Table 2.

Zeigler-Nichols tuning rules.


Table 3. Effect of the control action of the PID.

Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

Figure 5. The PID control system.

#### Figure 6.

measurements and set points. The PID variables are iteratively adjusted until the steady-state error from the output signal is eliminated. This control action is done by adjusting the controller gain with resulting decrease in the rise time and percent increase in the overshoot, which makes the system go unstable. This rise time is further reduced with the integral control action. Finally, the derivative action is introduced to compensate for the offset. This reduces the percent overshoot and

Eq. (27) gives the expression for the control action of the PID controller:

ðt

e tð Þdt þ KdTd

de tð Þ

dt (27)

0

S/N Type of controller Kp Ki Kd P 0.5 Kcr ∞ 0 PI 0.45 Kcr 0.83 Pcr 0 PID 0.6 Kcr 0.5 Pcr 0.125 Pcr

S/N Controller response Rise time Overshoot Setting time 1 Kp Decrease Increase Small change 2 Ki Decrease Increase Increase 3 Kd Small change Decrease Decrease

The Nichols-Ziegler tuning rules employed for tuning the PID control as well as the summary of the effects of its control action on the PID are presented in Tables 2

The signal (U) which passes through the controller computes the derivative and integral of error signal. The signal error is thereafter sent to the system in order to obtain the system's output (Y). The PID controller was designed in the MATLAB Simulink 2018 environment to generate a continuous time control. Using the Nichols-Ziegler rules, the tuning of PID controller was done by generating the

Ti

settling time, thus making the system stable over time.

and 3, respectively.

Table 2.

Table 3.

126

Zeigler-Nichols tuning rules.

Effect of the control action of the PID.

Figure 4.

The block diagram of the PID controller.

Noise and Vibration Control - From Theory to Practice

uc <sup>¼</sup> Kpe tðÞþ Ki

The PID control and the rail car systems.

system's transfer function and subsequent importation of the parameters obtained into the linear time-invariant system.

The PID control system and its connection to the rail car system are shown in Figures 5 and 6, respectively.

The aim of the control design is to keep the system variables close to the reference in order to compensate for the effect of load and rail disturbances. The system requirement is to check unpleasant motion and ensure rail car stability by reaching a compromise between the stiff primary suspension and soft secondary suspension system.

The actively controlled suspension system can be activated via the use of solenoid, hydraulic, electromagnetic means or through a magnetorheological damper. This system is designed to use the solenoid actuators because of its lightweight, simplicity in structure, ease of installation and short response time, which makes it highly sensitive to disturbances.

## 3. Results and discussion

Figure 7 shows the step response before the iterative adjustment of the PID control. The amplitude of oscillation, which is a function of the percent overshoot, is 2 mm, and the system could not return to the equilibrium position after 3 s. The shape of the plot represents a system that is underdamped, which signifies the need for damping to minimize unwanted motion. The system whose step response is depicted in Figure 7 is relatively unstable as vibration will reduce the system's and ride performance.

Figure 8 shows the step response from the controlled system. When compared to Figure 1, the amplitude of oscillation has reduced to 1.15 mm and settling time 0.5 s under the effect of the PID control action. The system is relatively stable as the

Figure 7. Step response for the uncontrolled system.

Figure 8.

plot gives an indication of a critically damped system. This implies that under the effect of the PID control system, the system could regain its stability and equilibrium position after encountering some level of disturbances.

to noise measurement. The negligible value of the steady-state error as shown by the degree of agreement between the tuned output response of the system and the baseline (reference) indicates the high sensitivity and compensation of the system's

Vibration Analysis and Control in the Rail Car System Using PID Controls

DOI: http://dx.doi.org/10.5772/intechopen.85654

Figure 11 shows the closed-loop system response to load disturbance. The plot represents a step disturbance from the system's input. The differences between the tuned response and the baseline response, which is a function of the steady-state error, are significantly large. The implication of this is that the control system is insensitive to the rejection of load and other input disturbances, which is capable of offsetting the balance of the system. The large steady-state error resulting from input disturbance rejection stems from the fact that a single PID may not be able to satisfy all the design requirements at the same time; hence, there is always a performance trade-off amongst the reference tracking, percent overshoot and input disturbance rejection. However, the use of Fuzzy PID or ISA-PID controller can be used to meet the design requirements significantly. This will improve the response of the reference tracking with the provision of an additional tuning parameter,

control to the noise measurement.

Plot of output disturbance rejection.

Figure 9.

Figure 10.

129

Step plot for reference tracking.

Figure 9 shows the response of the PID controller in order to determine whether the control system meets the design requirements. The tuned response represents the actual measurement of the process variables, while the baseline response represents the reference or set point. The differences between the tuned response and the baseline response give the steady-state error. The nature of the plot also indicates that the system is critically damped; thus, the damping ratio ξ ¼ 1:

Figure 9 shows the reference tracking of the tuned response and the reference (baseline). The design requirement of the control system includes set-point tracking; hence, this plot shows how the closed-loop system responds to a step change in set point. From the plot, the steady-state error is minimal, thus indicating the effectiveness of the control system.

Figure 10 shows the closed-loop step response to a step disturbance at the system's output. This is important in analyzing the sensitivity of the control system Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

Figure 9. Step plot for reference tracking.

Figure 10. Plot of output disturbance rejection.

to noise measurement. The negligible value of the steady-state error as shown by the degree of agreement between the tuned output response of the system and the baseline (reference) indicates the high sensitivity and compensation of the system's control to the noise measurement.

Figure 11 shows the closed-loop system response to load disturbance. The plot represents a step disturbance from the system's input. The differences between the tuned response and the baseline response, which is a function of the steady-state error, are significantly large. The implication of this is that the control system is insensitive to the rejection of load and other input disturbances, which is capable of offsetting the balance of the system. The large steady-state error resulting from input disturbance rejection stems from the fact that a single PID may not be able to satisfy all the design requirements at the same time; hence, there is always a performance trade-off amongst the reference tracking, percent overshoot and input disturbance rejection. However, the use of Fuzzy PID or ISA-PID controller can be used to meet the design requirements significantly. This will improve the response of the reference tracking with the provision of an additional tuning parameter,

plot gives an indication of a critically damped system. This implies that under the effect of the PID control system, the system could regain its stability and equilib-

Figure 9 shows the response of the PID controller in order to determine whether the control system meets the design requirements. The tuned response represents the actual measurement of the process variables, while the baseline response represents the reference or set point. The differences between the tuned response and the baseline response give the steady-state error. The nature of the plot also indicates

Figure 9 shows the reference tracking of the tuned response and the reference (baseline). The design requirement of the control system includes set-point tracking; hence, this plot shows how the closed-loop system responds to a step change in set point. From the plot, the steady-state error is minimal, thus indicating the

Figure 10 shows the closed-loop step response to a step disturbance at the system's output. This is important in analyzing the sensitivity of the control system

rium position after encountering some level of disturbances.

that the system is critically damped; thus, the damping ratio ξ ¼ 1:

effectiveness of the control system.

Figure 7.

Figure 8.

128

Step response for the controlled system.

Step response for the uncontrolled system.

Noise and Vibration Control - From Theory to Practice

Figure 11. Plot of input disturbance rejection.


#### Table 4.

Summary of the performance of the PID control.

which allows for independent control of the effect of the reference signal on the proportional action.

The robustness and performance of the PID control are directly proportional to the degree of stability of the rail car, ride comfort and performance of the rail car. The summary of the performance of the PID control system is presented in Table 4.

be 83<sup>o</sup> at a frequency of 16.6 rad/s for the tuned response compared to the baseline response with a phase margin and frequency of 60.4<sup>o</sup> and 17.8 rad/s, respectively. The relationship between the phase margin and percent overshoot is inversely proportional. Hence, the high value of the phase margin results in significant decrease in the percent overshoot, thus improving the rail car stability. Also, the higher closed-loop bandwidth results in faster rise time. The rise time was found to be 0.07 s for the tuned response and 0.0847 s for the baseline response. This implies that for the baseline response, the percent overshoot is still significant to offset the

Figures 13 and 14 show the result of the linearization of the rail car system. This

is to determine the dynamics of the system in real time and within time and frequency domains. Figure 13 shows the impulse response of the rail car system, which is the degree of rail car body displacement as a result of load or rail disturbances. The maximum amplitude oscillation is 0.02 mm, which is negligible and insufficient to offset the rail car balance. In addition, the settling time (less than 1 s)

stability of the rail car system.

Impulse response of the actively controlled rail car.

Figure 12.

Figure 13.

131

Bode plot for the controller effort.

Vibration Analysis and Control in the Rail Car System Using PID Controls

DOI: http://dx.doi.org/10.5772/intechopen.85654

From Table 4, the tuned response has better performance and robustness compared to the baseline due to periodic iterative adjustment to eliminate the steadystate error for each step input. The rise time of the tuned response indicates a fast response time of the control system to disturbances or changes when compared to the baseline. This signifies some degree of delay in the time it takes the system to respond to fluctuations. In addition, comparing the settling time for both responses, the tune response settles faster after some disturbances compared to the baseline response. This explains the ability of the rail car to regain its stability after encountering some level of disturbances with the active control system. Also, the percent overshoot for the tuned response is still within the range of the permissible oscillation (5%) for critically damped system which indicates that the rail car system is relatively stable amidst load and rail disturbances. An increase in the phase margins implies a significant reduction in the percent overshoot and bandwidth.

Figure 12 shows the bode response of the control system. This is the plot of the frequency and phase response of the control system. The phase margin was found to Vibration Analysis and Control in the Rail Car System Using PID Controls DOI: http://dx.doi.org/10.5772/intechopen.85654

Figure 12. Bode plot for the controller effort.

Figure 13.

which allows for independent control of the effect of the reference signal on the

Parameter Tuned response Baseline (reference) response

Rise time 0.07 s 0.0847 s Settling time 0.499 1.94 s Overshoot 4.76% 10.3% Peak 1.05 1.1 mm Phase margin 83o 60.4<sup>o</sup> Frequency 16.6 17.8 rad Closed loop Stable Stable

implies a significant reduction in the percent overshoot and bandwidth.

Figure 12 shows the bode response of the control system. This is the plot of the frequency and phase response of the control system. The phase margin was found to

The robustness and performance of the PID control are directly proportional to the degree of stability of the rail car, ride comfort and performance of the rail car. The summary of the performance of the PID control system is presented in Table 4. From Table 4, the tuned response has better performance and robustness compared to the baseline due to periodic iterative adjustment to eliminate the steadystate error for each step input. The rise time of the tuned response indicates a fast response time of the control system to disturbances or changes when compared to the baseline. This signifies some degree of delay in the time it takes the system to respond to fluctuations. In addition, comparing the settling time for both responses, the tune response settles faster after some disturbances compared to the baseline response. This explains the ability of the rail car to regain its stability after encountering some level of disturbances with the active control system. Also, the percent overshoot for the tuned response is still within the range of the permissible oscillation (5%) for critically damped system which indicates that the rail car system is relatively stable amidst load and rail disturbances. An increase in the phase margins

proportional action.

Summary of the performance of the PID control.

Figure 11.

Table 4.

130

Plot of input disturbance rejection.

Noise and Vibration Control - From Theory to Practice

Impulse response of the actively controlled rail car.

be 83<sup>o</sup> at a frequency of 16.6 rad/s for the tuned response compared to the baseline response with a phase margin and frequency of 60.4<sup>o</sup> and 17.8 rad/s, respectively. The relationship between the phase margin and percent overshoot is inversely proportional. Hence, the high value of the phase margin results in significant decrease in the percent overshoot, thus improving the rail car stability. Also, the higher closed-loop bandwidth results in faster rise time. The rise time was found to be 0.07 s for the tuned response and 0.0847 s for the baseline response. This implies that for the baseline response, the percent overshoot is still significant to offset the stability of the rail car system.

Figures 13 and 14 show the result of the linearization of the rail car system. This is to determine the dynamics of the system in real time and within time and frequency domains. Figure 13 shows the impulse response of the rail car system, which is the degree of rail car body displacement as a result of load or rail disturbances. The maximum amplitude oscillation is 0.02 mm, which is negligible and insufficient to offset the rail car balance. In addition, the settling time (less than 1 s)

Figure 14. Bode plot for the actively controlled rail car.

is also within the permissible range indicating the ability of the system to regain its stability within a second as a result of load or rail disturbances.

Figure 14 shows the bode plot of the rail car system; the phase margin was found to be 48<sup>o</sup> at a frequency of 20.2 rad/s which signifies significant reduction in the percent overshoot due to the compensation for steady-state error by the derivative action of the PID.
