Current sensorless position-tracking control with angular acceleration error observers for hybrid-type stepping motors | Scientific Reports

Scientific Reports volume 12, Article number: 14993 (2022) Cite this article

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This paper exhibits an advanced observer-based position-tracking controller for hybrid-type stepping motors with consideration of parameter and load uncertainties. As the main contribution, a current sensorless observer-based pole-zero cancellation speed controller is devised for the outer loop position-tracking controller including the convergence rate boosting mechanism. The features of this study are summarized as follows; first, the pole-zero cancellation angular acceleration error observer for the inner loop speed controller, second, the pole-zero cancellation speed control forcing the order of the controlled speed error dynamics to be 1, and, third, the outer loop position control incorporating the first-order target tracking system with its convergence rate booster. The resultant effectiveness is verified on a 10-W stepping motor control system.

The major advantages of stepping motors are the elimination of brushes and the use of a simple position regulation method to count the pulse numbers. These allow various industrial position control applications, such as computerized numerical control (CNC) machines, nuclear reactor control rods, robot arms, and printers1,2,3,4,5,6,7.

Implementing position and speed regulation with stepping motors is possible without any feedback sensors by counting the pulse numbers and adjusting the pulse frequency; however, their precision is predominantly reliant on the teeth numbers. At high-speeds, a stepping motor can experience mechanical problems such as step-out, resonance, and reversal of speed8. To overcome these, a micro-stepping technique with a partial closed-loop structure was proposed that determines the voltage update law statically while assuming the current-loop transfer function as 1. The corresponding closed-loop control precision and performance are dependent on the current controller. A proportional-integral-derivative (PID) control constitutes the current-loop for each phase with a well-tuned feedback gain using Bode and Nyquist techniques8,9,10,11,12. To maintain the desired performance across a large operation range, the resultant feedback gain must be magnified by increasing the motor speed, which is proportional to the back-electromotive force (EMF) disturbance. Parameter-dependent feed-forward compensators deal with this problem by canceling the motor-speed-dependent disturbance, which can achieve significant performance improvement in the high-speed mode13. A novel current-control technique was proposed based on the incorporation of a disturbance observer (DOB) in the sliding-mode control (SMC) to improve the feed-forward terms by reducing the parameter dependence; the proof of closed-loop convergence was presented by the Lyapunov stability theorem14. Another recent study established the elimination of current feedback sensors by combining feedback-linearization (FL) control and a passive observer driven by the position error, which included closed-loop stability analysis15. The level to which parameters depend on these techniques can be lowered by using the novel online parameter identifiers as in16,17,18,19,20,21. Interestingly, the position dynamics were considered, which transformed the entire machine dynamics into a linear-time varying system that could be stabilized by an \(H_2\) controller with a passive observer22.

Unlike the aforementioned approaches (designed in the a-b axis), the introduction of a rotational d-q transformation simplifies the controller design task considerably by removing of the nonlinearities of the model that rely on the motor position23. This method also enables to enlarge the operation range by controlling the negative d-axis current24. Moreover, several recent techniques for three-phase permanent machines, as in25,26,27,28,29,30, can be considered as stabilizing solutions. In FL methods31,32, the position regulation task was transformed into a third-order nonlinear dynamics stabilization problem that required inverse dynamics with perfect machine parameter knowledge. Parameter updating mechanisms have been incorporated into an FL controller to reduce the regulation errors with the closed-loop system order increment33. The energy-shaping approach combining the two techniques of flatness and passivity alleviates the dependence on inverse dynamics and parameters, using the function of open-loop energy34. Sliding mode control (SMC) that forces closed-loop dynamics into the desired surface with the suppression of disturbances from model-plant mismatches is available; here, a discontinuous function with a conservative gain is used in the feedback loop35,36,37. Closed-loop performance improvement can be achieved by incorporating a learning part for the feedback-loop in the back-stepping controller that minimizes the cost-function using a learning algorithm38. Another learning control was suggested with compensation for the q-axis current reference using a repetitive space-learning technique. The tracking performance improvement for sinusoidal references was only observed from numerical simulations39. The recent DOB-based proportional-type positioning technique tried to robustly provide boosting to the closed-loop cut-off frequency in its transient periods, which requires current feedback and could limit the closed-loop performance limitation due to the absence of integral actions40,41.

From this literature survey, the accurate machine model information and current feedback remain as the practical problems to be handled in this study. The machine parameters are decomposed as their nominal and variations to lower the system model dependence for controller design task. The acceleration error observer removes the requirement of the current feedback loop without any machine model information. The contributions of this study are as follows:

The proposed observer estimates the angular acceleration error by adopting the specially structured observer gain to invoke the first-order pole-zero cancellation for the estimation error dynamics, independent from the machine model information,

The inner loop controller robustly stabilizes the speed and estimated acceleration errors in accordance the first-order dynamics, involving the order reduction property obtained from the pole-zero cancellation through the active damping compensation, and

The outer loop adopts the convergence rate booster to reinforce the position-tracking performance by increasing the target system feedback gain proportional to the tracking error.

The prototype control system including a commercial 10-W hybrid-type stepping motor validates the effectiveness of the proposed technique in various scenarios.

The stator of the stepping motors includes a- and b-phase, whose phase current and voltage are denoted as \(i_x\) and \(v_x\), \(x=a,b\), respectively. Applying the orthogonal coordinate transformation with the rotor position \(\theta\) and each phase teeth number \(N_r\), respectively, it holds that9,10,42

\(\forall t \ge 0\), with the state variables : \(\theta\) - rotor position (rad), \(\omega\) - rotor speed (rad/s), and \(i_x\), \(x=d,q\) - current (A) and control variable : \(v_x\), \(x=d,q\) - voltage (V). The output torque \(T_e (i_q)\) (Nm) is proportional to the q-axis current as \(T_e(i_q) \,{:=}\, K_m i_q\) with the torque coefficient \(K_m\). The load torque \(T_L\) (Nm) acts as the mismatched disturbance depending on the operation conditions. The disturbances \(p_x (i_d,i_q,\omega )\) in modeling the back-EMF effect are defined as \(p_d (i_d, i_q, \omega ) \,{:=}\, L N_r \omega i_q\) and \(p_q (i_d, i_q, \omega ) \,{:=}\, -(L N_r i_d + K_m)\omega\) with the stator inductance L (H). The remaining machine parameters are given by: J - inertia moment of the rotor (kgm\(^2\)), B - viscous friction (Nm/rad/s), and R - stator resistance (\(\Omega\)).

To deal with the variations of parameter and load torque, nominal parameters denoted as \((\cdot )_0\) are introduced for the speed and current dynamics (2)–(3) to be expressed as

\(x=d,q\), with \(T_{e,0}(i_q) \,{:=}\, T_e(i_q)|_{K_m = K_{m,0}}\), \(p_{x,0}(i_d,i_q,\omega ) \,{:=}\, p_{x}(i_d,i_q,\omega )|_{L= L_0, K_m = K_{m,0}}\), and lumped disturbances \({\bar{d}}_\omega\), \({\bar{d}}_d\), and \({\bar{d}}_q\). The following section presents the development of the position regulation law with the dynamics (4) and (5).

This study adopts the target position tracking behavior denoting \(\theta ^*\) (different from the actual position measurement \(\theta\)) as the first-order system given by

for the error \(\tilde{\theta }^* \,{:=}\, \theta _{ref}-\theta ^*\), reference trajectory \(\theta _{ref}\), and convergence rate \(\omega _{pc} > 0\) (constant). The system (6) accomplishes the tracking objective; that is \(\lim _{t\rightarrow \infty }\theta ^* = \theta _{ref}\), exponentially for any reference trajectory \(\theta _{ref}\) according to the convergence specification \(\omega _{pc}\). Therefore, the tracking controller is designed to guarantee the control objective: \(\lim _{t\rightarrow \infty } \theta = \theta ^*\), exponentially, which is proved by analyzing the closed-loop dynamics in “Analysis” section.

The time -varying convergence ratio \({\hat{\omega }}_{pc}\) replaces its constant version \(\omega _{pc}\) in (6) as

where \({\hat{\omega }}_{pc}(0) = \omega _{pc}\), \(\tilde{\omega }_{pc} \,{:=}\, \omega _{pc} - {\hat{\omega }}_{pc}\), and two design parameters \(\gamma _{pc} > 0\) and \(\rho _{pc} > 0\) determine the convergence rate booting and restoring natures, respectively. The time-varying nature of \({\hat{\omega }}_{pc}\) by the rule (8) makes the stability issue questionable, which is addressed in “Analysis” section with the boundedness property \({\hat{\omega }}_{pc} \ge \omega _{pc}\), \(\forall t \ge 0\).

The manipulation \(\omega = \omega _{ref} - \tilde{\omega }\) with \(\tilde{\omega } \,{:=}\, \omega _{ref} - \omega\) and error \(\tilde{\theta } \,{:=}\, \theta ^* - \theta\) give the error dynamics as \(\dot{\tilde{\theta }} = - \omega _{ref} + \tilde{\omega } + {\dot{\theta }}^*\), \(\forall t \ge 0\), whose stabilization can be established by the proportional-type stabilizing law:

with the adjustable convergence rate \(\lambda _{pc} > 0\). Note that the compensation term \({\dot{\theta }}^*\) is obtainable from the implementations of (7) and (8) such that \({\dot{\theta }}^* = {\hat{\omega }}_{pc}\tilde{\theta }^* + {\dot{\theta }}_{ref}\). The proposed stabilizing solution (9) results in the controlled position dynamics:

through the substitution of (9) to the open-loop error dynamics \(\dot{\tilde{\theta }} = - \omega _{ref} + \tilde{\omega } + {\dot{\theta }}^*\), which is used in “Analysis” section to analyze the whole system stability and convergence properties considering all control dynamics in “Outer loop” and “Inner loop speed control” sections.

This section presents the stabilizing solution for the second-order speed error dynamics given by

with the coefficient \(c_{\omega ,0} \,{:=}\, \frac{J_0 L_0}{K_{m,0}}\) (known) and lumped disturbance \(d_\omega \,{:=}\, R_0 i_q + (L_0 N_r i_d + K_{m,0})\omega - {\bar{d}}_q + \frac{B_0 L_0}{K_{m,0}}{\dot{\omega }} - \frac{L_0}{K_{m,0}}\dot{{\bar{d}}}_\omega + \frac{J_0 L_0}{K_{m,0}}\ddot{\omega }_{ref}\), which is obtained using the open-loop speed and current dynamics (4)–(5). The stabilization of the open-loop dynamics (11) requires angular acceleration error (\({\tilde{a}} \,{:=}\, {\dot{\omega }}_{ref} - {\dot{\omega }}\)) feedback; however, this is not available online without the direct differentiation associated with high-frequency noise magnification. Therefore, an angular acceleration error observer is proposed without requiring any plant true parameter values.

It follows from the definition \({\tilde{a}} \,{:=}\, {\dot{\omega }}_{ref} - {\dot{\omega }}\) that \(\dot{\tilde{\omega }} = {\tilde{a}}\), \(\forall t \ge 0\), where the uncertain acceleration error \({\tilde{a}}\) is decomposed as its DC (\({\tilde{a}}_0\)) and AC (\(\Delta {\tilde{a}}\)) components such that \({\tilde{a}} = {\tilde{a}}_0 + \Delta {\tilde{a}}\). This representation yields the open-loop system in the chain form, independent from the machine model (2)–(3):

where \(w \,{:=}\, \Delta \dot{{\tilde{a}}}\) and \(|w| \le w_{max}\), \(\forall t \ge 0\), which corresponds to the genuine idea of this work to solve the model dependence problem in the observer design task. Defining the observer errors \(e_{\tilde{\omega }} \,{:=}\, \tilde{\omega } - \tilde{\omega }_{obs}\) and \(e_{{\tilde{a}}} \,{:=}\, {\tilde{a}} - {\tilde{a}}_{obs}\) with their estimates \(\tilde{\omega }_{obs}\) and \({\tilde{a}}_{obs}\), an acceleration error observer is proposed as

with observer gains \(l_{obs,d} > 0\) (for disturbance attenuation level) and \(l_{obs,c} > 0\) (for estimation error convergence rate), whose pole-zero cancellation property results in the exponential convergence property \(\lim _{t\rightarrow \infty }{\tilde{a}}_{obs} = {\tilde{a}}\) according to the first-order dynamics \({\dot{e}}_{{\tilde{a}}} = - l_{obs,c} e_{{\tilde{a}}}\) with a sufficient large \(l_{obs,d} > 0\). See “Analysis” section for details specifying the admissible range for \(l_{obs,d}\).

A proposed stabilizing solution for the second-order open-loop dynamics of (11) is given by

with the two design factors \(k_d > 0\) and \(\lambda _{sc} > 0\). The disturbance estimate \({\hat{d}}_\omega\) comes from the observer-based DOB:

\(\forall t \ge 0\), with design factor \(l > 0\). Figure 1 shows the controller structure.

Controller structure.

Lemma 1 proves that the time-varying subsystem (7) is stable by augmenting the convergence rate boosting system (8).

The convergence rate booster (7) and (8) guarantees the stability and exponential convergence \(\lim _{t\rightarrow \infty }\theta ^* = \theta _{ref}\).

The equivalent form of (7) given by

and the update rule (8) turn the Lyapunov function candidate

with \({\textbf {x}}_m \,{:=}\, \left[ \begin{array}{cc} \tilde{\theta }^*&\tilde{\omega }_{pc} \end{array}\right] ^T\) and \({\textbf {P}}_m \,{:=}\, \mathrm{diag}\{1, \frac{1}{\gamma _{pc}}\}\) (\(\gamma _{pc} > 0\) for the update rule (8)) into

with \({\textbf {Q}}_m \,{:=}\, \mathrm{diag}\{ \omega _{pc}, \rho _{pc} \}\) (\({\hat{\omega }}_{pc}(0) = \omega _{pc} > 0\) and \(\gamma _{pc} > 0\) for the update rule (8)) and \(\alpha _m \,{:=}\, \frac{2\lambda _{min}({\textbf {Q}}_m)}{\lambda _{max}({\textbf {P}}_m)}\) (\(\lambda _{min}((\cdot ))\) and \(\lambda _{max}((\cdot ))\) representing the minimum and maximum eigenvalues of the square matrix \((\cdot )\), respectively), which completes the proof. \(\square\)

Lemma 2 shows that the subsystem comprising (7) and (8) increases the convergence ratio \({\hat{\omega }}_{pc}\) from its initial value \({\hat{\omega }}_{pc}(0) = \omega _{pc} (>0)\), which provides the rationale for a better tracking behavior than the original tracking system (6).

The convergence rate booster (7) and (8) achieves the lower bound on its initial value \(\omega _{pc}\). i.e.,

Consider the another form of (8) given by

whose solution obtained by the both side integration above has a lower bound as

which completes the proof. \(\square\)

Lemma 3 analyzes the observer error behavior used for the closed-loop convergence analysis.

The acceleration observer driven by (14) and (15) guarantees the exponential convergence

where the target trajectory \(e^*_{\tilde{\omega }}\) satisfies

for a given \(l_{obs,c} > 0\) and a sufficient large \(l_{obs,d} > 0\) such that \(\frac{2w_{max}}{l_{obs,d}}\approx 0\).

Consider the vector form of observer errors defined as \({\textbf {e}}_{obs} \,{:=}\, \left[ \begin{array}{cc} e_{\tilde{\omega }}&e_{{\tilde{a}}}\end{array}\right] ^T\), which results in the state-space representation for the observer error dynamics using the open-loop system (12)–(13) and the proposed observer (14)–(15) as

where \(r \,{:=}\, 0\), \(w = \Delta \dot{{\tilde{a}}}\), \({\textbf {A}}_{obs} \,{:=}\, \left[ \begin{array}{cc} -(l_{obs,d} + l_{obs,c}) &{} 1 \\ - l_{obs,d} l_{obs,c} &{} 0 \end{array}\right]\), \({\textbf {b}}_1 \,{:=}\, \left[ \begin{array}{cc} l_{obs,c}&l_{obs,d}l_{obs,c} \end{array}\right] ^T\), and \({\textbf {b}}_2 \,{:=}\, \left[ \begin{array}{cc} 0&1 \end{array}\right] ^T\). This state-space representation and the definition of the observer output with respect to \({\textbf {c}}_{obs} \,{:=}\, \left[ \begin{array}{cc} 1&0 \end{array}\right]\) as \(y_{obs} \,{:=}\, {\textbf {c}}_{obs} {\textbf {e}}_{obs} (=e_{\tilde{\omega }})\), \(\forall t \ge 0\), yields that

and

where the proposed observer gain structure causes the pole-zero cancellation in the first calculation result above. These two calculation results give the Laplace transformed observer error output \(y_{obs}\) as

with \(Y_{obs}(s)\), R(s), and W(s) being the Laplace transforms of \(y_{obs}\), \(r (=0)\), and w, respectively, and \(W_F(s) \,{:=}\, \frac{1}{s + l_{obs,d}}W(s)\). The inverse Laplace transform obtains the time domain expression as

with \(w_{F}\) denoting the inverse Laplace transform of \(W_F(s)\), which yields the performance error for \(\varepsilon _{\tilde{\omega }} \,{:=}\, e^*_{\tilde{\omega }} - e_{\tilde{\omega }}\) as (using (18))

Define the Lyapunov function candidate as

which gives along the performance error dynamics (19) using Young’s inequality (e.g., \(xy \le \frac{\varepsilon }{2}x^2 + \frac{1}{2\varepsilon }y^2\), \(\forall \varepsilon > 0\)) as

with a positive constant \(w_{max}\) satisfying \(|w| \le w_{max}\), \(\forall t \ge 0\). Then, the selection of \(\kappa _{w_F} \,{:=}\, \frac{1}{l_{obs,d}} (\frac{1}{l_{obs,c}} + 1)\) removes the indefinite term in the upper bound as

where \(\alpha _e \,{:=}\, \min \{ l_{obs,c}, \frac{1}{\kappa _{w_F}} \}\) subject to a large gain setting \(l_{obs,d}\) satisfying \(\frac{2w_{max}}{l_{obs,d}}\approx 0\), which completes the proof by the comparison principle in43. \(\square\)

The inequality \({\dot{V}}_\varepsilon < 0\) reveals that

for some setting \(l_{obs,d} > 0\). This leads to the reasoning process using the subsystem dynamics of (15) given by

which concludes that

verifying the exponential acceleration error estimation convergence used for the remaining analysis.

Lemma 4 analyzes the disturbance estimate behavior from DOB used for the closed-loop convergence analysis using the result of Lemma 3.

The DOB (17) ensures that

for some \(\gamma > 0\), where \({\tilde{d}}_\omega \,{:=}\, d_\omega - {\hat{d}}_\omega\), \(\forall t \ge 0\).

It follows from the output of DOB in the right side of (17) that

where the DOB (17) yields the second equation and the last equation is obtained from the relationships (11) and (21), which completes the proof. \(\square\)

Interestingly, the proposed controller results in the order reduction of the closed-loop speed dynamics by the stable pole-zero cancellation, which is proven in Lemma 5.

The proposed voltage updating law of (16) allows the speed error to be satisfied:

with the perturbations \({\tilde{d}}_{\omega ,F}\) and \(e_{{\tilde{a}},F}\) such that

for some \(b_i > 0\), \(i=1,2,3\).

The controlled speed error system is obtained as

with the combination of (11) and (16) and \(\tilde{\omega }_{ref} \,{:=}\, 0\). Taking the Laplace transform, it holds that

which yields that

where the pole-zero cancellation from the factorization

is applied and \({\tilde{D}}_{\omega ,F}(s) = \frac{\frac{1}{c_{\omega ,0}}}{s + \frac{k_d}{c_{\omega ,0}}}{\tilde{D}}_\omega (s)\) and \(E_{{\tilde{a}},F}(s) = \frac{\frac{k_d + c_{\omega ,0}\lambda _{sc}}{c_{\omega ,0}}}{s + \frac{k_d }{c_{\omega ,0}}} E_{{\tilde{a}}}(s)\), which completes the proof. \(\square\)

Finally, Theorem 1 asserts the main result.

The proposed controller comprising (14)–(17) ensures the exponential convergence (control objective)

for \(l > 0\) such that \(\frac{2d_{max}}{l}\approx 0\), where \(|{\dot{d}}_\omega | \le d_{max}\), \(\forall t \ge 0\).

The vector \({\textbf {x}} \,{:=}\, \left[ \begin{array}{cccc} \tilde{\theta }&\tilde{\omega }&{\tilde{d}}_{\omega ,F}&e_{{\tilde{a}},F} \end{array}\right] ^T\) leads to the augmented system given by

where \({\textbf {A}}_x \,{:=}\, \left[ \begin{array}{cccc} -\lambda _{pc} &{} 1 &{} 0 &{} 0 \\ 0 &{} -\lambda _{sc} &{} 1 &{} 1 \\ 0 &{} 0 &{} -b_1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -b_1 \end{array}\right]\), \({\textbf {b}}_{x,d} \,{:=}\, \left[ \begin{array}{cccc} 0 \\ 0 \\ b_2 \\ 0 \end{array}\right]\), and \({\textbf {b}}_{x,e} \,{:=}\, \left[ \begin{array}{cccc} 0 \\ 0 \\ 0 \\ b_3 \end{array}\right]\). The stability of \({\textbf {A}}_x\) picks an unique solution \({\textbf {P}}_x > {\textbf {0}}\) such that \({\textbf {A}}_x^T {\textbf {P}}_x + {\textbf {P}}_x {\textbf {A}}_x = - {\textbf {I}}\), which defines the Lyapunov function candidate defined as

The above augmented system and (22) gives its time derivative along the trajectories as

with its upper bound by Young’s inequality (e.g., \({\textbf {p}}^T {\textbf {q}} \le \frac{\varepsilon }{2}\Vert {\textbf {p}}\Vert ^2 + \frac{1}{2\varepsilon }\Vert {\textbf {q}}\Vert ^2\), \(\forall \varepsilon > 0\)):

Then, the selections of \(\kappa _d \,{:=}\, \frac{2}{l} ( \frac{\Vert {\textbf {P}}_x\Vert ^2 b^2_2}{4} + 1 )\) and \(\kappa _a \,{:=}\, \frac{1}{l_{obs,c}}(\frac{\Vert {\textbf {P}}_x\Vert ^2 b_3^2}{4} + \frac{\kappa _d^2 l^2 \gamma ^2}{2} + \frac{1}{2})\) rearrange the upper bound of \({\dot{V}}\) such that

where \(\alpha \,{:=}\, \min \{ \frac{2}{3\lambda _{min}({\textbf {P}}_x)}, \frac{1}{\kappa _d}, \frac{1}{\kappa _a} \}\), completing the proof. \(\square\)

Based on the above analysis results, this remark finalizes this section by suggesting a tuning process of the proposed controller comprising the speed (inner) and position (outer) loops shown in Fig. 1 as follows:

(speed loop for steps 1-4) Using well-working speed controller, e.g., PI controller with a constant speed reference \(\omega _{ref}\), tune the observer gains \(l_{obs,c}\) and \(l_{obc,d}\) for a rapid estimation error convergence in accordance with Remark 1; for example, first, choose \(l_{obs,c}\) such that \(l_{obs,c} \ge 50\) for \({\dot{e}}_{\tilde{\omega }} = -l_{obs,c}e_{\tilde{\omega }}\) and, second, increase \(l_{obs,d}\) holding \(l_{obs,d}\gg l_{obs,c}\).

Tune the DOB gain \(l > 0\) to assign the cut-off frequency (\(l = 2\pi f_l\) rad/s, equivalently, \(f_{l} = \frac{l}{2\pi }\) Hz) for the transfer function \(\frac{{\hat{D}}_\omega (s)}{D_{\omega }(s)} = \frac{l}{s + l}\) (obtained from (22) under the condition \(e_{{\tilde{a}}} = 0\)); for example, choose \(f_{l} \ge 2\) Hz such that \(l \ge 2\pi f_l = 12.6\) rad/s.

Using the proposed speed controller (16) with a constant speed reference \(\omega _{ref}\) (for step 3 and 4), select \(f_{sc} \in [10, 30]\) yielding \(\lambda _{sc} \in [2\pi 10 (=\lambda _{sc,min}), 2\pi 30 (=\lambda _{sc,max})]\) (e.g., \(\lambda _{sc} = 2\pi f_{sc}\) rad/s and \(f_{sc} = \frac{\lambda _{sc}}{2\pi }\) Hz); the maximum interval value may be increased depending on the hardware specification.

Increase the active damping coefficient \(k_d\) (for example, \(k_d \ge 0.001\)) for an acceptable speed tracking response \({\dot{\omega }} \approx \lambda _{sc} \tilde{\omega }\) (some iteration between step 3 and 4 may be required).

(position loop for steps 5-7) Using the proposed position controller (9) with a constant position reference \(\theta _{ref}\), set \(\gamma _{pc} = \rho _{pc} = 0\) and select \(f_{pc} \in [0.1, 5]\) yielding \([2\pi 0.1 (=\omega _{pc,min}), 2\pi 5 (=\omega _{pc,max})]\); the maximum interval value may depend on the hardware specification.

Increase \(\lambda _{pc}\) (for example, \(\lambda _{pc} \ge 10\)) until an acceptable position tracking response \({\dot{\theta }} \approx \omega _{pc}\tilde{\theta }\) is obtained (some iteration between step 5 and 6 may be required).

Increase \(\gamma _{pc}\) and \(\rho _{pc} = \frac{\kappa _{pc}}{\gamma _{pc}}\) with \(\kappa _{pc} > 0\) until the peak value and restoration rate of the convergence rate are acceptable; for example, choose \(\gamma _{pc} \ge 1\) and \(\kappa _{pc} \ge \frac{\gamma _{pc}}{2}\).

This process results in the controller tuning values used in “Experimental results” section.

This section experimentally demonstrates the position-tracking performance improvements by comparison with an extant controller. A 10-W stepping motor embedding an encoder for position feedback (model:NK266E-02AT) and Texas Instrument (TI) LAUNCHXL-F28069M (digital signal processor) were used for experimental setup shown in Fig. 2 (see41 for more detailed configuration).

Experimental setup.

The controller tuning results are summarized as (convergence rate booster) \(f_{pc} = 0.2\) Hz such that \({\hat{\omega }}_{pc}(0) = \omega _{pc} = 2\pi 0.2\) rad/s, \(\gamma _{pc} = 2\), \(\rho _{pc} = 0.5/\gamma _{pc}\), (control gain) \(\lambda _{pc} = 1.25\), \(\lambda _{sc} = 125.6\), \(k_d = 0.1\), (observer gain) \(l_{obs,c} = 100\), \(l_{obs,d} = 500\), and (DOB gain) \(l = 20\). This study chooses the FL controllers (in30) for the comparison study, including the active damping and feed-forward term, given by: \(v_x = K_{P,cc} {\tilde{i}}_x + K_{I,cc} \int _0^t {\tilde{i}}_x d\tau - p_{x,0}\), \(i_{q,ref} = \frac{1}{K_{m,0}} ( - k_d \omega + K_{P,sc}\tilde{\omega } + K_{I,sc} \int _0^t \tilde{\omega }d\tau ),~(i_{d,ref} = 0)\), \(\omega _{ref} = k_{pc}\tilde{\theta }\), \(\forall t \ge 0\), \(x=d,q\), with feedback gains \(K_{P,cc} = L_0 \lambda _{cc}\), \(K_{I,cc} = R_0 \lambda _{cc}\), \(K_{P,sc} = J_0 \lambda _{sc}\), \(K_{I,sc} = (k_d + B_0) \lambda _{sc}\), and carefully tuned value \(k_d = 0.01\). The current cut-off frequency was tuned to \(\lambda _{cc} = 314\) by applying the same settings for the position and speed loops in the proposed controller.

For the stair position reference, Fig. 3 demonstrates an improved tracking performance from the convergence rate boosting mechanism and performance recovery property proved in Theorem 1. Figure 4 presents the d-q axis current and observer error responses. Figure 5 shows the DOB and convergence rate booster responses. The intended convergence rate behavior improves the tracking performance as shown in Fig. 3.

Position response comparison from tracking task.

d–q axis current and angular acceleration estimation error responses from tracking task.

DOB and convergence rate responses from tracking task.

The proposed controller robustly forces the position motion to be the first-order tracking error system (7) by the beneficial capability shown in Theorem 1. This section verifies this advantage for the sinusoidal reference signals 0.1, 0.2, and 0.3 Hz. Figure 6 shows that the proposed controller provides the desired position-tracking behavior without any magnitude and phase distortion unlike the FL controller.

Position responses for sinusoidal references of 0.1, 0.2, and 0.3 Hz.

To evaluate the regulation performance, a load torque of \(T_L = 0.1\) Nm was abruptly applied to the closed-loop system (by suddenly attaching the rotating wheel to the rotor) operating at \(\theta _{ref,Deg} = 90^{\circ }\) under the three load conditions, such as light-, medium-, and heavy-sized fan. Figure 7 presents that the proposed technique accomplishes drastic regulation performance improvement under different loads compared with the FL controller that provides the magnified undershoots with oscillations and performance inconsistency for different load conditions. The corresponding q-axis current responses are compared in Fig. 8, which exhibits the improved current regulation performance by the proposed controller despite in the absence of current feedback.

Position response comparison under regulation task for light-, medium-, and heavy-sized fan loads.

q-axis current response comparison under regulation task for light-, medium-, and heavy-sized fan loads.

The proposed current sensorless feedback system was driven by a PD-type controller incorporating the novel techniques, such as a convergence rate booster, angular acceleration error observer (model-free), and DOB without requiring the true motor parameters. This study has both proved the beneficial closed-loop properties and experimentally confirmed the practical advantages for tracking tasks. However, an acceptable setting for numerous design parameters should be identified through a systematic process, which is will be conducted in a future study.

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Lü, L., Xiong, W. & Hu, C. Equivalent magnetic circuit method of estimating iron losses in induction motor spindles. Sci. Rep. 12, 9509 (2022).

Article PubMed CAS PubMed Central Google Scholar

Wang, Y. et al. Control method of stepping motor for spaceborne solar irradiance spectrometer. Sci. Rep. 12, 2873 (2022).

Article ADS CAS PubMed PubMed Central Google Scholar

Saleh, S. M. & Hassan, A. Y. Sensorless based SVPWM-DTC of AFPMSM for electric vehicles. Sci. Rep. 12, 9023 (2022).

Article CAS PubMed PubMed Central Google Scholar

Zhou, S., Li, B., Wang, J. & Xu, D. A modified modular multilevel converter for motor drives capable of high-torque operation at zero/low motor speeds. IEEE Trans. Circuits Syst. II Express Briefs 68, 2493–2497 (2021).

Google Scholar

Fang, L. et al. Design of wireless individual-drive system for variable-reluctance stepping motor. IEEE Trans. Circuits Syst. II Express Briefs 69, 2141–2145 (2022).

Google Scholar

Kim, S.-K., Park, J. K. & Ahn, C. K. Learning and adaptation-based position-tracking controller for rover vehicle applications considering actuator dynamics. IEEE Trans. Ind. Electron. 69, 2976–2985 (2022).

Article Google Scholar

Kim, S.-K. & Ahn, C. K. Active-damping speed tracking technique for permanent magnet synchronous motors with transient performance boosting mechanism. IEEE Trans. Ind. Inf. 18, 2171–2179 (2022).

Article Google Scholar

Yeadon, A. W. Handbook of Small Electric Motors (McGraw-Hill, 2001).

Google Scholar

Acarnley, P. P. Stepping Motors: A Guide to Modern Theory and Practice (Peregrinus, 1982).

Google Scholar

Kenjo, T. Stepping Motors and Their Microprocessor Control (Clarendon, 1984).

Google Scholar

Bellini, A., Concari, C., Franceschini, G. & Toscani, A. Mixed mode PWM for high-performance stepping motors. IEEE Trans. Ind. Electron. 54, 3167–3177 (2007).

Article Google Scholar

Ghafari, A. S. & Behzad, M. Investigation of the micro-step control positioning system performance affected by random input signals. Mechatronics 15, 1175–1189 (2005).

Article Google Scholar

Shin, D., Kim, W., Lee, Y. & Chung, C. C. Phase compensated microstepping for permanent-magnet stepper motors. IEEE Trans. Ind. Electron. 60, 5773–5780 (2013).

Article Google Scholar

Kim, W., Shin, D. & Chung, C. C. Microstepping using a disturbance observer and a variable structure controller for permanent-magnet stepper motors. IEEE Trans. Ind. Electron. 60, 2689–2699 (2013).

Article Google Scholar

Kim, W., Shin, D. & Chung, C. C. Lyapunov-based controller with a passive nonlinear observer to improve position tracking performance of microstepping in permanent magnet stepper motors. Automatica 48, 3064–3074 (2012).

Article MathSciNet MATH Google Scholar

Li, Y., Zhu, Q. & Zhang, J. Distributed adaptive fixed-time neural networks control for nonaffine nonlinear multiagent systems. Sci. Rep. 12, 8459 (2022).

Article CAS PubMed PubMed Central Google Scholar

Mukaddim, R. A. et al. Murine cardiac fibrosis localization using adaptive Bayesian cardiac strain imaging in vivo. Sci. Rep. 12, 8522 (2022).

Article PubMed CAS PubMed Central Google Scholar

Yang, Z. et al. Full-state time-varying asymmetric constraint control for non-strict feedback nonlinear systems based on dynamic surface method. Sci. Rep. 12, 10469 (2022).

Article CAS PubMed PubMed Central Google Scholar

Kim, S.-K., Kim, K. S. & Ahn, C. K. Order reduction approach to velocity sensorless performance recovery PD-type attitude stabilizer for 2-DOF helicopter applications. IEEE Trans. Ind. Inf. 1, 1–8. https://doi.org/10.1109/TII.2022.3143204 (2022).

Article Google Scholar

Kim, S.-K. & Ahn, C. K. Proportional-derivative voltage control with active damping for DC/DC boost converters via current sensorless approach. IEEE Trans. Circuits Syst. II Express Briefs 68, 737–741 (2021).

Google Scholar

Kim, K. S., Kim, S.-K. & Ahn, C. K. Decentralized current sensorless speed synchronization with feedback-loop adaptation for multi-servo system applications. IEEE Trans. Ind. Electron. 1, 1–8. https://doi.org/10.1109/TIE.2022.3176295 (2022) (in press).

Article Google Scholar

Lee, Y., Shin, D., Kim, W. & Chung, C. C. Nonlinear \({H}_2\) control for a nonlinear system with bounded varying parameters: Application to PM stepper motors. IEEE/ASME Trans. Mechatron. 22, 1349–1359 (2017).

Article Google Scholar

Park, R. H. Two-reaction theory of synchronous machines generalized method of analysis. AIEE Trans. 48, 716–727 (1929).

Google Scholar

Bodson, M., Chiasson, J., Novotnak, R. & Rekowski, R. High performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Trans. Control Syst. Technol. 1, 5–15 (1993).

Article Google Scholar

Kim, S.-K., Lee, K.-G. & Lee, K.-B. Singularity-free adaptive speed tracking control for uncertain permanent magnet synchronous motor. IEEE Trans. Power Electron. 31, 1692–1701 (2016).

Article ADS Google Scholar

Kim, S.-K., Lee, J.-S. & Lee, K.-B. Offset-free robust adaptive back-stepping speed control for uncertain permanent magnet synchronous motor. IEEE Trans. Power Electron. 10, 7065–7076 (2016).

Google Scholar

Kim, S.-K., Lee, J.-S. & Lee, K.-B. Self-tuning adaptive speed controller for permanent magnet synchronous motor. IEEE Trans. Power Electron. 32, 1493–1506 (2017).

Article ADS Google Scholar

Kim, S.-K. Robust adaptive speed regulator with self-tuning law for surfaced-mounted permanent magnet synchronous motor. Control Eng. Pract. 61, 55–71 (2017).

Article Google Scholar

Kim, S.-K. & Ahn, C. K. Offset-free proportional-type self-tuning speed controller for permanent magnet synchronous motors. IEEE Trans. Ind. Electron. 66, 7168–7176 (2019).

Article Google Scholar

Sul, S.-K. Control of Electric Machine Drive Systems Vol. 88 (Wiley, 2011).

Google Scholar

Zribi, M. & Chiasson, J. N. Position control of a PM stepper motor by exact linearization. IEEE Trans. Autom. Control 36, 620–625 (1991).

Article Google Scholar

Bodson, M., Chiasson, J., Novotnak, R. & Rekowski, R. High performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Trans. Control Syst. Technol. 1, 5–14 (1993).

Article Google Scholar

Marino, R., Peresada, S. & Tomei, P. Nonlinear adaptive control of permanent magnet step motors. Automatica 31, 1595–1604 (2007).

Article MathSciNet MATH Google Scholar

Sira-Ramirez, H. A passivity plus flatness controller for the permanent magnet stepper motor. Asian J. Control 2, 1–9 (2000).

Article Google Scholar

Zribi, M., Sira-Ramirez, H. & Ngai, A. Static and dynamic sliding mode control schemes for a permanent magnet stepper motor. Int. J. Control 74, 103–117 (2001).

Article MATH Google Scholar

Defoort, M., Nollet, F., Floquet, T. & Perruquetti, W. A third order sliding-mode controller for a stepper motor. IEEE Trans. Ind. Electron. 56, 3337–3346 (2009).

Article Google Scholar

Zhu, Q. Complete model-free sliding mode control (CMFSMC). Sci. Rep. 11, 22565 (2021).

Article ADS CAS PubMed PubMed Central Google Scholar

Bifaretti, S., Tomei, P. & Verrelli, C. M. A global robust iterative learning position control for current-fed permanent magnet step motors. Automatica 47, 227–234 (2011).

Article MathSciNet MATH Google Scholar

Verrelli, C. M., Tomei, P., Consolini, L. & Lorenzani, E. Space-learning tracking control for permanent magnet step motors. Automatica 73, 223–230 (2016).

Article MathSciNet MATH Google Scholar

Kim, S.-K. & Ahn, C. K. Position regulator with variable cut-off frequency mechanism for hybrid-type stepper motors. IEEE Trans. Circuits Syst. I Regul. Pap. 67, 3533–3540 (2020).

Article MathSciNet MATH Google Scholar

Kim, S.-K. & Ahn, C. K. Variable-performance positioning law for hybrid-type stepper motors via active damping injection and disturbance observer. IEEE Trans. Circuits Syst. II Express Briefs 68, 1308–1312 (2021).

Google Scholar

Defoort, M., Nollet, F., Floquet, T. & Perruquetti, W. A third-order sliding-mode controller for a stepper motor. IEEE Trans. Ind. Electron. 56, 3337–3346 (2009).

Article Google Scholar

Khalil, H. K. Nonlinear Systems (Prentice Hall, 2002).

MATH Google Scholar

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This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1A6A1A03026005), in part by the NRF grant funded by the Korea government (Ministry of Science and ICT) (No. NRF-2021R1C1C1004380), and in part by the NRF grant funded by the Korea government (Ministry of Science and ICT) (No. NRF-2020R1A2C1005449).

Department of Creative Convergence Engineering, Hanbat National University, Daejeon, 341-58, Korea

Seok-Kyoon Kim

School of Electrical Engineering, Korea University, Seoul, 136-701, Korea

Kwan Soo Kim, Dong Kyu Lee & Choon Ki Ahn

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S.-K.K. and C.K.A. wrote the main manuscript text and prepared all figures. D.K.L. and K.S.K. performed the experimental study and collected the resultant actual data. All authors reviewed the manuscript.

Correspondence to Choon Ki Ahn.

The authors declare no competing interests.

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Kim, SK., Kim, K.S., Lee, D.K. et al. Current sensorless position-tracking control with angular acceleration error observers for hybrid-type stepping motors. Sci Rep 12, 14993 (2022). https://doi.org/10.1038/s41598-022-19250-0

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Received: 27 June 2022

Accepted: 26 August 2022

Published: 02 September 2022

DOI: https://doi.org/10.1038/s41598-022-19250-0

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