Extremum-seeking Control Approach Research Articles

Perturbation-based Extremum Seeking Control Design for a Class of Nonlinear Systems

In this paper, we try to address the problem of output (performance) function by applying the Perturbation-based Extremum Seeking Control (PESC) approach to the observer of the Single-Input Single-Output (SISO) nonlinear systems. In this work, PESC is applied so that the performance function can reach its maximum value. We apply tow controllers design that will take care of maximizing of the cost function. First controller is designed in the availability of full and unknown variables which are fed to the objective function, and the second controller is designed when the full state availability is removed, and these variables become known by applying the High Gain Observer (HGO) model to estimate the system variables. The construction of a seeking algorithm is used to drive the system variables and the observer output to the desired set-points that maximize the value of an objective (performance) function. In addition, Lyapunov's stability theorem and the perturbation theory including the averaging method are used in the design of the extremum seeking controller structure to check the stability of the system. Finally, the simulation results show the performance of the proposed procedures.

Improving the Performance of a Hierarchical Traffic Flow Control Framework Using Lyapunov-Based Switched Newton Extremum Seeking

Abstract The primary aim of this research paper is to enhance the effectiveness of a two-level infrastructure-based control framework utilized for traffic management in expansive networks. The lower-level controller adjusts vehicle velocities to achieve the desired density determined by the upper-level controller. The upper-level controller employs a novel Lyapunov-based switched Newton extremum seeking control approach to ascertain the optimal vehicle density in congested cells where downstream bottlenecks are unknown, even in the presence of disturbances in the model. Unlike gradient-based approaches, the Newton algorithm eliminates the need for the unknown Hessian matrix, allowing for user-assignable convergence rates. The Lyapunov-based switched approach also ensures asymptotic convergence to the optimal set point. Simulation results demonstrate that the proposed approach, combining Newton’s method with user-assignable convergence rates and a Lyapunov-based switch, outperforms gradient-based extremum seeking in the hierarchical control framework.

Perturbation observer design based on extremum seeking control: An analytical insight

In this paper, an optimal extended state observer (ESO) is designed for nonlinear integral chain system. For the first time, in this paper, proportional–integral (PI) extremum seeking control (ESC; PIESC) approach is applied to design a perturbation observer by minimizing the estimation error. The PI type structure of the ESC provides fast transient response for the closed-loop system to attain optimum equilibrium point. A mathematical proof is presented to show that the average error dynamic asymptotically converges to zero while the estimation error is minimized. The simulation results verify that the estimation error achieved by the proposed PIESC-based ESO (PIESC-ESO) is found significantly less than that achieved by the classic ESO and PSO-based ESO. Moreover, if slow changes occur within the system or external to it, the PIESC-ESO is still able to modify the observer gains such that the estimation error is minimized again. Therefore, this may help PIESC-ESO to be used in practical applications.

Extremum seeking control to optimize mineral recovery of a flotation circuit using peak air recovery

A flotation circuit is controlled in simulation using an extremum seeking control (ESC) approach to keep the cells operating at the optimal operating point, as represented by peak air recovery. It is assumed that optimal performance is achieved at this operating point where the froth layer is stable, and the mineral recovery of the flotation cell is maximized. Two gradient-based ESCs, a classical perturbation-based ESC and a time-varying ESC, as well as a non-gradient-based direct search Nelder–Mead simplex ESC, are compared on the flotation circuit to steer the plant through an unknown static map towards the peak in air recovery. The three ESCs can respectively optimize the flotation circuit and find the peak air recovery operating point. The simplex ESC can converge quickly to the optimum but does not adapt to changing conditions. The gradient-based ESCs can track the time-varying peak air recovery operating point and adapt to an external disturbance. Although the three ESC methods are not dependent on a process model to optimize the plant, their convergence times are relatively slow. The ESCs are ideally suited for model-independent long-term automated optimization of a flotation circuit with a slow time-varying optimal operating point.

Cooperative distributed extremum seeking control for coupled multiagent systems based on distributed identification‐gradient tracking

SummaryThis paper proposes a cooperative distributed extremum seeking control (ESC) approach for solving distributed optimization problems in static and dynamical multiagent systems, where the agents' coupled dynamics are unknown and only local cost values can be measured. The approach begins with each agent estimating the local gradient information of the unknown local cost function through recursive identification using measured local cost values. Subsequently, the distributed identification‐gradient tracking (DIGT) algorithm is employed to enable each agent to track the global gradient information of the global cost function, utilizing the estimated local gradient information from itself and its neighbors in the network. The proposed cooperative distributed ESC method enables the multiagent system to reach the extreme point of the global cost function under model‐free scenarios. Furthermore, the convergence properties of the approach are established for both static and dynamical multiagent systems. Finally, numerical examples are presented to demonstrate the effectiveness and stability of the proposed approach.

Learning generalized Nash equilibria in monotone games: A hybrid adaptive extremum seeking control approach

In this paper, we solve the problem of learning a generalized Nash equilibrium (GNE) in merely monotone games. First, we propose a novel continuous semi-decentralized solution algorithm without projections that uses first-order information to compute a GNE with a central coordinator. As the second main contribution, we design a gain adaptation scheme for the previous algorithm in order to alleviate the problem of improper scaling of the cost functions versus the constraints. Third, we propose a data-driven variant of the former algorithm, where each agent estimates their individual pseudogradient via zeroth-order information, namely, measurements of their individual cost function values. Finally, we apply our method to a perturbation amplitude optimization problem in oil extraction engineering.

Quasi-Newton extremum seeking control based on incremental recursive identification

This paper proposes a novel quasi-Newton extremum seeking control method based on incremental recursive identification for a real-time optimization problem. The optimization of static map and dynamical system are both considered where the dynamics of the nonlinear system and the cost function to be minimized are unknown. Firstly, the proposed method only employs the measured function values to effectively estimate the time-varying gradient of the unknown cost function by using incremental recursive least square identification. Then, an extremum seeking control (ESC) approach based on the quasi-Newton direction is developed to improve convergence speed, which only requires the estimated gradient of the objective function without Hessian information. Furthermore, the convergence properties of the proposed ESC method are established for static map and dynamical system. Finally, we demonstrate the superiority and effectiveness of the proposed method using three examples.

Disturbance Rejection Extremum Seeking: A Sliding Mode Control Approach

This paper investigates the sliding mode extremum seeking control problem for nonlinear systems with disturbances to improve the disturbance rejection capability, delivering promising searching accuracy and efficiency under disturbances. The existing results in sliding mode extremum seeking control suffer from the drawback of unsatisfactory transient performance and slow convergence speed when disturbances, especially unmatched disturbances, are injected into the system. To this end, a disturbance rejection sliding mode extremum seeking control approach is proposed in this paper. A nonlinear disturbance observer is developed for disturbance estimation, and then a disturbance rejection sliding mode extremum seeking control algorithm is proposed by exploiting the disturbance estimation. Under the proposed approach, the disturbances can be online estimated and compensated. As a result, the cost function is less disturbed by unknown disturbances, and faster convergence speed as well as shorter recovery time can be obtained. The proposed method is applied for directional antenna automatic tracking system. Simulation results show that the newly proposed searching control approach outperforms the baseline extremum searching approaches in terms of much more reliable and high efficiency searching performance under disturbances.

Traffic Congestion Control Using Distributed Extremum Seeking and Filtered Feedback Linearization Control Approaches

This letter presents a hierarchical infrastructure-based control algorithm to manage mainstream traffic flow on freeways. At the upper level, a distributed Extremum-Seeking control approach is employed to determine the optimal density of vehicles in a congested cell. The local objective function is defined such that the average flow within the target cell is maximized to resolve the congestion, and the flow difference with its upstream cell is minimized to prevent back-propagating the congestion. At the lower level, a distributed Filtered Feedback Linearization controller is used to update the suggested velocity communicated to the vehicles so that the desired density determined by the upper level can be achieved in each cell. We adopted the METANET model to describe the aggregated dynamics of the traffic network. We tested the performance of these controllers via a MATLAB-VISSIM COM interface. The results demonstrate that the designed distributed controllers can achieve the desired closed-loop performance despite unknown disturbances in an uncertain large-scale traffic network.

Real-time energy-efficient operation of a dedicated mechanical subcooling based transcritical CO2 heat pump water heater via multi-input single-output extreme seeking control

In the field of transcritical CO2 systems applied research, dedicated mechanical subcooling (DMS) is one of the most recently investigated and promising solutions, especially for space heating. However, this kind of system brings with it an intrinsic complexity, and to pursue effective and efficient operation, it requires suitable control and optimization strategies. The paper considers a combined transcritical CO2 and R134a -based DMS system and depicts the benefits of using a Multi-Input-Single-Output (MIMO) Extremum Seeking Control (ESC) approach. The real-time optimization strategy was obtained through a Computer-Aided Control System Design (CACSD) tool, exploiting an ad hoc Dymola model of the DMS system that has been validated and tested by using experimental data. The performance of the proposed approach has been compared with standard solutions relying on physical correlations. The MISO-ESC strategy always performed better in terms of system Coefficient of Performance (COP), under both fixed and variable ambient temperature conditions. Finally, the influence of the different refrigerants in the combined DMS system was analyzed from both heating capacity and performance points of view.

A Gradient-Based Extremum Seeking Control of a Substrate- and Product-Inhibited Alcoholic Fermentation Process

In this paper we avail of an ESC approach to maximise ethanol concentration in a fermentation process carried out in a CSTR. Ethanol is aimed to be used as a fuel, so the reactor is fed with a stream of large concentration of sugar; hence, the model considers fermentation inhibition due to large concentrations of sugar and ethanol. By considering the minimisation of substrate concentration, to indirectly maximise the ethanol concentration, we apply our recent extremum seeking control (ESC) approach. The results in this paper show that the fermentation process considered in this work is suitable for ESC techniques, and that our control strategy is capable of finding the minimum value of the substrate concentration in numerical simulations.

Designing Traffic Management Strategies for a Heterogeneous Traffic Network

This paper focuses on modeling and controlling a non-signalized heterogeneous traffic network consisting of Human-Driven Vehicles (HDVs) and Autonomous Vehicles (AVs) at a macroscopic level. To describe the traffic network's behavior, we introduce an extended heterogeneous METANET model wherein each vehicle class's density and velocity dynamics are described. To develop traffic control policies, we propose a two-level control structure. At the higher level, a decentralized Extremum-Seeking (ES) control approach is designed to determine an optimal density of human and autonomous vehicles so that the average flow within the cell is maximized. At the lower level, a filtered feedback linearization control approach is designed to determine the velocity of autonomous and human-driven vehicles such that the density of each type of vehicle reaches the values set by the upper-level macroscopic controller. Numerical simulation demonstrates the effectiveness of the proposed approach in managing the traffic flow of a heterogeneous system.

Implementation of Extremum Seeking Control in an Experimental Lab-Rig

Extremum seeking control (ESC) is an adaptive control method that can be used for production optimization. In the standard ESC approach, the process optimum is found by slowly adapting the inputs based on plant gradient approximations. Since ESC “learns” the system characteristics online using only plant measurements, it requires an appropriate time-scale separation between the system response, the excitation signal used for gradient approximation, and the dynamics to be optimized. Krishnamoorthy et al. (2019) proposed a dynamic ESC that identifies a local linear dynamic model using transient measurements and applies this model to compute the plant gradients. Consequently, the time-scale separation between the system response and the excitation signal is not required and the scheme converges to the optimum faster than the standard ESC. Albeit attractive, some practical challenges have been identified, such as gradient computation and constraint handling. The main contribution of this paper is to investigate how to overcome these challenges in a more realistic setting, in which dynamic ESC is implemented on a lab-scale plant that emulates a subsea oil well network. We compare the performance of two gradient computation approaches and present a constraint handling strategy for the system of interest, where multiple units compete for limited resources. The results show that dynamic ESC is able to drive the system to its optimum without constraint violations.

Extremum Seeking With Enhanced Convergence Speed for Optimization of Time-Varying Steady-State Behavior of Industrial Motion Stages

Recently, an extremum-seeking control (ESC) approach has been developed for optimization of generically time-varying steady-state responses of nonlinear systems. A generic filter structure was introduced, the so-called dynamic cost function, which has been instrumental in facilitating the use of ESC in the more generic, time-varying context. However, the dynamic cost function must operate sufficiently slow compared to the time-varying nature of the system responses, thereby compromising the convergence speed of the ESC scheme. In this work, a modified ESC approach is proposed that incorporates explicit knowledge about the user-defined dynamic cost function, able to enhance the convergence speed of the ESC scheme. Moreover, we provide a stability analysis for this extended approach. The main contribution of this work is the experimental demonstration of both ESC approaches for the performance optimal tuning of a variable-gain control (VGC) strategy employed on a high-accuracy industrial motion stage setup, exhibiting generically time-varying steady-state responses. VGC is able to enhance the system performance by balancing the typical linear control tradeoff between low-frequency disturbance suppression properties and sensitivity to high-frequency disturbances in a more desirable manner. We experimentally show that, for the unknown disturbance situation at hand, the variable-gain controller can be automatically tuned using both ESC approaches to achieve the optimal system performance. In addition, enhanced convergence speed with the modified ESC approach is evidenced experimentally.

Real-time optimization of an ellipsoidal trajectory orientation using muscle effort with Extremum Seeking Control

We present an approach for real-time model-free optimization of the orientation of the elliptical trajectory. The performance is evaluated in simulation and experimental stages. Our model-free approach is based on the use of Extremum Seeking Control (ESC) as the real-time optimizer. The experimental stage is performed using a 4 degrees-of-freedom robot and its impedance control system to create advanced exercise protocols whereby the user is asked to follow a path against the machine’s neutral path and resistance. Another model-free approach based on the use of the global optimizer Biogeography-based optimization (BBO) was previously reported for simulation results. This last framework has a good performance as a result of exhaustive searches but with a high computational cost limiting its use on real-time experiments. The performance of the ESC approach was validated by comparing the results with those of BBO using five different arm models representing real human arms. In the real-time experiments, muscle activations representing the participation of each muscle in the training activity were measured with electromyography sensors (EMG) and real-time processed from raw signals. The muscle objective can be professionally selected by a therapist to emphasize or de-emphasize certain muscle groups. The robot establishes a zero-effort circular path, and the subject is asked to follow an elliptical trajectory. The control system produces a user-defined stiffness between the deviations from the neutral path and the force/torque applied by the subject. The results show that the framework was able to successfully find the optimal ellipsoidal orientation converging to similar solutions in short period trials of 50 s.

Maximum-likelihood extremum seeking control of microalgae cultures

Abstract This paper proposes a model-free extremum seeking control (ESC) approach to optimize the productivity of continuous cultures of microalgae, considering the dilution rate and the light intensity as manipulated variables, and the biomass concentration as single measurement. The resulting two-input single-output optimization problem is first solved using a recursive least-squares strategy based on the representation of the process by a Hammerstein block-oriented model. In order to face the presence of noise on the regressor variables (input and output signals), the problem is then reformulated as a maximum-likelihood estimation problem, which is solved on a moving horizon. Simulation results demonstrate the method performance.

Finite‐time extremum seeking control for a class of unknown static maps

SummaryThis article proposes an extremum‐seeking control design that achieves finite‐time stability of the optimum of an unknown measured cost function. The finite‐time extremum‐seeking control technique is shown to achieve finite‐time practical stability of the optimum of the cost function. The main characteristic of the proposed extremum seeking control approach is that the corresponding target averaged system achieves finite‐time stability. A simulation study is presented to demonstrate the effectiveness of the approach.

A Retrofit Hierarchical Architecture for Real-Time Optimization and Control Integration

To achieve the optimal operation of chemical processes in the presence of disturbances and uncertainty, a retrofit hierarchical architecture (HA) integrating real-time optimization (RTO) and control was proposed. The proposed architecture features two main components. The first is a fast extremum-seeking control (ESC) approach using transient measurements that is employed in the upper RTO layer. The fast ESC approach can effectively suppress the impact of plant-model mismatch and steady-state wait time. The second is a global self-optimizing control (SOC) scheme that is introduced to integrate the RTO and control layers. The proposed SOC scheme minimizes the global average loss based on the approximation of necessary conditions of optimality (NCO) over the entire operating region. A least-squares regression technique was adopted to select the controlled variables (CVs) as linear combinations of measurements. The proposed method does not require the second order derivative information, therefore, it is numerically more reliable and robust. An exothermic reaction process is presented to illustrate the effectiveness of the proposed method.

Finite-time extremum seeking control for a class of unknown static maps

This paper proposes an extremum-seeking control design that achieves finite-time stability of the optimum an unknown measured cost function. The finite-time extremum-seeking control technique is shown to achieve finite-time practical stability of the optimum of the unknown cost function. The main characteristic of the proposed extremum seeking control approach is that the target averaged system considered achieves finite-time stability of the unknown optimum. A simulation study is presented to demonstrate the effectiveness of the approach.

The design and performance analysis of multivariate fractional-order gradient-based extremum seeking approach

In this paper, a novel multivariate fractional-order (FO) Gradient-based extremum seeking control (Gradient-based ESC) approach is developed for the optimization of multivariable dynamical systems. The proposed Gradient-based ESC, utilizing FO operators, is presented to not only speed up the convergence rate and enhance the control accuracy but also improve the search efficiency of the extrema by regulating the fractional-order. For multivariable dynamical systems, the stability analysis of the proposed multivariate FO Gradient-based ESC is presented in details to guarantee the convergence performance of multi-input optimization problems. Simulation and experimental results are given to demonstrate the effectiveness and advantages of the proposed approach by comparing with the corresponding integer-order (IO) Gradient-based ESC.