Practical Asymptotic Stability Research Articles

Uniform Practical Asymptotic Stability for Position Control of Underwater Snake Robots

Uniform Practical Asymptotic Stability for Position Control of Underwater Snake Robots

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

Overcoming local extrema in torque-actuated source seeking using the divergence theorem and delay

We study the problem of seeking the source of a scalar signal with a force- and torque-controlled unicycle. The vehicle does not have the capability to measure or track its position. The only information about the vehicle’s position relative to the source is given by measurements of the signal. We propose a feedback law that induces a uniform forward motion and an almost uniform rotational motion. This in turn leads to an almost circular motion of the unicycle; i.e., a motion along the boundary of a disk. An application of the divergence theorem reveals that the measurements of the signal along the boundary of the disk provide access to the gradient of an averaged signal, where the average is taken over the entire disk. Such a local average of the sensed signal can eliminate undesired local extrema, which in turn is beneficial for the goal of reaching a global extremum. We prove practical asymptotic stability for the closed-loop system with respect to extrema of the averaged signal. Our theoretical analysis takes L∞-measurement errors of the signal into account. We compare the performance of the proposed control law to another source seeking method in the presence of local extrema through numerical simulations.

Visual Observer-Based State-Feedback Autonomous Motion Control on SE(3)

This paper introduces a visual observer-based state-feedback control tailored for autonomous motion on SE(3) to address challenges posed by modeling uncertainties and measurement noise. The proposed approach unfolds in two fundamental phases. In the initial stage, the visual observer, based on modeling information, visual sensors, and pre-deployed landmarks, along with the state-feedback controller, are developed independently, underpinning their individual semi-global practical asymptotic (SPA) stability. This modular approach ensures a robust foundation for the subsequent synthesis. The latter stage applies the well-established Small Gain Principle to regulate observer and controller parameters to guarantee the SPA stability of the closed-loop system with the visual-observer based state feedback control. The effectiveness of the proposed method is validated through simulation and experiments.

Practical asymptotic stability for time-varying nonlinear systems

This paper is concerned with the global uniform practical asymptotic and exponential stability for a large class of time-varying systems by using the concept of practical scalar functions. A new converse stability theorem is established which is used to give some sufficient conditions that guarantee the global uniform practical asymptotic and exponential stability of a class of perturbed systems. Furthermore, some examples are presented.

Linearly discounted economic MPC without terminal conditions for periodic optimal operation

In this work, we study economic model predictive control (MPC) in situations where the optimal operating behavior is periodic. In such a setting, the performance of a standard economic MPC scheme without terminal conditions can generally be far from optimal even with arbitrarily long prediction horizons. Whereas there are modified economic MPC schemes that guarantee optimal performance, all of them are based on prior knowledge of the optimal period length or of the optimal periodic orbit itself. In contrast to these approaches, we propose to achieve optimality by multiplying the stage cost by a linear discount factor. This modification is not only easy to implement but also independent of any system- or cost-specific properties, making the scheme robust against online changes therein. Under standard dissipativity and controllability assumptions, we can prove that the resulting linearly discounted economic MPC without terminal conditions achieves optimal asymptotic average performance up to an error that vanishes with growing prediction horizons. Moreover, we can guarantee practical asymptotic stability of the optimal periodic orbit under the additional technical assumption that dissipativity holds with a continuous storage function. We complement these qualitative guarantees with a quantitative analysis of the transient and asymptotic average performance of the linearly discounted MPC scheme in a numerical simulation study.

A distributed strategy for games in Euler–Lagrange systems with actuator dead zone

This paper explores a distributed Nash equilibrium (NE) seeking problem for games in which the involved players are of Euler–Lagrange (EL) dynamics with actuator dead zone. To find NE of such games, a novel distributed algorithm with a dynamical dead-zone compensator is proposed. The compensating module in this new strategy is regarded as a fast dynamics that rapidly addresses the effect caused by actuator dead-zone characteristic. The other module of this algorithm is a slow optimizer that enables the actions of EL players to approach the NE. Since the designed strategy is a two-time-scale model, semi-global practical asymptotic stability of this strategy can be obtained by utilizing the generalized singular perturbation method. Finally, an electricity market game with six turbine-generator dynamics is utilized to validate the theoretical result of the proposed method.

Practical Input-to-State Stability of Nonlinear Systems with Time Delay

An investigation of the input-to-state practical stability (ISpS) and the integral input-to-state practical stability (iISpS) of nonlinear systems with time delays (NSWTDs) is presented in this paper. The ISpS and iISpS of the systems are obtained by using a continuously differentiable Lyapunov–Krasovskii functional (LKF) within-definite derivative, which generalizes the classical LKF with a positive definite derivative. As a result, the uniform practical asymptotic stability (UPAS) criteria of NSWTD were also given by using the proposed LKF.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

A generalization of practical stability of nonlinear impulsive systems

Abstract In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

Source Seeking With a Torque-Controlled Unicycle

We propose a novel source seeking method for a torque-controlled unicycle. Our control law leads to a positive constant forward velocity and an oscillatory rotational velocity with non-zero mean. Under suitable assumptions, we prove that the unicycle tends to a circular motion around the source. An implementation of the proposed method only requires real-time measurements of the unknown scalar signal. The feedback law for the torque involves a periodic large-amplitude high-frequency perturbation signal. A suitable averaging argument for mechanical systems under vibrational control reveals that the closed-loop system approximates the behavior of a certain averaged system. The motion of the averaged system is determined by a symmetric product of a vector field from the approximating closed-loop system. This symmetric product causes a torque which is given by the gradient of the unknown scalar signal. The gradient-based torque can lead to asymptotic stability for the averaged system, which in turn implies practical asymptotic stability for the approximating closed-loop system.

Nonlinear Adaptive Observers for an SIS System Counting Primo-infections

Observation and identification are important issues for the practical use of compartmental models of epidemic dynamics. Usually, the state and parameters of the epidemic model are evaluated based on the number of infected individuals (the prevalence) or the newly infected cases (the incidence). Other estimation techniques, for example, based on the exploitation of the proportion of primo-infected individuals (easily retrievable data), are rarely considered. We are thus interested in a general question: may the measure of the number of primo-infected individuals and the prevalence improve simultaneous state and parameter estimation? In this paper, we design a nonlinear adaptive observer for a simple infection model with waning immunity and consequent reinfections to answer this question. The practical asymptotic stability of the estimation errors is then proved using the Lyapunov function method. Finally, the convergence of the observer is illustrated in simulations.

Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.

Practical Stability Analysis and Switching Controller Synthesis for Discrete-time Switched Affine Systems via Linear Matrix Inequalities

This paper considers the practical asymptotic stabilization of adesired equilibrium point in discrete-time switched affine systems.The main purpose is to design a state feedback switching rule forthe discrete-time switched affine systems whose parameters can beextracted with less computational complexities. In this regard,using switched Lyapunov functions, a new set of sufficientconditions based on matrix inequalities are developed to solve thepractical stabilization problem. For any size of the switched affinesystem, the derived matrix inequalities contain only one bilinearterm as a multiplication of a positive scalar and a positivedefinite matrix. It is shown that the practical stabilizationproblem can be solved via a few convex optimization problems,including Linear Matrix Inequalities (LMIs) through gridding of ascalar variable interval between zero and one. The numericalexperiments on an academic example and a DC-DC buck-boost converter,as well as comparative studies with the existing works, prove thesatisfactory operation of the proposed method in achieving betterperformances and more tractable numerical solutions.

Safe model‐based reinforcement learning for nonlinear optimal control with state and input constraints

AbstractSafety is a critical factor in reinforcement learning (RL) in chemical processes. In our previous work, we had proposed a new stability‐guaranteed RL for unconstrained nonlinear control‐affine systems. In the approximate policy iteration algorithm, a Lyapunov neural network (LNN) was updated while being restricted to the control Lyapunov function, and a policy was updated using a variation of Sontag's formula. In this study, we additionally consider state and input constraints by introducing a barrier function, and we extend the applicable type to general nonlinear systems. We augment the constraints into the objective function and use the LNN added with a Lyapunov barrier function to approximate the augmented value function. Sontag's formula input with this approximate function brings the states into its lower level set, thereby guaranteeing the constraints satisfaction and stability. We prove the practical asymptotic stability and forward invariance. The effectiveness is validated using four tank system simulations.

Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks

This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results

Practical partial stability of time-varying systems

<p style='text-indent:20px;'>In this paper we investigate the practical asymptotic and exponential partial stability of time-varying nonlinear systems. We derive some sufficient conditions that guarantee practical partial stability of perturbed systems using Lyapunov's theory where a converse theorem is presented. Therefore, we generalize some works which are already made in the literature. Furthermore, we present some illustrative examples to verify the effectiveness of the proposed methods.</p>

On the practical output h-stabilization of nonlinear uncertain systems

This paper investigates the problem of output feedback $h$-stabilization of nonlinear uncertain systems. We construct an output feedback controller that guarantees global uniform practical $h$-stability of the closed-loop system. Our original results generalize well-known fundamental results: practical stability, practical asymptotic stability and practical exponential stability for nonlinear time-varying systems. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.

Optimal Bounded Ellipsoid Identification With Deterministic and Bounded Learning Gains: Design and Application to Euler-Lagrange Systems.

This article proposes an effective optimal bounded ellipsoid (OBE) identification algorithm for neural networks to reconstruct the dynamics of the uncertain Euler-Lagrange systems. To address the problem of unbounded growth or vanishing of the learning gain matrix in classical OBE algorithms, we propose a modified OBE algorithm to ensure that the learning gain matrix has deterministic upper and lower bounds (i.e., the bounds are independent of the unpredictable excitation levels in different regressor channels and, therefore, are capable of being predetermined a priori). Such properties are generally unavailable in the existing OBE algorithms. The upper bound prevents blow-up in cases of insufficient excitations, and the lower bound ensures good identification performance for time-varying parameters. Based on the proposed OBE identification algorithm, we developed a closed-loop controller for the Euler-Lagrange system and proved the practical asymptotic stability of the closed-loop system via the Lyapunov stability theory. Furthermore, we showed that inertial matrix inversion and noisy acceleration signals are not required in the controller. Comparative studies confirmed the validity of the proposed approach.

Hybrid Adaptive Control for the DC-DC Boost Converter

Abstract In this paper, we consider the problem of practically asymptotically stabilizing the DC-DC boost converter under parameter uncertainty. In particular, we propose an estimation algorithm that identifies the input voltage and output load of the converter in finite time. Using these estimates, we design a control algorithm that “unites” global and local control schemes. The global control scheme induces practical asymptotic stability of a desired output voltage and corresponding current, and the local control scheme maintains industry-standard PWM behavior during steady state. Stability properties for the resulting hybrid closed-loop system are established and simulation results illustrating the main results are provided.