The bandwidth of the speed loop

What impact does the bandwidth of the speed loop have on control performance? In the previous article, I discussed how the bandwidth of the current loop affects system performance and highlighted key considerations when designing the current loop bandwidth. In this article, I will focus on analyzing the closed-loop speed control system, starting with observable phenomena to intuitively illustrate the impact of speed loop bandwidth on the control system.

The bandwidth of the speed loop plays a crucial role in the control system, primarily impacting the following aspects:

1. System Response Speed: The larger the bandwidth of the speed loop, the faster the system's response to speed commands, allowing quick adjustments to meet setpoints. However, excessive bandwidth may lead to system overshoot or instability.
2. Disturbance Rejection: Higher bandwidth generally enhances the system's ability to suppress external disturbances. With an appropriate bandwidth, the system can quickly respond to load changes or external interferences, maintaining high control accuracy.
3. Load Handling Capability: When load variations are significant, higher bandwidth helps the system maintain consistent speed, improving steady-state performance.
4. .Steady-State Accuracy: Bandwidth is somewhat related to steady-state accuracy; a suitable bandwidth setting can reduce steady-state errors and improve control precision. However, a balance must be struck between bandwidth and noise sensitivity to avoid amplifying high-frequency noise.

When designing the control parameters for the speed loop, the current loop is often approximated as "1," meaning that the closed-loop transfer function of the current loop is approximately equal to 1. Due to factors like angle estimation delay and the varying complexity of estimation algorithms, sensorless control strategies typically adopt a smaller control bandwidth to ensure system stability. I designed the current loop bandwidth at 500Hz and for ease of display, the width of the bandwidth is chosen to be lower at 0.89~1.15 Hz. This frequency design aligns with the above assumption. Let’s analyze the impact of different bandwidths on system control performance from the following two aspects:

System Response Speed

Intuitively, with different PI controller parameters, the bandwidth of the speed loop changes—the larger the P parameter, the higher the bandwidth. The following figure shows the tracking performance of the speed command at different speed loop bandwidths. From the figure, you can clearly see the differences in speed response rate. I only changed the proportional parameter in the PI controller, keeping the integral parameter constant. This method allows us to focus on observing the impact of proportional gain on system response speed. By keeping the integral parameter constant and adjusting only the proportional parameter P. we can directly change the system bandwidth, improving the system's response speed to the speed command. Typically, this adjustment does not affect the steady-state error of the system but change the dynamic response speed.

Figure 1 shows a speed loop bandwidth of approximately 1.15 Hz, while Figure 2 shows a speed loop bandwidth of about 0.89 Hz. The red curve represents the command input, and the blue curve represents the speed feedback. Both figures demonstrate that the feedback speed can ultimately follow the speed command. In Figure 2, the control parameters were adjusted by only reducing the proportional parameter. The integral parameter, relative to the current bandwidth, appears relatively large, which results in overshoot.

However, just from observing the rise time, the bandwidths shown in the two figures are evidently different, and this alone is sufficient for our analysis. By comparing the rise times in the two figures, we can clearly see the effect of bandwidth on system response speed. In Figure 1, the higher bandwidth results in a significantly shorter rise time and faster response speed; in Figure 2, the lower bandwidth causes the rise time to increase, making the response relatively slower. This noticeable bandwidth difference is sufficient to analyze the dynamic performance of the system at different bandwidths, helping us understand how to select appropriate bandwidth parameters based on requirements in practical applications.

Disturbance Rejection

Next, let's examine the impact of bandwidth on disturbance rejection.

The bandwidth significantly affects the system's disturbance rejection capability. Generally, a higher bandwidth allows the system to respond more quickly to external disturbances, enabling it to counteract the effects of disturbances on control variables more effectively. This is because the higher the bandwidth, the faster the system can detect and correct deviations from the setpoint. However, excessively high bandwidth may also make the system more sensitive to high-frequency noise, potentially introducing unnecessary oscillations or amplifying noise.

In a low-bandwidth scenario, the system’s response speed is slower, and its disturbance rejection is relatively weaker, but it has better noise immunity, making it suitable for applications where fast dynamic response is not essential. Therefore, finding an appropriate balance between disturbance rejection and noise suppression is a crucial consideration when designing bandwidth parameters. The following two figures illustrate the disturbance rejection effects at different bandwidths. The dips in the graphs represent the system's corrective control process when disturbances are applied.

Clearly, Figure 3 shows stronger disturbance rejection capability, as the speed quickly returns to the command value after encountering the same disturbance. The quick recovery characteristic indicates a higher system bandwidth, enabling it to detect deviations rapidly and perform effective corrective control. This fast response allows the system to return to the setpoint immediately after a disturbance, demonstrating excellent disturbance rejection. In practical applications, this design is especially suitable for scenarios requiring high precision and stability, such as industrial automation or servo control systems. In contrast, a lower-bandwidth system shows a slower recovery, delaying its response to disturbances, making it more suitable for applications with lower dynamic performance requirements.

Finally, let's outline how to estimate bandwidth using waveform data from the host computer. Apply a step input to the system through the host computer, and observe the response waveform. Measure the rise time (the time it takes for the output to reach a certain percentage, often 63% or 90%, of the final value). The faster the rise time, the higher the estimated bandwidth. A commonly used approximation is: Bandwidth ≈ 0.35 / Rise?Time.

My control command rises from 418 radians per second to 628 radians per second. I chose the rise time from 10% to 90% of the amplitude as the parameter for bandwidth calculation. The rise time in Figure 5 is approximately 302 ms (5161 ms - 4859 ms), so the bandwidth is approximately 0.35 ÷ 0.302 = 1.16?Hz. Similarly, the bandwidth for Figure 2 can be calculated as approximately 0.89?Hz. To clarify once again, since the integral parameter has not been modified, please ignore the effects of overshoot and focus solely on the rise time.

I highly recommend using a host computer for waveform observation and analysis. I use VOFA+, an excellent host computer software that offers highly flexible features and supports high communication baud rates (up to 6 Mbps). Here is the link:

vofa+

Have fun.