Overshoot‐free nonholonomic source seeking in 3‐D

Krstić

et al. 2020

Summary We present a Newton‐based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time‐varying delays. Dealing with time‐varying delays has impact in the predictor design in terms of the transport PDE with variable convection speed functions, the backstepping transformation as well as the conditions imposed on the delay. First, the delay can grow at a rate strictly smaller than one but not indefinitely, the delay must remain uniformly bounded. Second, the delay may decrease at any uniformly bounded rate but not indefinitely, that is, it must remain positive. We incorporate a filtered predictor feedback with a perturbation‐based estimate for the Hessian's inverse using a differential Riccati equation, where 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. Furthermore, exponential stability and convergence to a small neighborhood of the unknown extremum point are achieved for locally quadratic derivatives by using backstepping transformation and averaging theory in infinite dimensions.

“…where d and d are constants. This is consequence of ( 10) and (11). Furthermore, from Assumption 2 we conclude that (t) is strictly increasing.…”

Section: Conditions On Time-varying Delaymentioning

confidence: 55%

Section: Conditions On Time-varying Delaymentioning

confidence: 99%

Newton‐based extremum seeking of higher‐derivative maps with time‐varying delays

Rušiti

Krstić

et al. 2020

“…In Reference 2 and over 180 references therein, a large number of real‐world applications is presented, to name a few, the design of anti‐lock braking systems, 3 source seeking, 4 autonomous vehicles, 5 internal combustion engines, 6 process control, 7 bioreactors, 8 and near‐optimal compressor operation 9 . Many problems that require feedback optimization are inherently multivariable 10 .…”

Section: Introductionmentioning

confidence: 99%

Multivariable extremum seeking control via cyclic search and monitoring function

Aminde

Hsu

2020

Summary This article addresses the design of a multivariable extremum‐seeking controller for a class of nonlinear static multi‐input systems. A method is proposed based on cyclic search directions and a monitoring function to guarantee the objective function to track an adequate reference trajectory via sliding‐mode control. The method allows achieving an arbitrarily small neighborhood of the desired optimal point (extremum) starting from any initial conditions. Simulation results illustrate the significant advantages of the proposed multivariable control strategy in terms of speed of convergence, global convergence and small residual errors.

“…Extremum seeking (ES) has received great attention in the control community, being recognized as one of the powerful methodologies in adaptive systems to face control problems where the plant is poorly modeled or its model is contaminated by severe uncertainties and unmodeled dynamics. [1][2][3][4][5][6] Despite of ES has been successfully employed to many engineering applications, [7][8][9][10] the authors in Reference 11 pointed out the presence of delay as one limiting factor in the application of ES in practical situations. Although for ES with distinct input and/or output delays many predictor-based control designs 12,13 have been developed since the paper, 11 it was not until the results in Reference 11 that a rigorous ES solution has appeared to systems described by partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

“…Despite of ES has been successfully employed to many engineering applications, 7-10 the authors in Reference 11 pointed out the presence of delay as one limiting factor in the application of ES in practical situations. Although for ES with distinct input and/or output delays many predictor‐based control designs 12,13 have been developed since the paper, 11 it was not until the results in Reference 11 that a rigorous ES solution has appeared to systems described by partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

Extremum seeking for unknown scalar maps in cascade with a class of parabolic partial differential equations

Feiling

Koga

et al. 2020

Self Cite

Summary We present a generalization of the scalar gradient extremum seeking (ES) algorithm, which maximizes static maps in the presence of infinite‐dimensional dynamics described by parabolic partial differential equations (PDEs). The PDE dynamics contains reaction‐advection‐diffusion (RAD) like terms. Basically, the effects of the PDE dynamics in the additive dither signals are canceled out using the trajectory generation paradigm. Moreover, the inclusion of a boundary control for the PDE process stabilizes the closed‐loop feedback system. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the output of the unknown map. In particular, our parabolic PDE compensator employs the same perturbation‐based (averaging‐based) estimate for the Hessian of the function to be maximized applied in the previous publications free of PDEs. We prove local stability of the algorithm, real‐time maximization of the map 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. Finally, we present the generalization to the scalar Newton‐based ES algorithm, which maximizes the map's higher derivatives in the presence of RAD‐like dynamics. By modifying the demodulating signals, the ES algorithm maximizes the nth derivative only through measurements of the own 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 and remove limitations of standard gradient‐based ES. Numerical examples support the theoretical results.