Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
TuA02-5
Nonlinear Dynamic Output Feedback Stabilization of Electrostatically Actuated MEMS D. H. S. Maithripala & Jordan M. Berg Department of Mechanical Engineering Texas Tech University Lubbock, TX 79409-1021, USA
W. P. Dayawansa Department of Mathematics & Statistics Texas Tech University Lubbock, TX 79409, USA
{sanjeeva.maithripala,jordan.berg}@ttu.edu
[email protected]
Abstract— Operating regions of electrostatically-actuated microelectromechanical systems are limited by a bifurcation phenomenon called “snap-through” or “pull-in”. It is known that charge feedback control can be employed to avoid this bifurcation. The performance of such controllers may be poor, especially if the natural damping of the system is very low or very high. This paper discusses two possible feedback control strategies that eliminate snap-through as well as improve performance. A serious drawback of these control laws is that they require the measurement of the device velocity. A reduced order observer is presented to overcome this. The observer is based on well known nonlinear observer design techniques, and can be assigned arbitrary linear error dynamics.
designed to operate in one of two possible configurations, exploit snap-through in this way [4], [12]. In others the gap is usually made at least three times as large as the required range of motion of the upper plate. The result is a more difficult fabrication and higher operating power requirements. Our goal is setpoint control from any point in the gap to any other point between the electrodes with minimal or no electrode contact. That is, to stabilize any desired equilibrium gap in such a way that the region of attraction contains a given set of initial conditions O (corresponding to normal operational conditions) and that for those initial conditions the performance result in no electrode contact. However we note that for sudden disturbances that can be modelled by initial conditions that lie outside the given set O, electrode contact may be unavoidable. In the case of electrode contact we assume that upon contact the movable electrode instantaneously loses some or all of its kinetic energy and that the system dynamics when the electrodes are in contact are governed only by the dynamics of the electrical system. An insulating layer is assumed to prevent charge loss to the bottom electrode. Further, if and when the spring force exceeds the electrostatic forces, the governing dynamics are once again to be those of the unconstrained system. It is well known that by placing an appropriately sized capacitor in series with the device, every point of the gap can be stabilized [16]. Several difficulties of such feedback due to parasitic capacitance are reported in [2]. Improvements to the method are also proposed in [2]. Two current drive control strategies are proposed in [14]. One of these is a feedback control strategy that feeds back charge, while the other is an open-loop charge control law. Both aim at directly controlling the amount of charge on the electrodes. In [9] we show that with capacitor charge as output, the system has uniform relative degree one and asymptotically stable zero dynamics corresponding to the dynamics of the mechanical subsystem. Input-output linearization of this system amounts to positive feedback of the voltage across the electrodes. For the inputoutput linearized system it can be easily shown that negative charge feedback, together with an appropriate constant bias voltage, globally stabilizes any point of the gap [9]. It is also shown in [9] that the current control laws of [14] are equivalent to that of charge feedback with respect to the input output linearized system. However the transient response of the system depends completely on the zero dynamics of the
I. I NTRODUCTION Of the enormous variety of actuation methods that have been developed for microelectromechanical systems (MEMS) (For a partial survey, see e. g. [8]), electrostatic devices are the most common. Electrostatic actuation makes use of the coulomb forces that develop between capacitivelycoupled conductors that differ in voltage. We use a simple 1-D model of a typical electrostatically actuated MEMS. The model might represent, for example, a single element of an array of rigid micromirrors [5], [6] or an electrostatically actuated microswitch [12]. Other configurations, such as deformable membranes or torsional mirrors, will display similar qualitative behavior to that considered here. The model consists of a parallel plate capacitor with a movable top plate and fixed bottom plate, with the top plate attached to a fixed spring. Viscous damping, corresponding to structural and squeeze film effects, is assumed. While some MEMS may have sufficient natural damping [13] others may not [3], [12]. Damping that is too low may produce long settling times or, particularly when stabilizing points low in the gap, cause electrode contact, such contact may or may not be acceptable on an occasional basis, but is almost certainly undesirable in the long term as it will reduce the lifetime of the device [12]. Furthermore the natural damping in some cases may be too high, resulting in unacceptable slow rise times. With constant voltage control the gap cannot be adjusted arbitrarily due to a saddle-node bifurcation phenomenon commonly known as “snap-through” or “pull-in,” [18], [16], [9]. Permanent damage due to snap-through can be avoided by coating the bottom plate with a thin layer of insulating material, and employing appropriate lubricants and environmental control. Some MEMS, particularly digital devices
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system which correspond to the dynamics of the mechanical system. Thus if the natural damping of the device is low the scheme suffers from large overshoots and relatively long settling times and on the other hand if the natural damping is very high it may result in unacceptable slow rise times. The authors have exploited [11] the input-output linearized system’s Port Controlled Hamiltonian structure with damping (PCHD). Based on the ideas presented in [15], [20], the Casimir functions of the PCHD are used to shape the total energy of the system such that it is positive definite in some sufficiently large neighborhood of the desired equilibrium. Consequently the use of damping control asymptotically stabilizes the system. The resulting feedback laws are shown to be charge feedback and it is also shown that the now familiar capacitive feedback control law of [16] also falls into this framework. However even in this scheme the rise time, overshoots and the settling time are governed by the natural damping of the mechanical subsystem. It is shown in [11] that this class of controls introduce damping only in the electrical subsystem. An inherent drawback of charge feedback control is that the transient response properties are primarily governed by the natural damping of the mechanical subsystem. In Section III we investigate several different control strategies that improve transient performance, especially in the case of low damping. First we treat the input output linearized system as the interconnection of two subsystems of which one is passive and the other is in a form that enables the feedback passivation of the composite system [20]. In this case the resulting storage function is quadratic and with damping injection, and one may globally asymptotically stabilize the desired equilibrium. Though this scheme yields low overshoots, in the case of low damping, it has a long settling time especially when moving from a setpoint in the lower part of the gap to a higher part of the gap. Next, for low damping, we show that by employing a linear state feedback law with appropriately chosen gains it is possible to achieve almost no overshoot and faster settling times with respect to system performance for trajectories that start from a set O. The set O corresponds to the set of equilibrium points in the gap. The implementation of the last two feedback laws require that we measure the velocity of the movable electrode. Unfortunately, while such measurements are possible in the laboratory [19], this velocity is extremely difficult to sense directly during normal operation of the device. In order to overcome this we show that by the measurement of the capacitance across the electrodes [1] and the voltage across the electrodes it is possible to construct a reduced order (one-dimensional) observer with arbitrary linear error dynamics. A version of this observer is also presented by the authors in [10]. Finally we implement the two state feedback laws by means of this state estimator and show that the composite systems preserve the stability properties of the state feedback laws.
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II. 1D M ODEL A typical electrostatic microactuator is schematically represented in the Figure 1. The system is actuated by controlling the input voltage v(t). The spring and the dashpot in the figure represent the flexibility and damping in the support assembly. Let Q(t) be the charge of the device, i(t) be the current through the resistor, l(t) be the air gap, v(t) be the input voltage, l0 be the zero voltage gap, A be the plate area, and be the permittivity in the gap. Then, the capacitance of the device is equal to A/l(t), the attractive electrostatic 2 force on the top plate is F (t) = Q(t) 2A , and the current through the input resistance r is 1 Q(t)l(t) i(t) = v(t) − . (1) r A Thus, the complete equations of motion are [18],
Fig. 1. 1D model of a electrostatic microactuator. Top plate of the MEMS is free to move and the bottom plate is held fixed.
m¨l(t)
=
˙ Q(t)
=
2 ˙ − k(l(t) − l0 ) − Q (t) , − bl(t) 2A Q(t)l(t) 1 (v(t) − ). r A
(2) (3)
˙ The state space of the system is Q = {(Q(t) l(t) l(t)) ∈ R | l ≥ δ0 }. Here δ0 is the thickness of the insulating material coated on the bottom plate. In general, the input current i(t) and the voltage across the device, Q(t)l(t)/A , are available for measurement at a relatively low cost. Note from (1) that it suffices to consider only one of them. We prefer the latter since it does not have direct control term and is easily measurable. Accurate measurement of capacitance across the device, A/l(t), is also possible [1]. It is clear that from these two measurements the charge across the device, Q(t), and the gap between the electrodes of the device, l(t) can be inferred as well. We consider the voltage, Q(t)l(t)/A and the reciprocal of the capacitance, l(t)/A, as the measured outputs of the system. The direct or otherwise measurement of the velocity of the moving electrode is not feasible at the moment and would typically involve a very high cost. Therefore we assume that velocity measurements are unavailable. Let σ be a positive constant. Performing a normalizing time scale change of τ = σt and a change of variables l = αˆl, 3
√ v = rσβν, Q = βq where α = Arσ, β = Aσ mrσ ˙ and letting the state vector x = [q, ˆl, ˆl]T and ωn2 = k/m, ωn 2τ ωn = b/m and ω = σ , the control system can be put into the state space form,
x˙ 1 x˙ 2 = x˙ 3 y1 = y2 =
−x1 x2 x3 −2τ ωx3 − ω 2 (x2 − ˆl0 ) − x1 x2 , x2 ,
with the state space X = {(x1 x2 x3 ) ∈ R3 | x2 ≥ δ} (δ0 = αδ > 0), y1 the voltage across the device divided by the constant rσβ, and y2 the reciprocal of the capacitance divided by the constant rσ. For a given constant voltage v¯ = rσβ ν¯, let the equilibrium points of (4) be x ¯ where x ¯ = [¯ x1 , x ¯2 , 0]T and ν¯ = x ¯1 x ¯2 . For a given x ¯2 , x ¯1 is given by x ¯21 = 2 ω 2 (ˆl0 − x ¯2 ). For a given constant bias voltage v¯ the open-loop system (4) has three equilibria. One of these lies below the bottom plate, and so is outside the operating region of the device (that is outside of X ). The two remaining, including the desired equilibrium at the origin, lie within X . It is known (see for instance [16], [9]) that the equilibrium points corresponding to a gap less than two thirds of the zero voltage gap, l0 , are unstable while the equilibrium point corresponding to a gap larger than two-thirds of the zero-voltage gap is stable. From the viewpoint of p bifurcation theory, for each input voltage below vpull = 8kl03 /27A there are two equilibrium values of l in the zero-voltage gap. A stable one lies in [0, 2l0 /3) and an unstable one lies in (2l0 /3, l0 ]. They coincide at vpull . Thus the point l = 2l0 /3 corresponds to a saddle-node bifurcation with respect to v¯. Therefore the operational gap of the device will have to be restricted to only one third of the zero voltage gap. The next section investigate possible control strategies available that will enable the use of the entire gap as well as improve the transient performance of the MEMS. Improvement of performance will be considered with respect to faster settling times and smaller overshoots. For the sake of convenience of designing and analyzing the various state feedback laws we perform a globally defined change of variables ζ1 = x1 − x ¯1 , ζ2 = x2 − x ¯2 , ζ3 = x3 , u = ν − ν¯, η1 = y1 − x ¯1 x ¯2 , η2 = y2 − x ¯2 , so that in the transformed system the desired equilibrium is shifted to the origin. The state space in these new coordinates is Z = {(ζ1 ζ2 ζ3 ) ∈ R3 | ζ2 ≥ −¯ x2 + δ}. Expressing the system (4)–(6) in these new coordinates, we have the control affine system ζ˙ = f (ζ)+g(ζ)u, η = h(ζ), where ζ = [ζ1 , ζ2 , ζ3 ]T and
f (ζ)
1 g(ζ) = 0 , 0 ζ1 ζ2 + x ¯ 2 ζ1 + x ¯ 1 ζ2 , h(ζ) = . ζ2
x21 2
ν + 0 (4) 0 (5) (6)
−ζ1 ζ2 − x ¯ 2 ζ1 − x ¯ 1 ζ2 , = ζ3 −2τ ωζ3 − ω 2 ζ2 − x ¯1 ζ1 − ζ12 /2
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III. C ONTROL OF THE MEMS Given any desired equilibrium gap x ¯2 we investigate the existence of a control law that will stabilize the corresponding equilibrium in such a way that its region of attraction contains a prescribed set of initial conditions O and has “good” transient performance. This has to be achieved only with the measurements of y1 and y2 . Let O = {(x1 x2 x3 ) ∈ R3 | δ ≤ x2 ≤ ˆl0 , x21 = 2ω 2 (ˆl0 − x2 ), x3 = 0.}. This is the set of initial conditions that correspond to equilibrium points that lie inside the gap. First we seek a coordinate transformation that will reveal more structural properties of the system and provide a greater degree of freedom in designing a suitable controller. As a first step we look for a new output for which the system has uniform relative degree. Consider the output y = hc (ζ) = ζ1 . With respect to this output the system (4) in ζ coordinates has uniform relative degree one and is naturally in the globally defined zero dynamic form of y˙ = Lf hc + Lg hc u, z˙ = q(z, y), where z = [ζ2 , ζ3 ]T , Lf hc = −ζ1 ζ2 − x ¯ 2 ζ1 − x ¯1 ζ2 = −η1 (t), Lg hc = 1 and ζ3 q(z, y) = . −2τ ωζ3 − ω 2 ζ2 − x ¯1 y − y 2 /2 Note that the zero dynamics of the system given by z˙ = q(z, 0) are globally exponentially stable or, what is equivalent, the system is said to be strongly minimum phase. Also note, however, that the zero dynamics are exactly those of the mechanical subsystem. If the damping is very low, the zero dynamics will die out very slowly. The feedback control law defined by u = η1 + u ˆ is globally smooth. Substituting this smooth feedback law in (4) we obtain the following inputoutput linearized system y˙ z˙
= =
u ˆ, q(z, y).
(7) (8)
Since Lf hc = η1 is the voltage deviation across the device with respect to the applied bias voltage x ¯1 x ¯2 and is a measurable quantity, it is possible to robustly linearize the original system. Note that the system appears least nonlinear in this form. Furthermore as shown in [9] the linear feedback law u ˆ = −ky globally asymptotically stabilizes the origin of (7)–(8). However if the zero dynamics of (8) are lightly damped the system suffers from longer settling times and large overshoots and if the zero dynamics are over damped the system may suffer from longer rise times. Large overshoots are not acceptable especially when stabilizing smaller gap lengths as it will cause the movable electrode
to repeatedly hit the fixed electrode and hence reduce the lifetime of the device [12]. A. Passivity-Based Controller Consider the system (7)–(8) as the interconnection of two subsystems. System (7) is passive with storage function S1 (y) = y 2 /2 and output y. System (8) can also be expressed as z˙ = q(z, 0) + r(y)y, where r(y) = [0 − (¯ x1 + y/2)]T . 2 Further if S2 (z) = ω 2 z12 /2 + z22 /2 then ∂S q(z, 0) ≤ 0. In ∂z such a case Theorem 5.2.1 of [20] shows that the control 2 u ˆ = − ∂S ˆ renders (7)–(8) passive with storage ∂z r(y) + v function S(y, z) = S1 (y) + S2 (z), output y and input vˆ. The function S(y, z) is globally proper and positive definite in Z. Assume that when the electrodes are in contact the system dynamics are governed only by that of the electrical system and that when the spring force acting on the electrode exceeds the electrostatic force the system switches back to its original form. That is the system (7)–(8) when restricted to the set {(ζ1 ζ2 ζ3 ) ∈ Z | ζ2 = −¯ x2 + δ} is governed by y˙ = u ˆ, z˙ = 0 and while in contact if and when 2ω 2 (ˆl0 − x2 ) ≥ x21 the system switches back to (7)–(8). Further assume that the velocity of the the moving electrode before and after contact satisfy the relation ζ3+ = − µ ζ3− where 0 ≤ µ ≤ 1 and ζ3− , ζ3+ are the velocities of the moving electrode just before and after contact respectively. The closed-loop system is zero state detectable with the output y(= ζ1 ), thus damping control vˆ = −ky (k > 0) guarantees that S(y, z) strictly reduce along the trajectories of (7)–(8). Therefore u ˆ = ζ3 (¯ x1 + ζ1 /2) − kζ1 globally asymptotically stabilizes the origin of (7)–(8). In terms of the coordinates x = [x1 x2 x3 ]T ,
law whose transient response depends on the damping of the mechanical system [9]. Thus in the case of low damping in order to obtain improved transient performance it is necessary that f3 be nonzero. For a suitably designed γ and f3 simulations indicate that the system has good transient performance as well as a large region of attraction. In terms of the coordinates x = [x1 x2 x3 ]T this control law is expressed by κ y1 ¯1 , (10) ν = ϕ2 (y1 , y2 , x3 ) = y1 − γ + x3 + γ x y2 x ¯1 and locally asymptotically stabilizes the point x ¯ = [¯ x1 x ¯2 0]T 2 2 ˆ of (4) where for a given x ¯2 , x ¯1 is given by x ¯1 = 2ω (l0 −x2 ). Although both these control laws exhibit good performance and stability properties a serious drawback is that their implementation requires the measurement of the velocity variable x3 . To overcome this obstacle we construct a reduced order state observer that estimates the the velocity variable x3 with arbitrary fast error dynamics. The construction of this observer is discussed in the next section. IV. R EDUCED O RDER O BSERVER WITH A RBITRARY L INEAR E RROR DYNAMICS
y1 y1 1 ν = ϕ1 (y1 , y2 , x3 ) = y1 + x3 (¯ x1 + ) − k + k¯ x1 (9) 2 y2 y2 globally asymptotically stabilizes the point x ¯ = [¯ x1 x ¯2 0]T 2 2 ˆ of (4) with for a given x ¯2 , x ¯1 is given by x1 = 2ω (l0 − x2 ). B. Linear State Feedback Controller Observe that the system (7)–(8) is linearly controllable. Thus it is possible to find an F such that the linear feedback control u ˆ = F ζˆ locally asymptotically stabilizes the origin of (7)–(8) with arbitrarily fast dynamics close to the origin. For a generic F = [f1 f2 f3 ] it is seen that the system has two equilibria. Thus this scheme can not globally stabilize the system. However with f2 = 0, the origin is the only equilibrium of the system and the Routh-Hurwitz test shows that all the eigenvalues of the linearized system lie in the strict left half complex plane for all f1 < 0 and f3 > 0 (the characteristic polynomial of the linearized system matrix is λ3 + (2τ ω − f1 )λ2 + (ω 2 − 2f1 τ ω + κ)λ − f1 ω 2 where κ = x ¯1 f3 ). Setting f1 = −γ, for some fixed positive γ, and with the aid of the root locus with respect to the gain κ it is possible to select the feedback gain f3 such that good transients are achieved for low damping. Note that setting f3 = 0 yields the globally stable linear charge feedback
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Fig. 2. Stabilizing a gap length of 10% of the zero voltage gap. Units are non-dimensional, corresponding to (4). Initially the movable electrode is at the equilibrium point (0 1 0). The solid curve corresponds to the linear feedabck law (10), the dashed-dotted curve corresponds to the passive based control law (9), and the dashed curve corresponds to charge feedback.
Observe that the system (4) with output yˆ = [y1 /y2 y2 ]T can be expressed as ˆ + Γ(ˆ x˙ = Ax y ) + Bν yˆ = Cx.
(11) (12)
where
0 0 0 Aˆ = 0 0 −ω 2
0 1 , −2τ ω
0 Γ(ˆ y) = 0 2 , y − 12 y12 2
Implementing (10) with ν = ϕ2 (y1 , y2 , x ˆ3 ) and the observer (13) – (14) we have the closed-loop system κ ˆ + Γ(ˆ e, (19) x˙ = Ax y ) + B ϕ2 (y1 , y2 , x3 ) − B ϑ x ¯1 e˙ = T e. (20) Thus the equilibrium point [¯ x1 x ¯2 00]T of (19) – (20) is locally asymptotically stable. Simulations show that with T = −0.2, γ = 5 and κ = 6.89 for any given x ¯2 and initial conditions in O the system response has virtually no overshoot and fast settling time even for a damping ratio of τ = 0.1. Figures (2) – (3) show the response. V. C ONCLUSIONS Fig. 3. Stabilizing a gap length of 80% of the zero voltage gap. Units are non-dimensional, corresponding to (4). Initially the movable electrode is at the equilibrium point (0.6325 0.1 0). The solid curve corresponds to the linear feedabck law (10), the solid dotted curve corresponds to the passive based control law (9), and the dotted curve corresponds to charge feedback.
1 B = 0 , 0
C=
1 0
0 1
0 0
.
For a system of the form (11)–(12) it is known that a full order observer with linear error dynamics can be designed, ˆ is observable these dynamics can [7]. Further since (C, A) be even made to be arbitrarily fast. Based on this concept we design a reduced order observer with linear error dynamics z˙
= T z + V K yˆ + V Γ(ˆ y) + V B u ˆ, yˆ x ˆ = Q−1 . z
(13) (14)
The area of microelectromechanical systems offer novel challenges to control theorists. An important one, treated in this paper, is how to to control an electrostaticallyactuated MEMS so that a) the device range of motion is the entire capacitive gap, b) supply voltages may be kept as low as possible, and c) contact between electrodes may be minimized. In this regard, nonlinear feedback control can play a significant role. Here we illustrate two approaches to design of an appropriate nonlinear state feedback controller, and implement them using a nonlinear observer. One of the two controls laws is shown to have a global region of attraction, but relatively slow convergence near the setpoint for low damping conditions, and the other can be designed to have relatively fast convergence—but only locally. Currently we are investigating ways to combine the two laws in order to produce a single globally stabilizing controller with fast rate of convergence to the setpoint. VI. ACKNOWLEDGEMENTS
Where z ∈ R and x ˆ is the estimated state. The reduced order observer parameters T, K and V are selected such that Q = [C T V T ]T is invertible, V (Aˆ − KC) = T V and the spectrum of T is pre-assigned [21] (that is T < 0). Setting e = (V x − z) a straightforward calculation shows that
This work was partially supported by NSF grant ECS0218245. Discussions with Mr. Pete van Kessel of Texas Instruments Digital Light Processing group are gratefully acknowledged.
e˙ = T e, x ˆ = x − Q−1 D e,
[1] F. Ayela, J. L. Bret, J. Chaussy, T. Fournier and E. Menegaz, “A Two-Axis Micromachined Silicon Actuator with Micrometer Range Electrostatic Actuation and Picometer Sensitive Capacitive Detection,” Review of Scientific Instruments, Vol. 71, Number 5, ppg 2211– 2218, May 2000. [2] E. K. Chan, R. W. Dutton, “Electrostatic micromechanical actuator with extended range of travel,”, Journal of Microelectromechanical Systems, Vol. 9, No. 3, pp. 321–328, 2000. [3] H. C. Larnaudie, F. Rivoirard and B. Jammes, “Analytical Simulation of a 1D Single Crystal Silicon Electrostatic Micromirror,” Proc. of the Second Int. Conf. on Modelling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Chicago, CA, ppg 628–631, April 1999.
(15) (16)
where D = [0 0 1]T . Note that Q−1 D is of the form [0 0 ϑ]T for some constant ϑ. Thus the velocity estimate is of the form x ˆ3 = x3 − ϑe. Note that this observer design does not depend on the desired equilibrium point. Implementing (9) with ν = ϕ1 (y1 , y2 , x ˆ3 ) and the observer (13) – (14) we have the closed-loop system ˆ + Γ(ˆ x˙ = Ax y ) + B ϕ1 (y1 , y2 , x3 ) − B ϑ ex1 , (17) e˙ = T e. (18) Theorem 4.7 of [17] guarantees that the equilibrium point [¯ x1 x ¯2 0 0]T of (17) – (18) is globally asymptotically stable. Figure (2) – (3) shows the performance of this scheme for T = −5 and k = 1 and a damping ratio of τ = 0.1.
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[19] H. Toshiyoshi, M. Mita, H. Fujita, “A MEMS Piggyback Actuator for Hard-Disk Drives,” J. of Microelectromechanical Systems, Vol. 11, No. 6, 2002, pp. 648– 654. [20] A. J. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, Springer-Verlag, London 2000. [21] W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd Edition, Springer-Verlag, New York 1985.
