Flow and Temperature Control of a Tank System by

Flow and Temperature Control of a Tank System by Backstepping Method

JinÐHua She Tokyo Engineering University, Hachioji, Tokyo, Japan, [email protected] Hiroshi Odajima Advanced Film Technology Inc., Musashino, Tokyo, Japan, [email protected] Hiroshi Hashimoto Tokyo Engineering University, Hachioji, Tokyo, Japan, [email protected] Minoru Higashiguchi Tokyo Engineering University, Hachioji, Tokyo, Japan, [email protected]

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 and with 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 3 / s) qo Rate of outflow of the water ( m 3 / s) θ i Temperature of the inflow (K) θ o Temperature of the outflow (K) θ a Air temperature (K) h Height of the water level (m)

A Φi R

Cross-sectional area of the tank ( m 2 ) Heater supplied (W) Equivalent heat resistance of the tank (K/W)

a ρ cp

Discharge coefficient of valve ( m 2.5 / s) Density of water ( kg / m 3 ) 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 qi − qo = , dt A qo = a h ,

(1) (2)

and qo ≥ q m > 0 .

(3)

Let Φ T be the heat that the water in the tank possesses, Φ o be the heat that the output water takes away, Φ S be the heat released to the air and Φ 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

Φ T = −Φ o − Φ s + Φ c + Φ i .

(4)

 qo = qo + qoe  θ o = θ o + θ oe ,

qi , qi

h

where qoe and θ oe are the errors between the real and the desired outputs. If we substitute (10) into (9) and transform the control inputs into

qo , q o

qi = (qo + qoe )(1 + uq )  θa 1 Φ i = {(qo + qoe )c p ρ + R}θ o − R  −(qo + qoe )c p ρθ i + uΦ , 

Fi FIGURE 1: The tank system. and

ΦT =

(10)

(11)

then the plant model becomes Aρc p 2

qo2

dθ o , dt

a Φ o = c p ρθ o qo , θ − θa θo − θa = , Φs = o R[h, A] R[qo ] Φ c = c p ρθ i qi .

(5) (6) (7) (8)

From the above relationships, the model of the tank can be written as  dqo a2 a2 1 qi =− +  2 A 2 A qo  dt  dθ o = − B( c p ρ + 1 )θ + Bθ a o  dt qo qo2 R qo2 R   Bc p ρθ i B + qi + 2 Φ i ,  2 qo qo 

 dqoe a 2 = uq  2A  dt Bc ρθ Bθ oe  dθ oe = − p oe −  dt qo + qoe (qo + qoe )2 R  Bc p ρθ i  B + uq + uΦ .  qo + qoe (qo + qoe )2 

(12)

If we let x := [qoe θ oe ] , T

(13)

and rewrite the model in the form: (9)

where B = a 2 /( Ac p ρ ) . It is clear from (9) that the tank system is a two-input two-output nonlinear system. If we let qo and θ 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

dx / dt = f ( x ) + g1 ( x )uq + g2 ( x )uΦ ,

(14)

then 0    Bθ oe  f ( x ) =  Bc p ρθ oe   − q + q − (q + q )2 R  oe o oe  o    2    a 0      B .   = g ( x ) g1 ( x ) =  Bc2 A 2 ρθ  (qo + qoe )2   p i     qo + qoe  

(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 ) + g1 ( x )uq + g2 ( x )uΦ + ∆  T   a2  1 B ∆ ∆ ∆ =   1 2 , (qo + qoe )2    2 A qo + qoe with

(16)

∆1 ≤ δ1 ,

∆2 ≤ δ 2 .

(17)

3. DESIGN OF BACKSTEPPING CONTROL LAWS We first consider a plant without uncertainties. If a Lyapunov function is defined as V ( x ) :=

1 T 1 2 1 2 x x = qoe + θ oe , 2 2 2

(18)

makes

(19)

(20)

θ + kθ { oe − c p ρθ i (qo + qoe )vq − vΦ } R Bθ oe − (qo + qoe )2

dVa / dt = dV / dt + (vq − α q )(uq − dα q / dt ) +(vΦ − α Φ )(uΦ − dα Φ / dt )

∂V ( f + g1α q + g2α Φ + g1zq + g2 zΦ ) ∂x ∂α q ( f + g1vq + g2 vΦ )] + z q [u q − ∂x ∂α + zΦ [uΦ − Φ ( f + g1vq + g2 vΦ )]. ∂x

are chosen.

So, if the control law is chosen to be

Now, backstepping the plant (14) gives

uq = − k1 (vq − α q ) +

dx / dt = f ( x ) + g1 ( x )vq + g2 ( x )vΦ   d  vq  =  uq .  dt vΦ  uΦ  

(21)

1 2 1 2 zq + zΦ , 2 2

= −( k1kq

(22)

∂x

( f + g1vq + g2 vΦ ) −

Bc p ρθ i 2 A a2 )qoe − ( k1 + kq )vq − + θ oe , qo + qoe a2 2 A

uΦ = − k2 (vΦ − α Φ ) +

∂α Φ ∂V g2 ( f + g1vq + g2 vΦ ) − ∂x ∂x

1 (qo + qoe )[2 kqθ i qoe + kθ (qo + qoe )θ oe ]} B − c p ρ{kqθ i (qo + 2 qoe ) + kθ (qo + qoe )θ oe}vq = − k2 {vΦ +

where  zq := vq − α q ( x ) .  zΦ := vΦ − α Φ ( x )

∂α q

∂V g1 ∂x Bc p ρθ i 2A a2 = − k1 (vq + kq 2 qoe ) − kq vq − qoe − θ oe 2A qo + qoe a

A new Lyapunov function is selected: Va := V ( x ) +

(24-2)

=

dV ∂V = ( f + g1α q + g2α Φ ) negative definite if dt ∂x

kq , kθ > 0

+ kθ (qo + qoe )θ oe}vq + kθ c p ρθ oe (qo + qoe )

guarantees the global asymptotic stability of the system (21) at (qo , θ o ) = (qo , θ o ) for any k1 , k2 > 0 . Proof:

it is clear that the following control law  uq   α q ( x )  −1  − kq qoe  u  := α ( x ) = [ g1 ( x ) g2 ( x )] − k θ   Φ  Φ   θ oe  2A   − kq 2 qoe   a =  1 − (qo + qoe )[2 kqθ i qoe + kθ (qo + qoe )θ oe ]   B

1 (qo + qoe )[2 kqθ i qoe B + kθ (qo + qoe )θ oe ]} − c p ρ{kqθ i (qo + 2 qoe )

uΦ = − k2 {vΦ +

(23)

+ kθ c p ρθ oe (qo + qoe )

θ Bθ oe + kθ { oe − c p ρθ i (qo + qoe )vq − vΦ } − , R (qo + qoe )2

Then we have the following lemma. Lemma 1: The control law

then 2 A a2 uq = −( k1kq 2 + )qoe − ( k1 + kq )vq 2A a Bc p ρθ i θ oe , − qo + qoe

(24-1)

dVa ∂V = ( f + g1α q + g2α Φ ) − k1zq2 − k2 zΦ2 < 0 . dt ∂x That guarantees the global asymptotic stability at (qoe , θ oe ) = (0, 0) , i.e. (qo , θ o ) = (qo , θ o ) . (QED)

Based on the above backstepping control law, a robust backstepping control law is derived for a plant with uncertainties (16): Lemma 2: The control law  2 A a2 )qoe − ( k1 + kq )vq uq = −( k1kq 2 + 2A a  Bc p ρθ i  δ1 , θ oe − k∆1kq sgn[ zq ]  − qo + qoe qo + qoe  u = − k {v + 1 (q + q )[2 k θ q 2 Φ o oe q i oe  Φ B  + kθ (qo + qoe )θ oe ]} − c p ρ{kqθ i (qo + 2 qoe )   + kθ (qo + qoe )θ oe}vq + kθ c p ρ (qo + qoe )θ oe (25)   θ + kθ { oe − c p ρθ i (qo + qoe )vq − vΦ }  R  Bθ oe  −  (qo + qoe )2   − k∆ 2 sgn[ zΦ ]{(c p ρkqθ i qoe + 2 qoe   1 δ1 + kθ δ 2} + kθ (qo + qoe ) θ oe )  qo + qoe  guarantees the global uniform boundedness of the system: dx / dt = f ( x ) + g1 ( x )vq + g2 ( x )vΦ + ∆   d  v q   uq   dt v  = u    Φ  Φ at (qo , θ o ) = (qo , θ o ) for any k1 , k2 > 0 and k∆1 , k∆ 2 > 0 . Proof: Defining  ∂α ∂α ∂α :=  ∂x  ∂x1 ∂x 2 sgn( y) := [sgn( y1 )

kq −

1 1 2 , kθ − 4 > 0, qm qm

  (α ∈ R)  sgn( y2 )] for y = [ y1

 a2 1 δ = δ1 A q qoe 2 + o 

(26)

 B 2 δ2  (qo + qoe ) 

y2 ] and

T

yields dVa / dt = dV / dt + (vq − α q )(uq − dα q / dt ) +(vΦ − α Φ )(uΦ − dα Φ / dt )

∂V ∂V ( f + g1α q + g2α Φ + g1zq + g2 zΦ ) + ∆ ∂x ∂x ∂α q ∂α q + zq {uq − ( f + g1vq + g2 vΦ )} − zq ∆ ∂x ∂x =

+ zΦ {uΦ −

∂α Φ ∂α Φ ( f + g1vq + g2 vΦ )} − zΦ ∆. ∂x ∂x

Using Young's Inequality

αβ ≤ λα 2 +

1 2 β , 4λ

where (α , β ) ∈ R 2 and l is any positive number, yields qoe θ oe ∂V a2 ∆= ∆1 + B ∆2 ∂x 2 A qo + qoe (qo + qoe )2 ≤

2 2 qoe θ oe a4 2 B2 2 + δ + + δ2 1 4 (qo + qoe )2 16 A 2 (qo + qoe ) 4

≤

2 2 qoe θ oe a 4 2 B2 2 δ1 + δ2 . + + 4 qm2 qm4 16 A 2

So, if the control law is chosen to be uq = − k1 (vq − α q ) + − k∆1 sgn[vq − α q ]

∂α q ∂x

∂α q ∂x

( f + g1vq + g2 vΦ ) −

∂V g1 ∂x

δ

2 A a2 + )qoe − ( k1 + kq )vq a2 2 A Bc p ρθ i δ1 − , θ oe − k∆1kq sgn[ zq ] qo + qoe qo + qoe = −( k1kq

∂α Φ ( f + g1vq + g2 vΦ ) ∂x ∂V ∂α − g2 − k∆ 2 sgn[vΦ − α Φ ] Φ δ ∂x ∂x 1 = − k2 {vΦ + (qo + qoe )[2 kqθ i qoe B + kθ (qo + qoe )θ oe ]} − c p ρ[kqθ i (qo + 2 qoe )

uΦ = − k2 (vΦ − α Φ ) +

+ kθ (qo + qoe )θ oe ]vq + kθ {c p ρθ oe (qo + qoe )

θ oe − c p ρθ i (qo + qoe )vq − vΦ } R (c p ρkqθ i qo + 2 qoe Bθ oe − 2 − k ∆ 2 sgn[ zΦ ]( qo + qoe (qo + qoe ) +

+

kθ (qo + qoe ) θ oe ) qo + qoe

δ1 + kθ δ 2 ),

then dVa ∂V = ( f + g1α q + g2α Φ ) − k1zq2 − k2 zΦ2 ∂x dt

+

∂α q

∂V ∆ − k∆1 sgn[ zq ] zq (δ + ∂x ∂x

∂α − k∆ 2 sgn[ zΦ ] Φ zΦ (δ + ∂x ≤ − ( kq − +

sgn[ zΦ

sgn[ zq

∂α q ∂x

]

k∆1

∆)

∂α Φ ] ∂x ∆ )

k∆ 2

1 2 1 2 )qoe − k1zq2 − ( kθ − 4 )θ oe − k2 zΦ2 qm2 qm

a 4 2 B2 2 δ2 . δ1 + 4 16 A 2

That implies that 4

dVa / dt

is negative when

2

a B 2 2 δ2 2 δ1 + T 4 X > 16 A , where X := qoe θ oe zq zΦ ζ 1 1 and ζ := min( kq − 2 , kθ − 4 , k1 , k2 ) , i.e. the qm qm state X is globally uniformly bounded. (QED)

[

]

4. NUMERICAL EXAMPLE Let the parameters of the tank system be  A = 0.2826 ( m 2 )  −3 2.5 a = 8.688 × 10 ( m / s)   R = 1000 ( K / W ) θ = 288.15 ( K )  a θ i = 289.15 ( K )

(27)

and qo (0) = 0.0013 ( m 3 / s)   θ o (0) = 279.15 ( K )

(28)

The desired outputs are qo = 0.001 ( m 3 / s)   θ o = 282.15 ( K )

(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  kq = kθ = 0.00473  6 k1 = k2 = 0.75 × 10

(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  ∆1 = 5.0 × 10 −5 sin(t )  −4 ∆2 = 1.0 × 10 sin(t )

(31)

The parameters of the controller (25) are chosen to be (30) and

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

FIGURE 3: Simulation results for a plant with uncertainties. k∆1 = k∆ 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.