# A Robust Direct Parameter Identification of Exponentially Damped Low-Frequency Oscillation in Power Systems

Authors: Zhaobi Chu; Yuanyuan Pan; Haiyan Tang; Min Zhu; Xueping Dong
Journal: Journal of Sensors
 (2020)

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## Abstract

Low-frequency oscillations in power systems can be modeled as an exponentially damped sinusoid (EDS) signal. Its frequency, damping factor, and amplitude are identified by the robust algorithm proposed in this paper. Under the condition of no noise, the exponentially convergent property of the proposed identification is proved by the use of time scale change, variable transformation, slow integral manifold, averaging method, and Lyapunov stability theorem in sequence. Under the condition of bounded additive noise, the antinoise performance of the identification of each parameter is investigated by the perturbed system theorem and error synthesis principle. The robustness of the proposed method is embodied in the following aspects: the exponential convergence for EDS signal with a wide range of frequency, especially with a rather low frequency; the boundary values of identification errors resulting from high-frequency sinusoidal noise of both frequency and damping factor can be adjusted by tuning the design parameters; and the different effects of the four design parameters on tracking performance and antinoise performance of each parameter identification. Simulation results demonstrate the performance of the algorithm and validate the conclusions.


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## Body

## 1. Introduction

Widespread used renewable and sustainable energy sources bring modern power systems electromechanical oscillations, synchronous, and subsynchronous oscillations [1–6]. The low-frequency oscillations (LFO) may result in unstable and unsecure operation of power systems, so the real-time identification of frequency, amplitude, and damping factor are still essential in recent years [7–10].In the parameter identification filed, there are mainly two challenges, model selection, and parameter estimation [11, 12]. When number of sinusoids, i.e., model order, is unknown, some techniques, such as the Bayesian method, have been developed to identify the model order [13].In a transient state of power systems, the electromechanical oscillations usually have amplitudes decaying as time and low oscillating frequency in range of 0.05-2.0 Hz [9, 10]. Because the oscillating frequency is much lower than the fundamental frequency of power systems, 50 Hz or 60 Hz, the LFOs in power systems are easily extracted from the measured voltage signals or current signals by low-pass filters. Furthermore, the most energy of oscillation usually concentrates on one frequency. So, the LFO can be modelled as low frequency exponentially damped sinusoid (EDS) signal. The key technique to identify the parameters of LFOs is now the parameter identification of one exponentially damped low-frequency sinusoid.When number of sinusoids, i.e., model order, is known, numerous studies have been developed in the signal processing field. Singular value decomposition (SVD) has been used to estimate the parameters of exponentially damped sinusoidal signals in noise [14]. In SVD method, many sampled data in a rather long period of time must be required, and heavy computation burden is implemented with the number of sinusoid components more than one.A type of methods to estimate the parameters of EDS signals are based on integration, such as a method combining Hankel singular value decomposition (HSVD) with the extended complex Kalman filter (ECKF) [2], an algorithm employing the DFT coefficients around the peaks of the components [3], a moving window discrete Fourier transform filter (MWDFT) in cooperation with a frequency-locked loop [8], and two novel interpolation iterative estimators based on DFT coefficients [10]. In these integral methods, the large computation restrains the online identification in a more wider engineering.Another large class of techniques has been presented based on the differential model instead of the integration model. These differential methods employ the concept of frequency estimator.In [6], the existing concept of the adaptive notch filter and second-order generalized integrator was modified for the estimation of the frequency, the magnitude, and the damping of low-frequency electromechanical oscillations with a direct component in an interconnected power system.Two types of frequency estimators on the basis of adaptive notch filter (ANF), the normalized estimator and the nonnormalized estimator, were investigated in [15]. The actual value of signal amplitude has little effect on the convergence speed in the normalized methods while it has a great influence on the convergence speed in the nonnormalized methods. The nonnormalized version was extended in [16] to estimate frequencies and amplitudes of multisinusoidal components. The normalized version was improved in [17] in order to gain exponential convergence property in a wide range of frequency, especially in a rather low frequency. It was then applied in [18] to cancel sinusoidal disturbance with unknown low frequency of linear time-invariant systems.Based on the adaptive identifier proposed in [19], an indirect algorithm was proposed in [20] for identifying the parameters of the EDS signal. The variables in this type of indirect algorithms converge to the value combining of the square of frequency and square of damping factor, which is inconvenient to provide the instantaneous estimate value of EDS parameters.The concept of enhanced phase-locked loop (EPLL) proposed in [21], a nonnormalized frequency estimation, was normalized and extended [9] for the estimation of EDS parameters. In the EPLL-based methods, it is inconvenient to obtain the instantaneous value of input EDS signal.Based on the adaptive internal model (AIM) controller, a normalized frequency estimator proposed in [22] was extended in [23] to identify the frequency and the damping factor of the EDS signal and then was applied in [24] to the power system for damping of electromechanical oscillations. In addition to the frequency and the damping factor, the AIM-based algorithms directly provide the instantaneous estimate values of both the in-phase component and the quadrature-phase component of the EDS signals.Compared with integral methods, the differential methods need a smaller number of data in a shorter period time and less computation burden.As been addressed in [17], the AIM methods may lost convergence property with its design parameters unchanged when the frequency value of EDS signal changes from some number to a rather smaller number. So, it is significant to bring some improvement on transient performance and antinoise performance of the parameter identification of LFOs modeled as EDS signals.Motivated by [15, 17, 23], the robustness of the new method proposed here includes exponential convergence property in a wide range of frequency under no noise and adjustable boundary of steady-state identification errors under bounded additive noise.The rest of the paper is arranged as follows: The framework of the new algorithm is described in Section2. The identification characteristics are investigated in Section 3. The simulation study is carried out in Section 4. Section 5 concludes the whole paper.


## 2. Materials and Methods

### 2.1. Framework of New Algorithm

A low-pass filtered LFO in the power system can be represented as EDS signal in timeτ:
(1)yτ=y0τ+y~τ=a0e−σ0τsinw0τ+δ0+y~τ,where y0 denotes exponentially damped oscillation and y~ represents the additive measure noise. The amplitude a0, the damping factor σ0, and the frequency ω0 are all unknown constant real positive numbers, and the initial phase −π<δ0<π.Motivated by the normalized updating law proposed in [15] and the robust frequency estimator proposed in [17], we change the AIM-based identification method proposed in [23] into the new parameter identification and write as the following close-loop dynamical system that consists of three differential equations and three algebraic equations
(2)dx^1dττdx^2dττ=−σ^τω^τ−ω^τ−σ^τx^1τx^2τ+μω^τ0eτ,(3)dω^dττ=εμω^2τx^2τeτκ+x^12τ+x^22τ,(4)dσ^dττ=−εγμω^2τx^1τeτκ+x^12τ+x^22τ,(5)y^τ=10x^1τx^2τ=x^1τ,(6)eτ=y0τ−y^τ=y0τ−x^1τ,(7)a^τ=eσ^τx^21τ+x^22τ.In the equations abovementioned,μ, ε, γ, and κ are four positive real design parameters to adjust the performance of identification, x^1τ, x^2τ, ω^τ, and σ^τ are four variables, y^τ, eτ, and a^τ represent the identification value of EDS signal y0τ, and the signal tracking error and the identification value of amplitude a0, respectively. The two variables, ω^τ and σ^τ, are just right the identification values of the frequency ω0 and the damping factor σ0, respectively.We may assume that the four variables have initial values expressed as(8)x^10x^20ω^0σ^0=x^10x^20ω^0σ^0.Generally, all the parameters of the EDS signaly0τ have finite values. So, the four differential variables are bounded for τ≥0 and can be assumed in a compact set defined as
(9)Dτ=x^1τx^2τω^τσ^τx~1≤amaxx~2≤amax0<ωmin≤ω^τ≤ωmax0<σ^τ≤σmax,where all the boundary values, amax, ωmin, ωmax, and σmax, can be experimentally preset.


## 2.1. Framework of New Algorithm

A low-pass filtered LFO in the power system can be represented as EDS signal in timeτ:
(1)yτ=y0τ+y~τ=a0e−σ0τsinw0τ+δ0+y~τ,where y0 denotes exponentially damped oscillation and y~ represents the additive measure noise. The amplitude a0, the damping factor σ0, and the frequency ω0 are all unknown constant real positive numbers, and the initial phase −π<δ0<π.Motivated by the normalized updating law proposed in [15] and the robust frequency estimator proposed in [17], we change the AIM-based identification method proposed in [23] into the new parameter identification and write as the following close-loop dynamical system that consists of three differential equations and three algebraic equations
(2)dx^1dττdx^2dττ=−σ^τω^τ−ω^τ−σ^τx^1τx^2τ+μω^τ0eτ,(3)dω^dττ=εμω^2τx^2τeτκ+x^12τ+x^22τ,(4)dσ^dττ=−εγμω^2τx^1τeτκ+x^12τ+x^22τ,(5)y^τ=10x^1τx^2τ=x^1τ,(6)eτ=y0τ−y^τ=y0τ−x^1τ,(7)a^τ=eσ^τx^21τ+x^22τ.In the equations abovementioned,μ, ε, γ, and κ are four positive real design parameters to adjust the performance of identification, x^1τ, x^2τ, ω^τ, and σ^τ are four variables, y^τ, eτ, and a^τ represent the identification value of EDS signal y0τ, and the signal tracking error and the identification value of amplitude a0, respectively. The two variables, ω^τ and σ^τ, are just right the identification values of the frequency ω0 and the damping factor σ0, respectively.We may assume that the four variables have initial values expressed as(8)x^10x^20ω^0σ^0=x^10x^20ω^0σ^0.Generally, all the parameters of the EDS signaly0τ have finite values. So, the four differential variables are bounded for τ≥0 and can be assumed in a compact set defined as
(9)Dτ=x^1τx^2τω^τσ^τx~1≤amaxx~2≤amax0<ωmin≤ω^τ≤ωmax0<σ^τ≤σmax,where all the boundary values, amax, ωmin, ωmax, and σmax, can be experimentally preset.


## 3. Results and Discussion

### 3.1. Characteristics of the Identification

In this section, we characterize the new identification from two aspects: transient convergence property and steady-state antinoise performance.


#### 3.1.1. Convergence Property Analysis

The convergence property of the proposed identification algorithm can be summarized as the following theorem.Theorem 1.
When the measure noisey~τ=0, there exist positive numbers ε∗, μ∗, γ∗, and σ∗, such that if 0<ε<ε∗, 0<γ<γ∗, μ>μ∗, and σ0<σ∗, then all the identification values exponentially converge to their actual values, i.e.,
(10)x^1∞x^2∞ω^∞σ^∞a^∞⟶a0e−σ0τsinω0τ+δ0a0e−σ0τcosω0τ+δ0ω0σ0a0.Proof.
We use time-scale change, variable transformation, slow integral manifolds, and averaging method in sequence to prove the asymptotic convergence property of the proposed identification algorithm.
Firstly, we define the time scale change as(11)t=ω0τ+δ0,τ=ω0−1t−δ0and rewrite the EDS signal y0τ in time t as
(12)y1t=a1e−σ1tsint=a0e−σ0ω0−1t−δ0sintwhere
(13)σ1=ω0−1σ0,a1=a0eσ1δ0
Put (6) into (2), (3), (4), let x^̇=dx/dt then we rewrite the state-space representation in time t as
(14)x^̇1x^̇2=1ω0−σ^−μω^ω^−ω^−σ^x^1x^2+1ω0μω^0y1t,(15)ω^̇=εμ1ω0ω^2x^2y1t−x^1κ+x^12+x^22,(16)σ^̇=−εμγ1ω0ω^2x^1y1t−x^1κ+x^12+x^22,where the time t in the state variables is omitted for convenience of writing, the same below.
Then, we define the variable transformation as(17)x^1x^2ω^σ^=a1e−σ1tx1a1e−σ1tx2ω0θω0σ+σ0,x1x2θσ=a1−1eσ1tx^1a1−1eσ1tx^2ω0−1ω^ω0−1σ^−σ1.
Letx=x1x2T. After substituting (17) into (14) (15) (16), we obtain a nonlinear dynamic system consisting of two coupled 2-dimension differential equations:
(18)ẋ=Aθ,σx+bθsint,(19)θ̇σ̇=εθf1t,x,θ,σγf2t,x,θ,σ,where
(20)Aθ,σ=−σ−μθθ−θ−σ,bθ=μθ0,f1t,x,θ,σ=μθx2sint−x1a1−2κ+x12+x22,f2t,x,θ,σ=−μθx1sint−x1a1−2κ+x12+x22.
The definite domain of the transformed variables, corresponding to the compact setDτ, is expressed as
(21)D=x1x2θσa0−1amin≤x12+x22≤a0−1amaxω0−1ωmin<θ≤ω0−1ωmax0≤σ≤ω0−1σmax.
Now, the truth of (10) is equivalent to that the nonlinear system consisting of (18) and (19) has the asymptotical convergence property on the compact set D expressed as
(22)x1∞x2∞θ∞σ∞⟶sintcost10.Secondly, we prove (22) true by the schedule similar to that done in [16, 17].The dynamic system including (18) and (19) just accords with standard form of the system addressed in [25]. So, we use slow integral manifold proposed in [25] to decouple the updating law (19) from the state equation (18).Considering frozen parameter method, letε=0, the state equation (18) becomes a linear time-invariant (LTI) system which constant coefficient matrix Aσ,θ has two eigenvalues expressed as
(23)λ1,2θ,σ=−σ+0.5μθ±0.5θμ2−4The two eigenvaluesλ1,2θ,σ have both negative real parts if the following condition in 0,∞×D is always satisfied.
(24)σ+0.5μθ>0σ+0.5θμ−μ2−4>0μ≤2μ>2.So return to time domain ofτ it must be satisfied that
(25)σ0<0.5μω^τ+σ^τσ0<0.5μ−μ2−4ω^τ+σ^τμ≤2μ>2.This declares the existence ofμ∗ and σ∗ such that μ>μ∗ and σ0<σ∗ should be ensured.For the LTI system (18) when ε=0, the negative real parts of λ1,2θ,σ also reveal that its transient response to the initial value of x vanishes and its steady-state response forced by input signal sin t can be calculated by use of frequency characteristics.Letj2=1. Under the condition of initial relaxation, the frequency characteristics of (18) from the input signal, i.e., sint, to state variable vector x can be written as
(26)H1σ,θ,jH2σ,θ,j=jI−Aσ,θ−1bθ=μθ2σ+μθ−jθ2−1+σ2+μθσ1−jσjθ.We represent the complex denominator in the form of amplitude-frequency characteristics and phase-frequency characteristics expressed as(27)Hσ,θ=θ2−1+σ2+μθσ2+2σ+μθ2,φ=argθ2−1+σ2+μθσ2σ+μθ,where.
(28)cosφ=2σ+μθHσ,θ,sinφ=θ2−1+σ2+μθσHσ,θ.Now, the steady-state response of LTI system (18) is a 2π- period orbit represented by
(29)xt,θ,σ=x¯1t,θ,σx¯2t,θ,σ=H1θ,σ,jH2θ,σ,jsint=μθHσ,θsint+φ−σcost+φθcost+φ.Then, the steady-state identification error is(30)e¯t,θ,σ=sint−x¯1t,θ,σ=sint−μθsint+φ−σcost+φHσ,θ=2σsint+φ−θ2−1+σ2cost+φHσ,θ.It can be verified that all the following items(31)xt,θ,σ,∂x¯∂θt,θ,σ,∂x¯∂σt,θ,σ,are continuous, bounded, and Lipschitzian with respect to both θ and σin 0,∞×D. This ensues both Assumption 2.2 and Assumption 3.1 of [25]. From (25), Assumption 2.1 of [25] has been ensured by the negative real parts of λ1,2θ,σ.So, according to Theorem 3.1 of [25], there exists ε∗>0 and γ∗>0 such that if 0<ε<ε∗ and 0<γ<γ∗ then system combining (18) and (19) has ε-family slow integral manifolds.On these manifolds, the state variable vectorx can be expressed as
(32)xt,θ,σ,ε=x1t,θ,σ,εx2t,θ,σ,ε=x¯1t,θ,σ,εx¯2t,θ,σ,ε+εx~1t,θ,σ,εx~2t,θ,σ,ε.Then, the identification error is represented by(33)et,θ,σ,ε=sint−x1t,θ,σ,ε=e¯t,θ,σ−εx~1t,θ,σ,ε.Now, the parameter updating law (19) can be decoupled from the state equation (18) and expressed as two coupled almost periodical dynamic systems with respect to t.
(34)θ̇=εθF1t,θ,σ,ε=εθf1t,xt,θ,σ,ε,θ,σ=εθμθx2t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε,(35)σ̇=εθγF2t,θ,σ,ε=εθγf2t,xt,θ,σ,ε,θ,σ=−εθγμθx1t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε.Thirdly, we use the averaging method to prove the convergence property of (34) and (35).From (29), (30), (32), and (33), we let t1=t+φ and get
(36)F1t,θ,σ,0=μθx¯2t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=μ2θ3cost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θθ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+θκ¯θ,σ+1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+2θκ¯θ,σ+1a¯θ,σ+b¯θ,σcos2t1+β,where
(37)κ¯θ,σ=a1−2κH2θ,σμ−2θ−2,a¯θ,σ=2κ¯θ,σ+σ2+θ2+1,b¯θ,σ=θ2−1+σ22+4σ2,sinβ=2σb¯θ,σ,cosβ=θ2−1+σ2b¯θ,σ.Similarly, we also get(38)F2t,θ,σ,0=−μθx¯1t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=−μ2θ2sint1−σcost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θ1+θ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1+θ2−1+σ2sint1cost1−2σsin2t1+σκ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−σ+2σκ¯θ,σ+b¯θ,σsin2t1+βa¯θ,σ+b¯θ,σcos2t1+β.The integration of the following periodical function is zero.(39)1π∫−π2π22b¯θ,σcos2t+2φ+βdta¯θ,σ+b¯θ,σcos2t+2φ+β=−1π∫−ππdb¯θ,σcosta¯θ,σ+b¯θ,σcost=0.According to the integral formula, we get(40)1π∫−π−2φ−β2π−2φ−β22dta¯θ,σ+b¯θ,σcos2t+2φ+β=1π∫−π2π22dta¯θ,σ+b¯θ,σcos2t=1π∫−π2π2dta¯θ,σ+b¯θ,σcost=2a¯2θ,σ−b¯2θ,σ=1F¯0,where
(41)F¯0=κ¯θ,σ+1κ¯θ,σ+θ2+κ¯θ,σσ2=κ¯θ,σ+12+θ2−1κ¯θ,σ+1+κ¯θ,σσ2.According to Section 10.4 of [26], we get
(42)F¯1θ,σF¯2θ,σ=1π∫0πF1t,θ,σ,0F2t,θ,σ,0dt=1F¯0θF¯0−κ¯θ,σ−1σF¯0−κ¯θ,σ.Thus, the averaged system of updating law (18) is
(43)θ̇σ̇=εθF¯1θ,σγF¯2θ,σ=εθF¯0θF¯0−κ¯θ,σ−1γσF¯0−κ¯θ,σ.ConsideringF¯0>κ¯θ,σ, the point θσT=10T is the unique equilibrium point of the averaged system (43). Let Lyapunov candidate function be
(44)Vθ,σ=0.5Kσ2+0.5θ−12,where K is a sufficiently large positive real number. Its derivative along the trajectory of the system (43) is
(45)V̇θ,σ=Kσσ̇+θ−1θ̇−εθF¯0Kγσ2F¯0−κ¯σ,θ+θ−1θF¯0−κ¯σ,θ−1.Whenθ>1, then F0>κ¯θ,σ+1 and V̇θ,σ<0 obviously. When θ<1, it can be discussed as follows:
(1)
Ifκ¯θ,σσ2<1−θ2κ¯θ,σ+1, then F0<κ¯θ,σ+1 and V̇θ,σ<0(2)
Ifκ¯θ,σσ2>1−θ2κ¯θ,σ+1, then we can let K be enough large such that V̇θ,σ<0 resulting from Kγσ2F¯0−κ¯σ,θ>θ−1θF¯0−κ¯σ,θ−1So, the derivative functionV̇θ,σ is negatively defined and the point θσT=10T is the unique globally stable equilibrium point of the averaged system (43).Invoking Theorem 10.4 of [20], for sufficiently small positive real number ε, the two variables θ and σ in (34) and (35) are asymptotically convergent to their equilibrium point,respectively, i.e.,θσT⟶10T.In steady state, putθ∞=1 and σ∞=0 into (29), we get x1∞=sint and x2∞=cost then prove (22) true.Now, recalling the time scale change (11), the variable transformation (17), and the condition (25), the theorem and the conclusion of (10) have been proved true.


#### 3.1.2. Noise Characteristics Analysis

Let the high-frequency measure noise in the signal (1) be bounded and represented by
(46)y~τ=a~sinω~τ+δ~ω~≫ω0,where the amplitude a~, the frequency ω~, and the phase angle δ~ are constants.The noise characteristics of the new identification can be summarized asTheorem 2.
Under high-frequency noise of (46), if the design parameters, μ, ε, γ, and κ satisfy the conditions for Theorem 1, then the steady-state identification errors will be bounded by
(47)eω=ω^−ω0<εμκω0a~b¯1,(48)eσ=σ^−σ0<γεμκω0a~b¯2,(49)ea=a^−a0<a0eσt+μω0ω~−1a~eσ0t,where b¯1 and b¯2 are positive real constants in 0,∞×Dτ.Proof.
By use of the time scale change (11), the noise (46) is rewritten in time t as
(50)y~1t=a~sint~=a~sinω−10ω~t+δ~2.
Firstly, considerε=0, let x~1x~2T denote the state variable vector of the LTI system (14) forced only by noise y~1t instead of EDS signal y1t, we get
(51)x~̇1x~̇2=1ω0−σ^−μω^ω^−ω^−σ^x~1x~2+1ω0μω^0y~1t.
The frequency characteristics of the system (51) from y~1t to state variable vector x~1x~2T can be written as
(52)H~1jω~H~2jω~=μω^ω~−jσ^jω^ω~2σ^+μω^−jσ^σ^+μω^+ω^2−ω~2.
So the steady-state response of the LTI system (51) is
(53)x~1x~2=H~ω~a~ω~sint~+φ~−σ^cost~+φ~ω^cost~+φ~,where
(54)H~ω~=μω^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22,φ~=argω~2σ^+μω^+jσ^σ^+μω^+ω^2−ω~2.
In this instance, the identification error resulting fromy~1t is rewritten as
(55)e~t=y~1t−x~1=H~eω~a~sint~+φ~+β~,where β~ has some appropriate value and
(56)H~eω~=4σ^2ω~2+σ^2+ω^2−ω~22ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22
Letx~1x~2T denote the steady-state response of the LTI system (14) forced only by the EDS signal y1t. According to the superposition principle of LTI systems, the steady-state response of the LTI system (14) resulting from both the EDS signal y1t and the noise y~1t is represented by
(57)x~1x~2T=x^1x^2+x~1x~2.
Secondly, the decoupled updating laws (15) and (16) can be written as the following perturbed system:
(58)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1+εμω0‐1ω^2e~tκ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1.
Theorem1 abovementioned shows that the nominal system
(59)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1,is an exponentially stable system.
The perturbation items are bounded by(60)εμω0‐1ω^2x^2+x~2e~tκ+x^1+x~12+x^2+x~22≤εμω^2H~eω~a~2ω0κ,(61)εμγω0‐1ω^2x^1+x~1e~tκ+x^1+x~12+x^2+x~22≤εμγω^2H~eω~a~2ω0κ.
According to Lemma 9.2 of [26], there exit boundary values of b¯1 and b¯2 such that (47) and (48) hold true.
Thirdly, from (7), by error synthesis principle, we know
(62)ea=a^t−a0=∂a^∂σ^eσ+∂a^∂x^1x~1+∂a^∂x^2x~2=a^eσt+eσ^tx^1+x~1x~1+x^2+x~2x~2x^1+x~12+x^2+x~22≤a^eσt+eσ^tx~21+x~22≤a0eσt+H~aω~a~eσ0t,where
(63)H~aω~=μω^ω~2+ω~σ^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22.After the trajectory goes into the small neighborhood of the equilibrium point, we getω^≈ω~ and σ^≈σ~. If ω~≫ω^ and σ0≪1, then H~eσ^,ω^,ω~≈1, H~aσ^,ω^,ω~≈μω0ω~−1.Now, the Theorem2 has been consequently proved true.


#### 3.1.3. Performance Evaluation.

For the measured signal (1), the identification method proposed in [17] with two positive design parameters, Ka and Kbcan be written in time τ as
(64)dx1dττdx2dττ=−σ^τω^τ−ω^τ−σ^τx1τx2τ+01eτ,dω^dττ=−KaKbx1τeτx12τ+x22τ,dσ^dττ=−Kax2τeτx12τ+x22τ,eτ=−y0τ+x2τ.With time scale change (11) applied, it is written in time t as
(65)ẋ1ẋ2=1ω0−σ^ω^−ω^−σ^x1x2+1ω001−y1t−x2,ω^̇=−KaKbω0x1−y1t−x2x12+x22,σ^̇=−Kaω0x2−y1t−x2x12τ+x22τ.After time scale change (11), the identification algorithm proposed in [6] can be rewritten in time t as
(66)ẋ0ẋ1ẋ2=1ω0−k0ω^0−k0ω^0−σ^ω^−kω^−ω^−σ^−kω^x0x1x2−ω^ω0k00ky1t,ω^̇=−μωω0x1−y1t−x2−x0x12+x22,σ^̇=−μσω0x2−y1t−x2−x0x12τ+x22τ,where k0, k, μω and μσare four positive design parameters.Remark 3.
Whenω0 has a rather large value, both the algorithm (64) and the new algorithm exemplify exponential convergence property. When ω0 has a rather small value, the explicit reciprocal ω0−1 in (65) may bring the algorithm (64) unconvergence property. Contrarily, the reciprocal ω0−1 in (15) and (16) is compensated by the frequency estimate value ω^, which brings the new method better robustness for low-frequency signals.When being distorted by the noise (50), the perturbation terms of the algorithm (65) similar to (60) and (61) are
(67)KaKbω0−1x2e~tx12+x22≤KaKbH~eω~a~x12+x22,Kaω0−1x1e~tx12+x22≤KaH~eω~a~x12+x22.In a small neighborhood of the equilibrium point,x12+x22≈a1e−σ1t, so the boundary values of identification errors of frequency and damping factor in (65) can be represented as
(68)eω=ω^−ω0<KaKbω0−1a~a1eσ1tb¯4,eσ=σ^−σ0<Kaω0−1a~a1eσ1tb¯5.Remark 4.
When the signal is distorted by high-frequency sinusoidal noise with constant amplitude, the identification errors of frequency and damping factor in the algorithm (65) increase exponentially over time, while that in the new method have both bounded values governed by design parameter κ.Remark 5.
For the identification of both frequency and damping factor in the new method, a greater value of the design parameterκ brings smaller steady-state identification errors under noise and slower transient convergence speeds simultaneously.Remark 6.
The state variablex0 in (66) denotes the estimated value of the direct current component. Furthermore, compare (65) with (66), we know that the same conclusions as Remark 3 and Remark 4 can be drawn for the algorithm in [6].


## 3.1. Characteristics of the Identification

In this section, we characterize the new identification from two aspects: transient convergence property and steady-state antinoise performance.


### 3.1.1. Convergence Property Analysis

The convergence property of the proposed identification algorithm can be summarized as the following theorem.Theorem 1.
When the measure noisey~τ=0, there exist positive numbers ε∗, μ∗, γ∗, and σ∗, such that if 0<ε<ε∗, 0<γ<γ∗, μ>μ∗, and σ0<σ∗, then all the identification values exponentially converge to their actual values, i.e.,
(10)x^1∞x^2∞ω^∞σ^∞a^∞⟶a0e−σ0τsinω0τ+δ0a0e−σ0τcosω0τ+δ0ω0σ0a0.Proof.
We use time-scale change, variable transformation, slow integral manifolds, and averaging method in sequence to prove the asymptotic convergence property of the proposed identification algorithm.
Firstly, we define the time scale change as(11)t=ω0τ+δ0,τ=ω0−1t−δ0and rewrite the EDS signal y0τ in time t as
(12)y1t=a1e−σ1tsint=a0e−σ0ω0−1t−δ0sintwhere
(13)σ1=ω0−1σ0,a1=a0eσ1δ0
Put (6) into (2), (3), (4), let x^̇=dx/dt then we rewrite the state-space representation in time t as
(14)x^̇1x^̇2=1ω0−σ^−μω^ω^−ω^−σ^x^1x^2+1ω0μω^0y1t,(15)ω^̇=εμ1ω0ω^2x^2y1t−x^1κ+x^12+x^22,(16)σ^̇=−εμγ1ω0ω^2x^1y1t−x^1κ+x^12+x^22,where the time t in the state variables is omitted for convenience of writing, the same below.
Then, we define the variable transformation as(17)x^1x^2ω^σ^=a1e−σ1tx1a1e−σ1tx2ω0θω0σ+σ0,x1x2θσ=a1−1eσ1tx^1a1−1eσ1tx^2ω0−1ω^ω0−1σ^−σ1.
Letx=x1x2T. After substituting (17) into (14) (15) (16), we obtain a nonlinear dynamic system consisting of two coupled 2-dimension differential equations:
(18)ẋ=Aθ,σx+bθsint,(19)θ̇σ̇=εθf1t,x,θ,σγf2t,x,θ,σ,where
(20)Aθ,σ=−σ−μθθ−θ−σ,bθ=μθ0,f1t,x,θ,σ=μθx2sint−x1a1−2κ+x12+x22,f2t,x,θ,σ=−μθx1sint−x1a1−2κ+x12+x22.
The definite domain of the transformed variables, corresponding to the compact setDτ, is expressed as
(21)D=x1x2θσa0−1amin≤x12+x22≤a0−1amaxω0−1ωmin<θ≤ω0−1ωmax0≤σ≤ω0−1σmax.
Now, the truth of (10) is equivalent to that the nonlinear system consisting of (18) and (19) has the asymptotical convergence property on the compact set D expressed as
(22)x1∞x2∞θ∞σ∞⟶sintcost10.Secondly, we prove (22) true by the schedule similar to that done in [16, 17].The dynamic system including (18) and (19) just accords with standard form of the system addressed in [25]. So, we use slow integral manifold proposed in [25] to decouple the updating law (19) from the state equation (18).Considering frozen parameter method, letε=0, the state equation (18) becomes a linear time-invariant (LTI) system which constant coefficient matrix Aσ,θ has two eigenvalues expressed as
(23)λ1,2θ,σ=−σ+0.5μθ±0.5θμ2−4The two eigenvaluesλ1,2θ,σ have both negative real parts if the following condition in 0,∞×D is always satisfied.
(24)σ+0.5μθ>0σ+0.5θμ−μ2−4>0μ≤2μ>2.So return to time domain ofτ it must be satisfied that
(25)σ0<0.5μω^τ+σ^τσ0<0.5μ−μ2−4ω^τ+σ^τμ≤2μ>2.This declares the existence ofμ∗ and σ∗ such that μ>μ∗ and σ0<σ∗ should be ensured.For the LTI system (18) when ε=0, the negative real parts of λ1,2θ,σ also reveal that its transient response to the initial value of x vanishes and its steady-state response forced by input signal sin t can be calculated by use of frequency characteristics.Letj2=1. Under the condition of initial relaxation, the frequency characteristics of (18) from the input signal, i.e., sint, to state variable vector x can be written as
(26)H1σ,θ,jH2σ,θ,j=jI−Aσ,θ−1bθ=μθ2σ+μθ−jθ2−1+σ2+μθσ1−jσjθ.We represent the complex denominator in the form of amplitude-frequency characteristics and phase-frequency characteristics expressed as(27)Hσ,θ=θ2−1+σ2+μθσ2+2σ+μθ2,φ=argθ2−1+σ2+μθσ2σ+μθ,where.
(28)cosφ=2σ+μθHσ,θ,sinφ=θ2−1+σ2+μθσHσ,θ.Now, the steady-state response of LTI system (18) is a 2π- period orbit represented by
(29)xt,θ,σ=x¯1t,θ,σx¯2t,θ,σ=H1θ,σ,jH2θ,σ,jsint=μθHσ,θsint+φ−σcost+φθcost+φ.Then, the steady-state identification error is(30)e¯t,θ,σ=sint−x¯1t,θ,σ=sint−μθsint+φ−σcost+φHσ,θ=2σsint+φ−θ2−1+σ2cost+φHσ,θ.It can be verified that all the following items(31)xt,θ,σ,∂x¯∂θt,θ,σ,∂x¯∂σt,θ,σ,are continuous, bounded, and Lipschitzian with respect to both θ and σin 0,∞×D. This ensues both Assumption 2.2 and Assumption 3.1 of [25]. From (25), Assumption 2.1 of [25] has been ensured by the negative real parts of λ1,2θ,σ.So, according to Theorem 3.1 of [25], there exists ε∗>0 and γ∗>0 such that if 0<ε<ε∗ and 0<γ<γ∗ then system combining (18) and (19) has ε-family slow integral manifolds.On these manifolds, the state variable vectorx can be expressed as
(32)xt,θ,σ,ε=x1t,θ,σ,εx2t,θ,σ,ε=x¯1t,θ,σ,εx¯2t,θ,σ,ε+εx~1t,θ,σ,εx~2t,θ,σ,ε.Then, the identification error is represented by(33)et,θ,σ,ε=sint−x1t,θ,σ,ε=e¯t,θ,σ−εx~1t,θ,σ,ε.Now, the parameter updating law (19) can be decoupled from the state equation (18) and expressed as two coupled almost periodical dynamic systems with respect to t.
(34)θ̇=εθF1t,θ,σ,ε=εθf1t,xt,θ,σ,ε,θ,σ=εθμθx2t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε,(35)σ̇=εθγF2t,θ,σ,ε=εθγf2t,xt,θ,σ,ε,θ,σ=−εθγμθx1t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε.Thirdly, we use the averaging method to prove the convergence property of (34) and (35).From (29), (30), (32), and (33), we let t1=t+φ and get
(36)F1t,θ,σ,0=μθx¯2t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=μ2θ3cost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θθ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+θκ¯θ,σ+1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+2θκ¯θ,σ+1a¯θ,σ+b¯θ,σcos2t1+β,where
(37)κ¯θ,σ=a1−2κH2θ,σμ−2θ−2,a¯θ,σ=2κ¯θ,σ+σ2+θ2+1,b¯θ,σ=θ2−1+σ22+4σ2,sinβ=2σb¯θ,σ,cosβ=θ2−1+σ2b¯θ,σ.Similarly, we also get(38)F2t,θ,σ,0=−μθx¯1t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=−μ2θ2sint1−σcost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θ1+θ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1+θ2−1+σ2sint1cost1−2σsin2t1+σκ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−σ+2σκ¯θ,σ+b¯θ,σsin2t1+βa¯θ,σ+b¯θ,σcos2t1+β.The integration of the following periodical function is zero.(39)1π∫−π2π22b¯θ,σcos2t+2φ+βdta¯θ,σ+b¯θ,σcos2t+2φ+β=−1π∫−ππdb¯θ,σcosta¯θ,σ+b¯θ,σcost=0.According to the integral formula, we get(40)1π∫−π−2φ−β2π−2φ−β22dta¯θ,σ+b¯θ,σcos2t+2φ+β=1π∫−π2π22dta¯θ,σ+b¯θ,σcos2t=1π∫−π2π2dta¯θ,σ+b¯θ,σcost=2a¯2θ,σ−b¯2θ,σ=1F¯0,where
(41)F¯0=κ¯θ,σ+1κ¯θ,σ+θ2+κ¯θ,σσ2=κ¯θ,σ+12+θ2−1κ¯θ,σ+1+κ¯θ,σσ2.According to Section 10.4 of [26], we get
(42)F¯1θ,σF¯2θ,σ=1π∫0πF1t,θ,σ,0F2t,θ,σ,0dt=1F¯0θF¯0−κ¯θ,σ−1σF¯0−κ¯θ,σ.Thus, the averaged system of updating law (18) is
(43)θ̇σ̇=εθF¯1θ,σγF¯2θ,σ=εθF¯0θF¯0−κ¯θ,σ−1γσF¯0−κ¯θ,σ.ConsideringF¯0>κ¯θ,σ, the point θσT=10T is the unique equilibrium point of the averaged system (43). Let Lyapunov candidate function be
(44)Vθ,σ=0.5Kσ2+0.5θ−12,where K is a sufficiently large positive real number. Its derivative along the trajectory of the system (43) is
(45)V̇θ,σ=Kσσ̇+θ−1θ̇−εθF¯0Kγσ2F¯0−κ¯σ,θ+θ−1θF¯0−κ¯σ,θ−1.Whenθ>1, then F0>κ¯θ,σ+1 and V̇θ,σ<0 obviously. When θ<1, it can be discussed as follows:
(1)
Ifκ¯θ,σσ2<1−θ2κ¯θ,σ+1, then F0<κ¯θ,σ+1 and V̇θ,σ<0(2)
Ifκ¯θ,σσ2>1−θ2κ¯θ,σ+1, then we can let K be enough large such that V̇θ,σ<0 resulting from Kγσ2F¯0−κ¯σ,θ>θ−1θF¯0−κ¯σ,θ−1So, the derivative functionV̇θ,σ is negatively defined and the point θσT=10T is the unique globally stable equilibrium point of the averaged system (43).Invoking Theorem 10.4 of [20], for sufficiently small positive real number ε, the two variables θ and σ in (34) and (35) are asymptotically convergent to their equilibrium point,respectively, i.e.,θσT⟶10T.In steady state, putθ∞=1 and σ∞=0 into (29), we get x1∞=sint and x2∞=cost then prove (22) true.Now, recalling the time scale change (11), the variable transformation (17), and the condition (25), the theorem and the conclusion of (10) have been proved true.


### 3.1.2. Noise Characteristics Analysis

Let the high-frequency measure noise in the signal (1) be bounded and represented by
(46)y~τ=a~sinω~τ+δ~ω~≫ω0,where the amplitude a~, the frequency ω~, and the phase angle δ~ are constants.The noise characteristics of the new identification can be summarized asTheorem 2.
Under high-frequency noise of (46), if the design parameters, μ, ε, γ, and κ satisfy the conditions for Theorem 1, then the steady-state identification errors will be bounded by
(47)eω=ω^−ω0<εμκω0a~b¯1,(48)eσ=σ^−σ0<γεμκω0a~b¯2,(49)ea=a^−a0<a0eσt+μω0ω~−1a~eσ0t,where b¯1 and b¯2 are positive real constants in 0,∞×Dτ.Proof.
By use of the time scale change (11), the noise (46) is rewritten in time t as
(50)y~1t=a~sint~=a~sinω−10ω~t+δ~2.
Firstly, considerε=0, let x~1x~2T denote the state variable vector of the LTI system (14) forced only by noise y~1t instead of EDS signal y1t, we get
(51)x~̇1x~̇2=1ω0−σ^−μω^ω^−ω^−σ^x~1x~2+1ω0μω^0y~1t.
The frequency characteristics of the system (51) from y~1t to state variable vector x~1x~2T can be written as
(52)H~1jω~H~2jω~=μω^ω~−jσ^jω^ω~2σ^+μω^−jσ^σ^+μω^+ω^2−ω~2.
So the steady-state response of the LTI system (51) is
(53)x~1x~2=H~ω~a~ω~sint~+φ~−σ^cost~+φ~ω^cost~+φ~,where
(54)H~ω~=μω^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22,φ~=argω~2σ^+μω^+jσ^σ^+μω^+ω^2−ω~2.
In this instance, the identification error resulting fromy~1t is rewritten as
(55)e~t=y~1t−x~1=H~eω~a~sint~+φ~+β~,where β~ has some appropriate value and
(56)H~eω~=4σ^2ω~2+σ^2+ω^2−ω~22ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22
Letx~1x~2T denote the steady-state response of the LTI system (14) forced only by the EDS signal y1t. According to the superposition principle of LTI systems, the steady-state response of the LTI system (14) resulting from both the EDS signal y1t and the noise y~1t is represented by
(57)x~1x~2T=x^1x^2+x~1x~2.
Secondly, the decoupled updating laws (15) and (16) can be written as the following perturbed system:
(58)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1+εμω0‐1ω^2e~tκ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1.
Theorem1 abovementioned shows that the nominal system
(59)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1,is an exponentially stable system.
The perturbation items are bounded by(60)εμω0‐1ω^2x^2+x~2e~tκ+x^1+x~12+x^2+x~22≤εμω^2H~eω~a~2ω0κ,(61)εμγω0‐1ω^2x^1+x~1e~tκ+x^1+x~12+x^2+x~22≤εμγω^2H~eω~a~2ω0κ.
According to Lemma 9.2 of [26], there exit boundary values of b¯1 and b¯2 such that (47) and (48) hold true.
Thirdly, from (7), by error synthesis principle, we know
(62)ea=a^t−a0=∂a^∂σ^eσ+∂a^∂x^1x~1+∂a^∂x^2x~2=a^eσt+eσ^tx^1+x~1x~1+x^2+x~2x~2x^1+x~12+x^2+x~22≤a^eσt+eσ^tx~21+x~22≤a0eσt+H~aω~a~eσ0t,where
(63)H~aω~=μω^ω~2+ω~σ^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22.After the trajectory goes into the small neighborhood of the equilibrium point, we getω^≈ω~ and σ^≈σ~. If ω~≫ω^ and σ0≪1, then H~eσ^,ω^,ω~≈1, H~aσ^,ω^,ω~≈μω0ω~−1.Now, the Theorem2 has been consequently proved true.


### 3.1.3. Performance Evaluation.

For the measured signal (1), the identification method proposed in [17] with two positive design parameters, Ka and Kbcan be written in time τ as
(64)dx1dττdx2dττ=−σ^τω^τ−ω^τ−σ^τx1τx2τ+01eτ,dω^dττ=−KaKbx1τeτx12τ+x22τ,dσ^dττ=−Kax2τeτx12τ+x22τ,eτ=−y0τ+x2τ.With time scale change (11) applied, it is written in time t as
(65)ẋ1ẋ2=1ω0−σ^ω^−ω^−σ^x1x2+1ω001−y1t−x2,ω^̇=−KaKbω0x1−y1t−x2x12+x22,σ^̇=−Kaω0x2−y1t−x2x12τ+x22τ.After time scale change (11), the identification algorithm proposed in [6] can be rewritten in time t as
(66)ẋ0ẋ1ẋ2=1ω0−k0ω^0−k0ω^0−σ^ω^−kω^−ω^−σ^−kω^x0x1x2−ω^ω0k00ky1t,ω^̇=−μωω0x1−y1t−x2−x0x12+x22,σ^̇=−μσω0x2−y1t−x2−x0x12τ+x22τ,where k0, k, μω and μσare four positive design parameters.Remark 3.
Whenω0 has a rather large value, both the algorithm (64) and the new algorithm exemplify exponential convergence property. When ω0 has a rather small value, the explicit reciprocal ω0−1 in (65) may bring the algorithm (64) unconvergence property. Contrarily, the reciprocal ω0−1 in (15) and (16) is compensated by the frequency estimate value ω^, which brings the new method better robustness for low-frequency signals.When being distorted by the noise (50), the perturbation terms of the algorithm (65) similar to (60) and (61) are
(67)KaKbω0−1x2e~tx12+x22≤KaKbH~eω~a~x12+x22,Kaω0−1x1e~tx12+x22≤KaH~eω~a~x12+x22.In a small neighborhood of the equilibrium point,x12+x22≈a1e−σ1t, so the boundary values of identification errors of frequency and damping factor in (65) can be represented as
(68)eω=ω^−ω0<KaKbω0−1a~a1eσ1tb¯4,eσ=σ^−σ0<Kaω0−1a~a1eσ1tb¯5.Remark 4.
When the signal is distorted by high-frequency sinusoidal noise with constant amplitude, the identification errors of frequency and damping factor in the algorithm (65) increase exponentially over time, while that in the new method have both bounded values governed by design parameter κ.Remark 5.
For the identification of both frequency and damping factor in the new method, a greater value of the design parameterκ brings smaller steady-state identification errors under noise and slower transient convergence speeds simultaneously.Remark 6.
The state variablex0 in (66) denotes the estimated value of the direct current component. Furthermore, compare (65) with (66), we know that the same conclusions as Remark 3 and Remark 4 can be drawn for the algorithm in [6].


## 3.1.1. Convergence Property Analysis

The convergence property of the proposed identification algorithm can be summarized as the following theorem.Theorem 1.
When the measure noisey~τ=0, there exist positive numbers ε∗, μ∗, γ∗, and σ∗, such that if 0<ε<ε∗, 0<γ<γ∗, μ>μ∗, and σ0<σ∗, then all the identification values exponentially converge to their actual values, i.e.,
(10)x^1∞x^2∞ω^∞σ^∞a^∞⟶a0e−σ0τsinω0τ+δ0a0e−σ0τcosω0τ+δ0ω0σ0a0.Proof.
We use time-scale change, variable transformation, slow integral manifolds, and averaging method in sequence to prove the asymptotic convergence property of the proposed identification algorithm.
Firstly, we define the time scale change as(11)t=ω0τ+δ0,τ=ω0−1t−δ0and rewrite the EDS signal y0τ in time t as
(12)y1t=a1e−σ1tsint=a0e−σ0ω0−1t−δ0sintwhere
(13)σ1=ω0−1σ0,a1=a0eσ1δ0
Put (6) into (2), (3), (4), let x^̇=dx/dt then we rewrite the state-space representation in time t as
(14)x^̇1x^̇2=1ω0−σ^−μω^ω^−ω^−σ^x^1x^2+1ω0μω^0y1t,(15)ω^̇=εμ1ω0ω^2x^2y1t−x^1κ+x^12+x^22,(16)σ^̇=−εμγ1ω0ω^2x^1y1t−x^1κ+x^12+x^22,where the time t in the state variables is omitted for convenience of writing, the same below.
Then, we define the variable transformation as(17)x^1x^2ω^σ^=a1e−σ1tx1a1e−σ1tx2ω0θω0σ+σ0,x1x2θσ=a1−1eσ1tx^1a1−1eσ1tx^2ω0−1ω^ω0−1σ^−σ1.
Letx=x1x2T. After substituting (17) into (14) (15) (16), we obtain a nonlinear dynamic system consisting of two coupled 2-dimension differential equations:
(18)ẋ=Aθ,σx+bθsint,(19)θ̇σ̇=εθf1t,x,θ,σγf2t,x,θ,σ,where
(20)Aθ,σ=−σ−μθθ−θ−σ,bθ=μθ0,f1t,x,θ,σ=μθx2sint−x1a1−2κ+x12+x22,f2t,x,θ,σ=−μθx1sint−x1a1−2κ+x12+x22.
The definite domain of the transformed variables, corresponding to the compact setDτ, is expressed as
(21)D=x1x2θσa0−1amin≤x12+x22≤a0−1amaxω0−1ωmin<θ≤ω0−1ωmax0≤σ≤ω0−1σmax.
Now, the truth of (10) is equivalent to that the nonlinear system consisting of (18) and (19) has the asymptotical convergence property on the compact set D expressed as
(22)x1∞x2∞θ∞σ∞⟶sintcost10.Secondly, we prove (22) true by the schedule similar to that done in [16, 17].The dynamic system including (18) and (19) just accords with standard form of the system addressed in [25]. So, we use slow integral manifold proposed in [25] to decouple the updating law (19) from the state equation (18).Considering frozen parameter method, letε=0, the state equation (18) becomes a linear time-invariant (LTI) system which constant coefficient matrix Aσ,θ has two eigenvalues expressed as
(23)λ1,2θ,σ=−σ+0.5μθ±0.5θμ2−4The two eigenvaluesλ1,2θ,σ have both negative real parts if the following condition in 0,∞×D is always satisfied.
(24)σ+0.5μθ>0σ+0.5θμ−μ2−4>0μ≤2μ>2.So return to time domain ofτ it must be satisfied that
(25)σ0<0.5μω^τ+σ^τσ0<0.5μ−μ2−4ω^τ+σ^τμ≤2μ>2.This declares the existence ofμ∗ and σ∗ such that μ>μ∗ and σ0<σ∗ should be ensured.For the LTI system (18) when ε=0, the negative real parts of λ1,2θ,σ also reveal that its transient response to the initial value of x vanishes and its steady-state response forced by input signal sin t can be calculated by use of frequency characteristics.Letj2=1. Under the condition of initial relaxation, the frequency characteristics of (18) from the input signal, i.e., sint, to state variable vector x can be written as
(26)H1σ,θ,jH2σ,θ,j=jI−Aσ,θ−1bθ=μθ2σ+μθ−jθ2−1+σ2+μθσ1−jσjθ.We represent the complex denominator in the form of amplitude-frequency characteristics and phase-frequency characteristics expressed as(27)Hσ,θ=θ2−1+σ2+μθσ2+2σ+μθ2,φ=argθ2−1+σ2+μθσ2σ+μθ,where.
(28)cosφ=2σ+μθHσ,θ,sinφ=θ2−1+σ2+μθσHσ,θ.Now, the steady-state response of LTI system (18) is a 2π- period orbit represented by
(29)xt,θ,σ=x¯1t,θ,σx¯2t,θ,σ=H1θ,σ,jH2θ,σ,jsint=μθHσ,θsint+φ−σcost+φθcost+φ.Then, the steady-state identification error is(30)e¯t,θ,σ=sint−x¯1t,θ,σ=sint−μθsint+φ−σcost+φHσ,θ=2σsint+φ−θ2−1+σ2cost+φHσ,θ.It can be verified that all the following items(31)xt,θ,σ,∂x¯∂θt,θ,σ,∂x¯∂σt,θ,σ,are continuous, bounded, and Lipschitzian with respect to both θ and σin 0,∞×D. This ensues both Assumption 2.2 and Assumption 3.1 of [25]. From (25), Assumption 2.1 of [25] has been ensured by the negative real parts of λ1,2θ,σ.So, according to Theorem 3.1 of [25], there exists ε∗>0 and γ∗>0 such that if 0<ε<ε∗ and 0<γ<γ∗ then system combining (18) and (19) has ε-family slow integral manifolds.On these manifolds, the state variable vectorx can be expressed as
(32)xt,θ,σ,ε=x1t,θ,σ,εx2t,θ,σ,ε=x¯1t,θ,σ,εx¯2t,θ,σ,ε+εx~1t,θ,σ,εx~2t,θ,σ,ε.Then, the identification error is represented by(33)et,θ,σ,ε=sint−x1t,θ,σ,ε=e¯t,θ,σ−εx~1t,θ,σ,ε.Now, the parameter updating law (19) can be decoupled from the state equation (18) and expressed as two coupled almost periodical dynamic systems with respect to t.
(34)θ̇=εθF1t,θ,σ,ε=εθf1t,xt,θ,σ,ε,θ,σ=εθμθx2t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε,(35)σ̇=εθγF2t,θ,σ,ε=εθγf2t,xt,θ,σ,ε,θ,σ=−εθγμθx1t,θ,σ,εet,θ,σ,εa1−2κ+x12t,θ,σ,ε+x22t,θ,σ,ε.Thirdly, we use the averaging method to prove the convergence property of (34) and (35).From (29), (30), (32), and (33), we let t1=t+φ and get
(36)F1t,θ,σ,0=μθx¯2t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=μ2θ3cost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θθ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+θκ¯θ,σ+1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−θ+2θκ¯θ,σ+1a¯θ,σ+b¯θ,σcos2t1+β,where
(37)κ¯θ,σ=a1−2κH2θ,σμ−2θ−2,a¯θ,σ=2κ¯θ,σ+σ2+θ2+1,b¯θ,σ=θ2−1+σ22+4σ2,sinβ=2σb¯θ,σ,cosβ=θ2−1+σ2b¯θ,σ.Similarly, we also get(38)F2t,θ,σ,0=−μθx¯1t,θ,σe¯t,θ,σa1−2κ+x¯12t,θ,σ+x¯22t,θ,σ=−μ2θ2sint1−σcost12σsint1−θ2−1+σ2cost1/H2θ,σa1−2κ+μ2θ2sint1−σcost12+θ2cos2t1/H2θ,σ=−θ1+θ2−1+σ2cos2t1−σsin2t1κ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1+θ2−1+σ2sint1cost1−2σsin2t1+σκ¯θ,σ+1+θ2−1+σ2cos2t1−σsin2t1=−σ+2σκ¯θ,σ+b¯θ,σsin2t1+βa¯θ,σ+b¯θ,σcos2t1+β.The integration of the following periodical function is zero.(39)1π∫−π2π22b¯θ,σcos2t+2φ+βdta¯θ,σ+b¯θ,σcos2t+2φ+β=−1π∫−ππdb¯θ,σcosta¯θ,σ+b¯θ,σcost=0.According to the integral formula, we get(40)1π∫−π−2φ−β2π−2φ−β22dta¯θ,σ+b¯θ,σcos2t+2φ+β=1π∫−π2π22dta¯θ,σ+b¯θ,σcos2t=1π∫−π2π2dta¯θ,σ+b¯θ,σcost=2a¯2θ,σ−b¯2θ,σ=1F¯0,where
(41)F¯0=κ¯θ,σ+1κ¯θ,σ+θ2+κ¯θ,σσ2=κ¯θ,σ+12+θ2−1κ¯θ,σ+1+κ¯θ,σσ2.According to Section 10.4 of [26], we get
(42)F¯1θ,σF¯2θ,σ=1π∫0πF1t,θ,σ,0F2t,θ,σ,0dt=1F¯0θF¯0−κ¯θ,σ−1σF¯0−κ¯θ,σ.Thus, the averaged system of updating law (18) is
(43)θ̇σ̇=εθF¯1θ,σγF¯2θ,σ=εθF¯0θF¯0−κ¯θ,σ−1γσF¯0−κ¯θ,σ.ConsideringF¯0>κ¯θ,σ, the point θσT=10T is the unique equilibrium point of the averaged system (43). Let Lyapunov candidate function be
(44)Vθ,σ=0.5Kσ2+0.5θ−12,where K is a sufficiently large positive real number. Its derivative along the trajectory of the system (43) is
(45)V̇θ,σ=Kσσ̇+θ−1θ̇−εθF¯0Kγσ2F¯0−κ¯σ,θ+θ−1θF¯0−κ¯σ,θ−1.Whenθ>1, then F0>κ¯θ,σ+1 and V̇θ,σ<0 obviously. When θ<1, it can be discussed as follows:
(1)
Ifκ¯θ,σσ2<1−θ2κ¯θ,σ+1, then F0<κ¯θ,σ+1 and V̇θ,σ<0(2)
Ifκ¯θ,σσ2>1−θ2κ¯θ,σ+1, then we can let K be enough large such that V̇θ,σ<0 resulting from Kγσ2F¯0−κ¯σ,θ>θ−1θF¯0−κ¯σ,θ−1So, the derivative functionV̇θ,σ is negatively defined and the point θσT=10T is the unique globally stable equilibrium point of the averaged system (43).Invoking Theorem 10.4 of [20], for sufficiently small positive real number ε, the two variables θ and σ in (34) and (35) are asymptotically convergent to their equilibrium point,respectively, i.e.,θσT⟶10T.In steady state, putθ∞=1 and σ∞=0 into (29), we get x1∞=sint and x2∞=cost then prove (22) true.Now, recalling the time scale change (11), the variable transformation (17), and the condition (25), the theorem and the conclusion of (10) have been proved true.


## 3.1.2. Noise Characteristics Analysis

Let the high-frequency measure noise in the signal (1) be bounded and represented by
(46)y~τ=a~sinω~τ+δ~ω~≫ω0,where the amplitude a~, the frequency ω~, and the phase angle δ~ are constants.The noise characteristics of the new identification can be summarized asTheorem 2.
Under high-frequency noise of (46), if the design parameters, μ, ε, γ, and κ satisfy the conditions for Theorem 1, then the steady-state identification errors will be bounded by
(47)eω=ω^−ω0<εμκω0a~b¯1,(48)eσ=σ^−σ0<γεμκω0a~b¯2,(49)ea=a^−a0<a0eσt+μω0ω~−1a~eσ0t,where b¯1 and b¯2 are positive real constants in 0,∞×Dτ.Proof.
By use of the time scale change (11), the noise (46) is rewritten in time t as
(50)y~1t=a~sint~=a~sinω−10ω~t+δ~2.
Firstly, considerε=0, let x~1x~2T denote the state variable vector of the LTI system (14) forced only by noise y~1t instead of EDS signal y1t, we get
(51)x~̇1x~̇2=1ω0−σ^−μω^ω^−ω^−σ^x~1x~2+1ω0μω^0y~1t.
The frequency characteristics of the system (51) from y~1t to state variable vector x~1x~2T can be written as
(52)H~1jω~H~2jω~=μω^ω~−jσ^jω^ω~2σ^+μω^−jσ^σ^+μω^+ω^2−ω~2.
So the steady-state response of the LTI system (51) is
(53)x~1x~2=H~ω~a~ω~sint~+φ~−σ^cost~+φ~ω^cost~+φ~,where
(54)H~ω~=μω^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22,φ~=argω~2σ^+μω^+jσ^σ^+μω^+ω^2−ω~2.
In this instance, the identification error resulting fromy~1t is rewritten as
(55)e~t=y~1t−x~1=H~eω~a~sint~+φ~+β~,where β~ has some appropriate value and
(56)H~eω~=4σ^2ω~2+σ^2+ω^2−ω~22ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22
Letx~1x~2T denote the steady-state response of the LTI system (14) forced only by the EDS signal y1t. According to the superposition principle of LTI systems, the steady-state response of the LTI system (14) resulting from both the EDS signal y1t and the noise y~1t is represented by
(57)x~1x~2T=x^1x^2+x~1x~2.
Secondly, the decoupled updating laws (15) and (16) can be written as the following perturbed system:
(58)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1+εμω0‐1ω^2e~tκ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1.
Theorem1 abovementioned shows that the nominal system
(59)ω^̇σ^̇=εμω0‐1ω^2y1t−x^1κ+x^1+x~12+x^2+x~22x^2+x~2−γx^1+x~1,is an exponentially stable system.
The perturbation items are bounded by(60)εμω0‐1ω^2x^2+x~2e~tκ+x^1+x~12+x^2+x~22≤εμω^2H~eω~a~2ω0κ,(61)εμγω0‐1ω^2x^1+x~1e~tκ+x^1+x~12+x^2+x~22≤εμγω^2H~eω~a~2ω0κ.
According to Lemma 9.2 of [26], there exit boundary values of b¯1 and b¯2 such that (47) and (48) hold true.
Thirdly, from (7), by error synthesis principle, we know
(62)ea=a^t−a0=∂a^∂σ^eσ+∂a^∂x^1x~1+∂a^∂x^2x~2=a^eσt+eσ^tx^1+x~1x~1+x^2+x~2x~2x^1+x~12+x^2+x~22≤a^eσt+eσ^tx~21+x~22≤a0eσt+H~aω~a~eσ0t,where
(63)H~aω~=μω^ω~2+ω~σ^ω~22σ^+μω^2+σ^σ^+μω^+ω^2−ω~22.After the trajectory goes into the small neighborhood of the equilibrium point, we getω^≈ω~ and σ^≈σ~. If ω~≫ω^ and σ0≪1, then H~eσ^,ω^,ω~≈1, H~aσ^,ω^,ω~≈μω0ω~−1.Now, the Theorem2 has been consequently proved true.


## 3.1.3. Performance Evaluation.

For the measured signal (1), the identification method proposed in [17] with two positive design parameters, Ka and Kbcan be written in time τ as
(64)dx1dττdx2dττ=−σ^τω^τ−ω^τ−σ^τx1τx2τ+01eτ,dω^dττ=−KaKbx1τeτx12τ+x22τ,dσ^dττ=−Kax2τeτx12τ+x22τ,eτ=−y0τ+x2τ.With time scale change (11) applied, it is written in time t as
(65)ẋ1ẋ2=1ω0−σ^ω^−ω^−σ^x1x2+1ω001−y1t−x2,ω^̇=−KaKbω0x1−y1t−x2x12+x22,σ^̇=−Kaω0x2−y1t−x2x12τ+x22τ.After time scale change (11), the identification algorithm proposed in [6] can be rewritten in time t as
(66)ẋ0ẋ1ẋ2=1ω0−k0ω^0−k0ω^0−σ^ω^−kω^−ω^−σ^−kω^x0x1x2−ω^ω0k00ky1t,ω^̇=−μωω0x1−y1t−x2−x0x12+x22,σ^̇=−μσω0x2−y1t−x2−x0x12τ+x22τ,where k0, k, μω and μσare four positive design parameters.Remark 3.
Whenω0 has a rather large value, both the algorithm (64) and the new algorithm exemplify exponential convergence property. When ω0 has a rather small value, the explicit reciprocal ω0−1 in (65) may bring the algorithm (64) unconvergence property. Contrarily, the reciprocal ω0−1 in (15) and (16) is compensated by the frequency estimate value ω^, which brings the new method better robustness for low-frequency signals.When being distorted by the noise (50), the perturbation terms of the algorithm (65) similar to (60) and (61) are
(67)KaKbω0−1x2e~tx12+x22≤KaKbH~eω~a~x12+x22,Kaω0−1x1e~tx12+x22≤KaH~eω~a~x12+x22.In a small neighborhood of the equilibrium point,x12+x22≈a1e−σ1t, so the boundary values of identification errors of frequency and damping factor in (65) can be represented as
(68)eω=ω^−ω0<KaKbω0−1a~a1eσ1tb¯4,eσ=σ^−σ0<Kaω0−1a~a1eσ1tb¯5.Remark 4.
When the signal is distorted by high-frequency sinusoidal noise with constant amplitude, the identification errors of frequency and damping factor in the algorithm (65) increase exponentially over time, while that in the new method have both bounded values governed by design parameter κ.Remark 5.
For the identification of both frequency and damping factor in the new method, a greater value of the design parameterκ brings smaller steady-state identification errors under noise and slower transient convergence speeds simultaneously.Remark 6.
The state variablex0 in (66) denotes the estimated value of the direct current component. Furthermore, compare (65) with (66), we know that the same conclusions as Remark 3 and Remark 4 can be drawn for the algorithm in [6].


## 4. Simulation Results

In this section, the performance of the proposed algorithm is exemplified by simulation results employing Matlab/Simulink.


### 4.1. Performance Evaluation

Firstly, we let the EDS signal beyτ=4e−σ0τsinw0τ+0.1π.Atτ=70s, the frequency ω0 changes from 1.5rad/s down to 0.5rad/s, and the damping factor σ0 changes from 0.03 down to 0.01. All the time, the design parameters of our new identification are set to μ=1, ε=0.5, γ=0.5, and κ=1, the design parameters of the Browns algorithm proposed in [23] are ka=0.25 and kb=1. The design parameters of the Mojiri’s algorithm proposed in [6] are μω=0.2, μσ=0.6, k=1, and k0=0.2. The simulation results of the three algorithms are illustrated in Figure 1 for comparison.Figure 1
Simulation for tracking the performance of the three algorithms under no noise when the frequency of the EDS signal has different values.
(a)(b)In Figure1, the two black dashed lines of the algorithm in [6] and the two red solid lines of the algorithm in [23] show that both the frequency and the damping factor are accurately estimated when ω0=1.5rad/s and neither the frequency nor the damping factor are correctly identified when ω0=0.5rad/swith the same values of their design parameters. Contrarily, the two blue solid curves, results from the new algorithm, exemplify that both frequency and damping factor are exactly identified no matter what value ω0 is. This validates the conclusion of Remark 3: the convergence robustness of the new identification algorithm is better than that of the algorithm in [23] and the algorithm in [6].Secondly, let the EDS signalyτ=4e−0.02τsin0.5τ+0.1π be distorted by sinusoidal noise y~τ=0.1sin10τ. We set μ=1, ε=0.2, γ=1.5, and κ=0.5 in the new algorithm, and let ka=0.2 and kb=0.5 in algorithm of [23], and let μω=0.1, μσ=0.2, k=1.5, and k0=1.5 in algorithm in [16]. The simulation results of the three algorithms are illustrated in Figure 2.Figure 2
Simulation for anti-noise performance underyτ=4e−0.02τsin0.5τ+0.1π with noise y~τ=0.1sin10τ.In Figure2, the two red curves from the algorithm in [23] and the two black curves from the algorithm in [6] show that the identification errors of both frequency and damping factor, respectively, increase exponentially over time. In the two blue curves from the new algorithm, the boundary value of frequency identification error is ±0.0004 and the boundary value of identification error of the damping factor is ±0.0005. This fact verifies the conclusion of Remark 4: the antinoise performance of the new algorithm is far better than that of the algorithm in [23].


### 4.2. Amplitude Identification

Thirdly, the amplitude identification proposed in this paper is exemplified in Figure3. We let the EDS signal be yτ=2e−0.02τsin0.5τ+0.1π, let the sinusoidal noise be y~τ=0.1sin10τ, and let the design parameters be μ=1, ε=0.5, γ=0.5, and κ=0.5.Figure 3
Simulation for amplitude identification of the new algorithm under no noise (a) and under sinusoidal noise (b), respectively.
(a)(b)In Figure3, the subplot (a) shows that the proposed algorithm can exactly identify the amplitude of the EDS signal under no noise while the subplot (b) declares that the amplitude identification error resulting from high-frequency sinusoidal noise grows exponentially over time. These simulation results verify the conclusion of (49).


### 4.3. Effects of Design Parameters

Finally, we exemplify the effect of the design parameterκ on tracking performance and antinoise performance by simulation examples with different value of κ. We let the EDS signal be yτ=4e−0.02τsin1.2τ+0.1π and let the three design parameters be μ=1, ε=0.5, and γ=0.5.In each subplot of Figure4, from the same initial values of x^1x^2ω^σ^T=111.50T, the convergence speed of the red dashed curves corresponding to κ=2 is slower than that of the blue solid curves corresponding to κ=1, while faster than that of the black dashed curves corresponding to κ=8. This validates one conclusion of Remark 5: the larger value of the design parameter κ brings slower convergence speed to the identification of frequency, damping factor, and amplitude.Figure 4
Simulation for effects of design parameterκ on tracking performance with yτ=4e−0.02τsin1.2τ+0.1π.In Figure5, let the EDS signal in Figure 4 be distorted by the noise y~τ=0.1sin10τ. In the three frequency identification errors illustrated in the left three subplots, the boundary values of the blue curve, the red curve, and the black curve, corresponding to κ=1, κ=2, and κ=8 are ±0.0041, ±0.0025, and ±0.0009, respectively.Figure 5
Simulation for effects of design parameterκ on antinoise performance.In the three damping factor identification errors illustrated in the right three subplots, the boundary values of the blue curve, the red curve, and the black curve, corresponding toκ=1,κ=2, and κ=8, are ±0.0022, ±0.0012, and ±0.0005, respectively. So, a larger value of the design parameter κ brings smaller steady-state identification error resulting from sinusoidal noise to the identification of frequency and damping factor. This verifies the other conclusion of Remark 5.The design parameters should be tuned according to the compromise between transient performance and steady-state performance. From (43)–(49) as well as Figures 4 and 5, big values of the three parameters, μ, γ, andε, bring faster-tracking performance and worse antinoise performance while big value of κ results in slower tracking speed and better antinoise performance.


## 4.1. Performance Evaluation

Firstly, we let the EDS signal beyτ=4e−σ0τsinw0τ+0.1π.Atτ=70s, the frequency ω0 changes from 1.5rad/s down to 0.5rad/s, and the damping factor σ0 changes from 0.03 down to 0.01. All the time, the design parameters of our new identification are set to μ=1, ε=0.5, γ=0.5, and κ=1, the design parameters of the Browns algorithm proposed in [23] are ka=0.25 and kb=1. The design parameters of the Mojiri’s algorithm proposed in [6] are μω=0.2, μσ=0.6, k=1, and k0=0.2. The simulation results of the three algorithms are illustrated in Figure 1 for comparison.Figure 1
Simulation for tracking the performance of the three algorithms under no noise when the frequency of the EDS signal has different values.
(a)(b)In Figure1, the two black dashed lines of the algorithm in [6] and the two red solid lines of the algorithm in [23] show that both the frequency and the damping factor are accurately estimated when ω0=1.5rad/s and neither the frequency nor the damping factor are correctly identified when ω0=0.5rad/swith the same values of their design parameters. Contrarily, the two blue solid curves, results from the new algorithm, exemplify that both frequency and damping factor are exactly identified no matter what value ω0 is. This validates the conclusion of Remark 3: the convergence robustness of the new identification algorithm is better than that of the algorithm in [23] and the algorithm in [6].Secondly, let the EDS signalyτ=4e−0.02τsin0.5τ+0.1π be distorted by sinusoidal noise y~τ=0.1sin10τ. We set μ=1, ε=0.2, γ=1.5, and κ=0.5 in the new algorithm, and let ka=0.2 and kb=0.5 in algorithm of [23], and let μω=0.1, μσ=0.2, k=1.5, and k0=1.5 in algorithm in [16]. The simulation results of the three algorithms are illustrated in Figure 2.Figure 2
Simulation for anti-noise performance underyτ=4e−0.02τsin0.5τ+0.1π with noise y~τ=0.1sin10τ.In Figure2, the two red curves from the algorithm in [23] and the two black curves from the algorithm in [6] show that the identification errors of both frequency and damping factor, respectively, increase exponentially over time. In the two blue curves from the new algorithm, the boundary value of frequency identification error is ±0.0004 and the boundary value of identification error of the damping factor is ±0.0005. This fact verifies the conclusion of Remark 4: the antinoise performance of the new algorithm is far better than that of the algorithm in [23].


## 4.2. Amplitude Identification

Thirdly, the amplitude identification proposed in this paper is exemplified in Figure3. We let the EDS signal be yτ=2e−0.02τsin0.5τ+0.1π, let the sinusoidal noise be y~τ=0.1sin10τ, and let the design parameters be μ=1, ε=0.5, γ=0.5, and κ=0.5.Figure 3
Simulation for amplitude identification of the new algorithm under no noise (a) and under sinusoidal noise (b), respectively.
(a)(b)In Figure3, the subplot (a) shows that the proposed algorithm can exactly identify the amplitude of the EDS signal under no noise while the subplot (b) declares that the amplitude identification error resulting from high-frequency sinusoidal noise grows exponentially over time. These simulation results verify the conclusion of (49).


## 4.3. Effects of Design Parameters

Finally, we exemplify the effect of the design parameterκ on tracking performance and antinoise performance by simulation examples with different value of κ. We let the EDS signal be yτ=4e−0.02τsin1.2τ+0.1π and let the three design parameters be μ=1, ε=0.5, and γ=0.5.In each subplot of Figure4, from the same initial values of x^1x^2ω^σ^T=111.50T, the convergence speed of the red dashed curves corresponding to κ=2 is slower than that of the blue solid curves corresponding to κ=1, while faster than that of the black dashed curves corresponding to κ=8. This validates one conclusion of Remark 5: the larger value of the design parameter κ brings slower convergence speed to the identification of frequency, damping factor, and amplitude.Figure 4
Simulation for effects of design parameterκ on tracking performance with yτ=4e−0.02τsin1.2τ+0.1π.In Figure5, let the EDS signal in Figure 4 be distorted by the noise y~τ=0.1sin10τ. In the three frequency identification errors illustrated in the left three subplots, the boundary values of the blue curve, the red curve, and the black curve, corresponding to κ=1, κ=2, and κ=8 are ±0.0041, ±0.0025, and ±0.0009, respectively.Figure 5
Simulation for effects of design parameterκ on antinoise performance.In the three damping factor identification errors illustrated in the right three subplots, the boundary values of the blue curve, the red curve, and the black curve, corresponding toκ=1,κ=2, and κ=8, are ±0.0022, ±0.0012, and ±0.0005, respectively. So, a larger value of the design parameter κ brings smaller steady-state identification error resulting from sinusoidal noise to the identification of frequency and damping factor. This verifies the other conclusion of Remark 5.The design parameters should be tuned according to the compromise between transient performance and steady-state performance. From (43)–(49) as well as Figures 4 and 5, big values of the three parameters, μ, γ, andε, bring faster-tracking performance and worse antinoise performance while big value of κ results in slower tracking speed and better antinoise performance.


## 5. Conclusion

A robust algorithm is proposed for asymptotically identifying the frequency, the damping factor, and the amplitude of a LFO modeled as an exponentially damped sinusoidal signal. The transient convergence property under no noise and the boundary of the steady-state identification errors under sinusoidal noise are exhibited. The new algorithm exemplifies good convergence property for a rather wide range of frequency signal, especially for low-frequency signal, as well as bounded identification errors being adjusted by values of design parameters under bounded noise. The steady-state identification errors of frequency and damping factors are bounded with boundary values adjustable by the value of design parameters. The design parameters have different effects on transient performance and steady-state performance of the identification of frequency, damping factor, and amplitude, respectively. The proposed amplitude identification will be improved in antinoise performance in expectation.



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