A high gain observer coupled to a slidin

15th international conference on Sciences and Techniques of Automatic control A high gain observer coupled to a sliding mode technique for electropneumatic system control A.. T his pap

15th international conference on Sciences and Techniques of Automatic control

A high gain observer coupled to a sliding mode technique

for electropneumatic system control

A AYADIl, S HAJJIl, M SMAOUI2, A CHAARIl and M FARZA 3

Abstract- Nonlinear control laws become the most important

strategies to control electropnemnatic actuator To get high

accuracy and performance, we need the knowledge of aU state

variables T his paper focus on the design of high gain observer

which estimates the unmeasured states (velocity and pressure

in chamber N) from the measurements of the position and

pressure in chamber P

Simulations results are presented to test the effectiveness of our

high gain observer which is applied to sliding mode controller

in order to control tracking position and pressure

Keywords : electropneumatic actuator, high gain ob­

server, sliding mode controller

NO MENCLATURE

y, v, a Position(m),velocity(mI8), acceleration(mI82)

j lerk(mI83)

PP,N Pressure in the chamber P and N (Pa)

PE Exhaust pressure(Pa)

Up, UN Servodistributors voltages(V)

qm Mass flow rate provided from servodistributor to

cp(.)

'IjJ( )

b

Fext

k

l

M

S

T

r

cylinder chamber (kg/s)

Leakage polynomial function (kg I 8)

Polynomial function (kg I 8 IV)

Viscous friction coefficient (N 1m I 8 )

External force (N)

Polytropic constant

Length of stroke (m)

Total load mass (kg)

Piston P, N section

Temperature K

Perfect gas constant related to unit mass

(JlkgIK)

I INTRODUCTION Pneumatic control systems have become the focus of

several research due to their complexity and the presence

of non linearities The first classical controller applied to

the electropneumatic system is a fixed-gain linear controller

[2] However, it has the inconvenient of the limitation of

*This work was supported by the Ministry of Higher Education and

Scientific Research of Tunisia

1 A AYADI, S HAm and A CHAAR! are with (Lab-STA), National

School of Engineering of Sfax, University of Sfax PB 1173, 3038 Sfax,

Tunisia ayadiassil@yahoo fr

2 M SMAOUl is with AMPERE, lNSA de Lyon, Villeurbanne, France

mohamed.smaoui@insa-lyon.fr

3 M FARZA is with GRE YC, Universite de Caen, ENSICAEN, France

mongher.farza@unicaen.fr

the linear feedback controller for the nonlinearities adverse effect or parameter variations Thus, many research have developed in the nonlinear control such as feedback lineari­ zation [6], fuzzy control algorithms [10], adaptive control [7], robust linear control [9], robust differentiator controller [13], sliding mode control ([15],[14]) and higher sliding mode control([12],[5])

In this paper, a sliding mode controller designed in [1] is used The reaching law method is applied on a PD sliding surface form in order to reduce the chattering phenomenon and to minimize the tracking error However, for the appli­ cation of such control laws, all the state variables must be known In fact, the position measurement is still available and the pressure is not systematically So, we need an observer to reduce the number of sensors and to evaluate the disturbance The observation has several advantages such as disturbance reconstructing to increase the robustness and removing one

of pressure sensors to reduce the manufacturing cost Very few works have made in observation Only observability property has been studied in [8] High gain and sliding mode observers are developed in [4], the gain of the pro­ posed observer involves the computation of the jacobian inverse The originality of the current papers is to construct

a state observer for a class of MIMO non linear system under similar assumptions The main characteristics of the proposed observer lies in its simplicity and its capability in implementation which it does not need the inversion of any jacobian transformation

The paper is organized as follow In section 2, we present the model of the electropneumatic system Section 3 is devoted to observer's design where we give the class of the nonlinear MIMO system and the observer under investiga­ tion The high gain observer applied to our system is given

in section 4 The simulation results are presented to show the effectiveness of our observer in section 5 In the last section,

we present some conclusions

II ELECT RO PNEUMATIC MO DEL The electropneumatic system under interest is a double acting actuator (Fig.I) constituted by two chambers, denoted

P (as positive) and N (as negative) The air mass flow rates entering the two chambers are modulated by two three-way servodistributors controlled by a micro-controller with two electrical inputs The pneumatic jack horizontally moves a

load carriage of mass M The electropneumatic plant model

is obtained from three physical laws :

• the mass flow rate through a restriction,

• the pressure behavior in a chamber with variable vo­

lume,

• the fundamental mechanical equation

Fig 1 Electropneumatic system

In our case, the bandwidths of the Servotronic 10ucomatic

servodistributor and actuator are, respectively, about 200 and

2, 4 Hz Using the singular perturbation theory, the dynamics

of the servodistributors are neglected and their model can

be reduced to a static one, described by two relationships

qmP(up,pp) and qmN(UN,PN) between the mass flow rates

qmP and qmN, the input voltages Up and UN and the

output pressures P p and P N The pressure evolution law in

a chamber with variable volume is obtained assuming the

following assumptions [11] :

• air is a perfect gas and its kinetic energy is negligible,

• the pressure and the temperature are homogeneous in

each chamber,

• the process is polytropic and characterized by a coeffi­

cient,

• the temperature variation is negligible with respect to

average and equal to the supply temperature

Therefore, the following relations give the model of the

previous system :

dy

dt = v

dt = M[SPPP - SNPN - Ff(v) - Fextl

dt = Vp(y) [qmP(up,pp) - rTPpvl

dt = VN(y) [qmN(UN,PN) + rT PNVl

where

l

{ Vp(y) = Vp(O) + Spy { vP(O) = VDP + SP2

WIth VN(y) = VN(O) - SNY TT vN = vDN N2 (0) TT + S l

are the effective volumes of the chambers for the zero

position and VD[PorN]are dead volumes present at each

extremity of the cylinder

The mass flow rate qm is an algebraic function and is given as in [2] :

qm(U,p) = <p(P) + 'IjJ(P, sign(u))u (2) with <p(p) is a polynomial function of the pressure and 'IjJ(P, sgn(u)) is a polynomial function of both the pressure and of the input control From (2) the nonlinear affine model

is given by these following equations :

dy

dt = v

dt = M [SPPP - SNPN - Ff(v) - Fextl

dt = Vp(y) [<p(pp) - rTPpvl

+ Vp(y) 'IjJ(pp, szgn(up ))up

dt = VN(y) [<P(PN) + rT PNVl

+ VN(y) 'IjJ(PN, szgn(uN ))UN with two inputs up and UN, the nonlinear model of the electropneumatic system has the following form :

(4) with Jl = [ Y v pp PN f and U = [ up UN ]T are respectevely the state vector and the input control and f (x) and g(x) are nonlinear functions

Where

_ ( Ir[SPPP - SNP ; - Ff(v) - Fextl )

f(x) - vp krTp [In(p ) - kp vl J Y) y p rTp p

t:(:) [<p(PN) + :£'PNvl g(x) = (gl(X) g2(X))

= ( ;,,1,)

,pc.

: o ,sgn(up)) J':�)'IjJ(PN' sgn(uN)) ! )

III OBSERVERS DESIGN

A Class of nonlinear system

Consider the nonlinear MIMO systems :

with

xl x2 x=

xq-l

xq

{X = f(u,x)

y = ex = xl

; f(u, x) =

F(u, xl, x2) f2(U,X\X2,X3) r-l(u,x) r(u,x)

(5)

964

c = (Inb OnlXn2, OnlXn3, , OnlXnq)

we note that the state x E IRn with xk E IRnk, k = 1··· q

q and P = nl � n2 � � nq; L nk = n; the input

k=l u(t) E U the set of bounded absolutely continuous functions

with bounded derivatives from ]R+ into U a compact subset

of]R· ; f(u, x) E ]Rn with fk(u, x) E ]Rnk

Our aim consists in design an observer for the system (5)

Such a design needs some assumptions which will be stated

in due course

B Assumptions

At this step, one assumes the following :

Ai) Each function fk(u, x), k = 1, , q - 1 satisfies the

following rank condition :

( afk )

rang axk+l (u, k) = nkH Vx E ]Rn; Vx E U (6)

Moreover ::la, (3 > 0 such that for all k E {I, , q - I},

Vx E ]Rn;vu E U,

a2Ink+, � (a�Cl (u,k) ) T (a�Cl (u,k) ) � (32Ink+'

where Ink+, is the (nkH) x (nkH) identity matrix

A2) For 1 � k � q - 1; the function xk+ 1 t +

fk (u, xl, , xk, xkH) is one to one from ]Rnk+l into ]Rnk

Now, we propose to synthesize a nonlinear observer for

system (5)

C Observers synthesis

The synthesis of this observer is based on uniform obser­

vability propriety of (5) The class of observer is interesting

due its applicability to a large class of nonlinear systems

A candidate observer for systems (5) is given by the follo­

wing equation :

� = f(u,x) - OA+(u,x)tl.;lS-lOTC(x - x) (7)

where x � [��l E nt" withX' E nt"', k � 1" " , q;

tl.(J = diag [Inll 0-1 Inll , o-(p-l) In,] where 0 > 0 is a

real number

S is the unique solution of the algebraic Lyapunov equation :

w ere - q p, , q P WI n - (n-p)!p!

and 0 = [Inl, Onl, ,Onl]

A • Inl, �(U,X�'l�(U,X)8xlr (U,x),

A( u, X) = d�ag qn {)rk

( ) , a;;t+T U, X k=l

(8)

A(u,x) is a diagonal matrix and left invertible according

to assumption (AI) we note A+(u,x) its left inverse

Theorem Consider systems (5) and (7).Then,

::l00 > 0; VB > 00; ::l oX > 0; ::l/-t(J > 0;

Vu E U; Vx(O) E Rn,q;

we have:

Ilx(t) - x(t)11 � oXOq-le-JL9tllx(0) - x(O)11 where x(t) is the unknown trajectory of (5) associated to the input u, x(t) is any trajectory of system (7) associated

to (u, y) Moreover, we have lim /-t(J = +00

(J-+oo Proof: The proof of this theorem is given in [3]

IV HIGH GAIN OBSERVER APPLIED TO ELECTRO PNEUMATIC S YSTEM The electropneumatic system given by (3) can be written

as follow:

Xl = X2 X2 = � [SpX3 - SNX4 - Ff(X2) - Fext]

X3 = VP(Xl) [<p(X3) - rT X3X2]

+ VP(Xl) '!f;(X3, Up )up

X4 = VN(xd [<P(X4) + rT X4X2]

krT + VN(Xl) '!f;(X4, UN )UN with x = [Xl X2 X3 x4f = [Y v pp PN]T ;

U = [Up UN]T and y = [Xl X3f are respectively the state, the input and the output vectors, where x E ]R.4,

U E ]R.2 and y E ]R.2

We note that only the position and pressure in chamber

P is measured In this Step, we put the system (10) in the form of class (5) We define the new state vector :

xl = [Xl X3]T ,x2 = X2 and x3 = X4 and f(u,x) a nonlinear function given as follow :

1 0

o 1

o 0

o

o

1

o

o

o

A( u, x) = 0 0 - � VP(Xl) 0

o 0 0 SN

o 0 0 kS-;'"l!X3 MVP(Xl)

We define A + the left invertible matrix of A by the following

expression :

A+ = (ATA)-lAT

A high gain observer for (10) is defined as :

where () is the gain of the observer

A Simulations and results

The controller used in the following simulations is a

sliding mode controller designed in [1] It ensures a good

tracking for both actuator's position and pressure in chamber

P and it reduces the chattering phenomon The proposed

sliding surface ai(i = 1 , 2) defined as a 1 = k1ey + k 2ev +

k3ea + k 4 ea and a2 = k5ep + k6ep, where ex is the

tracking error X E {y,v,a,pp} (with ey = y - yd,ev =

v - vd,ea = a - ad and ep = pp - p� yd,vd,ad and

p� are respectively the position, velocity, acceleration and

pressure desired trajectories and ki are positives constants

with i = 1, ,6)

The reaching law is a differential equation which specifies

the dynamics of a switching function ai

It was selected as follows ai = -"l(ai + w signai ) with "l

and w are positives constants

The existence condition of sliding mode implies that both

ai and ai will tends to zero when t tend to infinity, which

means that the dynamic of the system will stay into the

sliding surface The existence condition of the sliding mode

is aiai < O

The control law given in [1] is deduced by the derivation of

sliding surface

One has [&1 &2f

We note that G(x) is F(x) + G(x) [up UN f

an invertible matrix thus there

are no singularity in the control law [ up UN f =

G(x)-l [F(x) + [&1 &2fJ

The control needs the knowledge of all state variables which implies, in the current case, the use of an observer The initial actual and estimated conditions have the following values: X1(0) = -0.12m , :1;1(0) = O.Om, X2(0) = O.Om/s,

:1;2(0) = O.Om/ s, X3(0) = 3bar, :1;3(0) = 2bar, X4(0) =

3bar, :1;4(0) = 1bar and the gain of the observer is () = 80

I Desired position

I-Estimetedposition

\

-O'1S0C -; -; -;-, -; 7 ;

Time (sec)

Fig 2 Desired and estimated position (m)

·

I Desiredpressu

I

-Estimetedpressure

,

"

Time (sec)

Fig 3 Desired and estimated pressure in chamber P (bar)

·

,

· i

i

·

\

\

·

, ,

\

, Time (sec)

)

I Simulated�ure -Estimated prvssure

r

Fig 4 Simulated and estimated pressure in chamber N (bar)

966

Simulated velocity

- Estimeted velocity

�� � � -,� � � �

Time (sec)

Fig 5 Desired and estimated velocity (m/s)

Fig 2 and Fig 3 shows the applicability and the efficiency

of the control law coupled to observer(7) since we obtained

good tracking responses of position and pressure in chamber

P

Estimated results are reported in Fig 4 and Fig 5 where

they are compared to the true value ( obtained by the model

of simulation) Fig 4 and Fig 5 clearly show the good

performance of the proposed observer Indeed, we remark

a good agreement between simulated and estimated curves

beyond 0.1s

v CONCLUSIONS High gain observer is designed for electropneumatic sys­

tem in order to be applied on sliding mode controller

to control actuator and pressure in chamber P Simulation

results are presented to show the applicability and the high

accuracy of our observer Future work concern to evaluate

this observer in experimentation and to compare it with

dynamic high gain observer

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