OBSERVER-BASED STATE FEEDBACK REMOTE CONTROL WITH

BOUNDED TIME-VARYING DELAYS

Imane Dilaneh and Laurent Laval

ECS–ENSEA

6, avenue du Ponceau, 95014 Cergy-Pontoise Cedex, France

Keywords:

Networked Control Systems, Time-varying delays, Lyapunov–Krasovskii functional, LMI.

Abstract:

This paper investigates the problem of remote stabilization via communication networks with uncertain, “non-

small”, time-varying, non-symmetric transmission delays affecting both the control input and the measured

output. More precisely, this paper focuses on a closed-loop Master-Slave setup with a TCP network as commu-

nication media, and an observer-based state-feedback control approach to deal with the stabilization objective.

First, we establish some asymptotic stability criteria regarding to a Lyapunov–Krasovskii functional derived

from a descriptor model transformation, in case of “non-small” delays (that are time-varying delays with non-

zero lower bounds). Then, some stability conditions are given in terms of Linear Matrix Inequalities which

are used, afterwards, to design the observer and controller gains. Finally, the proposed stabilizing approach is

illustrated through numerical and simulation results, related to the remote control of a “ball and beam” system.

1 INTRODUCTION

Over the past few years, the widespread development

of low-cost wired and wireless data networks has lead

to an increasing interest for Networked Control Sys-

tems (NCSs) (for instance, see (Yang, 2006; Tang

and Yu, 2007; Hespanha et al., 2007) and references

therein). Indeed, such networks seem to be suitable

for large scale control systems with sensors, actua-

tors and controllers that communicate over a shared

medium. However, most of common network physi-

cal conﬁgurations and communication protocols

1

lead

to transmission delays and even data losses. Then,

from a control viewpoint, it is well-known that such

undesirable features affect the overall NCS behav-

ior, leading possibly to poor performance and/or in-

stabilities (e.g. (Niculescu, 2001; Ge et al., 2007)).

This justiﬁes the increasing investigations on con-

trol strategies to insure both closed-loop stability

and good performance for time-delayed systems (see

(Tipsuwan and Chow, 2003; Richard, 2004) and ref-

erences therein). Following this, the present paper

then deals with the stabilization of a Networked Con-

trol System with consideration of TCP (Transmis-

sion Control Protocol) networking protocol for bi-

1

Such as User Datagram Protocol (UDP), Transfer Con-

trol Protocol (TCP), Medium Access Control protocols, etc.

directional communications between a Master sys-

tem (computing the control) and a Slave system (to

be controlled). In particular, we investigate the de-

sign of an observer-based (static) state-feedback con-

troller (located in the Master system) so as to insure

the asymptotic stability of the closed-loop NCS what-

ever the presence of time-varying, non-symmetric de-

lays in the control and feedback loops. In this pur-

pose, ﬁrst, we establish some stability conditions by

means of a Lyapunov–Krasovskii functional derived

from a descriptor model transformation (Fridman and

Shaked, 2002). These conditions are given in terms

of Linear Matrix Inequalities which are used after-

wards to design both controller and observer gains,

by means of LMI optimization. This design approach

is then illustrated through an example related to the

remote control of a “ball and beam” system.

This paper is organized as follows. Section 2 de-

scribes the Networked Control System under consid-

eration. Section 3 deﬁnes the observer-based control

law, while section 4 focuses on the design of both

state-feedback controller and observer gains. Section

5 presents a “ball and beam” system as remote con-

trolled plant for illustrating the proposed control strat-

egy. Then, some numerical and simulations results

related to the observer-based control of this system

are presented. Finally, some concluding remarks are

given in section 6.

136

Dilaneh I. and Laval L.

OBSERVER-BASED STATE FEEDBACK REMOTE CONTROL WITH BOUNDED TIME-VARYING DELAYS.

DOI: 10.5220/0002170201360142

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page

ISBN: 978-989-8111-99-9

Copyright

c

2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

2 SYSTEM DESCRIPTION

Regarding to Figure 1, the Networked Control System

under consideration consists in a Master-Slave setup,

with a TCP network as communication media linking

these two systems.

Figure 1: The Networked Control System (Master-Slave

conﬁguration).

• The exchanged data correspond respectively to

the control input (sent by the Master to the Slave),

and a measured output of the remote system (sent

by the Slave to the Master). Due to the net-

working protocol and communication lines prop-

erties, we consider some time-delays τ

1

and τ

2

,

respectively related to the Master-to-Slave and

Slave-to-Master transmissions. Moreover these

delays are assumed to be time-varying, uncertain

(with known lower and upper bounds), and non-

symmetric (that is τ

1

6= τ

2

).

Remark 1. The consideration of TCP networking

protocol insure that all transmitted data are received

in the emission order. Thus, when considering a ﬁrst

data packet emitted at time t

1

undergoing a delay τ

1

,

and a second data packet emitted at time t

2

undergo-

ing a delay τ

2

, the correct scheduling of data implies

that (see (Witrant et al., 2003)):

t

1

+ τ

1

< t

2

+ τ

2

⇔ −1 <

τ

2

− τ

1

t

2

− t

1

≃

dτ

dt

(1)

Therefore, the Master-to-Slave and Slave-to-

Master delays τ

i

(t) (with i = 1,2) can be expressed

as differentiable functions, and such that:

∀t ≥ 0, τ

i

(t) = h

i

+ η

i

(t),

with 0 ≤ η

i

(t) ≤ µ

i

,

˙

η

i

(t) ≤ d

i

< 1 (2)

where the τ

i

(t) (with i = 1,2) are considered as time-

varying bounded delays with non-zero lower bounds

h

i

> 0 (sometimes referred to as ”non-small delays”).

η

i

(t) is a differentiable function which characterizes

a (bounded) time-varying perturbation with bounded

time-derivative

˙

η

i

(t) < 1 (so that τ

i

(t) are commonly

referred to as slowly-varying delays – e.g. (Shustin

and Fridman, 2007)), and µ

i

and d

i

are strictly posi-

tive, constant upper-bounds (see (Fridman, 2004)).

Moreover, we can deﬁne τ

∗

i

= h

i

+ µ

i

as an upper-

bound for τ

i

(t), leading ﬁnally to h

i

≤ τ

i

(t) ≤ τ

∗

i

.

Remark 2. Such an assumption on non-zero lower

bounds h

i

of delays is realistic. Indeed, zero or

close to zero delays (corresponding to instantaneous

or quasi-instantaneous transmissions) are usually not

met in most of real networks (due to, at least, propa-

gation phenomena).

• The controlled system (within the Slave part),

is supposed to be linear, controllable and ob-

servable, with a known state-space representation

(A;B;C). By taking into account the time-delay

τ

1

(intrinsic to the Master-to-Slave transmission),

this Slave system is then given by:

˙x(t) = Ax(t) + Bu(t − τ

1

(t))

y(t) = Cx(t) (3)

where x(t) ∈ R

n

is the state vector, u(t) ∈ R

m

is

the delayed control input with an input time-delay

τ

1

(t) > 0 that we assume to be a differentiable

function satisfying to relation (2). y(t) ∈ R

p

is

the system output, and A, B and C are constants

matrices of appropriate dimensions.

• The Master system includes an observer which

aims at providing an estimation ˆx(t) of the full

state-vector x(t) of the Slave system, from the

output y(t) it receives after a delay τ

2

(t) (assum-

ing this delay also satisﬁes to relation (2)). From

this estimation ˆx(t), the Master then computes the

control and forwards it to the Slave.

3 THE OBSERVER-BASED

STATE-FEEDBACK CONTROL

3.1 The Full-state Observer

As already mentioned, this paper considers, from the

Master system viewpoint, a full-state reconstruction

of the Slave state-vector x(t) from the transmitted, de-

layed, scalar output y(t − τ

2

) coming from the Slave

system. As this last system is assumed to be linear,

we propose here to perform this full-state estimation,

by means of a Luenberger-typeobserver (Luenberger,

1971).

With respect to the NCS setup, the observer can

OBSERVER-BASED STATE FEEDBACK REMOTE CONTROL WITH BOUNDED TIME-VARYING DELAYS

137

then be deﬁned by:

·

ˆx(t) = Aˆx(t) + Bu(t − τ

1

(t))

−L[y(t − τ

2

(t)) − ˆy(t − τ

2

(t))]

ˆy(t) = Cˆx(t) (4)

where L is the observer gain which has to be designed

so as to ensure a sufﬁciently fast convergence of ˆx(t)

towards the true system state x(t), regardless of time-

varying delay τ

2

(t).

Remark 3. Delay τ

2

(t) is supposed to be time-

varying and uncertain. Nevertheless, we assume the

knowledge of an upper-bound τ

∗

2

= h

2

+ µ

2

≥ τ

2

(t).

3.2 The Control Law

Regarding to the literature, many control strategies

have been proposed to deal with the stabilization

problem of NCS with delays. In our case, as the

Luenberger-type observer is supposed to provide a

full-state reconstruction ˆx(t) of the Slave state-vector

x(t), we propose to investigate the use of a simple

state-feedback control u(t) of the following form:

u(t) = K ˆx(t) (5)

where K is the control gain to design so as to guar-

antee the closed-loop stability of the controlled sys-

tem (the Slave), regardless of the control input time-

varying delay τ

1

(t).

4 DESIGN OF CONTROLLER

AND OBSERVER GAINS

With respect to (4) and (5), this section is devoted

to the design of both controller and observer gains

that guaranty the closed-loop stabilization of the NCS

(despite of both input and output time-varying delays

τ

1

(t) and τ

2

(t)). In this aim, let us establish some

asymptotic stability criteria, by applying a Lyapunov-

Krasovskii methodology based on a descriptor model

transformation (see (Fridman and Shaked, 2002)).

4.1 Control Design

First, let us focus on the design of an ideal controller

u(t) = Kx(t) by considering a perfect observer (such

that ˆx(t) = x(t)), before to deal, in a later subsec-

tion, with the inﬂuence of the observation error on the

whole system stability.

Thus, ﬁrst, let us recall that the controlled system

is represented by a linear system with bounded, time-

varying, input delay (see Remark 1), whose dynamics

can be expressed as:

˙x(t) = Ax(t) + BKx(t − τ

1

(t))

x(θ) = ϕ(θ), θ ∈ [−τ

∗

1

,0] (6)

where τ

∗

1

= h

1

+ µ

1

is an upper-bound for time-delay

τ

1

(t). Then, following a similar approach as in

(Fridman, 2004), let us express a result that gives

some asymptotic stability conditions for system (6),

in terms of Linear Matrix Inequalities, for a given K.

Theorem 1. Given a gain matrix K, system (6) is

asymptotically stable if there exists n×n matrices 0 <

P

1

, P

2

, P

3

, S

1

, S

a

1

,Y

1i

,Y

a

1i

,Z

1k

,Z

a

1k

, and R

1

,R

a

1

satis-

fying the LMI conditions for i = 1,2 and k = 1,2,3:

Γ =

Ψ

1

P

T

0

BK

−Y

T

a

1

Y

T

a

1

−Y

T

1

∗ −(1− d

1

)S

a

1

0

∗ ∗ −S

1

< 0

(7)

and,

R

1

Y

1

∗ Z

1

≥ 0,

R

a

1

Y

a

1

∗ Z

a

1

≥ 0 (8)

with,

Y

1

= [Y

11

Y

12

] Y

a

1

= [Y

a

11

Y

a

12

]

Z

1

=

Z

11

Z

12

∗ Z

13

Z

a

1

=

Z

a

11

Z

a

12

∗ Z

a

13

where ∗ denotes the symmetric, and Ψ

1

is given by:

Ψ

11

= P

T

2

A+ A

T

P

2

+ S

1

+ h

1

Z

11

+Y

11

+Y

T

11

+ S

a

1

+µ

1

Z

a

11

Ψ

12

= A

T

P

3

+ P

T

1

− P

T

2

+ h

1

Z

12

+Y

12

+ µ

1

Z

a

12

Ψ

13

= −(P

3

+ P

T

3

) + h

1

(Z

13

+ R

1

) + µ

1

R

a

1

+ µ

1

Z

a

13

Proof — Representing (6) in an equivalent descriptor

form ((Fridman and Shaked, 2002)) leads to:

˙x(t) = z(t) (9)

0 = −z(t) + Ax(t) + BKx(t − τ

1

(t))

By posing ¯x(t) = col{x(t), z(t)} and E =

diag{I

n

,0}, then (9) can be rewritten as:

E

·

¯x =

"

˙x(t)

0

#

=

"

z(t)

−z(t) + Λx(t)

#

−

"

0

BK

#

Z

t

t−h

1

z(s)ds

−

"

0

BK

#

Z

t−h

1

t−h

1

−η

1

z(s)ds (10)

where Λ = A+ BK.

Now, considering a Lyapunov-Krasovskii func-

tional (LKF) of the form:

V(t) = V

n

(t) + V

a

(t); (11)

where V

n

(t) is a nominal LKF corresponding to the

nominal system (10) with h

1

6= 0 and η

1

(t) = 0, and

such that (see (Fridman, 2004)):

V

n

(t) = ¯x(t)

T

EP¯x(t) +

Z

0

−h

1

Z

t

t+θ

z(β)

T

R

1

z(β)dβdθ

+

Z

t

t−h

1

x(s)

T

S

1

x(s)ds (12)

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

138

and V

a

(t) is an additional term (which corresponds to

the perturbed system), which vanishes when the delay

perturbation approaches to 0 (that is when η

1

(t) ≃ 0)

and such that:

V

a

(t) =

Z

0

−µ

1

Z

t

t+θ−h

1

z(s)

T

R

a

1

z(s)dsdθ

+

Z

t

t−τ

1

(t)

x(s)

T

S

a

1

x(s)ds (13)

with P =

P

1

0

P

2

P

3

, P

1

, R

1

, R

a

1

, S

1

and S

a

1

> 0.

Noting that V

1

= ¯x(t)

T

EP¯x(t) = x(t)

T

P

1

x(t), then

differentiating this term in t along the trajectories of

the perturbed system (9) leads to:

dV

1

(t)

dt

= 2¯x(t)

T

P

T

˙x(t)

0

Then, replacing [ ˙x(t) 0]

T

by the right side of

(10), the derivative of V

n

(t) in t along the trajecto-

ries of the perturbed system (9) satisﬁes the following

relation:

˙

V

n

(t) = ¯x(t)

T

Ψ

0

¯x(t) + δ

1

(t) + δ

2

(t) + h

1

z(t)

T

R

1

z(t)

−

Z

t

t−h

1

z(s)

T

R

1

z(s)ds+ x(t)

T

S

1

x(t)

−x(t −h

1

)

T

S

1

x(t −h

1

) (14)

with Ψ

0

= P

T

0 I

n

Λ −I

n

+

0 I

n

Λ −I

n

T

P.

Moreover, it comes that δ

1

(t) and δ

2

(t) are given

by:

δ

1

(t) = −2¯x(t)

T

P

T

0

BK

Z

t

t−h

1

z(s)ds

δ

2

(t) = −2¯x(t)

T

P

T

0

BK

Z

t−h

1

t−h

1

−η

1

z(s)ds

Now, let us bound δ

1

(t) and δ

2

(t) by applying the

bounding given in (Moon et al., 2001), where, for any

a ∈ R

n

, b ∈ R

2n

, R ∈ R

n×n

, Y ∈ R

n×2n

, Z ∈ R

2n×2n

,

N ∈ R

2n×n

the following holds:

−2b

T

Na ≤

a

b

T

R Y − N

T

Y

T

− N Z

a

b

with

R Y

Y

T

Z

≥ 0.

By considering such bounding condition and tak-

ing N = P

T

[0 BK]

T

, a = z(s), b = ¯x(t), R = R

1

,

Z = Z

1

, Y = Y

1

such that:

R

1

Y

1

∗ Z

1

≥ 0

then, we can ﬁnd the following bound for δ

1

:

δ

1

(t) ≤

Z

t

t−h

1

z(s)

T

R

1

z(s)ds+ h

1

¯x(t)

T

Z

1

¯x(t)

+2(x(t)

T

− x(t − h

1

)

T

)

(Y

1

−

0 (BK)

T

P) ¯x(t) (15)

Similarly to δ

1

(t), by posing N = P

T

[0 BK]

T

, a =

z(s) , b = ¯x(t), R = R

a

1

, Z = Z

a

1

, Y = Y

a

1

, such that:

R

a

1

Y

a

1

∗ Z

a

1

≥ 0

the bound of δ

2

(t) is given by:

δ

2

(t) ≤

Z

t−h

1

t−h

1

−η

1

z(s)

T

R

a

1

z(s)ds+ µ

1

¯x(t)

T

Z

a

1

¯x(t)

+2(x(t − h

1

)

T

− x(t − h

1

− η

1

)

T

)

(Y

a

1

−

0 (BK)

T

P) ¯x(t) (16)

Then, the time-derivative of V

a

(t) is given by:

˙

V

a

(t) = µ

1

z(t)

T

R

a

1

z(t) −

Z

t−h

1

t−h

1

−η

1

z(s)

T

R

a1

z(s)ds

+x(t)

T

S

a

1

x(t)

−(1− d

1

)x(t −h

1

− η

1

)

T

S

a

1

x(t −h

1

− η

1

)

Substituting (15) and (16) into (14), we ﬁnd that the

derivative of V(t) along the trajectories of the per-

turbed system satisﬁes the following inequality:

˙

V(t) ≤ ζ(t)

T

Γζ(t) (17)

where ζ(t) = col{ ¯x(t),x(t − h

1

−η

1

),x(t − h

1

)}, and

Γ is a negative matrix given by (7). Thus

˙

V(t) is neg-

ative deﬁnite if conditions (7) and (8) are satisﬁed,

while V(t) ≥ 0. Therefore, system (6) is asymptoti-

cally stable, and the proof is achieved.

Note that conditions (7) and (8) are satisﬁed for

a given state-feedback gain K. However, in our case

(that is a stabilization problem involving a control law

u(t) = Kx(t)), K is an unknown control gain to be

designed so as to insure the closed-loop stability of

system (9). In such a case, the LMI condition (7)

contains a bilinear term coming from the product of

the LMI variable with K, leading (7) to be a Bilin-

ear Matrix Inequality. Therefore, to give rise to a

LMI condition for computing of gain K, we can ap-

ply the transformation given in (Suplin et al., 2004).

In this aim, let us deﬁne: P

3

= εP

2

where ε ∈ R is

a tuning scalar parameter. Moreover, let us note that

P

2

is nonsingular since the only matrix which can be

negative deﬁnite in the second block on the diago-

nal of Ψ

1

is −ε(P

2

+ P

T

2

). Therefore we can also

deﬁne:

¯

P = P

−1

2

. In addition, for any matrix V =

{P

1

,S

1

,S

a

1

,Y

1i

,Y

a

1i

,Z

1k

,Z

a

1k

,R

1

,R

a

1

}, for i = 1,2 and

OBSERVER-BASED STATE FEEDBACK REMOTE CONTROL WITH BOUNDED TIME-VARYING DELAYS

139

k = 1,2,3, let us deﬁne an other matrix

¯

V =

¯

P

T

V

¯

P. Then, by multiplying (7), from the right and

the left sides respectively, by ∆

4

= diag{

¯

P,

¯

P,

¯

P,

¯

P}

and its transpose ∆

T

4

, and multiplying (8) by ∆

3

=

diag{

¯

P,

¯

P,

¯

P} and its transpose ∆

T

3

, from the right and

the left sides respectively, and posing W = K

¯

P, the

proof of the following theorem is straightforward.

Theorem 2. Suppose that, for some positive number

ε, there exists a positive-deﬁnite matrix

¯

P

1

, n × n ma-

trices

¯

P,

¯

S

1

,

¯

S

a

1

,

¯

Y

1i

,

¯

Y

a

1i

,

¯

Z

1k

,

¯

Z

a

1k

,

¯

R

1

,

¯

R

a

1

and W ∈

R

m×n

satisfying the LMI conditions for i = 1,2 and

k = 1,2, 3:

Γ =

Ψ

2

BW

εBW

−

¯

Y

T

a

1

¯

Y

T

a

1

−

¯

Y

T

1

∗ −(1− d

1

)

¯

S

a

1

0

∗ ∗ −

¯

S

1

< 0 (18)

and,

¯

R

1

¯

Y

1

∗

¯

Z

1

≥ 0,

¯

R

a

1

¯

Y

a

1

∗

¯

Z

a

1

≥ 0 (19)

where,

¯

Y

1

= [

¯

Y

11

¯

Y

12

]

¯

Y

a

1

= [

¯

Y

a

11

¯

Y

a

12

]

¯

Z

1

=

¯

Z

11

¯

Z

12

∗

¯

Z

13

¯

Z

a

1

=

¯

Z

a

11

¯

Z

a

12

∗

¯

Z

a

13

and matrix Ψ

2

is given by:

Ψ

21

= A

¯

P+

¯

P

T

A

T

+ S

1

+ h

1

¯

Z

11

+

¯

Y

11

+

¯

Y

T

11

+

¯

S

a

1

+ µ

1

¯

Z

a

11

Ψ

22

= ε

¯

P

T

A

T

+

¯

P

1

−

¯

P+ h

1

¯

Z

12

+

¯

Y

12

+ µ

1

¯

Z

a

12

Ψ

23

= −ε(

¯

P+

¯

P

T

) + h

1

(

¯

Z

13

+

¯

R

1

) + µ

1

¯

R

a

1

+µ

1

¯

Z

a

13

Then, the gain,

K = W

¯

P

−1

(20)

asymptotically stabilizes the system (6) for delay

τ

1

(t) ≤ τ

∗

1

.

4.2 Observer Design

Since the pair (A;C) is assumed to be observable,

it is possible to determine, in the non-delayed case

(that is τ

2

= 0), a gain L such that the Luenberger-

type observer leads the estimation error to asymptot-

ically converge towards zero. Now, by taking into

account the variable delay τ

2

(t) on the Slave out-

put, then, from (3) and (4), the observation error

e(t) = ˆx(t) − x(t) is ruled by:

˙e(t) = Ae(t) + LCe(t − τ

2

(t)) (21)

We then express the following result which insures

that the observer state ˆx(t) converges sufﬁciently fast

towards the true system state x(t) despite of delay

τ

2

(t).

Theorem 3. Suppose that, for some positive scalar

ε, there exists n × n matrices 0 < P

1

, P, S

2

,

S

a

2

,Y

2i

,Y

a

2i

,Z

2k

,Z

a

2k

, R

2

,R

a

2

and X ∈ R

n×p

satisfying

the LMI conditions for i = 1,2 and k = 1, 2, 3:

Γ =

Ψ

1

XC

εXC

−Y

T

a

2

Y

T

a

2

−Y

T

2

∗ −(1− d

2

)S

a

2

0

∗ ∗ −S

2

< 0 (22)

and,

R

2

Y

2

∗ Z

2

≥ 0,

R

a

2

Y

a

2

∗ Z

a

2

≥ 0 (23)

where,

Y

2

= [Y

21

Y

22

], Y

a

2

= [Y

a

21

Y

a

22

]

Z

2

=

Z

21

Z

22

∗ Z

23

, Z

a

2

=

Z

a

21

Z

a

22

∗ Z

a

23

with matrix Ψ

1

is given by:

Ψ

11

= P

T

A+ A

T

P+ S

2

+ h

2

Z

21

+Y

21

+Y

T

21

+S

a

2

+ µ

2

Z

a

21

Ψ

12

= εA

T

P+ P

T

1

− P

T

+ h

2

Z

22

+Y

22

+ µ

2

Z

a

22

Ψ

13

= −ε(P+ P

T

) + h

2

(Z

23

+ R

2

) + µ

2

R

a

2

+µ

2

Z

a

23

Then, the gain

L = (P

T

)

−1

X (24)

leads the estimation error e(t) = ˆx(t)− x(t) to asymp-

totically converge towards zero.

Proof — Representing (21) in an equivalent descrip-

tor form (Fridman and Shaked, 2002):

E

·

¯e(t) =

z(t)

−z(t) + Ae(t) + LCe(t− τ

2

(t))

(25)

where ¯e(t) = col{e(t),z(t)}, E = diag{I

n

,0}.

Recalling that τ

2

(t) = h

2

+ η

2

(t),0 ≤ η

2

(t) ≤

µ

2

,

˙

η

2

(t) ≤ d

2

< 1, the proof of this theorem use the

same Lyapunov-Krasovskii as given by (11) with a

single delay τ

2

(t):

V(t) = V

n

(t) + V

a

(t)

where V

n

(t) is a nominal LKF corresponding to the

nominal system (25) with h

2

6= 0 and η

2

(t) = 0, and

such that (see (Fridman, 2004))

V

n

(t) = ¯e(t)

T

EP¯e(t) +

Z

0

−h

2

Z

t

t+θ

z(β)

T

R

2

z(β)dβdθ

+

Z

t

t−h

2

e(s)

T

S

2

e(s)ds (26)

and V

a

(t) is an additional term of the following form:

V

a

(t) =

Z

0

−µ

2

Z

t

t+θ−h

2

z(s)

T

R

a

2

z(s)dsdθ

+

Z

t

t−τ

2

(t)

e(s)

T

S

a

2

e(s)ds (27)

Then, by differentiating of V(t) along the trajectories

of system (25), and posing P = P

2

, P

3

= εP

2

, where

ε ∈ R is a tuning scalar parameter, then the proof is

achieved by noting that X = P

T

L.

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

140

5 ILLUSTRATIVE EXAMPLE

This section aims at illustrating the theoretical results

of section 4, through an example related to the re-

mote control of a “ball and beam” system. Regard-

ing to Figure 2, this plant mainly consists in a steel

ball rolling on two parallel tensioned wires. These are

mounted on a beam, pivoted at its center, such that the

beam angle may be controlled by a servo-motor and

sensed by transducers to provide measurements of the

beam angle and ball position.

Figure 2: The Ball and Beam system to be controlled.

Regarding to the control scheme of Figure 3, the

fast dynamics of the plant are regulated by two inner

loops (with PI and PD controllers located in the Slave

systems), so that the remaining control problem is to

regulate the ball position by varying the beam angle.

State

feedback

Controller

Elec.

Motor

Mecha.

Motor

Ball

Beam

+ PD + PI

Observer

Master

Inner loop 1

Inner loop 2

τ

1

τ

2

Figure 3: Control scheme of the Slave system.

According to this, the system dynamics to be con-

trolled by means of the remote observer-based state-

feedback controller, can then be deﬁned by:

˙x(t) =

0 1

0 0

x(t) +

0

k

b

u(t − τ

1

(t))

y(t) =

k

x

0

x(t) (28)

where x(t) = [x

x

(t) x

v

(t)]

T

∈ R

2

is the state-vector,

x

x

(t) and x

v

(t) correspond respectively to the position

and the speed of the ball. u(t − τ

1

(t)) is the control

input (with input delay τ

1

(t))), y(t) is the measured

output (corresponding to the ball position) which is

forwarded to the Master system. k

b

and k

x

are two

constant parameters (with k

b

= 6.1 ms

−2

rad

−1

and

k

x

= 7 V/m).

Now, let us consider non-symmetric delays

τ

1

(t) 6= τ

2

(t), with, according to (2): h

1

= 0.3s, h

2

=

0.25s, µ

1

= µ

2

= 0.1s (recalling that h

1

and h

2

are con-

stant values, while η

1

(t) and η

2

(t) are time-varying

perturbations bounded by µ

1

the and µ

2

respectively).

Moreover, let us consider d

1

= d

2

= 0.1. By applying

Theorem 2 to (6) for ε = 9 , we ﬁnd the LMI (18) is

feasible for symmetric, positive-deﬁnite matrices:

¯

P

1

=

1.98 0

0 1.98

¯

R

1

=

2.29 −0.64

−0.64 1.38

¯

R

a

1

=

2.13 −0.45

−0.45 1.48

¯

S

1

=

0.12 −0.08

−0.08 0.11

¯

S

a

1

=

0.1934 −0.1241

−0.1241 0.5671

and,

¯

P =

0.5138 −0.2327

−0.2327 0.3694

W

T

=

−0.0088

−0.0699

With respect to (20), the state-feedback controller

gain K is then given by,

K =

−0.1440 −0.2800

(29)

Now, by applying Theorem 3 to (21) for ε = 5.5

(tuned by trial and error), we ﬁnd the LMI (22) is fea-

sible for symmetric, positive-deﬁnite matrices:

P

1

=

8.27 0

0 8.27

R

2

=

1.05 −1.46

−1.46 10.29

R

a

2

=

4.22 −2.16

−2.16 15.84

S

2

=

0.62 −0.23

−0.23 0.70

S

a

2

=

0.157 −0.112

−0.112 0.373

with,

P =

0.963 −2.240

−2.240 9.964

X =

−0.069

−0.097

Then, from (24), we ﬁnally obtain the observer gain:

L = [−0.198 − 0.054]

T

(30)

By considering the control scheme of ﬁgure 3 (mean-

ing that the two inner loops are taking in account in

simulating the dynamical behavior of the closed-loop

Master-Slave system), and numerical results (29)–

(30) for the controller and observer gains respectively,

we then obtain the simulation results of ﬁgures 4 and

5 (for delays h

1

= 0.3s, h

2

= 0.25s). Figure (4) rep-

resents the ball position on the beam axis when deal-

ing with a step response of the closed-loop system

OBSERVER-BASED STATE FEEDBACK REMOTE CONTROL WITH BOUNDED TIME-VARYING DELAYS

141

0 2 4 6 8 10 12 14 16 18 20

0

0.02

0.04

0.06

0.08

0.1

0.12

t(time sec)

The ball position on the beam

axis : X(m)

Figure 4: Step response of the closed-loop system.

0 2 4 6 8 10 12 14 16 18 20

−8

−6

−4

−2

0

2

4

x 10

−3

t(time sec)

Estimation errors : e1(t) and e2(t)

e1(t)

e2(t)

Figure 5: Estimation error ˆx(t) − x(t).

with a step magnitude 0.1 m, while Figure (5) repre-

sents the observations errors e

1

(t) = ˆx

x

(t) − x

x

(t) and

e

2

(t) = ˆx

v

(t) − x

v

(t). Moreover, Figure (6) shows the

corresponding delayed control input.

By looking at these simulations results, we can see

that the Luenberger-type observer insure the asymp-

totic convergence of the estimation error towards

zero, while the state-feedback control guarantees the

asymptotic stability of the closed-loop system, what-

ever the presence non-symmetric delays τ

1

6= τ

2

in the

control and feedback loops.

6 CONCLUSIONS

This paper has dealt with the stabilization problem

of a Networked Control System with a TCP network

as communication media. In particular, our attention

was focusing on a Master-Slave setup with uncertain,

time-varying, ”non-small”, non-symmetric transmis-

sion delays affecting the Slave control input and its

transmitted (scalar) output. A main feature of our

work was the use of a Lyapunov-Krasovskii func-

tional derived from a descriptor model transforma-

tion, to give rise to some conditions for the design

of an observer-based state-feedback control. In future

works, we will study the stability of Networked Con-

trol Systems with both delays and packet dropping.

0 2 4 6 8 10 12 14 16 18 20

−0.01

−0.005

0

0.005

0.01

0.015

t(time sec)

The corresponding delayed control

Figure 6: The corresponding delayed control input.

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