Local Exponential Convergence Research Articles

A High-Gain Observer for a Wet Gas Centrifugal Compressor

Subsea wet gas compression is an enabling boosting technology for cost-efficient production of small and remote gas condensate fields. These wet gas compressors are capable of compressing gases containing up to 5% liquid, removing the need for pre-separation and thereby reducing the investment and maintenance costs. However, the presence of liquid in the gas significantly changes the compression performance from that of dry gas, resulting in a reduced process gain and an increased normal operating region for increasing liquid content. A novel backstepping process controller was previously proposed to track and stabilize a desired pressure reference inside the normal operating region, requiring feedback from several variables, including the mass flow. Mass flow is an inaccurate and expensive measurement. Therefore, a full-order high-gain observer is derived for the wet gas compression process with proven local exponential convergence. The performance of the derived observer is studied in simulations.

Gradient extremum seeking for static maps with actuation dynamics governed by diffusion PDEs

We design and analyze the scalar gradient extremum seeking control feedback for static maps with actuation dynamics governed by diffusion PDEs. Conceptually, a non-model based online optimization control scheme is paired with actuation dynamics which occur in chemistry, biology and economics. A learning-based adaptive control approach with known actuation dynamics is considered in this paper. In the design part, we first compensate the actuation dynamics in the dither signals. Secondly, we introduce an average-based actuation dynamics compensation controller via a backstepping transformation, which is fed by the perturbation-based gradient and Hessian estimates of the static map. The stability analysis of the error-dynamics is based on using Lyapunov’s method and applying averaging for infinite-dimensional systems to capture the infinite-dimensional state of the actuator model. Local exponential convergence to a small neighborhood of the optimal point is proven and illustrated by numerical simulations.

Design of Coupled Harmonic Oscillators for Synchronization and Coordination

Synchronization and coordination of coupled oscillators are fundamental behaviors in complex dynamical systems. This paper considers the design of coupled harmonic oscillators to generate an orbitally stable limit cycle of prescribed oscillation profile. Based on the Floquet theory and averaging techniques, necessary and sufficient conditions are obtained for nonlinear coupling functions to achieve local exponential convergence to a desired orbit. Unlike globally convergent methods based on contraction analysis, the result applies to oscillators without flow invariance properties. Insights into coordination mechanisms are gained through interpretation of the coupling structure as a directed graph. The theory is illustrated by simple tutorial examples.

Batch-to-Batch Finite-Horizon LQ Control for Unknown Discrete-Time Linear Systems Via Stochastic Extremum Seeking

We employ our recent discrete-time stochastic averaging theorems and stochastic extremum seeking to iteratively (batch-to-batch) optimize open-loop control sequences for unknown but reachable discrete-time linear systems with a scalar input and without known system dimension, for a cost that is quadratic in the measurable output and the input. First, for a multivariable gradient-based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open-loop control sequence. Second, to remove the convergence rate's dependence on the Hessian matrix of the cost function, which is unknown since the system's model (the system matrices $(A,B,C)$ ) is unknown, we develop a multivariable discrete-time Newton-based stochastic extremum seeking method, design the Newton-based algorithm for the iteration of the input sequence, and prove local exponential convergence to the optimal input sequence. Finally, two simulation examples are given to illustrate the effectiveness of the two methods.

Stochastic source seeking with forward and angular velocity regulation

This paper studies a stochastic extremum seeking method to steer a nonholonomic vehicle to the unknown source of a spatial field in a plane. The key challenge lies in the lack of vehicle’s position information and the distribution of the scalar field. Different from the existing stochastic strategy that keeps the forward velocity constant and controls only the angular velocity, we design a stochastic extremum seeking controller to regulate both forward and angular velocities simultaneously in this work. Thus, the vehicle decelerates near the source and stays within a small area as if it comes to a full stop, which solves the overshoot problem in the constant forward velocity case. We use the stochastic averaging theory to prove the local exponential convergence, both almost surely and in probability, to a small neighborhood near the source for elliptical level sets. Finally, simulations are included to illustrate the theoretical results.

A global observer for attitude and gyro biases from vector measurements

We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple “geometry-free” nonlinear observer with guaranteed uniform global asymptotic convergence and local exponential convergence; the stability analysis, which relies on a strict Lyapunov function, is rather simple. The excellent behavior of the observer is illustrated through a detailed numerical simulation.

Extremum Seeking for Static Maps With Delays

In this paper, we address the design and analysis of multi-variable extremum seeking for static maps subject to arbitrarily long time delays. Both Gradient and Newton-based methods are considered. Multi-input systems with different time delays in each individual input channel as well as output delays are dealt with. The phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation-based (averaging-based) estimate of the Hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point, even in the presence of delays. The stability analysis is carried out using backstepping transformation and averaging in infinite dimensions, capturing the infinite-dimensional state due the time delay. In particular, a new backstepping-like transformation is introduced to design the predictor for the Gradient-based extremum seeking scheme with multiple and distinct input delays. The proposed Newton-based extremum seeking approach removes the dependence of the convergence rate on the unknown Hessian of the nonlinear map to be optimized, being user-assignable as in the literature free of delays. A source seeking example illustrates the performance of the proposed delay-compensated extremum seeking schemes.

Angular velocity nonlinear observer from vector measurements

This paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of the linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. A high gain design allows to establish local uniform exponential convergence. Simulation results are provided to illustrate the method.

Adaptive disturbance rejection for unknown stable linear systems

The design of an adaptive output feedback compensator is addressed to reject biased multi-sinusoidal disturbances of unknown amplitudes, phases and frequencies, acting on unknown stable single-input single-output linear systems of unknown order and relative degree. A single biased sinusoidal disturbance is first considered. It is shown that the knowledge of the sign of the real part (or the imaginary part) of its transfer function at zero and at the disturbance frequency is sufficient for the explicit design of an adaptive compensator which guarantees local exponential convergence to zero of the system output and of the frequency estimation error. This result is generalized to the case of biased multi-sinusoidal disturbances. As a special case, an adaptive notch filter which provides the unknown frequencies contained in the disturbance is obtained.

Discrete-time Stochastic Source Seeking via Tuning of Forward Velocity

We apply the method of discrete-time stochastic extremum seeking to navigate an autonomous vehicle towards the maximum of an unknown, spatially distributed signal field. We assume that the information of the vehicle's position is unavailable. Only using the measurement of the signal at the vehicle's location, we design a control law with controlling the forward velocity and keeping the angular velocity constant. We firstly build a discrete-time vehicle model by discretizing the model of a continuous nonholonomic unicycle with the help of the definition of integration. Then, we design a discrete-time control law in which a discrete-time ergodic stochastic process is taken as the probing signal.We prove local exponential convergence, both almost surely and in probability, to a small neighborhood near the source. A numerical simulation is given to illustrate the effectiveness of the control law.

3-D Velocity Regulation for Nonholonomic Source Seeking Without Position Measurement

We consider a three-dimensional problem of steering a nonholonomic vehicle to seek an unknown source of a spatially distributed signal field without any position measurement. In the literature, there exists an extremum seeking-based strategy under a constant forward velocity and tunable pitch and yaw velocities. Obviously, the vehicle with a constant forward velocity may exhibit certain overshoots in the seeking process and can not slow down even it approaches the source. To resolve this undesired behavior, this paper proposes a regulation strategy for the forward velocity along with the pitch and yaw velocities. Under such a strategy, the vehicle slows down near the source and stays within a small area as if it comes to a full stop, and controllers for angular velocities become succinct. We prove the local exponential convergence via the averaging technique. Finally, the theoretical results are illustrated with simulations.

A Single Forward-Velocity Control Signal for Stochastic Source Seeking With Multiple Nonholonomic Vehicles

With a single stochastic extremum seeking control signal, we steer multiple autonomous vehicles, modeled as nonholonomic unicycles, toward the maximum of an unknown, spatially distributed signal field. The vehicles, whose angular velocities are constant and distinct, travel at the same forward velocity, which is controlled by the stochastic extremum seeking controller. To determine the vehicles’ velocity, the controller uses measurements of the signal field at the respective vehicle positions and excitation based on filtered white noise. The positions of the vehicles are not measured. We prove local exponential convergence, both almost surely and in probability, to a small neighborhood near the source and provide a numerical example to illustrate the effectiveness of the algorithm.

Bounded controllers for global path tracking control of unicycle-type mobile robots

This paper presents a design of bounded controllers with a predetermined bound for global path tracking control of unicycle-type mobile robots at the torque level. A new one-step ahead backstepping method is first introduced. The heading angle and linear velocity of the robots are then considered as immediate controls to force the position of the robots to globally and asymptotically track its reference path. These immediate controls are designed based on the one-step ahead backstepping method to yield bounded control laws. Next, the one-step ahead backstepping method is applied again to design bounded control torques of the robots with a pre-specified bound. The proposed control design ensures global asymptotical and local exponential convergence of the position and orientation tracking errors to zero, and bounded torques driving the robots. Experimental results on a Khepera mobile robot verify the proposed control controller.

Symmetry-Based Observers for Some Water-Tank Problems

In this paper, we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured. The fluid motion is described by shallow water equations in two horizontal dimensions. We build a simple nonlinear observer which takes advantage of the symmetries of fluid dynamics laws. As a result its structure is based on convolutions with smooth isotropic kernels, and the observer is remarkably robust to noise. We prove the convergence of the observer around a steady-state. In numerical applications local exponential convergence is expected. The observer is also applied to the problem of predicting the ocean circulation. Realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques.

Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs

We introduce a multi-input stochastic extremum seeking algorithm to solve the problem of seeking Nash equilibria for a noncooperative game whose N players seek to maximize their individual payoff functions. The payoff functions are general (not necessarily quadratic), and their forms are not known to the players. Our algorithm is a nonmodel-based approach for asymptotic attainment of the Nash equilibria. Different from classical game theory algorithms, where each player employs the knowledge of the functional form of his payoff and the knowledge of the other players' actions, a player employing our algorithm measures only his own payoff values, without knowing the functional form of his or other players' payoff functions. We prove local exponential (in probability) convergence of our algorithms. For nonquadratic payoffs, the convergence is not necessarily perfect but may be biased in proportion to the third derivatives of the payoff functions and the intensity of the stochastic perturbations used in the algorithm. We quantify the size of these residual biases. Compared to the deterministic extremum seeking with sinusoidal perturbation signals, where convergence occurs only if the players use distinct frequencies, in our algorithm each player simply employs an independent ergodic stochastic probing signal in his seeking strategy, which is realistic in noncooperative games. As a special case of an N-player noncooperative game, the problem of standard multivariable optimization (when the players' payoffs coincide) for quadratic maps is also solved using our stochastic extremum seeking algorithm.

Local Exponential Regulation of Nonholonomic Systems in Approximate Chained Form with Applications to Off-Axle Tractor-Trailers

Most of drift-less nonholonomic systems cannot be exactly converted to an nonholonomic chained form, a wealth of design tools developed for the control of nonholonomic chained form are thus not directly applicable to such systems. Nevertheless, there exists a class of systems that may be locally approximated by the nonholonomic chained form around certain equilibrium points. In this work, we propose a discontinuous and a smooth time-varying control laws respectively for the approximated nonholonomic chained form, guaranteeing local exponential convergence of state to the desired equilibrium point. An tractor towing off-axle trailers is taken as an example to illustrate the approaches.

Stochastic source seeking for nonholonomic unicycle

We apply the recently introduced method of stochastic extremum seeking to navigate a nonholonomic unicycle towards the maximum of an unknown, spatially distributed signal field, using only the measurement of the signal at the vehicle’s location but without the measurement of the vehicle’s position. Keeping the forward velocity constant and controlling only the angular velocity, we design a stochastic source seeking control law which employs excitation based on filtered white noise, rather than sinusoidal perturbations used in the existing work. We study stability with the help of stochastic averaging theorems that we recently developed for general nonlinear continuous-time systems with stochastic perturbations. We prove local exponential convergence, both almost surely and in probability, to a small neighborhood near the source. We characterize the convergence speed explicitly and provide design guidelines for maximizing it, as well as for minimizing the residual set near the source. We present a detailed simulation study, including a study of the effect of saturation on the steering input.

Nonholonomic Source Seeking With Tuning of Angular Velocity

We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle. The vehicle does not have the capability of sensing its position or the position of the source but is capable of sensing the scalar signal originating from the source. The signal field is assumed to decay away from the position of the source but the vehicle does not have the knowledge of the functional form of the field. We employ extremum seeking to steer the vehicle to the source. Our control strategy keeps the forward velocity constant and tunes the angular velocity, a setting suitable for most autonomous vehicles, including aerial ones. Because of the constant forward velocity constraint, after it has converged near the source, the vehicle exhibits extremely interesting and complex motions. Using averaging theory, we prove local exponential convergence to an ldquoorbit-likerdquo attractor around the source. We also present a thorough analysis of non-local behaviors and attractors that the vehicle can exhibit near the source. The richness and complexity of behaviors makes only some of them amenable to analysis, whereas others are illustrated through a carefully laid out simulation study.

Tracking and robustness analysis for controlled microelectromechanical relays

AbstractThe feedback control problem for microelectromechanical (MEM) relays is complicated by a quadratic nonlinearity in the dynamic model. We show that this nonlinearity imposes constraints on the reference trajectories that can be tracked and on the global convergence rate of the tracking error. Using a dynamic model that is applicable to both electrostatic and electromagnetic MEM relays, we introduce a new class of nonlinear tracking controls. In particular, we use Lyapunov theory to construct a state feedback that yields uniform global asymptotic stability and arbitrarily fast local exponential convergence of the tracking error. We then show how our control can be redesigned with partial‐state feedback under the assumption that only the movable electrode position and the electrical state (i.e. charge or flux) are fed back. Finally, we utilize input‐to‐state stability theory to quantify the robustness of our state feedback controller to parametric uncertainties. Our simulation results illustrate the good stability and tracking performance of the proposed control. They also illustrate how to craft a reference trajectory that satisfies the aforementioned constraints while being compatible with a typical relay operation. Copyright © 2007 John Wiley & Sons, Ltd.

Fast Parking Control of Mobile Robots: A Motion Planning Approach With Experimental Validation

This paper presents a solution to the general parking problem of nonholonomic mobile robots based on motion planning and tracking controller design. A new global tracking controller is first proposed to achieve global uniformly asymptotic stability and local exponential convergence. The parking problem is then transformed into a tracking one by adding a redesigned virtual trajectory to the original trajectory, thus guaranteeing practical stability with exponential convergence. Further improvement in parking performance is obtained through linearization and pole-placement methods. One feature of our approach is that fast convergence in parking and tracking can be treated at the same time without switching between two controllers. Moreover, a tuning function is used to enhance parking performance. With the proposed framework, various tracking controllers given in the literature can be adopted to handle parking problems. The effectiveness of the proposed methods is verified by several interesting experiments including parallel parking and back-into-garage parking.