The Difference and Connection Between First-Order LADRC and PI Controller in Sensorless Motor Control Applications

LADRC (Linear Active Disturbance Rejection Control) is a linearized version of the ADRC (Active Disturbance Rejection Control) algorithm, created by Professor Gao Zhiqiang. This algorithm has been widely applied in motor control systems and has shown promising results. In one of Professor Gao's papers, it was pointed out that the first-order LADRC can be equivalently represented as a series combination of a PI controller and a low-pass filter[1] [2].

As we know, the motor speed-torque equation can be approximated as a first-order system. When a PI controller is added into this loop, the speed system can be equivalently represented as a second-order system.

The LADRC controller internally includes an Extended State Observer (ESO), which plays a crucial role in enhancing the system's robustness, particularly in motor speed control applications. The ESO has two degrees of freedom in its output state: one for observing the load (or external disturbance) and the other for observing disturbances affecting the motor system. The ESO outputs both disturbance estimates and speed estimates, allowing the controller to compensate for disturbances through a feedforward mechanism, which effectively improves the system's performance.

So the question arises: Since LADRC can be equivalent to a series combination of a PI controller and a filter, what is the use of studying LADRC? It seems that using a PI controller directly would be sufficient to meet the control requirements. What is the significance of LADRC? Actually, some literature points out that to achieve the same control performance (same closeed-loop bandwidth), when LADRC is equivalent to a PI controller, the equivalent PI parameters are different from the PI parameters corresponding to the same control performance[1].

Let's analyze the advantages of LADRC compared to the PI controller based on this article, and provide an answer to the above question.

1. Speed Control Loop Modeling and PI Control

In sensorless applications, the current control loop bandwidth is usually not high, which is due to the nature of the sensorless estimation algorithm. This results in a low bandwidth for the speed control loop as well. Some articles suggest that although LADRC can be equivalent to a PI controller, this does not mean that the two controllers can be equivalent in terms of control bandwidth.

If the same controller gain parameters are used, LADRC can be seen as a PI controller with increased system bandwidth. This is the fundamental reason why LADRC has superior disturbance rejection performance.

We will demonstrate this point through simulation. First, we model the control loop and use the standard motor speed closed-loop PI parameter tuning method to implement a Simulink model with a current loop bandwidth of 500Hz and a speed loop bandwidth of 50Hz.

Modeling parameters:

• phase inductance: 0.0005 H
• phase resistance: 0.6 ohms
• Rotor Inertia: 0.000006 Kgm2
• Control method: Feedforward FOC control to cancel back electromotive force (EMF)

Using feedforward FOC control to cancel back electromotive force (BEMF), and the motor is equivalent to a first-order controlled system. Then discretization, closely aligned with the physical system, with a discrete sampling time of 1 μs. The loop model is established as shown in the following diagram.

From the open-loop Bode plot, it can be seen that discretization introduces some errors, but it does not affect the control performance. Next, we will verify whether the closed-loop bandwidth meets the design requirements. Generally, in a control system with unity negative feedback, the open-loop crossover frequency is close to the closed-loop bandwidth (with the closed-loop bandwidth being slightly higher than the open-loop crossover frequency).

From the closed-loop Bode plot, it can be seen that the closed-loop system bandwidth is slightly larger than the open-loop system crossover frequency, which meets the design requirements. This indicates that our parameter selection is reasonable and that the system is operating within the desired bandwidth, ensuring both stability and responsive control performance.

2. Implementing the ADRC control strategy for the speed loop

We directly provide the speed loop LADRC control model.

3. Comparison and Analysis of Disturbance Rejection Performance Between LADRC and PI Controller

3.1 closed-loop control performance

First, we verify whether the control parameters match for the closed-loop systems using PI and LADRC controllers. Specifically, we aim to ensure that the speed closed-loop system bandwidth is the same for both controllers.

As shown in Figure 6, we used both the LADRC and PI controllers to control the system, and obtained the same closed-loop bandwidth of approximately 400 rad/s for both controllers (LADRC 400 rad/s, PI 380 rad/s). This way, we can avoid the impact of different bandwidths on the verification of disturbance rejection performance. We know that, generally, the higher the bandwidth, the stronger the disturbance rejection capability.

However, from Figure 6, we observe that the closed-loop Bode plot of LADRC differs significantly from that of PI. The key reason has already been marked in the figure. This is because LADRC effectively uses a compensator of type II, which is cascaded with the PI controller. This compensator compensates for the zero of the PI controller, thereby suppressing overshoot. At the same time, introducing another pole in the system creates a second-order filter in conjunction with the system's existing dynamics. This combination of the new pole and the system's natural behavior forms a double pole at high frequencies, which helps to filter out high-frequency disturbances.

As observed, the closed-loop system after compensation exhibits no overshoot, which also explains why the step response of LADRC does not overshoot.

3.2 open-loop control performance

Next, let's examine the open-loop Bode plot, which more clearly reflects the correction performance of LADRC.

The PI controller has a pole at zero frequency, which, in combination with the motor's inherent pole at zero frequency, causes a -40 dB/dec slope in the frequency response. This results in the system having a higher phase lag at low frequencies. This phase lag reduces the system's ability to respond quickly to changes in the input or disturbances, especially in low-frequency ranges.

LADRC corrects the phase lag at low frequencies and compensates for the pole and zero of the PI controller. This ensures that the open-loop system at mid to low frequencies primarily reflects the system's inherent poles, maintaining phase margin in this frequency range.

Additionally, the pole of the compensator is shifted to higher frequencies, resulting in a -40 dB/dec gain slope at high frequencies. This behavior illustrates the filtering effect at high frequencies, allowing the system to suppress high-frequency disturbances more effectively.

By compensating for the PI controller's limitations at low frequencies and enhancing disturbance rejection at high frequencies, LADRC improves the overall system stability and robustness across the entire frequency spectrum. This leads to a smoother and more stable performance, particularly in the presence of disturbances.

3.3 Advantages of LADRC

Finally, to verify the advantages of LADRC control under the same speed loop bandwidth, we present the open-loop Bode plot.

See? Even though the closed-loop system bandwidth is the same, the crossover frequency of the open-loop system for LADRC is more than twice as high as that of the PI controller (PI 314 rad/s, LADRC 790 rad/s). This means that in order to maintain the closed-loop system bandwidth unchanged, the open-loop gain of LADRC needs to be set higher, possibly more than twice the proportional gain of the PI controller. This is because the ESO in LADRC compensates for the pole of the PI controller, which increases the open-loop crossover frequency. Meanwhile, the introduced filter reduces the closed-loop bandwidth, ensuring that the speed loop bandwidth remains approximately the same when using either controller.

In the LADRC controller, the open-loop crossover frequency is determined by the controller gain k. This gives us a new perspective on using the LADRC controller: by increasing the controller gain, we can achieve better disturbance rejection while ensuring that the closed-loop system bandwidth remains unchanged.

This is the advantage of LADRC over the PI controller. While increasing the proportional gain of a PI controller does improve disturbance rejection, it also leads to an increase in the closed-loop system bandwidth. Since in motor control applications, the speed loop bandwidth cannot be easily increased due to system constraints (such as limitations in system dynamics, power requirements, or stability concerns), this provides a significant opportunity for the application of LADRC.

3.4 Disturbance Performance Comparison

We added noise in the simulation to test the disturbance rejection capability of the PI controller and LADRC under the same speed loop bandwidth. The simulation results confirmed that LADRC has superior disturbance rejection capability compared to the PI controller.

The control command is set to 100. Figures 8 and 9 show the control effects of the PI and LADRC controllers, respectively, representing the speed feedback waveforms. Figures 10 and 11 display the speed tracking errors. The above figures clearly demonstrate that LADRC exhibits superior disturbance rejection capability compared to the PI controller.

Finally, please always remember our test condition: the same closed-loop bandwidth for the speed loop.

Note the above figure: our testing premise is that the closed-loop system bandwidth must be the same. Roughly speaking, this means that the rise time of the step response is essentially the same.

5. Conclusion

When using LADRC and PI controllers with the goal of achieving the same closed-loop control bandwidth, the corresponding open-loop system crossover frequencies are different. The advantage of LADRC lies in enhancing disturbance rejection capability by increasing the control gain, which accelerates the convergence speed of disturbance estimation and raises the open-loop crossover frequency, all while ensuring that the closed-loop bandwidth remains unchanged. In contrast, to improve disturbance rejection performance, the PI controller requires an increase in the closed-loop bandwidth. This provides a better solution for sensorless applications.

6. Reference

[1] Zhong, S. , Huang, Y. , & Guo, L. . (2020). A parameter formula connecting pid and adrc. Sciece China. Information Sciences, 63(9).

[2] Miklosovic, R. , & Gao, Z. . (2004). A robust two-degree-of-freedom control design technique and its practical application. IEEE.