only on system and input matrices are satisfied. A lower bound of the convergence rate is also provided. We figure out the proof by applying matrix analysis and linear functional analysis. A simulation example is presented to illustrate our theoretical findings.

Asymptotic properties of linear filter for deterministic processes

It is known that Kalman–Bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability. In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system, i.e., for deterministic processes. The earlier stability results cannot be applied for this case, as the system is not controllable. We further show that the optimal linear filter for a general class of non-Gaussian initial conditions is asymptotically proximal to Kalman–Bucy filter. It is also shown that the filter corresponding to non-zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of Gaussian initial conditions.

Adaptive Kalman filter for actuator fault diagnosis

An adaptive Kalman filter is proposed in this paper for actuator fault diagnosis in discrete time stochastic time varying systems. By modeling actuator faults as parameter changes, fault diagnosis is performed through joint state-parameter estimation in the considered stochastic framework. Under the classical uniform complete observability–controllability conditions and a persistent excitation condition, the exponential stability of the proposed adaptive Kalman filter is rigorously analyzed. In addition to the minimum variance property of the combined state and parameter estimation errors, it is shown that the parameter estimation within the proposed adaptive Kalman filter is equivalent to the recursive least squares algorithm formulated for a fictive regression problem. Numerical examples are presented to illustrate the performance of the proposed algorithm.

On stability of the Kalman filter for discrete time output error systems

The stability of the Kalman filter is classically ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of discrete time output error (OE) systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability is sufficient to ensure the stability of the Kalman filter applied to time varying OE systems, regardless of the stability of the OE systems. Though the continuous time case has been studied recently, the results on continuous time systems cannot be directly transposed to discrete time systems, because of a difficulty related to the observability of the discrete time filter error dynamics system.

Relaxed conditions for uniform complete observability and controllability of LTV systems with bounded realizations

Uniform complete observability and controllability are key concepts for the design of state observers and controllers for linear time-varying systems. While the definitions of these properties are simple, it is more often that not very difficult to show they hold for a particular system. This paper presents alternative simpler sufficient conditions. Examples are shown that illustrate the improvement over previous results, the fact that the new conditions are not necessary, and a practical application.

Stabilization by Output Feedback of Multivariable Invertible Nonlinear Systems

In this paper, the problem of global stabilization of a rather general class of MIMO nonlinear systems is addressed. The systems considered in the paper are invertible, have a trivial zero dynamics and possess a “normal form” in which certain multipliers are functions of the state vector of a special kind. While special structural dependence of such multipliers on the components state vector has been exploited before in the context of achieving stabilization (via full-state feedback, though), the novelty in the approach of this paper is that a peculiar structure is identified which happens to be intimately related to the property of uniform complete observability (thus making the design of observers possible) and to the property of uniform invertibility, relations never established before in the literature. As a result, for this class of MIMO nonlinear systems, a dynamic output feedback law can be designed, yielding semiglobal (and even global, under appropriate assumptions) asymptotic stability.

Stability of the Kalman filter for continuous time output error systems

The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of continuous time output error systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability alone is sufficient to ensure the asymptotic stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered systems themselves. The exponential or polynomial convergence of the Kalman filter is then further analyzed for particular cases of stable or unstable output error systems.

Persistence based convergence rate analysis of consensus protocols for dynamic graph networks

This paper deals with the consensus problem of agents communicating via time-varying communication links in undirected graph networks. The highlight of the current work is to provide practically computable rates of convergence to consensus that hold for a large class of time-varying edge weights. A novel analysis technique based on classical notions of persistence of excitation and uniform complete observability is proposed. The new analysis technique for consensus laws under time-varying graphs provides explicit bounds on rate of convergence to consensus for single integrator dynamics. In the case of double integrators a minor modification to the standard relative state feedback law is shown to guarantee exponential convergence to consensus under similar assumptions as the single integrator case. The consensus problem is re-formulated in the edge agreement framework to which persistence of excitation based results apply. A novel application of results from Ramsey theory allows for proof of consensus and convergence rate estimation under switching graph topology. The current work connects classical results from nonlinear adaptive control and combinatorics to the modern theory of consensus over multi-agent networks.

A consequence of invertibility on the normal form of a MIMO nonlinear systems

Abstract: In this paper, a relevant consequence of invertibility on the normal form of MIMO nonlinear systems is presented. Making full use of the definition of uniform invertibility in the sense of Singh, the multipliers δi+1(x)’s in the normal form are shown to possess a special property, that is to be triangularly dependent on the x2,i’s. As a matter of fact, this property turns out to be necessary and sufficient for uniform invertibility in the case of a two-input two-output systems having normal form. Moreover, we also show that invertibility, or this special property of the multipliers, is a sufficient condition of uniform complete observability.

Observability conservation by output feedback and observability Gramian bounds

Though it is a trivial fact that the observability of a linear state space system is conserved by output feedback, it requires a rigorous proof to generalize this result to uniform complete observability, which is defined with the observability Gramian. The purpose of this paper is to present such a proof. Some issues in existing results are also discussed. The uniform complete observability of closed loop systems is useful for the analysis of some adaptive systems and of the Kalman filter.

A comparative study of persistence based convergence rate estimates to consensus

This article presents a comparative study of convergence rate estimates for degenerate gradient flows in the context of multi-agent systems. A novel analysis methodology was introduced in (Roy Chowdhury and Srikant, 2014) based on the classical notions of persistence of excitation (PE) and uniform complete observability (UCO) to study the exponential stability of consensus protocol with time-varying agent dynamics. The new technique allowed precise computation of the convergence rate for single integrator agent dynamics. The purpose of this article is principally twofold. Firstly, we revisit the continuous time spanning edge agent dynamics proposed in (Roy Chowdhury and Srikant, 2014) and present a different analysis approach to study the convergence property for the same. Our analysis mimics the recent result by Brockett (Brockett, 2000) on single agent adaptive control. Secondly, the convergence rate estimates arrived at from the two approaches are compared. Simulations on different examples are provided to elucidate the main theoretical contribution.

Stability of the Kalman Filter for Output Error Systems

Optimality and numerical efficiency are well known properties of the Kalman filter, whereas its stability property, though equally classical and important in practice, is less often mentioned in the recent literature. The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. Such classical results cannot be applied to output error systems, in which the process noise is totally absent. It is shown in this paper that the uniform complete observability is sufficient to ensure the stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered system itself.

PRACTICAL OBSERVER FOR IMPULSIVE SYSTEMS

In this paper, we deal with the problem of practical observer design and the practical stabilization for a class of perturbed impulsive systems. We show that, under the classical conditions of uniform complete controllability and uniform complete observability of the nominal system without impulsive effects, it is possible to design an observer controller for a class of perturbed linear impulsive system when the origin is not an equilibrium point.

Deterministic learning from control of nonlinear systems with disturbances

In this paper, we investigate deterministic learning in environments with disturbances. We will show that for a class of uncertain nonlinear systems with bounded disturbances, by using an appropriately designed adaptive neural controller, the disturbances are attenuated and the system output tracks a periodic orbit in finite time. As radial basis function (RBF) neural networks (NN) are employed, this leads to the satisfaction of a partial persistence of the excitation (PE) condition. By using the uniform complete observability (UCO) technique, it is analyzed that partial estimated NN weights will converge to a neighborhood of zero, with the size of the neighborhood depending on the amplitude of disturbances as well as on the control gains. Locally-accurate approximation of unknown system dynamics can still be achieved in the stable NN control process. The approximation error level is influenced by the amplitude of disturbances. The obtained knowledge of system dynamics can be reused in another control process towards stability and improved performance. Simulation studies are included to demonstrate the effectiveness of the approach.