## 3 Dynamic Model of Incremental Step Shift of DIHPS
### BV Dynamic Model  
Depending on the position of the actuator piston relative to the outflow gaps, as shown in Figure 4, the flow rate \(q_{v}\) that bleeds through the BV into the tank results from the flow continuity equations as follows:  
\[\left\{\begin{array}{l}q_{v}=q_{v21}+q_{v22}\\ q_{v}=q_{v22}\\ q_{v}=q_{v11}+q_{v22}\\ q_{v}=q_{v11}\\ q_{v}=q_{v11}+q_{v12}\end{array}\right. \tag{15}\]  
The BV is either completely closed (_off_ state) or completely open (_on_ state). The BV is modeled as an orifice valve with turbulent flow. The flow rate of \(q_{v}\) through the BV with bleed-off into the tank is written in the following form:  
\[q_{v}=C_{d}\,A_{v}\,\sqrt{\frac{2}{\rho}\Delta p_{v}}=C_{d}\,A_{v}\,\sqrt{ \frac{2}{\rho}(p_{v}-p_{T})} \tag{16}\]  
where \(C_{d}\) is the flow rate coefficient, \(A_{v}\) is the cross-section area, \(  
In the BV, the local hydraulic resistance depends on the pressure drop and the flow rate. The pressure drop caused by local resistance in the control valve is calculated using a semi-empirical formula based on the nominal pressure and flow, which is usually provided in catalogues, data sheets, or hydraulic textbooks. The nominal flow rate \(q_{vm}\) at the nominal pressure drop \(\Delta p_{n}\) is defined as follows:  
\[q_{vm}=C_{d}\,A_{vm}\,\sqrt{\frac{2}{\rho}\Delta p_{n}} \tag{18}\]  
From comparisons of (17) and (18), it follows that,  
\[q_{v}=\frac{A_{v}}{A_{vm}}\frac{q_{vm}}{\sqrt{\Delta p_{n}}}\,\sqrt{p_{v}}= \frac{A_{v}}{A_{vm}}K_{vm}\,\sqrt{p_{v}} \tag{19}\]  
where \(K_{vm}\) is the coefficient of the nominal parameters,  
\[K_{vm}=\frac{q_{vm}}{\sqrt{\Delta p_{n}}} \tag{20}\]  
The nominal flow rate is a measure of the volumetric flow that passes through the valve under a specified operating condition. For flow control valves, such as BV, the nominal flow rate \(q_{vm}\) is determined at the nominal pressure drop \(\Delta p_{n}\), which is typically 10 bar.  
The pressure result from (19) is as follows:  
\[p_{v}=\frac{1}{K_{vm}^{2}}\bigg{(}\frac{A_{vm}}{A_{v}}\bigg{)}^{2}q_{v}^{2}=R _{v}(A_{v})\,q_{v}^{2} \tag{21}\]  
where \(R_{v}\) is the hydraulic resistance of the BV,  
\[R_{v}(A_{v})=\frac{1}{K_{vm}^{2}}\bigg{(}\frac{A_{vm}}{A_{v}}\bigg{)}^{2} \tag{22}\]  
Figure 6 shows the relationship between pressure drop and flow rate (\(p_{v}-q_{v}\)) for 100% opening the BV when \(A_{v}=A_{vm}\).  
Numerical calculations performed during the simulation use the relationship between pressure drop and flow rate through BV.  
Opening and closing of BV are related to revealing and covering the cross-section \(A_{v}\), which is the result of shifting the valve slider:  
\[A_{v}=\pi\,d_{n}\,y \tag{23}\]  
where \(d_{n}\) is the nominal diameter, \(y\) is the shift of valve spool depending on the voltage \(u_{v}\) on the solenoid coil, \(0\leq y\leq y_{\text{max}}\) for \(0\leq u_{v}\leq u_{v}\)s,  
\[\frac{y}{y_{max}}=\frac{u_{v}}{u_{v}}=\,u \tag{24}\]  
where \(y_{\text{max}}\) is the maximum shift of the valve spool (100% open BV), \(u_{v}\) is the voltage input signal, \(u_{v}\)s is the set voltage input signal, and \(u\) is the binary input signal,  
\[y=0\text{ for }u=0\,\,\,y=y_{\text{max}}\text{ for }u=1 \tag{25}\]  
The dynamic model of the solenoid-controlled BV can be presented as a first-order differential equation as follows [33]:  
\[T\,\frac{dy}{dt}+y=K\,u\,\Rightarrow\,\,\frac{dy}{dt}=-\frac{1}{T}y+\frac{K} {T}\,u \tag{26}\]where \(T\) is the time constant of the valve, and \(K\) is the gain factor of the valve.  
The BV is controlled by a modulation frequency ranging from 200 to 1000 Hz. Figure 7 shows the dynamic response of the BV during the setting of the binary input signal of two frequencies.