Nonlinear Positioning Technique via Dynamic Current Cut-off Frequency and Observer-based Pole-zero Cancellation Approaches for MAGLEV Applications

This article solves the problem caused by high level current feedback gain setting for fast responsiveness of magnetic levitation systems considering the current dynamics and presents advanced nonlinear positioning technology without plant parameter information. The main features of this study are summarized as follows: First, current control demonstrates current ripple reduction and overall performance guarantee through a low feedback gain in the steady state, including a dynamic feedback loop increased by an error variable magnitude in the transient period. Second, the plant parameter information-free velocity observer replaces the observer output error integral action with the disturbance estimation action to improve the closed-loop performance. The simulation results reveal the practical advantages derived from the contributions of this study.


I. INTRODUCTION
M ASS positioning tasks can be accomplished by the electromagnetic force caused by the coil current, which paly a vital role in magnetic levitation (MAGLEV) technique-based industrial trains. MAGLEV trains have been considered an alternative to conventional engine-based systems because of their decreased pollution and noise levels and increased durability [1]- [5].
The current and position dynamics are coupled with a nonlinear relationship in the presence of mismatched disturbances by the sudden increase/decrease in the number of passengers, which makes it nontrivial to solve the positioning problem of MAGLEV systems [6]- [8]. Moreover, variations in the coil inductance and resistance value are also problematic, leading to inconsistent closed-loop performance over a wide operating region. The linearization technique can transform nonlinear dynamics into an unstable linear system with limited admissibility, which helps in solving the positioning problem using a simple proportional-integral (PI) controller [9]. State-feedback control with feed-forward compensation terms has been presented as another linearization-based solution with improved closed-loop performance assignability via the pole placement technique [9], [10]. The resultant closed-loop performance obtained from these linear controllers can be limited when considering the feasibility of parameter-dependent linearized system dynamics for a given operating point. The gain scheduler requiring online membership tests can be considered a solution to this problem [10]. Additional advanced mechanisms, including optimization [11], [12] and adaptation [13] have been adopted for the feedback gains and feed-forward terms used for the statefeedback controller. The recent online parameter estimation techniques as in [14]- [17] can alleviate the parameter dependence level of these results.
In addition to linearization techniques, nonlinear techniques, including fuzzy [18], sliding mode [19], backstepping [20], adaptation [21], and coordinate transformation [22] have been applied to solve the positioning problem by handling the nonlinearity without operating point dependency. This has required the additional feedback and feed-forward loops to incorporate the sensors, which can be addressed by utilizing the observers as in [23]- [26]. The disturbance observer (DOB) estimates the lumped disturbances from the deviation between the system model and the actual system such that it yields the feed-forward compensation terms to secure improved closed-loop robustness [27]. The combination of simple proportional-type control was presented through a back-stepping process, forming a multi-loop structure [28]. Active damping-based multiloop controllers incorporating DOBs solved the system parameter dependence problem [29]. The elimination of the velocity sensor was accomplished using a recent DOBbased proportional-derivative (PD) controller, including the parameter-independent velocity observer [30] with convergence and closed-loop behavior analysis.
The above-mentioned results must set the current dynam-ics to be sufficiently fast to secure an acceptable positioning performance during both the transient and steady-state periods. During the transition periods, high current feedback gains are required for fast responsiveness. However, this high-level current feedback gain setting expands the undesirable current ripple and reduces the relative stability margin. This study considers this practical concern as the main problem in this work and proposes a solution to this problem with a few contributions given by • Dynamic current feedback mechanism in closed form without numerical retrieval to ensure the desired performance by boosting and restoring the feedback gain value; • Improvement of closed loop performance by using a model-free proportional-type velocity observer including a disturbance estimator; which constitutes the novel multi-loop positioning controller adopting the active damping to the inner loop for orderreduction from the pole-zero cancellation. Section II introduces a nonlinear induction equation for MAGLEV, and Section III presents the dynamic current cutoff frequency techniques and controller design for inner/outer loops. Section IV presents the pole-zero cancellation approaches and the stability analysis of the inner and outer loops, and Section V validates the practical benefits of simulating with MATLAB/Simulink through various scenarios for position tracking and regulation. Section VI presents conclusions and future work. Fig. 1 illustrates the MAGLEV system configuration including the actuator provided by the current controller. This system is designed to maintain the desired gap (denoted as p in Fig. 1) between the magnet (attached to the train body) and the rail, which can be accomplished by the magnetic force triggered by the coil current magnitude proportional to the coil voltage. Consequently, from the perspective of system engineering, the coil voltage and position correspond to the input and output, respectively. Specifically, there is a set of state variables, p (position in m), v p (:=ṗ) (velocity in m/s), and i c (current in A) and control input, u (coil voltage in V), which satisfies the nonlinear dynamical relationship [13]:

II. MAGLEV NONLINEAR MOTION EQUATIONS
where the air spring causes the load force w p (in N ) to act as an unknown mismatched disturbance. Meanwhile, the system parameters M , M L , g, K, R c , and L c represent the masses of the electromagnet and load (in kg), gravitational acceleration (9.8 in m/s 2 ), electromagnet force coefficient (in N · m 2 /A 2 ), and coil resistance and inductance (in Ω and H), respectively. Their known nominal parameter values from manufacturing are denoted as M 0 , K 0 , R c,0 , and L c,0 . The application of nominal parameters to the original system dynamics (1) and (2) leads to the adoption of uncertain 2 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
which is used as the basis for devising the control law in the following section.

III. CONTROL LAW
This study adopts a low-pass filter (LPF) [31] as the performance index given by which are considered to be the desired dynamics for position and current. This derives the main control objective as exponential convergence: Note that the feedback gain ω c acting as the (inner) current loop cutoff frequency must be tuned sufficiently to enlarge the admissible range of the outer loop cutoff frequency. The large constant inner-loop cutoff frequency can magnify the unnecessary current ripple and degrade the closed-loop robustness. The proposed technique alleviates this limitation by incorporating a dynamic current feedback gain mechanism into the control action.

A. OUTER LOOP 1) Velocity Observer
The stabilization of the second-order position dynamics (1) requires the feedback of the velocity (v p =ṗ). The timederivative process of the measurement p could extract the velocity information perturbed by high-frequency measurement noise, which could degrade the closed-loop accuracy. This study attempts to devise an advanced velocity observer with two merits: plant parameter independence and disturbance estimator as a replacement for the observer output error integral action, which is given bẏ with respect to the observer state variablesp and z vp , output v p , and error e p = p −p, the velocity estimation error is defined as e v := v p −v p . The two design parameters l v,i > 0 and i = 1, 2 determine the state update rates, the roles of which are revealed in Section IV.

Consider an equivalent expression of the open-loop position dynamics (3) given bÿ
with the known coefficient c p := K0 M0 , the newly defined lumped disturbance d p := 1 M0w p , the nonlinear function , and the design variable z ref to be used as the control input to stabilize the open-loop dynamics (12). The control law for updating z ref is proposed as: ∀t ≥ 0, which attempts to stabilize the errorp := p ref − p in accordance with the setting of the design parameters b d,p > 0 and ω p > 0. The observer-based DOB yields the disturbance estimated p such thaṫ z dp = −l dp z dp − l 2 dpvp + l dp (c p where gain l dp > 0 determines the disturbance estimation rate. Note that the observer-based feed-forward compensation term b d,pvp injects additional damping to the closed loop, where the gain b d,p adjusts this artificial damping intensity. Moreover, the cooperation of b d,p and ω p renders the order of the closed-loop position dynamics as1 (i.e., order reduction) via pole-zero cancellation without involving any uncertainty problems. For details, see Section IV. VOLUME 4, 2016

B. INNER LOOP
The proposed inner loop aims to devise an exponential convergent current controller, such that lim t→∞ i c = i * c leadsto lim t→∞ φ(i c , z ref ) = 0. For this purpose, we define the coil current reference using the outer loop control law (13)

1) Dynamic Feedback Loop
To implement the dynamic feedback loop, consider a slight modification of (7) aṡ subject to a feedback loop update mechanism [32]: where ∆i * c := i c,ref − i * c ; the gains κ ωc > 0 and ς ωc > 0 determine the feedback gain boosting and restoring rates. The initial condition is given byω c (0) = ω c for a constant base cutoff frequency ω c . Issues related to stability (owing to the nonlinear term (∆i * c ) 2 ) and the cutoff frequency boosting propertyω c ≥ ω c , ∀t ≥ 0, are addressed in Section IV.

2) Controller
The error ∆i c := i * c − i c yields the dynamics: where the newly defined lumped disturbance d c :=i * c + Rc,0 Lc,0 i c − 1 Lc,0w c . The control law for updating the coil voltage u is proposed as: (19) ∀t ≥ 0, which attempts to stabilize the error ∆i c = i * c − i c in accordance with the setting of the design parameters b d,c > 0 and k c > 0. The DOB continuously adjusts the variabled c to exponentially estimate the actual disturbance d c such thaṫ where gain l dc > 0 determines the disturbance estimation rate. Fig. 2 illustrates the proposed cascade feedback system structure.

Remark 1
The feed-forward compensation term −b d,c i c injects additional damping to the closed-loop system where the gain b d,c adjusts this artificial damping intensity. Moreover, the combination of the design parameters b d,c and k c renders the order of closed-loop position dynamics as1 (i.e., order reduction) via pole-zero cancellation without involving any uncertainty problems. See Section IV for further details. ♦ This section shows the accomplishment of the control objective by proving the exponential convergence (8) by considering the closed-loop error and auxiliary system dynamics. To this end, Section IV-A begins with an inner loop analysis.

A. INNER LOOP
In this section, we prove the accomplishment of the control objective for the current loop (lim t→∞ i c = i * c ) and exponential convergence lim t→∞ i c = i c,ref such that lim t→∞ φ = 0.
Result (22) plays an important role in proving the exponential convergence lim t→∞ ∆i * c = i c,ref in Theorem 2. Lemma 2 analyzes the closed-loop stability of a time-varying system (16).

Lemma 2
The subsystem comprising (16) and (17) guarantees that there exists a i > 0, i = 1, 2, such that This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  (16) and (17) as follows: , ∀t ≥ 0, and Lyapunov function candidate where α 1 := min{ω c , 2ς ωc ω c }, which completes the proof based on the comparison principle in [33].
Remark 2 Considering the cutoff frequency magnification property (22), it is reasonable to assume that 2δi c,ref ωc ≈ 0 yieldingV 1 ≤ −α 1 V 1 , ∀t ≥ 0, for some settings of κ ωc and ς ωc used for the update rule (17), which is employed in the following convergence analysis. ♦ Lemma 3 clarifies the disturbance estimation behavior from DOB (20) and (21) by investigating its output dynamics.

Lemma 3
The DOB comprising (20) and (21) ensures that: ♦ Proof: Consider the time derivative of the output (21) using (20), such thaṫ where the last equality is obtained using the equation d c = ∆i c + 1 Lc,0 u from (18). This completes this proof.
with w dc :=ḋ c and |w dc | ≤ δ dc , ∀t ≥ 0, which is used in the following convergence analysis. ♦

2) Control Law
As can be seen from the combination of (18) and (17), the inner loop system seems to be governed by secondorder dynamics owing to the first-order integral action of the control law (19). Interestingly, the combination of the active damping coefficient b d,c and the design parameter structure results in first-order closed-loop dynamics owing to the order reduction property of active damping. For details, refer to Lemma 4.

Lemma 4
The inner-loop system shown in Fig. 2 controls the coil current, such that with filtering dynamicṡ e dc,F = −a c,1 e dc,F − a c,2 e dc , ∀t ≥ 0, for some a c,i > 0, i = 1, 2. ♦ Proof: Substituting (19) into (18) yields the closed-loop current dynamics: where r = 0 and e dc := d c −d c . Its vector form for  (28) and (29) as where the combination of design parameters b d,c and k c results in order reduction given as follows:  Fig. 2 guarantees that there exist b i > 0, i = 1, 2,

Theorem 1 The inner-loop system shown in
♦ Proof: Consider the positive definite function V e dc := 1 2 e 2 d c,F + ζ dc 2 e 2 dc with ζ dc > 0 whose time derivative is obtained by (using (25), (27), and Young's inequality (e.g., where α e dc := min{a c,1 , 1 ζ dc }, The result (26) and inequality (32) render the positive definite function V ∆ic := 1 2 ∆i 2 c + η e dc V e dc with η e dc > 0 as follows: where α ∆ic := min{k c , 1 ηe dc }. This confirms the result of this theorem by using the comparison principle in [33].
The result (31) shows exponential convergence (control objective (8)): with the condition 2δ dc l dc ≈ 0 (DOB gain setting), which concludes the control objective (8) and is assumed to derive the useful inequality from (33): for the remaining convergence analysis.
Theorem 2 proves the exponential convergence of the actual coil current errorĩ c := i c,ref − i c based on the inequality (34), which acts as the rationale for assuming that lim t→∞ φ = 0. Fig. 2 guarantees the property:

Theorem 2 The inner-loop system shown in
exponentially. ♦
for some α φ > 0, corresponding to one of the main results in this subsection. ♦

B. WHOLE LOOP
This section aims to prove the accomplishment of the main control objective (lim t→∞ p = p * ) by analyzing the outer loop dynamics and employing the main inner loop analysis result (37).

VOLUME 4, 2016
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

1) Outer Loop Auxiliary Systems
Lemma 5 clarifies the velocity estimation behavior of the observer (9)-(11) by investigating its output dynamics.

Lemma 5
The observer comprising (9)-(11) ensures that: ♦ Proof: Consider the time derivative of the observer output (11) along (9) and (10) This completes the proof. s+lv,2 with V p (s) and V p (s) representing the Laplace transforms of v p andv p , respectively, which indicates that the observer l v,2 can be tuned as the cutoff frequency (l v,2 = 2πf v,2 rad/s) of LPF from the input v p to the outputv p . After this setting for l v,2 , the remaining observer gain l v,1 should be adjusted for e p and e v to be convergent as fast as possible. lv,2 . This validates the exponential convergence lim t→∞vp =v * p (performance recovery) with the observer gain setting 2δv p lv,2 ≈ 0. Therefore, it is reasonable to assume thaṫ by the proposed observer (9)- (11), which is used in the remaining convergence analysis.

♦
Similar to the proof of Lemma 3, Lemma 6 derives the disturbance estimation behavior of the observer-based DOB (14) and (15) using dynamics (39).

Lemma 6
The DOB driven by (14) and (15) ensures that: ♦ Proof: The proof is omitted because it is identical to the proof of Lemma 3 using the outputs (15), (14), and d p = x + c p i 2 c p 2 − g (from (12), and (39)).

Remark 6
Two implications can be derived from the result (40) by setting e v = 0.
• (DOB gain tuning)D p (s) Dp(s) = l dp s+l dp with D p (s) and D p (s) representing the Laplace transforms of d p andd p , respectively, which indicates that the DOB gain can be tuned as the cutoff frequency (l dp = 2πf dp rad/s) of LPF from the input d p to the outputd p . • (estimation error dynamics) the disturbance estimation error dynamics for e dp := d p −d p : e dp = −l dp e dp − l dp l v,2 e v + w dp , ∀t ≥ 0, with w dp :=ḋ p and |w dp | ≤ δ dp , ∀t ≥ 0, which is used in the following convergence analysis. ♦

2) Whole System Dynamics
As can be seen from the combination of (12) and (13), the outer-loop system seems to be governed by secondorder dynamics. Interestingly, the combination of the active damping coefficient b d,p and the design parameter structure results in first-order closed-loop dynamics, owing to the order reduction property. For details, refer to Lemma 7.

Lemma 7
The proposed outer-loop system, shown in Fig. 2 controls the position such thaṫ with filtering dynamicṡ e F = −a p,1 e F + a p,2 (qe v + φ + e dp ), ∀t ≥ 0, for some a p,i > 0, i = 1, 2, and q > 0. ♦ Proof: Substituting (13) into (12) yields the closed-loop position dynamics: ∀t ≥ 0, where q := b d,p + ω p and e dp := d p −d p , which shows that: dτ +w ev + w φ + w e dp , ∀t ≥ 0, with w ev := t 0 qe v dτ , w φ := t 0 φdτ , and w e dp := t 0 e dp dτ . This gives an equivalent vector form for x p := p ζ p T with ζ p := b d,p ω p t 0p dτ : This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. to the proof of Lemma 4 using the representation of (44) and (45). Theorem 3 concludes this section by proving a closed-loop property related to the main control objective (8) based on the results of (37), (42), and (43).

V. SIMULATIONS
This section demonstrates the performance improvement from the closed-loop analysis results in Section IV using numerical simulations based on MATLAB/Simulink. The nonlinear differential equations (1) and (2) emulated the MA-GLEV system dynamics for the position, velocity, and coil current through Simulink programming in a continuous time setting, where the system coefficients are set to M = 725 kg, M L = 1000 kg (initial load), K = 5.45 × 10 −3 N · m 2 /A 2 , R c = 4.4 Ω, and L c = 908 mH. These values were obtained from an actual experimental test bed in [13]. The control algorithms were coded using C programming in the S-function environment, which was executed for each sampling and control period 1 ms and implemented using the nominal system parameter setting M 0 = 0.7M , K 0 = 1.5K, R c,0 = 1.3R c , and L c,0 = 0.5L c .
The design parameter tuning results for the proposed controller are summarized as: (outer loop) f p = 0.5 Hz (for ω p = 2π0.5 rad/s), b d,p = 2000, f dp = 300 Hz (for l dp = 2π300 rad/s), f v,1 = 200 Hz (for l v,1 = 2π200 rad/s), f v,2 = 1000 Hz (for l v,2 = 2π1000 rad/s), (inner loop) f c = 8 Hz (for ω c (0) = ω c = 2π8 rad/s), b d,c = 2000, k c = 1900, f dc = 1000 Hz (for l dc = 2π1000 rad/s), κ ωc = 5, and ς ωc = 1 κω c . A comparison analysis was conducted to clarify the practical advantage with the back-stepping controller (BSC) compensated by the active damping terms and DOBs such that: ,0 ), and d c = z c + l dc i c . Numerous shared design parameters, such as ω p , ω c , l dp , l dc , b d,v , and b d,c were chosen to be the same as those of the proposed controller. The velocity cutoff frequency was set as f v = 5 Hz (for ω v = 2π5 rad/s) for the best performance. 8 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. The load force w p was set to be sinusoidal such that w p = 2 × 10 3 sin(2π3t), ∀t ≥ 0, for all simulations to verify the disturbance rejection performance.

A. POSITION TRACKING COMPARISON
This subsection tests the closed-loop improvement under the position-tracking mission for the pulse reference with a minimum 1 cm and a maximum 3 cm, which were performed three times with increasing position-loop cutoff frequencies of f p = 0.5, 2, and 3 Hz. Fig. 3 presents the closedloop position responses driven by the proposed and BSC techniques; the proposed control action successfully eliminates the overshoots while maintaining the desirable closedloop performance given as the cutoff frequency f p , which comes from the dynamic feedback gain behavior presented in the right side of Fig. 5. Fig. 4 compares the coil current responses under this tracking mission. Unlike the BSC, the proposed controller featuring the dynamic cutoff frequency mechanism removes current ripples, unlike the BSC. This merit would lead to a power efficiency improvement in actual implementation during steady-state operation. The proposed observer estimates the actual velocity with satisfactory estimation error elimination behavior, which is presented on the left side of Fig. 5. The DOB responses are shown in Fig. 6, and their rapid disturbance estimation performance contributes to this significant improvement in the closed-loop performance.   This study incorporated a dynamic feedback loop mechanism into the control law to derive practical merits by increasing and decreasing the current-loop feedback gain according to the operating mode. Moreover, a plant parameterinformation-free velocity observer was devised to enable the implementation of a pole-zero cancellation control action with active damping. The numerical simulation results confirmed the practical advantages of the proposed technique. However, there were numerous design parameters for the introduced auxiliary subsystems, which will be automatically determined through the offline optimization process developed in a future study. Furthermore, we will extend our study based on the neuro-adaptive control method combining neural networks to demonstrate robust performance against complex model uncertainties and nonlinearities. He worked for LG Electronics as a senior research engineer from 2015 to 2016 and joined the Department of Creative Convergence Engineering at Hanbat National University, Daejeon, Korea, in 2017. His research interests include the tracking problem of power electronic applications, such as power converters and motor drives, the power system stabilization problem, and the development of control theory, such as passivity-based control and nonlinear adaptive control.
12 VOLUME 4, 2016 This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3187003