Event-Triggered Active Disturbance Rejection Control of DC Torque Motors

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1 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME Event-Triggered Active Disturbance Rejection Control of DC Torque Motors Dawei Shi, Jian Xue, Lixun Zhao, Junzheng Wang, and Yuan Huang Abstract Physical servo systems are affected by disturbances, uncertainties and resource restrictions. In this work, an eventtriggered active disturbance rejection control approach is proposed to achieve position tracking of DC torque motors. An event-based sampler together with an extended state observer is introduced, which allows the joint observation of the system state and the total disturbance induced by model uncertainty and intermittent sampling. Based on the observation results, closed-loop control is achieved with guaranteed stable tracking performance. To quantify the effect of event-based sampling, a quantitative relationship between the asymptotic upper bound of tracking error and the parameters in the event-based sampler is developed. The control strategy is applied to a DC torque motor system, and comparative experimental results indicate that for different reference signals, the proposed event-based control strategy can achieve satisfactory tracking performance with reduced sampling cost. Index Terms Event-Triggered Control, Active Disturbance Rejection Control, Extended State Observer, DC Torque Motor. I. INTRODUCTION With technological developments of power electronics and advanced control strategies, servo systems occupy an increasingly important role in industrial manufacturing processes, and control problems for servo systems have been extensively investigated in the recent years [] [6]. Among different kinds of servo systems, motion control of electrical motor systems has received a lot of attention in academia and industry. A number of control strategies have been investigated for high performance in motor systems, e.g., PID control [7], fuzzy logic control (FLC) [8], active disturbance rejection control (ADRC) [9], and adaptive robust control (ARC) []. Terzic and Jadric [] introduced a method to estimate the speed and rotor position for a brushless DC motor (BLDCM) based on the application of the extended Kalman filter. Lim and Krishnan [] introduced a novel current controller for a linear switched reluctance motor based on an extended state observer (ESO) and a nonlinear proportional controller. Chen et al. [3] studied precision motion control of linear motors combined with parameter variations and disturbances, and proposed an ARC algorithm with simultaneous compensation for all significant nonlinearities. A high-accuracy motion control approach for motors was introduced in [], and a highaccuracy tracking control approach based on a output feedback This work was supported in part by the National Natural Science Foundation of China under Grant J. Xue and L. Zhao contribute equally to this work. (Corresponding author: Dawei Shi.) D. Shi, J. Xue, L. Zhao, J. Wang, and Y. Huang are with Key Laboratory of Intelligent Control and Decision of Complex Systems; School of Automation, Beijing Institute of Technology, Beijing, 8, P.R. China ( dawei.shi@outlook.com). robust controller with an ESO for motors was introduced in [4]. In [5], Gubara et al. considered PID and FLC in motor speed control, and concluded that FLC provided better performance characteristics than that of PID. On the other hand, similar to many other engineering applications, it is sometimes unnecessary to update the control signal when a motor system is working desirably in the absence of significant events (e.g., abrupt reference change, occurrence of disturbance). Moreover, in application scenarios with limited communication resources (e.g., when motors are controlled via wireless communication channels), transmitting the signal in real time may cause a huge amount of cost. To avoid unnecessary communication resource consumption under the premise of satisfactory control performances, an approach named event-triggered control has been recently introduced [6] and investigated for many control problems [7] [4]; in such a control scheme, signals are only transmitted at certain instants determined by a pre-designed event-triggering condition. In [7], Tabuada investigated a simple event-triggered scheduler based on a certain feedback paradigm. Heemels and Donkers [8] proposed advanced event-triggering mechanisms to reduce communication in certain channels. Meng and Chen [9] applied the event based control approach to multi-agent networks. In [], the event based triggering approach is applied to economic model predictive control. Li and Shi [] and He and Shi [] applied the event-triggered approach to model predictive control (MPC) for continuous-time nonlinear systems combined with disturbances. In [3], an eventtriggered state estimation problem for complex networks with time delays was investigated. In this work, an event-triggered position tracking problem for a DC torque motor system is considered. To deal with different kinds of disturbances and uncertainties in the system together with the additional disturbance induced by the intermittent event-triggering sampling scheme, an active disturbance rejection control (ADRC) approach is employed in controller design. ADRC has special advantage of dealing with internal uncertainties and external disturbances [5] [9], and has been successfully applied in a number of control systems [3] [34]. In [3], Liu et al. applied the ADRC approach to a BLDCM system based on back-propagation neural network. Sira-Ramírez et al. [33] presented an ADRC scheme to achieve angular velocity trajectory tracking on a permanent magnet synchronous motor combined with uncertainties and disturbances. In [34], Du et al. designed an active disturbance rejection controller, and applied it to the sensorless control of internal permanent-magnet synchronous motors. The key problems considered in this work, however, are ) to figure (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME Fig. : Schematic of the event-triggered ADRC scheme. out how the event-triggering condition can be designed and how the ADRC scheme can be adapted to achieve closedloop position tracking and ) to implement the developed event-triggered system and experimentally evaluate whether the actual experimental performance is consistent with the theoretical analysis. The main contributions in this work are summarized as follows: ) In combination with an easy-to-implement eventtriggering mechanism, an event-triggered ADRC (ET- ADRC) scheme is proposed. Since the states of the DC motor cannot be obtained directly, the event-triggering mechanism only needs the output of the system. Based on this condition, the output of the DC motor system is controlled to track a desired trajectory. ) The effects that the event-triggered control mechanism brings to the system are investigated. The observation error is proved to be asymptotic bounded, and the asymptotic boundedness of the tracking error is guaranteed. The upper bounds of the observation error and the tracking error in steady-state are provided and their relationship with the parameters in the event-triggering condition is developed. 3) The experimental performance of the developed results are evaluated on a DC torque motor platform. For different reference signals and different parameter choices in the event-triggering condition, satisfactory tracking performance (in comparison with time-triggered ADRC) is achieved at reduced average sampling rate. The remainder of the paper is organized as follows: In Section II, the mathematical model of the DC torque motor and the problem formulation are introduced. The performance of the event-triggered ADRC scheme is analyzed in Section III. Comparative experimental results are presented to evaluate the actual performance of the system in Section IV. Some concluding remarks are summarized in Section V. II. PROBLEM FORMULATION AND DYNAMIC MODELS In this section, the mathematical model of the DC motor and the ET-ADRC scheme are introduced, and the tracking problem is formulated. To aid the description, the explicit form of the event-triggering condition is deferred to the next section. Consider the event-triggered motor control scheme in Fig.. The dynamics of DC torque motor can be described as Mÿ = F m (u) Bẏ + F friction (ẏ) + F cogging (y) + F dis, () where y represents the displacement, u is the control signal, M is the inertia, B is the viscous friction coefficient, F m (u) is the electromagnetic driving force of the linear motor, F friction (ẏ) represents the Coulomb friction, F cogging (y) represents the position dependent cogging force, and F dis represents the lumped modeling errors and external disturbances. According to [35], the Coulomb friction compensation can be expressed as F friction (ẏ) = A f S f (ẏ), () where A f represents the Coulomb friction coefficient and S f (ẏ) is the approximation of the sign function sgn(ẏ). In practice, we can define S f (ẏ) as S f (ẏ) = arctan(βẏ) (3) for some sufficiently large constant β. Considering the periodical properties of F cogging (y) with a known constant P >, F cogging (y) can be approximated as F cogging (y) = n i= ( Si sin( iπ P y) + C i cos( iπ P y)) (4) through Fourier expansion for a sufficiently large n, where S i and C i are some constants. According to [36], we approximate F m (u) by F m (u) = A u (5) for some A >. Thus, combining the results in ()-(5), if we define θ := y, θ := ẏ, (6) then the dynamics of DC motor in () can be expressed as θ = θ, θ = B M θ + n ( Si iπ i= M sin( P θ ) + Ci iπ M cos( P θ ) ) A f M arctan(βθ ) + F dis M + A M u. (7) For notional brevity, we define γ = B M, γ 4 = A f M, γ 5 = M, ρ = A π M,ω = P, and write γ,i = Si M, γ 3,i = Ci M for i {,..., n}, then the dynamics of the DC motor system can be rewritten as θ (t) =θ (t), θ (t) =f(θ(t)) + w(t) + ρu(t), (8) where system function f(θ) and disturbance w(t) are defined as f(θ) = γ θ + n i= (γ,i sin(iωθ ) + γ 3,i cos(iωθ )) γ 4 arctan(βθ ), w(t) = γ 5 F dis. Through a simple computation we have f θ = ω n i= iγ 3,i sin(iωθ ) + ω n i= iγ,i cos(iωθ ), f θ = γ γ4β +β θ If we choose b as. (9) b = max{ω n i= ( iγ + iγ 3 ), γ + γ 4 β}, i {, }, then the system function f(θ) satisfies f θ j b, j {, }. () (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

3 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME 3 As the parameters of the system function f(θ) cannot be exactly known, we treat f(θ) as the uncertainties of the system. Equations (9)-() guarantee that the uncertainties of the motor system satisfy the boundedness assumption, that is, f θ j is bounded. Moreover, considering the saturation phenomenon in practical applications, there exists some constants C w > and C θ > such that w + ẇ C w, θ C θ. () In this work, we consider the scenario that the reference trajectory θ (t) of the DC torque motor satisfies θ (t) = θ (t); θ (t) = r(t), () where a, a, a 3 are some constants such that the matrix J J := a a (8) a 3 is Hurwitz and ϕ(θ) is defined by ϕ(θ) = Define function P : R 3 R as 4, θ < π, 4 sinθ, π θ π, (9) 4, θ > π. P (z) = P z, z + z where the matrix P > satisfies ϕ(s)ds, () where ṙ(t) C r holds for some C r >, and r(t) C r for some C r >. In order to investigate the tracking performance of the DC motor system in (8), we define the tracking error as x(t) := θ(t) θ (t), (3) where x := [x, x ] T, θ := [θ, θ ] T, θ := [θ, θ ] T, then the dynamics of the tracking error x i has the following form ẋ (t) = x (t); ẋ (t) = f(θ(t)) r(t) + w(t) + ρu(t). (4) Note that our goal in this work is to guarantee the boundedness of the tracking error. However, the external disturbances together with internal disturbances may deteriorate the tracking performance of the system. To estimate and compensate these disturbances, we define an extended state according to the standard ESO formulation [5], [37] as x 3 (t) := f(θ(t)) r(t) + w(t), (5) where x 3 represents the uncertainties and disturbances of the system, and introduce an event-triggered extended state observer of the following form ( ) ˆx (t) = ˆx (t) + εg ξ(t) ˆx(t) ε, ˆx (t ) = ˆx, ( ) ˆx (t) = ˆx 3 (t) + g ξ(t) ˆx(t) ε + ρu(t), ˆx (t ) = ˆx, ( ) ˆx 3 (t) = ε g ξ(t) ˆx(t) 3 ε, ˆx 3 (t ) = ˆx 3, (6) where ˆx := [ˆx, ˆx, ˆx 3 ] T R 3 is the observer state, ε is a high-gain parameter, ˆx(t ) := [ˆx, ˆx, ˆx 3 ] T R 3 is the initial value and ξ(t) is the output measurement which is given by { x (t ξ(t) = k ), if =, (7) x (t), otherwise, where is the triggering condition which will be presented and t k is the transmission instant determined by the eventtriggering condition. Only when = will the value of ξ(t) be updated. In this observer scheme, g i ( ) are chosen as g (z ) = a z + ϕ(z ), g (z ) = a z, g 3 (z ) = a 3 z, P J + J T P = I. () By choosing the functions g, g, g 3 and ϕ in this form, there exists a nonnegative-definite function Θ : R 3 R such that λ z P (z) λ z, () λ 3 z Θ(z) λ 4 z, (3) j= P P z z j (z j+ g j (z )) P x 3 g 3 (z ) Θ(z), (4) τ z, (5) where λ, λ, λ 3, λ 4 and τ are some positive constants. The choice of ϕ(θ) is not unique as long as it satisfies the requirement above. The form of ϕ(θ) defined in (8) basically acts a smooth approximation of the saturation nonlinearity. In this work, we design the control signal u as u = k ρ ˆx k ρ ˆx ρ ˆx 3, (6) where k j are some constants such that the matrix H [ ] H := k k (7) is Hurwitz. Obviously, there exists a matrix Q > such that then we define Q(z) as and we have QH + H T Q = I. (8) Q(z) = z T Qz, (9) ς z Q(z) ς z, Q z τ z, (3) for some positive constants ς, ς and τ (for instance, we can choose ς = λ min ( Q), ς = λ max ( Q) and τ = λ max ( Q)). To describe the performance of the observer proposed in (6), we define x j and e j as x j := x j ˆx j, e j := xj ε, j {,, 3}. (3) 3 j Then we define the sampling error σ(t) as σ(t) := x(t k) x (t) ε, (3) (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

4 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME 4 where t [t k, t k+ ), and for notional brevity, we define α j (e, σ) as α j (e, σ) := g j (e + σ) g j (e ), j {,, 3}. (33) In this work, we mainly investigate the following problems: ) Considering the event-triggered ESO and the DC motor, can we propose an event-triggering condition to guarantee the boundedness of the observation error? ) Based on the proposed event-triggering condition, can we control the system to track a desired trajectory, and guarantee the boundedness of the tracking error? III. MAIN RESULTS In this section, the problems stated in the previous section are investigated through theoretic analysis. An event-triggering condition that guarantees the performance of state observation and closed-loop control is proposed. From (6), the control signal u in this work can be rewritten as u = k ρ x k ρ x ρ ˆx 3 + k ρ (x ˆx ) + k ρ (x ˆx ) = j= k j ρ x j + j= For notational brevity, we define K as k j ρ ε3 j e j ρ ˆx 3. (34) K = max{k j, k j ε 3 j, }, j {, }. (35) Then we present the following theorem. Theorem : Consider the closed-loop system in (8), the ESO in (6) and the control signal in (6). For a given Ψ > and any initial values of x and ˆx, there exist ε > and an eventtriggering condition { 3, if = j= α j(e, σ) Ψε, (36), otherwise, such that for any ε (, ε ), it holds that lim sup x j ˆx j ε 3 j τ ε 3 λ (F t γ + Ψ), j {,, 3}, (37) lim sup x t τ ε 3 ς γ (F + Ψ). (38) Proof: For the closed-loop system proposed in (8), (6) and (6), considering the definition of the observation error in (3), we have ( ) x j (t) = x j+ (t) ε j g ξ(t) ˆx(t) j ε j {, } ( ) = x j+ (t) ε j g x(t) ˆx (t) j ε + ξ(t) x(t) ε = x j+ (t) ε j g j (e (t)) ε j α j (e (t), σ(t)), ( ) x 3 (t) = d (f(θ(t)) r(t) + w(t)) ε g ξ(t) ˆx(t) 3 ε = d g3(e(t)) (f(θ(t)) r(t) + w(t)) ε α3(e(t),σ(t)) ε. Then, through a simple computation, we obtain the dynamic of e i in the following form ė j (t) = ε (e j+(t) g j (e (t)) α j (e (t), σ(t))), j {, }, ė 3 (t) = ẋ 3 (t) ε g 3(e (t)) ε α 3(e (t), σ(t)). (39) Define a nonnegative-definite function V (x, e) as V (x, e) = Q(x) + P (e), (4) where Q(x) and P (e) are provided in () and (9), respectively. We observe that dv = dq + dp = Q x x + Q x (x 3 + ρu) + 3 j= P ė j. (4) From the dynamic of e i in (39), we have dv = Q x x + Q x (x 3 j= k jx j + j= k jε 3 j e j ˆx 3 ) + j= P [ ε (e j+ g j (e ) α j (e, σ))] + P e 3 [ẋ3 ε g 3(e ) ε α 3(e, σ) ] Q x x Q x j= k jx j + Q x ( j= k jε 3 j e j + e 3 ) [ ] + ε j= P (e j+ g j (e )) P e 3 g 3 (e ) + P e 3 ẋ3 3 ε j= P α j (e, σ). (4) According to the definition of Q( ) in (9), we observe Q x x Q x j= k jx j = x T QHx = x T ( QH + H T Q)x = x. (43) Then, considering the function P ( ) we have ε [ j= P (e j+ g j (e )) P e 3 g 3 (e )] λ3 ε Θ(e) ε e. (44) According to inequality (3), we obtain that Q x ( j= k jε 3 j e j + e 3 ) τ x j= k jε 3 j e j + e 3 [ = ε τ ε 3 x ε 3 ( ] j= k jε 3 j e j + e 3 ) 3 ε 3 τ x + τ P e ẋ3 3 τ e ẋ 3 ε ( j= k j ε 3 j + ε 3 ) e, (45) = ε τ ε 3 e ε 3 ẋ 3 ε τ (ε 3 e + ε 4 3 ẋ 3 ). (46) Considering the control signal u provided in (6), we have ẋ 3 = f(θ) ṙ + ẇ = f θ θ + ẇ ṙ + f θ (e 3 j= k jx j + j= k jε 3 j e j + r) b C θ + b (K e + K x + e + C r ) + C w +C r (A x + B e + C), (47) where A, B and C are defined as A := b K, B := b (K + ), C := b (C θ + C r )+C r + C w. Then, define F := C + AC + BC, F := A + AB + AC, F := B + AB + BC, such that ẋ 3 F + F x + F e. (48) For the inequalities (46)-(48), we have P e ẋ3 3 τ ε [ε 3 e + ε 4 3 (F + F x + F e )] τε 3 e + τε 3 F + τε 3 F x + τε 3 F e ( τε 3 + τε 3 F ) e + τε 3 F x + τε 3 F. (49) (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

5 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME According to P z τ z and the event-triggering condition (36), we obtain that 3 ε j= P σ j (e, σ) 3 ε j= P σ j (e, σ) ε τ e Ψε ( ε 3 e + ε 3 )Ψτ. (5) Combining the results, and according to (4)-(45), (49) and (5), we have dv x λ3 ε e + 3τ ε 3 x + Ψτ ( ε 3 e + ε 3 ) + τε 3 F + τε 3 F x + ( τε 3 + τε 3 F ) e + τ ε ( j= k j ε 3 j + ε 3 ) e ( 3τ ε 3 τ ε 3 F ) x + τε 3 F + Ψτε 3 ε [λ 3 τ ( j= k j ε 3 j + ε 3 ) ( τ ε 3 Then, write + τ ε 4 3 F ) ε 3 Ψτ ] e. (5) and define ε as := λ 3 τ ( j= k j ε 3 j + ε 3 ) (Ψτ ε 3 + τ ε 3 + τ ε 4 3 F ), := 3τ ε 3 τ ε 3 F, (5) ε = max{ε R (ε), (ε), and ε > }. (53) Note that (ε) and (ε) are monotone decreasing functions of ε, that lim ε (ε) >, lim ε (ε) >, and that (ε) <, (ε) < when ε becomes sufficiently large. Thus ε is well-defined. For ε (, ε ), from inequality (5), we obtain dv x ε e + τ ε 3 F + Ψτ ε 3 ς Q(x) ελ P (e) + τ ε 3 F + Ψτ ε 3. (54) Then, we further define γ, δ(t, t ) as Therefore, γ = min{ ς, ελ }, δ(t, t ) = exp( γ(t t )). dv (x,e) dv (x,e) Then, we obtain that V (x, e) can be rewritten as γv (x, e) + τ ε 3 F + Ψτ ε 3. (55) V (x(t ), e(t ))δ(t, t ) + ( τε 3 F + Ψτε 3 ) t t δ(t, ϑ)dϑ V (x(t ), e(t ))δ(t, t ) + ( τε 3 F γ + Ψτε 3 γ )( δ(t, t )). When t, δ(t, t ), and we have hence V (x, e) τ γ ε 3 F + Ψτε 3 γ, (56) P (e) + Q(x) τ γ ε 3 F + Ψτε 3 γ. (57) From (3) and (57), we obtain that when t ς x + λ e V τ γ ε 3 F + Ψτε 3 γ, (58) then we have and obtain the conclusion that x τε 3 ς γ F + Ψτε 3 ς γ, (59) lim sup x t Considering (58), we obtain that when t τ ε 3 ς γ (F + Ψ). (6) λ e τ γ ε 3 F + Ψτε 3 γ. (6) Hence, for observation error e, it is guaranteed that lim sup e t τ ε 3 λ γ (F + Ψ), (6) and from the definition of e in (3), we obtain for j {,, 3}, lim sup x j ˆx j ε 3 j τ ε 3 λ (F t γ + Ψ). (63) This completes the proof. Remark : For the torque motor system in (8), the ESO in (6) and the control signal in (6), the event-triggering mechanism proposed in (36) guarantees the asymptotic boundedness of the observation error and the tracking error. Specifically, equation (6) guarantees the boundedness of the steady-state tracking error, and equation (63) guarantees the boundedness of the steady-state observation error. The parameterization of the functions g i, i =,, 3 and ϕ(θ) chosen in Section II satisfy the requirements in ()-(4), which helps ensure the input-to-state stability of the ESO. The function ϕ(θ) is the nonlinear part of the proposed ESO. If we choose ϕ(θ) =, the ESO reduces to a linear one. With the proposed functions, we can infer from (63) that the estimated state variables will converge to the real physical states at steady state for properly chosen values of ε and Ψ. The proposed event-triggering condition is based on the value of the sampling error σ(t), and presents an upper bound of σ(t). Once σ(t) reaches the predefined bound, the value of ξ(t) will be updated. From the event-triggering condition, the system performance can be adjusted by changing the value of the parameter Ψ; this point is further analytically characterized in (38), where quantitative relationships of the observation and tracking performance with the tuning parameter Ψ in the event-triggering condition are provided. Remark : By assumption, the extended state x 3 (t) in equation (5) is not directly known; however, it evolves in a unique (but unknown) way determined by the realization of the external disturbance signal and the values of the unknown parameters. From equations (4)-(5) and the definition of ξ(t) in (7), the information of x 3 (t) is reflected in the event-triggered output measurement ξ(t); through the extended state observer in (6), ˆx 3 (t) tries to learn the determined but unknown behavior of x 3 (t) by exploring this information. In other words, taking advantage of the knowledge of ξ(t), the dynamic behavior of ˆx 3 (t) mimics that of x 3 (t). In particular, equation (37) of Theorem indicates that the difference between ˆx 3 (t) and x 3 (t) can be sufficiently small as t for ε (, ε ) (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

6 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME 6 Fig. : The motor load test bench. IV. EXPERIMENTAL PERFORMANCE EVALUATION In this section, the introduced event-triggered control scheme is evaluated on a DC torque motor platform. The experimental platform mainly consists of a motor load test bench and a embedded measurement and control test bench. The motor load test bench contains a permanent-magnet DC torque motor (3LYX3), a magnetic brake, a speed reducer, an inertial loading mechanism, and a speed sensor (JN338). By regulating the input current, the magnetic brake can simulate the adjustable sliding friction load. The embedded measurement and control test bench is regarded as a controller, since it can generate signals to control the motor and process the received signals. In this section, we choose two different kinds of reference trajectory inputs, which are squarewave input and multitone sinusoid input, to carry out the experiments. Apart from the ET-ADRC controller, a time-triggered continuoustime ADRC (CT-ADRC) controller is also implemented for comparison purpose. In these experiments, the ADRC controllers are implemented by difference approximation with the sampling time t s =.5s. In order to evaluate the tracking performance of the system, we define the tracking error (E T ) and the average sampling time (T A ) as follows: E T = T T θ (t) θ (t), (64) { 5, for CT-ADRC schemes, T A = (65) for ET-ADRC schemes, T N s, where we recall θ (t) represents the reference trajectory, θ (t) is the output of the controlled system, T represents the experimental time in milliseconds, N s is the total triggering counts in one experiment. A. Experimental Results with Squarewave Type Input In this subsection, the tracking performance of the system with squarewave type reference input is investigated. For the ESO in (6), g i is given as g (z ) = 3z + ϕ(z ), g (z ) = 3z, g 3 (z ) = z, (66) and the high-gain parameter ε =.. The control law ρu (t) is designed as ρu (t) = 4ˆx ˆx ˆx 3, (67) and the reference trajectory θ (t) is a squarewave signal processed by a low pass filter, so that r(t) and ṙ(t) are guaranteed to be bounded. Since the output of the system is controlled to track the processed signal, adding a low pass filter will not affect the accuracy of the estimated state variables. In addition, low pass filters are often adopted to process signals in many practical applications [38], since it can make the response smoother and reduce overshoot. Consider the closed-loop system and the event-triggering condition proposed in (36), we choose different Ψ so that we can obtain experimental results with different average communication rates. For comparison purpose, two groups of experiments are carried out. One group shows the noload performance of the system, and the other group shows the performance of the system with the load of.7nm. The tracking performance of the system without triggering condition (namely, CT-ADRC) is also investigated for each group. These experimental results are shown in Fig. 3 Fig. 6. The tracking error (E T ) and the average sampling time (T A ) are summarized in Table I. 6.67π.33π.33π.33π.33π 6.67π.33π.33π.33π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 3: No-load tracking performance with Ψ = π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 4: Tracking performance with the load of.7nm and Ψ = (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

7 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME π.33π.33π.33π.33π 6.67π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 5: No-load tracking performance with Ψ =.4. is preferred in reducing resource consumption, it may create negative effects on control performance. TABLE I: Performance for squarewave-type input Experimental schemes Average sampling time Tracking error Time-triggered Event-triggered Ψ =.35 Event-triggered Ψ =.4 No-load 5ms 4.7 Load 5ms 4.4 No-load 69ms 4.67 Load 7.4ms 4.64 No-load 67.8ms Load 66.7ms B. Experimental Results with Multitone Sinusoid Input In this subsection, the tracking performance of the system with multitone sinusoid input is investigated. Consider the ESO in (6) with g i still given in (66) and ε =.. The control law ρu (t) is designed as.33π.33π.33π.33π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 6: Tracking performance with the load of.7nm and Ψ =.4. From the experimental results, we observe that with the event-triggering control mechanism, the system can be controlled to track the desired trajectory with tracking error around 4. Based on the obtained tracking performance, we can infer that the uncertainties x 3 can be properly compensated in the feedback loop. Since the tracking performances of the ET-ADRC schemes are almost the same as the results of the time-triggered schemes, the CT-ADRC curves in the figures cannot be seen very clearly. The curves of x and ˆx partially show that the estimated states can converge to the real physical states. Compared with the performance of CT- ADRC schemes, the event-triggering approach has an obvious decrease in average sampling rate, and the proposed eventtriggering schemes maintain a similar level of performance in terms of tracking error. The tracking performance can be adjusted by changing the average sampling rate, which is implemented by the triggering parameter Ψ. When Ψ is set larger, the average sampling rate will obviously become lower, but the tracking error will be more likely to be bigger, which can be seen in Table I. Although a lower average sampling rate ρu (t) = 4ˆx ˆx ˆx 3. (68) The reference trajectory θ (t) is a multitone sinusoid signal. The experimental results are shown in Fig. 7 Fig.. The tracking error (E T ) and the average sampling time (T A ) of the different experimental schemes are shown in Table II. 6π 6π.67π.67π.67π Angle.rad.67π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 7: No-load tracking performance with Ψ =.35. Similar to the squarewave-type input experimental results, the proposed ET-ADRC scheme has a satisfactory performance in controlling the system tracking the desired trajectory, as the response of the ET-ADRC almost overlaps that of the CT- ADRC. Compared with the squarewave-type input experimental results, multitone sinusoid input experimental results have better tracking performances (lower tracking errors) but higher average sampling rates, as the reference signal keeps changing for this case. The tracking errors of response with load are bigger than that without load, and the effect of the load is more obvious compared with squarewave-type input. From this we can obtain that the event-triggering approach has more advantages when the reference trajectory is given smoother (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

8 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME 8 6π TABLE II: Performance for multitone sinusoid input 6π.67π.67π.67π.67π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 8: Tracking performance with the load of.7nm and Ψ =.35. 6π 6π.67π.67π.67π.67π 6π Angel,rad 6π.67π Angel,rad.67π.67π Angel,rad.67π CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. 9: No-load tracking performance with Ψ = CT-ADRC x CT-ADRC ˆx ET-ADRC x ET-ADRC ˆx Fig. : Tracking performance with the load of.7nm and Ψ =.4. Experimental schemes Average sampling time Tracking error Time-triggered Event-triggered Ψ =.35 Event-triggered Ψ =.4 No-load 5ms.5833 Load 5ms.946 No-load 5.5ms.58 Load 5.3ms.847 No-load 67.8ms.79 Load 69.5ms.466 and steadier. Finally, we note that compared with the CT- ADRC schemes, the event-triggered schemes with two kinds of inputs can both maintain a similar level of performance in terms of tracking error, but the average sampling rate can be obviously reduced. Since the internal interference cannot be measured and ADRC approach does not consider the uncertainty explicitly but takes all uncertainties and disturbances as total disturbances, the internal interference suppression effect cannot be validated directly in this work. However, the obtained tracking results show that the proposed ADRC schemes can achieve a good performance in compensating total disturbances, including internal interferences and external disturbances. Moreover, the CT-ADRC schemes have less external disturbances (in terms of sampling errors induced by event trigger) than ET-ADRC schemes, and therefore the fact that the ET-ADRC schemes maintain a similar level of control performance (in terms of tracking error) to that of CT-ADRC schemes helps us to indirectly validate the effect of the internal interference suppression for ET-ADRC schemes. V. CONCLUSION In this work, an event-triggered ADRC approach is proposed to control a DC torque motor, and the effects of the triggering scheme are investigated. An event-triggering mechanism which only relies on the output of the controlled system is designed. With the proposed ET-ADRC scheme, the observation error of the ESO and the tracking error of the system can be guaranteed to be asymptotic bounded. The actual performance of the ET-ADRC scheme is extensively evaluated through comparative experiments on a DC torque motor platform for different load conditions and reference signals. REFERENCES [] Y. Lin, Y. Shi, and R. Burton, Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system, IEEE/ASME, vol. 8, no., pp., Feb 3. [] W. Sun, H. Gao, and O. Kaynak, Vibration isolation for active suspensions with performance constraints and actuator saturation, IEEE/ASME, vol., no., pp , April 5. [3] L. Zhang, S. Wang, H. R. Karimi, and A. Jasra, Robust finitetime control of switched linear systems and application to a class of servomechanism systems, IEEE/ASME, vol., no. 5, pp , Oct 5. [4] X. Wang, H. Gao, O. Kaynak, and W. Sun, Online deflection estimation of X-axis beam on positioning machine, IEEE/ASME Transactions on Mechatronics, vol., no., pp , Feb (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

9 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME 9 [5] H. An, J. Liu, C. Wang, and L. Wu, Approximate back-stepping fault-tolerant control of the flexible air-breathing hypersonic vehicle, IEEE/ASME, vol., no. 3, pp , June 6. [6] Y. Shi, C. Shen, and B. Buckham, Integrated path planning and tracking control of an AUV: A unified receding horizon optimization approach, IEEE/ASME. [7] S. W. Khubalkar, A. S. Chopade, A. S. Junghare, and M. V. Aware, Design and tuning of fractional order PID controller for speed control of permanent magnet brushless DC motor, in 6 IEEE First International Conference on Control, Measurement and Instrumentation (CMI), Jan 6, pp [8] S. L. Ghalehpardaz and M. 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Contreras-Ordaz, On the control of the permanent magnet synchronous motor: An active disturbance rejection control approach, IEEE Transactions on Control Systems Technology, vol., no. 5, pp , Sept 4. [34] B. Du, S. Wu, S. Han, and S. Cui, Application of linear active disturbance rejection controller for sensorless control of internal permanentmagnet synchronous motor, IEEE Transactions on Industrial Electronics, vol. 63, no. 5, pp , May 6. [35] Z. Chen, B. Yao, and Q. Wang, Adaptive robust precision motion control of linear motors with integrated compensation of nonlinearities and bearing flexible modes, IEEE Transactions on Industrial Informatics, vol. 9, no., pp , May 3. [36] L. Xu and B. Yao, Adaptive robust precision motion control of linear motors with negligible electrical dynamics: theory and experiments, IEEE/ASME, vol. 6, no. 4, pp , Dec. [37] J. Han, The extended state observer of a class of uncertain systems, Control and Decision, 995. [38] N. He, D. Shi, M. Forbes, J. Backström, and T. Chen, Robust tuning for machine-directional predictive control of MIMO paper-making processes, Control Engineering Practice, vol. 55, pp., 6. Dawei Shi received his B.Eng. degree in Electrical Engineering and Automation from the Beijing Institute of Technology in 8. He received his Ph.D. degree in Control Systems from the University of Alberta, Canada, in 4. In December 4, he was appointed as an Associate Professor at the School of Automation, Beijing Institute of Technology, China. In February 7, he joined Harvard John A. Paulson School of Engineering and Applied Sciences USA, as a Postdoctoral Fellow. His research interests include event-based control and estimation, model predictive control and parameter autotuning, and anomaly detection in control systems. Dr. Shi is a reviewer for a number of international journals, including IEEE Transactions on Automatic Control, Automatica, and Systems & Control Letters. In 9, he received the Best Student Paper Award in IEEE International Conference on Automation and Logistics. Jian Xue was born in Shandong, China, in 993. He received the B.Eng. degree from the School of Automation, Beijing Institute of Technology, Beijing, China, in 6. He is currently pursuing the M. Sc. degree in control engineering at the Beijing Institute of Technology. His research interests include event-triggered active disturbance rejection control and high gain control (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

10 This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI.9/TMECH , IEEE/ASME Lixun Zhao was born in Hebei, China, in 99. He received the B.Eng. degree (Hons.) from the College of Automation, Taiyuan Institute of Technology, Taiyuan, China, in 5. He is currently pursuing the M. Sc. degree in Control Engineering at the Beijing Institute of Technology. His research interests include Active Disturbance Rejection Control, servo control system design. Junzheng Wang was born in Shanxi Province, China, in 964. He received the Ph.D. degree from the Beijing Institute of Technology, Beijing, China, in 994. He is the Deputy Director of the Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing Institute of Technology, where he is a Professor and Ph.D. Supervisor. His current research interests include motion control, static and dynamic performance testing of electric and electric hydraulic servo system and dynamic target detection and tracking based on image technology. Prof. Wang is a Senior Member of the Chinese Mechanical Engineering Society and the Chinese Society for Measurement. He was the recipient of the Second Award of the National Scientific and Technological Progress (No. ) in. Yuan Huang was born in Heilongjiang, China, in 99. He received the bachelor s degree from School of Chemical Engineering and Environment, Beijing Institute of Technology, Beijing, China, in 4. He is currently pursuing the Ph.D. degree in automation at Beijing Institute of Technology. His research interests include event-triggered control, nonlinear control and networked system control (c) 7 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.