A Parameter Tuning Method of Adaptive Dynamic PID Controller for Time Delay Integral System

PID controllers are widely used in the control of integral systems. In the practical application of integral systems, the control effect is usually affected by the delay time, and at the same time, the large variation of the controlled integral parameters will also affect the control effect of the controller. This paper proposes an adaptive dynamic PID controller parameter tuning method suitable for time delay integral systems, considering the influence of parameters such as the voltage of the integral system, the variation range of the controlled object, and delay time, making the relationship between the parameters of the PID controller and the control object of the integral system more explicit. At the same time, this paper considers the tuning of the differential filter parameters in practical applications. A non-linear system with a 10-fold variation of the controlled integral parameters is simulated in Simulink, and the dynamic control of the non-linear system is realized using the method proposed in this paper, achieving a better control effect compared to the traditional control with constant PID parameters.


Introduction
The motor drive system consists of a drive motor, power converter, and controller [1][2].The winding of the drive motor can be regarded as a combination of resistance and inductance in the calculation and analysis.When the winding is energized, the excitation voltage changes and the process of current change in the winding is an integral process.To keep the winding output current stable without overshoot or oscillation, PID current controllers are most commonly used.PID controllers have the advantages of high reliability, robustness, and simple control algorithms [3][4].Therefore, it is important to study the parameter tuning method of the PID controller [5].
The existing PID parameter tuning methods mainly include fuzzy PID control, intelligent algorithm PID control, and fractional order PID control.Zhang, et al. [6] use fuzzy PID controllers to design brushless DC motor control systems to achieve high-accuracy speed control.Lin and Wang [7] propose an improved particle swarm optimization fuzzy PID algorithm for the control of brushless DC motors, which has a faster dynamic response and higher stability than conventional PID control.Kommula et al. [8] propose a fractional-order PID controller based on the MFA-PSO algorithm to reduce torque fluctuations of brushless DC motors.
According to the analysis of existing papers, there are few papers considering the effect of delay time on PID parameter setting, but in actual motor drive systems, measurement systems and sensors always lead to time delay [9], which will affect the accuracy of the control.At the same time, for some nonlinear, controlled object parameters vary in a wide range of control systems, such as the switched reluctance motor control system, its inductance is highly non-linear, and the range of inductance value changes more than ten times [10], the existing PID parameter tuning methods cannot achieve good control effect on them.
Given the above situation, this paper proposes an adaptive dynamic PID controller parameter tuning method suitable for time delay integral systems, which considers the influence of the voltage of the integral system, the value of the controlled object, and the delay time, making the relationship between the parameters of the PID controller and the control object of the integral system clearer, and when the delay time or the parameters of the controlled object change, the value of the PID controller parameters can be calculated quickly and accurately.A non-linear system with 10-fold variation was simulated in Simulink, and the method proposed in this paper was realized fast, without oscillation and overshoot control effect.

PID Controller Parameters Tuning Considering Delay Time
The winding of a motor can be considered a combination of resistance and inductance.A pure inductive circuit has a greater phase angle than an impedance circuit, which means that the impedance circuit is more stable than the purely inductive circuit [11].Therefore, this paper takes the pure inductance current as the research object to investigate the parameter tuning method of the time delay integral system PID controller.In a pure inductance circuit, the time constant of the control object is =L/U, where L is the value of the inductance, U is the DC voltage applied to the inductance, and the open-loop transfer function of the control object can be expressed as: The system block diagram of the whole PID controller is shown in Figure 1, and its corresponding transfer function is shown in Equation (2), where K p is the proportional coefficient, K i is the integral coefficient, K d is the derivative coefficient, and DELAY is the system delay time, expressed as t d .
Assuming the time constant =0.001 and the delay time t d =0.0004 s, the parameters of the PID controller are tuned by manual parameter adjustment, so that the inductor current responds quickly and has no overshoot.The parameters of the PID controller can be obtained as K p =1.4, K i =17.66,K d =0.000165, and the step response of the current is shown in Figure 2. It can be seen from Figure 2 that using the obtained parameters of the PID controller, the current response time remains within 2.5 ms, and there is no overshoot.With a constant time constant of  = 0.001, the delay time t was varied between 0.4 ms and 4 s and the parameters of the PID controller were tuned using manual adjustment, resulting in a rapid and overshoot-free inductance current response.The obtained parameters of the PID controller are shown in Table 1 It can be seen from Table 1 that under the condition of constant time constant, the value of K p is inversely proportional to the delay time, the value of K i is inversely proportional to the square of the delay time, and K d is independent of the delay time.
In the case of delay time t d =0.0004, change the value of the time constants , and use manual tuning to adjust the parameters of the PID controller.The obtained parameter values are shown in Table 2.According to the analysis of Table 1 and Table 2, the equations for K p , K i and K d can be obtained as follows: where a, b and c are constants.By using the data in Tables 3 and 4, a, b and c can be calculated as a=1.78, b=352.47,and c=0.17, which can be brought into Equations ( 3)-( 5), and the general equation for PID parameter tuning can be obtained.Based on the time constants and delay times of the control system, the parameters of the PID controller can be calculated directly, and the obtained parameters can make the response of the inductance current rapid and without overshoot.

Tuning of The Derivative Filter Coefficient
The derivative action can produce advanced control on deviation and significantly improve control performance [12], but in practice, direct derivative action is rarely used.Mainly because the derivative action is very sensitive to noise, too strong derivative action is not good for the system anti-interference, and in the actual circuit, the pure derivative link cannot be realized through circuit elements, so a lowpass filter is usually added to the derivative item to form a derivative filter, which can then be used to limit high-frequency noise and gain, as shown in Figure 3.After adding the derivative filter, the corresponding transfer function of the control system is as follows: Among them, K n is the coefficient of the derivative filter.The existing PID parameter tuning methods rarely solve the tuning problem of the derivative filter coefficient.Most of them only rely on the operator's experience and the calculated parameters of the PID controller.Without a reliable calculation formula, it is very complicated to apply the derivative filter to practice, so this paper will derive the tuning method of the derivative filter coefficient K n .
Assuming the delay time t d =0.0004 s and the time constant =0.001, the parameters of the PID controller can be tuned according to Equations ( 3)-( 5) as follows: K p = 1.40,K i = 17.73 and K d = 0.00017.
On the basis of K p , K i , and K d , the parameter K n is tuned.When K n =16500, the step response is fast, and there is no overshoot and oscillation, which meets the requirements of the control system, as shown in Figure 4.  Since the parameters of the PID controller are always tuned based on the time constant and delay time of the control system, we can obtain different values of K n for different time constants and delay times.In the case of time constant =0.001,changing the value of the delay time td, we can obtain the values of the derivative filter shown in Table 3.In the case of delay time t d =0.0004 s, changing the value of the time constant , we can obtain the derivative filter values shown in Table 4.
Table 3.The corresponding derivative filter values for different delay times  3 and 4, it can be seen that the value of K n is only related to the delay time and is independent of the time constant of the control system.Based on the relationship between K n and the delay time, the equation for the tuning of K n can be derived as shown below: So far, we have derived the tuning formulas of the PID controller and derivative filter coefficients, which show that for integral systems with a fixed delay time and fixed inductance, the PID controller and derivative filter coefficients are fixed.However, for some time delay integral systems, such as switched reluctance motor drive systems, the minimum and maximum values of winding inductance differ by a factor of ten or more, and it changes non-linearly with the change of the excitation current ( and the rotor position angle (), at this time the time constant  ,  =L/U also changes non-linearly.Therefore, by substituting  ,  into the tuning formula, the parameters of the PID controller that change adaptively and dynamically with the time constant can be obtained, and then the inductance current can be controlled dynamically.The tuning formula is as follows:   2 , 352.47   0.17 , 6 where the excitation current () and the rotor position angle () are functions of time (t) during rotor rotation, and the time constant  ,  can also be converted to a function of time   .

Simulation Study of PID Control under Large Inductance Variation
In this paper, the variation curve of winding inductance with rotor position and winding current in a switched reluctance motor driven by an asymmetric half-bridge power converter is approximated as a sinusoidal waveform.The sinusoidal variation of inductance can reflect the non-linear characteristics of the switched reluctance motor, which is used to verify the effectiveness of the tuning method proposed in this paper, the simulation block diagram in MATLAB/SIMULINK is shown in Figure 5.In Figure 5,   =L(t)/U, where U is the value of the inductor voltage, set to 1 V, and L(t) is the inductance function with a frequency of 12.5 Hz, varying sinusoidally with time, used to simulate the non-linearly varying inductance waveform of a switched reluctance motor, with a minimum value of 1mH and a maximum value of 10mH to reflect larger inductance variations.DELAY is the delay time, set to 0.000 4 s.The input current is a rectangular waveform with an amplitude of 1A and a frequency of 50 Hz, which is used to observe the current output response corresponding to the non-linear inductance under PID control.For example, we use the PID controller parameters corresponding to the lowest inductance 1 mH to control the current, then K p =1.40, K i =17.73,K d =1.70x10 -4 , and K n =16 500, and the control effect is shown in Figure 6. Figure 6.The control effect of the PID controller parameters corresponding to the lowest inductance As can be seen from Figure 6, the tuning effect is poor with constant PID parameters.At low inductance, the response is fast and stable and there is no overshoot, but at larger values of inductance, the output current cannot follow the input current and the response becomes very slow.If the PID controller parameters corresponding to the inductance greater than 1 mH are used to control the current, when the actual inductance is less than 1mH, the inductance current will be out of balance.Therefore, using constant PID controller parameters, the control effect on the inductance current is not ideal.
Based on Equations ( 8)-( 11), the parameters of the PID controller are tuned, so that the parameters of the PID controller change adaptively with the change of the inductance, and realize the dynamic control of the inductance current, the control effect is shown in Figure 7.As can be seen from Figure 7, the inductance current is controlled using the PID controller parameter tuning method proposed in this paper, the response time is rapid and there is no overshoot or oscillation, meeting the basic requirements of the control system.Therefore, the adaptive dynamic PID controller parameter tuning method proposed in this paper can be well applied to systems with non-linear changes of the controlled object.

Conclusion
This paper proposes an adaptive dynamic PID controller parameter tuning method suitable for time delay integral systems, which makes the relationship between PID parameters and the voltage of the integration system, the value of the controlled object, and the delay time more explicit.At the same time, this method is used to achieve dynamic control of non-linear systems in Simulink, showing a better control effect compared to constant PID parameter control, which proves the effectiveness of this method.

Figure 1 .
Figure 1.Block diagram of PID controller for time-delay integral system

3 Figure 2 .
Figure 2. The corresponding step response of the time delay integral system under the PID controller

Figure 3 .
Figure 3. Block diagram of time delay integral system PID controller (including derivative filter)

Figure 4 .
Figure 4.When K n =16500, the corresponding step response of the time delay integral system under the PID controller (including derivative filter)

Figure 5 .
Figure 5. Block diagram of PID controller system corresponding to non-linear varying inductance

Figure 7 .
Figure 7.The control effect corresponding to the parameter tuning method of adaptive dynamic PID controller

Table 1 .
. PID controller parameters corresponding to different delay time

Table 2 .
PID controller parameters corresponding to different time constants It can be seen from Table2that when the delay time is constant, the values of K p , K i and K d are proportional to the time constant.

Table 4 .
The corresponding derivative filter values for different time constants