3.1. Physical Model

1

Pilot valve model (I and II): These pilot valves consist of a solenoid, its armature, and a pre-loaded spring. Equations (3.6) and (3.7) in [11] describe a third-order coupled system, which, in this case, is the dynamics of the armature and the electrical circuit of the coil. We assumed that the time-constants of the armature’s motion are much smaller than those of the spindle’s motion. Therefore, we neglect the inductance of the coil’s circuit, the linear momentum, the viscous force, the flow forces acting on the armature, and the induced electromagnetic force from the armature’s motion. The last is allowed because the electromagnetic force to the armature is greater; as such, we obtain

$$k_i{y}_i=\frac{{\alpha }_i{v}_{max}}{R_i}{v}_i-F_{0,i}$$

(1)

for valve i, where $i=\left\{1,...,m\right\}$ corresponds to the m pilot valves of Group (I), and $i=\left\{m+1,...,m+n\right\}$ to the n of Group (II). In this case, $m=n=3$ (see Figure 1); $k_i$ is the spring rate of the spring; $y_i$ is the armature position; ${\alpha }_i$ is the magnetic coupling coefficient, which provides the constant of proportionality for the electromagnetic force; $v_{max}$ is the maximum supplied voltage applied to the coil; $R_i$ is the resistance of the electrical circuit of the coil; $v_i$ is the supplied voltage with respect to its maximum value applied to the coil; and $F_{0,i}$ is the pre-load force acting on the armature by the spring when $y_i=0$. By setting $v_i=v_0$ for $y_i=0$, we obtain

$$y_i\left({\overline{v}}_i\right)=\frac{{\alpha }_i{v}_{max}}{R_i{k}_i}{\overline{v}}_i$$

(2)

where ${\overline{v}}_i=v_i-v_0$. ${\overline{v}}_i$ assumes $v_0$ as the offset of $v_i$, which is assumed to be the same for all i (see Figure 3b).

2

Volumetric flow rate of the pilot valves (I and II): we use the standard equation (4.1) in [11] for describing the flow (which assumes the fluid to be incompressible with a high Reynold number (i.e. laminar flow)). As such, we obtain

$$Q_i=c_{d}A\left({\overline{v}}_j\right)\sqrt{\frac{2}{\rho }\left(P_i\right)},$$

(3)

where $c_d$ is the discharge coefficient; $A_i\left({\overline{v}}_j\right)=w{y}_i\left({\overline{v}}_j\right)$ is the discharge flow area when assuming a rectangular section, which depends on the voltage applied to the coil of the pilot valve i; and $\rho$ is the fluid density and $P_i$ is the pressure drop across the valve, which is $P_i=(P_z-P_c)$ for $i=\left\{,2,...,m\right\}$, and $P_i=(P_c-P_r)$ for $i=\left\{m+1,...,m+n\right\}$. We substitute j by subindexing $in$ for $i=\left\{1,2,...,m\right\}$, as well as by subindexing $out$ for $i=\left\{m+1,...,m+n\right\}$.

3

The deformable volume (X) of the main valve is chosen as the control volume, as indicated by the solid blue line in Figure 2. The change in the control volume is understood as a function of the spindle’s position. For an incompressible fluid, the continuity Equation (5.5) in [32] becomes

$$A_B\dot{x}=Q_1+...+Q_i+...+Q_m-Q_{m+1}-...-Q_{m+n},$$

(4)

where $A_B$ is the cross-sectional area of the lower chamber subtracting $A_L$, which is assumed to be constant along x; $\dot{x}$ is the derivative of the spindle’s position; and $Q_i$ is the volumetric flow rate of valve i (see Figure 1 and Figure 2).

4

The spindle (IV) equation of motion of the main valve determines the motion of the main valve’s spindle, which is

$$M\ddot{x}+B\dot{x}=A_B{P}_c\left(t\right)-A_T{P}_z-F_{fm}sgn\left({\dot{x}}_m\right)-F_{flow}+A_L{P}_L-Mg,$$

(5)

when ${\overline{v}}_{in}>0$ or ${\overline{v}}_{in}>0$, and where M is the main valve’s spindle mass, and B is the viscous drag coefficient. For a marginally stable valve, as the one under study, there is no spring; hence, no term is proportional to x. $F_{fm}$ is the coefficient of the Coulomb friction, and $F_{flow}$ is the force exerted by the flow going from (VIII) to (IX) in Figure 1. An analysis of the flow forces allows for the term $A_L{P}_L$ to appear in the equation. $Mg$ is the force given by the main valve’s spindle mass. We assumed that the Coulomb friction term can be neglected in contrast to $A_B{P}_c\left(t\right)$ and $A_T{P}_z$ due to a difference of 3–5 orders of magnitude when assuming friction between steel and rubber, i.e., the seal’s material. Furthermore, as we had values of $P_L\ll P_z$ (c.f. scale of Figure 3 (c) and (d)) and $A_L\ll A_T$ (c.f. Figure 2), we thus obtained $A_L{P}_L\ll A_T{P}_z$, as well as $Mg\ll A_T{P}_z$. The flow forces can be described by

$$F_{flow}=\rho L{K}_q\dot{x}+\rho L{K}_c{P}_L+K_{fq}(x-x_0)+K_{fc}(P_L-P_{L_0})$$

(6)

which can be obtained after deriving and linearizing with respect to x and $P_L$ (see (4.20) in [11]), where $K_q$ is the flow gain coefficient, $K_c$ is the pressure-flow coefficient, $K_{fq}$ is the flow force gain, and $K_{fc}$ is the pressure flow force. Considering the control volume, as illustrated in Figure 4–21 by dashed lines and the dimension given in (4.122) in [11], as well as the operating conditions of the hydraulic valve under study, we consider $F_{flow}$ to be negligible in contrast to $A_B{P}_c\left(t\right)$ and $A_T{P}_z$. This was performed because there was a difference of two orders of magnitude. More information about how to derive the expression of the coefficients in (6) can be found in Chapter 4, more specifically in Section 4.3 and Section 4.6, in [11]. Finally, we obtained the main valve’s spindle equation of motion, which is

$$\left\{\begin{array}{cc}M\ddot{x}+B\dot{x}=A_B{P}_c\left(t\right)-A_T{P}_z\hfill & \mathrm{if}{\overline{v}}_{in}\mathrm{or}{\overline{v}}_{out}>0\hfill \\ \dot{x}=0,\hfill & \mathrm{if}{\overline{v}}_{in}={\overline{v}}_{out}=0\hfill \end{array}\right..$$

(7)

These equations are valid for $x\in [0,x_{max}]$, where $x_{max}$ is the maximum position of the spindle. The case of ${\overline{v}}_{in}\left(t\right)={\overline{v}}_{out}\left(t\right)=0$ describes the system when $P_c\left(t\right)$ cannot be derived from (3) and (4).

5

Approximation of $P_c$ in the main valve (X): Since $P_c$ is not measured, it is derived from (3) and (4). Furthermore, assuming that the pilot parallel valves are identical, and knowing that ${\overline{v}}_{in}\left(t\right)$ is applied to all of the coils of pilot valves (I) and ${\overline{v}}_{out}\left(t\right)$ to (II) (see Figure 1 and (4)), we obtain

$$A_B\dot{x}=m{Q}_{in}-n{Q}_{out},$$

(8)

where $Q_{in}=Q_1=...=Q_m$ and $Q_{out}=Q_{m+1}=...Q_{m+n}$. For $Q_{in}$, we use the auxiliary function

$$f_{in}\left(P_c\right)=\sqrt{(2/\rho )(P_Z-P_c)}$$

(9)

and, for $Q_{out}$, we use

$$f_{out}\left(P_c\right)=\sqrt{(2/\rho )\left(P_c\right)},$$

(10)

which are derived from (3). We approximate $f_{in}$ and $f_{out}$ by a Taylor series in $P_c$ around the operating point $P_c=P_{c,0}=\frac{A_T}{A_B}{P}_z$, which is the equilibrium point of (7). By performing this, keeping the linear terms, and substituting these in (8), we obtain

$$\begin{array}{cc}\hfill P_c\left(t\right)& =P_{c,0}+g({\overline{v}}_{in},{\overline{v}}_{out})\times \hfill \\ \hfill & (f_{in}(P_{c,0}{\overline{R}}_1{\overline{v}}_{in}\left(t\right)-f_{out}\left(P_{c,0}\right){\overline{R}}_2{\overline{v}}_{out}\left(t\right)-A_B\dot{x}),\hfill \end{array}$$

(11)

where

$$g({\overline{v}}_{in},{\overline{v}}_{out})=\frac{1}{f_{out}^{\prime }\left(P_{c,0}\right){\overline{R}}_2{\overline{v}}_{out}\left(t\right)-f_{in}^{\prime }\left(P_{c,0}\right){\overline{R}}_1{\overline{v}}_{in}\left(t\right)},$$

(12)

In the above, $f_{in}^{\prime }\left(P_{c,0}\right)$ is the first derivative of $f_{in}\left(P_c\right)$, which is evaluated at $P_{c,0}$, $f_{out}^{\prime }\left(P_{c,0}\right)$ is the one of $f_{out}$, ${\overline{R}}_1=m{c}_d{w}_{in}\frac{{\alpha }_{in}{v}_{max}}{k_{in}{R}_{in}}$, and ${\overline{R}}_2=n{c}_d{w}_{out}\frac{{\alpha }_{out}{v}_{max}}{k_{out}{R}_{out}}$.

6

The input–output system model is obtained by substituting (11) in (7), where

$$\begin{array}{cc}\hfill M\ddot{x}& +B\dot{x}=A_{B}g({\overline{v}}_{in},{\overline{v}}_{out})\times \hfill \\ \hfill & (f_{in}\left(P_{c,0}\right){\overline{R}}_1{\overline{v}}_{in}\left(t\right)-f_{out}\left(P_{c,0}\right){\overline{R}}_2{\overline{v}}_{out}\left(t\right)-A_B\dot{x}).\hfill \end{array}$$

(13)

We approximate the auxiliary function $g(\overline{v_{in}},\overline{v_{out}})$ by a Taylor series in ${\overline{v}}_{in}\left(t\right)$ and ${\overline{v}}_{out}\left(t\right)$ around ${\overline{v}}_{in,0}$ and ${\overline{v}}_{out,0}$, respectively, where ${\overline{v}}_{in,0}={\overline{v}}_{out,0}={\overline{v}}_0\ne 0$. We keep the linear terms and (13) is approximated by

$$\begin{array}{c}\hfill {\gamma }_1\ddot{x}+{\gamma }_2\dot{x}={\varphi }_1{\overline{v}}_{in}\left(t\right)+{\varphi }_2{\overline{v}}_{out}\left(t\right)+{\varphi }_3{\overline{v}}_{in}^2\left(t\right)+{\varphi }_4{\overline{v}}_{out}^2\left(t\right)+...\\ \hfill ...+{\varphi }_5{\overline{v}}_{in}\left(t\right){\overline{v}}_{out}\left(t\right)+{\varphi }_6{\overline{v}}_{in}\left(t\right)\dot{x}+{\varphi }_7{\overline{v}}_{out}\left(t\right)\dot{x},\end{array}$$

(14)

where

$$\left\{\begin{array}{c}{\gamma }_1=\frac{M}{A_B}\ge 0\hfill \\ {\gamma }_2=\frac{B}{A_B}+\overline{g}{A}_B\ge 0\hfill \\ {\varphi }_1=\overline{g}{f}_{in}\left(P_{c,0}\right){\overline{R}}_1\ge 0\hfill \\ {\varphi }_2=-\overline{g}{f}_{out}\left(P_{c,0}\right){\overline{R}}_2\le 0\hfill \\ {\varphi }_3=g_{{\overline{v}}_{in}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)f_{in}\left(P_{c,0}\right){\overline{R}}_1\le 0\hfill \\ {\varphi }_4=-g_{{\overline{v}}_{out}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)f_{out}\left(P_{c,0}\right){\overline{R}}_2\ge 0\hfill \\ {\varphi }_5=g_{{\overline{v}}_{out}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)f_{in}\left(P_{c,0}\right){\overline{R}}_1-g_{{\overline{v}}_{in}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)f_{out}\left(P_{c,0}\right){\overline{R}}_2\in \mathcal{R}\hfill \\ {\varphi }_6=-g_{{\overline{v}}_{in}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)A_B\ge 0\hfill \\ {\varphi }_7=-g_{{\overline{v}}_{out}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)A_B\ge 0\hfill \end{array}\right.,$$

(15)

where $\overline{g}=g({\overline{v}}_0,{\overline{v}}_0)-g_{{\overline{v}}_{in}}^{\prime }({\overline{v}}_0,{\overline{v}}_0){\overline{v}}_0-g_{{\overline{v}}_{out}}^{\prime }({\overline{v}}_0,{\overline{v}}_0){\overline{v}}_0$, $g_{{\overline{v}}_{in}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)$ is the derivative of $g({\overline{v}}_{in},{\overline{v}}_{out})$ with respect to ${\overline{v}}_{in}$, and $g_{{\overline{v}}_{out}}^{\prime }({\overline{v}}_0,{\overline{v}}_0)$ is the derivative of $g({\overline{v}}_{in},{\overline{v}}_{out})$ with respect to ${\overline{v}}_{out}$. Indeed, both are evaluated at ${\overline{v}}_{in}={\overline{v}}_{out}={\overline{v}}_0$. The sign of each of the parameters is specified explicitly in (15), and this is performed in accordance with the physics of the device described in (1-13). ${\varphi }_5$ is positive or negative depending on $A_T/A_B$. Notice that (14) is a non-linear input–output model and linear in the parameters. The non-linearities are related to ${\overline{v}}_{in}^2$, ${\overline{v}}_{out}^2$, ${\overline{v}}_{in}\dot{x}$, and ${\overline{v}}_{out}\dot{x}$. These terms are obtained from the square root in $\sqrt{\frac{2}{\rho }\left(P_i\right)}$ and the product $A\left({\overline{v}}_j\right)\times \sqrt{\frac{2}{\rho }\left(P_i\right)}$ in (3).

3.2. Parameter Estimation

3.3. Model Validation

4.1. Physical Model

Table 1. Estimated parameter values for the input–output system model in (16) and (17). ${\widehat{\theta }}_1$ and ${\widehat{\theta }}_2$ are least squares estimates, and ${\widehat{\theta }}_3$ was obtained by solving (21). $-1\le \theta \left(1\right)\le 0$, $\theta \left(2\right)\ge 0$, $\theta \left(3\right)\le 0$, $\theta \left(4\right)\le 0$, and $\theta \left(5\right)\ge 0$ are based on the physical constraints of the device.

Figure 4. Modeling results with the validation data: a) the measured output $x_{exp}$ (position of IV) and the modeled output ${\widehat{x}}_3$; b) the normalized input signals ${\overline{v}}_{in}$ and ${\overline{v}}_{out}$; and c) the error signal obtained by subtracting ${\widehat{x}}_3$ from $x_{exp}$.

Figure 4. Modeling results with the validation data: a) the measured output $x_{exp}$ (position of IV) and the modeled output ${\widehat{x}}_3$; b) the normalized input signals ${\overline{v}}_{in}$ and ${\overline{v}}_{out}$; and c) the error signal obtained by subtracting ${\widehat{x}}_3$ from $x_{exp}$.

4.2. Modeling Errors

4.3. Aging

Supplementary Materials