## 3 Proposed VPDPI Speed Regulator  
### The Design of the VPDPI Regulator  
There is no doubt that the proportion link can quickly amplify the speed error signal, and it can quickly adjust the speed to the reference value. Unfortunately, the single proportion link will make the system have a certain steady-state error, which can not meet the performance requirements of the most control systems. Fortunately, the integration link is extremely effective in eliminating static errors. However, the control system has a new integral saturation problem since the accumulation of error signals, which leads to the increase of overshoot and even out of control. In order to ensure the control performance of PI controller, it is highly necessary to eliminate the integral saturation phenomenon in the integral link. At present, the anti-saturation ideas is widely used to suppress integral saturation phenomenon, which weakens the negative impact of saturation problem but has not been fundamentally solved. Different from the traditional controller design ideas, a new regulator designed in this paper is based on thoroughly eliminating integral saturation phenomenon and the actual performance requirements of fast response. The ideal response curve with fast response and no overshoot is shown in Figure 6. Therefore, a novel VPDPI regulator is further designed. In details, the dynamic response process of the controlled system can be simply divided into three stages according to the ideal response of the system (i.e., Stage A, B and C). The Stage A is called as the fast rising zone, in which the speed error signal is large and the integration process rapidly accumulates the speed error. Therefore, this stage will be the main process leading to the phenomenon of integral saturation. The Stage B is defined as the steady-state regulation zone, in which the system response transitions from dynamic regulation to steady-state process, and the integral regulator in the steady-state regulation zone still accumulates a certain speed error, which can play the role of the integral link in this stage to eliminate the system deviation. The Stage C is generally considered to be the stability zone, in which the system response is within the error band and runs stably around a given speed. In order to realize the actual output tracking in terms of the ideal output curve, the primary tasks in Stage A have to ensure that the integral will not fall into saturation state. Hence, the logical threshold method is used to judge the speed error in real time, and the feedback compensation coefficient \(\gamma\) (\(\gamma<0\)) is introduced in Stage A. In Stage B, the rotation speed should be adjusted smoothly to the steady-state interval, and the traditional PI control idea can be adopted without additional compensation process. Stage C is in the steady-state interval, and the response speed can be appropriately improved at this stage. On the one hand, the effect of integral link is not obvious due to too much desaturation, and on the other hand, the response speed of the system can be significantly improved. The structure diagram of VPDPI control method proposed in this paper is shown in Figure 7.  
Figure 6: The curve of the ideal response. Where \(K_{p1}\) and \(K_{p2}\) are proportional coefficients, respectively. \(K_{i}\) is the integral coefficient. In terms of the ideal response curve, \(c\) is the threshold used to determine whether to change the proportionality coefficient. Meanwhile, the threshold \(\varphi\) is used to judge the saturation condition. In practical application, the control parameter configuration of the proposed VPDPI regulator can usually be divided into the following steps.