Local Exponential Convergence Research Articles

Exponential Convergence of Primal-Dual Dynamics Under General Conditions and Its Application to Distributed Optimization.

In this article, we establish the local and global exponential convergence of a primal-dual dynamics (PDD) for solving equality-constrained optimization problems without strong convexity and full row rank assumption on the equality constraint matrix. Under the metric subregularity of Karush-Kuhn-Tucker (KKT) mapping, we prove the local exponential convergence of the dynamics. Moreover, we establish the global exponential convergence of the dynamics in an invariant subspace under a technically designed condition which is weaker than strong convexity. As an application, the obtained theoretical results are used to show the exponential convergence of several existing state-of-the-art primal-dual algorithms for solving distributed optimization without strong convexity. Finally, we provide some experiments to demonstrate the effectiveness of our results.

Automatic reinforcement for robust model‐free neurocontrol of robots without persistent excitation

SummaryModel‐based adaptive control suffers over parametrization from the many adaptive parameters compared to the order of system dynamics, leading to sluggish tracking with a poor adaptation transient without robustness. Likewise, adaptive model‐free neurocontrol that relies on the Stone–Weierstrass theorem also suffers from similar problems in addition to over‐fitting to approximate inverse dynamics. This article proposes a novel reinforced adaptive mechanism to guarantee a transient and robustness for the model‐free adaptive control of nonlinear Lagrangian systems. Inspired by the symbiosis of Actor‐Critic (AC) architecture and integral sliding modes, the reinforced stage neural network, analogous to the critic, injects excitation signals to reinforce the parametric learning of the adaptive stage neural network, analogous to the actor to improve the approximation of inverse dynamics. The underlying integral sliding surface error drives improved learning onto a low‐dimensional invariant manifold to guarantee local exponential convergence of tracking errors. Lyapunov stability substantiates the robustness with an improved transient response. Our proposal stands for a hybrid approach between AC and neurocontrol, where the reinforced stage does not require a value function nor reward to provide automatic reinforcement to the adaptive stage parametric adaptation. Dynamic simulations are presented for a nonlinear robot manipulator under different conditions.

A Novel Actor—Critic Motor Reinforcement Learning for Continuum Soft Robots

Reinforcement learning (RL) is explored for motor control of a novel pneumatic-driven soft robot modeled after continuum media with a varying density. This model complies with closed-form Lagrangian dynamics, which fulfills the fundamental structural property of passivity, among others. Then, the question arises of how to synthesize a passivity-based RL model to control the unknown continuum soft robot dynamics to exploit its input–output energy properties advantageously throughout a reward-based neural network controller. Thus, we propose a continuous-time Actor–Critic scheme for tracking tasks of the continuum 3D soft robot subject to Lipschitz disturbances. A reward-based temporal difference leads to learning with a novel discontinuous adaptive mechanism of Critic neural weights. Finally, the reward and integral of the Bellman error approximation reinforce the adaptive mechanism of Actor neural weights. Closed-loop stability is guaranteed in the sense of Lyapunov, which leads to local exponential convergence of tracking errors based on integral sliding modes. Notably, it is assumed that dynamics are unknown, yet the control is continuous and robust. A representative simulation study shows the effectiveness of our proposal for tracking tasks.

Velocity-Aided IMU-Based Tilt and Attitude Estimation

This paper addresses the problem of estimating the tilt and more generally the attitude of a rigid body that is subject to high accelerations and equipped with inertial measurement units (IMU) and a sensor providing the body velocity (expressed in the reference frame attached to the body). In the absence of a magnetometer, tilt estimation is proposed through a two-step observer: a global-exponentially stable pre-estimation given to a manifold-constrained complementary filter. In the presence of a magnetometer, the presented observer allows to reconstruct the full attitude and tune how much the estimation of the tilt is influenced by the magnetometer, depending on the level of confidence given to the measurements of the magnetometer. All state estimators are proposed with proofs of almost global asymptotic stability and local exponential convergence. Finally, these estimators are compared with state-of-the-art solutions in clean and noisy simulations, allowing recommended solutions to be drawn for each case.

Local Stability and Convergence Analysis of Neural Network Controllers With Error Integral Inputs.

This article investigates the local stability and local convergence of a class of neural network (NN) controllers with error integrals as inputs for reference tracking. It is formally proved that if the input of the NN controller consists exclusively of error terms, the control system shows a non-zero steady-state error for any constant reference except for one specific point, for both single-layer and multi-layer NN controllers. It is further proved that adding error integrals to the input of the (single- and multi-layers) NN controller is one sufficient way to remove the steady-state error for any constant reference. Due to the nonlinearity of the NN controllers, the NN control systems are linearized at the equilibrium points. We provide proof that if all the eigenvalues of the linearized NN control system have negative real parts, local asymptotic stability and local exponential convergence are guaranteed. Two case studies were explored to verify the theoretical results: a single-layer NN controller in a 1-D system and a four-layer NN controller in a 2-D system applied to renewable energy integration. Simulations demonstrate that when NN controllers and the corresponding generalized proportional-integral (PI) controllers have the same eigenvalues, all control systems exhibit almost the same responses in a small neighborhood of their respective equilibrium points.

Entropic optimal transport: Geometry and large deviations

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of c-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.

A new Bernard–Praly-like observer for sensorless IPMSMs

A slight extension of the global Bernard–Praly gradient adaptive observer – originally presented for nonsalient-pole surface Permanent Magnet Synchronous Motors (PMSMs) and recently proved to own local exponential convergence properties under well-known observability conditions – is here designed and proposed for Interior Permanent Magnet Synchronous Motors (IPMSMs) (with relatively small saliency) in the case in which the q-axis current is constant. It guarantees local exponential estimation – from stator current and voltage measurements – of both the stator fluxes and the sinusoidal/co-sinusoidal functions of the motor electrical angle, with no modification of the previous persistency of excitation condition requirements. Such an extension complements – at least locally – a previous analysis concerning IPMSMs under constant d-axis current. Applications to the sensorless control of IPMSMs (with realistic simulations) are included.

Synchronization and pinning control of stochastic coevolving networks

Network dynamical systems are often characterized by the interlaced evolution of the node and edge dynamics, which are driven by both deterministic and stochastic factors. This manuscript offers a general mathematical model of coevolving network, which associates a state variable to each node and edge in the network, and describes their evolution through coupled stochastic differential equations. We study the emergence of synchronization, be it spontaneous or induced by a pinning control action, and provide sufficient conditions for local and global convergence. We enable the use of the Master Stability Function approach for studying coevolving networks, thereby obtaining conditions for almost sure local exponential convergence, whereas global conditions are derived using a Lyapunov-based approach. The theoretical results are then leveraged to design synchronization and pinning control protocols in two select applications. In the first one, the edge dynamics are tailored to induce spontaneous synchronization, whereas in the second the pinning edges are activated/deactivated and their weights modulated to drive the network towards the pinner’s trajectory in a distributed fashion.

Extremum Seeking for Stefan PDE with Moving Boundary and Delays

This paper presents the design and analysis of the extremum seeking for static maps with input governed by a parabolic partial differential equation (PDE) of the diffusion type defined on a time varying spatial domain described by an ordinary differential equation (ODE). This is the first effort to pursue an extension of extremum seeking from the heat PDE to the Stefan PDE. We compensate the average-based actuation dynamics by a controller via backstepping transformation for the moving boundary, which is utilized to transform the original coupled PDE-ODE into a target system whose exponential stability is proved. The local exponential convergence to a small neighborhood of the optimal point is proven by means of backstepping methodology, Lyapunov functional and averaging in infinite dimensions. The extension for the delay-compensated extremum seeking control of the Stefan problem is also discussed.

Extremum seeking boundary control for PDE–PDE cascades

The goal of this paper is to shed light on new opportunities for extremum seeking through extensions to locally quadratic nonlinear maps with actuator dynamics modeled by cascades of partial differential equations (PDEs). First, we deal with PDEs with input delays such as, for example, the notoriously difficult problem of a wave PDE with input delay where, if the delay is left uncompensated, an arbitrarily short delay destroys the closed-loop stability. Then, we move forward to an even more challenging class of problems for parabolic–hyperbolic cascades of PDEs, coping with a heat equation at the input of a wave PDE. The treatment of such systems with PDE–PDE cascades is performed by means of boundary control. The proposed approach yields (small-gain) loops that make the control design constructive and enables stability analysis with quantitative estimates. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are rigorously guaranteed. This result is achieved by using backstepping transformation and averaging in infinite dimensions. Although we keep the presentation using the classical Gradient extremum seeking, the generalization of the results for the Newton-based approach is also indicated. A numerical example is given to illustrate the effectiveness of the proposed extremum seeking boundary control for compensation of PDE–PDE cascades.

Robustness to delay mismatch in extremum seeking

In this paper, we generalize our previous results of Newton-based extremum seeking of higher derivatives via predictors with averaging-based estimates for uncertain delays. We show necessary conditions for the robustness to small delay uncertainties of the proposed extremum seeking controller with predictor feedback. Local stability and exponential convergence to a small neighborhood of the unknown extremum can still be guaranteed. A numerical simulation example is presented to illustrate the performance of the predictor-based control scheme for uncertain time-delay compensation.

An Adaptive Filter for Parameter Estimation of Damped Sinusoidal Signals

Abstract In this paper, we introduce a continuous-time adaptive algorithm for online estimation of the parameters of exponentially damped sinusoidal signals. The proposed algorithm can also reconstruct the original signal from its corrupted measurements in real-time and without extra computational cost. For sufficiently slow adaptation gains, we establish the local stability of the filter and local exponential convergence of the estimated parameters to their true values. We also obtain a linear approximation of the adaptation laws which can be utilized for tuning the adaptation gains. Simulation studies are also provided to corroborate our claims.

Lyapunov-Stable Orientation Estimator for Humanoid Robots

In this paper, we present an observation scheme, with proven Lyapunov stability, for estimating a humanoid's floating base orientation. The idea is to use velocity aided attitude estimation, which requires to know the velocity of the system. This velocity can be obtained by taking into account the kinematic data provided by contact information with the environment and using the IMU and joint encoders. We demonstrate how this operation can be used in the case of a fixed or a moving contact, allowing it to be employed for locomotion. We show how to use this velocity estimation within a selected two-stage state tilt estimator: (i) the first which has a global and quick convergence (ii) and the second which has smooth and robust dynamics. We provide new specific proofs of almost global Lyapunov asymptotic stability and local exponential convergence for this observer. Finally, we assess its performance by employing a comparative simulation and by using it within a closed-loop stabilization scheme for HRP-5P and HRP-2KAI robots performing whole-body kinematic tasks and locomotion.

Multivariable Extremum Seeking for PDE Dynamic Systems

In this article, we address the design and analysis of multivariable Newton-based extremum seeking (ES) for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations (PDEs). Multi-input systems with distinct diffusion coefficients in each individual input channel are dealt with. The phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation-based (averaging-based) estimate of the Hessian's inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point. The stability analysis is carried out using backstepping transformation and averaging in infinite dimensions, capturing the infinite-dimensional state due the diffusion PDEs. In addition, the generalization of the results for different classes of parabolic (reaction-advection-diffusion) PDEs, wave equations and/or first-order hyperbolic (transport-dominated) PDEs is also discussed. As for ordinary differential equations case, the proposed Newton approach removes the dependence of the algorithm's convergence rate on the unknown Hessian of the nonlinear map to be optimized, being user-assignable unlike the gradient algorithm. A numerical example illustrates the performance of our multivariable ES for compensating PDE-based systems.

Continuous Control for Fully Damped Mechanical Systems With Input Constraints: Finite-Time and Exponential Tracking

A motion continuous control scheme for fully damped mechanical systems with constrained inputs is proposed. It gives the freedom to choose among finite-time and (local) exponential convergence through a simple design parameter. The control objective is achieved from any initial conditions, for desired trajectories that can be physically tracked avoiding actuator saturation and loss of motion error dissipation, globally induced through the aid of the natural damping terms explicitly considered in the open-loop dynamics. The stability analysis is based on a strict Lyapunov function and is formally developed within an appropriate analytical framework that takes into account the time-varying character naturally adopted by the closed loop. Simulation tests are further included.

Gradient Extremum Seeking With Nonconstant Delays

This paper proposes a gradient extremum seeking method to address locally quadratic static maps in the presence of time-varying delays. Accommodating nonconstant delays has a strong impact in the predictor construction in terms of the associated transport partial differential equation (PDE) with variable convection speeds beyond the restrictions imposed on the delay regarding its arbitrary duration but bounded variation. A novel predictor design using perturbation-based estimates of the unknown Gradient and Hessian of the map must be introduced to handle this variable nature of the delays, which can arise both in the input and output channels of the nonlinear map to be optimized. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are guaranteed. This technical result is assured by using backstepping transformation and averaging theory in infinite dimensions. Implementation aspects of the presented predictor for variable delays as well as the extension to state-dependent delays are also discussed. At last, we introduce the first results in the topic of extremum seeking control for cascades of transport PDEs that are interconnected through boundary conditions. Such a PDE-PDE cascade is useful to represent simultaneous time- and state-dependent delays. A simulation example illustrates the effectiveness of the proposed predictor-based extremum seeking approach for time-delay compensation.

Trajectory Tracking Control of AUVs via Adaptive Fast Nonsingular Integral Terminal Sliding Mode Control

This article aims to develop an effective control method that can improve the convergence rate over the existing adaptive nonsingular integral terminal sliding mode control (ANITSMC) method for the trajectory tracking control of autonomous underwater vehicles (AUVs). To achieve this goal, an adaptive fast nonsingular integral terminal sliding mode control (AFNITSMC) method is proposed. First, considering that the existing nonsingular integral terminal sliding mode (NITSM) has slow convergence rate in the region far from the equilibrium point, a fast NITSM (FNITSM) is proposed, which guarantees fast transient convergence both at a distance from and at a close range of the equilibrium point, and therefore increases the convergence rate over the existing NITSM. Then, using this FNITSM and adaptive technique, an AFNITSMC method is designed for AUVs. It yields local finite-time convergence of the velocity tracking errors to zero and then local exponential convergence of the position tracking errors to zero, without requiring any a priori knowledge of the upper bounds of the uncertainties and disturbances. Compared with the existing ANITSMC method, the salient feature of the proposed AFNITSMC method is that it provides AUV dynamics a faster convergence rate. Finally, simulation results demonstrate the efficiency of the proposed AFNITSMC method and its superiority over the existing ANITSMC method.

Adaptive Second-Order Fast Nonsingular Terminal Sliding Mode Tracking Control for Fully Actuated Autonomous Underwater Vehicles

This paper focuses on the design of an adaptive second-order fast nonsingular terminal sliding mode control (ASOFNTSMC) scheme for the trajectory tracking of fully actuated autonomous underwater vehicles (AUVs) in the presence of dynamic uncertainties and time-varying external disturbances. First, a second-order fast nonsingular terminal sliding mode (SOFNTSM) manifold is designed to achieve a faster convergence rate than the existing second-order nonsingular terminal sliding mode (SONTSM) manifold. Then, by using this SOFNTSM manifold, an ASOFNTSMC scheme is developed for the fully actuated AUVs to track the desired trajectory. The designed SOFNTSM manifold yields local exponential convergence of the position and attitude tracking errors to zero, and the developed ASOFNTSMC scheme ensures that the error trajectories always move toward the SOFNTSM manifold and once they hit the manifold, remain on it in the presence of dynamic uncertainties and time-varying external disturbances. By deriving the expression of the bounding function of the system uncertainty and using adaptive technique to estimate the unknown parameters of the system uncertainty bounds, the ASOFNTSMC scheme does not require the prior knowledge of the upper bound of the system uncertainty. Meanwhile, through involving the discontinuous sign function into the time derivative of the control input, the ASOFNTSMC scheme eliminates the chattering without reducing the tracking precision. Compared with the existing adaptive SONTSM control (ASONTSMC) scheme, the proposed ASOFNTSMC scheme offers a faster convergence rate for the trajectory tracking control of fully actuated AUVs. Two comparative simulation cases performed respectively on two fully actuated AUVs demonstrate the superiority of the ASOFNTSMC scheme over the ASONTSMC scheme.

Attitude observer on the special orthogonal group with Earth velocity estimation

This paper proposes a new nonlinear attitude observer based on high-grade rate gyros and single body-fixed vector measurements of a constant inertial vector, in contrast with typical solutions that require two of these vectors. The structure is cascaded, where in the first block a vector that is related to the angular velocity of the Earth is estimated and in the second block the attitude itself is obtained. The attitude is directly estimated on the special orthogonal group and the estimation error is shown to converge to zero, with a region of convergence that is best described as semi-global, with local exponential convergence. Simulation results illustrate the achievable performance of the proposed solution and the robustness to sensor noise.

Stochastic Averaging for Time-Varying Systems and Its Applications to Distributed Stochastic Source Seeking

We investigate stochastic averaging theory for time-varying nonlinear systems and its applications to convergence analysis of a distributed source seeking algorithm with time-varying topology. First, we present continuous-time stochastic averaging theorems for time-varying nonlinear systems with stochastic perturbations. Based on the method of stochastic extremum seeking, we propose a distributed stochastic source seeking algorithm with switching topology. We navigate multiple vehicles to seek the source of an unknown signal field, using the measurements of the signal field at their own positions and relative positions to their neighbors. Then, we apply our extended stochastic averaging theory to prove the local exponential convergence, both almost surely and in probability, to a small neighborhood near the source. Finally, a numerical example is included to illustrate the effectiveness of the algorithm.