Newton-based Extremum Seeking Research Articles

Filtered High Gain Observer for an Electric Vehicle's Electro-hydraulic Brake: Design and Optimization Using Multivariable Newton-Based Extremum Seeking

Filtered High Gain Observer for an Electric Vehicle's Electro-hydraulic Brake: Design and Optimization Using Multivariable Newton-Based Extremum Seeking

Inverse optimal extremum seeking under delays

We establish the inverse optimality in the average sense of our earlier Gradient- and Newton-based extremum seeking algorithms for maximizing unknown locally quadratic maps in the presence of constant delays. To compensate the delay, these algorithms employ a predictor feedback with a perturbation-based estimate for the unknown Hessian (or its inverse). The algorithm’s inverse optimality is the result of running the predictor through a simple first-order filter with a fast enough pole.

Extremum seeking control for real-time optimization of high temperature heat pump systems incorporating vapor injection

In order to minimize the power consumption of a vapor-injected high temperature heat pump system with R245fa as refrigerant, a control strategy, namely Extremum seeking control, is proposed. It can implement real-time model-free optimization control of plants with concave and convex properties. Modelica-based dynamic simulation model of the proposed system is developed. The Newton-based Extremum seeking control is used to optimize the intermediate pressure of the proposed system to obtain the minimum power consumption of the compressor at a fixed heating capacity (200 kW). The adopted ESC strategy requires four measured quantities, namely, injection pressure, condenser outlet water temperature, evaporator outlet superheat and compressor power consumption, to achieve optimal control of the system in real time. Two operation scenarios are simulated: stable and freely variable heat source temperature. For both scenarios, the condenser outlet water temperature was set to 140 °C. The simulation results show that at stable heat source temperature, the power consumption is reduced by 5.2% and the COP is increased by 7.7% compared to a fixed initial intermediate pressure. Under the free-variable heat source condition, the power consumption can be reduced by up to 5 kW compared to a constant input. A total of 283 kWh of power was saved during the 48 h simulation. Thus, the Newton-based extremum seeking control strategy can optimize the intermediate pressure to achieve the minimum power consumption while maintaining excellent transient performance, and demonstrate the superiority of the ESC strategy for real-time optimization..

Illumination-Resilient Lane Detection by Threshold Self-Adjustment Using Newton-Based Extremum Seeking

The ability to detect lane markings under varying lighting conditions is essential for autonomous mobile robots and automatic vehicle driving assistance systems. Because the object color information is subject to illumination variation, this article presents a novel and computationally efficient algorithm based on the extremum-seeking method to achieve illumination-resilient lane detection and path-following tasks for autonomous driving. Lane detection is performed in the hue-saturation-value color space by distinguishing the colored lane marks from the background. The system’s inputs are the upper and lower thresholds in each of the hue, saturation, and value channels. We define a cost function as the combination of detection accuracy and lane coverage to evaluate the algorithm performance. Two extremum-seeking schemes, one with fixed dither amplitudes and another with adaptive dither amplitudes, are designed to adjust the system inputs to minimize the cost function. The two proposed methods are validated and compared through video-based simulation studies and scaled-car field experimental tests.

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.

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.

Fixed-Time Newton-Like Extremum Seeking

In this paper, we present a novel Newton-based extremum seeking controller for the solution of multivariable model-free optimization problems in static maps. Unlike existing asymptotic and fixed-time results in the literature, we present a scheme that achieves (practical) finite time convergence to a neighborhood of the optimal point, with a convergence time that is independent of the initial conditions and the Hessian of the cost function, and therefore can be arbitrarily assigned a priori by the designer with an appropriate choice of parameters in the algorithm. The extremum seeking dynamics exploit a class of fixed time convergence properties recently established in the literature for a family of Newton flows, as well as averaging results for perturbed dynamical systems that are not necessarily Lipschitz continuous. The proposed extremum seeking algorithm is model-free and does not require any explicit knowledge of the gradient and Hessian of the cost function. Instead, real-time optimization with fixed-time convergence is achieved by using real time measurements of the cost, which is perturbed by a suitable class of periodic excitation signals generated by a dynamic oscillator. Numerical examples illustrate the performance of the algorithm.

Multidimensional Fractional-Order Newton-Based Extremum Seeking for Online Light-Energy Saving Technique of Lighting System

This article proposes a real-time light-energy saving control technique to reduce unnecessary light-energy consumption (LEC) for varying lighting environment, through regulating brightness of illuminative lamps while providing the expect illumination level. Considering the influence of environmental conditions, a fractional-order Newton-based extremum seeking control (ESC) method in the proposed technique is developed to enhance the tracking accuracy and convergence speed. It can result in the high efficient lighting system and the rapid track of minimum LEC. Moreover, it significantly reduces the impact of selecting an arbitrary Hessian matrix on tracking speed of the algorithm, which is the main drawback in the most conventional ESC algorithms. Experimental results reveal that the proposed method has the ability to track the available minimum LEC under changing environmental conditions.

Newton-based extremum seeking: A second-order Lie bracket approximation approach

In this paper, we present novel multi-variable Newton-based extremum seeking systems, based on Lie bracket approximation methods. More precisely, we consider cost functions with an unknown mathematical description, but whose value can be measured on-line. We propose extremum seeking systems that approximate the Newton-based optimization law, by combining the on-line measurement of the cost with time-periodic excitation signals. The inversion of the Hessian matrix is avoided by introducing a first order dynamical system, whose output approximates the Newton step. This provides practical robustness with respect to ill-conditioned Hessian matrices. Semi-global stability properties of the proposed schemes are demonstrated both for static cost functions and for cost functions associated with a general non-linear dynamical system. The effectiveness of the approach is shown in simulations.

Extremum Seeking for Maximizing Higher Derivatives of Unknown Maps in Cascade with Reaction-Advection-Diffusion PDEs

We present a generalization of the scalar Newton-based extremum seeking algorithm, which maximizes the map’s higher derivatives in the presence of dynamics described by Reaction-Advection-Diffusion (RAD) equations. Basically, the effects of the Partial Differential Equations (PDEs) in the additive dither signals are canceled out using the trajectory generation paradigm. Moreover, the inclusion of a boundary control for the RAD process stabilizes the closed-loop feedback system. By properly demodulating the map output corresponding to the manner in which it is perturbed, the extremum seeking algorithm maximizes the n-th derivative only through measurements of the own map. The Newton-based extremum seeking approach removes the dependence of the convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance and remove limitations of standard gradient-based extremum seeking. In particular, our RAD compensator employs the same perturbation-based (averaging-based) estimate for the Hessian’s inverse of the function to be maximized provided by a differential Riccati equation applied in the previous publications free of PDEs. We prove local stability of the algorithm, maximization of the map sensitivity and convergence to a small neighborhood of the desired (unknown) extremum by means of backstepping transformation, Lyapunov functional and the theory of averaging in infinite dimensions.

Stochastic Extremum Seeking for Dynamic Maps With Delays

We address a Newton-based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of known time delays. Different from all previous works on this topic, we employ stochastic instead of periodic perturbations, allow arbitrarily long output delays and consider a dynamic map to be optimized. We incorporate a novel predictor feedback for delay compensation and show local exponential stability along with convergence to a small neighborhood of the unknown extremum point. For the purpose of the proof, we apply a backstepping transformation and averaging theory in infinite dimensions for stochastic systems. Moreover, simulations highlight the effectiveness of the proposed predictor-feedback scheme.

Maximizing Map Sensitivity and Higher Derivatives Via Extremum Seeking

We present a generalization to the scalar Newton-based extremum seeking algorithm which, through perturbation-induced measurements of an unknown steady-state input-to-output map, maximizes the map's higher derivatives. As with other extremum seeking (ES) problems, the scheme relies on derivative estimators and learning dynamics operating in different time scales. We provide analysis for the typical deterministic (sinusoidal) perturbation of the optimal parameter estimate. Also, we alternatively include a stochastic method where the map is perturbed via the sinusoid of Brownian motion about the boundary of a circle. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the $n{\rm th}$ derivative only through measurements of the map. The Newton-based ES approach removes the dependence of the convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance over standard gradient-based ES. Our design stems from the existing multivariable Newton-based ES algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, whereas employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear equilibrium profiles of dynamic maps and compare the Newton-based method against the gradient-based method. We also extend this abstraction to multiplayer noncooperative games but limit our attention to a two-player game for notational simplicity and length considerations.

Deterministic and Stochastic Newton-based extremum seeking for higher derivatives of unknown maps with delays

We present a Newton-based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time delays using deterministic perturbations. Different from previous works about extremum seeking for higher derivatives, arbitrarily long input-output delays are allowed. We incorporate a predictor feedback with a perturbation-based estimate for the Hessian’s inverse using a differential Riccati equation. As a bonus, the convergence rate of the real-time optimizer can be made user-assignable, rather than being dependent on the unknown Hessian of the higher-derivative map. Averaging method for arbitrary shaped derivatives under delays is presented. Exponential stability and convergence to a small neighbourhood of the unknown extremum point are achieved for locally quadratic derivatives by using a backstepping transformation and averaging theory in infinite dimensions. Furthermore, we give a brief introduction into stochastic Newton-based Extremum Seeking for constant output delays, where we show the differences and similarities with respect to the deterministic case. We also present illustrative numerical examples in order to highlight the effectiveness of the proposed predictor-based extremum seeking for time-delay compensation applying both deterministic and stochastic perturbations.

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.

A Multivariable Newton-Based Extremum Seeking Control for Condenser Water Loop Optimization of Chilled-Water Plant

This paper presents a multivariable Newton-based extremum seeking control (ESC) scheme for efficient operation of a chilled-water plant. A modelica-based dynamic simulation model of the chilled-water plant consists of one screw chiller and one counter-flow cooling tower was adopted for evaluation of proposed two-input Newton-based ESC controller. The ESC controller takes the total power of the chiller compressor, the cooling-tower fan, and the condenser water (CW) pump as feedback signal and discovers the optimum outputs of cooling-tower fan speed and the condenser-loop water flow rate to maximize the power efficiency in real time with the cooling load being satisfied. Remarkable energy saving is observed for several testing conditions.

Newton-based Extremum Seeking under Actuator and Sensor Delays

In addition to the gradient based extremum seeking for delay systems presented in a companion paper, we derive here the control design and analysis for scalar Newtonbased extremum seeking subject to input-output delays. differently from the previous work, a new predictor with a perturbation-based estimate of the Hessian's inverse is incorporated in the closed-loop such that the convergence rate of the real-time controller can be made userassignable, rather than being dependent on the unknown Hessian of the static nonlinear map to be optimized. Exponential stability and convergence to a small neighborhood of the unknown extremum point can be obtained. This result is rigorously achieved by using backstepping transformation and averaging in infinite dimensions. A numerical example is given to illustrate the effectiveness of the proposed predictor based extremum seeking for time-delay compensation.

Power Optimization for Photovoltaic Microconverters Using Multivariable Newton-Based Extremum Seeking

Extremum seeking (ES) is a real-time optimization technique that has been applied to maximum power point tracking (MPPT) design for photovoltaic (PV) microconverter systems, where each PV module is coupled with its own dc/dc converter. Most of the existing MPPT designs are scalar, i.e., employ one MPPT loop around each converter, and all designs, whether scalar or mutivariable, are gradient based. The convergence rate of gradient-based designs depends on the Hessian, which in turn is dependent on environmental conditions, such as irradiance and temperature. Therefore, when applied to large PV arrays, the variability in environmental conditions and/or PV module degradation results in nonuniform transients in the convergence to the maximum power point (MPP). Using a multivariable gradient-based ES algorithm for the entire system instead of a scalar one for each PV module, while decreasing the sensitivity to the Hessian, does not eliminate this dependence. We present a recently developed Newton-based ES algorithm that simultaneously employs estimates of the gradient and Hessian in the peak power tracking. The convergence rate of such a design to the MPP is independent of the Hessian, with tunable transient performance that is independent of environmental conditions. We present simulation as well as the experimental results that show the effectiveness of the proposed algorithm in comparison with the existing scalar designs, and also to multivariable gradient-based ES.

A Newton-Based Extremum Seeking MPPT Method for Photovoltaic Systems with Stochastic Perturbations

Microcontroller based maximum power point tracking (MPPT) has been the most popular MPPT approach in photovoltaic systems due to its high flexibility and efficiency in different photovoltaic systems. It is well known that PV systems typically operate under a range of uncertain environmental parameters and disturbances, which implies that MPPT controllers generally suffer from some unknown stochastic perturbations. To address this issue, a novel Newton-based stochastic extremum seeking MPPT method is proposed. Treating stochastic perturbations as excitation signals, the proposed MPPT controller has a good tolerance of stochastic perturbations in nature. Different from conventional gradient-based extremum seeking MPPT algorithm, the convergence rate of the proposed controller can be totally user-assignable rather than determined by unknown power map. The stability and convergence of the proposed controller are rigorously proved. We further discuss the effects of partial shading and PV module ageing on the proposed controller. Numerical simulations and experiments are conducted to show the effectiveness of the proposed MPPT algorithm.

Multivariable Newton-based extremum seeking

We present a Newton-based extremum seeking algorithm for the multivariable case. The design extends the recent Newton-based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the inverse of the Hessian matrix that removes the difficulty with the possible singularity of a possible direct estimate of the Hessian matrix. The estimator of the inverse of the Hessian has the form of a differential Riccati equation. We prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations. In comparison with the standard gradient-based multivariable extremum seeking, the proposed algorithm removes the dependence of the convergence rate on the unknown Hessian matrix and makes the convergence rate, of both the parameter estimates and of the estimates of the Hessian inverse, user-assignable. In particular, the new algorithm allows all the parameters to converge with the same speed, yielding straight trajectories to the extremum even with maps that have highly elongated level sets, in contrast to curved “steepest descent” trajectories of the gradient algorithm. Simulation results show the advantage of the proposed approach over gradient-based extremum seeking, by assigning equal, desired convergence rates to all the parameters using Newton’s approach.