Nano and Microelectromechanical Systems P6

varying current radiates electromagnetic waves radiated electromagneticfields.. We Time-use the vectors of the electric field intensity E, electric flux density D, magnetic field intensi

CHAPTER 4 CONTROL OF NANO- AND MICROELECTROMECHANICAL

SYSTEMS 4.1 FUNDAMENTALS OF ELECTROMAGNETIC RADIATION AND

ANTENNAS IN NANO- AND MICROSCALE

ELECTROMECHANICAL SYSTEMS

The electromagnetic power is generated and radiated by antennas varying current radiates electromagnetic waves (radiated electromagneticfields) Radiation pattern, beam width, directivity, and other majorcharacteristics can be studied using Maxwell’s equations, see Section 2.2 We

Time-use the vectors of the electric field intensity E, electric flux density D, magnetic field intensity H, and magnetic flux density B The constitutive

equations are

E

D = ε and B = µ H

where ε is the permittivity; µ is the permiability

It was shown in Section 2.2 that in the static (time-invariant) fields,electric and magnetic field vectors form separate and independent pairs That

is, E and D are not related to H and B, and vice versa However, for

time-varying electric and magnetic fields, we have the following fundamentalelectromagnetic equations

t

t z y x t

z y

,

t

t z y x t

z y x t

z y

ε

ρ ( , , , ) )

, ,

,

t z

J = σ ; ρv is the volume charge density

The total current density is the sum of the source current JS and theconduction current density σ E (due to the field created by the source JS).Thus,

E J

dv dt

Therefore, the net outflow of current from a closed surface results indecrease of the charge enclosed by the surface.

The electromagnetic waves transfer the electromagnetic power That is,the energy is delivered by means of electromagnetic waves Using equations

∂

∂ µ

) ( ) ( )

In a media, where the constitute parameters are constant (time-invariant),

we have the so-called point-function relationship

2 1 2 2 1

E

s H

E

of presence

in the power dissipatedohmic

2

field magnetic and field electric the rate of change of energy stored intime

2 2 1 2 2

H E

in the volume v The pointing vector, which is a power density vector,

represents the power flows per unit area, and

H E s

s

dv dv

w w t d

∂

∂

volume enclosed the

ρσ = = is the ohmic power density

The important conclusion is that the total power transferred into a closed

surface s at any instant equals the sum of the rate of increase of the stored

electric and magnetic energies and the ohmic power dissipated within the

be studied Hence, the phasor analysis is applied For example,

z z y y x

r

E ( ) = ( ) + ( ) + ( )

The electric field intensity components are the complex functions Inparticular,

Im Re

)

E r = + , Ey( r ) = EyRe+ jEyIm, Ez( r ) = EzRe+ jEzIm.For the real electromagnetic field, we have

t E

t E

σ

j j

• Gauss’s law

ε ω

j is the complex permittivity However, for simplicity we will

use ε keeping in mind that the expression for the complex permittivity

functions of spatial coordinates

From the equation (4.1.1) taking the curl of ∇ × E = − j ωµ H, which isrewritten as ∇ × E = − j ω B, and using ∇ × H = j ω D + J, one obtains

J E

1

ε µ

π π

kv

v

2 2

0

Using the magnetic vector potential A, we have B = ∇ × A.

Hence,

0 )

E + j ω = −∇ ,

where Ë is the scalar potential

To guarantee that ∇ × H = j ω D + J holds, it is required that

J E A

A A

J Ë A

J A

To derive E, one must have A The Laplacian for A in different coordinate

systems can be found For example, we have

x x

1 )

, ' ( )

,

r r

r r r

r r

.Thus, we have the expressions for the phasor retarded potentials

dv e

)

(

'

r J r r A

r r r

π

µ

,

dv e

r r

Example 4.1.1.

Consider a short (dl) thin filament of current located in the origin, see

Figure 4.1.1 Derive the expressions for magnetic vector potential andelectromagnetic field intensities

Figure 4.1.1 Current filament in the spherical coordinate system

Solution.

The magnetic vector potential has only a z component, and thus, from

J A

A k

Taking note of the spherical symmetry, we conclude that the magnetic

vector potential A z is not a function of the polar and azimuth anglesθandφ Inparticular, the following equation results

z k A r

d

has two

components In particular, ejk v r (outward propagation) and e−jk v r (inwardpropagation) The inward propagation is not a part of solution for the filamentlocated in the origin Thus, we have

r jk t

j v

ae

r

ψ ( , ) (outward propagating spherical wave)

In free space, we have

) / (

r

a r A

ω

−

= )

z s

d r z v

c d d r A dv

2 2

r

r

a r c

j r

A A

d d ae

r c

d

r c j d

r

idl r

e r

From az = arcos θ − aθsin θ, we have

) sin cos

( 4

r j

e r

E = − + ∇∇ ⋅

.Then, one finds

φ

ω

ω π

θ

r j

e r cr

j r

idl r

sin )

4

1 cos

4

)

(

3 2 2

2 0

0

3 2 0

0

θ ω

ω

ω ω θ πω

ε

µ

ω θ πω

ε µ

a

a E

c r j

r c r j

e r cr

j r c

cidl

j

e r cr j cidl

Near-field and far-field electromagnetic radiation fields can be found,

simplifying the expressions for H(r) and E(r).

For near-field, we have

e cr

idl j r

sin )

( 1

)

(

4 )

0

θ

θ π

ω ε

µ

a E

r c

cidl j

The complex Pointing vector can be found as

) ( )

12 12

) ( )

(

2 2 0 2 2 0 0 0 2

* 2

d r

The real quality is found, and the power dissipated in the sense that it

travels away from source and cannot be recovered

Example 4.1.2.

Derive the expressions for the magnetic vector potential andelectromagnetic field intensities for a magnetic dipole (small current loop)which is shown in Figure 4.1.2

Figure 4.1.2 Current loop in the xy plane

e r

The distance between the source element dl and point O ( r , θ ,π2) is

denoted as r’ It should be emphasized that the current filament is lies in the xy

plane, and

φ φ φ

2 /

0 0

'

sin 2

π π

ir0φ

) , ,

O

dl

sin 4 sin

sin sin 1 2

'

sin 2

2

2 0 0 2

/

2 /

0 0

0

2 /

2 /

0 0

θ µ

φ φ φ θ π

µ

φ

φ π

µ

φ π

π φ

π π φ

r

ir d

r

r r

ir

d r ir

a a

2 ( 4

sin

2 0 0 2

2 0

r

ir r

ir

.Taking note of the expression for the magnetic dipole moment

4

sin

µ θ

i

'

1 4

dl r

e i

' 4

' 0

φ

ω ω

sin 1

4 '

)]

' ( 1 [

0

c j c l

r c j

r

M dl

r

e r r j

ω

ω ω

3 0

2 2

r c j c

c

e r r j c

M j

ε µ

0

0 2

3 0

3 3 2

2

r c j c

c

r j r c

M j

θ π

ε µ

ω

ω ω

ω

4

3 2

0

0 2

3 0

3 3 2

2

r c j c

c c

e r j r r j c

M j

The electromagnetic fields in near- and far-fields can be straightforwardlyderived, and thus, the corresponding approximations for theEφ, Hr and Hθ

Let the current density distribution in the volume is given as J ( r0), andfor far-field from Figure 4.1.3 one has r ≈ r ' − r0

Figure 4.1.3 Radiation from volume current distribution

The formula to calculate far-field magnetic vector potential is

dv e e

jk r

Z jk

v

jk r

r r jk v

Source

Figure 4.1.4 Half-wave dipole antenna

Solution.

The wavelength is given as

µε ω

π π

kv

v

2 2

Z jk

v

jk r

r r jk v

r l r r jk v

( 4

2

) cos cos(

cos ) cos ( 4

)

(

2 1 0

cos 0

4

4

θ

θ λ

λ

θ π

θ π

θ π

a

a a

E r

r jk v

z jk v z r

r jk v v

v

v v

v v

e r

i jZ

dz ze

k e

r

i Z jk

θ π

a a

r E a

r

ji H

=

sin 2

) cos cos(

) ( )

1 0

,the power flux per unit area is

φ

2 1 2 0 2 0

* 2 1

* 2

1

sin 8

) cos ( cos )

( )

(

Re

r

Z i H E

and integrating the derived expression over the surface

φ θ θ θ

θ π π

ππ

d d Z

i

sin sin

) cos ( cos

8

2

0 0

2 2 1 2

E j

ε ωµε

E H

The solutions are

r r J

jk r jk

jk r jk

E sin v −2

:Field

0

l

Slot

l s

d d

j dl

the boundary conditions for the magnetic current sheet are found as

H n

θ π ωµε

−

=

sin 2

) cos cos(21 0 0

,

sin

2

) cos cos(21 0

φ

θ π

θ π

s

jk r jk

4.2 DESIGN OF CLOSED-LOOP NANO- AND

MICROELECTROMECHANICAL SYSTEMS USING THE

LYAPUNOV STABILITY THEORY

The solution of a spectrum of problems in nonlinear analysis, structuralsynthesis, modeling, and optimization of NEMS and MEMS lead to thedevelopment of superior high-performance NEMS and MEMS In this section,

we address introductory control issues Mathematical models of NEMS andMEMS were derived, and the application of the Lyapunov theory is studied asapplied to solve the motion control problem.

It was illustrated that NEMS and MEMS must be controlled Nano- andmicroelectromechanical systems augment a great number of subsystems, and tocontrol microscale electric motors, as discussed in previous chapters, poweramplifiers (ICs) regulate the voltage or current fed to the motor windings.These power amplifiers are controlled based upon the reference (command),output, decision making, and other variables Studying the end-to-end NEMSand MEMS behavior, usually the output is the nano- or microactuator linearand angular displacements There exist infinite number of possible NEMS andMEMS configurations, and it is impossible to cover all possible scenarios.Therefore, our efforts will be concentrated on the generic results which can beobtained describing NEMS and MEMS by differential equations That is, usingthe mathematical model, as given by differential equations, our goal is developcontrol algorithms to guarantee the desired performance characteristicsaddressing the motion control problem (settling time, accuracy, overshoot,controllability, stability, disturbance attenuation, et cetera)

Several methods have been developed to address and solve nonlineardesign and motion control problems for multi-input/multi-output dynamicsystems In particular, the Hamilton-Jacobi and Lyapunov theories are found to

be the most straightforward in the design of control laws

The NEMS and MEMS dynamics is described as

u x B d r x

F

t

x & ( ) = ( , , ) + ( ) ,y = H ( x ),umin ≤ ≤ u umax, ( ) x t0 = x0,

(4.2.1)where x∈X⊂c is the state vector; u∈U⊂m is the bounded control vector;

r∈R⊂band y∈Y⊂bare the measured reference and output vectors; d∈D⊂s

is the disturbance vector; F(⋅):c×b×s→c and B(⋅):c→ c × m are jointlycontinuous and Lipschitz; H(⋅):c→b is the smooth map defined in theneighborhood of the origin, H(0) = 0.

Before engaged in the design of closed-loop systems, which will be basedupon the Lyapunov stability theory, let us study stability of time-varyingnonlinear dynamic systems described by

) , (

Consider the system described by nonlinear differential equations

) , (

V t

V dt

dx x

V t

3

) (

1( t ) x x x

7 2

( )

) ( ,

8 2 2 2 2 1 4 1 7 2 2

2 2 1 3

V x

x x

x

V

x F x

V dt

dx x

V dt

.The total derivative of V ( x1, x2) > 0 is negative definite

Therefore, the equilibrium state is uniformly asymptotically stable

Example 4.2.2.

Study stability of the time-varying nonlinear system modeled by thefollowing differential equations

3 2 1

1( t ) x x

3 2 2 2 2 1 10

,

4 2

10

2

1

3 2 2 2 2 1 10 2

3 2 1 1

V x

x x

V t

V dt

−

∂

∂ +

.Hence, the equilibrium state is uniformly asymptotically stable

( 2)

2 2 2 1 2 2 1 1 2

(

x x x x x x x dt

Hence, the equilibrium state is uniformlyasymptotically stable, and the quadratic function ( 2)

2 2 1 2 1 2

Consider a microdrive actuated by permanent-magnet synchronous motor

if T L=0 In drive applications, using equations (3.5.12), three nonlineardifferential equations in the rotor reference frame are

3 2

1

r m r qs m r

J

B i J

m r

qs m ls

s

r

L L

i L L

r dt

di

ω ω

+

− +

−

=

2 3 2

r

r qs

r ds m ls

s

r

L L

r dt

di

ω

+ +

J

B i J

+

− +

−

=

0 0

8 3

0 0

0 )

(

2

2 3

2 3 2

3

r

r qs

r

r ds

r

r ds

r qs

m m

m ls s

m ls

m m

ls s

i

i i

i

J

B J

P

L L r

L L L

L

r t

ω

ω ψ

.8

38

),

,

(

2 2

2 2

r r qs ss

m ss m

r m r ds r

L P J

J

B i i

Consider the closed-loop system

To guarantee the balanced operation we let

r ds r m ls

m r

qs m ls

s

r

L L

i L

L

i L L

r dt

di

ω ω

ω

ψ

ω

2 3 2

3 2

3

1 +

−

− +

− +

−

r

r qs

r ds m ls

s

r

L L

r dt

di

ω

+ +

J

B i J

−

+

+

− +

−

=

0 0

8

3

0 0

0 )

(

2

2 3

2 3 2

3

r

r qs

r

r ds

r

r ds

r qs

m m

m ls s

m ls

m m

ls

s

i

i i

i

J

B J

P

L L r

L L

k L

Taking note of the quadratic positive-definite Lyapunov function

8

3 8

) ,

,

(

2 2

2 2

r

r qs ss

m ss m

r m r

L P k J

J

B i

ω

ω

ω − +

<

dt

i i

In Example 4.2.4 it was shown that dynamic systems can be controlled toattain the desired transient dynamics, stability margins, etc Let us study how

to solve the motion control problem with the ultimate goal to synthesizetracking controllers applying Lyapunov’s stability theory

Using the reference (command) vector r(t) and the system output y(t), the

tracking error (which ideally must be zero) is

) ( ) (

)

The Lyapunov theory is applied to derive the admissible control laws

(voltages and currents are bounded, and therefore the saturation effect is alwaysthe reality) That is, the admissible bounded controller should be designed as

continuous function within the constrained rectangular control set

U={u∈m : umin ≤ u ≤ umax, umin < 0, umax > 0}⊂m

Making use of the Lyapunov candidate V ( t , x , e ), the boundedproportional-integral controller with the state feedback extension is expressedas

e x t V B t G x

e x t V x B t

G

e i T

e e T

sat max

min

(4.2.3)where G x(⋅):≥0→m × m, G e(⋅):≥0→m × m

Figure 4.2.1 Bounded control, umin ≤ ≤ u umax

For closed-loop NEMS and MEMS (4.2.1)–(4.2.3) with

X0={x0∈c}φX⊂c,u∈U⊂m, r∈R⊂b and d∈D⊂s, it is straightforward tofind the evolution set X(X0, U, R, D)⊂c Furthermore, using the outputequation, one has X  →H Y Thus, the system (4.2.1)-(4.2.3) evolves in

XY(X0,U, R, D)={(x,y)∈X × Y: x0∈X0, u∈U, r∈R, d∈D, t∈[t0,∞)}⊂c × b.The tracking error

) ( ) (

Lemma.

Consider the closed-loop systems (4.2.1) – (4.2.3).

1 Solutions of system are uniformly ultimately bounded;

2 equilibrium point is exponentially stable in the convex and compact state evolution set X(X0, U, R, D)⊂c;

3 tracking is ensured and disturbance attenuation is guaranteed in the error evolution set XE(X0, E0, U, R, D)⊂c× b,

state-if there exists a C κ function V(t,x,e) in XE such that for all x∈X, e∈E, u∈U,

Here, ρ1(⋅):≥0→≥0, ρ2(⋅):≥0→≥0, ρ3(⋅):≥0→≥0and ρ4(⋅):≥0→≥0 are the

K∞-functions; ρ5(⋅):≥0→≥0and ρ6(⋅):≥0→≥0are the K-functions

The major problem is to design the Lyapunov candidate functions.

Let us apply a family of nonquadratic Lyapunov candidates

.)(

)()

()

2

1 2

0 ) 1 ( 2 1 2

0 ) 1 ( 2 1 2

1 2 1 1

2 1

1 2 1 1

2 1 1

2 1 1

2 1

+ + +

+ +

+ +

ς β β η

γ γ

µ µ

β β β

β γ

γ γ

γ

i

si T i

i

ei T i

xi T

i i

i i

i i

i i

e t K e

e t K e x

t K x e

1 ) ( )

( diag

)

(

) ( diag

) ( ) (

0 0

0

1 2 1 1

2 1

2 1 1

2

1 2 1 1

2 max







 +

− +

+ +

−

+ + +

−

σ ς

η

µ µ µ

µ β

β β

β

γ γ γ

γ

T e i

u

u

i i

i i

i i

e t K e s

B t G e t K e B

t

G

x t K x x

B t

G

u sat

(4.2.7)Here, K xi(⋅):≥0→c × c, K ei(⋅):≥0→b × b and K si(⋅):≥0→b × bare the matrix-functions

It is evident that assigning the integers to be zero, the well-knownquadratic Lyapunov candidate results, and

) ( )

( )

( )

,

,

0 2

1 0

2

1x K t x e K t e e K t e e

+

s t K B t G e t K B t G x t K x B t G

e i e

T e e x

T x

Example 4.2.5.

Consider a micro-electric drive actuated by a permanent-magnet DC motorwith step-down converter, see Figure 4.2.2 Find the control algorithm

i a i L

+ -

Using the Kirchhoff laws and the averaging concept, we have the

following nonlinear state-space model with bounded control

L c

L t L d t

L d

r a L a

m a a a a a a

L

L L

r u

L V

i i u

J

B J k L

k L

r L

L

C C

000

00

01

0001

0110

max max

ω ω

,

uc∈ [ 0 10 ] V

A bounded control law should be synthesized

From (4.2.6), letting ς σ = = 1 and β µ η γ = = = = 0, one finds thenonquadratic function V ( e , x ) In particular, we apply the followingLyapunov candidate

,]

[)

,

2 1 4 1 4 1 2 0 2 1 4 1 4 1 2

++

+

=

r a L a x r a L a ei

ei e

e

i i

u K i

i u e k e k e k e

where Kx0 ∈4 × 4

Therefore, from (4.2.7), one obtains

,0for

,100

for

,10for

10

3 3

r a L a

c

k i k i k u k dt e k edt k e k

u u

++