Uniform quasi-convex optimisation via Extremum Seeking

The paper deals with a well-known extremum seeking scheme by proving uniformity properties with respect to the amplitudes of the dither signal and of the cost function. Those properties are then used to show that the scheme guarantees the global minimiser to be semi-global practically stable despite the presence of local saddle points. To achieve these results, we analyse the average system associated with the extremum seeking scheme via arguments based on the Fourier series

Experimental Evaluation for an Extremum Seeking Control Strategy based on Input-output Correlation with a Mini-split Air Conditioning System

Extremum Seeking Control (ESC) has emerged as a model-free real-time optimization framework, typically based on dither-demodulation driven gradient estimation. However, such conventional ESC suffers from slow convergence. Salsbury et al. have recently proposed an input-output correlation based ESC (IOC-ESC) strategy anchored on a statistical analysis. The IOC-ESC algorithm is less sensitive to changes in its internal parameters because of the use of a normalized correlation coefficient in the feedback loop. The design goal of the algorithm is to have only two tunable parameters: (1) a time scale parameter that relates to the time open loop time constant of the system; and (2) the amplitude of the dither signal. A suitable set of generic internal parameters is still in the process of being identified as more test data become available from different system types. For the work reported here, the feedback gain (referred to as the tuning factor) with the IOC-ESC was also tuned for optimal performance. This study aims to conduct an experimental evaluation for the IOC-ESC strategy with a ductless mini-split air conditioning system, compared with conventional ESC (CON-ESC). The system features variable-capacity compressor operation and variable-speed operation for the evaporator and condenser fans. In this study, both single-input and two-input ESC scenarios are tested. The manipulated inputs include the evaporator and condenser fan speeds, while the total power consumption is used as feedback for all cases. The experimental setup is developed with a 9000 BTU variable-speed mini-split AC system serving a 4’x8’x6’ insulated chamber, and an electrical fan heater is used to provide an artificial heat load. The data acquisition and control algorithms are implemented on a National Instruments CompactRIO platform. Both IOC-ESC and CON-ESC are tested with the same setup. For single-input scenario, the manipulated input is the condenser fan speed. The testing results of five trials of IOC-ESC are used to evaluate the impact of the two tuning parameters, i.e. dither frequency and tuning factor, on the ESC performance. IOC-ESC#1, IOC-ESC#4 and IOC-ESC#5 have the same dither frequency but different tuning factors, while IOC-ESC#1, IOC-ESC#2 and IOC-ESC#3 have the same tuning factor but different dither frequencies. The testing results of two trials of CON-ESC are then compared with the IOC-ESC results. Both CON-ESC and IOC-ESC can effectively reduce the power consumption of the mini-split system without sacrificing zone temperature regulation. Moreover, the settling time of IOC-ESC ranges from 300 to 600 seconds, while the settling time of CON-ESC ranges from 900 to 1200 seconds. Overall, the IOC-ESC converges faster than the CON-ESC. For two-input scenario, the manipulated inputs are condenser fan speed and evaporator fan speed. The testing results of the two-input IOC-ESC are compared with the result of a two-input CON-ESC trial by Yan et al. with the same system. The settling times for CON-ESC and IOC-ESC are about 1800 and 1200 seconds, respectively. In summary, both CON-ESC and IOC-ESC can optimize the condenser fan speed and evaporator fan speed for energy efficient operation, while the IOC-ESC converges faster and has fewer tuning parameters

Extremum seeking control for truck drag reduction

The aerodynamic drag on a heavy truck tractor and semi-trailer combination can be reduced by means of a wind deflector installed on the roof of the tractor cab. The drag reduction is dependent upon the height and shape of the deflector. A variable height deflector has been constructed and tested in a wind-tunnel and on-road. In this paper, an extremum-seeking control scheme is proposed to adjust on-line the deflector height to minimize the aerodynamic drag. The effectiveness of the scheme is evaluated by simulation and its practicality is evaluated

Modélisation dynamique et commande optimale d'un système de réfrigération à base d'éjecteur

Recently, the ejector-based refrigeration system (ERS) has been widely used in the cooling industry as an appropriate alternative to the compressor-based cooling systems. However, the advantages of ERS such as the reliable operation and low operation and maintenance costs are overshadowed by its low efficiency and design complexity. In this context, this thesis presents the efforts to develop a control model enabling the ERS to operate in its optimal operational conditions. The extensive experimental studies of ERS revealed that at a fixed condenser inlet condition, there exists an optimal primary stream mass flow rate (generating pressure) that simultaneously maximizes the compression ratio (Cr) and exergy efficiency and minimizes the evaporating pressure. Then, the steady state models of the heat exchangers were developed and used to investigate the influence of the increase in generating pressure on the coefficient of performance (COP) of the system and it showed that increasing the generating pressure reduces the COP, linearly. In order to predict the choking regime of the ejector and explain the reasons of observed physical phenomenon, the 1D model of a fixed geometry ejector installed within an R245fa ERS was developed. The developed model demonstrated that the ejector operates in the subcritical mode when the generating pressure is below the Cr optimum point, while it operates in critical mode at or above the optimum generating pressure. Next, a dynamic model of the ERS was built to evaluate the ERS transient response to an increase in the primary stream mass flow rate. Since the ERS dynamics is mainly dominated by the thermal dynamics of the heat exchangers, the dynamic models of the heat exchangers were developed using the moving boundary approach and connected to the developed models of the ejector and steady state models of the pump and expansion valve to build a single dynamic model of the system. The built dynamic model of an ERS was used to estimate the time response of the system in the absence of accurate experimental data of the system’s dynamics. Finally, a control model was designed to drive an ERS towards its optimal operation condition. A self-optimizing, model-free control strategy known as Extremum seeking control (ESC) was adopted to minimize evaporating pressure in a fixed condenser thermal fluid inlet condition. The innovative ESC model named batch phasor ESC (BPESC) was proposed based on estimating the gradient by evaluating the phasor of the output, in batch time. The simulation results indicated that the designed BPESC model can seek and find the optimum evaporating pressure with good performance in terms of predicting the steady state optimal values and the convergence rates.Récemment, le système de réfrigération à éjecteur (SRE) a été largement utilisé dans l'industrie du refroidissement en tant que solution de remplacement appropriée aux systèmes de refroidissement à compresseur. Cependant, les avantages du SRE, tels que le fonctionnement fiable et les faibles couts d'exploitation et de maintenance, sont éclipsés par son faible rendement et sa complexité de conception. Dans ce contexte, ce projet de recherche de doctorat a détaillé les efforts déployés pour développer une stratégie de commande permettant au système de fonctionner dans ses conditions opérationnelles optimales. Les études expérimentales approfondies du SRE ont révélé que, dans une condition d'entrée de condensateur constante, il existe un débit massique optimal du flux primaire (générant une pression) qui maximise simultanément le taux de compression (Cr) et l'efficacité exergétique, et minimise la pression d’évaporation. Ensuite, les modèles à l’état d’équilibre des échangeurs de chaleur ont été développés et utilisés pour étudier l’influence de l’augmentation de la pression générée sur le coefficient de performance (COP) du système et il en ressort que l'augmentation de la pression génératrice réduit le COP de manière linéaire. Afin de prédire le régime d'étouffement de l'éjecteur et d'expliquer les raisons du phénomène physique observé, le modèle 1D d'un éjecteur à géométrie fixe installé dans un système SRE R245fa a été développé. Le modèle développé a démontré que l'éjecteur fonctionne en mode sous-critique lorsque la pression génératrice est inférieure au point optimal de Cr, alors qu'il fonctionne en mode critique à une pression égale ou supérieure à la pression génératrice optimale. Ensuite, un modèle dynamique du SRE a été développé pour étudier la réponse transitoire du SRE lors d’une augmentation du débit massique du flux primaire. Puisque la dynamique du SRE est principalement dominée par la dynamique thermique des échangeurs de chaleur, les modèles dynamiques des échangeurs de chaleur ont été développés à l'aide de l'approche des limites mobiles et connectés aux modèles développés de l'éjecteur et des modèles à l'état stationnaire de la pompe et de la vanne un seul modèle dynamique du système. En l’absence de données expérimentales précises sur la dynamique d’un système SRE, le modèle dynamique développé du SRE a été simulé numériquement pour étudier sa réponse temporelle. Enfin, une stratégie de commande extrêmale (ESC) a été élaboré pour régler automatiquement le SRE à ses conditions de fonctionnement optimales, c’est-à-dire pour trouver la vitesse de la pompe qui minimise la pression dans des conditions d'entrée de condenseur fixes. Afin de proposer une ESC implémentable en temps discret sur une installation réelle sujette à un bruit de mesure important et un traitement hors-ligne par trame, une nouvelle commande extrémale basée sur une approche par phaseur avec une procédure de traitement de signal par trame (BPESC) a été développée et simulée avec le modèle numérique. Les résultats de la simulation ont indiqué que le modèle BPESC peut trouver la vitesse optimale de la pompe avec de bonnes performances en termes de précision et de vitesse de convergence

An adaptive and energy-maximizing control of wave energy converters using extremum-seeking approach

In this paper, we systematically investigate the feasibility of different extremum-seeking (ES) control schemes to improve the conversion efficiency of wave energy converters (WECs). Continuous-time and model-free ES schemes based on the sliding mode, relay, least-squares gradient, self-driving, and perturbation-based methods are used to improve the mean extracted power of a heaving point absorber subject to regular and irregular waves. This objective is achieved by optimizing the resistive and reactive coefficients of the power take-off (PTO) mechanism using the ES approach. The optimization results are verified against analytical solutions and the extremum of reference-to-output maps. The numerical results demonstrate that except for the self-driving ES algorithm, the other four ES schemes reliably converge for the two-parameter optimization problem, whereas the former is more suitable for optimizing a single-parameter. The results also show that for an irregular sea state, the sliding mode and perturbation-based ES schemes have better convergence to the optimum, in comparison to other ES schemes considered here. The convergence of PTO coefficients towards the performance-optimal values are tested for widely different initial values, in order to avoid bias towards the extremum. We also demonstrate the adaptive capability of ES control by considering a case in which the ES controller adapts to the new extremum automatically amidst changes in the simulated wave conditions

Nondisturbing extremum seeking control for multi-agent industrial systems

Industrial applications of extremum seeking control (ESC) can be a hit and miss affair. Although a gain in performance can be achieved, the dither applied to excite the system causes unwanted fluctuations in the performance of the system. The fluctuations in systems with a single extremum seeking loop are generally small. However, for systems with many extremum seeking loops, the fluctuations in each loop may add up to an intolerable amount of fluctuation in the total performance. In this article, we propose a method to cancel the dither-induced fluctuations in the overall system performance to a large extent by smartly constructing the dither signals in each extremum seeking loop using a centralized coordinator. The novelty of our method lies in the direct calculation of the dither signals that avoids the heavy computations required by other methods. Moreover, we provide a solvability analysis for the problem of cancelling dither-induced fluctuations in the total performance of the system. Furthermore, a complete stability analysis of the overall ESC scheme with dither coordination is given.publishedVersio

Lyapunov based optimal control of a class of nonlinear systems

Optimal control of nonlinear systems is in fact difficult since it requires the solution to the Hamilton-Jacobi-Bellman (HJB) equation which has no closed-form solution. In contrast to offline and/or online iterative schemes for optimal control, this dissertation in the form of five papers focuses on the design of iteration free, online optimal adaptive controllers for nonlinear discrete and continuous-time systems whose dynamics are completely or partially unknown even when the states not measurable. Thus, in Paper I, motivated by homogeneous charge compression ignition (HCCI) engine dynamics, a neural network-based infinite horizon robust optimal controller is introduced for uncertain nonaffine nonlinear discrete-time systems. First, the nonaffine system is transformed into an affine-like representation while the resulting higher order terms are mitigated by using a robust term. The optimal adaptive controller for the affinelike system solves HJB equation and identifies the system dynamics provided a target set point is given. Since it is difficult to define the set point a priori in Paper II, an extremum seeking control loop is designed while maximizing an uncertain output function. On the other hand, Paper III focuses on the infinite horizon online optimal tracking control of known nonlinear continuous-time systems in strict feedback form by using state and output feedback by relaxing the initial admissible controller requirement. Paper IV applies the optimal controller from Paper III to an underactuated helicopter attitude and position tracking problem. In Paper V, the optimal control of nonlinear continuous-time systems in strict feedback form from Paper III is revisited by using state and output feedback when the internal dynamics are unknown. Closed-loop stability is demonstrated for all the controller designs developed in this dissertation by using Lyapunov analysis --Abstract, page iv