Extremum Seeking Problem Research Articles

Extremum seeking control for the trajectory tracking of a skid steering vehicle via averaged sub-gradient integral sliding-mode theory

This study introduces the development of a novel extremum-seeking controller designed to stabilize the tracking trajectory of a fully actuated four-wheeled skid steering vehicle. The controller design employs a class of averaged sub-gradient optimization strategies for a suitable convex function that depends on the enforced tracking trajectory achieved through an integral sliding mode implementation. The application of this controller necessitates a nonlinear coordinate transformation of the skid steering vehicle dynamics, resulting in a perturbed multi-input–multi-output linear system. Using the proposed controller, a multidimensional sliding surface remains remarkably stable despite tracking errors. The surface design defines a manifold that aligns with the sub-gradient of a convex function, contingent upon the integral of the tracking error. This connection establishes a link between the extremum-seeking problem and the sliding mode control design. Employing the cascade design strategy, commonly referred to as backstepping, enables the calculation of torques for the four wheels of the mobile vehicle using a sequence of auxiliary sliding surfaces, each with its corresponding complementary function for every stage. The dynamic formulation for the wheel torques is derived from the integral sliding mode methodology. A series of numerical evaluations are conducted to assess the controller’s effectiveness in terms of functional performance. A comparison between the control action based on sliding mode and the traditional state feedback formulation validates the efficacy of the proposed design. This comparison demonstrates advancements in robustness in the face of modeling uncertainties and optimization of the performance function, which are theoretically supported by the suggested control design. Implementing the proposed controller in an experimental platform for a wheeled skid steering vehicle reaffirms the control design methodology presented in this study. This platform includes the application of an image processing algorithm to determine the non-inertial position of the autonomous vehicle using integral sliding mode control. This information completes the necessary data for implementing the on-site version of the proposed controller.

Cueing end‐effector acceleration of a two‐link robotic arm by dynamic averaged sub‐gradient integral sliding mode control

AbstractThe primary objective of this research is to determine if the dynamic version of an averaged sub‐gradient (ASG) integral siding mode (ISM) controller can effectively handle the end‐effector acceleration tracking problem for a two‐link robotic arm avoiding singularity conditions. The proposed scheme solves a nonlinear extremum seeking problem that minimizes a non‐strictly convex function that depends on the acceleration tracking error defined by the difference between the desired acceleration trajectory and the cueing end‐effector acceleration of a two‐link robotic arm in a 2D‐plane. A second‐order differential equation drives the proposed dynamic ASG with nonlinear coefficients and a discontinuous right‐hand side term, which allows for the achievement of the necessary ISM regime in the presence of considerable errors in the mathematical model description. A special switch between modes allows the controller to track accelerations even in extreme positions (near to the active boundaries of the joint articulations) effectively.

Output feedback averaged sub-gradient integral sliding mode control to regulate the tridimensional autonomous motion of autonomous submersible vehicles

This study presents the development of an output feedback control for regulating the movement of an autonomous submersible vehicles in the tridimensional space. The controller formulation applies the Averaged Sub-Gradient Integral Sliding Mode which is fed with the estimation of the translational velocity states by a distributed super-twisting algorithm. The structure of the autonomous submersible vehicles dynamics permits the implementation of the ASG algorithm using the estimates of translation velocity. The proposed scheme solves a non-linear extremum seeking problem that aims to minimize a non-strictly convex function that depends on the tracking error defined by the difference of some suitable reference trajectories and the coordinates of the submarine center of mass. The desired reference trajectories were designed to force the autonomous submersible vehicles three-dimensional motion to a continuous circuit in an oscillating shape over the horizontal plane combined with a submersion on the z-axis. The comparison between the proposed controller, the ASG with complete knowledge of the states and the state feedback controller is presented. This comparison confirms the output feedback ASG controller forced the autonomous submersible vehicles to the desired trajectory with a notable difference in the magnitude of the control given the estimation of the states. These outcomes justify the potential contributions of the suggested ASG integrated with the super-twisting algorithm to obtain the local minimization of the evaluated functional depending on the tracking error. The results justify the potential contributions of the suggested ASG with the super-twisting algorithm to regulate the position of the autonomous submersible vehicles to the desired trajectory.

Extremum seeking control and gradient estimation based on the Super-Twisting algorithm

This article addresses the problem of extremum seeking of a continuous-time dynamical system with a single input and a single output. A Super-Twisting-based optimization algorithm is proposed to compute the input that leads to the extremum value of an unknown convex objective function. Our optimization algorithm requires the input–output gradient of the system’s response at steady state, which we compute throughout a Super-Twisting-based differentiator. Feasibility of the proposed extremum seeking strategy is demonstrated by two simulation examples. The first one is an example of interest in academy. The second one is a novel biohydrogen production process.

Data‐enabled extremum seeking: A cooperative concurrent learning‐based approach

SummaryThis paper introduces a new class of feedback‐based data‐driven extremum seeking algorithms for the solution of model‐free optimization problems in smooth continuous‐time dynamical systems. The novelty of the algorithms lies on the incorporation of memory to store recorded data that enables the use of information‐rich datasets during the optimization process, and allows to dispense with the time‐varying dither excitation signal needed by standard extremum seeking algorithms that rely on a persistence of excitation (PE) condition. The model‐free optimization dynamics are developed for single‐agent systems, as well as for multi‐agent systems with communication graphs that allow agents to share their state information while preserving the privacy of their individual data. In both cases, sufficient richness conditions on the recorded data, as well as suitable optimization dynamics modeled by ordinary differential equations are characterized in order to guarantee convergence to a neighborhood of the solution of the extremum seeking problems. The performance of the algorithms is illustrated via different numerical examples in the context of source‐seeking problems in multivehicle systems.

Extremum seeking control of nonlinear dynamic systems using Lie bracket approximations

SummaryIn this article, we consider extremum seeking problems for a general class of nonlinear dynamic control systems. The main result of the article is a broad family of control laws which optimize the steady‐state performance of the system. We prove practical asymptotic stability of the optimal steady‐state and, moreover, propose sufficient conditions for the asymptotic stability in the sense of Lyapunov. The results generalize and extend existing results which are based on Lie bracket approximations. In particular, our approach does not rely on singular perturbation theory, as commonly used in extremum seeking of nonlinear dynamic systems.

Extremum Seeking Under Distributed Input Delay

In this paper, we propose an extremum seeking scheme for single-parameter static maps in the presence of distributed input delay. A probing signal is newly developed so that the delayed signal has a conventional form in standard extremum seeking problems. An update law for an estimate of the unknown argument of the extremum is designed based on the idea of the predictor feedback law. We prove the convergence of the estimation error to a neighborhood of the origin by means of the method of averaging. The effectiveness of the proposed scheme is confirmed by a numerical simulation.

Extremum Seeking for Nonlinear Uncertain Systems: A Small-Gain Synthesis

This paper studies the extremum seeking problem for static maps with the inputs of the maps generated by a nonlinear uncertain system. A new small-gain approach is developed which uses an extremum seeking strategy to generate a reference signal, and employs a control law for reference-tracking of the nonlinear uncertain systems. The notions of input-to-state stability (ISS) and input-to-output stability (IOS) are used to characterize the interconnection between the extremum seeking strategy and the reference-tracking controller, and the nonlinear small-gain theorem is employed to guarantee the stability of the closed-loop extremum seeking system. With the proposed approach, the extremum seeking problem for a complex nonlinear system is solvable as long as one can design a proper reference-tracking controller for the system. Examples are given to show the feasibility of the proposed approach, and a numerical simulation is employed to show the effectiveness of the proposed design.

On unified stability for a class of chemostat model with generic growth rate functions: Maximum yield as control goal

A unified criterion for stability is dealt for a class of chemostat model which includes a generic growth rate function representing substrate, biomass, and product inhibitions. The stability criterion is unified at sense that the growth kinetic rate takes a general form into the vector field and includes both the maintenance coefficient and the death rate parameters. Sufficient conditions are provided for multiplicity of the steady states. In addition, from a local analysis, an expression is proposed as a unified criterion for the local stability and for local bifurcations existence. Hence, a control design relies the unified criterion to account local stability and bifurcation with independence of the specific growth rate governing a chemostat. The criterion is based on partial derivatives which allows us to reduce the calculation needed to determine the equilibrium point stability and the bifurcation points existence. Analytical features of the chemostat were investigated when the maximum of productivity was set as the control objective. Results are illustrated through bifurcation diagrams for the specific growth rate models and parameter values reported in the scientific literature, and with examples for which the maximum of productivity is structurally stable. In this sense, the unified criterion is a tool for the formulation of the extremum seeking problem as well. Finally, some examples for applicability of the results are shown for chemostat that might be governed by different growth rates.

Partial Stability Concept in Extremum Seeking Problems

The paper deals with the extremum seeking problem for a class of cost functions depending only on a part of state variables of a control system. This problem is related to the concept of partial asymptotic stability and analyzed by Lyapunov’s direct method and averaging schemes. Sufficient conditions for the practical partial stability of a system with oscillating inputs are derived with the use of Lie bracket approximation techniques. These conditions are exploited to describe a broad class of extremum-seeking controllers ensuring the partial stability of the set of minima of a cost function. The obtained theoretical results are illustrated by the Brockett integrator and rotating rigid body.

Extremum Seeking Algorithms based on Non-Commutative Maps

In this work, discrete-time extremum seeking algorithms for unconstrained optimization problems are developed. A general class of non-commutative maps and one- and two point function evaluation polices are presented to approximate a gradient-descent algorithm, suitable for extremum seeking problems. Moreover, adaptive step size rules are discussed to achieve faster convergence and vanishing steady state oscillations.

On extremum seeking controllers based on the Lie bracket approximation in domains with obstacles

AbstractIn this paper, we consider the problem of extremum seeking in the presence of obstacles. The analytical expression of the cost function as well as locations and shapes of the obstacles are assumed to be partially or completely unknown. We describe a broad family of control algorithms for a unicycle type system, which provides a solution of the above problem based on Lie bracket approximation ideas. The obtained controllers generalize some known results and allow to construct new control strategies. Furthermore, it is shown that gradient‐free control algorithms can be used for excluding the attraction of trajectories to undesirable equilibriuma in obstacle avoidance problems.

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.

On a class of generating vector fields for the extremum seeking problem: Lie bracket approximation and stability properties

In this paper, we describe a broad class of control functions for extremum seeking problems. We show that it unifies and generalizes existing extremum seeking strategies which are based on Lie bracket approximations, and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks. The second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems. It provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability. In contrast to many known results, we also prove asymptotic and exponential stability in the sense of Lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields.

Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms

This paper formulates the problem of Extremum Seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization of cost functions defined on smooth Riemannian manifolds. This problem falls within the category of online optimization methods. We introduce the notion of geodesic dithers which is a perturbation of the optimizing trajectory in the tangent bundle of the ambient state manifolds and obtain the extremum seeking closed loop as a perturbation of the averaged gradient system. The main results are obtained by applying closeness of solutions and averaging theory on Riemannian manifolds. The main results are further extended for optimization on Lie groups. Numerical examples on Riemannian manifolds (Lie groups) SO(3) and SE(3) are presented at the end of the paper.

Distributed Extremum Seeking in Multi-Agent Systems with Arbitrary Switching Graphs

This paper studies the problem of averaging-based extremum seeking in dynamical multi-agent systems with time-varying communication graphs. We consider a distributed consensus-optimization problem where the plants and the controllers of the agents share information via time-varying graphs, and where the cost function to be minimized corresponds to the summation of the individual response maps generated by the agents. Although the problem of averaging-based extremum seeking control in multi-agent dynamical systems with time-invariant graphs has been extensively studied, the case where the graph is time-varying remains unexplored. In this paper we address this problem by making use of recent results for generalized set-valued hybrid extremum seeking controllers, and the framework of switched differential inclusions and common Lyapunov functions. For the particular consensus-optimization problem considered in this paper, a semi-global practical stability result is established. A numerical example in the context of dynamic electricity markets illustrates the results.

Extremum Seeking for Time-Varying Functions using Lie Bracket Approximations

The paper presents a control algorithm that steers a system to an extremum point of a time-varying function. The proposed extremum seeking law depends on values of the cost function only and can be implemented without knowing analytical expression of this function. By extending the Lie brackets approximation method, we prove the local and semi-global practical uniform asymptotic stability for time-varying extremum seeking problems. For this purpose, we consider an auxiliary non-autonomous system of differential equations and propose asymptotic stability conditions for a family of invariant sets. The obtained control algorithm ensures the motion of a system in a neighborhood of the curve where the cost function takes its minimal values. The dependence of the radius of this neighborhood on the bounds of the derivative of a time-varying function is shown.

Shahshahani gradient-like extremum seeking

In this paper we present novel extremum seeking controllers designed to achieve online constrained optimization of uncertain input-to-output maps of a class of nonlinear dynamical systems. The algorithms, inspired by a class of evolutionary dynamics that emerge in the context of population games, generate on-line Shahshahani gradient-like systems able to achieve extremum seeking under simplex-like constraints on the positive orthant. The method attains semi-global practical convergence to the optimal point, even when the initial conditions are not in the feasible set, and it can be naturally adapted to address distributed extremum seeking problems in multi-agent systems where an all-to-all communication structure is not available. Potential applications include problems on distributed dynamic resource allocation, congestion and flow control in networked systems, as well as portfolio optimization in financial markets. Via simulations, we illustrate our results under different scenarios.

A time-varying extremum-seeking control approach

This paper considers the solution of a real-time optimization problem using adaptive extremum seeking control. It is assumed that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown and that the objective function is measured. The main contribution of the paper is to formulate the extremum-seeking problem as a time-varying estimation problem. The proposed approach is shown to avoid the need for averaging results which minimizes the impact of the choice of dither signal on the performance of the extremum seeking control system. Several examples are used to illustrate the effectiveness of the proposed technique.

Multi-agent source seeking via discrete-time extremum seeking control

Recent developments in extremum seeking theory have established a general framework for the methodology, although the specific implementations, particularly in the context of multi-agent systems, have not been demonstrated. In this work, a group of sensor-enabled vehicles is used in the context of the extremum seeking problem using both local and global optimisation algorithms to locate the extremum of an unknown scalar field distribution. For the former, the extremum seeker exploits estimates of gradients of the field from local dithering sensor measurements collected by the mobile agents. It is assumed that a distributed coordination which ensures uniform asymptotic stability with respect to a prescribed formation of the agents is employed. An inherent advantage of the frameworks is that a broad range of nonlinear programming algorithms can be combined with a wide class of cooperative control laws to perform extreme source seeking. Semi-global practical asymptotically stable convergence to local extrema is established in the presence of field sampling noise. Subsequently, global extremum seeking with multiple agents is investigated and shown to give rise to robust practical convergence whose speed can be improved via computational parallelism. Nonconvex field distributions with local extrema can be accommodated within this global framework.