An adaptive sliding mode controller design to cope with unmatched uncertainties and disturbance in a MEMS voltage reference source

Ehsan Ranjbar 1 · Mostafa Yaghoubi 1 · Amir Abolfazl Suratgar

Abstract Tunable micro-electro-mechanical systems (MEMS) capacitors as the fundamental parts are embedded in MEMS AC voltage reference sources (VRS). Being concerned with the accuracy of the output voltage in the reference sources, it gets important to address uncertainties in the physical parameters of the MEMS capacitor. The uncertainties have the great inevitable potentiality of bringing about output voltage perturbation. The output deterioration is more remarkable when the uncertainties are accompanied by disturbance and noise. Manufacturers have been making great attempts to make the MEMS adjustable capacitor with desired rigorous physical characteristics. They have also tried to mitigate physical parameter veracity. However,ambiguity in the values of the parameters inescapably occurs in fabrication procedures since the micro-machining process might itself suff er from uncertainties. Employing a proportional integral (PI) adaptive sliding mode controller (ASMC),both terms of matched and unmatched uncertainties as well as the disturbance, are addressed in this work for the MEMS AC VRS so that a strict voltage is stabilized while the system is simultaneously subjected into uncertainties and exogenous disturbance. Cross-talk, some inertial forces, and electrostatic coercions may appear as matched and unmatched disturbances.Alteration in stiff ness and damping coeffi cients might also take place as matched uncertainties due to variations in the fabrication process or even working environment. The simulation results in the paper are persuasive and the controller design has shown a satisfactory tracking performance.

Keywords Matched and unmatched disturbance · Uncertainty · MEMS · Adaptive sliding mode controller · Proportional and integral sliding surface · Capacitive plates

1 Introduction

Basic principles of AC and DC MEMS (micro-electromechanical systems)-based voltage reference source (VRS),used in a simple circuit to generate a constant voltage, were initially presented by [ 1] in which a micro-parallel-moving plate capacitor was proposed as the primary working element. Precise output voltage, between plates of a MEMS tunable capacitor, is obtained due to the capacitor characteristic established upon noticeable mechanical stability of single-crystalline silicon. MEMS tunable capacitors are popular devices usable in electronic circuits due to costeffi ciency and tiny size. Being concerned with the MEMS tunable capacitor characteristics as the leading element of the voltage reference sources, other researchers developed the VRS to enhance their veracity. Some articles, such as[ 2- 7], have concentrated over micro-machining of MEMS adjustable capacitors. Not only have they mainly targeted to propose and produce novel schemes of tiny adjustable capacitors, but they also have tried to enhance the quality of the f inal made MEMS VRS.

Fig. 1 a Lumped model of the MEMS tunable capacitor. b Movable plate reaches one third of its initial distance to the f ixed plate around the pull-in point [ 23, 28]

Most probably, pull-in point is the most important common term used in association with MEMS adjustable capacitors of VRS. It is a sensitive point and determines a peculiar quality which stems from electromechanical coupling between the applied electrostatic force and the spring reaction force. The operational principle is put into a nutshell via illustrative f igures of Fig. 1 (a), (b). As depicted,the electrostatic force increases while the electrical charge between the capacitor plates gets accumulated. Therefore,the movable plate is progressively repelled towards the stationary one up to the threshold point. The threshold point,where the spring force does not tolerate the electrostatic coercion any more, is entitled as the pull-in point. The voltage changes between the capacitor plates proportionally function the square of def lection in the moving plate around the pull-in point. Hence, the VRS would give a regulated output being operated about the pull-in point, and the MEMS VRS is ultimately working adjacent to the pull-in point of its MEMS adjustable capacitor. About the pull-in point, some instability is engendered by mechanical stress and electrical charge over dielectric layer. K?rkk?inen et al.[ 8] innovated MEMS AC VRS and substantiated that if AC voltage is employed instead of DC voltage in MEMS adjustable capacitor, the charge accumulation over the dielectric layer will get diminished. Here, the MEMS AC VRS operation is considered.

Micro-fabrication technology has advanced in such a way that its industry is capable of fabricating MEMS tunable capacitors with substantially rigorous tiny dimensions and diminished mechanical stress. However, demanding permanent stability and critical rigorousness of reference voltage require design and implementation of controllers for MEMS voltage references or tunable capacitors. The recent control engineering state-of-the-art reveals that competent control methods are used for MEMS, robotic and other mechatronic systems to diminish error and augment performance [ 9- 14]. The pertinent application is sometimes extended. Being concerned with micro-displacement such as one in our work here, one might point out control designs to drive micro-stepper motors. The drive has been carried out in such a way that its setup for computer numerical control (CNC) machining application,where material removal takes place in a precise manner according to the trajectory ofX-Ypositioning stages, is quite successful [ 15]. Furthermore, some of the recent control research, involving adaptive and robust control techniques, have been applied regarding modeling defects in MEMS, parametric uncertainties and even unstructured dynamics [ 16- 20].

Mehrnezhad et al. [ 21] introduced a state space model to exploit disparate modern control methods for MEMS adjustable capacitors practically. The controllers in Refs. [ 22- 24]are employed to drive the movable plate towards the pull-in point and sustain its stability while the system is subjected to parameter uncertainty or slow variation such as stiff ness variation of the plate cantilevers. It should be notif ied that the pull-in stability is reached using a plain pole-placement state feedback controller but the ineluctable capacitor parametric uncertainties necessitate design of more complex controllers. Although unknownness of exact values for spring and damping coeffi cients might not ruin the pull-in stability and regulation totally on its own, the stability condition gets exacerbated when electrical measurement and mechanical noises as well as parameter variation are deemed as practical factors having impact over system dynamics and performance. Environmental changes, such as temperature, also have inf luence over the adjustable capacitor structure, and consequently impose considerable impact on stiff ness and damping factors. On the other hand, damping and stiff ness coeffi cients are also prone to alteration due to mechanical noise, vibration harmonics and cross-talk.

Our concern in this work is design of an adaptive sliding mode controller (ASMC), considering the system subjected not only to matched and unmatched disturbance but also uncertainty in the dynamics of the plates of the adjustable capacitor. A proportional integral (PI) sliding surface is considered for design of the ASMC to take benef it of the pull-in merits. The controller simultaneously confronts both matched and unmatched terms in the dynamics of the MEMS adjustable capacitive plates, and in particular makes the movable plate track or regulate around the pull-in point while displacing without showing detrimental motion overshoots. Actually, this article contributes to management and rejection of uncertainty and disturbance, respectively,of both matched and unmatched varieties in the dynamical equation of the moving micro-plate. Instead of a conventional f irst-order sliding surface used in Ref. [ 23], a PI sliding surface is considered in this article. Having simulated the proposed controller, it is persuasively concluded that suggested ASMC design in this paper shows a better performance in comparison to the previous ASMC design simulated in Ref. [ 23]. The method in this article is similar to Ref. [ 25]; however, the uncertainties and disturbance have known values for their upper bounds which makes us adopt another Lyapunov function candidate with less arguments and a more simplif ied structure.

For more clarif ication on the appearance of the matched and unmatched terms in the system dynamics, it should be acknowledged that unmatched disturbance and uncertainty terms entail stiff ness cross-talk, damping cross-talk,exogenous inertial coercion, and gravity weight coercion.The matched disturbance and uncertainties correspondingly involve electrostatic coercion, stiff ness mismatch and damping mismatch. It should be mentioned that not only the stability of the proposed ASMC has been mathematically proved while considering both matched and unmatched uncertainty, but also the simulation results in association with estimation of stiff ness and damping factors encourage one of the stability due to boundedness of the corresponding estimation and tracking errors.

The main highlights and contributions of this paper are described in the following items:

· Unlike previous works [ 26, 27] on modeling and control of MEMS capacitive plates with sinusoidal oscillations,this article concentrates on control of displacement of MEMS capacitive plates without no overshoot and in a very precise scale less than micro-meter, regulating around an ultimate point which is called the pull-in point,in existence of parameter uncertainties and disturbance.

· Contrary to previous works [ 21, 24], various kinds of disturbances, including electrostatic parasitic force, gravity force enforcing the movable plate, external inertial force and lumped modeling nonlinearities are considered in the tunable capacitor system dynamics. The described disturbance, nonlinearities as well as stiff ness and damping parametric uncertainties are separated as matched and unmatched terms to be addressed appropriately in the design of the controller. The proposed controller copes with both matched and unmatched terms. This separation has not been considered in previous works such as[ 21- 24].

· Unlike previous work [ 25], the matched and unmatched lumped uncertainties and disturbance are regarded bounded and the proposed Lyapunov function owns a more simple structure.

· The ASMC takes advantage of PI sliding surface. Actually, the adaptive PI-based ASMC presents a quick convergence of the sliding surface. Additionally, the implementation of the proposed controller requires tuning of the PI parameters without acquiring the knowledge of bounds on system uncertainties. The sliding surface in the similar previous works [ 23, 24] is not a PI surface and the convergence time is longer. The convergence time in the proposed control system is also a little bit smaller in comparison with [ 28].

· The proposed control method has been applied to the MEMS tunable capacitor for voltage regulation purposes for the f irst time.

We have structured the paper in the following format: the operation essentials and modeling for the MEMS adjustable capacitor is addressed in Sect. 2. Preliminary mathematics for design of the controller are declared afterwards in Sect. 3. Next, Sect. 4 concedes simulation results. Conclusions and future prospectives are ultimately presented to the reader in Sect. 5.

2 MEMS tunable capacitor operating principles

An adjustable capacitor essentially entails a pair of unmovable and movable plates. In this study, one of the typical adjustable capacitors, illustrated in Fig. 1 ( a), (b), is conversed. Actually, the capacitor is MEMS parallel plates,encompassing a movable plate which is capable of modulating the inter-plate gap. In this work, regarding both matched and unmatched terms of parameter uncertainty and disturbance in the adjustable capacitor, we will try to enhance tracking performance of the moving plate which is charged.The plate displacement is considerably stiff ened and it is prevented from settling at the pull-in point [ 23].

Sometimes, models suggested for tunable capacitor exploit simplifying assumptions which cause uncertainty in some parameters appearing in the dynamics of the plate such as stiff ness factor. It transpires as one of the most important properties of the movable plate suspension system. Additionally, device aging and change in working status exhaust desired mechanical properties which itself brings about parameter uncertainty. Neither should it be neglected that manufacturing procedures themselves suff er from uncertainty in process planning and implementation which play the role of another source of uncertainty in parameters of the fabricated device. For instance, plate mass, stiff ness and damping factors are all subjected to uncertainty in their values. That is the reason controllers equipped with appropriate control algorithms, which are competent of managing parameter uncertainty as well as rejecting disturbance,should be employed with these micro-devices or within their electronic boards.

Capacitance for a MEMS adjustable capacitor is presented by Parameters in Eq. ( 1), includingε,d,xandA, are the permittivity, the initial distance between the plates, the movable plate displacement and the plate area. Actuation of the capacitor is done by an alternating current represented by

Electrodes are captivated towards each other by the electrostatic force symbolized withFE. Spring force, which is equal tokx, hinders the plates from getting close to each other. These two forces are expressed by Eq. ( 3) regarding Fig. 1( a) [ 29, 30]:The corresponding RMS equations of the pull-in voltage and currents are presented in Ref. [ 28]. Small variation in the input current brings about a small alteration in the pull-in voltage, supporting the solution of exploitation of adjustable capacitor as the major element in a VRS circuitry. The proof mathematics are given in Ref. [ 21, 28]. It is just stated as the result referring to the corresponding references for mathematical proof. Iffacis assumed to be much larger thanfm,the electrostatic force in Eq. ( 3) can transform into the electrostatic force mean value, resulting in Eq. ( 5) [ 21, 25, 28],

One of the main problematic issues in exploitation of this device is emergence of cross stiff ness and damping forces along thexdirection. The reason is the slight diff erence among the lengths of the three cantilevers, which are displayed on Fig. 1 asL1,L2andL3and are suspending the moveable plate of the adjustable cap. Actually they lead the plate mass center to deviate fromxdirection intoyandzdirections. The deviations are partial; however, they could be deemed in the plant dynamics to design a controller to cope with them, increasing tracking performance accuracy(Fig. 2). Consequently, in a more real world, the dynamical model could be re-expressed by the following equation [ 28]:

Fig. 2 Components of the vector of the gravity force [ 28]

In Eq. ( 6),dEdsymbolizes the bounded electrostatic disturbance while Δkand Δbare, respectively, bounded uncertainties of stiff ness and damping coeffi cients. Additionally,the term ofmgcosψcosθis the component of the weight force of the adjustable cap along thexdirection. It should be explained that the angle ofψis the sweeping angle between the weight force vector and the plane ofXY. On the other hand,θis the angle between the previous mentioned component of the weight force vector (mgcosψ) andxdirection.The last term, which does not deserve of being neglected, is any inertial force which might come to existence when the VRS is used in a displacing accelerated system. It should be reminded thatkxy,kxz,dxyanddxzare stiff ness and damping crosstalk factors betweenX-and-Ydirection as well asX-and-Z. Approaching our goal stated in Sect. 1, we are concerned with designing a controller to make the displaceable capacitor track a reference model so that it could be placed at the pull-in point. The displacement quality should be managed in such a way that the overshoots are controlled and no damage would occur to the cantilever micro-structure during the transient trajectory of the distance travel. This target gets controversial when one considers the system being prone to described uncertain and disturbing condition. The reference model, which the movable plate has to track, determines the displacement quality and is given by

wherearandbrare constants vouching for the reference model stability. The mentioned constants as well ascrare responsible for adjustment of transient and steady-state response in the reference model behavior.

If the state space vector of the system is adopted as

the dynamics of the adjustable capacitive plate can be reexpressed using state-space approach:

andU=u(t) . The system description is a single input depiction in our study. Likewise, if the state space vector for the reference model is adopted as

3 Mathematics for design of the controller

Having considered parameter uncertainties as well as disturbance and obtained Eq. ( 14), one may comprehensively re-write dynamics of the adjustable capacitor with the following equational structure:

whereA,X=[x˙x]T,B, andu(t) are, respectively, the system matrix, the state space vector, the input manipulation matrix,and the system input. Unknown parameter uncertainties as well as disturbance, which is imposed on the system in both unmatched and matched manners, are, respectively, symbolized by ΔAanddin Eq. ( 15). These terms are in association with input manipulation matrix. The reference model for tuning the pull-in point in the adjustable MEMS cap is given by in whichAr,Xr=[xr˙xr]T,Br, andr(t) are, respectively, the reference model system matrix, the desired trajectory vector,the input matrix, and the reference input. The mathematical description of matched and unmatched uncertainty and disturbance is given by

In Eq. ( 19),lmandlumare the lumped matched and unmatched uncertainty and disturbance, respectively. Actually, they are expressed by the following equations:

In this article, it is assumed that matched and unmatched terms of uncertainties and disturbances are knowingly bounded:

should all be put into one’s consideration to calculate time-derivative of the tracking error with the subsequent work-out:

A proportional and integral structure of the tracking error is deemed for the sliding surface vector, which is symbolized withs(t), here in this work in contrast with [ 23]. In other terms,

in whichλ1,λ2andλ＞0 . Additionally,λ1andλ2are chosen so that [ΛB] is nonsingular. Time derivative of the sliding surface will be equal to

Equivalent control law to set time-derivative of the sliding surface to zero is obtained by

in which＾Φ1and＾Φ2are estimates ofΦ1andΦ2, respectively.It is notable that

This leads˙sto get expressed by

in which tr stands for trace of a square matrix andΓ1∈?n×nas well asΓ2∈?m×mare constant symmetric positive def inite matrixes (m=1 andn=2 ). Note thatnis the number of the system states. Time derivative of the Lyapunov candidate function will be calculated subsequently. Based on matrix transpose characteristic [ 34],˙Vwill be obtained in the consequent manner:

Equation ( 49) is comprised of all separate scalar terms so it is restated as

Trace properties [ 34] cause Eq. ( 50) to change into

Again, applying trace properties on Eq. ( 51) successively leads us to have

If one regardstrace property [ 34], he will f ind out timederivative of the Lyapunov function in the following format:

As guessed, our approach is making˙Vnegative semi-def inite, regarding stagnant alteration of the parameters. Achieving this aim, one may try to establish two update laws running in the control scheme,

ΛBis a scalar. Consequently, Eq. ( 55) can be restated in the following format:

wherec1andc2are scalar values. Ifc1≥c2, then the system will be stable. However, for a strong proof, for generalizing the proposed control method, and for making it applicable to similar dynamical systems with higher degrees, another proof of stability is demonstrated subsequently whileΛBis not a scalar. Note that [ΛBη]∈?m×mis an invertible matrix.One may write the inequality of ( 57) using linear algebra knowledge [ 35- 37],whereσis the eigenvalue of [ΛBη] . The inequality of ( 57)could be developed more,

Taking into account Eqs. ( 55) and ( 59),

The sliding condition is given by Eq. ( 69) which leads to

because˙V(s,～Φ1,～Φ2) gets negative semi-def inite. Applying an integral fromt1=0 tot2=tto Eq. ( 69) yields the following inequality:SinceVis positive def inite and it is non-increasing, one may state that Now, we are going to investigate if˙s(t)∈L∞. All terms, in Eq. ( 46), have been proved or assumed to be bounded except for the terms containing states ofXsuch asΛB～Φ1X. It is needed to prove boundedness ofXwhich itself gets back to boundedness ofesincee=X-Xm. Equations in ( 54) state that

Since～Φ1and～Φ2are bounded terms, the integrals over the right side of Eqs. ( 75) will be bounded. As a consequence,the term ofX(t) will be bounded due to boundedness ofs(τ)and all other terms inside the integrals except forX(τ) . The reason is that, according to [ 38] and Riemann Integrability

Table 1 Simulation values of the main parameters of the adjustable capacitor in the MEMS AC voltage reference source

Theorem,X(t) would be bounded in any def inite f inite time interval. Hence,Exploiting Eq. ( 74), the f irst subequation of ( 70), and ( 76)as necessary suppositions for Barbalat’s Lemma, zero-convergence ofs(t) is vouched for regarding [ 37],

4 Control system simulation

Parameter values of Table 1 are utilized for simulating the control scheme. The overall ASMC control system is depicted in Fig. 3 as well. It is easily observable that an ASMC design may entail four main parts: model reference,sliding surface generator, parameter update law, and control law generator. The simulation block diagram of the suggested ASMC is displayed in Fig. 4 accompanied by the associated table below which explains equations inside blocks.

An uncertainty with 0.05 divergence from the nominal value of the damping and stiff ness factors was adopted. For instance, they were adopted asb=1.05bnandk=1.05kn,wherebnandknare, respectively, the nominal values of damping and stiffness factors. A constant external disturbance entitled bydEdand equal to 1mN , is considered to be imposed on the system. The cross damping and stiffness are considered asdxy=dxz=0.001 N · s/m andkxy=kxz=1 N/m. Furthermore, weight force and crosstalk forces are also applied over the mass as additive terms inside Block No. 1. Angle ofθis approximately deemed by 15°. Respectively, cross displacements and velocities are put into consideration via adoption of the following values:˙y=˙z=0.1μm/s andy=z=0.1μm, in simulation scheme.Parameters, used in the simulation f ile for the controller,are set with the subsequent values:Γ1=1000I2,Γ2=1000 ,Λ=0.000001I2, andη=50,000 . Reference model parameters are introduced to the simulation scheme adoptingar=10,000 ,br=25,000,000 andcr=25,000,000 while the initial conditions are considered as zero.

Fig. 3 The diagram for design of a PI adaptive sliding mode controller

Fig. 4 Simulation scheme of the PI ASMC design considering matched and unmatched disturbance as well as uncertainties

Figure 5 depicts the results of the designed ASMC simulation. It is so convincing that one may welcome the new controller performance as a terrif ic outcome because the f inal goal of tracking of the desired trajectory is achieved in a precise manner. The displacement status is also satisfactory since the moveable plate placement at the pull-in point is obtained via a transient trajectory which is quite non-destructive. As it is observed in Fig. 5( a), the plate displacement is exactly 0.667μm far from its initial distance.The red and blue curves, in Fig. 5( a), are, respectively, the desired and real trajectories of the moveable plate displacement calledxr(t) andx(t). The blue and red curves shown in other diagram, as the f irst plot in Fig. 5( b), are the plate displacement and velocity errors, respectively. The second plot in Fig 5( b) depicts the sliding surface error close to zero while the third plot displays the control eff ort and disturbance foisted upon the system. By and large, Fig. 5 demonstrates zero-convergence of the tracking errors, representative of a perfect tracking performance in our design.Boundedness of＾Φ1and＾Φ2, which promote mathematically proven system stability, are displayed on Fig. 6.

As it is seen in Fig. 5,x(t) is leadingxr(t) for 5% uncertainty in the parameters of the stiffness and damping( Δk=0.05kn, and Δb=0.05bn). For other values of uncertainty,x(t) will not be leadingxr(t) anymore. The sweeping values of 45% , 35% , 25% , 15% and 5% for the parameter uncertainty are adopted for re-simulation of the controller.The resulted tracking performance ofx(t) is illustrated in Fig. 7 which demonstrates that it will take more time for the controller to regulate around the pull-in point when the parameter uncertainty gets exacerbated.

We have set the initial values for＾Φ1(0) and＾Φ2(0) as following in all the previous simulations:

Fig. 5 Results of the ASMC design simulation-tracking performance. a Convergence of x( t) towards xr(t) . b Control eff ort signal as well as the sliding and tracking errors which are dramatically close to zero

Fig. 6 Estimated values of the def ined parameters

Fig. 7 Tracking performance for various values of parameter uncertainty in the stiff ness and damping factors. Δb and Δk are swept from 5 to 45% of the nominal values of the damping and stiff ness factors.Although the movable plate displacement trajectories f inally reach the pull-in trajectory, they will not be leading the desired pull-in trajectory as the parameter uncertainty grows

Fig. 8 Results of the ASMC design simulation-tracking performance for another set of initial values in the equations of the parameter update laws. a Convergence of x( t) towards xr(t) ( x( t) is not leading xr(t) ). b Control eff ort signal as well as the sliding and tracking errors which are close to zero

Fig. 9 Estimated values of the def ined parameters for another set of initial values in the equations of the parameter update laws

x(t) will not be leadingxr(t) neither, while setting the uncertainty equal to 5% of the nominal values. The results are displayed in Figs. 8 and 9. As it is seen, the movable plate displacement trajectory is precisely regulating around 0.667μm.

For conf irmation of the perfect performance of the adaptive laws, some other new sets of initial values are considered. The f irst set is＾Φ1(0)=[00] and＾Φ2(0)=0 while the parameter uncertainty is swept from 45 to 5% . The result is demonstrated in Fig. 10( a), (b). The second set of initial values for conf irmation of the perfect performance of the adaptive laws is＾Φ1(0)=[2.3352×10-6889×10-9]and＾Φ2(0)=0 while the parameter uncertainty is swept from -5 to -55% . The corresponding yields are given in Fig. 11( a), (b). The third set of initial values for validation of the perfect performance of the adaptive laws is＾Φ1(0)=[7.0057×10-67.1120×10-4] and＾Φ2(0)=9.99×10-6subjected to the parameter uncertainty sweeping from -5 to -55% . The simulation results is presented in Fig. 12( a), (b). As it is demonstrated, the controller achieves perfect tracking while adopting various initial values in the adaptive laws and choosing diff erent amounts of parameter uncertainties.

Fig. 10 Results of the ASMC design simulation for ＾Φ1(0)=0 and＾Φ2(0)=0 while the parameter uncertainty is swept from 45 to 5% . a Tracking performance. b The estimated parameters

In the mathematics to prove the stability, we have adopted the control law by Eq. ( 35). This and the parameter update laws, adopted by Eq. ( 54), guarantee the zero-convergence of the sliding surface, proven by the Barbalat’s Lemma. However,avoiding the chattering eff ect, one may use an approximation of the function sgns(ors∕‖s‖ ), such as sat(s∕a1) , tanh(s∕a2) ,arctan(s∕a3) in the control law, which shows a smooth transience with respect to the variation of the sliding surface sign.This approximation causes the sliding surface not to exactly converge to zero and brings about a bounded deviation from zero. The more accurate the approximation is, the smaller this deviation will get. For our simulation purposes, we have used tanhs. Note that the sliding surface is not zero but very close to zero on diagrams. The tracking errors are not exactly zero neither but they are so much close to zero on the plots.

Fig. 11 Results of the ASMC design simulation for ＾Φ1(0)=[2.3352×10-6889×10-9] and ＾Φ2(0)=0 while the parameter uncertainty is swept from -5 to -55% . a Tracking performance. b The estimated parameters

For a positive def inite Lyapunov candidate, the negative semi-definiteness or negative definiteness of the time-derivative of the Lyapunov functionV(s) just guarantees boundedness of the sliding surface error. When the time-derivative of the Lyapunov function is negative semi-def inite and the Barbalat’s Lemma conditions (or Invariance Theorem conditions) are held true as well, the sliding surface error will converge to zero that we have approximately achieved in our study with acceptable precision due to approximation of the control eff ort switching function. However, for zero-convergence of the parameter estimation errors, it is also required to hold the persistently excitation (PE) condition true, according to “Parameter Convergence Analysis” discussed in Ref. [ 37] (pages 331-335, Chapter 8) and “On-line Parameter Estimation”conversed in Ref. [ 39] (pages 177-180, Section 4.3). Similar control works on MEMS gyroscopes are presented in[ 9- 11, 16, 40, 41], in which the PE condition is realized due to the richness of the regressor matrix and the goal parameter (angular velocity) is estimated correctly as a consequence. In those studies, the reference trajectories of the capacitive plates are sinusoidal oscillations with nonequal frequencies which realize PE condition (equation). In our study, the regressor matrix, which is comprised of the movable plate displacement trajectory, does not contain sinusoidal components nor entail richness of various frequencies since the reference trajectory of the plate movement is nearly close to a step signal. Consequently, it does not satisfy the PE condition. To illustrate that the parameter estimation error does not blast up and get assured about stability of the system in terms of the plate displacement tracking error, we have increased simulation time from 0.1 s to 1000 s. The results are given in Figs. 13( a), (b), and 14 which demonstrate the system stability in simulation for a broader time range; see Fig. 13( a), (b). The variation of the parameters in the large time interval of 1000 s is either zero (for 2nd entry of＾Φ1)or an extremely small value (for＾Φ2and 1st entry of＾Φ1),as seen in Fig. 14. In implementation scenarios, one may adjust the initialization of the parameter update laws to manage the variation of the parameters in a lower range so that memory usage in the control hardware is handled optimally such as Fig. 6. It is worthwhile being mentioned that our goal in this article, similar to Refs. [ 42, 43], is not identif ication of the parameters of the MEMS tunable capacitor. It is just precision positioning of the movable plate. This could be considered as one of the research prospectives, also mentioned in Sect. 5.

Fig. 12 Results of the ASMC design simulation for ＾Φ1(0)=[7.0057×10-67.1120×10-4] and ＾Φ2(0)=9.99×10-6 while the parameter uncertainty is swept from -5 to -55% . a Tracking performance. b The estimated parameters

Fig. 13 Results of the ASMC design simulation and the tracking of performance for an extra-large simulation time interval while the parameter uncertainty is -5% and external disturbance is 0.00015

At the end, it should be mentioned that some of the partial errors and deviations from the expected values may originate from the solver methods for diff erential equations within the simulation schemes which are expectable in numerical simulation techniques. For instance, in the solver options, we have used “variable-step” Type and“ode45(Dormand-Prince)” solver in our simulation.

Fig. 14 Estimated values of the def ined parameters for an extra-large simulation time interval

5 Conclusions and future work

MEMS tunable capacitors are regarded as the major elements in AC voltage reference sources. The movable plate of the tunable capacitor should be moved towards the pullin point in order to obtain the desired output voltage. The uncertainties and disturbance which are imposed on the system dynamics in both matched and unmatched manners cause problematic issues, in particular, while one makes eff orts to achieve perfect displacement tracking of the moveable plate towards the ultimate pull-in point non-destructively, utilizing either a simple pole placement controller or even a conventional model reference adaptive controller(MRAC). Although great attempts are carried out to manipulate fabrication process in order to build MEMS tunable capacitors with the desired physical characteristics, the uncertainties, such as Δband Δk, absolutely prevail in the MEMS VRS. On the other hand, matched and unmatched disturbances such as electrostatic forces, cross-talk stiff ness and damping coercions, external inertial and weight force may ruin the desired dynamics while putting the device into work at real conditions. The suggested designed proportional integral ASMC is competent of f lawless tracking of desired displacement trajectory of the movable plate in existence of separate lumped nonlinearities, both matched and unmatched parametric uncertainties, as well as matched and unmatched external disturbance. Hence, the AC MEMS voltage reference source is capable of producing the required output voltage. Drawing the article big picture, one issue,which requires one’s prospective attention is parameter identif ication working along side the adaptive controller to estimate stiff ness and damping factors for the device catalogue purposes.