Fuzzy dynamic self-tuning based linear active disturbance rejection control for PMSM speed control

3.1. The overall control structure of FDS-LADRC for the PMSM speed loop

From Eq. (6), it can be seen that the mathematical model of the PMSM speed loop can be expressed in the form of a first-order differential equation. Typically, the order of the LADRC corresponds to the order of the controlled object. Therefore, for the PMSM speed loop, a first-order LADRC can be effectively employed for control. The first-order LADRC mainly consists of three linear components: the first-order linear tracking differentiator (FOLTD), the second-order linear extended state observer (SOLESO), and the LSEF. By rewriting Eq. (6) in a general form, we have:

where $f\left(*\right)=\frac{n_p}{J}\left(∆T_e-T_l+∆T_l-\frac{B_r}{n_p}{\omega }_r\right)+∆d_n$ represents the sum of internal parameter perturbations and external load disturbances in the PMSM speed loop. This term is referred to as the “total disturbance” within the LADRC framework and is estimated and compensated by the SOLESO. Additionally, $b=\frac{3n_p}{2J}{\mathrm{\Psi }}_{eff}$ denotes the current gain coefficient, and $U=I_q$ represents the controller output control quantity, which is the q-axis current.

By defining the state variables for Eq. (7) as $x_1={\omega }_r$, $x_2=f\left({\omega }_r,∆d_n,t\right)$, and $o=\dot{f}\left({\omega }_r,∆d_n,t\right)$, Eq. (7) can be rewritten in the form of a state-space equation:

Designing a SOLESO for the controlled object represented by Eq. (8), we have:

where $z_1$ and $z_2$ represent the estimated values of the state variables $x_1$ and $x_2$, respectively. $b_0\approx b$ is the control gain compensation factor, and ${\alpha }_1$ and ${\alpha }_2$ are the feedback coefficients of the SOLESO. Typically, for any order of LESO, the feedback coefficients can be set at the observer bandwidth ${\omega }_o$ using pole placement methods [10], ensuring that the observer is bounded-input bounded-output (BIBO) stable, that is:

From Eq. (10), it can be seen that, in the SOLESO, the bandwidth ${\omega }_o$ is the only adjustable parameter and has a strong physical significance. Additionally, for the FOLTD, we have:

where $\mu$ is the derivative time constant and $Ref$ is the reference input signal.

However, at the current stage, the theoretical development of LADRC is relatively mature. In industrial control, the methods for handling system transient processes are also well-established. Meanwhile, to avoid high-frequency oscillations in practical industrial control applications, the vast majority of LADRC controllers no longer employ the LTD. Instead, the reference input signal $Ref$ is directly used as the input to the LSEF. Therefore, the LSEF can be designed as:

where $k_p$ represents the proportional coefficient. According to the bandwidth-based parameter tuning rule [18], $k_p$ can be set at the controller bandwidth ${\omega }_c$, i.e., $k_p={\omega }_c$. Consequently, the control law of the system can be derived as:

Fig. 1The overall control structure of FDS-LADRC

As indicated by Eq. (10) and Eq. (12), once the observer bandwidth ${\omega }_o$ and the controller bandwidth ${\omega }_c$ are set, their values remain fixed throughout the control process, making it difficult to simultaneously balance the trade-offs between system dynamic performance, steady-state characteristics, and disturbance rejection capabilities. Moreover, as discussed in Section 2, the PMSM systems are subject to various external disturbances in practical control applications. And the presence of internal parameter variations further complicates the control problem by introducing internal parameter uncertainties. This results in a lack of adaptability in both the controller and the SOLESO. To address these challenges, this paper proposes an enhanced LADRC method based on the fuzzy dynamic self-tuning mechanism, i.e., FDS-LADRC. To be specific, the fuzzy dynamic self-regulators are designed to adaptively optimize the parameters of the controller and the bandwidth of the SOLESO, respectively, thereby obtaining the adaptive fuzzy controller and the adaptive LESO, which can enhance the system’s adaptability and flexibility throughout the control process. This approach aims to achieve optimal coordination between the controller, SOLESO, and the controlled plant, resulting in improved control performance while avoiding the cumbersome and repetitive manual parameter tuning process. The overall control structure of FDS-LADRC is illustrated in Fig. 1. In the next subsection, the detailed design processes of fuzzy dynamic self-regulators are elaborated.

3.2. The design of fuzzy dynamic self-regulators

Fuzzy system, also known as fuzzy logic system, was proposed by Zadeh in 1965 [41]. It is an intelligent computing system based on fuzzy linguistic variables, fuzzy logic reasoning, and fuzzy set theory. The most significant feature of fuzzy systems is their ability to represent human prior experience in the form of IF-THEN linguistic rules, mimicking the human thought process, reasoning, and decision-making behaviors. Moreover, fuzzy logic can be employed to handle uncertainties within the system [42]. The basic components of a fuzzy system include the fuzzifier, knowledge base, rule base, fuzzy inference engine, and defuzzifier. The schematic diagram of a generic fuzzy system is shown in Fig. 2.

Above all, the specific configurations of fuzzy dynamic self-regulators are elaborated, with a Mamdani-type fuzzy structure adopted. On one hand, for the adaptive fuzzy controller’s fuzzy dynamic self-regulator, the error $e=Ref-y$ and its derivative $\dot{e}$ of the closed-loop system serve as input variables, while the controller parameter variation $∆k_p$ is the output. Note that the universes of discourse for the input and output variables depend on practical control scenarios. Three fuzzy subsets $\{N,Z,P\}$ are selected with respective semantic information, namely, negative, zero, and positive. Correspondingly, their membership functions are Z-shaped, triangular, and S-shaped functions, respectively. Moreover, the singleton fuzzifier, product inference engine, and center-of-sets defuzzification method are employed. An example of the membership functions for $∆k_p$ is illustrated in Fig. 3.

Fig. 2The schematic diagram of a classic fuzzy system

Fig. 3The membership function of ∆kp

Fig. 4The membership function of ∆ωo

On the other hand, for the adaptive fuzzy LESO’s fuzzy dynamic self-regulator, the error $e$ of the closed-loop system is also chosen as the input variable and the variation of the bandwidth ${\omega }_o$ serves as the output. The fuzzy subsets and corresponding membership functions for $e$ and ${∆\omega }_o$ are the same as those used for tuning ${∆k}_p$, i.e., $\{N,Z,P\}$. The singleton fuzzifier, product inference engine, and center-of-sets defuzzification method are employed as well. An example of the membership function for ${∆\omega }_o$ is illustrated in Fig. 4.

Since the controller is the most dominant component that affects the system performance, its parameters need to be precisely adjusted. Therefore, the tuning principle for $∆k_p$ is as follows: When the system response is in the rising phase ($e$ is $P$), $∆k_p$ should be $P$, meaning that $k_p$ should be increased. If the system response is in an overshoot state ($e$ is $N$), in this case, $∆k_p$ should be $N$, meaning that $k_p$ should be decreased. When the system response is in the steady-state ($e$ is $Z$), three scenarios are considered: (a) If $\dot{e}$ is $N$, indicating a decreasing trend in the overshoot, $∆k_p$ should be $N$; (b) If $\dot{e}$ is $Z$, indicating that the system response is already stable, $∆k_p$ should be $Z$; (c) If $\dot{e}$ is $P$, indicating that the error is increasing, $∆k_p$ should be $P$. In summary, the fuzzy rule table for the tuning of $∆k_p$ is shown in Table 1.

In addition to the controller, the bandwidth ${\omega }_o$ of the SOLESO also affects the system performance to a large extent. Specifically, when ${\omega }_o$ is sufficiently large, the observer can respond more quickly to changes in system states, thereby accelerating the convergence rate of state estimation and enhancing the system’s response speed. However, an excessively high ${\omega }_o$ may lead to a high overshoot in the system response, causing the observer’s state estimates to significantly exceed the actual state values. Moreover, it will amplify measurement noise, reduce the accuracy of state estimation and consequently degrade the control performance of the system. In severe cases, it may even lead to observer instability. Therefore, the tuning principle for ${∆\omega }_o$ is as follows: When the system response is in the rising phase ($e$ is $P$), ${∆\omega }_o$ should be $P$ to enable the system to track the reference input more rapidly. When the system response is in an overshoot state ($e$ is $N$), ${∆\omega }_o$ should be $N$ to ensure that ${\omega }_o$ does not become excessively large. When the system response is in a steady state ($e$ is $Z$), ${∆\omega }_o$ should also take a value tending towards zero to maintain system stability. The fuzzy rule table for the tuning of ${∆\omega }_o$ is shown in Table 2.

To sum up, the pseudocode of FDS-LADRC is shown in Table 3.

4.1. The stability analysis of the SOLESO

Let the state error of the system be defined as $e_i=x_i-z_i (i=1, 2)$. By combining Eq. (8) and Eq. (9), the state-space equation for the state error can be derived as:

Thus, the characteristic equation of Eq. (14) is:

Given that the bandwidth ${\omega }_{\mathrm{o}}$ of the SOLESO is always greater than zero, it can be inferred from Eq. (15) that all poles are located at $-{\omega }_{\mathrm{o}}$. Consequently, the SOLESO is BIBO stable.

4.2. The stability analysis of closed-loop system

According to Eq. (14) and Eq. (15), the control law can be rewritten as:

Substituting $z_1=x_1+e_1$ and $z_2=x_2+e_2$ into Eq. (16) and then into Eq. (8), the dynamics of $x_1$ becomes:

Defining the tracking error as ${\tilde{x}}_1=x_1-Ref$, Eq. (17) can be rewritten as:

Consider the following positive definite Lyapunov function candidate:

Differentiating $V$ with respect to time yields:

According to Eq. (14), the error dynamics is:

Substituting Eq. (18) and Eq. (21) into Eq. (20) gives:

Applying Young’s inequality [43] to bound the cross terms in Eq. (22):

Substituting Eq. (23-27) into Eq. (22), we obtain:

Define the following constants: $c_1=\frac{k_p-1}{2}$, $c_2={\alpha }_1-\left(\frac{1+k_p}{2}\right)-\frac{{\alpha }_2}{2}$, and $c_3=\frac{{\alpha }_2+3}{2}$. If $k_p>1$, ${\alpha }_1>\frac{1+k_p+{\alpha }_2}{2}$, and ${\alpha }_2>0$, then $c_1>0$, $c_2>0$, and $c_3>0$. Thus, Eq. (28) can be rewritten as:

Assuming the rate of change of the disturbance $o$ is bounded, i.e., $\left|o\right|\le H$, it follows that:

where $c_0=\mathrm{m}\mathrm{i}\mathrm{n}\{c_1,c_2,c_3\}$.

By the comparison lemma [44], the solution of Eq. (30) satisfies:

This indicates that the system states are ultimately uniformly bounded (UUB), Furthermore, if $H=0$ (i.e., in the absence of disturbance variation), the system is asymptotically stable.

5.1. Speed comparison under no-load startup conditions (simulation)

In this subsection, the simulations of speed comparison under no-load startup conditions of different methods are reported. At the beginning, the reference speed was assumed to be 500 rpm. Subsequently, the reference speed was increased to 1000 rpm at 0.2 s. The speed responses and the transient performance metrics are shown in Fig. 6 and Table 4, respectively.

As can be seen from Fig. 6 and Table 4, when compared with alternative control methods, the proposed FDS-LADRC demonstrates satisfactory transient performance under no-load startup conditions. It is capable of tracking the reference signal swiftly and smoothly. From a quantitative point of view, FDS-LADRC exhibits an overshoot of 13.87%, a peak time of 0.0096 s, and a settling time of 0.0219 s. These transient performance metrics outperforms those of its counterparts. Specifically, compared with nonlinear ADRC and LADRC, both of which employ fixed parameters, the adaptive fuzzy controller and the adaptive fuzzy LESO in FDS-LADRC have the ability to dynamically adjust the corresponding parameters in real time. This adaptive feature leads to its superior transient performance. In contrast to IT2FDS, although FDS-LADRC utilizes Type-1 fuzzy systems for parameter adjustment, it still achieves a more favorable control effect. In comparison with FSFOADRC and SMC, FDS-LADRC continues to deliver more desirable transient performance. Notably, due to the chattering phenomenon, the response of SMC is subject to certain fluctuations, whereas the response of FDS-LADRC exhibits little fluctuations.

Fig. 6The speed responses under no-load startup conditions (simulation)

In addition, the control quantities of all the methods under consideration are depicted in Fig. 7.

Fig. 7The control quantities under no-load startup conditions

As shown in Fig. 7, the control quantity of FSFOADRC is significantly larger than that of its counterparts and exhibits noticeable chattering. We speculate that this phenomenon is attributed to the application of fractional calculus. On the other hand, compared with other methods, the control quantity of FDS-LADRC is relatively small and demonstrates a relatively smooth profile. This observation implies that the control cost associated with FDS-LADRC is comparatively lower. Consequently, we consider that FDS-LADRC has certain potential to be deployed in scenarios with limited computational resources, such as certain applications running on embedded hardware.

Moreover, the outputs of fuzzy dynamic self-regulators under no-load startup conditions are shown in Fig. 8 and Fig. 9, respectively.

Please note that due to space limitations, the control quantities of different methods and the outputs of fuzzy dynamic self-regulators are demonstrated in this case.

Fig. 8The output values of ∆kp under no-load startup conditions (simulation)

Fig. 9The output values of ∆ωo under no-load startup conditions (simulation)

5.2. Speed comparison under loaded startup conditions (simulation)

In this subsection, unlike in Section 5.1, the simulations of speed comparison under loaded startup conditions of different methods are reported. Specifically, a load torque of 2 N∙m was applied to the rotor of the PMSM at the start stage. Meanwhile, the reference speed was assumed to be 500 rpm, and then the speed was increased to 1000 rpm at 0.2 s. The speed responses and transient performance metrics of different methods under loaded startup conditions are illustrated in Fig. 10 and Table 5, respectively.

Fig. 10The speed responses under loaded startup conditions (simulation)

As shown in Fig. 10 and presented in Table 5, FDS-LADRC also obtains satisfactory control performance under loaded startup conditions. To be specific, the overshoot, peak time, and settling time of FDS-LADRC are 10.54 %, 0.0123 s, and 0.0245 s, respectively. These transient performance metrics are, for the most part, more outstanding than those of its competing methods. In addition, when comparing the results under loaded startup conditions with those under no-load startup conditions, it can be observed that the overshoots of both methods have decreased, while the settling time and peak time of the system have increased to a certain degree. We consider that that the following factors may account for these results: When the PMSM is started under loaded conditions, the inertia and damping effects exerted by the load can potentially elevate the damping ratio of the system. As a consequence, this leads to a reduction in the speed overshoot, but simultaneously increases the system’s response time.

5.3. Speed comparison under sudden loading and unloading conditions (simulation)

In this subsection, the simulation models the scenario where the PMSM suddenly experiences a load disturbance. The simulation procedure is as follows: First, the PMSM was started under no-load conditions with a given speed of 500 rpm. At 0.2 s, a sudden load disturbance of 2 N∙m was applied to the motor. Subsequently, at 0.3 s, this load was suddenly released. The speed responses of different methods under this disturbance scenario are shown in Fig. 11.

Fig. 11Speed responses under sudden loading and unloading conditions (simulation)

As shown in Fig. 11, FDS-LADRC demonstrates remarkable disturbance rejection capabilities. This enables the output to quickly resynchronize with the reference speed after experiencing sudden increases or decreases in load disturbances. Specifically, at 0.2 s, when a load disturbance is abruptly introduced, the outputs of the traditional LADRC, ADRC, IT2FDS, and FSFOADRC drop to approximately 487 rpm, 484 rpm, 487 rpm, 490 rpm, and 478 rpm, respectively. Moreover, they require about 0.05 s, 0.05 s, 0.05 s, 0.025 s, and 0.04 s, respectively, to recover to the reference speed. In contrast, the output of FDS-LADRC only decreases to about 494 rpm and takes approximately 0.025 s to readjust back to the reference speed. Similarly, at 0.3 s, when the load disturbance is removed, FDS-LADRC demonstrates superior anti-disturbance ability compared to its counterparts. Through comparative analysis of these results, it is evident that FDS-LADRC exhibits superior performance and adaptability in resisting load disturbances.

5.4. Speed comparison under no-load startup conditions with measurement noise

In this subsection, to assess the anti-noise ability, we carried out a simulation under no-load conditions. Specifically, the Gaussian white noise with a noise power of 0.005 and a sample time of 0.001 s was introduced into the system output. The speed responses under this noisy condition are presented in Fig. 12.

Fig. 12Speed comparison under no-load startup conditions with measurement noise

As shown in Fig. 12, when the measurement noise is introduced, the response of FDS-LADRC remains confined within approximately [480 rpm, 520 rpm]. In contrast, other control methods exhibit wider fluctuation ranges. For instance, the traditional LADRC fluctuates between about [450 rpm, 550 rpm], while FSFOADRC varies within about [470 rpm, 530 rpm]. These results demonstrate that despite the detrimental effects of measurement noise, the fuzzy dynamic self-regulators in FDS-LADRC can autonomously and collaboratively adjust parameters under uncertainty, enabling superior noise rejection performance.

6.1. Speed comparison under no-load startup conditions (experiment)

Similar to the process of Section 5.1, in this subsection, the PMSM was started under no-load conditions, with a reference speed set as 50 rpm at 20 s. Subsequently, the reference speed was raised to 100 rpm at 30 s. The speed responses of the two methods are shown in Fig. 14 and Fig. 15, respectively.

Fig. 14The speed response of the traditional LADRC under no-load startup conditions (experiment)

As shown in Fig. 14 and Fig. 15, under no-load startup conditions, the traditional LADRC takes approximately 0.1325 s to reach a steady state at 50 rpm. In contrast, FDS-LADRC achieves steady state in just about 0.0485 s. This indicates that FDS-LADRC has a response time approximately 0.084 s faster than traditional LADRC, significantly enhancing the system’s dynamic response capability. Furthermore, the traditional LADRC method exhibits an overshoot of approximately 83.27 %, whereas FDS-LADRC shows an overshoot of only about 32.44 %, which is approximately 50.83 % less than that of traditional LADRC. This demonstrates that FDS-LADRC achieves higher stability during the control process, effectively reducing the risk of system oscillations and instability caused by excessive adjustments, and enabling more precise control of motor speed.

6.2. Speed comparison under loaded startup conditions (experiment)

Similar to the process of Section 5.2, in this subsection, the PMSM was started under loaded conditions, with a reference speed set as 50 rpm at 20 s. Subsequently, the reference speed was raised to 100 rpm at 30 s. The load imposed on the motor is equivalent to 10 % of its rated load, i.e., 0.032 N∙m. The speed responses of the two methods are shown in Fig. 16 and Fig. 17, respectively.

Fig. 15The speed response of FDS-LADRC under no-load startup conditions (experiment)

Fig. 16The speed response of the traditional LADRC under loaded startup conditions (experiment)

As shown in Fig. 16 and Fig. 17, under loaded startup conditions, the traditional LADRC method exhibits an overshoot of about 64.90 % and takes around 0.1425 s to reach steady state. In contrast, FDS-LADRC exhibits an overshoot of approximately 24.04 % and reaches steady state in just about 0.0545 s. Compared to the no-load startup conditions, both methods show a decrease in overshoot when starting under loaded conditions. Specifically, the overshoot of the traditional LADRC method decreases by about 18.37 %, while that of FDS-LADRC decreases by approximately 8.40 %. Additionally, the time taken to reach steady state increases by about 0.01 s for traditional LADRC and 0.006 s for FDS-LADRC. These results align with the phenomena observed in the simulations. Overall, FDS-LADRC outperforms traditional LADRC in terms of both overshoot and time to reach steady state, demonstrating superior transient performance.

6.3. Speed comparison under sudden loading and unloading conditions (experiment)

Similar to the process of Section 5.3, in this subsection, the PMSM was started under no-load conditions, with a reference speed set as 100 rpm. Subsequently, a load torque equivalent to 25 % of its rated load, i.e., 0.064 N∙m, was suddenly applied at 20 s. This load torque was then abruptly removed at 30 s. The speed responses of the two methods under this load disturbance condition are shown in Fig. 18 and Fig. 19, respectively.

Fig. 17The speed response of FDS-LADRC under loaded startup conditions (experiment)

Fig. 18The speed response of the traditional LADRC under sudden loading and unloading conditions (experiment)

As shown in Fig. 18 and Fig. 19, there is a significant difference in recovery time to the steady state between the traditional LADRC method and FDS-LADRC when subjected to load disturbances. Specifically, when a sudden load disturbance is added, the traditional LADRC method takes about 4.5 s to return to the steady state, while FDS-LADRC only requires approximately 4 s, demonstrating faster recovery. Similarly, when this load disturbance is suddenly removed, the traditional LADRC method takes about 6 s to recover, whereas FDS-LADRC again only needs around 4 s to return to the steady state. It is noted that although FDS-LADRC exhibits slightly larger speed variations during both the addition and removal of the load disturbance, we believe this is primarily due to inherent limitations in the hardware (such as sensors and mechanical components) during the experiments, which are normal phenomena in a real-world experimental environment. Therefore, according to these experimental results, FDS-LADRC shows superior disturbance rejection performance when dealing with load disturbances.

Fig. 19The speed response of FDS-LADRC under sudden loading and unloading conditions (experiment)