## 3 Case Studies
### Results and Discussion  
#### 3.2.1 Step Responses to Reference Inputs  
In order to investigate the dynamic response characteristics of LVPC systems, experiments in step responses to reference inputs for different weight ratios \(K_{cp}\) are carried out, as shown in Figure 7. The experimental condition is described as follows: the initial voltage of the pump \(u_{p0}=7.2\) V, the initial voltage of the valve \(u_{v0}=-5.8\) V, the system pressure \(P_{h}=4.2\)-\(5\) MPa and PI parameters of valve control circuit and pump control circuit are set as in Table 2. Taking \(K_{cp}=1\) as the axis of symmetry, on the left side of the axis, there is \(K_{cp}>1\) and the combined system is mainly dominated by the PTV and when \(K_{v}:K_{p}=1.0\), the combined system is totally controlled by the valve; on the right of the axis, there is \(K_{cp}<1\) and the combined system is mainly dominated by the VSP and when \(K_{v}:K_{p}=0.1\), the combined system is totally controlled by the pump. The following rules can be obtained:  
(1) The step response to reference inputs are of pyramid form, both ends respond slowly, the middle responds fast and LVPC responds faster than the single valve control and the single pump control. That is due to the weight factor \(K_{p}\) or \(K_{v}\) being equal to 1 in the experiment and \(K_{sl}>K_{sv}\)_or_\(K_{sl}>K_{sp}\) is maintained in Equation (14), when the bigger \(K_{p}+K_{v}\), the larger the \(K_{sl}\), the faster the LVPC system responds.  
Figure 7: Step response to reference inputs under different weight ratios \(K_{cp}\). (**a**) Motor speed response; (**b**) Pump and valve voltage variation; (**c**) System pressure and flow response.  
(2) For the same sum of weight factors, the response mainly based on valve control is greater than that based on pump control. That is because \(K_{sv}>K_{sp}\), for the same of sum of weight factor (\(K_{v}+K_{p}\)) and the synthetic open-loop gain of the symmetrical axis on the left side (mainly based on valve control) is larger than that on the right side (mainly by pump control).  
(3) The valve-pump weight ratio determines the flow distribution relationship of the valve and the pump. This can be verified from Figure 7b,c, from \(K_{v}:K_{p}\) = 1:0 to \(K_{v}:K_{p}\) = 0:1, \(K_{vp}\) gradually decreases, the valve control effect decreases as pump control effect increases, so valve flow gradually decreases and pump flow gradually increases.  
The step response experiments under the same weight ratio \(K_{vp}\) are also carried out, as shown in Figure 8 and the following rules can be obtained.  
(1) Under the same weight ratio, the response speed of the LVPC system is proportional to the weight factors of the valve control and the pump control. For example, when \(K_{v}:K_{p}\) = 0:2:0.2, the LVPC system responds slowest and when \(K_{v}:K_{p}\) = 1:1, the LVPC system responds fastest but the overshoot will be larger and stability is degraded when \(K_{v}\) and \(K_{p}\) are too great. That is because, according to equation (14), \(K_{v}\) and \(K_{p}\) determine \(K_{sl}\)_and_\(K_{sl}\) determines the response speed of the combined system.  
(2) Under the same weight ratio, the weight factor does not affect the valve flow and pump flow in the stable state and the ratio between valve flow and pump flow remains constant but great  
Figure 8: Step response to reference inputs when \(K_{vp}=1\). (a) Motor speed response; (b) Pump and valve voltage variation; (c) System pressure and flow response.  
weight factor will cause a high-pressure response. For example, from \(K_{v}:K_{p}=0.2\):\(0.2\) to \(K_{v}:K_{p}=1\):\(1\), the stable flow in pump control is always \(28\) L/min and stable leakage flow in the valve control is always \(7.5\) L/min.