Fuzzy control with random delays using invariant cones and its application to control of energy processes in microelectromechanical motion devices

Fuzzy control with random delays using invariant cones and its application to control of energy processes in microelectromechanical motion devices

Energy Conversion and Management 46 (2005) 1305–1318 www.elsevier.com/locate/enconman Fuzzy control with random delays using invariant cones and its ...

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Energy Conversion and Management 46 (2005) 1305–1318 www.elsevier.com/locate/enconman

Fuzzy control with random delays using invariant cones and its application to control of energy processes in microelectromechanical motion devices A.S.C. Sinha a

a,*

, S. Lyshevski

b

Department of Electrical Engineering, Purdue University at Indianapolis, Indianapolis, IN 46202-5132, USA b Rochester Institute of Technology, Rochester, NY 14623, USA Received 1 August 2003; received in revised form 9 March 2004; accepted 24 June 2004 Available online 26 August 2004

Abstract In this paper, a class of microelectromechanical systems described by nonlinear differential equations with random delays is examined. Robust fuzzy controllers are designed to control the energy conversion processes with the ultimate objective to guarantee optimal achievable performance. The fuzzy rule base used consists of a collection of r fuzzy IF-THEN rules defined as a function of the conditional variable. The method of the theory of cones and Lyapunov functionals is used to design a class of local fuzzy control laws. A verifiably sufficient condition for stochastic stability of fuzzy stochastic microelectromechanical systems is given. As an example, we have considered the design of a fuzzy control law for an electrostatic micromotor.  2004 Elsevier Ltd. All rights reserved. Keywords: MEMS; Control; Optimization; Energy conversion

*

Corresponding author. E-mail address: [email protected] (A.S.C. Sinha).

0196-8904/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2004.06.026

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Nomenclature Section I X fuzzy state, an n · 1, X 2 Rn R real number field C denotes Banach space of all continuous real functions on J = [s, 0] X member of probability space fX; I; P g set of square-integrable functions on X L2 u control input matrix of system, uðtÞ 2 U  Rn ; U an admissible control l membership functions of fuzzy model A(t, l) system matrix, a function of t and l B(l) fuzzy set in input space U / initial function that belongs to L2 n(t) random delay, a discontinuous process Z conditional variables denotes lth approximation inference rule Rl denotes closed convex sets (l = 1, . . . , r) Sl K(t) feedback control gain V Lyapunov functional Section III R, C resistance and capacitance e permittivity of material A area of plate g gap between plates W, L width and length of plates U, V electrical potential and voltage applied to parallel conducting plates u supply voltage n number of overlapping electrodes t rotor and stator thickness r1, r2 radii of rotor and stator electrodes T, TL motor torque; load torque J rotor inertia x angular velocity friction coefficient Bm

1. Introduction Many problems of dynamical systems are nonlinear in nature. In particular, nonlinear random delay differential equations arise in various disciplines whenever we model dynamical systems that involve some randomness due to ignorance or uncertainties [13]. Tanaka et al. [14,15] and Zhang and Feng [16], among many others, have introduced a fuzzy model of the system. Fuzzy model

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based analysis methods are introduced into the fuzzy logic control area. The stability analysis methods in Refs. [1,14,15] are based on the Lyapunov theory. However, all methods are model based controller design methods for time invariant fuzzy models. The major drawback of the fuzzy control system is the lack of a systematic method for its analysis and design. Many authors have studied the fuzzy control system when a common positive definite Lyapunov matrix P can be found to satisfy the Lyapunov equation. The methods tend to give a conservative controller design. This method was modified to obtain a set of PÕs to construct a Lyapunov function for stability analysis. Each P corresponds only to one subsystem, which is easier to find. Often, the Riccati equation is used to obtain each P for the local subsystems. However, all the above methods are model based controller design methods. They require a fuzzy model of the system. It is well known that the conventional fuzzy control methods for model free controller design are given by Zadeh [18] and Yager and Filev [17], among others. The question of fuzzy control logic for the fuzzy model described by constant coefficients is resolved if the Lyapunov matrix P can be determined from the Riccati equation for each subsystem. Another way to obtain fuzzy controller design and stability has been introduced for a class of nonlinear time invariant systems in Ref. [12] where the controllers were analytically designed instead of using a decision table. In this paper, we consider the general case in which the fuzzy coefficients are not constant and the delay n(t) is homogeneous to a Markov random process. These conditions bear on the case of constant fuzzy coefficients and constant delays. The method is based on the use of cone theory [9] and the Lyapunov functional developed by Krasovskii [8]. The paper gives a set of Ôeasily verifiableÕ sufficient conditions for a class of fuzzy control systems with random delays using invariant cones of positive initial functions and Lyapunov functionals. Our results are motivated by its application to electromechanical systems, which is governed by a fuzzy model with random disturbances and lags. They are often referred to as stochastic microelectromechanical systems with transport delays. Such microelectromechanical motion devices that can be fabricated using surface micromachining technologies have been extensively studied in Refs. [2–4]. The method introduced is applicable to communication problems when the system is described by a fuzzy model, and tape delays are introduced to sequence the signals. These lags in communication problems are often stochastic in nature. These results provide the foundations for the study of stochastic microelectromechanical systems and energy conversion in these systems.

2. Preliminary results and assumptions In this section, we study nonlinear microelectromechanical systems that are described by the following stochastic equations. We investigate a class of fuzzy stochastic control systems of the form dX ðtÞ ¼ ½A0 ðt; lÞX ðtÞ þ BðlÞuðtÞ þ f ðl; X ðtÞÞ dt þ

m X q¼1

2

n

X 0 ¼ / 2 L ðX; CðJ ; R ÞÞ;

J ¼ ½s; 0

Aq ðtÞX ðt  nðtÞÞdt

ð1Þ

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where the l are the membership functions of the fuzzy model. The matrices A0(t, l) are assumed to be a continuous function of t and of the membership function l. The matrices Aq(t), q = 1, . . . , m are also continuous functions in t. Since, in many physical problems, the coefficient terms associated with the stochastic delay process are small, the matrices Aq(t), q = 1, . . . ,m are not considered function of l. All matrices are assumed to be of appropriate dimensions. The nonlinear function f(l, X(t)) is assumed to be Lipschitzian in X(t). The fuzzy rule base used consists of a collection of r fuzzy IF-THEN rules defined as a function of the conditional variable z(t)  Z. Here, Z is the fuzzy set in the conditional space Z of the system, which includes components of the input variables, output variables, state variables and other variables of the system. The fuzzy set Z is defined as follows. We consider the following conventional fuzzy control law R‘ : IF z1 ðtÞ is A‘1 ; z2 ðtÞ is A‘2 zn ðtÞ is A‘n ‘ ; ‘ ¼ 1; . . . ; r THEN uðtÞ ¼ u

ð2Þ

where zðtÞ ¼ ðz1 ðtÞ; z2 ðtÞ; . . . ; znðtÞ ÞT 2 Z ¼ Z 1 Z 2 Z n

ð3Þ

The conditional variables of the system Al ¼ Al1 Al2 Aln 2 Z are the fuzzy set in the conditional space Z. u(t)2U is the control input of the system. ul is the center of the fuzzy set Bl in the input space U. Rl denotes the lth approximation inference rule. If we define r local regions in the state space as follows S ‘ ¼ fX ðtÞ j l‘ ðzðtÞÞ P li ðzÞ; i ¼ 1; 2; . . . ; r; i 6¼ ‘g;

‘ ¼ 1; 2; . . . ; r;

ð4Þ

then, the state space of Eq. (1) can be expressed as dX ðtÞ ¼ ½A0‘ ðt; lÞX ðtÞ þ B‘ ðlÞuðtÞ þ fl ðl; X ðtÞÞ dt þ

m X

Aq‘ ðtÞX ðt  nðtÞÞ dt

q¼1

ð5Þ

X ðtÞ 2 S ‘ Next, we define the local regions Sl, l = 1,2, . . . ,r as follows. Let lA‘i ðzi Þ and lB‘ ðul Þ be the membership functions of the fuzzy sets A‘i and Bl, respectively, then, using a singleton fuzzifier we have i ðzÞ ¼ l

n Y

lA‘i ðzi Þ;

lBl ð ul Þ ¼ 1

ð6Þ

i¼1

 ðzÞ l l‘ ðzÞ , Pr ‘ ‘ ðzÞ ‘¼1 l

ð7Þ

The commonly used local fuzzy control law is given by uðtÞ ¼ K l ðtÞX ðtÞ

if X ðtÞ 2 S ‘ ;

‘ ¼ 1; 2; . . . ; r

ð8Þ

Define r local regions in the state space as in Ref. [6] S ‘ ¼ fX ðtÞ j l‘ ðzðtÞÞ P li ðzÞ; i ¼ 1; 2; . . . ; n; i 6¼ ‘g;

‘ ¼ 1; 2; . . . ; r;

ð9Þ

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Then, in every local region of Eq. (9), we can rewrite Eq. (5) in the form m X Aq‘ ðtÞX ðt  nðtÞÞ dt dX ðtÞ ¼ ½A0‘ ðt; lÞX ðtÞ þ B‘ ðlÞuðtÞ þ fl ðl; X ðtÞÞ dt þ q¼1

A0‘ ðt; lÞ ¼

r X

ll ðtÞA0 ðtÞ;

B‘ ðlÞ ¼

l¼1

r X

ll ðtÞBl

ð10Þ

l¼1

lðtÞ ¼ ðl1 ðtÞ; l2 ðtÞ; . . . ; ln ðtÞÞ X ðtÞ 2 S ‘ The system Eq. (10) can be expressed as dX ðtÞ ¼ s½A0l ðtÞ þ DA0l ðt; lÞ X ðtÞ þ ½½Bl þ DBl ðlÞ uðtÞ tdt þ ½fl ðl; X ðtÞÞ dt m X sAql ðtÞX ðt  nðtÞÞtdt þ

ð11Þ

q¼1

X ðtÞ 2 S ‘ ;

‘ ¼ 1; 2; . . . ; r

where A0l ðt; lÞ , ½A0l ðtÞ þ DA0l ðt; lÞ r X li ðtÞDA0li ðtÞ; DA0l ðt; lÞ ¼

DA0li ðtÞ ¼ A0i ðtÞ  A0l ðtÞ

ð12Þ

i¼1;i6¼l

and B‘ ðlÞ , Bl þ DBl ðlÞ r X li ðtÞDBli ; DBl ðlÞ ¼

ð13Þ

DBli ¼ Bi  Bl

i¼1;i6¼l

We rewrite Eq. (11) in the form dX ðtÞ ¼ s½A0l ðtÞX ðtÞ þ Bl uðtÞ þ f~ l ðl; X ðtÞ tdt þ

m X

Aql ðtÞX ðt  nðtÞÞdt

q¼1

X ðtÞ 2 S ‘ ;

ð14Þ

‘ ¼ 1; 2; . . . ; r

where the redefined uncertainty function in Eq. (14) is expressed as f~ l ðl; X ðtÞÞ , ½DA0l ðt; lÞ þ DBl ðlÞK ‘ X ðtÞ þ fl ðl; X ðtÞ

ð15Þ

The nonlinear function is assumed to satisfy kf~ l ðl; X ðtÞÞk 6 LðkX ðtÞkÞ;

L>0

ð16Þ

The unknown constant L is assumed to be bounded, which will be defined later. The random delays n(t) are assumed to be purely discontinuous processes, which assume values inside the closed interval ½0; s , ðs > 0Þ. Let the random function n(t) take a finite number of values fn0 ; n1 ; . . . ; nm g 2 ½0; s . Let the solution of system (14) be defined by the initial curve {X0(t0, /), n0},

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where / is an initial function and n0 is the initial value of the process. Then, stochastic system (14) with the initial curve {X0(t0, /), n0} generates a probability process. It can be shown (see e.g. Refs. [7,11]) that the solution process denoted by X(t) = {X(t, t0, /), n}, t P t0 in the sequel is unique for any initial data ðt0 ; /Þ 2 Rþ C. Following the results in Ref. [11], we develop sufficient conditions for the stochastic delay differential Eq. (14) to be stable in probability. If, for every fixed arbitrary delay fn0i gmi¼1  ½0; s , the delay differential equation "" ## m X dX ðtÞ ¼ ½A0l ðtÞX ðtÞ þ Bl uðtÞ þ f~ l ðl; X ðtÞ þ Aql ðtÞX ðt  n0q Þ dt q¼1

X ðtÞ 2 S ‘ ;

ð17Þ

‘ ¼ 1; 2; . . . ; r

X 0 ¼ / 2 L2 ðX; CðJ ; Rn ÞÞ;

J ¼ ½s; 0

is stable, then the stochastic delay differential Eq. (14) is also stable in probability following the results developed in Ref. [11]. Next, we develop the criterion for the stability of the stochastic delay differential Eq. (14). Let us assume, without loss of generality, that the matrices (A0l(t), Bl) are controllable. Using local fuzzy control law Eq. (8), we obtain, in every local region, the closed loop system Eq. (17) as ## "" m X c 0 ~ ðtÞX ðtÞ þ f~ l ðl; X ðtÞ þ Ajl ðtÞX ðt  nk Þ dt dX ðtÞ ¼ ½A 0l ð18Þ k¼1 X ðtÞ 2 S ‘ ;

‘ ¼ 1; 2; . . . ; r

where 3T 2 32 l k n ðtÞ 0 6 76 k l ðtÞ 7 6 7 0 0 1 0 7 7 6 0 76 6 6 n1 7 7 6 7 6 6 .. 7 6.7 .. 7 ~ c ðtÞ , Ac ðtÞ þ Bl K l ðtÞ ¼ 6 7 A 6 . 7 þ 6 .. 76 0l 0l 6 7 6 76 . 7 6 7 6 7 6 7 l 0 0 0 1 7 5 4 0 56 4 4 k 2 ðtÞ 5 l l l l a0;n ðtÞ a0;n1 ðtÞ a0;n2 ðtÞ a0;1 ðtÞ 1 k l1 ðtÞ 3 2 0 1 0 0 6 0 0 1 0 7 7 6 7 6 .. 7 6 ,6 . 7 7 6 6 0 0 0 1 7 5 4 l l l l ~ a0;n ðtÞ ~ a0;n1 ðtÞ ~a0;n2 ðtÞ ~a0;1 ðtÞ 2

0

1

0



0

3

ð19Þ ~al0;j ðtÞ

al0;j ðtÞ

k lj ðtÞ

The feedback elements kj(t), j = 1, . . . ,n are chosen such that ,  > 0 for each l ~ j = 1, . . . , n. Next, we show that, if the coefficients a0;j ðtÞ > 0, then the controlled linear subsystem c

~ ðtÞyðtÞ _ ¼A yðtÞ 0l

ð20Þ

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is stabilized. The method of the theory of cones is used to obtain the sufficient conditions for the stability of the system Eq. (20). Define a cone } as follows. Let m1, m2, . . ., mn  1 be certain constants and let us consider the polynomials Di ðkÞ ¼ ðk  m1 Þðk  m2 Þ ðk  mi1 Þ ¼ ki1 þ ci;i1 ki2 þ ci;i2 ki2 þ þ ci;1 ði ¼ 2; . . . ; nÞ

ð21Þ

Constructing the matrix C from Eq. (21) as 3 2 1 0 0 0 7 6 0 0 7 6 c21 1 7 6 7 6 C ¼ 6 c31 c32 1 0 7 7 6 . 7 6 4 .. 5 cn1

cn2

cn3



ð22Þ

1

The cone }+ with nonnegative coordinates will have the form }þ ¼ fy : Cy þ P 0g

ð23Þ

The characteristic equation of the linear controlled system Eq. (20) is given by a1 ðtÞmn1 þ ~ a2 ðtÞmn2 þ þ ~an ðtÞ Pðt; mÞ , mn þ ~

ð24Þ

We denote K0 ðt; m1 Þ ¼ P ðt; m1 Þ P ðt; m1 Þ  P ðt; m2 Þ m1  m2 P ðt; m1 ; m2 Þ  P ðt; m2 ; m3 Þ K2 ðt; m1 ; m2 ; m3 Þ ¼ m1  m3 .. . K1 ðt; m1 ; m2 Þ ¼

Kn2 ðt; m1 ; . . . ; mn1 Þ ¼

ð25Þ

P ðt; m1 ; . . . ; mn2 Þ  P ðt; m2 ; . . . ; mn1 Þ m1  mn1

The following theorem gives conditions for the stability of Eq. (20). Theorem 1. [12] There exists negative constants m1,m2 . . . mn1 such that ðiÞ Kj ðt; m1 ; . . . ; mn1 Þ 6 0 for j ¼ 0; 1; . . . ; n  2 ðiiÞ m0 ¼ maxðmi Þ

ð1 6 i 6 n  1Þ

ðiiiÞ P ðt; m0 Þ P 0;

t P t0 :

ð26Þ

Then, the cone }+ defined by Eq. (23) remains invariant under the operator Y(t, s), and consequently, solutions of Eq. (20) are exponentially stable.

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Proof. Using the substitution xðtÞ ¼ CyðtÞ

ð27Þ

Eq. (20) takes the form c

e ðtÞC 1 yðtÞ _ ¼ CA yðtÞ 0l

ð28Þ

Direct computation shows that 2 1 m1 6 0 m2 6 6 6 ~ c ðtÞC 1 ¼ 6 CA 6 0l 6 0 0 6 6 4 K0 ðt; m1 Þ K1 ðt; m1 ; m2 Þ

0



1

.. .

0



K2 ðt; m1 ; m2 ; m3 Þ



3

0

7 7 7 7 7 7 7 1 7

 7 n1 P 5  ~a0;1 ðtÞ þ mi 0

i¼1

ð29Þ Condition (i) of theorem 1 implies that the Cauchy matrix Y(t, s) of Eq. (28) transform the cone }+ into itself. h The remainder of the proof for exponential stability follows from Refs. [10,11] under the assumptions of theorem 1. Next, the stochastic stability of the fuzzy controlled system Eq. (18) in the sense of Mohammed [11] is examined. m Suppose that, for every choice of fixed deterministic delays fn0q gq¼1  ½0; s , the fuzzy system of delay differential Eq. (20) is asymptotically stable. Then, we show that the stochastic fuzzy system of the stochastic delay differential ## "" m X c ~ ðtÞX ðtÞ þ f~ l ðl; X ðtÞ þ Ajl ðtÞX ðt  n0k Þ dt dX ðtÞ ¼ ½A 0l ð30Þ k¼1 X ðtÞ 2 S ‘ ;

‘ ¼ 1; 2; . . . ; r

is stable for all fixed deterministic delays fn0q gmq¼1  ½0; s . 0 Consider the system Eq. (30) and let Xi ðtÞ ¼ ed t Y i ðtÞ, i = 1, . . ., n, where d0 is a fixed real number. Substituting into Eq. (30), we obtain the system "" ##  m X 0 ~ 0l ðtÞ  dIÞY l ðtÞ þ dY l ðtÞ ¼ ðA Aql ðtÞednq Y l ðt  n0q Þ þ f~ l ðl; edt Y l ðtÞÞ dt ð31Þ q¼1 Y ðtÞ 2 S ‘ ;

‘ ¼ 1; 2; . . . ; r

Consider a Lyapunov functional V l ðtÞ ¼

n X i¼1

Y 2l;i ðtÞ

þ

n X m Z X i¼1

q¼1

t tn0q

Y 2l;i ðsÞ

ð32Þ

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Along the trajectories of Eq. (31), the right hand upper derivative of Eq. (32) yields n X n n X m X X l l l ~ _ ½ð2d  mÞdij  2A0ij Y i ðtÞY j ðtÞ þ Y 2il ðt  n0q Þ V l ðtÞ ¼ i¼1

2

j¼1

i¼1

n X n X m X i¼1

j¼1

q¼1

0

Alqij ðtÞednq Y li ðtÞY lj ðt  n0q Þ þ 2

q¼1

n X

Y li ðtÞf~ l;i ðl; edt Y l ðtÞÞ

ð33Þ

i¼1

For fixed real d0, the solution Yl(t) ” 0, is stable in probability if the quadratic form in n(m + 1) variables Y l;i ðtÞ; Y l;i ðt  n0j Þ, i = 1, . . ., n, j = 1, . . ., m is positive definite for all t P 0 for each fixed deterministic delay fn0q gmq¼1  ½0; s . The matrix associated with the quadratic form Eq. (33) can be written in block form as n X ~ l Þ2 ðtÞ Y li ðtÞf~ i ðl; edt Y l ðtÞÞ 6 ðY lt ÞT M l ðt; dÞðY lt Þ þ 2LðY ð34Þ V_ l ¼ ðY lt ÞT M l ðt; dÞðY t Þ þ 2 i¼1

where

2

~ c ðt; dÞ A 0l

6 6 Al1 ðtÞT edn01 6 6 T dn0 2 M l ðt; dÞ ¼ 6 6 Al2 ðtÞ e 6 .. 6 . 4 0 Alm ðtÞT ednm

0

0

0

Al1 ðtÞedn1

Al2 ðtÞedn2

Alm ðtÞednm

In

0



0

0 .. .

In .. .

.. .

0 .. .

0

0

0

In

3 7 7 7 7 7 7 7 7 5

ð35Þ

and ~  ðt; dÞ ¼ ð2d  mÞI l  ðA ~ 0l ðtÞ þ A ~ 0l ðtÞT Þ A n 0l where I ln is an identity matrix. These results can be summarized into the following theorem. Theorem 2. If, for each t > t0, the determinant      m X 0 ~ T 2dnq  ðt; dÞ þ A ðtÞA ðtÞe Dl ðt; dÞ ¼ A ¼0 ql ql   0l q¼1

ð36Þ

has real roots, let c(t) denote the largest of these. If cðtÞ 6 m0 ¼ max ðmi Þ < 0, then the trivial solu1 6 i 6 n1

tion Y(t) ” 0, of the fuzzy delay differential system Eq. (31) for each Y(t) 2 S‘; ‘ = 1, 2, . . . ,r, is asymptotically stable in probability. Proof. Since the matrix Ml(t, d) in Eq. (35) is anti-symmetric, then the roots of jMl(t, d)kIj = 0 shall be real and positive for fixed t > t0 and sufficiently large k. For sufficiently small k, the quadratic form in Eq. (35) shall be zero, since the roots of jMl (t, d)kIj = 0 are continuous functions of k. At least one root shall be zero for some value of k. These values of k satisfy the determinant of jMl(t, d) j = 0. It is easily seen that    m  m Y X 0 ~ ð37Þ M l ðt; dÞ ¼ Aql ðtÞATql ðtÞe2dnq  A0l ðt; dÞ þ   q¼1 q¼1

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Let c(t) be the largest invariant root of the determinant      m X 0 ~ T 2dnq  ðt; dÞ þ A ðtÞA ðtÞe Dl ðt; dÞ ¼ A ¼0 ql ql  0l  q¼1 such that cðtÞ 6 m0 ¼ max ðmi Þ < 0. Then, the quadratic form shall be positive if k = m0 for t > t0. 1 6 i 6 n1 Using the Lipschitzian condition stated in Eq. (16), the quadratic form Eq. (33) yields V_ l ðtÞ 6 m0 kY t k2  2LkY t k2 > 0

if L < m0 =2

This proves that there exists an upper bound for the unknown constant 0 6 L < m0/2 for each Y(t) 2 S‘; ‘ = 1,2, . . . ,r of the fuzzy delay differential system Eq. (31). Therefore, the system Eq. (31) is asymptotically stable in probability, and consequently, the fuzzy delay differential system Eq. (30) is asymptotically stable. Since, for every choice of fixed deterministic delays fn0q gmq¼1  ½0; s in Eq. (30) is asymptotically stable, then the fuzzy system of delay differential Eq. (20) is also asymptotically stable following the results in Ref. [11]. h

3. Energy conversion and control of an electrostatic micromotor Consider an electrostatic micromotor [5,7,15]. The cross section view of the designed electrostatic motor is shown in Fig. 1. The micromotor is fabricated on a layer of 1 lm nitride, which is deposited on a layer of 100 nm silicon dioxide on a silicon wafer. The thin oxide layer below the nitride layer is used to reduce the stress between the silicon nitride and the substrate. Below the polysilicon rotor, there is also a thin 300 nm layer of polysilicon, which is devised to shield the electric field between the rotor and the substrate (therefore, there are no vertical electrostatic forces on the rotor) that results in frictional forces between the rotor and the hub. The rotor is supported by a flanged hub (minimizing the frictional forces between the rotor and the substrate). The first problem to be solved is to derive the mathematical model of the studied micromotor that is needed to be controlled.

Fig. 1. Top and cross section view of the electrostatic micromotor.

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As the voltage V is applied to the parallel conducting rotor and stator plates, the charge is ; A is the area of the plate; e is the permittivity Q = CV, where C is the capacitance, C ¼ e Ag ¼ e WL g of the media between the plates (in vacuum, e0 = 8.85 · 1012 C2/N m2 = 8.85 · 1012 F/m); g is the gap distance between the plates; W and L are the width and length of the plates, respectively. The energy associated with the electrical potential is W ¼ 12 CV 2 . ¼  12 Hence, the electrostatic force is found as F el ¼ oW og

eWL 2 V . g2 1 e oðWLÞ 2 V , 2 g ox

¼ where x is the direction in The tangential force due to misalignment is F t ¼ oW ox ¼ 12 eLg V 2 . which misalignment occurs. If the misalignment occurs in the width direction, F t;w ¼ oW ox The capacitance of a cylindrical capacitor is needed to be found. The voltage between the cylinders can be obtained by integrating the electric field. The electric field at a distance r from q , where q a conducting cylinder has only a radial component denoted as Er. We have, Er ¼ 2per is the linear charge density, and Q = qL. Hence, the potential difference is found as Z b Z b Z r2 q 1 q r2 ~ ~ E dl ¼ Er dr ¼ dr ¼ ln : DV ¼ V a  V b ¼ 2pe r1 r 2pe r1 a a Q ¼ 2peL Thus, C ¼ DV r . The capacitance per unit length is ln 2 r1

C L

¼ DVq ¼ ln2per2 . r1

Therefore, using the stator–rotor electrodes (plates) overlap, for the studied rotational electrostatic micromotor, the capacitance can be expressed as a function of the angular displacement. In particular, Cðhr Þ ¼ N ln2per2 hr , where N is the number of overlapping stator–rotor electrodes (plates); r1 and r1 and r2 are the radii of the rotor and stator electrodes, respectively. Other expressions for the capacitances can be found. For example, accounting for second order effects, more detailed expressions for the capacitance can be derived. Our goal is to demonstrate the basic features, and correspondingly, a simple expression as Cðhr Þ ¼ N ln2per2 hr is used. r1 The electrostatic torque developed is found as T el ¼

1 oCðhr Þ 2 pe V ¼ N r2 V 2 : 2 ohr ln r1

The torsional–mechanical equations of motion are found using NewtonÕs law. We have ! dxr 1 1 pe 2 N r2 V  Bm xr  T L ¼ ðT el  Bm x  T L Þ ¼ J J ln r1 dt

ð38Þ

ð39Þ

dhr ¼ xr dt These differential equations describe the dynamics of the studied electrostatic micromotor. The parameters of the fabricated micromotor can be estimated using the micromotor size (to obtain N, r1 and r2). The fabrication technologies and processes significantly influence the micromotor dimensions and coefficients. For example, one can have N = 12, r1 = 100 lm and r2 = 105 lm. Using this sizing data, as well as the density of materials, the moment of inertia J can be found. It must be emphasized that motors rotate only if Tel > TL. Correspondingly, taking note of the loads, the rated electrostatic torque must be examined to guarantee Tel > TL. This results in the

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specific micromotor dimensions and voltage applied. In fact, one uses the equation T el ¼ N lnper2 V 2 r1 to examine the rated electrostatic torque developed. Though fabrication of electrostatic micromotors may appear to be simpler than that of electromagnetic motion devices, the facts that the high voltage must be supplied to the stator plates as well as that the rotor must be grounded (voltage may be supplied to the rotor plates) significantly limit the application of the electrostatic rotational micromachines. It also must be emphasized

Fig. 2. Dynamics of the microscale servo system.

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that the torque density of electrostatic micromachines is significantly lower than the torque density of electromagnetic motion microdevices. The electromagnetic micromotors can also be straightforwardly fabricated and assembled using axial windings and pre-made permanent magnets. The differential Eqs. (33) describe the dynamics of the studied electrostatic micromotors. The parameters of the fabricated micromotor are: Bm = 3 · 109 N m s/s; the polysilicon resistivity is q = 0.01 X cm, the winding resistance is R = 250 X; the radius of the rotor is r1 = 50 lm; the radius of the stator is r2 = 51 lm; the number of rotor electrodes (plates) is n = 4; the rotor thickness is t = 10 lm; and J = 8 · 1020 kg m2. The saturated PI controller u = sat(kpe + kie dt) is designed and implemented using the procedure reported in Ref. [7]. The micromotor dynamics in the servo-system application is illustrated in Fig. 2a for kp = 9 and ki = 4.1 if the command angular displacement is 0.2 rad. It must be emphasized that the initial load torque is applied to the microscale servosystem, and correspondingly the negative initial displacement is observed. This can be eliminated using advanced kinematics. The settling time for the closed loop microsystem with the saturated PI controller is 0.0024 sec. Our goal is to design high performance systems and improve the dynamics. Making use the procedure reported, the fuzzy logic control law, as given by Eq. (3), is designed and verified. Fig. 2b illustrates the servosystem dynamics for the reference command 0.2 rad. It is evident that the studied microscale servo system guarantees excellent dynamics. The settling time is 0.0008 s. We conclude that the reported design paradigm was verified.

4. Conclusions Microelectromechanical systems, described by nonlinear stochastic differential equations with random delays, were studied in order to design fuzzy controllers that guarantee superior achievable performance. The cone theory was used, and viable sufficient conditions for the stochastic stability of the fuzzy stochastic microelectromechanical systems was derived. The results are demonstrated by studying an electrostatic micromotor. Acknowledgement The authors wish to thank the reviewers for their valuable suggestions, which have improved the presentation of the paper.

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