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Research Papers

Robust Voltage Control for an Electrostatic Micro-Actuator OPEN ACCESS

[+] Author and Article Information
Prasanth Kandula

Mem. ASME
Department of Electrical Engineering
and Computer Science,
Cleveland State University,
2121 Euclid Avenue,
Cleveland, OH 44115
e-mail: p.kandula@vikes.csuohio.edu

Lili Dong

Department of Electrical Engineering
and Computer Science,
Cleveland State University,
2121 Euclid Avenue,
Cleveland, OH 44115
e-mail: L.Dong34@csuohio.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 5, 2017; final manuscript received November 14, 2017; published online December 22, 2017. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 140(6), 061012 (Dec 22, 2017) (7 pages) Paper No: DS-17-1238; doi: 10.1115/1.4038493 History: Received May 05, 2017; Revised November 14, 2017

When a parallel-plate electrostatic actuator (ESA) is driven by a voltage source, pull-in instability limits the range of displacement to one-third of the gap between plates. In this paper, a nonlinear active disturbance rejection controller (NADRC) is originally developed on the ESA. Our control objectives are stabilizing and increasing the displacement of an ESA to 99.99% of its full gap. Most of the reported controllers in literature are based on linearized models of the ESAs and depend on detailed model information of them. However, the ESA is inherently nonlinear and has model uncertainties due to the imperfections of microfabrication and packaging. The NADRC consists of a nonlinear extended state observer (NESO) and a feedback controller. The NESO is used to estimate system states and unknown nonlinear dynamics for the ESA. Therefore, it does not require accurate model. We simulate the NADRC on a nonlinear ESA in the presences of external disturbance, system uncertainties, and noise. The simulation results verify the effectiveness of the controller by successfully extending the travel range of ESA beyond pull-in point. They also demonstrate that the controller is robust against both disturbance and parameter variations, and has low sensitivity to measurement noise. Furthermore, the stability for the control system with NADRC is theoretically proved.

FIGURES IN THIS ARTICLE
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Electrostatic actuators (ESA) are miniaturized actuating devices that rely on the force between two conducting electrodes when a voltage or a charge is applied between them. The ESA is one of the most commonly used components in the applications of micro-electromechanical systems, such as accelerometers, micromirrors, micro switches, microresonators, and optical gratings [1]. A one degree-of-freedom (DOF), parallel-plate micro-actuator consists of a movable plate and a fixed plate in an electric field. When the movable plate is displaced from its original position, the capacitance formed between the two plates will be varying. The displacement of the movable plate is usually changed through controlling the gap of the capacitor. However, as the gap between the two plates is reduced by one-third of original gap, a pull-in phenomenon will cause the instability of the system and drag the movable plate immediately to the fixed plate, causing the failure of operation of the actuator. Therefore, extending the traveling range of the movable plate beyond the pull-in limit has become a central topic in the control of ESAs. Both charge and voltage controls are employed to the ESA [211]. Compared to charge control, voltage control is simple to implement and easy to model. Therefore, in this paper, we focus on a voltage-controlled ESA system. Besides pull-in instability, fabrication imperfections and surrounding disturbances could degrade the performance of ESA and consequently limit the traveling range of it. Mechanical compensation could partially enhance the functionality of the ESA. One straightforward method is to fabricate the gap so large that the actuator can output a desirable displacement. However, the mechanical method is time-consuming, costly, and hard to implement on a nail-size micro device. In addition, the mechanical wear and environmental change cause the system uncertainties of the ESA. Mechanical and electronic noises also result in imprecise movement. Therefore, a robust voltage controller is crucial for increasing the displacement of ESA despite of the presence of external disturbances, model uncertainties, and noises.

Different kinds of voltage control methods have been presented in the recent literature. A linearized model of ESA is utilized in most of the reported controllers such as feedback linearization control [8], gain-scheduling H-infinity control [12], and linear active disturbance rejection control (LADRC) [9]. A linear parameter varying technique is applied to the quasi-static model of ESA [10,11], which is a semilinearized system. Be that as it may, in reality, the ESA is naturally nonlinear. According to Ref. [5], “it is recognized to be a very complex task to incorporate the mechanical factors (such as damping coefficients) into the design of the algorithms in linear control theory, and compromising the optimality of the system is inevitable.” Thus, it is only reasonable to design a nonlinear controller for a nonlinear model of ESA. In Ref. [13], a switching control method is employed on the nonlinear ESA. The controller has achieved the travel range of 89% of ESA's full gap with mechanical stoppers and 100% of full gap without stoppers in the structure. However, the robustness of this switching control against system uncertainties is not investigated in Ref. [13]. In Ref. [6], a flatness-based technique (FBT) is developed on the nonlinear ESA as well. The FBT requires a physical observer to estimate the displacement of movable plate. The addition of an extra observer to the controller complicates the whole ESA control system. Nevertheless, the FBT can broaden the travel scope of ESA to full gap. It is also robust against noise and system uncertainties. But external disturbance is not discussed in Ref. [6].

In this paper, a nonlinear active disturbance rejection controller (NADRC) is developed and applied to the nonlinear model of the ESA. As stated previously, the nonlinear model of ESA includes comprehensive information of the actuator without excluding the effects of mechanical factors. Therefore, it makes the controller design practical. The NADRC consists of a nonlinear extended state observer (NESO) and a feedback controller. The NESO can estimate not only systems states, but the nonlinear unknown dynamics of ESA. The feedback controller then compensates the unknown dynamics in real time. Since the NADRC is independent of an accurate mathematical model of ESA, it is very robust against parameter variations and external disturbances, and has low sensitivity to noise. In addition, the system stability is theoretically justified using Lyapunov approach.

The organization of this paper is as follows: The modeling of a parallel-plate ESA is presented in Sec. 2. The NADRC for the ESA is developed in Sec. 3. The theoretical proof of stability is described in Sec. 4. Simulation results are shown in Sec. 5. Conclusions are made in Sec. 6.

Figure 1 illustrates the electromechanical model of ESA with one degree-of-freedom [9,14]. The actuator includes two plates, one of which is movable and the other of which is fixed. The movable plate is attached to a rigid frame through a spring and a damper. The voltage source Vs induces an electrostatic force between plates and drives the ESA to move. The spring constant is k, damping constant is b, the resistance of the resistor that is in series with the source voltage and movable plate is R, the current flowing through the resistor is Is, the gap between the fixed and movable plates is g, the mass of one plate is m, and the vertical displacement of the movable plate is X. From this figure, we can see that the ESA is operating in both electrical and mechanical domains. Hence, its mathematical model is based on both circuit and force analyses. As long as the ESA is charged in an electrical field, the electrostatic force is generated between its two plates. The electrostatic force is inversely proportional to the area of each plate which is very small (in square millimeter or square micrometer) for a micro-actuator. So the electrostatic force would be large and have a great effect on the ESA. The gravitational force on the movable plate is negligible due to the small mass of it.

The dynamic motion of the system and the circuit equation can be represented by Display Formula

(1)mX¨+bX˙+kX=Q22εAQ˙=VsRQ(gX)εAR

where Q is the charge on each plate, A is the area of the plate, ε is the permittivity of the medium between two plates, and the right side of the first equation in Eq. (1) is electrostatic force.

From Fig. 1, we can tell stability exists between the upward spring force and downward electrostatic force. As the voltage is increased, the charge between plates will be increased which would lead to a large electrostatic force. So there is a particular voltage where the stability of the equilibrium is lost. This specific voltage is defined as pull-in voltage (Vpu), and the corresponding charge is pull-in charge (Qpu). As the stability is lost, the large electrostatic force would pull the movable plate all the way to the bottom, resulting in failure of operation. This specific phenomenon is called pull-in phenomenon. The pull-in gap would be two-thirds of full gap. For simplifying controller design and performance analyses, we have to normalize the ESA model in Eq. (1) as follows. The displacement is normalized with full gap, voltage with pull-in voltage, charge with pull-in charge, and time with the natural frequency (ωn=k/m). Normalized variables are listed in the following equation: Display Formula

(2)x=Xg,v=VsVpu,q=QQpu,t=ωnT

Applying normalization to Eq. (1) and converting the system into a state-space model, we get Display Formula

(3)x˙1=x2x˙2=13x32x12ςx2x˙3=1r(1x1)x3+23v

where x1 = x, x2=x˙, x3 = q, ς=b/2mωn, and r=Rωn(εA/g).

Theoretically, there are two kinds of active disturbance rejection controller (ADRC): LADRC and NADRC which depend on the type of ESO utilized in the design. A linear ESO (LESO), which is very similar to traditional Luenberger observer, utilizes linear functions to observe system states and estimates the generalized disturbance with an extended state, while the nonlinear ESO utilizes nonlinear functions to do the same. The theoretical proof for stability analyses of the LADRC with LESO has been discussed in Refs. [1518]. In Refs. [1517], the convergence of a tracking differentiator in LADRC and the capability of LESO to estimate uncertainties are presented. The conditions for exponential stability of LADRC based on singular perturbation analyses are investigated in Ref. [18]. A survey of the LADRC is documented in Ref. [19].

The advantages of NADRC are its enhanced robustness and reduced settling time compared to LADRC when same observer and controller bandwidths are utilized. As mentioned earlier, a NADRC mainly consists of a NESO and an observer-based feedback controller. The NESO observes not only system states but the generalized disturbance which contains both unknown dynamics and external disturbance. The convergence of observer error for NESO is theoretically justified in Refs. [20] and [21] for single-input and single-output systems and in Refs. [22] and [23] for multiple-input and multiple-output systems. Although the proof in Refs. [2023] is mathematically rigorous, it is lengthy and extremely complicated. In this paper, we develop a completely different and relatively simple Lyapunov approach to prove that the estimation error of NESO is converging to zero, particularly for the application of ESA. The performance of NADRC is dependent on the accurate estimations of system states (including generalized disturbance) by NESO.

The ADRC has high level of robustness against disturbances and system uncertainties. LADRC has been applied to automobiles [24,25], power systems [26], chemical processes [27], medical areas [28], micro-electromechanical systems [9,29], and so on. Additional information regarding LADRC can be found in Refs. [17,3034]. However, the applications of NADRC to real physical systems are rarely reported in literature. In this paper, we originally develop the NADRC on a nonlinear ESA system. The controller development is given as follows.

A NADRC is designed on the normalized ESA model represented by Eq. (3). Let a1= x1, a2=x˙1, and a3=(x32/3)x12ςx2. Equation (3) can be rewritten as Display Formula

(4)a˙1=a2a˙2=a3a˙3=f+bpuy=a1

where f=a22ςa3(2/r)(1a1)(a3+a1+2ςa2)+((43a1+6ςa2+3a36)/9r)u and bp=(2/3r).

In Eq. (4), a1 is still a normalized displacement and a2 is a normalized velocity. But a3 is not a normalized charge any more.

Applying the NESO in Ref. [21] to the third-order ESA system, we have Display Formula

(5)NESO:{oe=yz1ωo3z˙1=z2+ωo2(3oe+φ(oe))z˙2=z3+3ωooez˙3=z4+3oe+b0uz˙4=1ωooe

where oe is the observer error, b0 is a constant which is approximately equal to bp, z1, z2, and z3 are observed states of the system, z4 is observed generalized disturbance, and ωo is observer bandwidth. The definition of ϕ(oe) is given as follows: Display Formula

(6)φ(oe):{14π,oe(,π/2)14πsin(oe),oe(π/2,π/2)14π,oe(π/2,)

Let the control law be defined as Display Formula

(7)u=u0z4b0

and Display Formula

(8)u0=kp(Rz1)+kd(z2)+kdd(z3)

where kp, kd, and kdd are controller gains which depend on the controller bandwidth (ωc) as listed in Table 1, and R is reference signal.

The block diagram of the ESA control system is given in Fig. 2, where Ref is a desired displacement output (i.e., Ref = R), n is measurement noise, and d is external disturbance. The disturbance is added to the input of the ESA system, while the measurement noise is added to the output of the system. The system parameter values of ESA are listed in Table 2.

In this section, we originally develop a Lyapunov approach to prove the convergence of observer error and the stability of this ESA control system. The free system of error equation for the NADRC with the NESO in Eq. (5) is defined as Display Formula

(9)e˙=Ee(φ(e1),0,0,0)T

where error vector e = [e1, e2, e3, e4]T with e1=a1-z1, e2=a2-z2, e3=a3-z3, and e4=f-z4. The matrix E is defined as follows:

E=[3100301030011000]

The Lyapunov-like positive definite function for error system is selected as Display Formula

(10)V(e)=Pe,e+0e1φ(s)ds

where ϕ is a function of observer error, s is the variable of this function, and matrix P is presented as follows:

P=[4.50.540.50.540.54.540.54.50.50.54.50.59.5]

Differentiating Eq. (10) yields Display Formula

(11)dV(e)dt=dPe,edt+φ(e1)e˙1

where Pe=[4.5e10.5e24e3+0.5e40.5e1+4e20.5e34.5e44e10.5e2+4.5e30.5e40.5e14.5e20.5e3+9.5e4].

Expanding the right side of (11), we have Display Formula

(12)dV(e)dt=e˙TPe+eTpe˙+φ(e1)e˙1=e(ETP+PE)eT2(φ(e1),0,0,0)Pe+φ(e1)e˙1=e2(9e1e28e3+e)4φ(e1)+(3e1+e2φ(e1))φ(e1)=(e12+e22+e32+e42)(12e12e28e3+e4+φ(e1))φ(e1)

According to the definition of ϕ(oe) in Eq. (6), we obtain Display Formula

(13)|φ(e1)||e1|4π|e1|12,e1φ(e1)0,2e2φ(e1)e222+2(φ(e1))2,8e3φ(e1)e322+32(φ(e1))2,e4φ(e1)e424+(φ(e1))2

Combining Eq. (12) with Eq. (13), we get Display Formula

(14)dV(e)dt=(e12+e22+e32+e42)(12e12e28e3+e4+φ(e1))φ(e1)(e12+e22+e32+e42)+e222+2(φ(e1))2+e322+32(φ(e1))2+e424+(φ(e1))2+(φ(e1))2e12e222e3223e424+36(φ(e1))2e12e222e3223e424+36(e12144)(34e12+e222+e322+3e424)

From Eq. (14), we can see that the derivative of Lyapunov function is negative definite. It implies that the terms e1, e2, e3, e4, e1ϕ(e1), e2ϕ(e1), e3ϕ(e1), e4ϕ(e1), and (ϕ(e1))2 are bounded. From Eq. (9), we can derive that the derivative of e is bounded. Therefore, V¨ is bounded. So V˙ is uniformly continuous. Invoking Barbalat's lemma [35], the estimation errors e1, e2, e3, and e4 will converge to zero as time goes to infinity. So the observed states would approximate the real states. Particularly we have z1 ≈ y, and z4 ≈ f. This proves the effectiveness of the NESO.

We replace f with z4 in Eq. (4), and substitute Eqs. (7) and (8) in Eq. (4). Then we have Display Formula

(15)a˙1=a2a˙2=a3a˙3=u0y=a1

where u0 is defined in Eq. (8). Suppose z1=a1, z2=a2, and z3=a3. We can rewrite Eq. (15) as Display Formula

(16)a˙1=a2a˙2=a3a˙3=kp(Ra1)+kd(a2)+kdd(a3)y=a1

Hence, the state matrix of the closed-loop system is

A=[010001kpkdkdd]

We replace kp, kd, and kdd with the values of ωc as shown in Table 1, where ωc is a positive real number. Then the poles of system (16) would become −ωc. Therefore, the ESA control system is stable.

The NADRC is implemented on a nonlinear ESA model in Matlab/Simulink. The system parameters ζ and r are chosen as 0.0387 and 0.1, respectively, as given in Ref. [10], where the natural frequency of ESA is 4517 Hz. The initial (or full) gap of ESA is chosen as 4 μm as listed in Table 2. In this paper, we suppose the insulation layer is embedded into the structural layer (or electrodes) of ESA. Therefore, the length of 4 μm represents air gap. The other parameter values of ESA are listed in Table 2 as well. Using these parameter values, we can calculate the pull-in voltage as 1.311 v. Because the control input, state variables, and time are normalized (as shown in Eq. (2)), they do not have units. However, in order to show the displacement output and control effort in real-world situation, we scale back these two variables to their original dimensions in our simulation results. The normalized simulation time is 80, which is equivalent to 17.7 ms according to Eq. (2). The controller bandwidth ωc and observer bandwidth ωo are selected as 2 rad/s and 0.005 rad/s, respectively.

Tracking Performance of Nonlinear Active Disturbance Rejection Controller.

We choose the reference signal as a step input with different magnitudes of 50% and 99.99% of full gap at 0 and 40 normalized time units, respectively. The reference signal and displacement are scaled back to their original dimensions by multiplying them with full gap (as shown in Eq. (2)). The step responses of NADRC-controlled ESA are shown in Fig. 3, where xs is the scaled displacement output of ESA, v is normalized velocity, and q is normalized charge. The magnitude of 1.333 μm represents pull-in position. From Fig. 3, we can see that the NADRC can overcome the pull-in limit and successfully extend the travel range of ESA to 99.99% of full gap. The control signal is shown in Fig. 4, where the maximum control effort is 2.822 V. The scaled control signal is obtained my multiplying normalized control signal with pull-in voltage (1.311 V).

Robustness Against Parameter Variations.

In this section, the robustness of the NADRC against system uncertainties is tested. The system parameters ζ and r are varied from −25% to 100% of their nominal values. The same step input which is used in Fig. 3 is chosen as the reference signal. The scaled displacement output is shown in Fig. 5. From the figure, we can observe that there is no overshoot in the displacement output despite the parameter variations. However, if we increase the variation percentage above 100% or below −25%, overshoot will appear in the displacement. So we can conclude that the NADRC is robust against the parameter variations between −25% and 100% of their nominal values for this nonlinear ESA. The control signal for the parameter variations is given in Fig. 6. From this figure, the maximum control effort is 4.1236 V, which is practically achievable.

Noise Attenuation.

In an ESA system, the actuation requires energy which in turn produces heat. According to Ref. [14], mechanical–thermal noise (or white noise) is a major noise source for ESA. In our simulation, a band limited white noise with noise power of 0.1 W and sample time of 1 s is intentionally added to the output of ESA and considered as measurement noise for the system. The mechanical–thermal noise is shown in Fig. 7. This measurement noise introduces both positive and negative variations to the displacement output of ESA. Figure 8 shows the displacement output of ESA with measurement noise. From Fig. 8, we can see that the output is following the reference signal without overshoot despite the presence of noise. The corresponding control signal is given in Fig. 9 where the maximum control effort is 9.61 V. For noise attenuation, the controller gain ωc is chosen to be 2.85 rad/s and observer gain ωo is chosen to be 0.02 rad/s.

Disturbance Rejection.

As indicated in Fig. 2, a unit step disturbance is added to the ESA system at t = 35 normalized time units. Figure 10 shows the scaled displacement output of ESA in the presence of external disturbance. From this figure, we can see that the NADRC can drive the movable plate of ESA to reach 99.99% of its full gap when disturbed with minimal fluctuation. Next we increase the disturbance magnitude to 10 V and add it to the system at t = 35 normalized time units. Figure 11 shows the scaled displacement output of ESA with the 10 V disturbance. When applied to ESA, the NADRC demonstrates robustness against the external disturbance. However, one volt input disturbance could produce a maximum displacement of 28 nm from reference level (see Fig. 10). Then a step disturbance with the magnitude of 14.2 mV would lead to a displacement of 0.4 nm, which is 0.01% of full gap. So as the reference signal is set as 99.99% of full gap, the NADRC could compensate a disturbance no larger than 14.2 mV.

The scaled control signals for the ESA system in the presence of unit and 10 V step disturbances are shown in Fig. 12. From this figure, we can conclude that although there is a negligible effect on output by these disturbances, there is a significant change in control signal at t = 35 normalized time units. Also from the same figure, we can see that the minimum normalized control efforts are −1.3138 V and −14.3288 V, respectively, and the maximum effort is 2.8178 V, which are feasible.

In this paper, a NADRC, as a voltage controller, is originally developed on the nonlinear model of a parallel-plate ESA system. The nonlinear ESA with NADRC is simulated in Matlab/Simulink. Simulation results verify the effectiveness of the NADRC in successfully extending ESA's travel range to 99.99% of its full gap with and without the disturbances, despite the presences of system uncertainties and measurement noise. The control effort is shown to be practically achievable. Moreover, the theoretical proof is developed for the convergence of observer error for NESO. In the future, the hardware implementation of the NADRC on a real ESA is to be conducted. We also plan to apply the NADRC to the ESA with multi- degrees-of-freedom.

The authors would like to thank Dr. Zhiliang Zhao for his valuable inputs in proving the convergence of observer error for NESO.

Hsu, T. R. , 2008, MEMS and Microsystems: Design, Manufacture, and Nanoscale Engineering, 2nd ed., Wiley, Hoboken, NJ.
Guardia, R. A. , Dehe, A. , Aigner, R. , and Castaner, L. M. , 2002, “ Current Drive Methods to Extend the Range of Travel of Electrostatic Microactuators Beyond the Voltage Pull-in Point,” J. Micro-Electro-Mech. Syst., 11(3), pp. 255–263. [CrossRef]
Seeger, J. I. , and Boser, B. E. , 2003, “ Charge Control of Parallel-Plate, Electrostatic Actuators and the Tip-In Instability,” J. Micro-Electro-Mech. Syst., 12(5), pp. 656–671. [CrossRef]
Wickramasinghe, I. P. M. , Maithripala, D. H. S. , Kawade, B. D. , Berg, J. M. , and Dayawansa, W. P. , 2009, “ Passivity-Based Stabilization of 1-DOF Electrostatic MEMS Model With a Parasitic Capacitance,” IEEE Trans. Control Syst. Technol., 17(1), pp. 249–256. [CrossRef]
Zhu, G. , Levine, J. , and Praly, L. , 2005, “ Improving the Performance of an Electrostatically Actuated MEMS by Nonlinear Control: Some Advances and Comparisons,” 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Seville, Spain, Dec. 12–15, pp. 7534–7539.
Zhu, G. , Levine, J. , Praly, L. , and Peter, Y. A. , 2006, “ Flatness-Based Control of Electrostatically Actuated MEMS With Application to Adaptive Optics: A Simulation Study,” J. Micro-Electro-Mech. Syst., 15(5), pp. 1165–1174. [CrossRef]
Zhu, G. , Praly, L. , and Levine, J. , 2007, “ Stabilization of an Electrostatic MEMS Including Uncontrollable Linearization,” 46th IEEE Conference on Decision and Control (CDC), New Orleans, LA, Dec. 12–14, pp. 2433–2438.
Maithripala, D. H. S. , Berg, J. M. , and Dayawansa, W. P. , 2004, “ Control of an Electrostatic Micro-Electro-Mechanical System Using Static and Dynamic Output Feedback,” ASME J. Dyn. Syst. Meas. Control, 127(3), pp. 443–450. [CrossRef]
Dong, L. , and Edward, J. , 2010, “ Closed-Loop Voltage Control of a Parallel-Plate MEMS Electrostatic Actuator,” American Control Conference (ACC), Baltimore, MD, June 30–July 2, pp. 3409–3414.
Shirazi, F. A. , Velni, J. M. , and Grigoriadis, K. M. , 2011, “ A LPV Design Approach for Voltage Control of an Electrostatic MEMS Actuator,” J. Micro-Electro-Mech. Syst., 20(1), pp. 302–311. [CrossRef]
Alwi, H. , Zolotas, A. , Edwards, C. , and Grigoriadis, K. , 2012, “ Sliding Mode Control Design of an Electrostatic Micro-Actuator Using LPV Schemes,” American Control Conference (ACC), Motreal, QC, Canada, June 27–29, pp. 875–880.
Vagia, M. , and Tzes, A. , 2009, “ Modeling Aspects and Gain Scheduled H∞ Controller Design for an Electro-Static Micro-Actuator With Squeezed Gas Film Damping Effects,” American Control Conference (ACC), St. Louis, MO, June 10–12, pp. 4805–4810.
Rocha, L. A. , Cretu, E. , and Wolffenbuttel, R. F. , 2006, “ Using Dynamic Voltage Drive in a Parallel-Plate Electrostatic Actuator for Full-Gap Travel Range and Positioning,” J. Micro-Electro-Mech. Syst., 15(1), pp. 69–83. [CrossRef]
Senturia, S. D. , 2001, Microsystem Design, Springer Science and Business Media, New York.
Xue, W. , and Huang, Y. , 2015, “ Performance Analysis of Active Disturbance Rejection Tracking Control for a Class of Uncertain LTI Systems,” ISA Trans., 58, pp. 133–154. [CrossRef] [PubMed]
Huang, Y. , Xue, W. , Zhiqiang, G. , Sira-Ramirez, H. , Wu, D. , and Sun, M. , 2014, “ Active Disturbance Rejection Control: Methodology and Theoretical Analysis,” ISA Trans., 53(4), pp. 963–976. [CrossRef] [PubMed]
Chen, W. H. , Yang, J. , Guo, L. , and Li, S. , 2016, “ Disturbance Observer-Based Control and Related Methods: An Overview,” IEEE Trans. Ind. Electron., 63(2), pp. 1083–1095. [CrossRef]
Shao, S. , and Gao, Z. , 2017, “ On the Conditions of Exponential Stability in Active Disturbance Rejection Control Based on Singular Perturbation Analysis,” Int. J. Control, 90(10), pp. 1–13. [CrossRef]
Gao, Z. , 2015, “ Active Disturbance Rejection Control: From an Enduring Idea to an Emerging Technology,” Tenth IEEE International Workshop on Robot Motion and Control (RoMoCo), Poznan, Poland, July 6–8, pp. 269–282.
Guo, B. , and Zhao, Z. , 2011, “ On Convergence of an Extended State Observer for Nonlinear Systems With Uncertainty,” Syst. Control Lett., 60(6), pp. 420–430. [CrossRef]
Guo, B. , and Zhao, Z. , 2016, “ On Convergence of Nonlinear Active Disturbance Rejection for SISO Nonlinear Systems,” J. Dyn. Control Syst., 22(2), pp. 385–412. [CrossRef]
Guo, B. , and Zhao, Z. , 2012, “ On Convergence of the Nonlinear Extended State Observer for MIMO Systems With Uncertainty,” IET Control Theory Appl., 6(15), p. 2375.
Guo, B. , and Zhao, Z. , 2013, “ On Convergence of the Nonlinear Active Disturbance Rejection Control for MIMO Systems,” SIAM J. Control Optim., 51(2), pp. 1727–1757. [CrossRef]
Dong, L. , Kandula, P. , Gao, Z. , and Wang, D. , 2010, “ On a Robust Control System Design for an Electric Power Assist Steering System,” American Control Conference (ACC), Baltimore, MD, June 30–July 2, pp. 5356–5361.
Dong, L. , Kandula, P. , Wang, D. , and Gao, Z. , 2010, “ Active Disturbance Rejection Control for an Electric Power Assist Steering System,” J. Intell. Control Syst., 15(1), pp. 18–24. https://pdfs.semanticscholar.org/efcc/1a3232e47b10be4b8213fa037c705b78b2c1.pdf
Dong, L. , Zhang, Y. , and Gao, Z. , 2012, “ A Robust Decentralized Load Frequency Controller for Interconnected Power Systems,” ISA Trans., 51(3), pp. 410–419. [CrossRef] [PubMed]
Chen, Z. , Zheng, Q. , and Gao, Z. , 2007, “ Active Disturbance Rejection Control of Chemical Processes,” IEEE International Conference on Control Applications (CCA), Singapore, Oct. 1–3, pp. 855–861.
Zhang, H. , Sun, Y. , Gao, Z. , and Wang, Y. , 2014, “ A Disturbance Rejection Framework for the Study of Traditional Chinese Medicine,” Evidence-Based Complementary Altern. Med., 2014(6), p. 787529.
Dong, L. , and Avanesian, D. , 2009, “ Drive-Mode Control for Vibrational MEMS Gyroscopes,” IEEE Trans. Ind. Electron., 56(4), pp. 956–963. [CrossRef]
Gao, Z. , 2006, “ Active Disturbance Rejection Control: A Paradigm Shift in Feedback Control System Design,” American Control Conference (ACC), Minneapolis, MN, June 14–16, pp. 2399–2406.
Tian, G. , and Gao, Z. , 2007, “ Frequency Response Analysis of Active Disturbance Rejection Based Control System,” IEEE International Conference on Control Applications (CCA), Singapore, Oct. 1–3, pp. 1595–1599.
Dong, L. , Zheng, Q. , and Gao, Z. , 2008, “ On Control System Design for the Conventional Mode of Operation of Vibrational Gyroscopes,” IEEE Sens. J., 8(11), pp. 1871–1878. [CrossRef]
Han, J. , 2009, “ From PID to Active Disturbance Rejection Control,” IEEE Trans. Ind. Electron., 56(3), pp. 900–906. [CrossRef]
Gao, Z. , 2014, “ On the Centrality of Disturbance Rejection in Automatic Control,” ISA Trans., 53(4), pp. 850–857. [CrossRef] [PubMed]
Slotine, J. J. E. , and Li, W. , 1991, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ.
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References

Hsu, T. R. , 2008, MEMS and Microsystems: Design, Manufacture, and Nanoscale Engineering, 2nd ed., Wiley, Hoboken, NJ.
Guardia, R. A. , Dehe, A. , Aigner, R. , and Castaner, L. M. , 2002, “ Current Drive Methods to Extend the Range of Travel of Electrostatic Microactuators Beyond the Voltage Pull-in Point,” J. Micro-Electro-Mech. Syst., 11(3), pp. 255–263. [CrossRef]
Seeger, J. I. , and Boser, B. E. , 2003, “ Charge Control of Parallel-Plate, Electrostatic Actuators and the Tip-In Instability,” J. Micro-Electro-Mech. Syst., 12(5), pp. 656–671. [CrossRef]
Wickramasinghe, I. P. M. , Maithripala, D. H. S. , Kawade, B. D. , Berg, J. M. , and Dayawansa, W. P. , 2009, “ Passivity-Based Stabilization of 1-DOF Electrostatic MEMS Model With a Parasitic Capacitance,” IEEE Trans. Control Syst. Technol., 17(1), pp. 249–256. [CrossRef]
Zhu, G. , Levine, J. , and Praly, L. , 2005, “ Improving the Performance of an Electrostatically Actuated MEMS by Nonlinear Control: Some Advances and Comparisons,” 44th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Seville, Spain, Dec. 12–15, pp. 7534–7539.
Zhu, G. , Levine, J. , Praly, L. , and Peter, Y. A. , 2006, “ Flatness-Based Control of Electrostatically Actuated MEMS With Application to Adaptive Optics: A Simulation Study,” J. Micro-Electro-Mech. Syst., 15(5), pp. 1165–1174. [CrossRef]
Zhu, G. , Praly, L. , and Levine, J. , 2007, “ Stabilization of an Electrostatic MEMS Including Uncontrollable Linearization,” 46th IEEE Conference on Decision and Control (CDC), New Orleans, LA, Dec. 12–14, pp. 2433–2438.
Maithripala, D. H. S. , Berg, J. M. , and Dayawansa, W. P. , 2004, “ Control of an Electrostatic Micro-Electro-Mechanical System Using Static and Dynamic Output Feedback,” ASME J. Dyn. Syst. Meas. Control, 127(3), pp. 443–450. [CrossRef]
Dong, L. , and Edward, J. , 2010, “ Closed-Loop Voltage Control of a Parallel-Plate MEMS Electrostatic Actuator,” American Control Conference (ACC), Baltimore, MD, June 30–July 2, pp. 3409–3414.
Shirazi, F. A. , Velni, J. M. , and Grigoriadis, K. M. , 2011, “ A LPV Design Approach for Voltage Control of an Electrostatic MEMS Actuator,” J. Micro-Electro-Mech. Syst., 20(1), pp. 302–311. [CrossRef]
Alwi, H. , Zolotas, A. , Edwards, C. , and Grigoriadis, K. , 2012, “ Sliding Mode Control Design of an Electrostatic Micro-Actuator Using LPV Schemes,” American Control Conference (ACC), Motreal, QC, Canada, June 27–29, pp. 875–880.
Vagia, M. , and Tzes, A. , 2009, “ Modeling Aspects and Gain Scheduled H∞ Controller Design for an Electro-Static Micro-Actuator With Squeezed Gas Film Damping Effects,” American Control Conference (ACC), St. Louis, MO, June 10–12, pp. 4805–4810.
Rocha, L. A. , Cretu, E. , and Wolffenbuttel, R. F. , 2006, “ Using Dynamic Voltage Drive in a Parallel-Plate Electrostatic Actuator for Full-Gap Travel Range and Positioning,” J. Micro-Electro-Mech. Syst., 15(1), pp. 69–83. [CrossRef]
Senturia, S. D. , 2001, Microsystem Design, Springer Science and Business Media, New York.
Xue, W. , and Huang, Y. , 2015, “ Performance Analysis of Active Disturbance Rejection Tracking Control for a Class of Uncertain LTI Systems,” ISA Trans., 58, pp. 133–154. [CrossRef] [PubMed]
Huang, Y. , Xue, W. , Zhiqiang, G. , Sira-Ramirez, H. , Wu, D. , and Sun, M. , 2014, “ Active Disturbance Rejection Control: Methodology and Theoretical Analysis,” ISA Trans., 53(4), pp. 963–976. [CrossRef] [PubMed]
Chen, W. H. , Yang, J. , Guo, L. , and Li, S. , 2016, “ Disturbance Observer-Based Control and Related Methods: An Overview,” IEEE Trans. Ind. Electron., 63(2), pp. 1083–1095. [CrossRef]
Shao, S. , and Gao, Z. , 2017, “ On the Conditions of Exponential Stability in Active Disturbance Rejection Control Based on Singular Perturbation Analysis,” Int. J. Control, 90(10), pp. 1–13. [CrossRef]
Gao, Z. , 2015, “ Active Disturbance Rejection Control: From an Enduring Idea to an Emerging Technology,” Tenth IEEE International Workshop on Robot Motion and Control (RoMoCo), Poznan, Poland, July 6–8, pp. 269–282.
Guo, B. , and Zhao, Z. , 2011, “ On Convergence of an Extended State Observer for Nonlinear Systems With Uncertainty,” Syst. Control Lett., 60(6), pp. 420–430. [CrossRef]
Guo, B. , and Zhao, Z. , 2016, “ On Convergence of Nonlinear Active Disturbance Rejection for SISO Nonlinear Systems,” J. Dyn. Control Syst., 22(2), pp. 385–412. [CrossRef]
Guo, B. , and Zhao, Z. , 2012, “ On Convergence of the Nonlinear Extended State Observer for MIMO Systems With Uncertainty,” IET Control Theory Appl., 6(15), p. 2375.
Guo, B. , and Zhao, Z. , 2013, “ On Convergence of the Nonlinear Active Disturbance Rejection Control for MIMO Systems,” SIAM J. Control Optim., 51(2), pp. 1727–1757. [CrossRef]
Dong, L. , Kandula, P. , Gao, Z. , and Wang, D. , 2010, “ On a Robust Control System Design for an Electric Power Assist Steering System,” American Control Conference (ACC), Baltimore, MD, June 30–July 2, pp. 5356–5361.
Dong, L. , Kandula, P. , Wang, D. , and Gao, Z. , 2010, “ Active Disturbance Rejection Control for an Electric Power Assist Steering System,” J. Intell. Control Syst., 15(1), pp. 18–24. https://pdfs.semanticscholar.org/efcc/1a3232e47b10be4b8213fa037c705b78b2c1.pdf
Dong, L. , Zhang, Y. , and Gao, Z. , 2012, “ A Robust Decentralized Load Frequency Controller for Interconnected Power Systems,” ISA Trans., 51(3), pp. 410–419. [CrossRef] [PubMed]
Chen, Z. , Zheng, Q. , and Gao, Z. , 2007, “ Active Disturbance Rejection Control of Chemical Processes,” IEEE International Conference on Control Applications (CCA), Singapore, Oct. 1–3, pp. 855–861.
Zhang, H. , Sun, Y. , Gao, Z. , and Wang, Y. , 2014, “ A Disturbance Rejection Framework for the Study of Traditional Chinese Medicine,” Evidence-Based Complementary Altern. Med., 2014(6), p. 787529.
Dong, L. , and Avanesian, D. , 2009, “ Drive-Mode Control for Vibrational MEMS Gyroscopes,” IEEE Trans. Ind. Electron., 56(4), pp. 956–963. [CrossRef]
Gao, Z. , 2006, “ Active Disturbance Rejection Control: A Paradigm Shift in Feedback Control System Design,” American Control Conference (ACC), Minneapolis, MN, June 14–16, pp. 2399–2406.
Tian, G. , and Gao, Z. , 2007, “ Frequency Response Analysis of Active Disturbance Rejection Based Control System,” IEEE International Conference on Control Applications (CCA), Singapore, Oct. 1–3, pp. 1595–1599.
Dong, L. , Zheng, Q. , and Gao, Z. , 2008, “ On Control System Design for the Conventional Mode of Operation of Vibrational Gyroscopes,” IEEE Sens. J., 8(11), pp. 1871–1878. [CrossRef]
Han, J. , 2009, “ From PID to Active Disturbance Rejection Control,” IEEE Trans. Ind. Electron., 56(3), pp. 900–906. [CrossRef]
Gao, Z. , 2014, “ On the Centrality of Disturbance Rejection in Automatic Control,” ISA Trans., 53(4), pp. 850–857. [CrossRef] [PubMed]
Slotine, J. J. E. , and Li, W. , 1991, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ.

Figures

Grahic Jump Location
Fig. 1

Electromechanical model of ESA [9]

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Fig. 2

Block diagram of NADRC with ESA

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Fig. 3

Step responses of ESA: (a) scaled displacement output, (b) normalized velocity, and (c) normalized charge

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Fig. 4

Scaled control signal for ESA

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Fig. 5

Scaled displacement outputs of ESA with parameter variations

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Fig. 6

Scaled control signals with parameter variations

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Fig. 8

Scaled displacement output of ESA with measurement noise

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Fig. 9

Scaled control signal with measurement noise

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Fig. 10

Scaled displacement output of ESA with unit step disturbance

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Fig. 11

Scaled displacement output of ESA with large step disturbance

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Fig. 12

Scaled control signals in the presence of step disturbances

Tables

Table Grahic Jump Location
Table 1 Controller gains
Table Grahic Jump Location
Table 2 Parameter values of ESA [10]

Errata

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