Dynamic Event-Triggered Disturbance Rejection Control for Speed Regulation of Networked PMSM

This article investigates the robust control problem for speed regulation of networked permanent magnet synchronous motor subject to the limited communication bandwidth. To handle this, a new sampled-data disturbance rejection control method is developed via a well-designed discrete-time dynamic event-triggered mechanism (DETM). First, a predictor-based generalized proportional integral observer is introduced to estimate the lumped disturbances, when only the sampled-data output is available. Then, a composite proportional feedback controller is formed by fully utilizing disturbance estimation. The composite controller updates only when the designed discrete-time DETM is violated, resulting in remarkable communication and computation resource savings while maintaining the desirable disturbance rejection ability. The designed DETM can be applied to digital computers easily due to the discrete-time detection. Simulations and experiments are carried out to validate the feasibility and effectiveness of the proposed control scheme.

synchronous motors (PMSMs) have received much research attentions from the control community and are widely used in practical systems [1], [2], [3], [4], [5], [6].For example, the networked PMSMs driving autonomous electric vehicles (AEVs) can achieve distributed cooperative longitudinal control for intellectualization and safety [7], [8], [9].A reliable control strategy is fundamental to ensure the robust performance in facing the changing conditions of AEVs, such as starting, climbing, heavy loading, and high-speed overtaking and cruising.However, multisource disturbances widely exist in the PMSM systems, such as cogging torque, parameter uncertainties, and unknown load torque, which cause low accuracy in speed regulation under traditional linear control methods [10].Nowadays, various elegant approaches have been investigated, such as adaptive control [11], [12], sliding mode control [13], [14], and model predictive control [15], [16].These techniques have enhanced the dynamic performances of PMSMs from different aspects.
Among various sophisticated control methods, disturbance/uncertainty estimation and attenuation (DUEA) technique has attracted a great deal of attention due to its performance and usefulness [17].The basic concept that makes up the DUEA rule is that an observer is created to estimate disturbances, and then appropriate compensation is applied by using the disturbance estimation [17].In DUEA, extended state observer (ESO) [18], [19], generalized proportional integral observer (GPIO) [20], [21], and other useful techniques have been used to precisely estimate the disturbances.Among them, the GPIO can not only asymptotically estimate the constant disturbance, but also estimate the fast time-varying disturbance.Compared to GPIO, the ESO can only accurately estimate the former, and the estimation performance of time-varying disturbances, such as harmonics or polynomials, is poor [20], [21].In PMSM systems, both fast and slow time-varying disturbances coexist [22].Hence, the DUEA method should be developed to compensate for the undesirable influence of different types of disturbances in order to get the high-precision tracking property of PMSMs.
It should be pointed out that the aforementioned control techniques are often implemented based on a time-triggered mechanism in the digital platforms, where the control law updates with a fixed sampling period [23].Despite being simple to implement, such a control method inevitably leads to a waste of communication and computation resources, which are limited and costly in practical networked control systems [24], [25], [26].Fortunately, these issues can be effectively addressed by a novel resourceaware control technique known as event-triggered control, which can reduce communication and computation resources while retaining the required control performance [27].In event-triggered control, the event-triggered mechanism (ETM) is applied when the transmission signal exceeds a set threshold, that is to say, the system state is only updated when a given event-triggered condition is met [27].Several event-triggered control approaches have been proposed in networked PMSM systems with the limited communication bandwidth, such as periodic event-triggered method [3], self-triggered method [4], and so on.However, all of the aforementioned results presupposed that the output or state was known in the form of continuous time, which is inappropriate to apply in digital computers [28].A dynamic event-triggered sliding mode control method is proposed for speed regulation of PMSM in [5], while the proposed event-triggering condition needs to be continuously detected, which limits its applications in digital platforms.
In this article, we propose a dynamic event-triggered DUEA technique to handle the speed regulation issue of PMSMs in face of multisource disturbances, which reduces the limited communication and saves computation resources while maintaining the desired control performance.First, for the sake of dynamic performance and robustness of the control strategy, a novel GPIO is constructed to estimate the multisource disturbances where an output predictor is tactfully designed to compensate the influence of sampling.The disturbance estimation is combined with a proportional feedback baseline controller to formulate the composite controller in the speed loop.Second, we propose a new dynamic ETM (DETM), which can be detected in the discrete time and determines when to update the proposed composite controller.Robust stability analysis of the closed-loop control system is finally presented, and the efficiency of the proposed method is demonstrated by simulations and experimental tests.The main difficulty of this work is how to obtain the lost speed information within each sampling interval when the sampling period of the networked PMSM system is greater than its control period.Moreover, the asynchronous characteristics caused by different control frequencies and sampling frequencies bring challenges to stability analysis and controller design.The primary contributions of this article are given as follows.
1) Compared with the traditional DUEA method where the control input updates periodically [20], a novel DETM under the proposed control framework is designed to determine when to update the DUEA control law, which facilitates to save the communication and computation resources.2) Different from the static and dynamic ETMs developed in [3], [5], where the event-triggering conditions are detected continuously.This article proposes a novel discrete-time DETM, which can be easier to apply on digital platforms.
3) The proposed predictor-based GPIO is a dexterous combination of GPIO and predictor, which compensates for the influence of output sampling.Compared with traditional control methods based on ESO [18], [19], the GPIO-based event-triggered control method has the advantage of attenuating the influence of time-varying disturbances.

II. MATHEMATICAL MODEL AND PROBLEM ANALYSIS
The ideal mathematical model of surface-mounted PMSM in the rotating dq coordinate system is expressed as where ω is the mechanical angular velocity, ω e is the electrical angular velocity and ω e = n p ω. n p is the number of motor pairs.i d,q and u d,q are dq-axis currents and voltages, respectively.R s and L s are the stator resistance and inductance, respectively.ψ f , B, J, T L are the rotor flux linkage, viscous frictional coefficient, inertia and load torque, respectively.The electromagnetic torque T e is presented as T e = K t i q and K t = 1.5n p ψ f .The control objective is to achieve high-precision speed tracking performance under cascade structure while considerably reducing the communication times in face of multisource disturbances.Denote ω ref as the reference signal.The control-oriented model can be expressed as where e ω = ω − ω ref is the tracking error, i * q is the control signal, and d is the lumped disturbance.
A mild assumption is given as follows.
Assumption 1: The first n-order derivatives of disturbance d(t) exists, and its n-order derivative h(t) = d (n) (t) is uniformly bounded for all t ≥ 0.
Remark 1: The multisource fast and slowly time-varying disturbances widely exist in PMSM, including cogging torque, flux harmonics, unmodeled dynamic, and unknown load torque [22].Hence, from a practical point of view, it is appropriate to model the multisource disturbances as high-order polynomials.

III. MAIN RESULTS
In this section, a predictor-based GPIO is initially designed to estimate the lumped disturbance in system (2) when just the sampled-data output is known.Then, a DETM based composite speed controller is presented to implement speed tracking issues.

A. GPIO Design
To achieve better dynamic performance and estimate disturbance information, we design a predictor-based GPIO as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where T is the sampling period, z 1 is the estimation of e ω , z 2 , . . ., z n+1 are the estimations of d and their first (n − 1)th derivatives.ω is the prediction value of e ω .The parameters l 1 • • • l n+1 are observer gains to be designed.Remark 2: When only the slow-varying disturbances are considered in PMSM, including unknown load torque, parameter uncertainties, the GPIO can be simplified to ignore the highorder information of the disturbance.In order to facilitate the actual implementation, the structure of ( 3) is known as ESO under n = 1 [29].
Remark 3: Compared with the conventional GPIO [20], the predictor-based GPIO ( 3) is a special combination of GPIO and predictor, where the predictor compensates for the speed information during the sampling interval of the sensor and prepares necessary signals to be used in the GPIO.

B. Dynamic Event-Triggered Control Design
Let t m denote the (m + 1)th event-triggering instant, and t 0 = 0.The event-triggered composite controller is designed as follows: where the term kz 1 (t m ) is to stabilize the control system and the term − J K t z 2 (t m ) is to compensate the undesirable influence of the time-varying disturbance.k is the control gain to be determined.The proposed DETM-based DUEA technique is shown in Fig. 1.
To develop the DETM, an error function is first defined as where t m ≤ σT < t m+1 , m ∈ N. Hence, the composite controller (4) can be rewritten as where According to the error function ( 5), a dynamic rule is defined as where δ is a nonnegative constant and ξ is a nonnegative dynamic threshold parameter.We design the discrete-time DETM as The dynamic threshold parameter ξ is modified in accordance with an adaptive rule, which can be described as where β 1 , β 2 ≥ 0. The initial condition of ξ(0) ≥ 0. A discretetime adaptive rule obtained by integrating system (9) from (σ − 1)T to σT is shown as The detail diagram of the proposed control method is shown in Fig. 2. Remark 4: The communication frequency can be adjusted by selecting the parameter δ.The larger δ is, the fewer the transmission number is, and vice versa.The proposed DETM (8) has the benefit of providing a better balance between transmission times and control performance compared to static ETMs with constant values [24].
As observed from ( 8) and (10), the parameter ξ is greater and the likelihood that the proposed DETM will be violated decreases as the error term ζ becomes smaller, and vice versa.According to the proposed DETM (8), it is clear to infer Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.that ∀σ ∈ N, there holds ||η(σT Remark 5: It should be noted that the proposed DETM (8) with the well-designed adaptive rule (10) can be simplified into the static ETMs [24] and typical time-triggered mechanisms [31] as special examples.The smaller parameter β 1 or the larger parameter β 2 leads to a faster change of dynamic threshold ξ.As a result, it is more adaptable in terms of controlling the network transmission rate.

TABLE I CONTROL PARAMETERS IN THE SIMULATIONS
In order to estimate sup 0≤s≤t (e τ 0 s |ϕ(s)|), we recall system (11) as Select τ 0 such that τ 0 < B J .Multiplying both sides of ( 19) by e τ 0 t , one has where and According to Assumption 1 and Theorem 1, it is finally conclude from (23) that the system (11) converges to a bounded region . Finally, we give the stability analysis of the PMSM system (2) under the designed DETM based DUEA method (4).
Theorem 2: Suppose that Assumption 1 is satisfied for the PMSM system (2).The tracking error e ω is uniformly bounded and converges to a bounded region as t → ∞ under the DETM based DUEA method (4), on condition that the control parameters are selected as where g = − B J and Π = e gT + K t J k T 0 e gs ds.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Proof: Collecting (2) and the proposed controller (4) together, one obtains Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Based on Theorem 1, we have that the term Δ(σT ) is uniformly bounded for all σ ∈ N. Hence, when |Π| + T 0 e gs ds K t J || k|| ξ < 1, the tracking error e ω (σT ) converges to a bounded region, which indicates that e ω converges to a bounded region as well.This completes the proof.
Remark 6: In Theorem 2, it can be seen from ( 28) that the term T 0 e gs ds K t J || k||δ is caused by the proposed DETM (8) and disappears when δ = 0.The term Δ(σT ) is resulted in by the disturbance estimation error and the sampled-data control.We can tune the observer gains and the sampling period to regulate the upper bound of the disturbance estimation error, which leads to a decrease in the upper bound of the tracking error e ω .In addition, if the observer gains are chosen large enough or the sampling period is small enough, e ω will be arbitrarily small.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Remark 7: In fact, expression (28) is a sufficient but unnecessary condition for the closed-loop system stability.Thus, the parameter selection is conservative.In real PMSM applications, the control parameters are fairly easy to be tuned to achieve the satisfactory performance instead of satisfying the inequality (28).

IV. SIMULATION RESULTS
In this part, by comparing with the P+ESO (DETM), P+GPIO without DETM and P+ESO without DETM methods, the advantages of the proposed P+GPIO (DETM) method in speed regulation performance and computation resources saving are verified by simulations.To ensure fair comparisons, the speed tracking profiles of four controllers are calibrated so that they have similar nominal performance characteristics during the no-load startup phase.The parameters of PMSM are listed as: Specifically, the load torque is expected to have a step and sinusoidal changes from 0 N•m to 0.4 N•m at t = 3 s and 0.2sin(4πt) + 0.2 N•m at t = 6 s, respectively.The reference speed ω ref is 500 rpm.In the comparative simulations, the dq-axis current loops adopt PI controllers with the same control parameters K cp = 20 and K ci = 1500.Control parameters in speed loop are listed in Table I.
Fig. 3(a)-(c) depicts the response curves of speed ω, q-axis current i q , and disturbance estimation error e 2 under four controllers, respectively.To further assess the control performances of four controllers, the performance indices are considered as follows: overshoot (OS), settling time (ST), maximum steadystate error (MSE), and root-mean-square error (RMSE) are used for the comparisons at no-load startup phase.At the step load torque phase, speed drop (SD), recovery time (RT), MSE, and RMSE are employed for comparisons.In order to make comparisons during the sinusoidal load torque phase, speed fluctuation (SF) and RMSE are utilized.The results are shown in Table II.
Combining Fig. 3(a) and Table II, it can obtain that speed responses under the two types of methods with and without DETM are similar.The dynamic performances under P+GPIO (DETM) and P+GPIOare similar to those of P+ESO (DETM) and P+ESO controllers at the no-load startup phase.When step load torque is applied, the responses of P+GPIO (DETM) and P+GPIO have lower SD and shorter RT than that of P+ESO (DETM) and P+ESO.The speed tracking performances of the four methods after returning to the steady state are similar.When sinusoidal load torque is applied, P+GPIO (DETM) and P+GPIO generate less SF and RMSE than the P+ESO (DETM) and P+ESO.According to Fig. 3(c), it is concluded that the disturbance rejection performance of P+GPIO (DETM) and P+GPIO are better than that of P+ESO (DETM) and P+ESO.
Fig. 4 gives the response curves of event number under the four control methods, respectively.As shown in Fig. 4, it is worth mentioning that the event-triggered numbers of control input under P+GPIO (DETM) and P+ESO (DETM) are less than P+GPIO and P+ESO under time-triggered mechanism, while satisfying the speed regulation performance and robustness in the whole process.The results of ξ(qT )|e ω (qT )| + δ, ||η(qT )||, and the interevent intervals under P+GPIO (DETM) and P+ESO (DETM) controllers are displayed in Fig. 5(a)-(h).From Fig. 5(a)-(h), one can see ξ(qT ) under two controllers are dynamically changing, which conforms to (10).The inequality ||η(qT )|| ≤ ξ(qT )|e ω (qT )| + δ satisfies in every time interval.In addition, the interevent intervals can correspond to the error under P+GPIO (DETM) and P+ESO (DETM) controllers.

V. EXPERIMENT RESULTS
In order to further verify the advantages of the proposed control method in speed tracking performance and computation resources saving, experimental results are given in this section.As shown in Fig. 6, the experimental setup (Teknic M-2310P-LN-04 K) is connected with host PC by ETL-PCI-662, and the sampling time is 0.001 s.The control parameters in the speed loop are shown in Table III.
In the experimental tests, ω ref = 500 rpm at t = 0 s.The load torque is expected to have a step and sinusoidal changes from 0 to 0.4 N•m at t = 10 s and 0.2sin(π/2 t) + 0.2 N•m at t = 20 s, respectively.Fig. 7(a)-(f) presents the curves of ω and i q under startup, step load torque and sinusoidal load torque, respectively.The same performance index results as the comparisons are shown in Table IV.
From Fig. 7 and Table IV, it can observer that the four controllers achieve comparable dynamic performance at no-load startup phase which indicates that the system properties under the four control strategies are quite similar.However, the dynamic performances under the P+GPIO (DETM) and P+GPIO control methods are significantly better those under P+ESO Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As a result, the advantages of the proposed control method are confirmed by the experiments, which can simultaneously reduce the communication numbers of control input, while maintaining satisfactory speed tracking performance.

VI. CONCLUSION
This article has developed the DETM-based DUEA strategy to handle the speed regulation problem of PMSM in face of lumped disturbances, which reduces the limited communication and saves computation resources while ensuring desirable disturbance rejection ability.The proposed DETM takes form of discrete-time, and is simpler to implement in digital operation.In addition, the proposed DUEA method is designed based on GPIO, which has advantages on attenuating the time-varying disturbances.To ensure the stability of the closed-loop control systems, comprehensive stability analysis have been given.Finally, the outcomes of the simulation and experiment have demonstrated the viability of the proposed control approach.

Fig. 1 .
Fig. 1.Schematic diagram of the speed regulation of networked PMSM under the proposed control method.

Fig. 2 .
Fig. 2. Detail diagram of the proposed control method.

Fig. 7 .
Fig. 7. Experimental results.(a) Speed ω at startup.(b) Q-axis current i q at startup.(c) Speed ω at step load torque.(d) Q-axis current i q at step load torque.(e) Speed ω at sine load torque.(f) Q-axis current i q at sine load torque.

Fig. 8 .
Fig. 8. Responses curves of disturbance estimation d in the experiments.

Fig. 9 .
Fig. 9. Responses curves of event number in the experiments.