Collocated Observer Design Based on Continuum Backstepping for a Class of Hyperbolic PDEs

(1)

Collocated Observer Design Based on Continuum

Backstepping for a Class of Hyperbolic PDEs

Xiaoguang Li

School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

Email: [email protected]

Jinkun Liu

School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

Email: [email protected]

Abstract—A collocated observer design approach is

addressed, which is based on the continuum backstepping algorithm for partial differential equations (PDEs) with boundary feedback. This observer for a class of hyperbolic PDEs is constructed through an integral form with undetermined kernel functions, combining with the “inverse” Volterra transformation and using the position and velocity of the collocated end as feedback through the boundary input with the Neumann type. The error system is transformed into the exponentially stable transformed system in the mapping space. The exponential stability of the source system is achieved with norm restrictions between the original and the transformed systems.

Index Terms—continuum backstepping approach for PDEs,

hyperbolic partial differential equations, boundary feedback, collocated observer, Volterra transformation.

I. INTRODUCTION

The continuum backstepping approach for PDEs was presented by Miroslav Krstic, Andrey Smyshlyaev in 2004 for the first time, rebooming the aspect of boundary control for distributed parameter systems, especially for 1-D parabolic partial differential equations (PDEs) [1]. This approach has been developed widely and has been applied on the plants of counter-convection equations [2], wave equations [3] and time-delay equations [4], because the approach has excellent features, such as the exponential stability and the explicit solutions of the closed-loop systems without model reduction.

So far, most of existing results have been focused on the mathematical models as parabolic PDEs, wave PDEs and first-order hyperbolic PDEs. As common second-order hyperbolic PDEs can be considered as the generalization of wave equations, the corresponding research should have more common sense. However, the

only study for common hyperbolic PDEs was reported in [5] based on state feedback. The further research is supposed to be needed.

Moreover, the continuum backstepping approach always relies on state feedback unless the exponentially stable observer is available. Andrey Smyshlyaev solved this problem by designing an infinite dimensional continuum backstepping observer for a class of 1-D parabolic PDEs based on boundary feedback [6], resulting in both of the observer and the closed-loop system exponentially stable. Then Miroslav Krstic, Bao-Zhu Guo, Andras Balogh, and Andrey Smyshlyaev stabilized an unstable wave equation based on output feedback [7], applying the approach on the special hyperbolic PDEs for the first time. Rafael Vazquez and Miroslav Krstic extended the observer for a 2-D fluid convection plant [8]. Even a nonlinear viscous burgers equation was discussed for the boundary control based on output feedback [9]. And the shock-like equilibrium was reached finally.

In this paper, based on the approach of continuum backstepping, an infinite dimensional observer is proposed for a class of common second-order hyperbolic PDEs. Using an iterative method for the inverse mapping， this observer is constructed with boundary feedback from the controlled end. Additionally this paper could be considered as extension of the result in [5].

We start in Section II with the plant of a class of hyperbolic PDEs and the transformed system. Then in Section III, we present the stability study of the transformed system. In Section IV, kernel functions are analyzed. Finally we prove the exponential stability of the original observer in Section V.

II. PROBLEMS STATEMENTS

In this paper, we consider the following class of common second-order hyperbolic PDEs with boundary conditions as the plant:

(2)

( )

( ) ( )

1 2

3

,

tt xx

t

u

x t

a u

x t

a

x u x t

a

x u x t

=

+

, (1)

( )

0,

0 u

t

=

, (2)

( )

1,

( )

x

u

t

=

f t

, (3)

where

x

∈

[ ]

0,1 ,

t

≥

0,

a

₁

>

0

. And

f t

( )

is the control at the end

x

=

1

. The displacement and velocity of the controlled end are measured as boundary outputs. In (1) – (3),

u x t u

_t

( ) ( )

,

_tt

x t

,

denotes

( )

,

u x t

t

∂

and

∂

2

u x t

( )

,

∂

t

2 respectively. And the rest of the paper will be taken this representation.

We propose the observer for the plant (1) – (3) as follows:

( )

( ) ( )

( ) ( ) ( )

( ) ( )

( )

1 2

3 1

2

ˆ

,

ˆ

,

ˆ

,

ˆ

,

1,

ˆ

1,

ˆ

1,

tt xx

t

t t

u

x t

a u

x t

a

x u x t

a

x u x t

p x

u

t

u

t

p

x

u

t

u

t

=

+

⎡

_⎣

−

⎤

_⎦

+

⎡

_⎣

−

⎤

_⎦

, (4)

( )

ˆ 0,

0 u

t

=

, (5)

( )

( ) ( )

( )

3

4

ˆ

1,

ˆ

1,

ˆ

1,

x

t t

u

t

f

p u

t

u

t

p

u

t

u

t

= −

⎡

_⎣

−

⎤

_⎦

−

⎡

_⎣

−

⎤

_⎦

, (6)

where

p p

₃

,

₄ are undermined constant coefficients, with

p x

₁

( ) ( )

,

p

₂

x

as undermined functions.

Take the error as

e x t

( )

,

u x t

( ) ( )

,

−

u x t

ˆ

,

and substitute it into the observer (4) – (6). Comparing with the plant (1) – (3), we obtain the error system PDEs:

( )

( ) ( )

1 2

3 1

2

,

1,

tt xx

t

e

x t

a e

x t

a

x e x t

a

x e x t

p x e

t

p

x e

t

=

+

−

, (7)

( )

0,

0 e

t

=

, (8)

( )

1,

3

( )

1,

4

( )

1,

x t

e

t

=

p e

t

+

p e

t

. (9) The convergence problem of the observer is converted into the problem of the stability of the error system (7) – (9).

We take the transformation in the Volterra-like form:

( )

( ) ( )

1

,

x

t x

e x t

h x z x t

k x y z y t dy

l x y z

y t dy

=

−

∫

,(10)

where

0 ≤ ≤ ≤

x

y

1

,

h x

( )

>

0

,

k x y l x y

( ) ( )

,

are undermined kernel functions, to transform (7) – (9) into the following PDEs:

( )

( ) ( )

1 1

2

,

tt xx

t

z

x t

a z

x t

b x z x t

b

x z x t

=

−

, (11)

( )

0,

0 z

t

=

(12)

( )

1,

0

x

z

t

=

(13)

where

min

_⎣

⎡

b x b

₁

( ) ( )

,

₂

x

_⎦

⎤

> >

ε

0

, with

ε

as any positive constant, and

b

₂

( )

x

+

a

₃

( )

x

≠

0

.

III. STABILITY OF TRANSFORMED SYSTEM

In this paper, we use the following domains and norms:

( )

{

2

}

1

x y

,

: 0

x

y

1 Ξ

∈

≤ ≤ ≤

,

( )

{

2

}

2

x t

,

: 0

x

1, 0

t

Ξ

∈

≤ ≤

≤

,

( )

( ) ( )

1

1 1

2 1 2

0 0

1 1

1 2

0 0

1 ,

,

2

2 ,

,

2

t x

t

a

z x t

z

x t dx

z

x t dx

b x

z

x t dx

δ

z x t z x t dx

Ψ

+

∫

,

( )

( ) ( )

2

1 1

2 1 2

0 0

1

1 2

0

1 ,

,

2

2 ,

2

t x

a

z x t

z

x t dx

z

x t dx

b x

z

x t dx

Ψ

+

∫

,

( )

3

1 1

2 2

0 0

1 2 0

1

1 ,

,

2

1 ,

2

t x

z x t

z

x t dx

z

x t dx

z

x t dx

Ψ

+

∫

,

( )

4

1 1

2 2

0 0

1

2 2

0

1

1 ,

,

2

1

1 ,

1,

2

t x

z x t

z

x t dx

z

x t dx

z

x t dx

z

t

Ψ

+

∫

(3)

where

z x t

( )

,

∈

C

2

( )

Ξ

₂ ,

δ

as a constant, with

a

₁,

( )

1

b x

mentioned in (11).

Lemma 1: Let a constant

δ

≤

min 1,

⎡

_⎣

b x

₁

( )

⎤

_⎦

, There exist constants

( )

(

0, min 1

,1

/

1

m

∈

⎡ −

_⎣

δ

−

δ

b x

⎤

_⎦⎦

⎤

, (14)

( )

)

1

max 1

,1

/

,

n

∈

⎡

_⎣

⎡ +

_⎣

δ

+

δ

b x

⎤ +∞

_⎦

, (15)

making

( )

2 1 2

0 ≤

m z x t

,

_Ψ

≤

z x t

,

_Ψ

≤

n z x t

,

_Ψ . Since Lemma 1 can be obtained through Young Inequality [10] directly, the proof is just omitted.

Theorem 1: Let a positive constant

δ

satisfy

0 <

δ

( )

2

( )

_{( )}

1

2

2 min 1,

,

2 b

x

b x

b

x

⎧

⎫

⎪

<

_⎨

_⎬

+

⎪

⎩

⎭

; let

m n

,

satisfy (15),

(16); and let ₁

( )

2

( )

2 b

x

b x

>

ε

hold, where

ε

is any positive constant. For

z x t

( )

,

in (11) – (13), there exists a positive constant

ϑ

=

min

{

_⎣

⎡

2 δ δ

−

b

₂

( ) ( )

x

b x

₁

⎤

_⎦

,

( )

}

2 2

2 , 2

δ

b

x

−

2 δ δ

−

b

x

yielding

( )

2 2

,

n

, 0

nt

z x t

z x

e

m

ϑ −

Ψ

≤

Ψ .

Proof: Differentiating

( )

1 2

,

z x t

_Ψ

z x t

_Ψ , and combining with (12), (13), we obtain

( )

1

1 ₂

1 0 1

2

2 2

0

1 ₂

1 2

0

,

/

,

2 ,

x

t

d z x t

dt

a z

x t dx

b

x

b

x

z

x t dx

b x

b

x

z

x t dx

δ

δ δ

δ

Ψ

≤ −

−

⎡

_⎣

− −

⎤

_⎦

−

⎡

_⎣

−

⎤

_⎦

∫

.

Note the definition of

ϑ

, then

( )

1

2 1

,

d z x t

z x t

dt

n

ϑ

Ψ

Ψ Ψ

≤ −

.

Consider Lemma 1, the following result is obtained:

( )

2 1 1

2

,

, 0

t n

m z x t

z x t

z x

e

n z x

e

ϑ

−

Ψ Ψ Ψ

−

Ψ

≤

where

m n

,

satisfy (15), (16). And the proof is completed.

IV. KERNEL FUNCTION ANALYSIS

In this section, we discuss the kernel functions and the undermined functions in (7), (9) and (10).

Differentiate the transformation (10) to

x t

,

, and compare (7) – (9) with (11) – (13). Because all

z x t

( )

,

can be verified in this comparison, we get

( )

_{( ) ( )}

( )

( ) ( )

2 3

2 1

1

2

,

2

1

2

2 ,

2

y

b

x

a

x

dk x x

l

x x

dx

a

x

b x

h x

a

b

x

l x x

+

⎡

⎤

⎣

⎦

=

+

⎡

⎤

⎣

⎦

−

+

&&

&

, (16)

( )

( ) ( )

( )

_{( ) ( )}

( ) ( )

2

3 2

2 1

1

3 1 1 2

1

1 2

,

2 ,

,

1 ,

xx yy y

yy

k

x y

k

x y

b

y l

x y

a

x

b

y

l

x y

a

x

b y

k x y

a

x b y

b y b

y

a

a b

y

l x y

−

= −

−

⎡

_⎣

+

⎤

_⎦

+

⎡

⎤

⎣

⎦

−

+

_⎡⎣

+

⎤

−

_⎦

&

&&

, (17)

( )

( ) ( )

( )

( ) ( )

( )

_{( ) ( )}

3 2

1 2

2 1 2

2 3

1

1 ,

,

xx yy

l

x y

l

x y

a

x b

y

a

x

b y

b

y

l x y

b

y

a

x

k x y

a

−

=

_⎡⎣

⎤

−

_⎦

+

⎡

⎤

⎣

⎦

−

, (18)

( )

2

( )

3

( ) ( )

,

2 b

x

a

x

h x

&

= −

⎡

⎣

+

⎤

⎦

l x x

, (19)

( )

2

( )

3

_{( ) ( )}

1

,

2 b

x

a

x

dl x x

h x

dx

a

+

⎡

⎤

⎣

⎦

= −

, (20)

( )

_{( )}

( )

( ) ( )

1 1 1 2

1 2 1 3

1 ,1

1

1 ,1

,1

y

y y

p x

a k

x

a b

l x

h

a b

l

x

a a

x l

x

⎡

= −

_⎣

−

⎤

−

_⎦

&

(4)

( )

_{( ) ( )}

1

2

,1

1

y

a

p

x

l

x

h

= −

, (22)

( ) ( )

( )

3

1 1,1

1 h

k

p

h

⎡

+

⎤

⎣

⎦

=

&

, (23)

( )

4

1,1

1 l

p

h

=

, (24)

( ) ( )

0,

0 k

y

=

l

y

=

, (25)

( )

1,

( ) ( )

1 1,

e

t

=

h

z

t

, (26)

( )

1,

( ) ( )

1 1,

t t

e

t

=

h

z

t

. (27)

Consequently,

p x

₁

( ) ( )

,

p

₂

x

,

p p

₃

,

₄ can be expressed in (21) – (24) through

h x k x y l x y

( ) ( ) ( )

,

, so that the key problem of the observer is converted into the problem of solutions of the kernel functions.

With the observation of (19), (20), we get

( )

{

( )

}

1

0 ,

h

sinh

, 0

l x x

x

a

ρ

= −

, (28)

( )

0 cosh

{

( )

, 0

}

h x

=

h

ρ

x

, (29)

where

( )

₂

( )

₃

( )

1

1 ,

2

x

y

x y

b

a

d

a

ρ

_∫

⎡

_⎣

τ

+

τ

⎤

_⎦

τ

and

( )

0 h

as an arbitrary positive constant.

Equation (25) results in

l

_y

( )

0,

y

=

0

. Note that

( )

_{( )}

,

2 ,

x y

xx yy

y

d l

x x

l

x x

l

x x

l

x x

dx

dl x x

d

l

x x

dx

⎡

−

⎤

⎣

⎦

−

=

⎡

⎤

−

⎢

⎥

⎣

⎦

=

And taking

x

=

y

into (18), with (16) considerd, we observe that

( )

{

(

)

}

( )

{

(

)

}

2 0

1 ₀ 1

,

cosh

,

sinh

,

x

k x x

Q

x

d

a

Q

x

d

σ

ρ

σ

ρ

σ

=

+

∫

, (30)

where

( )

( ) ( )

( )

_{( )}

2

1 2

2

3 2 2 1 2

1

,

1

2 ,

2 d l x x

Q x

dx

a

x b

x

a

x

b x

b

x

l x x

a

⎡

−

⎤

⎣

⎦

−

,

( )

( ) ( )

( )

_{( ) ( )}

2 2

2 1

1

1 ,

2

2 b

x

Q

x

h x

l x x

a

x

b x

h x

a

−

+

⎡

⎤

⎣

⎦

−

&

&&

.

With the Theorem 3.1,

k x y

( )

,

and

l x y

( )

,

for (17), (18), (25), (28), (29) and (30) can be solved uniquely by an iterative method exactly. So the functions in the transformation are all figured out.

V. STABILITY OF ERROR SYSTEM

In this section, the bidirectional restrictions between the transformed system and the error system are obtained. Consequently, the stability of the error system is proved.

A. Restriction of Inverse Mapping

Note that (12), with Agmon Inequality and Young Inequality [10], then

( )

2 2

1 ₂ 1 ₂

0 0

1,

0,

,

1

1 ,

,

2

x

L L

x

z

t

z

t

z x t

z

x t

z

x t dx

z

x t dx

≤

+

≤

_∫

+

_∫

With the observation of (26), we get

e

( ) ( )

1,

t

,

z

1,

t

is limited.

Consider the partial derivatives of (10), we rewrite the inverse mapping in the vector form:

( )

1

= Λ

r

E

Z

,

Where

1

:

Λ Ω → Ω

,

( ) ( ) ( ) ( )

,

_x

,

_t

,

1,

T

e x t

e

x t

e x t

e

t

⎡

⎤

⎣

⎦

r

E

,

( ) ( ) ( ) ( )

,

_x

,

_t

,

1,

T

z x t

z

x t

z x t

z

t

⎡

⎤

⎣

⎦

r

Z

,

with the denotation of

[ ] [

)

{

}

{

[ ] [

)

}

[

)

1 2

2

0,1

0,

0,1

0,

L

H

L

Ω

×

+∞ ×

×

+∞

(5)

Then

( )

4 1 4

,

e x t

_Ψ

≤

D z x t

_Ψ , with the denotation of

3

1 1

D

Λ

_Ψ . So we obtain:

( )

3 3

3

2

1 2

2 1

,

1,

,

2 e x t

D z x t

D z

t

D

z x t

Ψ Ψ

Ψ

≤

+

⎛

⎞

≤

_⎜

+

_⎟

⎝

⎠

, (31)

where

D

₂

≥

max

_⎣

⎡

D

₁

−

h

2

( )

1 , 0

⎤

_⎦

.

B. Restriction of Forward Mapping

Let

v

= −

1 x

,

ϕ

= −

1 y

. We rewrite the mapping using

Z v t

( ) ( )

,

z x t

,

instead. Note the partial derivatives of (10), and then

( )

( ) ( )

1 ₀ 1

2 0

,

v

t

Z v t

v t

v

Z

t d

v

Z

t d

ϖ

ι

ϕ

ι

ϕ

=

+

∫

, (32)

( )

( ) ( )

2 ₀ 3

4 0

,

v

t

Z

v t

v

Z

t d

v

Z

t d

ϖ

ι

ϕ

ι

ϕ

=

+

∫

,(33)

( )

( ) ( )

3 ₀ 5

6 0

,

v

t

v

t

Z v t

v t

v

Z

t d

v

Z

t d

ϖ

ι

ϕ

ι

ϕ

=

+

∫

,(34)

with denotations as

( )

(

₍

₎

)

1

1 ,

,

1 e

v t

h

v

ϖ

−

,

( ) (

) (

_{( )}

)

(

_{( )}

)

(

) (

)

(

) ( )

(

) (

_{( )}

)

(

) (

_{( )}

)

(

) (

)

( )

2 2 2

1

2

2 2

1

3

1 ,1

1 ,

1

1 ,1

1

1 ,1

1 ,

1

1 ,1

1,

1

y

v t

y

k

v

h

v

v t

e

v t

h

a l

v

v l

v

h

v h

h

v

l

v

e

v t

e

v t

h

a l

v

l

v

e

t

h

ϖ

−

⎡

_⎢

−

⎢⎣

⎤

−

+

_⎥

−

_⎦

−

+

−

+

−

&

,

( ) (

)

(

_{( )}

)

(

) (

)

(

) ( )

(

)

( ) (

)

(

( )

)

(

)

(

) (

)

( )

1

3 2

1

2

1

2 2

1

3

1 ,1

,

1 ,

1

1 ,1

1

1 ,1

1 ,

1

1 ,1

1 1,

1

y

t v

y

a l

v

v t

e

v t

h

a l

v

k

v

h

v

h

v h

h

v

a l

v

e

v t

e

v t

h

a l

v

h

v

e

t

h

ϖ

−

⎧

⎪

_⎨

−

⎪⎩

⎫

⎡

⎤

−

_⎣

−

− −

−

_⎦

_⎪

+

_⎬

−

_⎪⎭

−

+

−

&

,

( )

(

₍

₎

)

1

1 ,1

,

1 k

v

h

v

ϕ

ι

ϕ

−

,

( )

(

₍

₎

)

2

1 ,1

,

1 l

v

h

v

ϕ

ι

ϕ

−

,

( )

(

) (

_{( )}

)

(

) (

)

(

) ( )

(

) (

)

( )

(

) (

)

( )

(

) (

)

( )

1

3 2

1

2

1

2

1 ,1

,

1

1 ,1

1

1 ,1

1

1 ,1

1

1 ,1

1

v

a l

v

v l

v

h

a l

v

v l

v

v k

v

h

v h

b

l

v

v l

v

h

k

v

k

v

h

v

h

k

v

h

v

h

ϕϕ

ϕ

ι

ϕ

−

⎡

⎤

−

_⎣

−

− −

−

_⎦

+

−

+

&

,

( )

(

) (

_{( )}

)

(

) (

)

( )

(

) (

)

(

) ( )

(

) (

)

( )

(

) (

)

( )

4 2

2

1

2

1 ,1

,

1

1 ,1

1

1 ,1

1

1 ,1

1

1 ,1

1

1 ,1

1

v

k

v

l

v

h

b

l

v

v l

v

h

a l

v

v l

v

v l

v

h

v h

k

v

h

v

l

v

h

l

v

h

v

h

ϕ

ι

ϕ

−

⎡

−

− −

−

⎤

−

⎣

⎦

+

−

+

&

(6)

( )

(

) (

_{( )}

) (

)

(

) (

)

(

) ( )

(

)

(

)

(

) (

_{( )}

)

(

) (

)

( )

(

) (

)

( )

1 5 2 1 2 1 2 1 2 1 2

1

1 ,1

,

1

1 ,1

1

1 ,1

1

1 ,1

1

1 ,1

1

v

b

h

v l

v

h

a l

v

v k

v

k

v

h

v h

a h

v l

v

h

v

h

a l

v

v k

v

h

a l

v

v k

v

h

ϕϕ ϕ

ϕ

ι

ϕ

−

_⎡⎣

−

⎤

−

_⎦

−

+

−

&

,

( )

(

) (

_{( )}

) (

)

(

) (

)

(

) ( )

(

) (

)

( )

(

) (

)

( )

(

) (

)

( )

2 6 2 2 1 2 1 2 1 2 2

1

1 ,1

,

1

1 ,1

1

1 ,1

1

1 ,1

1

1 ,1

1

v

b

h

v l

v

h

a l

v

k

v

h

v

h

v h

a l

v

v l

v

h

a l

v

v l

v

h

k

v

h

v

h

ϕ

ι

ϕ

−

⎡

⎤

−

_⎣

−

− −

−

_⎦

−

+

−

&

.

We take the iterative method for (32) – (34). Denote

( )

{

}

max

_i

,

1, 2, 3

L

M

ϖ

v t

i

∞

=

,

( )

{

2 1 2

}

max

,

1, 2, 3

j _L j _L

N

v y

j

ι

₋

ι

∞ ∞

⎡

₊

⎤

⎣

⎦

=

. And let

( )

0 0 1 2 0 3

,

v t

Z v t

v t

Z

v t

Z

v t

ϖ

=

. Supposing that for

n

≥

1

, we have

( )

,

( )

,

( )

,

!

n n

n n n

v t

MN v

Z v t

Z

v t

Z

v t

n

≤

Then

( )

( ) ( )

1 1 0 2 0

,

n v v t

Z v t

v

Z

t d

v

Z

t d

ι

ϕ

ι

ϕ

+

=

+

∫

,

( )

( ) ( )

1 3 0 4 0

,

n v v v t

Z

v t

v

Z

t d

v

Z

t d

ι

ϕ

ι

ϕ

+

=

+

∫

,

( )

( ) ( )

1 5 0 6 0

,

n v t v t

Z

v t

v

Z

t d

v

Z

t d

ι

ϕ

ι

ϕ

+

=

+

∫

, And then

( )

₍

₎

(

)

1 1 2 0 1 1 2 1 1

,

!

,

1 !

n n n v L L n n L L n n

Z v t

MN v

v y

dy

n

MN v

v y

n

MN

v

n

ι

+ ∞ ∞ + ∞ ∞ + +

⎡

⎤

≤

_⎣

+

_⎦

⎡

⎤

=

_⎣

+

_⎦

+

≤

+

∫

,

There exist the following results in the same way:

( )

₍

1

₎

1

,

1 !

n n n v

MN

v

Z

v t

n

+ + +

≤

+

,

( )

(

)

1 1 1

,

1 !

n n n t

MN

v

Z

v t

n

+ + +

≤

+

.

Finally, we have the iterative form expressions as follows:

( )

( ) ( )

( )

1 1 1

,

, ,

,

n n v v n n n t t n

Z v t

Z v t Z

v t

Z

v t

Z v t

Z

v t

+∞ +∞ = = +∞ =

=

∑

.

Comparing with (26), we have Lemma 2 in the similar way.

Lemma 2: let

l x y

( )

,

∈

C

2

( )

Ξ

and

k x y

( )

,

( )

0

C

∈

Ξ

. There exists a map

Λ Ω → Ω

₂

:

making

( )

2

= Λ

r

Z

E

. And

( )

4 3 4

,

e x t

_Ψ

≤

D z x t

_Ψ

holds, where

3

3 2

D

Λ

_Ψ .

Consequently, we obtain

( )

3 3 4 3

,

2 D

z x t

_Ψ

≤

⎛

_⎜

D

+

⎞

_⎟

e x t

_Ψ

(7)

C. Restriction between Norms of

Ψ

₂and

Ψ

₃

As

b x

₁

( )

∈

C

0

[ ]

0,1

and

b x

₁

( )

>

ε

, we denote

( )

1

inf

⎡

_⎣

b x

⎤

_⎦

=

α

and

sup

⎡

_⎣

b x

₁

( )

⎤

_⎦

=

β

, such that

( )

3 2 3

1

z x t

,

z x t

,

2

z x t

,

λ

_Ψ

≤

_Ψ

≤

λ

_Ψ ,(36) where

λ

₁

min 1,

[

a

₁

,

α λ

]

,

₂

max 1,

[

a

₁

,

β

]

.

D. Exponential Stability of Error System

With the observation of (31), (35), (36) and Theorem 1, we obtain

( )

3 3

2

2 1

4

1 3

2 ,

, 0

2

t n

D

n

D

e x t

e x

e

D

m

D

ϑ

λ

−

Ψ Ψ

⎛

₊

⎞

⎜

⎟

⎝

⎠

≤

⎛

₊

⎞

⎜

⎟

⎝

⎠

The exponential stability of the whole error system gets proved finally. So this result about the observer is stated in the following Theorem 2.

Theorem 2: Let

b x a x a

₁

( ) ( ) ( )

,

₁

,

₂

x

∈

C

0

[ ]

0,1

;

( )

2

[ ]

2

0,1

b

x

∈

C

;

b x

₁

( )

>

b

₂

( )

x

2 >

ε

;

( )

2 3

b

x

≠

a

x

. The collocated observer (4) – (6),

corresponding to (21) – (24), exponentially convergents to the hyperbolic PDEs (1) – (3), in the norm of

Ψ

₃.

VI. CONCLUSION

We propose an infinite dimensional observer for common second-order hyperbolic PDEs based on the approach of continuum backstepping. The output is located at the controlled end. And its exponential stability is obtained. This work can be considered as the supplements of the result of [5].

There exists two possible directions for further research. One is to prove the exponential stability of the boundary control based on output feedback, instead of the stability of the single controller or the single observer. The other is to find the link between this kind of observer and the differentiator theory, as the derivative of time is used for measurement.

REFERENCES

[1] A. Smyshlyaev and M. Krstic, “Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations,” IEEE Trans. Auto. Control, vol. 49(12), pp. 2185 – 2202, 2004

[2] R. Vazquez and M. Krstic, “Explicit integral operator feedback for local stabilization of nonlinear thermal convection loop PDEs,” Systems & Control Letters, vol. 55, pp. 624 – 632, 2006

[3] A. Smyshlyaev and M. Krstic, “Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary,” Systems & Control Letters, vol. 58, pp. 617 – 623, 2009

[4] M. Krstic and A. Smyshlyaev, “Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays,” Systems & Control Letters, vol. 57, pp. 750 – 758, 2008.

[5] A. Smyshlyaev, E. Cerpa and M. Krstic, “Boundary stabilization of a 1-D wave equation with in-domain antidamping,” SIAM Journal Control Optim., vol. 48, pp. 4014–4031, 2011.

[6] A. Smyshlyaev and M. Krstic, “Backstepping observers for a class of parabolic PDEs,” Systems & Control Letters, vol. 54, pp. 613 – 625, 2005.

[7] M. Krstic, B. Guo, A. Balogh and A. Smyshlyaev, “Output-feedback stabilization of an unstable wave equation,” Automatica, vol.44, pp. 63 – 74, 2008.

[8] R. Vazquez and M. Krstic, “Boundary observer for output-feedback Stabilization of thermal-tluid convection loop,”

IEEE Trans. on Control Systems Tech., vol. 18, pp. 789 – 797, 2010.

[9] M. Krstic, L. Magnis and R. Vazquez, “Nonlinear Control of the Viscous Burgers Equation: Trajectory Generation, Tracking, and Observer Design,” Journal of Dynamic Systems, Measurement, and Control, vol. 131, pp. 021012 1 – 8, 2009.

[10]M. Kristic and A. Smyshlyaev, Boundary control of PDEs. Philadelphia: SIAM, 2008.

Xiaoguang Li received the B.S. degrees in the major of automation in Northeastern University, Shenyang, China. He is currently working toward the Ph.D. degree in School of Automation Science and Electrical Engineering, Beihang University, Beijing, China.

His research interests focus on boundary control of distributed parameter systems, escpecially on the hyperbolic partial differential equations.

Jinkun Liu received the Ph.D degree in control theory and control engineering from Northeastern University, Shenyang, China.