An adaptive change detection scheme for a nonlinear beam model

An Adaptive Change Detection Scheme for a

Nonlinear Beam Model

M. A. Demetriou

B. G. Fitzpatrick

[email protected] [email protected]

(919) 515-6544

(919) 515-7552

Center for Research in Scientic Computation

Department of Mathematics

North Carolina State University

Raleigh, NC 27695-8205

Fax: (919) 515-3798

December 11, 1995

Abstract

In this paper, we consider parameter estimation techniques for detecting changes of a nonlinear nature in an Euler-Bernoulli beam model. The nonlinear stiness used provides a very simple model of damage, and the adaptive estimation algorithm is used to track the onset of the nonlinearity. Using Lyapunov redesign methods, extended and applied to innite dimensional systems, a stable learning scheme is developed. The resulting parameter adaptation rule is able to \sense" the instance of the fault occurence. In addition, it identies the location and the shape of the fault where the beam is persistently excited. Simulation studies are used to illustrate the applicability of the theoretical results.

Keywords:

On-Line Estimation, Failure Detection, Nonlinear Beam Models.

1 Introduction

An important aspect of many structural vibration problems is the online (nondestructive) detec-tion of structural changes, particularly changes that indicate impending failure. In recent years there has been a great deal of research into adaptive parameter estimation techniques, which provide a means for online model calibration. We consider in this paper an extension of these ideas for a simple model of structural change. We use an Euler-Bernoulli model of a vibrating beam, with the change being an abrupt transition from linear to nonlinear stiness. The idea is that the nonlinear stiness provides a simple model of damage to the beam. Our goal is to investigate adaptive algorithms which can detect the occurence of this nonlinearity.

This problem involves approximation at several levels. The plant is modeled by a partial dierential equation, which must be solved numerically. The identication algorithm considered

here requires \full state feedback" which means we must estimate the innite dimensional state from nite dimensional observations. We give a complete description of our algorithm, along with a numerical example which illustrates its utility.

The paper is organized as follows. In Section 2 we give the basic problem statement. In Section 3, we outline the adaptive algorithm and give some well-posedness results. State and parameter convergence is the topic of discussion of Section 4, and in Section 5, we give our numerical results. Comments with future extensions follow in Section 6.

2 Problem Statement

In mathematically modeling the exible beam, we assume that it is of length

l

with uniform rectangular cross section of height

h

and width

b

. We let

w

(

t;x

) denote the transverse displace-ment of the beam at position

x

along its span at each time

t

. This is measured relative to the

x

-axis in the coordinate frame determined by the longitudinal axis of the beam in its undeformed state with origin located at the beam's xed end. We assume a cantilevered Euler Bernoulli beam with Kelvin-Voigt damping for the modeling of the dynamics and dissipation, see [3], [5], [10], [11], [14], and [26] . It is assumed that the beam undergoes only small deformations (i.e.

j

w

(

t;x

)j

<< l

, and j(

@w=@x

)(

t;x

)j

<<

1). The Euler-Bernoulli theory including Kelvin-Voigt

viscoelastic damping yields the partial dierential equation

b

w

tt+

@

2

@x

2

M

(

t;x

) =

f

(

t;x

)

;

0

< x < l;

t >

0

;

(2.1)

with the boundary conditions

w

(

t;

0) =

w

x(

t;

0) =

w

(

t;l

) =

w

x(

t;l

) = 0

;

t >

0

;

(2.2)

where

b is the linear mass density,

M

(

t;x

) is the internal moment and

f

is the external applied

force. For an uncontrolled beam with Kelvin-Voigt damping, the moment is given by (see [11])

M

(

t;x

) =

EIw

xx(

t;x

) +

c

D

Iw

txx(

t;x

)

;

0

< x < l; t >

0

;

where

E

is Young's modulus,

I

is the cross sectional moment of inertia, and

c

D is the damping

modulus. For actuation, a piezoceramic patch is attached to the beam as shown in Figure 2.4. This patch is excited in such a way so as to produce a pure bending moment, see [11]. If

H

0 is

used to denote the Heaviside function with unit step at

x

= 0, the model for the beam is then given by

b

w

tt"(

t;x

) + [

EIw

xx(

t;x

) +

c

D

Iw

txx(

t;x

)]xx = (2.3)

EIK

B

d

31

u

(

t

)[

H

0(

x

?

a

1) ?

H

0(

x

?

a

2)] #

xx

;

0

< x < l; t >

0

;

where

u

(

t

) is the voltage applied to the patch at time

t

,

K

B _{is a parameter which depends on}

the geometry and piezoceramic materal properties,

is the patch thickness,

a

1 and

a

2 denote

the position of the patch and

d

31is the piezoceramic strain constant (see [4, 11]). Equation (2.3)

% % %

% %

% %

% %

% %

% %

% %

% %

% %

e e

e e

e e

e e

e e

e e

e e

e e

e e

e

piezoceramic patch

width

?

6

-Z Z Z Z }

thickness

6 ?

length

Figure 2.1: Cantilevered beam with piezoceramic actuator patch.

We now consider the nonlinear Euler-Bernoulli beam with Kelvin-Voigt viscoelastic damping

b

w

tt(

t;x

) + [

EI

(

t;w

xx(

t;x

)

;x

)+

c

D

Iw

txx(

t;x

)]xx=

f

(

t;x

) (2.4)

with boundary conditions given by

w

(

t;

0) =

w

x(

t;

0) =

w

(

t;l

) =

w

x(

t;l

) = 0

and initial conditions given by

w

(0

;

) =

w

0(

)2

H

2

0(0

;l

)

;

w

t(0

;

) =

w

1( )2

L

2(0

;l

)

:

The nonlinear stiness term

EI

(

t;w

xx(

t;x

)

;x

) is given by

EI

(

t;w

xx(

t;x

)) =

EI

0

w

xx+

H

0(

t

?

t

)

[x1;x2](

x

)

g

(

w

xx)

=

EI

0

w

xx+

H

0(

t

?

t

)

[x1;x2](

x

) (

EI

1

w

xx(

t;x

) if

w

xx(

t;x

)

>

0

EI

2

w

xx(

t;x

) otherwise

;

which consists of the nominal (linear) term

EI

0

w

xx(

t;x

) (same as the one in (2.3)) and the

nonlinear term

g

(

w

xx), depicted in Figure 2.2 and given by the equation below

g

(

) =

EI

1

sign(

) +

EI

2

(1

?sign(

)) (2.5)

The latter is zero before failure and at the failure (i.e. at time

t

t

) depends on the curvature

of the beam

w

xx(

t;x

) and acts on part of the beam (i.e. the characteristic function

[x1;x2](

x

)).

Specically, the beam stiness parameter prior to the failure (i.e. for

t < t

) is given by

EI

(

t;w

xx(

t;x

)

;x

) =

EI

0

w

xx(

t;x

)

;

0

x

l;

and after the failure (i.e. for

t

t

) by

EI

(

t;w

xx(

t;x

)

;x

) =

(

EI

0

w

xx(

t;x

) +

[x1;x2](

x

)

EI

1

w

xx(

t;x

) if

w

xx(

t;x

)

>

0

EI

0

w

xx(

t;x

) +

[x1;x2](

x

)

EI

2

w

xx(

t;x

) otherwise

It is assumed, for the sake of simplicity, that the nominal beam stiness

EI

0 is known. In

? ? ? ? ? ? ?

w

xx

EI

1

EI

2

g

(

w

xx)

Figure 2.2: Nonlinear stiness term

g

(

w

xx).

here is to detect the failure modelled by the nonlinear term

g

(

w

xx) and identify the beam

stiness parameters

EI

1 and

EI

2 adaptively. The failure time

t

is unknown and it is desired

to detect (implicitly) this failure time by monitoring the system, via an appropriately chosen state observer. The proposed state estimator will detect changes in the system which would be an indication of the failure and thus

t

would be identied. It is assumed that the beam

displacement

w

(

t;x

) and velocity

w

t(

t;x

) are available for measurement at each time

t

. In the

next section, we present our algorithm for detecting failures and estimating the beam parameters from state observations.

3 Abstract Formulation

Before we proceed with the abstract formulation of the beam model, we need to provide some details on the abstract spaces involved. We consider the Hilbert space

L

2(0

;l

) as the state space.

We also consider the Sobolev space

H

2

0(0

;l

) as the space of test functions, see [1]. Using the

fact that the Sobolev space

H

2

0(0

;l

) is embedded densely and continuously in the Hilbert space

L

2(0

;l

), [25, 27], it follows that

H

2 0(0

;l

)

,

!

L

2(0

;l

)

,

!

H

?2

(0

;l

)

;

(3.1)

where

H

?2(0

;l

) denotes the continuous dual of

H

2

0(0

;l

), see [1, 13, 27]. In particular, we assume

that there exists an embedding constant

K

emb

>

0 such that j

'

jL2

K

embj

'

jH20,

'

2

H

2 0(0

;l

).

Here we useh

;

ito denote the usual duality product obtained as the extension by continuity

of the

L

2(0

;l

)- inner product from

L

2(0

;l

)

H

2

0(0

;l

) to

H

?2(0

;l

)

H

2

The

L

2(0

;l

)-inner product (energy inner-product) is given by

h

;

iL2= Z l

0

b (

x

)

(

x

)

dx;

;

2

L

2(0

;l

)

;

whereas the

H

2

0(0

;l

)-inner product is given by

h

;

iH20= Z l

0

xx(

x

)

xx(

x

)

dx

;

2

H

2 0(0

;l

)

:

The latter is the standard

H

2-inner product with the (clamped) boundary conditions imposed

on its elements; thus

H

2

0(0

;l

) = n

'

2

H

2(0

;l

) :

'

(

x

) =

'

x(

x

) = 0

; x

= 0

;l

o

:

In order to simplify equation (2.4) above, we rewrite (2.5) as

g

(

w

xx) =

EI

1

w

xx(

t;x

)

(

t;x

)+

EI

2

w

xx(

t;x

)(1

?

(

t;x

))

;

(3.2)

where we dene the indicator function

(

t;x

) (the sign function) by

(

t;x

) =

(

1 if

w

xx(

t;x

)

>

0

0 otherwise

:

In the rest of this note, we suppress the dependence of

w

(

t;x

),

w

t(

t;x

),

w

tt(

t;x

),

w

xx(

t;x

),

w

txx(

t;x

) on

t

and

x

and denote them simply by

w

,

w

t,

w

tt,

w

xx, and

w

txxrespectively. Similarly,

for

[x1;x2](

x

) and

(

t;x

), we write instead

I and

.

We now write the beam equation in weak or variational form (i.e. multiply by a \test function" and integrate over the spatial domain of interest, or take the inner product with a test function)

h

w

tt(

t

)

;'

iL2 + h

EI

0

w

(

t

)

;'

iH20+h

c

D

Iw

t(

t

)

;'

iH20 (3.3)

+

H

0(

t

?

t

) h

EI

1

I

(

t

)

w

(

t

) +

EI

2

I(1

?

(

t

))

w

(

t

)

;'

iH20 =h

Bu

(

t

)

;'

iL2

;

where the test function

'

2

H

2

0(0

;l

) and the input operator

B

2L(

U;H

?2(0

;l

)) is given by the

right hand side of equation (2.3) and it is assumed to be known. The space

U

is called the input or control space. The initial conditions associated with (3.3) are

w

(0

;

) =

w

0(

)2

H

2

0(0

;l

)

;

w

t(0

;

) =

w

1( )2

L

2(0

;l

)

:

(3.4)

The function

H

0(

t

?

t

) that represents the time prole of the failure is assumed to be the

Heaviside function given by

H

0(

t

?

t

) = 0 for

t < t

and

H

0(

t

?

t

) = 1 for

t

t

.

When the beam is actuated by a centered piezoceramic patch, we can rewrite the above equation (3.3) explicitly in terms of integrals over the spatial domain by

Z l 0

b

w

tt(

t;x

)

'

(

x

)

dx

+ Z l

0

EI

0

w

xx(

t;x

)

'

xx(

x

)

dx

+ Z l

0

c

D

Iw

txx(

t;x

)

'

xx(

x

)

dx

+

H

0(

t

?

t

) Z l

0

[

EI

1

(

t;x

) +

EI

2(1

?

(

t;x

))]

[x1;x2](

x

)

w

xx(

t;x

)

'

xx(

x

)

dx

= Z l 0

KB

[a1;a2](

x

)

'

xx(

x

)

dx

!

u

(

t

)

;

'

2

H

2

where

u

(

t

) 2

U

is the voltage applied to the patch, KB =

EI

K Bd

31

, and

[a1;a2](

x

) is the

characteristic function over the interval [

a

1

;a

2], see [8, 12, 13] for additional details on the

modeling equations of a beam with piezoceramic actuators.

Since in this note we deal with two constant parameters, then the parameter space

Q

is identied with the Euclidean space RI2, i.e.

Q

= RI2. Before we proceed with the state and

parameter estimator, we must impose a boundedness condition on the state of the beam.

Assumption 3.1 (Boundedness of plant)

A plant is a triple(

EI

1

;EI

2

;w

) with

w

a solution

to the initial-value problem (3.3) with

w

2

H

2

0(0

;l

), a.e.

t >

0, for which there exists a constant

>

0 such that

h(

EI

1

(

t;x

) +

EI

2(1

?

(

t;x

)))

I(

x

)

w

(

t;x

)

;

iH20

EI

1

EI

2

R2

j

jH20

for almost every

t >

0 and all

2

H

2 0(0

;l

).

Remark 3.2

The well posedness of the plant equation has been treated in the paper by Banks, Gilliam and Shubov in [6, 7].

4 Estimator and Convergence

We now proceed with the state estimator given in the form of an initial value problem

h

v

tt(

t

)

;'

iL2+h

EI

v

(

t

)

;'

iH20+h

c

D

I

v

t(

t

)

;'

iH20+h

EI

0

w

(

t

)

;'

iH20

+h

c

D

Iw

t(

t

)

;'

iH20 ?h

EI

w

(

t

)

;'

iH20?h

c

D

I

w

t(

t

)

;'

iH20

+h c

EI

1(

t

)

I

(

t

)

w

(

t

) + c

EI

2(

t

)

I(1

?

(

t

))

w

(

t

)

;'

iH20 =h

Bu

(

t

)

;'

iL2

;

(4.1)

where the parameters

EI

and

c

D

I

are some tuning parameters , (see, for example, [18]); i.e.

they are values of the stiness and damping parameters chosen to aect the convergence of the estimator. The initial conditions for the state observer are taken to be the same as the ones for the plant, namely

v

(0

;

) =

w

(0

;

)

v

t(0

;

) =

w

t(0

;

)

:

(4.2)

The parameters

EI

c

1(

t

) and c

EI

2(

t

) in (4.1) are the adaptive estimates of the unknown parameters

EI

1 and

EI

2. By denoting the state error

v

?

w

, by

e

=

v

?

w

we then have that, using the

Lyapunov redesign method, the unkown parameters can be adjusted via

d

dt

EI

c

1(

t

) =

1

h

I

(

t

)

w

(

t

)

;e

t(

t

) +

e

(

t

)iH20 (4.3)

d

dt

EI

c

2(

t

) =

2

h

I(1?

(

t

))

w

(

t

)

;e

t(

t

) +

e

(

t

)iH20 (4.4)

for some

>

0 and with initial conditions given by

c

EI

1(0) =

c

EI

2(0) = 0

:

(4.5)

We denote the parameter errors by

r

1(

t

) = c

EI

1(

t

)

?

EI

1 and

r

2(

t

) = c

EI

2(

t

)

?

EI

2,

respec-tively. Using the denition of the state error, and using (3.3), (4.1), (4.3) and (4.4), we arrive at the (state and parameter) error equations

h

e

tt(

t

)

;'

iL2+h

EI

e

(

t

)

;'

iH20+h

c

D

I

e

t(

t

)

;'

iH20

+h[

r

1(

t

)

(

t

) +

r

2(

t

)(1

?

(

t

))]

I

w

(

t

)

;'

iH20 = 0

;

(4.6)

d

dtr

1(

t

) =

1

h

I

(

t

)

w

(

t

)

;e

t(

t

) +

e

(

t

)iH20 (4.7)

d

dtr

2(

t

) =

2

h

I(1?

(

t

))

w

(

t

)

;e

t(

t

) +

e

(

t

)iH20 (4.8)

with initial conditions given by

e

(0) =

e

t(0) =

r

1(0) =

r

2(0) = 0

:

(4.9)

Remark 4.1

The parameter errors

r

1(

t

) and

r

2(

t

) are given by

r

1(

t

) =

c

EI

1(

t

)

?0

;

r

2(

t

) =

c

EI

2(

t

)

?0 for

t < t

;

r

1(

t

) =

c

EI

1(

t

)

?

EI

1

; r

2(

t

) = c

EI

2(

t

)

?

EI

2 for

t

t

;

since the unknown parameters

EI

1

;EI

2 are zero prior to the (unknown) failure time

t

.

Before we present any convergence results, we dene the energy functional by

V

(

t

) =

n h

EI

e

(

t

)

;e

(

t

)

iH20+j

e

t(

t

)j 2 L2

o

+ 2h

e

(

t

)

;e

t(

t

)iL2

+h

c

D

I

e

(

t

)

;e

(

t

)

iH20+

r

2 1(

t

) +

r

2

2(

t

)

;

(4.10)

where the constant

>

0 will be dened below. With no additional assumptions we have the following convergence result.

Theorem 4.2

Assume that the plant satises the boundedness condition given byAssumption 3.1. If the constant

satises

>

max

(

K

emb

; K

EI

emb

; K

2 emb

c

D

I

)

;

(4.11)

then, for

t < t

we have

V

(

t

) =j

e

(

t

)jH20 =

r

1(

t

) =

r

2(

t

) = 0

;

and for

t

t

we have

j

e

(

t

)j 2

H₂₀+j

e

t(

t

)j 2 L2 +

r

2

1(

t

) +

r

2 2(

t

) +

Z t t

n j

e

(

)j

2

H₂₀ +j

e

(

)j 2 H₂₀

o

d

n j

e

(

t

) j

2

H₂₀+j

e

t(

t

)

j 2 L2 +

r

2

1(

t

) +

r

2

2(

t

)

o

Proof :

Using the fact that for

t < t

, the term

H

0(

t

?

t

) in (3.3) is zero, we have that

EI

1=

EI

2 = 0 for that time interval and thus

r

1(0) =

c

EI

1(0)

?0 = 0,

r

2(0) =

c

EI

2(0)

?0 = 0.

When the time derivative of equation (4.10) is calculated, it yields

d

dtV

(

t

) = ?2

c

D

I

j

e

t(

t

)j 2

H₂₀+ 2j

e

t(

t

)j 2

L2 ?2

EI

j

e

(

t

)j 2 H₂₀

?

2

c

D

I

?2

K

2

j

e

t(

t

)j 2

H20?2

EI

j

e

(

t

)j 2

Using the fact that

V

(

t

)

0

n j

e

(

t

)j

2

H20+j

e

t(

t

)j 2 L2+

r

2

1(

t

) +

r

2 2(

t

)

o

, for some

0

>

0 and by

integrating the above equation from 0 to

t < t

we obtain

j

e

(

t

)j 2

H₂₀+j

e

t(

t

)j 2 L2+

r

2

1(

t

) +

r

2 2(

t

) +

Z t 0

n j

e

(

)j

2

H₂₀+j

e

(

)j 2 H₂₀

o

d

n j

e

(0)j

2

H₂₀+j

e

t(0)j 2 L2+

r

2

1(0) +

r

2 2(0)

o 0

;

for

; >

0, which yields the desired result. After the failure, i.e. for

t

t

, we integrate

equation (4.12) from

t

to some

t > t

to obtain

j

e

(

t

)j 2

H₂₀ +j

e

t(

t

)j 2 L2 +

r

2

1(

t

) +

r

2 2(

t

) +

Z t t

n j

e

(

)j

2

H₂₀+j

e

(

)j 2 H₂₀

o

d

n j

e

(

t

)j 2

H₂₀+j

e

t(

t

)j 2 L2+

r

2

1(

t

) +

r

2 2(

t

)o

;

where we used the fact that

V

(

t

)

1

n j

e

(

t

)j

2

H₂₀+j

e

t(

t

)j 2 L2+

r

2

1(

t

) +

r

2 2(

t

)

o

, for some

1

>

0.

This then concludes the proof of the theorem. 2

From the above theorem, only a boundedness condition can be established for

t

t

, i.e. we

have that the state error

e

(

t

) satises

e

2

L

2(

t

;t

;

H

2 0(0

;l

))

\

L

1(

t

;t

;

H

2 0(0

;l

))

;

with its derivative satisfying

e

t2

L

2(

t

;t

;

H

2 0(0

;l

))

\

L

1(

t

;t

;

L

2(0

;l

))

;

while the parameter errors satisfy

r

1

;r

2 2

L

1(

t

;t

;RI)

:

Using arguments similar to those for establishing Barbalats lemma, see [19, 20, 23] for nite dimensional systems, and [9] for innite dimensional systems, we can establish the convergence of the state error to zero; see [17] for similar results for the adaptive parameter identication of second order distributed parameter systems.

Theorem 4.3

Assume that the boundedness condition given by Assumption 3.1 is satised and that

satises condition (4.11). If the adaptation (4.3), (4.4) is used with the observer (4.1), then we have that for

t > t

lim

t!1

j

e

(

t

)jH20 = lim t!1

j

e

t(

t

)jL2 = 0

:

Proof :

The proof is identical to the case of parameter identication for second-order

dis-tributed parameter systems in [17] and it is therefore omitted. 2

Because of the structure of the failure assumed, the conditions required for parameter conver-gence, i.e. limt!1

r

1(

t

) = 0 and limt!1

r

2(

t

) = 0 are identical to those used for the parameter

Denition 4.4 ([17, 18])

A plant is said to be persistently excited if there exists

T

0

;

0

;

0

>

0,

and a sequence of positive real numbers f

t

kg 1

k=1 with limt

!1

t

k =

1, such that for each

p

=

(

p

1

;p

2) 2RI

2 with

j

p

jR2 = 1 and each positive integer

k

, there exists a

t

k 2[

t

k

;t

k+

T

0] such that

sup

jj H20 1 Z t k

+0 t

k h[

p

1

(

) +

p

2(1

?

(

))]

I

w

(

)

;

iH20

d

0

:

The above can be written explicitly as

0

sup

jj H20 1 Z t k

+0 t

k

Z l 0

[

p

1

(

;x

) +

p

2(1

?

(

;x

))]

[x1;x2](

x

)

w

xx(

;x

)

xx(

x

)

dx

!

d

= sup

jj H20 1 Z t k

+0 t

k

Z x

2

x1 [

p

1

(

;x

) +

p

2(1

?

(

;x

))]

w

xx(

;x

)

xx(

x

)

dx

d

We can now prove parameter convergence by imposing the persistence of excitation condition onto the plant information.

Theorem 4.5

If the plant is persistently excited then we have lim t!1

r

1(

t

)

r

2(

t

)

!

R2 = 0

Proof. The proof of this theorem is similar to the one given for the linear case in [17]. 2

5 Numerical Results

In this section we describe the implementation scheme and present some of our numerical nd-ings. Using the Galerkin scheme outlined in [15, 16, 17], we discretize the beam in terms of spline expansions (see [24]). Modied cubic splines on the interval (0

;l

) with respect to the uniform mesh f0

;

1l n

;

2l

n

;:::;l

g were used to approximate (4.1) - (4.4). We denote the 1-D cubic splines

byf

B

nig n?1

i=1 and the approximating subspace

H

n_{= span}f

B

nig n?1

i=1. For each

n

= 1

;

2

;:::;

let

P

n

denote the orthogonal projection of

L

2(0

;l

) onto

H

n _{and set}

v

_n=

P

n

v

_{. We also let}

P

_{n be the}

orthogonal projection of

H

2

0(0

;l

) onto

H

n _{with respect to the}

H

2

0(0

;l

) inner product, and set

w

n=

P

n

w

. As was noted in [17] we have

h

P

n

;

niH20 =h

;

niH20

;

n2

H

n

and by letting

w

n(

t

) =

P

n

w

(

t

) =n ?1 X j=1

W

nj(

t

)

B

nj(

x

)

;

where

W

n₍

t

₎2RIn

?1 is the coordinate vector for

w

n(

t

) with respect to the spline basisf

B

njgn ?1 j=1,

we have that

W

n₍

t

_{) = (}

K

n₎?1 Z l

0

w

xx(

t;x

) h

j

= 1

;

2

;:::;n

?1 where

K

n is the (

n

?1)(

n

?1) stiness matrix and is given by

K

n_{= [}

K

n_{]ij =}Z l

0

[

B

ni(

x

)]_xx h

B

nj(

x

)i xx

dx:

Now we let

V

n₍

t

₎2RIn

?1 be the vector representation of the state estimator

v

n(

t

),

v

n₍

t

_{) =}n?1

X j=1

V

nj(

t

)

B

nj(

x

)

:

Then the nite dimensional state estimator equation corresponding to (4.1) is given by

M

n

D

2

t

V

n(

t

) +

c

D

I

K

n

D

t(

V

n(

t

)?

W

n₍

t

_{)) +}

EI

K

n(

V

n(

t

) ?

W

n₍

t

_{)) +}

c

_D

IK

n

D

_t

V

n₍

t

₎

+

EI

0

K

n

V

n₍

t

_{) + (}

EI

c n 1(

t

)

n₊

EI

cn 2(

t

)(1

?

n))

K

n

W

n(

t

) =KB

F

n(

t

) (5.1)

where the (

n

?1)(

n

?1) mass matrix

M

n is given by

M

n_{= [}

M

n_{]ij =}Z l

0

B

ni(

x

)

B

nj(

x

)

dx;

and

F

n₍

t

_{) is given by}

F

n₍

t

_{) = [}

F

n_]i= "

Z l 0

[a1;a2](

x

)[

B

ni(

x

)]xx

dx

#

u

patch(

t

)

= Z a

2

a1 [

B

ni(

x

)]xx

dx

u

patch(

t

)

:

The parameter estimator equation corresponding to (4.3), (4.4) are given by _

c

EI

1n(

t

) =

1[

W

n₍

t

_)]T

K

n

n₍

t

₎

G

_n(

t

₎

;

_(5.2)

_

c

EI

2n(

t

) =

2[

W

n₍

t

_)]T

K

n₍₁?

n(

t

))

G

n(

t

)

;

(5.3)

where

G

n(

t

) is given by

G

n(

t

) = h

E

n(

t

) +

E

_n(

t

) i

with

E

n(

t

) =

W

n(

t

)?

V

n(

t

) and

1

;

2 are

positive constants acting as adaptive gains (see [17]).

For our numerical simulations we assumed that the nonlinear stiness term (

EI

(

w

xx) =

g

(

w

xx)) is given by

g

(

w

xx(

t;x

)) = (

0

w

xx(

t;x

) if

w

xx(

t;x

)

>

0 ?5

w

xx(

t;x

) otherwise

;

for 0

x

l

,

t >

0, the nominal damping parameter is

c

D

I

(

x

) = 0

:

005

Nm

2

sec

, 0

x

l

, the

nominal stiness parameter is

EI

0(

x

) = 15

Nm

2, and the linear mass density is

b= 1

:

35

kg=m

3.

The tuning parameters (see [18])

EI

and

c

D

I

are chosen to be

EI

(

x

) = 20

; c

D

I

(

x

) = 0

:

01

;

0

x

l:

The adaptive gains

1

;

2 in (5.2), (5.3) are

1 =

2= 1

10 5

;

the parameter

in (5.1) is

= 110

and the plant and estimator states are

w

(0

;x

) =

v

(0

;x

) = 210 ?3

x

2(

x

?

l

) 2

;

w

t(0

;x

) =

v

t(0

;x

) = 110

?2

sin2

(2

x=l

)cos(2

x=l

)

;

for 0

< x < l

. The beam length is

l

= 0

:

60

m

and the (centered) patch covers a half of the beam length, i.e.

x

1 = 0

:

15

m

and

x

2 = 0

:

45

m

. The piezoceramic constant is

KB = 0

:

002331655 and

the patch voltage is

u

patch(

t

) = 10[sin(150

t

) + sin(650

t

)+ sin(400

t

)+ sin(800

t

)]

:

We now summarize the implemented stiness for our numerical simulations. We simulated the plant (3.5) with

EI

(

t;w

xx(

t;x

)

;x

) = 15

w

xx(

t;x

) 0

t <

2,

EI

(

t;w

xx(

t;x

)

;x

) = 15

w

xx(

t;x

) +

(

0

w

xx(

t;x

)

[0:30;0:36](

x

) if

w

xx(

t;x

)

>

0 ?5

w

xx(

t;x

)

[0:30;0:36](

x

) otherwise

2

t

5.

The above integrals for the matrices and input vectors were computed numerically using a Gauss quadrature. Both the plant and the state estimator were approximated using a 16 cubic spline nite element method. They were also integrated using the ODE solver rkf45, a Fehlberg fourth-fth order Runge-Kutta method solver.

We run two sets of simulations, namely one where the state initial conditions are assumed known and another one where the plant initial conditions were unknown. The latter, is often encounterd in actual cases as it is seldom the case that initial conditions are known exactly. This in a way, tests the robustness of the adaptive estimator.

Case (i): Zero initial conditions of the state error.

In this part, we simulated the plant and its estimator with the same initial conditions, namely

w

(0

;x

) =

v

(0

;x

) = 210 ?3

x

2(

x

?

l

) 2

w

(0

;x

) =

v

(0

;x

) = 110

?2sin2(2

x=l

)cos(2

x=l

)

:

This stiness simulates a plant that initially (0

t <

2) has a linear stiness parameter that

becomes nonlinear for 2

t

5 and assumes dierent values depending on the sign of the

curvature (

(

t

) = 1 if

w

xx

>

0). In Figure 5.3 we plot the actual (dashed) values of the

parameters (

EI

1 =

EI

2 = 0, for

t <

2,

EI

1 = 0,

EI

2 =

?5, for

t

2), and their estimates

(solid). We observe that both parameters

EI

c 1 and

c

EI

2 are identied and that the time (

t

= 2)

that the nonlinearity occurs is sensed by the estimator. In addition, the evolution of the state error is depicted in Figure 5.4. There, we observe that both

e

xx(

t

) and

e

t(

t

) assume a large

value at

t

= 2 and then converge to zero around

t

= 3 seconds.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −3

−2 −1 0 1 2

Time (sec)

EI1(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −6

−4 −2 0 2 4

Time (sec)

EI2(t)

Figure 5.3: Evolution of the parameter estimates

EI

c

1(

t

) and c

EI

2(

t

).

Theorem 4.2 we have that for

t < t

we have

j

e

(

t

)j 2

H₂₀ +j

e

t(

t

)j 2 L2 +

r

2

1(

t

) +

r

2 2(

t

) +

Z t 0

n j

e

(

)j

2

H₂₀+j

e

(

)j 2 H₂₀

o

d

n j

e

(0)j

2

H₂₀+j

e

t(0)j 2 L2

o

6

= 0

This can be observed in Figures 5.5 and 5.6, where we plot the time evolution of the two parameters and the state errors. In Figure 5.5 we observe that initially, both parameters start at non zero value, converge to zero around

t

= 0

:

5 sec and for

t

2 they start converging to the

true values.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

1 2x 10

−5

Time (sec)

norm |e(t)|

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.5 1 1.5 2 2.5

x 10−5

Time (sec)

norm |De(t)|

Figure 5.4: Evolution of state errorsj

e

(

t;x

)jH20 andj

e

t(

t;x

)jL2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −3

−2 −1 0 1 2

Time (sec)

EI1(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −6

−4 −2 0 2 4

Time (sec)

EI2(t)

Figure 5.5: Evolution of the parameter estimates

EI

c

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

1 2x 10

−5

Time (sec)

norm |e(t)|

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.5 1 1.5 2 2.5

x 10−5

Time (sec)

norm |De(t)|

Figure 5.6: Evolution of state errorsj

e

(

t;x

)jH20 andj

e

t(

t;x

)jL2.

6 Comments and Future Extensions

A simple model for a nonlinear stiness in an Euler-Bernoulli beam with Kelvin-Voigt viscoelastic damping was utilized to test an online detection scheme. The failure was actually modeled as a nonlinear function of the beam's stiness occuring abruptly at some unknown failure time

t

. A

state observer in the form of an adaptive estimator was used to rst monitor any changes in the system's dynamics, and thus identifying implicitly the failure time

t

, and second, to identify the

nonlinear stiness parameters. The proposed state estimator assumed the same initial conditions as the plant (beam) and sensed the failure time

t

(time when stiness abruptly changed from

linear to nonlinear function). In addition it identied the nonlinear parameters. When the state estimator was simulated with dierenent initial conditions, it still detected the failure time

t

.

This in a way tested the robustness of the state estimator with respect to initial conditions. In this case, the state error already converged to zero much before the abrupt change in the system's dynamics occured. Of course, if the change occurs before this pre-failure state error evolution has settled down, we may not see convergence of the state error with nonzero initial conditions. Further numerical results are needed in this regard.

One might choose the tuning parameters

EI

and

c

D

I

to aect the convergence properties

of both j

e

(

t

)jH20 and

e

t(

t

)jL2, see [18]. If for certain values of the tuning parameters, both

errors (j

e

(

t

)jH20 and j

e

t(

t

)jL2) are not converged to zero prior to the failure time

t

, then the

estimator might not be able to sense the time of failure

t

. This robustness property is currently

investigated along with some more general cases of estimators which can not only identify the stiness parameters

EI

1 and

EI

2, but also the location that the nonlinearities act on the beam

nonlinear part of the stiness parameter might be given by

EI

(

t;w

xx(

t;x

)) =

[x1;x2](

x

) [

EI

1(

x

)

w

xx(

t;x

)sign(

w

xx) +

EI

2(

x

)

w

xx(

t;x

)(1

?sign(

w

xx))]

where the parameters

EI

1 and

EI

2 are not constants, but functions of the spatial variable

x

that vanish outside the interval [

x

1

;x

2]. A further goal is to propose an estimator that will also

identify the interval [

x

1

;x

2] and

EI

1(

x

),

EI

2(

x

).

The persistence of excitation condition, needed for parameter convergence, might be hard to check in systems governed by nonlinear hyperbolic p.d.e.'s. For the linear case this was presented in [18], but even in linear systems with spatially varying parameters this condition might be dicult to prove. Perhaps, by taking advantage of the nonlinear nature of the system, the persistence of excitation condition given by Denition 4.4 might be something that can be concluded by imposing simple conditions on the input patch voltage. This warrants additional theoretical studies.

References

[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2] H. T. Banks and J. Burns, Introduction to Control of Distributed Parameter Systems,

Birkhauser, Boston, 1995.

[3] H. T. Banks, J. M. Crowley, and I. G. Rosen, Methods for the identication of

material parameters in distributed models for exible structures, Mathematica Aplicada e Computacion, 5 (1986), pp. 139{168.

[4] H. T. Banks, W. Fang, R. J. Silcox, and R. C. Smith, Approximation methods for

control of structural acoustics models with piezoceramic actuators, Journal of Intelligent Material Systems and Structures, 4 (1993), pp. 98{116.

[5] H. T. Banks, S. S. Gates, I. G. Rosen, and Y. Wang, The identication of a

dis-tributed parameter model for a exible structure, SIAM J. Control and Optimization, 4 (1988), pp. 743{762.

[6] H. T. Banks, D. S. Gilliam, and V. I. Shupov, Well-posedness for a one dimensional

nonlinear beam, CRSC Report 94-18, Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC 27695-8205, October 1994.

[7] , Global solvability for damped abstract nonlinear hyperbolic systems, CRSC Report 95-25, Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC 27695-8205, August 1995.

[8] H. T. Banks, K. Ito, and Y. Wang, Well posedness for damped second order systems

with unbounded input operators, Dierential and Integral Equations, 8 (1995), pp. 587{606.

[9] H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems,

[10] H. T. Banks and I. G. Rosen, Computational methods for the identication of spa-tially varying stiness and damping in beams, Control-Theory and Advanced Technology, 3 (1987), pp. 1{31.

[11] H. T. Banks, R. J. Silcox, and R. C. Smith, The modeling and control of

acous-tic/structure interaction problems via piezoceramic actuators: 2-d numerical examples, ASME Journal of Vibration and Acoustics, 116 (1994), pp. 386{396.

[12] H. T. Banks, R. C. Smith, and Y. Wang, Modeling aspects for piezoceramic patch

activation of shells, plates and beams, Tech. Report 92-12, Center for Research in Scien-tic Computation, North Carolina State University, Raleigh, NC 27695, November, 1992; Quarterly of Applied Mathematics, vol LIII, no. 2, June 1995, pp. 353-381.

[13] H. T. Banks, Y. Wang, D. J. Inman, and J. C. Slater, Approximation and parameter

identication for damped second order systems with unbounded input operators, CRSC Re-port 93-9, Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC 27695-8205, May, 1993; Control: Theory and Advanced Techology, 10 (1994), pp. 873-892.

[14] R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1975.

[15] M. A. Demetriou and B. G. Fitzpatrick, Results on the adaptive estimation of stiness

in nonlinear beam models, in Proc. of the 3rd IEEE Mediterranean Symposium on New Directions in Control and Automation, Limassol, CYPRUS, July 11-13 1995.

[16] M. A. Demetriou and I. G. Rosen, Adaptive estimation of a exible beam via

piezoce-ramic actuation, in Proc. of the 1st IEEE Mediterranean Symposium on New Directions in Control Theory and Applications, Chania, Crete, GREECE, June 21-23 1993.

[17] , Adaptive identication of second-order distributed parameter systems, Inverse Prob-lems, 10 (1994), pp. 261{294.

[18] , On the persistence of excitation in the adaptive identication of distributed parameter systems, IEEE Transactions on Automatic Control, 39 (1994), pp. 1117{1123.

[19] P. A. Ioannou and J. Sun, Robust Adaptive Control, Prentice Hall, Englewood Clis,

NJ, 1995.

[20] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice Hall,

En-glewood Clis, NJ, 1989.

[21] M. M. Polycarpou and A. J. Helmicki, Automated fault detection and accomodation:

A learning systems approach, IEEE Trans. on Systems, Man and Cybernetics, 25 (1995), pp. 1447{1458.

[22] M. M. Polycarpou and A. T. Vemuri, Learning methodology for failure detection and

accomodation, Control Systems Magazine, special issue on Intelligent Learning Control, 15 (1995), pp. 16{24.

[24] M. H. Schultz, Spline Analysis, Prentice Hall, Englewood Cls, N.J., 1973.

[25] R. E. Showalter, Hilbert Space Methods for Partial Dierential Equations, Pitman,

Lon-don, 1977.

[26] J. Storch and S. Gates, Transverse vibration and buckling of a cantilevered beam with

tip body under axial acceleration, J. Sound and Vibration, 99 (1985), pp. 43{52.