This paper presents spectral analysis of an electrohydraulic system. For a linear system, spectral analysis using a frequency response function (FRF) offers great insight into system dynamics and controls. The objective of this paper is to extend such benefits to the nonlinear electrohydraulic system. To achieve the objective, generalized frequency response functions (GFRFs) and output spectra of the electrohydraulic system are analyzed in frequency domain. In this paper, two different approaches are proposed to derive the GFRFs. In the first approach, the analytic GFRFs are derived from physical dynamics of the electrohydraulic system. Thus, the dynamic features of the electrohydraulic system can be explored with respect to the physical parameters explicitly in frequency domain. In the second approach, the experimental GFRFs are identified from frequency response data. Although the explicit relationship with the physical parameters is not available, they can predict the output spectrum without a priori knowledge of the electrohydraulic system. The proposed approaches are applied to derive the GFRFs analytically and experimentally for spectral analysis of an electrohydraulic system. Spectral analysis reveals the critical dynamic features of the electrohydraulic system in frequency domain, and it turns out to be crucial for system design, identification, and controls of the electrohydraulic system.

## Introduction

Given the high power density, large force capacity, and mechanical flexibility over other power transmission means, electrohydraulic systems have been used in a broad range of applications: construction machinery, agriculture equipment, manufacturing machines, industrial robots, aerospace actuators, and automobile components [14]. However, the inherent nonlinear dynamics of the electrohydraulic system has complicated its practical use. Therefore, modeling, analysis, and control design of the electrohydraulic system have been actively studied since the past decades.

In the vicinity of the equilibrium, the linear models of the electrohydraulic system were developed and used for control design while regarding nonlinearities as uncertainties [58]. A beauty of the linear model lies in spectral analysis using the FRF (i.e., linear transfer function in frequency domain), since it offers a physical interpretation of a system and a great tool for system identification and control design in frequency domain. However, since the electrohydraulic system includes nonlinear dynamics, the linear model cannot capture all the critical features. As a quasi-linearization method, the describing function was employed to develop the transfer function depending on the input amplitude [9]. But, it cannot retain frequency interference of a nonlinear system such as harmonics and intermodulation [10].

To improve model accuracy, the nonlinear models were used. The essential nonlinear factor is the orifice flow rate, which is proportional to the orifice area and the square root of the pressure drop. Another factor is the friction force, such as stiction and Coulomb friction, exerting on the contact surfaces of the spool and the piston. These nonlinearities were taken into account for modeling, parameter estimation, and nonlinear control design in Refs. [1114]. One of the drawbacks of a nonlinear model is the absence of universal analysis tool except numerical simulation using a computational machine. Such a technical difficulty motivates the need of informative spectral analysis for studying dynamic features of a nonlinear system.

For a wide class of nonlinear systems, input/output relationship can be represented by convergent Volterra series. Volterra series is a functional expansion of a nonlinear system with a kernel in time domain. Since the theoretical foundations were established [1518], Volterra series has been extensively studied for spectral analysis of nonlinear systems [19,20]. The GFRF is the multivariable Fourier transform of the kernel. Provided the GFRF is available, the output spectrum, which is the multivariable Fourier transform of the output, can be computed analytically. In spite of such great utility in spectral analysis, few applications to actual mechanical systems have been made due to its complex derivation [21]. For fault detection and parametric identification of the electrohydraulic system, Volterra series was employed to describe the system in time domain [22,23], but spectral analysis using the GFRF has not been studied yet.

In this paper, the GFRFs of the electrohydraulic system are derived in two approaches. In the first approach, motivated by Refs. [17] and [21], the GFRF is analytically derived from physical dynamics of the electrohydraulic system. Because of the explicit connection with the physical parameters, the analytic GFRFs can be characterized by the physical parameters. In the second approach, the GFRF is experimentally identified with the assumption of the block-oriented models such as the Wiener- and Hammerstein models. Frequency domain identification of these models has been actively studied for their broad application to complex and undetermined systems [2426]. In this paper, further improvement from the former work [27] has been made for efficient and consistent identification of the block-oriented models.

With the derived GFRFs, the output spectrum is estimated and analyzed to explore the nonlinear dynamic features of the electrohydraulic system in frequency domain. First, spectral analysis with the analytic GFRF reveals that the nonlinear behavior of the electrohydraulic system is dominant near the resonance frequency. And the effect of the input amplitude on the output spectrum is investigated with respect to the working range of the state variables. Second, the output spectrum of the electrohydraulic system is estimated using the experimental GFRFs with good accuracy. And it turns out that the Wiener model is better suited than the Hammerstein model to describe the given electrohydraulic system. In addition, spectral analysis offers a versatile tool for control design of the electrohydraulic system in frequency domain.

The rest of the paper is organized as follows: In Sec. 2, theoretical background of spectral analysis using Volterra series is briefly introduced. In Secs. 3 and 4, the GFRFs of the nonlinear electrohydraulic system are derived analytically and experimentally, respectively. Then, Sec. 5 presents spectral analysis to investigate the dynamic features of the electrohydraulic system in frequency domain using the GFRFs. The concluding remarks are given in Sec. 6.

## Theoretical Background

Theoretical background is briefly introduced to clarify the technical objective of the paper. Input/output representation of a nonlinear system can be described by Volterra series
$y(t)=∑n=1Nyn(t)$
(1)
$yn(t)=∫−∞∞⋯∫−∞∞hn(τ1,…,τn)∏i=1nu(t−τi)dτi$
(2)

where u(t), y(t), and $hn(τ1,…,τn)$ are the input, the output, and the nth order Volterra kernel, respectively. The Volterra kernel is the counterpart of the impulse response function of a linear system.

Applying (multivariable) Fourier transform to the output and the Volterra kernel yields the output spectrum and the nth order GFRF
$Yn(jω)=∫−∞∞yn(τ)e−jωτdτ$
(3)
$Hn(jω1,…,jωn)=∫−∞∞⋯∫−∞∞hn(τ1,…,τn)∏i=1ne−jωiτidτi$
(4)
It is noted that a symmetric GFRF of which all permutations are identical is used for uniqueness without explicit notice hereafter [17]. Multivariable Fourier transform is the generalization of Fourier transform from single-frequency domain to multifrequency domain. Similarly with the Volterra kernel, the GFRF is the counterpart of the FRF of a linear system. The output and the nth order Volterra kernel can be rewritten by inverse transform
$yn(t)=12π∫−∞∞Yn(jω)ejωtdω$
(5)
$hn(τ1,…,τn)=1(2π)n∫−∞∞⋯∫−∞∞Hn(jω1,…,jωn)∏i=1nejωiτidωi$
(6)
Substituting Eq. (6) into Eq. (2) and reordering integrals give
$yn(t)=1(2π)n∫−∞∞⋯∫−∞∞Hn(jω1,…,jωn)U(jω1)⋯U(jωn)ej(ω1+⋯+ωn)tdω1⋯dωn$
(7)
$U(·)$ is the input spectrum defined similarly with the output spectrum. Then, by equating Eqs. (5) and (7) and by denoting $ω=ω1+⋯+ωn≥0$, the output spectrum can be computed from the input spectrum and the GFRFs such as
$Yn(jω)=1(2π)n−1∫ω=ω1+⋯+ωnHn(jω1,…,jωn)U(jω1)⋯U(jωn)dω1⋯dωn−1$
(8)
$Y(jω)=∑n=1NYn(jω)$
(9)
With the single frequency input $u(t)=A sin(ωot+θ)$ (A: amplitude, ωo: frequency, and θ: initial phase angle), using Eqs. (8) and (9), the output spectrum is given by
$Yn(jω)=12n−1∑ωk1+⋯+ωkn=ωHn(jωk1,…,jωkn)U(jωk1)U(jωk2)⋯U(jωkn)$
(10)
$Y(jω)=∑n=1NYn(jω)$
(11)

where $ωki∈{+ωo,−ωo}$ for $i∈{1,…,n}$. $U(−jωki)$ is a complex conjugate of $U(jωki)$. Nonnegative ω is the output frequency which is not necessarily same with the input frequency ωo. It is noted that spectral analysis with steady-state data is available for a stable system in the sense of bounded input and bounded output (BIBO). In other words, the magnitude of the GFRFs should be bounded to yield bounded output spectrum [28].

For a stable nonlinear system, provided GFRFs and input spectrum are available, output spectrum can be estimated. Or provided input- and output spectra are measurable, GFRFs can be experimentally identified. Accordingly, the GFRFs of the nonlinear electrohydraulic system will be derived in these two approaches.

## Analytic Generalized Frequency Response Functions Derivation

In this section, the GFRFs of the electrohydraulic system are analytically derived from physical dynamics with the three steps: (1) physical model development, (2) system augmentation through Carleman bilinearization, and (3) GFRFs derivation by applying the growing exponential method. For mathematical details of Carleman bilinearization and the growing exponential method, please see Ref. [17].

### Physical Dynamics.

Figure 1 depicts the schematics of the electrohydraulic system for camless engine valve actuation. $xp(t), pc(t)$, and $xs(t)$ indicate the piston position, cylinder pressure, and spool position, respectively. $qo(t), qli(t)$, and $qle(t)$ are the orifice flow rate, internal and external leakage, respectively. The leakages are due to radial clearance in the spool and piston. i(t) is the solenoid current. ps and pr are the supply and return pressure, respectively. Positive directions of $xp(t), xs(t), qo(t), qli(t)$, and $qle(t)$ are given by the arrows in the figure. The whole system consists of three main components: hydraulic pump, spool valve, piston. The pump supplies high pressure oil. The spool valve is of a three-way type and it is assumed to be critically centered [5]. The spool is controlled by the voice coil motor and it controls hydraulic flow to actuate the piston. Thereby, the piston operates the poppet valve. This type of structure is quite common in many electrohydraulic applications.

Fig. 1
Fig. 1
Close modal
Physical dynamics of the electrohydraulic system is given by
$x¨p(t)=1mp(Acpc(t)−kpxp(t)−bpx˙p(t)−Fo−Ff(t))$
(12)
$p˙c(t)=βVc(t)(qo(t)−qli(t)−qle(t)−Acx˙p(t))$
(13)
$x¨s(t)=1ms(kmi(t)−ksxs(t)−bsx˙s(t)−Fss(t)−Ftr(t))$
(14)

In Eq. (12), $kp, bp$, and mp denote the spring constant, viscous damping coefficient, and lumped mass of the piston and the poppet valve. Ac is the cross section area of the hydraulic cylinder. Since the piston and the poppet valve are not physically connected, the poppet valve is heavily preloaded by the spring (Fo) for continuous contact with the piston [8,29]. The last term is the friction including stiction, Coulomb, and viscous friction. The friction is one of the main contributors to nonlinear behaviors of the electrohydraulic system [2,13,30]. However, this paper concerns the viscous friction only and it will be lumped into the viscous damping in the third term hereafter. This is because stiction and Coulomb friction can be greatly reduced by design and oil lubrication [31]. In addition, nonsmooth functions to represent stiction and Coulomb friction make derivation of analytic GFRFs tedious and inefficient.

In Eq. (13), β is the bulk modulus of the oil. Vc(t) is the cylinder volume including the dead volume in the pipe line and it is assumed to be constant as Vco because of a small range of the piston position, that is, $Vc(t)=Acxp(t)+Vco≈Vco$ [5]. The orifice flow rate is given by
$qo(t)={Coxs(t)(ps−pc(t)), if xs(t)≥0Coxs(t)(pc(t)−pr), if xs(t)<0$
(15)
where $Co=CdW2/ρ$. Cd, W, and ρ are the discharge coefficient, area gradient of the spool valve, and oil density, respectively. The internal and external leakages assumed to be laminar flows are given by
$qli(t)={Cli(pc(t)−pr), if xs(t)≥0Cli(pc(t)−ps), if xs(t)<0$
(16)
$qle(t)=Cle(pc(t)−pr)$
(17)

where Cli and Cle are the corresponding constants.

In Eq. (14), km is the magnetic force sensitivity of the voice coil motor. And ks,bs, and ms are defined similarly with the piston. The last two terms indicate the steady-state and transient flow forces which act in direction to close the spool valve [5,7]. Since these flow forces are much smaller than spring and damping forces, they can be neglected in general.

From Eqs. (12)(14), the state, the input, and the output of the electrohydraulic system are defined by
$x(t)=[xp(t), x˙p(t), pc(t), xs(t), x˙s(t)]T∈ℜ5×1u(t)=i(t)∈ℜy(t)=xp(t)∈ℜ$

For existence of the convergent Volterra series and convenient derivation, the physical dynamics given by Eqs. (12)(17) is revised regarding Volterra theory and the unique feature of the engine valve application.

#### Variable Shift.

The deviation from the equilibrium is defined as $x¯(t)=x(t)−xeq$. Only the cylinder pressure and piston position have nonzero equilibrium, and they are related as $pceq=(kpxpeq+Fo)/Ac$. By variable shift, the constant input of Fo is removed for algebraic convenience. The revised dynamics renders a general form of the electrohydraulic system as used in Refs. [57,1114].

#### Smoothness.

For existence of convergent Volterra series, the physical dynamics is smoothened. To be specific, the nonsmooth orifice equation and internal leakage are approximated by the hyper-tangent functions such as
$q¯o(t)=Coxs(t)((pseq−p¯c(t))(12+12tanh(kxs(t)))+(preq+p¯c(t))(12−12tanh(kxs(t))))$
(18)
$q¯li(t)=Cli((preq+p¯c(t))(12+12tanh(kxs(t)))+(−pseq+p¯c(t))(12−12tanh(kxs(t))))$
(19)

where $pseq=ps−pceq≥0$ and $preq=pceq−pr≥0$. k is the parameter which should be determined appropriately regarding approximation accuracy and the convergent radius of Taylor series for the hyper-tangent function.

#### Stability.

As presented in Sec. 2, the GFRFs should be bounded at all frequencies for frequency domain analysis. However, the magnitude of the first order GFRF derived from Eqs. (12)(14) is infinite at zero frequency (see Refs. [5] and [32] for the linear transfer function of the electrohydraulic system). Therefore, the electrohydraulic system is stabilized by the feedback loop as shown in Fig. 2 [32]. Then, the solenoid current is $i(t)=kst(r(t)−xp(t))$, where kst is the proportional control gain.

Fig. 2
Fig. 2
Close modal
Finally, by setting $r(t)=ro+r¯(t)$ and ro = xpeq, the dynamic equations of Eqs. (12)(14) are revised as
$x¯¨p(t)=1mp(Acp¯c(t)−kpx¯p(t)−bpx¯˙p(t))$
(20)
$p¯˙c(t)=βVco(q¯o(t)−q¯li(t)−q¯le(t)−Acx¯˙p(t))$
(21)
$x¯¨s(t)=1ms(ksvr¯(t)−ksvx¯p(t)−ksx¯s(t)−bsx¯˙s(t))$
(22)
where $ksv=kstkm$ and $q¯le(t)=Cle(preq+p¯c(t))$. The new system definition is given by
$x(t)=[x¯p(t), x¯˙p(t), p¯c(t), x¯s(t), x¯˙s(t)]T∈ℜ5×1$
(23)
$u(t)=r¯(t)∈ℜ$
(24)
$y(t)=x¯p(t)∈ℜ$
(25)

### Carleman Bilinearization.

Equations (20)(22) are approximated by multivariable Taylor series at the equilibrium up to Nth order such as
$x˙(t)=∑i=1NAkx(k)(t)+B0u(t)$
(26)
$y(t)=C1x(t)$
(27)
x(k)(t) is the Kronecker product vector: $x(k)(t)=x(t)⊗⋯⊗x(t)$. A1, B0, and C1 (used for the first‐order GFRF) are shown below:
$A1=[01000−a21−a22a23000−a32−a33a34000001−a5100−a54a55],B0=[0000a51]C1=[10000]$
The elements aij are given below:
$a21=kpmp,a22=bpmp,a23=Acmp,a32=βAcVco,a33=βVco(Cli+Cle),a34=β2Vco(Co(pseq+preq)−kCli(pseq+preq))a51=kvsms,a54=ksms,a55=bsms,$

Nonzero elements of Ak for k > 1 are due to the orifice flow rate and internal leakage flow rate, which are both the nonlinear functions of the cylinder pressure and spool position. Ak for k > 1 are omitted for the limited space.

By differentiating the Kronecker product vector x(j)(t), the differential equation of x(j)(t) can be obtained such as
$ddtx(j)(t)=∑k=1N−j+1Aj,k x(k+j−1)(t)+Bj,0 x(j−1)u(t)$
(28)
where Aj,k and Bj,0 are given by
$Aj,k=Ak⊗In⊗⋯⊗In+In⊗Ak⊗⋯⊗In+⋯+In⊗⋯⊗In⊗Ak$
(29)
$Bj,0=B0⊗In⊗⋯⊗In+In⊗B0⊗⋯⊗In+⋯+In⊗⋯⊗In⊗B0$
(30)
In is the 5 × 5 identity matrix. For the augmented system
$x⊗(t)T=[x(1)(t)Tx(2)(t)T⋯x(N)(t)T]$
the dynamics is also augmented in a bilinear form
$ddtx⊗(t)=Ax⊗(t)+Dx⊗(t)u(t)+Bu(t)$
(31)
$y(t)=Cx⊗(t)$
(32)
Bilinearization matrices A, B, C, and D are given below
$A=[A1,1A1,2⋯A1,N0A2,1⋯A2,N−100⋯A3,N−2⋮⋮⋱⋮00⋯AN,1],D=[00·00B2,00·000B3,0·00⋮⋮⋱⋮⋮00⋯BN,00]BT=[B1,0T00⋯0],C=[C10⋯0]$

### Growing Exponential Method.

In the growing exponential method, the input and the state are assumed to be growing exponential functions. Then, by applying them into dynamic equations and equating same exponentials, the GFRFs can be computed. Detailed mathematics can be found from Ref. [17]. In this section, the final results of the analytic GFRFs are given.

For the bilinear system given by Eqs. (31) and (32), the growing exponential method yields the following analytic GFRFs:
$H1(jω)=C(jωI−A)−1B$
(33)
$H2(jω1,jω2)=12!∑π(·)C(j(ω1+ω2)I−A)−1D(jω1I−A)−1B$
(34)
$H3(jω1,jω2,jω3)=13!∑π(·)C(j(ω1+ω2+ω3)I−A)−1D(j(ω1+ω2)I−A)−1D(jω1I−A)−1B$
(35)
$⋮$
$HN(jω1,…,jωN)=1N!∑π(·)C(j(ω1+⋯+ωN)I−A)−1D(j(ω1+⋯+ωN−1)I−A)−1×⋯×D(j(ω1+ω2)I−A)−1D(jω1I−A)−1B$
(36)

$∑π(·)$ indicates summation of all permutations for symmetricity of the GFRFs. Since the bilinearization matrices A, B, C, and D are functions of the physical parameters of the electrohydraulic system, the analytic GFRFs can be characterized by the physical parameters.

## Experimental Generalized Frequency Response Functions Derivation

In this section, the experimental GFRFs of the electrohydraulic system are identified from frequency response data. The input/output definition is same with Sec. 3.1: $u(t)=r¯(t),y(t)=x¯p(t)$. In this paper, the block-oriented nonlinear models such as the Wiener and the Hammerstein models are assumed for a structure of the electrohydraulic system without a priori knowledge about the system.

### Model Structure.

The Wiener model has a linear dynamic block, which is followed by a nonlinear static block as shown in Fig. 3(a). On the contrary, a static nonlinear block precedes a linear dynamic block in the Hammerstein model like Fig. 3(b). u(t), x(t), and y(t) are the input, the internal state, and the output, respectively. System identification of these models are challenging because the internal state is not measurable.

Fig. 3
Fig. 3
Close modal
For both models, the linear dynamic block and the nonlinear static block are assumed to be a stable FRF and a polynomial equation of Nth order and they are given by
$G(jω)=B(jω)A(jω)=bm(jω)m+bm−1(jω)m−1⋯+b1(jω)+b0(jω)n+an−1(jω)n−1+⋯+a1(jω)+a0$
(37)
$f(x)=c1x(t)+c2x(t)2+⋯+cNx(t)N$
(38)
where n ≥ m for causality. $P=[a0,…,an−1;b0,…,bm]$ and $Q=[c1,…,cN]$ are the real model parameters of the linear dynamic block and the nonlinear static block, respectively. By applying the growing exponential method, the GFRFs of the Wiener and the Hammerstein models are given by (in order) [26]
$Hn(jω1,…,jωn)=cnG(jω1)⋯G(jωn)$
(39)
$Hn(jω1,…,jωn)=cnG(jω1+⋯+jωn)$
(40)

Therefore, parameter estimation of $P$ and $Q$ replaces identification of the experimental GFRFs of the electrohydraulic system.

### Parameter Estimation.

Let the electrohydraulic system be excited by the input of different q frequencies
$u(t)=A sin(ωkt+θ),k∈{1,…,q}$
(41)
By substituting Eqs. (39) and (40) into Eqs. (10) and (11), the harmonic output spectra up to the four times of the input frequency can be analytically computed: Eqs. (42)(46) for the Wiener model and Eqs. (47)(51) for the Hammerstein model. Here, the GFRFs up to the fourth‐order are taken into account and higher‐order terms are omitted.
$Y(0)=c2G(jωk)G(−jωk)U(jωk)U(−jωk)+3c44G(jωk)2G(−jωk)2U(jωk)2U(−jωk)2$
(42)
$Y(jωk)=c1G(jωk)U(jωk)+3c34G(jωk)2G(−jωk)U(jωk)2U(−jωk)$
(43)
$Y(2jωk)=c22G(jωk)2U(jωk)2+c42G(jωk)3G(−jωk)U(jωk)3U(−jωk)$
(44)
$Y(3jωk)=c34G(jωk)3U(jωk)3$
(45)
$Y(4jωk)=c48G(jωk)4U(jωk)4$
(46)
$Y(0)=c2G(0)U(jωk)U(−jωk)+3c44G(0)U(jωk)2U(−jωk)2$
(47)
$Y(jωk)=c1G(jωk)U(jωk)+3c34G(jωk)U(jωk)2U(−jωk)$
(48)
$Y(2jωk)=c22G(2jωk)U(jωk)2+c42G(2jωk)U(jωk)3U(−jωk)$
(49)
$Y(3jωk)=c34G(3jωk)U(jωk)3$
(50)
$Y(4jωk)=c48G(4jωk)U(jωk)4$
(51)

### Linear Dynamic Block Identification.

The fundamental output spectra of the two models (i.e., Eqs. (43) and (48)) are rewritten as (in order)
$Y(jωk)=G(jωk)U(jωk)(c1+3c34G(jωk)G(−jωk)U(jωk)U(−jωk))$
(52)
$Y(jωk)=G(jωk)U(jωk)(c1+3c34U(jωk)U(−jωk))$
(53)
Because all terms in the brackets are real, the phase angles of the linear dynamic blocks are same and given by
$ϕ(ωk)=∠Y(jωk)−∠U(jωk)$
(54)
The linear dynamic block parameter of $P$ cannot be uniquely solved using the phase angle measurements only. Hence, $P$ is additionally constrained with the monic numerator, i.e., bm = 1. However, this constraint is compensated by the nonlinear static block, since any pair of $(αG(jω),f(·)/α)$ shows a same system gain with nonzero α for the Wiener and the Hammerstein models. Linear dynamic block identification is faced with finding the optimal $P$ of the nonlinear least squares problem
$argPmin∑k=1q(ϕ̂(ωk)−ϕ(ωk))2$
(55)

where $ϕ̂(ωk)$ is the estimated phase angle from Eq. (37), and $ϕ(ωk)$ is the measured phase angle from Eq. (54) at each frequency. In Ref. [27], the above problem is solved using the iterative Newton–Gauss type algorithm [33]. However, the solution needs the good initial guess as in a general nonlinear least squares problem. In this paper, motivated from the pole/zero estimation from the phase angle measurements [34], the efficient iterative linear least squares method is proposed.

Let us define the even and odd functions of G(jω)
$Ge(jω)=12(G(jω)+G(−jω))=M(jω)A(jω)A(−jω)$
(56)
$Go(jω)=12(G(jω)−G(−jω))=N(jω)A(jω)A(−jω)$
(57)
where M(jω) and N(jω) are defined below.
$M(jω)=A(−jω)B(jω)+A(jω)B(−jω)2$
(58)
$N(jω)=A(−jω)B(jω)−A(jω)B(−jω)2$
(59)
And from Eqs. (58) and (59)
$M(jω)+N(jω)=A(−jω)B(jω)$
(60)
$M(jω)−N(jω)=A(jω)B(−jω)$
(61)
Then, the phase angle function of the linear dynamic block is determined by
$tan ϕ(ω)=Go(jω)jGe(jω)=N(jω)jM(jω)$
(62)
The new FRF is defined and rewritten using Eqs. (60)(62)
$T(jω)=G(jω)G(−jω)=A(−jω)B(jω)A(jω)B(−jω)=1+j tan ϕ(ω)1−j tan ϕ(ω)$
(63)
The poles of T(jω) include the poles and negative zeros of G(jω). Similarly, the zeros of T(jω) include the zeros and negative poles of G(jω). The magnitude of T(jω) is uniformly one. If G(jω) is estimated from the phase angle measurement, T(jω) can be estimated either. Therefore, linear dynamic block identification is replaced by solving the following problem:
$argPmin∑k=1q||T̂(jωk)−T(jωk)||2$
(64)

where $T̂(jωk)$ is the complex function of $P$, and T(jωk) is the measurement at each frequency. T(jω) is identified using the iterative linear least squares method (SK iteration) [33]. Then, by appropriate assignment of the poles and zeros of T(jω), P is estimated. In this paper, the minimum phase and stable linear dynamic block is assumed. It is worth noting that the Wiener and the Hammerstein models share the common linear dynamic block with the same output measurement.

### Nonlinear Static Block Identification.

Provided the linear dynamic block is identified, the nonlinear static blocks can be identified for both models. Since the harmonic output spectra are linear in the nonlinear static block parameter $Q$, Eqs. (42)(46) (Wiener model) and Eqs. (47)(51) (Hammerstein model) can be rewritten in matrix forms
$Π(ωk)=ΞW(ωk)XW$
(65)
$Π(ωk)=ΞH(ωk)XH$
(66)

where $Π(ωk)=[Y(0),Y(jωk),Y(2jωk),Y(3jωk),Y(4jωk)]T∈C5×1$ is the complex vector of the measured harmonic output spectra and $X=[c1, c2, c3, c4]T∈ℜ4×1$ is the real vector of the $Q$ to be estimated. $Ξ(ωk)∈C5×4$ is the complex regressor matrix determined appropriately from Eqs. (42)(46) and Eqs. (47)(51), respectively. The subscripts “W” and “H” represent the Wiener and the Hammerstein models.

Nonlinear static block identification is finding the optimal $Q$ minimizing the cost function given by
$argQmin∑k=1q||Π̂(ωk)−Π(ωk)||2$
(67)

$Π̂(ωk)$ is the estimated harmonic output spectra from Eqs. (65) and (66), and $Π(ωk)$ is the measured harmonic output spectra at each frequency. Then, the linear least squares problem for each model structure is solved analytically [27]. Finally, the experimental GFRFs for the Wiener and the Hammerstein models are computed using Eqs. (39) and (40). It is noted that the experimental GFRFs do not have explicit connection with the physical parameters.

## Spectral Analysis of Electrohydraulic System

In this section, spectral analysis of the electrohydraulic system for camless engine valve actuation is conducted using the GFRFs and the harmonic output spectra.

### Spectral Analysis With Analytic GFRFs.

With the physical parameters given in Table 1, the analytic GFRFs are computed up to the fourth‐order.

Table 1

Physical parameters of electrohydraulic system

SymbolValueUnitSymbolValueUnit
mp0.120kgms0.025kg
bp7.5N ⋅ s/mbs15N ⋅ s/m
kp20 × 103N/mks2.5 × 103N/m
Ac44.18 × 10−6m2Fo75N
Vco0.5 × 10−6m3km9.79N/A
β1200 × 106PaCd0.6(-)
ps3.5 × 106PaCli1 × 10−14m
pr0.1 × 106PaCle2 × 10−14m
W5 × 10−4mρ833kg/m3
SymbolValueUnitSymbolValueUnit
mp0.120kgms0.025kg
bp7.5N ⋅ s/mbs15N ⋅ s/m
kp20 × 103N/mks2.5 × 103N/m
Ac44.18 × 10−6m2Fo75N
Vco0.5 × 10−6m3km9.79N/A
β1200 × 106PaCd0.6(-)
ps3.5 × 106PaCli1 × 10−14m
pr0.1 × 106PaCle2 × 10−14m
W5 × 10−4mρ833kg/m3

Figure 4 shows the magnitude of GFRFs up to the third‐order. From Fig. 4(a), the resonance of the electrohydraulic system is found near 25 Hz. In Fig. 4(b), $H2(jω1,jω2)$ has the largest peaks near (25 Hz, −25 Hz) and (−25 Hz, 25 Hz). Therefore, $H2(jω1,jω2)$ has the considerable impact on the zero frequency output spectrum Y(0) at the resonance frequency (see Eqs. (10) and (11)). Similarly, Fig. 4(c) shows that $H3(jω1,jω2,jω3)$ has the largest peaks near (25 Hz, −25 Hz, 25 Hz) and (−25 Hz, 25 Hz, 25 Hz); thus, its impact on the fundamental output spectrum Y (jω) is the largest at the resonance frequency.

Fig. 4
Fig. 4
Close modal

The harmonic output spectra up to four times of the input frequency are estimated using the analytic GFRFs. For comparison, simulations are carried out using the physical dynamics given by Eqs. (12)(17). The input of $u(t)=A sin(ωkt)$ (A = 1.25 mm and $ωk∈{1,2,…,100}$ in Hz) is used. The discrete Fourier transform (DFT) of the simulated output is compared with the estimated harmonic output spectra using the analytic GFRFs and Fig. 5 shows their good correspondence. As expected from Fig. 4(b), the zero frequency output spectrum has the peak near the resonance frequency. Actually, similar peaks are observed at all harmonics. Spectral analysis reveals that the nonlinear behavior of the electrohydraulic system is significant near the resonance frequency.

Fig. 5
Fig. 5
Close modal

To analyze the effect of the input amplitude on the output spectrum, the normalized fundamental output spectra by the input amplitude are compared in Fig. 6. The analytic GFRFs show good output spectrum estimation for all input amplitudes. Near the resonance frequency, the normalized fundamental output spectrum becomes smaller as the input amplitude gets larger. This results from the orifice flow rate, which is the most significant nonlinear factor. In Fig. 7, as the input amplitude increases, the working range of the spool stroke and the cylinder pressure moves in the arrow directions. Consequently, the orifice flow rate grows slowly (i.e., gradient decreases). This means that the spool valve efficiency gets lower due to the increased nonlinear hydraulic damping as the working range increases. This phenomenon is dominant at the resonance frequency where the working range is the largest. And such a nonlinear damping effect can be retained by the given analytic GFRFs. As discussed from Fig. 4(c), the effect of $H3(jω1,jω2,jω3)$ on the fundamental output spectrum is the strongest near the resonance frequency, but in negative direction.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

### Spectral Analysis With Experimental GFRFs.

The prototype electrohydraulic system for camless engine valve actuation shown in Fig. 8 is used for experimental GFRFs identification. The piston position is measured by the noncontact sensor (Microstrain NC-DVRT 1.0). For I/O interface, PCI-DAS1602-16 DAQ card is used and the data are processed by the matlab xpc target real-time system. And the voice coil motor (BEI Kimco Magnetics: LA13-12-00A) and the high bandwidth power amplifier (Advanced Motion Controls: 12A8) are used.

Fig. 8
Fig. 8
Close modal

The same input signal with the simulation is applied to the stabilized electrohydraulic system. And the experimental GFRFs of the Wiener and the Hammerstein models are identified from frequency response data. Figure 9 shows the identified linear dynamic block of which the phase angle agrees well with the measured phase angle. The estimated GFRFs of the Wiener and Hammerstein models are shown in Figs. 10 and 11, respectively. As can be seen, their magnitude shows quite different aspects due to the distinct nonlinear static blocks though both models share the common linear dynamic block. From Figs. 10(b) and 10(c), $H2(jω1,jω2)$ and $H3(jω1,jω2,jω3)$ of the Wiener model show quite similar patterns including the peaks and the ridges. For the Hammerstein model, $H2(jω1,jω2)$ and $H3(jω1,jω2,jω3)$ have the similar ridges without the peaks, but the ridge of $H3(jω1,jω2,jω3)$ is shifted. $H2(jω1,jω2)$ turns out to have the uniform impact on the zero frequency output spectrum.

Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal
Fig. 9
Fig. 9
Close modal

The harmonic output spectra are calculated using the identified GFRFs and they are compared with the measurements (i.e., DFT of the measured output) as shown in Fig. 12. From the comparison, the Wiener model is found to show better performance in estimating the output spectra except at the three times of the input frequency. Compared to the analytic GFRFs, the experimental GFRFs outperform the analytic ones. However, physical interpretation is not available since the explicit connection with the physical parameters is absent.

Fig. 12
Fig. 12
Close modal

## Conclusions

In this paper, spectral analysis of the nonlinear electrohydraulic system is addressed to investigate its nonlinear dynamic features in frequency domain. The GFRFs are derived in the two different approaches based on Volterra series representation of the electrohydraulic system. In the first approach, the analytic method is proposed for GFRFs derivation from the physical dynamics of the electrohydraulic system. In the second approach, the experimental GFRFs are identified from the frequency response tests with the assumption of the block-oriented nonlinear systems: Wiener and Hammerstein models.

From analysis with the analytic GFRFs, it turns out that the considerable nonlinear dynamic features are found near the resonance frequency. Therefore, regarding the operating frequency range, different control strategies can be determined. Such nonlinear behavior near the resonance is further analyzed with respect to the input amplitude. As the input amplitude grows, the normalized fundamental output spectrum decreases due to the nonlinear hydraulic damping. And the given analytic GFRFs can capture such nonlinear behavior of the electrohydraulic system in a wide working range.

Although the analytic GFRFs offer great insight into system dynamics, its utility may be limited if the accurate physical dynamics is not available. In this case, the experimental GFRFs can be employed for spectral analysis with the appropriate choice of the model structure. Though the direct link to physical dynamics is deficient, they enable to predict the output spectrum with acceptable accuracy. Spectral analysis using GFRFs has great potential in system design, identification, and controls for nonlinear systems in frequency domain. Robust control design of the nonlinear electrohydraulic system based on spectral analysis has been reported as a separate work [35].

## Acknowledgment

This work was supported by the National Science Foundation under Grant No. CMMI-1150957.

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