Back Stepping Method CSTR

8/3/2019 Back Stepping Method CSTR

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ABSTRACT

The backstepping method was applied to a tank system

to control the flow and temperature at the exit. A

backstepping control law and a robust backstepping

control law were developed for the plant without andwith uncertainties, respectively. The designed control

system is asymptotically stable for the plant without

uncertainties and globally uniformly bounded for the

plant with uncertainties.

1. INTRODUCTION

Over the past few years, a considerable number of studies

have been devoted to a new control design methodology:

backstepping ([1]). Unlike feedback linearization,

backstepping can avoid the cancellation of useful

nonlinearities. So, it offers the prospect of a more

practicable nonlinear control law.

This paper describes the application of the backstepping

control strategy to a tank system to control the flow and

temperature at the exit. Mathematical models of the

system are first derived. Then, a backstepping controller

is designed for the nominal plant. However, there are

usually some uncertainties in the mathematical model

of the plant. To achieve robustness for the control system,

a robust backstepping controller is designed by improving

the backstepping control law to guarantee global uniform

boundedness.

Nomenclature

qi Rate of inflow of the water ( / )m s3

qo Rate of outflow of the water ( / )m s3

i Temperature of the inflow (K)

o Temperature of the outflow (K)

a Air temperature (K)

h Height of the water level (m)

Flow and Temperature Control of a Tank System

by Backstepping Method

JinHua SheTokyo Engineering University, Hachioji, Tokyo, Japan, [email protected]

Hiroshi OdajimaAdvanced Film Technology Inc., Musashino, Tokyo, Japan, [email protected]

Hiroshi HashimotoTokyo Engineering University, Hachioji, Tokyo, Japan, [email protected]

Minoru HigashiguchiTokyo Engineering University, Hachioji, Tokyo, Japan, [email protected]

A Cross-sectional area of the tank ( )m2

i Heater supplied (W)

R Equivalent heat resistance of the tank (K/W)

a Discharge coefficient of valve ( / ).m s2 5

Density of water ( / )kg m3

cp Specific heat of water ( / / ) J kg K

2. MATHEMATICAL MODEL OF THE TANK

SYSTEM

The tank system studied here is shown in Fig. 1. In this

system, cold water is sent to the tank from a waterworks.

The water is heated in the tank, and then sent out. The

system has two control inputs: the rate of inflow and the

heater supply. The rate and temperature of the inflow,

the rate of the outflow, and the temperature of the water

in the tank are known.

The rate of outflow is given by

dh

dt

q q

A

i o=

, (1)

q a ho = , (2)

and

q qo m > 0 . (3)

Let T be the heat that the water in the tank possesses,

o be the heat that the output water takes away, S be

the heat released to the air and c be the heat that the

inflow brings in, and assume that the temperature of the

water in the tank is uniform. Then we have

T o s c i= + + . (4)


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FIGURE 1: The tank system.

and

Tp

oo

A c

aq

d

dt=

2

2 , (5)

o p o oc q= , (6)

so a o a

oR h A R q=

=

[ , ] [ ], (7)

c p i ic q= . (8)

From the above relationships, the model of the tank can

be written as

dq

dt

a

A

a

A qq

d

dtB

c

q q R

B

q R

Bc

qq

B

q

o

oi

o p

o oo

a

o

p i

oi

oi

= +

= + +

+ +

2 2

2 2

2 2

2 2

1

1

( )

,

(9)

where

B a Acp=2 /( ) .

It is clear from (9) that the tank system is a two-input

two-output nonlinear system. If we let qo and o be

the desired outputs, then the control objective is to make

the outflow and temperature track them. In the design

of such control systems, the flow sub-system and the

temperature sub-system are usually considered

separately, and the correlation between the controlled

outputs is ignored. Linear control theory is mainly used

for control system design. In particular, PID controllers

are generally designed for each linearized sub-system

([2]).

In this study, we used the MIMO nonlinear model directly

so as to take the nonlinearties of the plant and the

correlation between the controlled outputs into account.

Decomposing the outputs yields

qi , qi

qo , qoh

Fi

q q qo o oe

o o oe

= +

= +

,

(10)

where qoe and oe are the errors between the real and

the desired outputs. If we substitute (10) into (9) and

transform the control inputs into

q q q u

q q cR R

q q c u

i o oe q

i o oe p oa

o oe p i

= + +

= + +

+ +

( )( )

{( ) }

( ) ,

1

1

(11)

then the plant model becomes

dq

dt

a

Au

d

dt

Bc

q q

B

q q R

Bc

q q u

B

q q u

oeq

oe p oe

o oe

oe

o oe

p i

o oeq

o oe

=

= +

+

++

++

2

2

2

2

( )

( ) .

(12)

If we let

x qoe oeT

:= [ ] , (13)

and rewrite the model in the form:

dx dt f x g x u g x uq/ ( ) ( ) ( )= + +1 2 , (14)

then

f x Bc

q q

B

q q R

g x

a

ABc

q q

g x B

q q

p oe

o oe

oe

o oe

p i

o oe

o oe

( )

( )

( ) ( )( )

.

= +

+

=

+

=+

0

20

2

1

2

22

(15)

(14) is the nominal model of the plant. Since it is hard

to obtain an exact model of a real plant, it is useful to

include uncertainties in the plant model as follows:

dx dt f x g x u g x u

a

A q q

B

q q

q

o oe o oe

T

/ ( ) ( ) ( )

( ),

= + + +

=+ +

1 2

2

1 2 22

1

(16)

with


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1 1 2 2 , . (17)

3. DESIGN OF BACKSTEPPING CONTROL

LAWS

We first consider a plant without uncertainties. If a

Lyapunov function is defined as

V x x x qT

oe oe( ) := = +1

2

1

2

1

2

2 2 , (18)

it is clear that the following control law

u

u

x

xg x g x

k q

k

kA

aq

Bq q k q k q q

q q q oe

oe

q oe

o oe q i oe o oe oe

=

= [ ]

=

+ + +

:

( )

( )( ) ( )

( )[ ( ) ]

1 2

1

2

2

12

(19)

makes

dV

dt

V

x f g gq= + +

( )1 2 negative definite if

k kq , > 0 (20)

are chosen.

Now, backstepping the plant (14) gives

dx dt f x g x v g x v

d

dt

v

v

u

u

q

q q

/ ( ) ( ) ( )

.

= + +

=

1 2

(21)

A new Lyapunov function is selected:

V V x z za q: ( )= + +1

2

1

2

2 2, (22)

where

z v x

z v x

q q q: ( )

: ( )

=

=

. (23)

Then we have the following lemma.

Lemma 1: The control law

u k kA

a

a

Aq k k v

Bc

q q

q q oe q q

p i

o oeoe

= + +

+

( ) ( )

,

1 2

2

1

2

2

(24-1)

u k vB

q q k q

k q q c k q q

k q q v k c q q

kR

c q q v v

B

o oe q i oe

o oe oe p q i o oe

o oe oe q p oe o oe

oe p i o oe q

oe

= + +

+ + +

+ + + +

+ +

2

12

2

{ ( )[

( ) ]} { ( )

( ) } ( )

{ ( ) }

(( )q qo oe+ 2

(24-2)

guarantees the global asymptotic stability of the system

(21) at ( , ) ( , )q qo o o o = for any k k1 2 0, > .Proof:

dV dt dV dt v u d dt

v u d dt

V

xf g g g z g z

z u

x

f g v g v

z ux

f g v g

a q q q q

q q

q q

q

q

q

/ / ( )( / )

( )( / )

( )

[ ( )]

[ (

= +

+

= + + + +

+ + +

+ + +

1 2 1 2

1 2

1 2vv)].

So, if the control law is chosen to be

u k vx

f g v g vV

xg

k v kA

aq k v

a

Aq

Bc

q q

k kA

a

a

A

q k k vBc

q

q q qq

q

q q oe q q oep i

o oeoe

q oe q qp i

= + + +

= + +

= + +

1 1 2 1

1 2

2

1 2

2

1

2

2

2

2

( ) ( )

( )

( ) ( )

oo oe

oe

q+

,

u k vx

f g v g vV

xg

k vB

q q k q k q q

c k q q k q q v

k c

q

o oe q i oe o oe oe

p q i o oe o oe oe q

p oe

= + + +

= + + + +

+ + +

+

2 1 2 2

2

12

2

( ) ( )

{ ( )[ ( ) ]}

{ ( ) ( ) }

(

qq q

kR

c q q v vB

q q

o oe

oe p i o oe q

oe

o oe

+

+ + +

)

{ ( ) }( )

,

2

then

dV

dt

V

xf g g k z k za q q= + + 1 1

02 4

, ,

k k1 2 0, > and k k 1 2 0, > .Proof: Defining

x x xR: ( )=

1 2

sgn( ) : sgn( ) sgn( ) y y y= [ ]1 2 for y y y= [ ]1 2 and

=+ +

a

A q q

B

q qo oe o oe

T2

1 2 22

1

( )

yields

dV dt dV dt v u d dt

v u d dt

V

xf g g g z g z

V

x

z ux

f g v g v zx

a q q q q

q q

q qq

q qq

/ / ( )( / )

( )( / )

( )

{ ( )}

= + +

= + + + + +

+ + +

1 2 1 2

1 2

+ + + z ux

f g v g v zx

q

{ ( )} .

1 2

Using Young's Inequality

+2 21

4,

where ( , ) R2 and l is any positive number, yields

V

x

a

A

q

q qB

q q

q

q q

a

A q q

B

q

q q

a

A

B

oe

o oe

oe

o oe

oe

o oe

oe

o oe

oe

m

oe

m

=+

++

+

+ ++

+

+ + +

2

1 2 2

2

2

4

2 12

2

4

2

22

2

2

2

4

4

2 12

2

22

2

16 4

16 4

( )

( ) ( )

.

So, if the control law is chosen to be

u k vx

f g v g vV

xg

k vx

k kA

a

a

Aq k k v

Bc

q qk k z

q q

q q qq

q

q qq

q oe q q

p i

o oeoe q q

o oe

= + + +

= + +

+

+

1 1 2 1

1

1 2

2

1

11

2

2

( ) ( )

sgn[ ]

( ) ( )

sgn[ ] ,

u k v

x

f g v g v

V

xg k v

x

k vB

q q k q

k q q c k q q

k q

q

o oe q i oe

o oe oe p q i o oe

o

= + + +

= + +

+ + +

+ +

2 1 2

2 2

2

12

2

( ) ( )

sgn[ ]

{ ( )[

( ) ]} [ ( )

(

qq v k c q q

Rc q q v v

B

q qk z

c k q q

q q

k q q

q qk

oe oe q p oe o oe

oe p i o oe q

oe

o oe

p q i o oe

o oe

o oe oe

o oe

) ] { ( )

( ) }

( )sgn[ ](

(

( ) )),

+ +

+ +

+

+

+

++

++

2 2

1 2

2

then

dV

dt

V

xf g g k z k za q q= + +

( )1 2 1

22

2


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+ +

+

+

V

xk z

xz

zx

k

k zx

z

zx

k

kq

q k z k q

k z

a

A

qq

q

qq

qm

oe qm

oe

11

22

2

2

1

2

4

2

2

2

4

2

1 1

16

sgn[ ] (

sgn[ ]

)

sgn[ ] (

sgn[ ]

)

( ) ( )

112

2

22

4+

B .

That implies that dV dt a / is negative when

X

a

A

B

>

+4

2 12

2

22

16 4

, where X q z zoe oe q

T:= [ ]

and : min( , , , )= kq

kq

k kqm m

1 12 4 1 2

, i.e. the

stateXis globally uniformly bounded. (QED)

4. NUMERICAL EXAMPLE

Let the parameters of the tank system be

A m

a m s

R K W

K

K

a

i

=

=

=

=

=

0 2826

8 688 10

1000

288 15

289 15

2

3 2 5

. ( )

. ( / )

( / )

. ( )

. ( )

.

(27)

and

q m s

K

o

o

( ) . ( / )

( ) . ( )

0 0 0013

0 279 15

3=

=

(28)

The desired outputs are

q m s

K

o

o

=

=

0 001

282 15

3. ( / )

. ( )

(29)

The simulation results are shown in Figs. 2 and 3.

In Fig. 2, the nominal plant is controlled by the control

law (24). The parameters of the controller are

k k

k k

q = =

= =

0 00473

0 75 101 26

.

.(30)

It can be seen that the designed control system is stable

and the outputs reach the desired values after 1000

minutes.

FIGURE 2: Simulation results for the nominal plant.

In Fig. 3, the plant is assumed to contain uncertainties,and 1 and 2 are white noise with the bounds

15

24

5 0 10

1 0 10

=

=

. sin( )

. sin( )

t

t(31)

The parameters of the controller (25) are chosen to be

(30) and


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FIGURE 3: Simulation results for a plant

with uncertainties.k k 1 2 1 0= = . (32)

The simulation results show that the designed control

system is globally uniformly bounded and the outputs

converge to the desired values after 1000 minutes.

5. CONCLUSIONS

In this paper, a mathematical model of a tank system is

first derived, and a backstepping control law is developed

for the nominal plant. Then, based on the control law, a

robust backstepping control law is developed for a plant

with uncertainties. The validity of these control laws is

demonstrated by simulations.

REFERENCES

1. M. Krstic, I. Kanellakopoulos and P. Kokotovic:

Nonlinear and Adaptive Control Design, John Wiley

& Sons, Inc., 1995

2. N. Suda, PID Control, Asakura Shoten, 1992

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Back Stepping Method CSTR