Attractive Ellipsoid Method Research Articles

Suboptimal adaptive control of dynamic systems with state constraints based on Barrier Lyapunov functions

This study designed a suboptimal output control strategy to characterise an attractive and invariant set for the state trajectories of perturbed linear systems with noisy measurements and state constraints. The state constraints were defined by a given polytope formed of by n-dimensional vectors. An adaptive linear controller enforced the existence of an attractive and invariant set (centred at the origin) for the trajectories of the perturbed system. A barrier Lyapunov function (BLF) and the attractive ellipsoid method (AEM) derived the adjustment law of the adaptive gain. The controller design used the linear matrix inequality technique to solve two optimisation problems. The first solution provided a maximal set where the initial conditions must belong without violating the state constraints. The second optimisation solution characterised the invariant minimal attractive set for the system trajectories. An academic example verified how the proposed adaptive control generated the system trajectories that converged to the minimal attractive ellipsoid while keeping them inside the polytope defining the state constraints. The simulation script showed the advantages of the adaptive BLF controller (ABLC) against classical AEM controller. A second numerical example considered a direct current motor showing the advantages of the ABLC against the sliding mode technique.

Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part I: Continuous-time Systems

Abstract This paper develops sufficient conditions for the constrained robust stabilization of continuous-time polytopic linear systems with unknown but bounded perturbations. The attractive ellipsoid method (AEM) is employed to determine a robustly controllable invariant set, known as attractive ellipsoid, such that the state trajectories of the system asymptotically converge to a small neighborhood of the origin despite the presence of non-vanishing perturbations. To solve the stabilization problem, we employ the Finsler’s lemma and derive new linear matrix inequality (LMI) conditions for robust state-feedback control design, ensuring convergence of state trajectories of the system to a minimal size ellipsoidal set. We also consider the state and control constrained problem and derive extended LMI conditions. Under certain conditions, the obtained LMIs guarantee that the attractive ellipsoid is nested inside the bigger ellipsoids imposed by the control and state constraints. Finally, we extend our AEM approach to the gain-scheduled state-feedback control problem, where the scheduling parameters governing the time-variant system are unknown in advance but can be measured in real-time. Two examples demonstrate the feasibility of the proposed AEM and its improvements over previous works.

Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part II: Discrete-time Systems

Abstract This paper presents sufficient conditions for the robust stabilization of discrete-time polytopic systems subject to control constraints and unknown but bounded perturbations. The attractive ellipsoid method (AEM) is extended and applied to cope with this problem. To tackle the stabilization problem, new linear matrix inequality (LMI) conditions for robust state-feedback control are developed. These conditions ensure the convergence of state trajectories of the system to a minimal size ellipsoidal set despite the presence of non-vanishing disturbances. The developed LMI conditions for the AEM are extended to deal with the problem of gain-scheduled state-feedback control, where the scheduling parameters governing the time-variant dynamical system are unknown in advance but can be measured in real-time. A feature of the obtained conditions is that the state-space matrices and Lyapunov matrix are separated. The desired robust control laws are obtained by convex optimization. Numerical simulations are given to illustrate the feasibility of the proposed AEM for robust disturbance rejection.

Robust Stabilizing Control for the Electromechanical Triple-Link Inverted Pendulum System

This paper presents the results of the survey on the stabilization of the Triple-Link Inverted Pendulum as underactuated system. The mathematical model of the coupled electromechanical system is presented in the standard robotics format, where the geometry of its links is taken into account, further, the DC motor is considered. For the pursued aim, Attractive Ellipsoid Method is used in order to design robust stabilizing controller for electromechanical systems in a general form. In order to compare the performance for the suggested methodology, different robust Sliding Mode Control strategies are designed. The obtained algorithms are tested in numerical simulation.

Attractive ellipsoid method controller under noised measurements for SLAM

This paper presents a novel algorithm for Simultaneous Localization and Mapping (SLAM) of mobile robots. The proposed algorithm, termed as SLAM using on Attractive Ellipsoid Method (AEM) and Luenberger filter-type application with nonlinear models, under the hypothesis that errors affecting all sensor’s measurements and robot’s motions are unknown-but-bounded. We suggest to select the best parameters of the suggested observer providing a minimal size of this attractive ellipsoid. It is shown that this optimization problem may be converted into the trace optimization of this attractive ellipsoid under some specific constraints of a set of LMI’s (Linear Matrix Inequalities). Simulated indoor experiments are used to demonstrate the performance of the proposed algorithm.

Quasi-minimal active disturbance rejection control of MIMO perturbed linear systems based on differential neural networks and the attractive ellipsoid method

This study addresses the problem of designing an output-based controller to stabilize multi-input multi-output (MIMO) systems in the presence of parametric disturbances as well as uncertainties in the state model and output noise measurements. The controller design includes a linear state transformation which separates uncertainties matched to the control input and the unmatched ones. A differential neural network (DNN) observer produces a nonlinear approximation of the matched perturbation and the unknown states simultaneously in the transformed coordinates. This study proposes the use of the Attractive Ellipsoid Method (AEM) to optimize the gains of the controller and the gain observer in the DNN structure. As a consequence, the obtained control input minimizes the convergence zone for the estimation error. Moreover, the control design uses the estimated disturbance provided by the DNN to obtain a better performance in the stabilization task in comparison with a quasi-minimal output feedback controller based on a Luenberger observer and a sliding mode controller. Numerical results pointed out the advantages obtained by the nonlinear control based on the DNN observer. The first example deals with the stabilization of an academic linear MIMO perturbed system and the second example stabilizes the trajectories of a DC-motor into a predefined operation point.

Robust Model-Free Software Sensors for the HIV/AIDS Infection Process

This paper considers the problem of the HIV/AIDS Infection Process filtering characterized by three compounds, namely, the number of healthy T-cells, the number of infected T-cells and free virus particles. Only the first and third of them can be measurable during the medical treatment process. Moreover, the exact parameter values are admitted to be also unknown. So, here we deal with an uncertain dynamic model that excludes the application of classical filtering theory and requires the application of robust filters successfully working in the absence of a complete mathematical model of the considered process. The problem is to estimate the number of infected T-cells based on the available information. Here we admit the presence of stochastic “white noise” in current observations. To do that we apply the Luenberger-like filter (software sensor) with a matrix gain, which should be adjusted at the beginning of the process in such a way that the filtering error would be as less as possible using the Attractive Ellipsoid Method (AEM). It is shown that the corresponding trajectories of the filtering error converge to an ellipsoidal set of a prespecified form in mean-square sense. To generate the experimental data sequences in the test-simulation example, we have used the well-known simplified HIV/ AIDS model. The obtained results confirm the effectiveness of the suggested approach.

Robust control of linear systems under input saturation using Barrier Lyapunov functions

This paper deals with the robust stabilization of input saturated uncertain linear systems with perturbed measurements. The controller designed as a linear output feedback allowed to characterize the region of attraction where the initial conditions must belong in order to guarantee the asymptotic convergence of the system trajectories into a ultimately bounded region around the origin. The stability analysis employed the concept of Barrier Lyapunov functions (BLF) and the attractive ellipsoid method (AEM) to find sufficient conditions in terms of linear matrix inequalities characterizing the ultimate bounded region of attraction. The methodology proposed is to use the BLF to estimate the region of attraction, while the AEM successfully characterized the ultimately bound. And optimization procedure maximized the set estimating (in some sense) the region of attraction and minimized the ultimate bounded set. Simulation results showed the accuracy in the estimation of the region of attraction and the minimal ellipsoid characterization.

Robust Control for State Constrained Uncertain Systems: Attractive Ellipsoid Method Approach

Abstract: A robust control feedback strategy is developed to solve the stabilization problem of constrained systems with uncertainties and output perturbations. The states are assumed to be constrained inside a given polytope and the perturbations bounded. The control law is developed using an extended version of the attractive ellipsoid method (AEM) approach, and a barrier Lyapunov function (BLF); this is a function whose value goes to infinity whenever its arguments approach to the boundary of a given set. The control parameters are obtained through the solution of some optimization problems related to the approximation of the constraints set and the characterization of a minimal ultimate bounded set for the system trajectories. The implementability of the resulting algorithm is supported by a numerical example and by the comparison with the regular AEM based on a quadratic Lyapunov function.

Adaptive-Robust Stabilization of the Furuta's Pendulum Via Attractive Ellipsoid Method

This paper focuses on the issue of adaptive-robust stabilization of the Furuta's pendulum around unstable equilibrium where the dynamical model is unknown. The control scheme lies at the lack of the dynamical model as well as external disturbances. The stabilization analysis is based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities. Even more, a modification of the AEM concept that permits to use online information obtained during the process is suggested here. This adjustment (or adaptation) is made only in some fixed sample times, so that the corresponding gain matrix of the robust controller is given on time interval too. Furthermore, under a specific “regularized persistent excitation condition,” the proposed method guarantees that the controlled system trajectories remain inside an ellipsoid of a minimal size (the minimal size is refereed to as the minimal trace of the corresponding inverse ellipsoidal matrix). Finally, the adaptive process describes a region of attraction (ROA) of the considered system under adaptive-robust nonlinear control law.

Finite-time attractive ellipsoid method: implicit Lyapunov function approach

ABSTRACTA finite-time version, based on implicit Lyapunov functions (ILFs), for the attractive ellipsoid method (AEM) is developed. Based on this, a robust control scheme is presented to ensure finite-time convergence of the solutions of a chain of integrators with bounded output perturbations to a minimal ellipsoidal set. The control parameters are obtained by solving a minimisation problem of the ‘size’ of the ellipsoid subject to a set of linear matrix inequalities (LMIs) constraints, and by applying the implicit function theorem. A numerical example is presented to support the implementability of these theoretical results.

Robust output stabilization for a class of nonlinear uncertain stochastic systems under multiplicative and additive noises: The attractive ellipsoid method

This work concerns the robust stabilization of a class of ``Quasi-Lipschitz'nonlinear uncertain systems governed by stochastic Differential Equations(SDE) subject to both multiplicative and additive stochastic noises modeledby a vector Brownian motion. The state-vector is admitted to benon-completely available, and be estimated by a Luenberger-type filter. Thestabilization around the origin is realized by a linear feedbackproportional to the current state-estimates. First, the class of feedbackmatrices and filter matrix-gains, providing the boundedness of thestochastic trajectories with probability one in a vicinity of the origin, isspecified. Then a corresponding ellipsoid, containing these trajectories, isfound. Its ``size' (the trace of the ellipsoid matrix) is derived as afunction of the applied gain matrices. To make this ellipsoid ``as small aspossible' the corresponding constrained optimization problem is suggested tobe solved. These constraints are given by a system of Matrix Inequalities(MI's) which under a specific change of variables may be converted into aconventional system of Bilinear Matrix Inequalities (BMI's). The last may beresolved by the standard MATLAB toolboxes such as ``penbmiTL, Tomlabtoolbox'. Finally, a numerical example, containing the arctangent-typenonlinearities, is presented to illustrate the effectiveness of thesuggested methodology.

PID Based on Attractive Ellipsoid Method for Dynamic Uncertain and External Disturbances Rejection in Mechanical Systems

This paper presents a stability analysis for LNDS (Lagrangian nonlinear dynamical systems) with dynamic uncertain using a PID controller with external disturbances rejection based on attractive ellipsoid methods, since the PID-CT (proportional integral derivative computed torque) compensator has been used for the nonlinear trajectory tracking of an LNDS, when there are external perturbations and system uncertainties. The global system convergence of the trivial solution has not been proved. In this sense, we propose an approach to find the gains of the nonlinear PID-CT controller to guarantee the boundedness of the trivial solution by means of the concept of the UUB (uniform-ultimately bounded) stability. In order to show the effectiveness of the methodology proposed, we applied it in a real 2-DoF robot system.

Switching robust control for ozone generators using the attractive ellipsoid method

This paper deals with a switching robust tracking feedback design for a corona-effect ozone generator. The generator is considered as a switched systems in the presence of bounded model uncertainties as well as external perturbations. Three nonlinear dynamic models under arbitrary switching mechanisms are considered assuming that a sample-switching times are known. The stabilization issue is achieved in the sense of a practical stability. We apply the newly elaborated (extended) version of the conventional attractive ellipsoid method (AEM) for this purpose. The same analysis was efficient to obtain the minimal size of region where the tracking error between the trajectories of the ozone generator and reference states converges. The numerically implementable sufficient conditions for the practical stability of systems are derived based on bilinear matrix inequalities (BMIs).

Quantised and sampled output feedback for nonlinear systems

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method.

‘KL’-gain adaptation for attractive ellipsoid method

‘KL’-gain adaptation for attractive ellipsoid method

On the practical stability of control processes governed by implicit differential equations: The invariant ellipsoid based approach

This paper deals with a new approach to the robust control design for a class of nonlinearly affine control systems. The dynamical models under consideration are described by implicit differential equations in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique. We call this extension the Attractive Ellipsoid (AE) method. The stability/robustness analysis of the resulting closed-loop system involves a modified descriptor approach associated with the usual Lyapunov-type methodology. The theoretic schemes elaborated in our contribution are finally illustrated by a simple computational example.

Practical Stability of Control Processes Governed by Semiexplicit DAEs

This paper deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamical models under consideration are described by a special class of structured implicit differential equations called semi-explicit differential-algebraic equations (of index one), in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique that we call the Attractive Ellipsoid (AE) method. The theoretic schemes elaborated in our contribution are illustrated by a simple computational example.

Drilling vibration reduction via attractive ellipsoid method

The oscillations occurring during the drilling process constitute a major problem to solve in the oil extraction industry. Drilling vibrations most frequently occurring in the field are torsional (stick-slip) and axial (bit-bounce). They may cause equipment failures which increase drilling operation costs. An important challenge to overcome is the modeling of drillstring dynamics and the design of effective controllers to reduce undesirable behavior. This paper introduces a neutral-type time-delay model that effectively describes the oscillatory behavior in the drilling system. It simplifies significantly the system analysis and simulation. The attractive ellipsoid method allows us develop an efficient strategy for the stabilizing controllers synthesis guaranteeing the elimination of drilling oscillations. In fact, this methodology can be applied to any system described by a time-delay equation subject to bounded disturbances. The effectiveness of this approach is validated through simulations.

Robust stochastic tracking for discrete-time models: designing of ellipsoid where random trajectories converge with probability one

We study the behaviour of stochastic discrete-time models controlled by an output linear feedback during a tracking process. The controlled system is assumed to be nonlinear satisfying the global ‘quasi-Lipschitz’ condition and subjected to stochastic input and output disturbances. Two gain matrices (in a feedback and in an observer) define an ‘averaged’ ellipsoid in the tracking-error space where all system's trajectories arrive ‘with probability one’. The selection of the ‘best’ gain matrices is realised numerically by application of the robust attractive ellipsoid method with the linear matrix inequality technique application. The suggested approach is illustrated by designing of a robust tracking controller for a benchmark example in the presence of stochastic noises both in the state dynamics and in the output observations.