Extremum Seeking for the First Derivative of Nonlinear Maps with Constant Delays via a Time-Delay Approach

Abstract

This paper introduces an extremum seeking (ES) scheme for the unknown map’s first derivative by tailoring a demodulation signal in which the closed-loop system is subject to constant transmission delays. Unlike most publications that manage delays using predictor-based methods, we are concerned with the delay-robustness of the introduced ES system via the newly developed time-delay approach. The original ES system is transformed to a nonlinear retarded-type plant with disturbances and the stability condition in the form of linear matrix inequalities is achieved. When the related bounds of the nonlinear map are not known, a rigorous practical stability proof is provided. Second, and more importantly, under the availability of prior knowledge about the nonlinear map, we are able to provide a quantitative calculation on the maximum allowable delay, the upper bound of the dither period, and the ultimate seeking error. Numerical examples are offered to exemplify the effectiveness of the proposed method.

1. Introduction

Extremum seeking (ES) is a feedback adaptive control strategy which is used to steer a dynamic plant toward the extrema of the unknown maps. The rigorous analytical framework of the ES scheme was achieved in the literature [1] in 2000. Since then, there have been many advances for the ES algorithm in theory (e.g., non-local ES stability [2], stochastic ES [3,4], sampled-data ES [5,6], Lie-bracket-based ES [7,8], distributed ES [9], ES with time delays [10,11]) as well as practical applications (e.g., [12,13,14,15,16] and references therein).

Most of the above publications are focused on searching for the extrema of an unknown objective function. However, refs. [17,18] studied the scheme in which the desired operating point of a refrigeration system is the maximum slope rather than the extremum for the objective function with sigmoid-like properties. Moreover, ref. [19] were concerned with seeking the slope by introducing a slope reference signal into a perturbation-based ES algorithm. A Newton-like ES system which employs the estimate of the second derivative of the unknown function was conducted by [20]. In addition, by generalizing the choice of demodulation signals to estimate the map’s nth derivatives on average, the authors of [21] proposed a scalar Newton-based ES system which maximizes the higher derivatives of unknown maps. These authors further extended their scheme to the case of stochastic ES in [22,23]. The paper [24] subsequently proposed the Newton-based ES scheme for maximizing the map’s higher derivatives in the presence of time delays by implementing a predictor that compensates the known and constant delay. Subsequent publications [25,26] further developed Newton-based ES for higher-derivative maps subject to time-varying and uncertain delays.

On the other hand, inspired by the research of [27], who introduced a novel time-delay approach to periodic averaging, ref. [28] developed a constructive time-delay approach for stability analysis of gradient-based ES control system. Recently, ref. [29] extended the time-delay approach for ES schemes from quadratic maps to general nonlinear maps. It should be noted that in comparison with the classical averaging method (e.g., [21,22,23,24,25,26]), the time-delay approach (refer to, e.g., [27,28,29]) is able to provide quantitative upper bounds on the parameter, that is, the dither period, which provides guidance for ES system design.

As is well known, in most cases the presence of time delays (due to computation, measurement, transmission, etc.) is a source of instability for a control system [30,31]. As a kind of real-time optimization control strategy, the impact of time delays on the ES system is particularly serious. Consequently, analyzing time-delay effects on extremum seeking for the nonlinear map’s first derivative is essential. The existing predictor-based methods (see, e.g., [10,24,25,26]) compensate for delays through various sorts of predictors, thereby accommodating arbitrarily large time delays. However, if the ES system is capable of bearing somewhat large delays, then there is no need to use predictors that introduce more complexity. In addition, predictor-based methods employ the classical averaging theory to prove the stability of ES systems. As a result, these ES systems are practically stable provided that the dither frequency is large enough while the dither magnitude is sufficiently small; in other words, it is a qualitative analysis.

Motivated by the above discussion, we focus on extremum seeking of the first derivative of nonlinear maps with constant time delays, and extend the time-delay approach to the delay-robustness analysis of the proposed ES system. First, the demodulation signal is designed to estimate the gradient of the map’s first derivative in the presence of constant delays. Second, by applying the time-delay approach to averaging, the original ES system is transformed into a neutral-type time-delay system and then further reformulated as a retarded-type model with disturbances. Finally, by employing the proposed Lyapunov functional, stability conditions in the form of linear matrix inequalities (LMIs) are rigorously derived. When the nonlinear map is unknown, we provide a rigorous practical stability analysis. In addition, we suggest a quantitative calculation on the maximum allowable delay and the upper bound of the dither period along with the ultimate seeking error if the related bounds of nonlinear maps are available. A preliminary conference paper version which considers the delay-free case has been presented in [32].

Notation 1.

Let $\mathbb{N}$ stand for the set of non-negative integers. The set of n-times continuously differentiable functions is denoted by $C^n$, with $n\in \mathbb{N}$. The notation $f^{(n)}(·)$ denotes the n-th derivative of the function $f(·)$. For a symmetric matrix X, the notation $X<0$ represents that it is a negative definite matrix.

2. Main Results

We consider the extremum seeking system for maximizing the map’s first derivative as depicted in Figure 1. The basic principle of the proposed system (Figure 1) is as follows: the periodic perturbation $S(t)$ is added to the signal $\widehat{\theta }$, which is the estimate of ${\theta }^{\ast }$. If $\widehat{\theta }$ is on either side of ${\theta }^{\ast }$, then the perturbation $S(t)$ will create a periodic response of y. With consideration of time delays, the designed demodulation signal $\mathrm{{\rm Y}}(t)$ estimates the gradient of the map’s first derivative from the output signal y. The integrator module is approximately the gradient update law, which is used to tune $\widehat{\theta }$ to ${\theta }^{\ast }$. To simplify analysis, the proposed scheme does not consider the filters. For more details, please refer to [1]. Note that the ES control scheme is subject to transmission input delay $D_{\mathrm{in}}$ and output delay $D_{\mathrm{out}}$, where $D_{\mathrm{in}}$ and $D_{\mathrm{out}}$ are known positive constants (refer to, e.g., [10,24]). The whole time delay of the closed-loop system is $D=D_{\mathrm{in}}+D_{\mathrm{out}}$. The scalar steady-state map is provided as follows:

$$y(t)=f\left(\theta (·)\right)$$

(1)

where $y(t)\in \mathbb{R}$ is the measurable output, $\theta \in \mathbb{R}$ is the scalar input, and $f(\theta )$ is a nonlinear function. The map (1) satisfies the following assumptions, which are adapted from [29].

Assumption 1.

There exist ${\theta }^{\ast }\in \mathbb{R}, \sigma >0$ and small $a>0$ such that $f(\theta )\in C^3({\theta }^{\ast }-\sigma -a,{\theta }^{\ast }+\sigma +a)$. Without loss of generality, we define

$${\theta }_{\max }=\left\{\theta \left|f^{(2)}(\theta )=0,f^{(3)}(\theta )<0\right.\right\}$$

(2)

${\theta }_{\max }$ to be the collection of maxima where $f^{(1)}(\theta )$ is locally concave. Assuming that ${\theta }^{\ast }\in {\theta }_{\max }$ and ${\theta }_{\max }\ne \varnothing$, the following relation holds:

$$f^{(2)}({\theta }^{\ast }+\Delta )·\Delta \le -\mu (\sigma )·{\Delta }^2<0,\forall 0<|\Delta |<\sigma$$

(3)

where $\mu (\sigma )>0$ is a known $\sigma$—dependent constant which decreases monotonically with respect to σ.

Assumption 2.

For any $|\Delta |<\sigma$ and a defined in Assumption 1, given $\xi \in [-1, 1]$, we have

$$\begin{array}{c}\left|f({\theta }^{\ast }+\Delta )\right|<f_0(\sigma ),\left|f^{(1)}({\theta }^{\ast }+\Delta )\right|<f_1(\sigma ),\left|f^{(2)}({\theta }^{\ast }+\Delta )\right|<f_2(\sigma ),\hfill \\ \left|f^{(3)}({\theta }^{\ast }+\Delta )\right|<f_3(\sigma ),\left|f^{(3)}({\theta }^{\ast }+\Delta +a\xi )\right|<f_3(\sigma ,a),\hfill \\ \left|f^{(2)}({\theta }^{\ast }+\Delta )-f^{(2)}({\theta }^{\ast })\right|<L|\Delta |,\hfill \end{array}$$

(4)

in which $f_0(\sigma ),f_1(\sigma ),f_2(\sigma ),f_3(\sigma )$, $f_3(\sigma ,a)$, and L are available positive constants.

Remark 1.

Note that in most of the literature based on the classical averaging method, the upper bounds involved in (4) are not necessary, making for a qualitative analysis approach. This paper also suggests a qualitative stability analysis when these upper bounds are unknown. Furthermore, if the corresponding bounds provided in (4) are available (that is, if we are faced with a “grey box” instead of “black box”), then the developed time-delay approach provides a quantitative analysis by which the maximum allowable delay, the upper bound on the dither period, and the ultimate seeking error can all be attained. It is evident that a tradeoff exists between the quantitative calculation using model knowledge and the qualitative analysis without the plant information.

Let $\widehat{\theta }(t)$ be the estimate of ${\theta }^{\ast }$. Then, the estimation error is defined as

$$\tilde{\theta }(t)=\widehat{\theta }(t)-{\theta }^{\ast }.$$

(5)

The ES regulator is stated as follows:

$$\begin{array}{c}\theta (t)=\widehat{\theta }(t)+S\left(t\right),\hfill \\ \dot{\widehat{\theta }}(t)=-k·\mathrm{{\rm Y}}(t)·y(t-D_{\mathrm{out}}),t\ge D\hfill \end{array}$$

(6)

in which the initial condition $\widehat{\theta }(t)\in \left[{\theta }^{\ast }-{\sigma }_0,{\theta }^{\ast }+{\sigma }_0\right]$ and $\dot{\widehat{\theta }}(t)\equiv 0$ for $t\in \left[0,D\right]$, i.e., the ES controller starts to work when the measured output signal arrives at the controller side at time $t=D$. Moreover, $k>0$ is the adaptation gain and the excitation signals are provided as follows:

$$\begin{array}{c}S(t)=a·\sin \left(\omega t\right),\hfill \\ \mathrm{{\rm Y}}(t)=\frac{8}{a^2}·\cos \left(2\omega (t-D)\right)\hfill \end{array}$$

(7)

where a and $\omega$ are the amplitude and frequency of the dither signal $S(t)$, respectively. The demodulation signal $\mathrm{{\rm Y}}(t)$ provided in (7) is designed to estimate the gradient of the map’s first derivative in the presence of constant delays, which is adapted from [23]. It should be noted that the only obtainable measurements are from the map rather than its derivative.

Based on (5)–(7), we consider the Taylor expansion of $y(t-D_{\mathrm{out}})$ such that

$$\begin{array}{c}y(t-D_{\mathrm{out}})\hfill \\ =f\left(\theta (t-D_{\mathrm{in}}-D_{\mathrm{out}})\right)=f\left(\theta (t-D)\right)\hfill \\ =f\left({\theta }^{\ast }+\tilde{\theta }(t-D)+a\sin (\omega (t-D))\right)\hfill \\ =f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)+f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)·a\sin (\omega (t-D))\hfill \\ +\frac{1}{2!}{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)·a^2{\sin }^2(\omega (t-D))+\frac{1}{3!}B(t)·a^3{\sin }^3(\omega (t-D)),\hfill \end{array}$$

(8)

where $B(t)=f^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)+\zeta a\sin (\omega (t-D))\right)$ with $\zeta \in (0,1)$. Substituting (7) and (8) into (6) and taking the time derivative of (5) along with the second formula of (6), the dynamics of the estimation error are governed by

$$\begin{array}{c}\dot{\tilde{\theta }}(t)=\frac{-8k}{a^2}·\cos \left(2\omega (t-D)\right)·y(t-D_{\mathrm{out}})\hfill \\ =\frac{-8k}{a^2}\cos \left(2\omega (t-D)\right)[f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)+f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)·a\sin (\omega (t-D))\hfill \\ +\frac{a^2}{2}{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)·\frac{1-\cos (2\omega (t-D))}{2}+\frac{a^3}{6}B(t)·{\sin }^3(\omega (t-D))]\hfill \\ =\frac{-8k}{a^2}f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(2\omega (t-D)\right)\hfill \\ +\frac{-8k}{a}{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(2\omega (t-D)\right)\sin (\omega (t-D))\hfill \\ -2k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(2\omega (t-D)\right)\hfill \\ +k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\hfill \\ +k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(4\omega (t-D)\right)\hfill \\ -\frac{4ka}{3}B(t)\cos \left(2\omega (t-D)\right){\sin }^3(\omega (t-D)).\hfill \end{array}$$

(9)

We re-scale the dither period as $\epsilon =\frac{2\pi }{\omega }$ and subsequently rearrange Equation (9) as follows:

$$\dot{\tilde{\theta }}(t)=k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)+k\mathcal{F}(t)-\frac{4}{3}ka\mathcal{R}(t)$$

(10)

where

$$\begin{array}{c}\mathcal{F}(t)\triangleq -\frac{8}{a^2}f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\hfill \\ -\frac{8}{a}{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\sin \left(\frac{2\pi }{\epsilon }(t-D)\right)\hfill \\ -2f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\hfill \\ +f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{8\pi }{\epsilon }(t-D)\right),\hfill \\ \mathcal{R}(t)\triangleq B(t)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right){\sin }^3\left(\frac{2\pi }{\epsilon }(t-D)\right).\hfill \end{array}$$

(11)

The stability analysis for the ES system in (10) is performed by employing the classical averaging method (refer to [33]) in the existing literature (see, e.g., [24]). To be specific, by setting the dither frequency $\frac{1}{\epsilon }$ large enough compared to the regulator gain k, the state $\tilde{\theta }(·)$ varies slowly relative to the high-frequency dither signals. Then, $f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)$, $f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)$, and $f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)$ are ‘frozen’ and we treat them as constants. In adddition, the last term $-\frac{4}{3}ka\mathcal{R}(t)$ on the right-hand side of (10) is neglected by choosing the parameter a to be sufficiently small. Therefore, we derive the corresponding averaged system

$$\begin{array}{c}{\dot{\tilde{\theta }}}_{av}(t)=k{f}^{(2)}\left({\theta }^{\ast }+{\tilde{\theta }}_{av}(t-D)\right),\hfill \end{array}$$

(12)

in which the following trigonometric integrals are utilized:

$$\begin{array}{c}\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau =0,\hfill \\ \frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau =0,\hfill \\ \frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau =0.\hfill \end{array}$$

Considering $D\equiv 0$ in (12), under (3) in Assumption 1 and $k>0$, if ${\tilde{\theta }}_{av}(·)>0$, then $f^{(2)}\left({\theta }^{\ast }+{\tilde{\theta }}_{av}(·)\right)<0$ such that the derivative ${\dot{\tilde{\theta }}}_{av}(t)<0$, which implies that ${\tilde{\theta }}_{av}(·)$ will converge to zero and vice versa. As a result, the delay-free averaged system is stable. For the case with $D>0$ in (12), we are interested in the robustness against the length of the delay such that the averaged nonlinear system in (12) is able to maintain stability. It is worth mentioning that for the scenario of quadratic maps involved in a linear time-delay model (see [34], Corollary 1), the designers can decrease the decay rate k with the purpose of handling any large delay D in principle. Here, we are faced with the comparable fact for the nonlinear time-delay system that we are also able to deal with any large delay, but with almost zero decay rate. However, the real-time convergence speed of the ES system should be guaranteed, meaning that the controller gain cannot be arbitrarily small in practice. Therefore, we should keep a balance between the convergence rate and the maximum allowable delay.

In the following, the time-delay approach (see [27,28]) is performed to cope with the original ES system in (10). Specifically, we integrate $\mathcal{F}(t)$ over the interval $[t-\epsilon ,t]$ for $t\ge \epsilon +D$ such that

$$\begin{array}{c}\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\mathcal{F}(\tau )\mathrm{d}\tau \hfill \\ =-\frac{8}{a^2}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^{t}f\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ -\frac{8}{a}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ -2·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ +\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau .\hfill \end{array}$$

(13)

For the first term on the right-hand side of (13), the following relation is true:

$$\begin{array}{c}-\frac{8}{a^2}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^{t}f\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ =\frac{8}{a^2}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\left[f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)-f\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\right]\mathrm{d}\tau \hfill \\ =\frac{8}{a^2}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right){\int }_{\tau -D}^{t-D}{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \hfill \end{array}$$

(14)

where we use the fact that ${\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau ·f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)=0$. To handle the second term on the right-hand side of (13), we obtain

$$\begin{array}{c}-\frac{8}{a}·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ =\frac{8}{a\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)\left[f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)-f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\right]\mathrm{d}\tau \hfill \\ =\frac{8}{a\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right){\int }_{\tau -D}^{t-D}{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau ,\hfill \end{array}$$

(15)

in which we have utilized ${\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau ·f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)=0$. Similarly, we manage the third term on the right-hand side of (13) as follows:

$$\begin{array}{c}-2·\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ =\frac{2}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\left[f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)-f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\right]\mathrm{d}\tau \hfill \\ =\frac{2}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right){\int }_{\tau -D}^{t-D}{f}^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \hfill \end{array}$$

(16)

where ${\int }_{t-\epsilon }^t\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau ·f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)=0$ is employed. The last term on the right-hand side of (13) is calculated as

$$\begin{array}{c}\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau \hfill \\ =-\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)\left[f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)-f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(\tau -D)\right)\right]\mathrm{d}\tau \hfill \\ =-\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right){\int }_{\tau -D}^{t-D}{f}^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau ,\hfill \end{array}$$

(17)

where we use the fact that ${\int }_{t-\epsilon }^t\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)\mathrm{d}\tau ·f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)=0$.

After that, by substituting (14)–(17) into (13) we obtain the following relation:

$$\begin{array}{c}\frac{1}{\epsilon }{\int }_{t-\epsilon }^t\mathcal{F}(\tau )\mathrm{d}\tau =\frac{8}{a^2}{Y}_1(t)+\frac{8}{a}{Y}_2(t)+2Y_3(t)+Y_4(t)\hfill \end{array}$$

(18)

where

$$\begin{array}{c}{Y}_1(t)\triangleq \frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau ,\hfill \\ Y_2(t)\triangleq \frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau ,\hfill \\ Y_3(t)\triangleq \frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)f^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau ,\hfill \\ Y_4(t)\triangleq -\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)f^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau .\hfill \end{array}$$

(19)

We introduce the variable as follows:

$$\begin{array}{c}G(t)=\frac{1}{\epsilon }{\int }_{t-\epsilon }^t(\tau -t+\epsilon )\mathcal{F}(\tau )\mathrm{d}\tau ,\hfill \end{array}$$

(20)

then the following equation holds:

$$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left[\tilde{\theta }(t)-kG(t)\right]=\dot{\tilde{\theta }}(t)-k\mathcal{F}(t)+\frac{k}{\epsilon }{\int }_{t-\epsilon }^t\mathcal{F}(\tau )\mathrm{d}\tau .\hfill \end{array}$$

(21)

By substituting (10) and (18) into (21), the closed-loop error system is provided by

$$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left[\tilde{\theta }(t)-kG(t)\right]\hfill \\ =k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)+\frac{8k}{a^2}{Y}_1(t)+\frac{8k}{a}{Y}_2(t)+2k{Y}_3(t)+k{Y}_4(t)-\frac{4}{3}ka\mathcal{R}(t),t\ge \epsilon +D,\hfill \end{array}$$

(22)

with $\dot{\tilde{\theta }}(t)$ defined by (10).

Note that based on Assumption 2, the map $f(·)$ along with its derivatives $f^{(1)}(·)$, $f^{(2)}(·)$, and $f^{(3)}(·)$ are of order $O(1)$, which indicates that $\dot{\tilde{\theta }}(t)$ is $O\left(\frac{1}{a^2}\right)$. We subsequently obtain the perturbation terms $G(t)$ and $Y_i(t),i=1,2,3,4$ of order $O\left(\frac{\epsilon }{a^2}\right)$, while the term $-\frac{4}{3}ka\mathcal{R}(t)$ is $O(a)$. All of these disturbance terms involved in (22) will decay to zero when $\epsilon$ is chosen to be sufficiently small and $a={\epsilon }^{\frac{1}{5}}$.

Remark 2.

It is worth mentioning that based on the classical averaging theory, the original ES system (10) is approximated to the averaged system (12), in which the terms $k\mathcal{F}(t)$ and $-\frac{4}{3}ka\mathcal{R}(t)$ provided in (10) are omitted. However, the term $\mathcal{F}(t)$ is equivalently transformed into the disturbance terms $Y_i(t),i=1,2,3,4$ and the term $-\frac{4}{3}ka\mathcal{R}(t)$ is preserved by applying the proposed time-delay approach. It is apparent that the transformation (10) → (22) does not employ any approximation, i.e., the resulting time-delay plant (22) is an accurate model for the original ES system (10). Consequently, the stability of the system (10) can be deduced from that of the system (22).

In order to simplify the stability analysis, we further introduce the following change of variables (refer to [35]):

$$z(t)=\tilde{\theta }(t)-kG(t)$$

(23)

then rewrite the neutral-type retarded system (22) as follows:

$$\begin{array}{c}\dot{z}(t)=k{f}^{(2)}\left({\theta }^{\ast }+z(t-D)+kG(t-D)\right)+\frac{8k}{a^2}{Y}_1(t)+\frac{8k}{a}{Y}_2(t)\hfill \\ +2k{Y}_3(t)+k{Y}_4(t)-\frac{4}{3}ka\mathcal{R}(t)\hfill \\ =k{f}^{(2)}\left({\theta }^{\ast }+z(t-D)\right)+k\Phi (t)+\frac{8k}{a^2}{Y}_1(t)+\frac{8k}{a}{Y}_2(t)\hfill \\ +2k{Y}_3(t)+k{Y}_4(t)-\frac{4}{3}ka\mathcal{R}(t),t\ge \epsilon +D\hfill \end{array}$$

(24)

where

$$\Phi (t)\triangleq f^{(2)}\left({\theta }^{\ast }+z(t-D)+kG(t-D)\right)-f^{(2)}\left({\theta }^{\ast }+z(t-D)\right).$$

(25)

Let us suppose that the overall bound of the estimation error satisfies the following relation:

$$\begin{array}{c}|\tilde{\theta }(t)|<\sigma ,t\ge 0.\hfill \end{array}$$

(26)

According to the definitions provided in (5)–(7), we then have ${\theta }^{\ast }-\sigma -a<\theta (t)={\theta }^{\ast }+\tilde{\theta }(t)+a\sin \left(\omega t\right)<{\theta }^{\ast }+\sigma +a$, which is identical to that of Assumption 1. Combining the initial condition underneath (6) and the LMI conditions (30) in Theorem 1, the overall bound (26) will always be guaranteed. Based on the bounds (4) in Assumption 2 and the overall bound (26), the upper bounds on $\mathcal{F}(t),\mathcal{R}(t)$ and $\dot{\tilde{\theta }}(t)$ are calculated from (11) and (10), respectively:

$$\begin{array}{c}\left|\mathcal{F}(t)\right|\le \left|\frac{8}{a^2}f\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\right|\hfill \\ +\left|\frac{8}{a}{f}^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\sin \left(\frac{2\pi }{\epsilon }(t-D)\right)\right|\hfill \\ +\left|2f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right)\right|\hfill \\ +\left|f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\cos \left(\frac{8\pi }{\epsilon }(t-D)\right)\right|\hfill \\ <\frac{8}{a^2}{f}_0(\sigma )+\frac{8}{a}{f}_1(\sigma )+3f_2(\sigma )\triangleq {\Delta }_{\mathcal{F}},\hfill \\ \left|\mathcal{R}(t)\right|=\left|B(t)\cos \left(\frac{4\pi }{\epsilon }(t-D)\right){\sin }^3\left(\frac{2\pi }{\epsilon }(t-D)\right)\right|\hfill \\ \le \left|B(t)\right|<f_3(\sigma ,a)\triangleq {\Delta }_{\mathcal{R}},\hfill \\ \left|\dot{\tilde{\theta }}(t)\right|\le \left|k{f}^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(t-D)\right)\right|+\left|k\mathcal{F}(t)\right|+\left|\frac{4}{3}ka\mathcal{R}(t)\right|\hfill \\ <k\left(f_2(\sigma )+{\Delta }_{\mathcal{F}}+\frac{4}{3}a{\Delta }_{\mathcal{R}}\right)\triangleq {\Delta }_{\theta }.\hfill \end{array}$$

(27)

Accordingly, from (19), (20), and (24), the following relations are satisfied:

$$\begin{array}{c}|G(t)|=\frac{1}{\epsilon }\left|{\int }_{t-\epsilon }^t(\tau -t+\epsilon )\mathcal{F}(\tau )\mathrm{d}\tau \right|\le \frac{1}{\epsilon }{\int }_{t-\epsilon }^t(\tau -t+\epsilon )\mathrm{d}\tau ·|\mathcal{F}(\tau )|<\frac{{\Delta }_{\mathcal{F}}}{2}\epsilon ,\hfill \\ |Y_1(t)|=\frac{1}{\epsilon }\left|{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)f^{(1)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \right|\hfill \\ <\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\mathrm{d}s\mathrm{d}\tau ·f_1(\sigma ){\Delta }_{\theta }=\frac{f_1(\sigma ){\Delta }_{\theta }}{2}\epsilon ,\hfill \\ |Y_2(t)|=\frac{1}{\epsilon }\left|{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)\sin \left(\frac{2\pi }{\epsilon }(\tau -D)\right)f^{(2)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \right|\hfill \\ <\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\mathrm{d}s\mathrm{d}\tau ·f_2(\sigma ){\Delta }_{\theta }=\frac{f_2(\sigma ){\Delta }_{\theta }}{2}\epsilon ,\hfill \\ |Y_3(t)|=\frac{1}{\epsilon }\left|{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{4\pi }{\epsilon }(\tau -D)\right)f^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \right|\hfill \\ <\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\mathrm{d}s\mathrm{d}\tau ·f_3(\sigma ){\Delta }_{\theta }=\frac{f_3(\sigma ){\Delta }_{\theta }}{2}\epsilon ,\hfill \\ |Y_4(t)|=\frac{1}{\epsilon }\left|{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\cos \left(\frac{8\pi }{\epsilon }(\tau -D)\right)f^{(3)}\left({\theta }^{\ast }+\tilde{\theta }(s)\right)\dot{\tilde{\theta }}(s)\mathrm{d}s\mathrm{d}\tau \right|\hfill \\ <\frac{1}{\epsilon }{\int }_{t-\epsilon }^t{\int }_{\tau -D}^{t-D}\mathrm{d}s\mathrm{d}\tau ·f_3(\sigma ){\Delta }_{\theta }=\frac{f_3(\sigma ){\Delta }_{\theta }}{2}\epsilon ,\hfill \\ |\dot{z}(t)|\le |k{f}^{(2)}\left({\theta }^{\ast }+z(t-D)+kG(t-D)\right)|+\left|\frac{8k}{a^2}{Y}_1(t)\right|\hfill \\ +\left|\frac{8k}{a}{Y}_2(t)\right|+|2k{Y}_3(t)|+|k{Y}_4(t)|+\left|\frac{4}{3}ka\mathcal{R}(t)\right|\hfill \\ <k\left(f_2(\sigma )+\frac{4f_1(\sigma ){\Delta }_{\theta }}{a^2}\epsilon +\frac{4f_2(\sigma ){\Delta }_{\theta }}{a}\epsilon +\frac{3f_3(\sigma ){\Delta }_{\theta }}{2}\epsilon +\frac{4}{3}a{\Delta }_{\mathcal{R}}\right)\triangleq {\Delta }_z.\hfill \end{array}$$

(28)

Additionally, based on the bounds in (4) and (28) and using the differential mean value theorem, (25) yields

$$\begin{array}{c}\left|\Phi (t)\right|\le \left|f^{(3)}\left({\theta }^{\ast }+z(t-D)+\zeta kG(t-D)\right)\right|·\left|kG(t-D)\right|<f_3(\sigma )\frac{k{\Delta }_{\mathcal{F}}}{2}\epsilon ,\hfill \end{array}$$

(29)

where $\zeta \in (0,1)$.

Theorem 1.

Consider the closed-loop system consisting of the scalar map (1) and the ES regulator (6) under Assumptions 1 and 2 as well as the initial condition $|\tilde{\theta }(0)|\le {\sigma }_0<\sigma$ for $t\in \left[0,D\right]$. Given tuning parameters ${\sigma }_0,D,k,a,\delta ,{\epsilon }^{\ast }>0$, where ${\epsilon }^{\ast }$, a, and D are small, let the scalar decision variables ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\lambda }_6,{\lambda }_7>0$, and $P\ge 1,Q>0,R>0$ satisfy the following LMIs:

$$\begin{array}{c}{\Omega }_0=\left[\begin{array}{cc}\Psi & \Xi \\ {\Xi }^{\mathrm{T}}& -R\end{array}\right]<0,\hfill \\ {\Omega }_1=\frac{\varpi }{2\delta }<{\left(\sigma -\frac{k{\Delta }_{\mathcal{F}}}{2}{\epsilon }^{\ast }\right)}^2\hfill \\ {\Omega }_2=(P+QD){\nu }^2+R{D}^3{\Delta }_z^2<{\left(\sigma -\frac{k{\Delta }_{\mathcal{F}}}{2}{\epsilon }^{\ast }\right)}^2\hfill \end{array}$$

(30)

where Ψ is the symmetric matrix composed of

$$\begin{array}{c}{\Psi }_{11}=-\left(2k\mu (\sigma )-2\delta \right)P+Q,{\Psi }_{13}=kLP,{\Psi }_{15}=kP,\hfill \\ {\Psi }_{16}=\frac{8k}{a^2}P,{\Psi }_{17}=\frac{8k}{a}P,{\Psi }_{18}=2kP,{\Psi }_{19}=kP,\hfill \\ {\Psi }_{1,10}=\frac{4}{3}kaP,{\Psi }_{22}=-Q{e}^{-2\delta D}+{\lambda }_1{L}^2,{\Psi }_{33}=-R{e}^{-2\delta D},\hfill \\ {\Psi }_{44}=-{\lambda }_1,{\Psi }_{55}=-\frac{{\lambda }_2}{{\epsilon }^{\ast }},{\Psi }_{66}=-\frac{{\lambda }_3}{{\epsilon }^{\ast }},{\Psi }_{77}=-\frac{{\lambda }_4}{{\epsilon }^{\ast }},\hfill \\ {\Psi }_{88}=-\frac{{\lambda }_5}{{\epsilon }^{\ast }},{\Psi }_{99}=-\frac{{\lambda }_6}{{\epsilon }^{\ast }},{\Psi }_{10,10}=-{\lambda }_{7}a,\hfill \end{array}$$

(31)

with the other terms being zero and with

$$\begin{array}{c}\Xi ={\left[0,0,0,k,k,\frac{8k}{a^2},\frac{8k}{a},2k,k,\frac{4}{3}ka\right]}^{\mathrm{T}}DR,\hfill \\ \nu ={\sigma }_0+{\Delta }_{\theta }{\epsilon }^{\ast }+\frac{k{\Delta }_{\mathcal{F}}}{2}{\epsilon }^{\ast },\hfill \\ \varpi =\frac{{\lambda }_2{k}^2{f}_3^2(\sigma ){\Delta }_{\mathcal{F}}^2}{4}{\epsilon }^{\ast }+\frac{{\lambda }_3{f}_1^2(\sigma ){\Delta }_{\theta }^2}{4}{\epsilon }^{\ast }+\frac{{\lambda }_4{f}_2^2(\sigma ){\Delta }_{\theta }^2}{4}{\epsilon }^{\ast }+\frac{{\lambda }_5{f}_3^2(\sigma ){\Delta }_{\theta }^2}{4}{\epsilon }^{\ast }+\frac{{\lambda }_6{f}_3^2(\sigma ){\Delta }_{\theta }^2}{4}{\epsilon }^{\ast }+{\lambda }_{7}a{\Delta }_{\mathcal{R}}^2.\hfill \end{array}$$

(32)

Then, for any $\epsilon \in (0,{\epsilon }^{\ast }]$, the estimation error satisfies

$$\begin{array}{c}|\tilde{\theta }(t)|<|\tilde{\theta }(D)|+{\Delta }_{\theta }(t-D)<\sigma ,t\in [D,\epsilon +D],\hfill \\ |\tilde{\theta }(t)|<{\left[\left((P+QD){\nu }^2+R{D}^3{\Delta }_z^2\right)e^{-2\delta (t-\epsilon -D)}+\left(1-e^{-2\delta (t-\epsilon -D)}\right)\frac{\varpi }{2\delta }\right]}^{\frac{1}{2}}\hfill \\ +\frac{k{\Delta }_{\mathcal{F}}}{2}\epsilon <\sigma ,t\in [\epsilon +D,\infty ),\hfill \end{array}$$

(33)

and is exponentially attracted to the set

$$\Theta =\left\{\tilde{\theta }\in \mathbb{R}:|\tilde{\theta }|<\sqrt{\frac{\varpi }{2\delta }}+\frac{k{\Delta }_{\mathcal{F}}}{2}\epsilon \right\},$$

(34)

which is adjustable through the tuning parameters ε and a. Furthermore, LMIs (30) are always feasible for sufficient small ${\epsilon }^{\ast }$, a, and D.

Proof of Theorem 1.

Please refer to Appendix A. □

It should be noted that the exponential convergence rate of $|\tilde{\theta }(t)|$ in (33) is $\delta$ for $t\ge \epsilon +D$, which is controlled by the adaptation gain k such that $0<\delta <k\mu (\sigma )$ in ${\Psi }_{11}$ of (31). It is also worth mentioning that the overall bound provided in (26) indicates that ES can be performed within this domain, whereas the ultimate bound (34) represents the ultimate seeking error of the ES system as $t\to \infty$, which can be adjusted via the parameters $\epsilon$ and a.

Remark 3.

Notice that LMIs (30) are always feasible for small enough ${\epsilon }^{\ast }$, a, and D, which means that we provide a rigorously practical stability analysis. In addition, Theorem 1 is able to suggest a quantitative estimation on the upper bound of the dither period and the maximum allowable delay when the bounds for the nonlinear maps in Assumption 2 are available.

3. Numerical Simulations

Consider the following nonlinear scalar map:

$$\begin{array}{c}f(\theta )=-{(\theta -0.5)}^3+\theta .\hfill \end{array}$$

(35)

This map (35) and its first derivative are depicted in Figure 2a. Obviously, it has a maximum at ${\theta }^{\ast }=0.5$ for the map’s first derivative. We can employ the ES regulator (6) with $k=0.01,a=0.1$ to search for the extrema of the first derivative of the map (35) with delays. For the map (35) with a given $\sigma$, the Real Bounds are from the exact bounds calculated by (4), while the Estimated Bounds are somewhat larger than the Real Bounds. In the case with $\sigma =1$, the corresponding bounds are as follows: Real Bounds: $f_0(\sigma )=0.5$, $f_1(\sigma )=2.0, f_2(\sigma )=6.0, f_3(\sigma )=6.0, f_3(\sigma ,a)=6.0, L=6.0, \mu (\sigma )=6.0$; Estimated Bounds: $f_0(\sigma )=1.0,f_1(\sigma )=2.2,f_2(\sigma )=6.3,f_3(\sigma )=6.2,f_3(\sigma ,a)=6.2,L=6.4,\mu (\sigma )=6.0$.

For the two groups of data, the results that attained from Theorem 1 are listed in

Table 1. Real bounds case.

	$\mathit{\epsilon }$	$\mathit{\delta }$	${\mathit{\sigma }}_0$	$\mathit{\sigma }$	D	UB
[32]	0.0001	0.03	0.9	1.0	-	0.236
Theorem 1	0.0001	0.03	0.9	1.0	1.0	0.268
[32]	0.0002	0.03	0.9	1.0	-	0.339
Theorem 1	0.0002	0.03	0.9	1.0	1.0	0.385

and

Table 2. Estimated bounds case.

	$\mathit{\epsilon }$	$\mathit{\delta }$	${\mathit{\sigma }}_0$	$\mathit{\sigma }$	D	UB
[32]	0.0001	0.03	0.9	1.0	-	0.329
Theorem 1	0.0001	0.03	0.9	1.0	1.0	0.380
[32]	0.0002	0.03	0.9	1.0	-	0.521
Theorem 1	0.0002	0.03	0.9	1.0	1.0	0.600

, respectively, where “UB” denotes the ultimate bound $\underset{t\to \infty }{lim}|\tilde{\theta }(t)|$, that is, the ultimate seeking error provided by (34). As shown in Table 1 and Table 2, a smaller dither period $\epsilon$ results in a tighter ultimate bound. It is also observed that more accurate map information results in a smaller ultimate bound. Note that existing predictor-based methods cannot yield quantitative analysis results. Figure 2 reveals the numerical simulation results, where $\epsilon =0.002, a=0.05$. As shown in Figure 2b, the system output practically converges to the extrema of the mapping function’s first derivative. It is evident from Figure 2b,c that increasing the time delay will lead to the system tending to be unstable. The results in Figure 2c,d confirm that we can reduce the controller gain to compensate for the effects of large time delays. Meanwhile, the controller gain cannot be arbitrarily small due to the requirement of guaranteeing the real-time convergence speed of the ES system.

4. Conclusions

In this article, an extremum seeking scheme which maximizes the first derivative of an unknown scalar map with constant delays is carried out in which the only available measurements are from the unknown map itself. The gradient of the mapping function’s first derivative is exactly extracted via the proposed demodulation signal. Subsequently, by applying the time-delay approach to averaging, the original ES system is reformulated as a retarded-type model with disturbances. Finally, the stability conditions are rigorously derived using the proposed Lyapunov functional. It is worth mentioning that under the assumption of some known bounds about the nonlinear map, we have obtained a quantitative analysis which is capable of calculating the ultimate seeking error, the upper bounds on the dither period, and the transmission delay that the original system is able to tolerate. Simulation results validate the conclusion from theoretical analysis that the time delay degrades the stability of the proposed system. Moreover, detrimental effects can be effectively mitigated through proper controller gain reduction. Extensions to time-varying delay cases will be explored in future studies.