Globally Exponentially Stable Research Articles

A Novel Globally Exponentially Stable Observer for Sensorless Control of the IPMSM via Kreisselmeier's Extension

In a recent paper Ortega et al. (2021a) the authors proposed the first solution to the problem of designing a globally exponentially stable (GES) flux observer for the interior permanent magnet synchronous motor. However, the establishment of the stability proof relies on the assumption that the adaptation gain is sufficiently small—a condition that may degrade the quality of transient behavior. In this paper we propose a new GES flux observer that overcomes this limitation ensuring a high performance behavior. The design relies on the use of a novel theoretical tool—the generation of a “virtual” invariant manifold—that allows the use of the more advanced Kreisselmeier's regression extension estimator, instead of a simple gradient descent one. We illustrate its superior transient behavior via simulations.

A Stabilizing Model Predictive Control Scheme With Arbitrary Prediction Horizon for Switched Linear Systems

This letter deals with the design and analysis of a stabilizing model predictive control (MPC) scheme for discrete-time switched linear systems with at least one stabilizable subsystem. An approach for pruning the switching sequences is proposed which is used for reducing the online computation and ensuring stability of the MPC scheme. With the pruning approach, the value function of the MPC scheme becomes an exponential Lyapunov function (ELF). Consequently, the MPC scheme results in globally exponentially stable (GES) closed-loop dynamics for switched linear systems for all prediction horizons. Feasibility of the MPC scheme is discussed and simulation results are given for illustrating the effectiveness of the MPC scheme.

Dynamic event-triggered control of networked control systems with uncertainty and transmission delay

This paper presents a method for designing dynamic event-triggered controller of networked control systems (NCSs) with uncertainty and time delays. Under the condition that the Lyapunov function of the system is allowed to increase at each jump point, the globally exponentially stable (GES) of the system can be achieved by using the Riccati differential equation and the principle of average dwell time (ADT). The minimum allowable inter-event interval is obtained by limiting the increment of the Lyapunov function within the transmission interval. Both the static event triggering and no transmission delay are included in the designed dynamic event triggering mechanism as special cases. A numerical example is given to verify the correctness and validity of the proposed method.

Global exponential angular velocity estimation of rigid-body spacecraft from quaternion and vector measurements

Two observer-based angular velocity estimation approaches for rigid-body spacecraft are presented in this paper, where quaternion and direct vector measurements are respectively adopted as available outputs. By transforming the estimation problem into the one in appropriate coordinate, the proposed observers are free from high-gain injection while being globally exponentially stable (GES) in the meantime. This unique feature is not possible in existing GES designs that are based on quaternion or vector measurements, and significantly improved performance under measurement noises are witnessed in numerical examples when compared with corresponding high-gain-based counterparts. Results in this paper also provide the first answer to the open problem on constructing truly GES angular velocity observer from vector measurements.

Dual-terminal decentralized event-triggered control for switched systems with cyber attacks and quantization

This article concerns dual-terminal event-triggered communication and decentralized control of switched systems that are the target of cyber attacks. Due to different properties of physical system, this system uses decentralized communication channels to transmit feedback data. In order to make efficient use of communication resources in each channel, the information sent by the sub-system needs to meet the given event-triggering conditions before it can be released. Moreover, the quantization is employed in both sides of the controller to further improve the data transmission efficiency. Then, considering that the triggered and quantified data are affected by dual-terminal cyber attacks, the event-triggered closed-loop switched (CLS) systems under attacks are derived. Furthermore, by utilizing average dwell time (ADT) technique and piecewise Lyapunov function (LF) method, sufficient conditions are given to ensure that the event-triggered CLS systems subject to dual-terminal cyber attacks are globally exponentially stable (GES). Accordingly, the design conditions for the gains of event-triggered dynamic output feedback (DOF) controllers and the parameters of decentralized event-triggering mechanisms (DETMs) are presented. Finally, simulations for verifying the system stability with and without cyber attacks are given.

Uniformly semiglobally exponential stability of vector field guidance law and autopilot for path-following

A uniform semiglobal exponential stability (USGES) proof for a time-varying vector field guidance law used for path-following control of vehicles is presented. A sliding mode control is introduced for heading autopilot design and Lyapunov methods are used to derive the control law. The equilibrium point of the autopilot error dynamics is proven to be globally exponentially stable (GES). The main result is a time-varying vector field guidance law in cascade with the autopilot. A theorem ensures that the equilibrium point of the cascaded system is uniformly semiglobally exponentially stable. Both straight-line and curved-path path following scenarios are considered in the presence of ocean currents. Simulation studies are carried out to verify the theoretical results. The time-varying guidance laws can also be applied to vehicles in general such as aircraft, underwater vehicles, drones and autonomous vehicles.

Kalman Filters for Air Data System Bias Correction for a Fixed-Wing UAV

This paper presents two Kalman filter approaches for correcting air data systems providing relative velocity measurements with an additive constant or slowly time-varying bias for fixed-wing unmanned aerial vehicles (UAVs). In addition to the air data system, both estimators rely on a standard sensor suite consisting of a GNSS receiver, an IMU, and a heading reference. Furthermore, the estimators are based on kinematics and do not require a model of the UAV. The two estimators are in the noise-free case proven to have globally exponentially stable (GES) equilibrium points of the error dynamics if provided with persistence of excitation (PE) of the angular velocity of the UAV. The estimators are verified through simulation and using experimental flight data. The relative velocity measurements in the experimental flight results are provided by an array of pressure sensors embedded in the surface of the UAV combined with a neural network algorithm. The results indicate that a certain amount of PE is needed in order to have converging bias estimates for turbulent wind conditions.

A globally exponentially stable filter for bearing-only simultaneous localization and mapping with monocular vision

Abstract This paper proposes a novel filter for sensor-based bearing-only simultaneous localization and mapping in three dimensions with globally exponentially stable (GES) error dynamics. A nonlinear system is designed, its output transformed, and its dynamics augmented so that the proposed formulation can be considered as linear time-varying for the purpose of observability analysis. This allows the establishment of observability results related to the original nonlinear system that naturally lead to the design of a Kalman filter with GES error dynamics. The performance of the proposed algorithm is assessed resorting to real experiments based on the Rawseeds dataset as well as further realistic simulations.

Design of a New Nonlinear Observer of Power Synchronous Generator Damper Currents

In this paper, a new nonlinear observer for the synchronous machine damper currents is developed. The observer design is based on the theory of Lyapunov of nonlinear systems. The stability of the proposed observer is shown to be Globally Exponentially Stable (GES) in the context of Lyapunov theory. The paper also describes how to design the observer gain, and includes computer simulation results to demonstrate the effectiveness of the proposed observer.

Lyapunov functions for discontinuous difference inclusions

Abstract: In this work we introduce the notion of finite-step Lyapunov functions for difference inclusions. First, we prove that the existence of a finite-step Lyapunov function is sufficient and necessary for the difference inclusion being KL -stable. Second, we establish four ways of constructing a Lyapunov function from a finite-step Lyapunov function. In contrast to Lyapunov functions obtained via converse Lyapunov theorems, the proposed Lyapunov functions do only depend on a finite sum resp. a maximum over finitely many terms. This crucial property allows to construct Lyapunov functions in principle, and to easily derive structure and regularity properties of the Lyapunov function obtained. Another advantage is that for all globally exponentially stable (GES) difference inclusions, norms are finite-step Lyapunov functions which yields a systematic approach for obtaining Lyapunov functions. Whereas all these finding are shown to hold also for discontinuous difference inclusion, we do also discuss robustness of the difference inclusion via upper semicontinuous Lyapunov functions.

Sensor-based globally exponentially stable range-only simultaneous localization and mapping

This paper proposes the design, analysis, and validation of a globally exponentially stable (GES) filter for tridimensional (3-D) range-only simultaneous localization and mapping. For observability analysis purposes, a nonlinear sensor-based dynamical system is formulated resorting only to exact linear and angular kinematics and a state augmentation is exploited that allows the proposed formulation to be considered as linear time-varying without linearizing the original nonlinear system. Constructive observability results can then be established, leading naturally to the design of a Kalman Filter with GES error dynamics. These results also provide valuable insight on the motion planning of the vehicle. Experimental results demonstrate the good performance of the algorithm and help validate the theoretical results presented. For completeness and to illustrate the necessity of a proper trajectory, simulation data are included as well.

Long baseline navigation with clock offset estimation and discrete-time measurements

This paper proposes a novel one-way-travel-time long baseline filtering system that includes the estimation of the clocks’ offset, which is assumed constant. Considering discrete-time pseudo-range measurements, in addition to the data provided by a Doppler velocity log and an attitude and heading reference system, an augmented system is derived that can be regarded as linear for observability and observer design purposes. Its observability is discussed and a Kalman filter provides the estimation solution, with globally exponentially stable (GES) error dynamics. Simulation results are presented to evaluate the proposed solution, which is also compared with the EKF, including Monte Carlo runs.

An alternative converse Lyapunov theorem for discrete-time systems

This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and for globally asymptotically stable systems, Lyapunov functions on a set [a,b] with 0<a<b<∞ are derived. Furthermore, for specific classes of systems, the developed converse Lyapunov theorem can be used to establish non-conservatism of existence of a particular type of Lyapunov functions. Most notably, a proof that the existence of conewise linear Lyapunov functions is non-conservative for GES conewise linear systems is given and, as a by-product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.

Sensor‐Based Long Baseline Navigation: Observability Analysis and Filter Design

AbstractThis paper presents a novel long baseline (LBL) position and velocity navigation filter for underwater vehicles based directly on the sensor measurements. The solution departs from previous approaches as the range measurements are explicitly embedded in the filter design, therefore avoiding inversion algorithms and allowing also the consideration of the cases of reduced numbers of readings, in particular when there are only two or three distance measurements. The nonlinear system dynamics are considered to their full extent and no linearizations are carried out whatsoever. The filter error dynamics are globally exponentially stable (GES) and it is shown, in a realistic simulation environment, that the filter achieves similar performance to the extended Kalman filter (EKF) and outperforms linear position and velocity filters based on algebraic estimates of the position obtained from the range measurements.

A GES attitude observer with single vector observations

This paper proposes a globally exponentially stable (GES) observer for attitude estimation based on a single time-varying reference vector, in inertial coordinates, and corresponding vector, in body-fixed coordinates, in addition to angular velocity readings. The proposed solution is computationally efficient and, in spite of the fact that the observer does not evolve on the Special Orthogonal Group SO(3), an explicit solution on SO(3) is also provided, whose error is shown to converge exponentially fast to zero for all initial conditions. The distinct roles of the inertial and the corresponding body-fixed vectors on the observability of the system are also examined and simulation results are shown that illustrate the performance of the proposed attitude observer in the presence of low-grade sensor specifications.

Stabilizing Nonlinear Control Using Damper Currents Sliding Mode Observer of Synchronous Generator

In this article, a new observer-based nonlinear controller for excitation control of synchronous generator is presented. First, a nonlinear sliding mode observer for the synchronous machine damper currents is developed. Second, the existence of a control law for the complete seventh order model of a generator, which takes into account the stator dynamics as well as the damper effects, is proved by using the theory of Lyapunov. The stabilizing feedback law for the power system is shown to be Globally Exponentially Stable (GES) in the context of Lyapunov theory. Simulation results, for a single-Machine-Infinite-Bus (SMIB) power system, are given to demonstrate the effectiveness of the proposed combined obsever-controller for the transient stabilization and voltage regulation.

GES Attitude Observers – Part I: Multiple General Vector Observations

AbstractThis paper proposes a set of observers for attitude estimation based on multiple general time-varying reference vectors, in inertial coordinates, and corresponding vectors, in body-fixed coordinates, in addition to angular velocity readings. The observer error dynamics are globally exponentially stable (GES) and problems such as singularities, unwinding phenomena, or topological limitations for achieving global asymptotic stability are absent. Although the observer estimates do not evolve on the Special Orthogonal Group SO(3), a final solution is provided, on SO(3), whose error is shown to converge to zero for all initial conditions. The proposed solutions are computationally efficient and simulation results are shown that illustrate the achievable performance.

Position and Velocity Navigation Systems for Unmanned Vehicles

This brief presents a new observer synthesis methodology for a class of kinematic systems with application to the estimation of linear motion quantities of mobile platforms (position and linear velocity), in three dimensions, that: 1) presents globally exponentially stable (GES) error dynamics, which are also input-to-state stable (ISS) with respect to angular quantities; 2) minimizes the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> induced norm from a generalized disturbance input to a performance variable; 3) provides a systematic design procedure, based upon robust control theory results, that allows for the use of frequency weights to shape the dynamic response of the observer. A practical application is presented, in the field of ocean robotics, that demonstrates the potential and usefulness of the proposed design methodology and simulation results are included that illustrate the observer achievable performance in the presence of both extreme environmental disturbances and realistic sensors' noise.

Global exponential stability of discrete-time neural networks for constrained quadratic optimization

A class of discrete-time recurrent neural networks for solving quadratic optimization problems over bound constraints is studied. The regularity and completeness of the network are discussed. The network is proven to be globally exponentially stable (GES) under some mild conditions. The analysis of GES extends the existing stability results for discrete-time recurrent networks. A simulation example is included to validate the theoretical results obtained in this letter.

A remark on "Nonlinear output feedback control of underwater vehicle propellers using feedback form estimated axial flow velocity"'

For original paper see T.I.Fossen and M.Blanke, ibid., vol.25, pp.241-55 (2000). In the work presented by Fossen and Blanke, a nonlinear observer for estimation of propeller axial flow velocity for UUVs was introduced. The proof of the convergence behavior of the observer was carried out with a Lyapunov analysis. In this technical communication, such an analysis was also briefly compared with the so-called contraction analysis. This communication goes further into the details of the comparison by proving - through a link between contracting and globally exponentially stable (GES) systems - that the observer is both contracting and GES, showing thus that contraction analysis might be regarded as an interesting alternative to Lyapunov-based designs. Furthermore, some qualitative advantages gained when using contraction analysis are given.