Optimization Based Approach for Constrained Optimal Servo Control of Integrating Process Using Industrial PI Controller

Optimization Based Approach for Constrained

Optimal Servo Control of Integrating Process

Using Industrial PI Controller

Nguyen Viet Ha, Moonyong Lee*

Abstract—Constrained optimal control can be remarkably challenging to control. In this paper, optimization based approach for constrained optimal control of the integrating process is investigated through inventory control from using of the PI controller in the servo problem. The Lagrangian multiplier method is applied to find the PI parameters by solving the constrained optimal problem. The developed method explicitly deals with the important control constraints as well as minimizes the optimal performance measure. The method can be applied for any general integrating process.

Keywords—Constrained optimal control, Integrating process, Inventory control, PI controller tuning, Servo control.

I. INTRODUCTION

Some processes where the streams are comprised of gases, liquids, powers, slurries and melts do not naturally settle out at a steady state operating level, which refer as the non self-regulating or more commonly, as integrating processes. In process industry, many of the level, temperature, pressure, pH and other loops have the characteristics of the integrating processes [1].

In practice, a liquid level control system with a pump attached to the outflow line contains the integrating process and can be represented as a typical integrating model [2]. Another example of an integrating process is pressure control for a vapor system. In modern control, the integrating process also appears in the some applications such as the space telescope control system, the lightweight robotic arm and pilot crane control system, etc., [3].

In most research, the disturbance change referred to as the regulator problem has been studied [4], [5]. However, in some cases such as the reactor liquid level in chemical process [6], the Hubble telescope pointing controls and the control system of the disk drive head reader [3], both the regulator and servo problems need to be concerned.

( ) sp

Y s c

I

K s

τ

D

k s

p

k s

c

K ( ) U s

( )

Y s

( )

D s

+

− + + +

+

( ) sp

Y s c

I

K s

τ

D

k s

p

k s

c

K ( ) U s

( )

Y s

( )

D s

+

− + + +

+

In this paper, an analytical design method for PI controller is developed for constrained optimal control of integrating process. The constrained optimal control associated with the three constraints is formulated and then converted into a simple form with two independent variables by using a proper variable transformation.

Manuscript received December 30, 2008. This work was supported by Yeungnam University. 2008. * Corresponding author

Nguyen Viet Ha and Moonyong Lee are with the School of Chemical Engineering and Technology, Yeungnam University, 214-1 Dae-dong, Gyeongsan , 712-749, South Korea.

(Tel/Fax: 053-810-3241/811-3262; Email: mynlee@yu.ac.kr )

The Lagrangian multiplier method is utilized to handle the constrained optimization and then the PI parameters are found from the analysis of the global optimal condition. Liquid level control system is used throughout the case study section as a typical example for the constrained optimal servo control of the integrating process. The proposed method has explicitly brought out the way to find the optimal PI parameters in inventory control.

II. CONTROL SYSTEM DESCRIPTION The control system presented in Fig. 1 is simply described as:

_{Y s}( ) kD_{D s}( ) kp_{U s}( )

s s

= + (1)

To avoid the proportional kick for a servo problem, a modified PI controller is widely accepted as a practical solution.

( ) ( ) c

(

( ) ( )

)

c s

I

K

p

K Y s Y s Y s s

τ

= + −

U s (2)

where

K

_c and

τ

_I denote the dimensionless proportional gain and the integral time constant, respectively.

The closed-loop transfer functions for the level control system are

2 2

1

(s)= ( ) ( )

1 1

D H I

sp

H I I H I I

k s

Y D s

s s s s Y s

τ τ

τ τ

+

τ

+ +

τ τ

+

τ

+ (3)

H

2 2

H H

1

1 1

( )

( ) D c H I ( ) c ( )

sp

I I I I

k K s K s

U s D s Y s

s s s s

+

= −

+ + + +

τ τ τ

τ τ τ τ τ τ (4)

where _H 1

p c

k K

−

=

τ

(5)

Fig.1. Block diagram of the closed loop feedback control of the integrating process

the damping factor of the closed-loop transfer functions is 2 1

exp( 1)

1

2

1

tanh

exp

1

2

for

x

x

for

x

−

−

=

=

⎛

⎞

−

=

_⎜

−

_⎟

>

⎝

⎠

ζ

ζ

H 1 2 I

τ

ζ

τ

= (6)

III. FORMULATION OF CONSTRAINED OPTIMAL

CONTROL where

2 1

0 1

x

ζ

for

ζ

ζ

−

= < ≤

The control objective in the optimal control is to minimize the rate of change of the manipulated variable, , and the controlled variable, , whereas operating under the following constraints: ' ( ) u t

( )

y t

2 1 1

x=

ζ

− for ζ >

ζ

Applying the Lagrangian multiplier with slack variables converts the constrained optimization problem in (8a)-(8d) into an unconstrained equivalent problem with an augmented objective function, as follows:

(1) The maximum allowable controlled variable, y_max; (2) The maximum allowable rate of change of the manipulated variable,

u

_max' ;

1 2 3 1 2 3

2 2

1 1

4

2 2

2 2 3

min ( , , , , , , , ) 1

(1 4 ) ( ( ) )

( ( ) ) ( ( )

H

H h

g H f

L

h

g f

τ ζ ϖ ϖ ϖ σ σ σ

2

3 )

ϖ τ γ ζ σ

ϖ γ ζ σ ϖ τ γ ζ σ

Η 3

Η

= ατ + ζ + β + − −

τ ζ

+ − − + − −

(14) (3) The maximum allowable manipulated variable,

u

_max.

Therefore, the constrained optimal control problem can be defined as finding the controller parameters that minimize the performance measure in (7a) for a given , subject to the constraints in (7b)-(7d):

/

sp

y

s

∆

where

ϖ

_i is the Lagrange multiplier (

ϖ

_i≤0), and

σ

_i is a slack variable.

(

)

2

( )

' 2 min y

δ

y t( ) dt u u t( ) dt

∞ ∞

0 0

Φ = ω

_∫

+ω

_∫

(7a)

The necessary condition for an optimum solution of Eq. (14) is:

subject to y t( ) ≤ y_max (7b)

u t

'( )

≤

u

'

max (7c) 2

1 3

4 4

3

(1 4 ) 2 H 0

H H

L

β

α

ζ

ϖ τ

ϖ

τ

τ ζ

∂

= + − + + =

∂ (15)

max ( )

u t ≤u (7d)

' ' '

1 2 3

3 5 4

8 H L ( ) ( ) K ( ) 0(16)

H

L

h g k

β

ατ ζ ϖ γ ζ ϖ ζ ϖ γ ζ

ζ τ ζ

∂ = − − − − = ∂ 2 2 h 1 1

( ) 0

H L

h

τ

γ

ζ

σ

ϖ

∂

= − − =

∂ (17)

where

δ

y t( )=y t( )−

y

sp

( )

t

; ω is a weighting factor. After some mathematical manipulations, the above constrained optimal control problem can be expressed in terms of

τ

_H and

ζ

as follows:

2

H 3 4

1

min ( , ) _H(1 4 )

H

τ ζ

ατ

ζ

β

τ ζ

Φ = + + (8a) ₂2

2

( ) 0 (18)

g L

g

∂

= − − =

∂

ϖ

γ

ζ

σ

2

f 3

3

( ) 0

H L

f

τ

γ

ζ

σ

ϖ

∂

= − − =

∂ (19)

subject to

γ

_g

≥

g

( )

ζ

(8b) 2

H h

. ( )

h

τ

≥

γ

ζ

(8c)

1 1 1

2 0

L

_{ϖ σ}

σ

∂

= − =

∂ ; 2 2

2

2 0

L

_{ϖ σ}

σ

∂

= − =

∂ ;

H f

. ( )

f

τ

≥

γ

ζ

(8d)

where 2

( )

2

y sp

y

α

=ω ∆ ;

2 32 sp u p y k

β

=ω ⎛_⎜⎜∆ ⎞

⎝ ⎠⎟⎟

(9)

3 3 3

2

L

_{ϖ σ}

σ

∂

0

= − =

∂ (20)

max g sp

y

y

γ

=

∆

; ' max h sp p

y

k u

γ

=

∆

;

max sp f p

y

k u

γ

=

∆

(10)

( )

g

ζ

,h( )

ζ

and f( )

ζ

are given by

( )

1 exp 0 1

1 g

x

for

⎛ ⎞

= + _⎜− _⎟ < <

⎝ ⎠ = π ζ ζ ζ 1 for ≥ (11)

The solutions of (15)-(20) for

ϖ

=0,

σ

=0,

ϖ

≠0, and 0

σ

≠ are associated with the possible optimum cases. Fig. 2 shows seven possible instances of a global optimum with the contour of the objective function and the constraints in (8a)-(8d).

2 1

( ) 0

4 hζ

ζ

= forζ > (12)

2 1

1 tan

( ) exp 0 1

2 x x f x − ⎛ ⎞ +

= _⎜− _⎟ <

⎝ ⎠

ζ for ζ < (13)

Notice that the constraint by (8b) is vertical line in the

( ,

ζ τ

_H

)

space and the constraints in (8c) and (8d) bring the curves with a similar shape. It is clear that, the feasible region given by constraints , and with any positive value is always available and unbounded. The global optimum of

max

y

u

'

max

u

max

Fig.2. Typical contour and constraints with seven possible cases of optimum location

Case A ( ): The global optimum,

denoted by

1 2 3 0

ϖ

=

ϖ

=

ϖ

=

† †

(

ζ τ

,

H

)

, is in the interior of the feasible region. This case occurs when [τ†_H ≥max( γ_hh(ζ†) ,γ_f f(ζ†)) ] and [

ζ

_min

≤

ζ

†]. The global optimum is calculated by using (15) and (16) as follows:

1 4 † 4

H

β

τ

α

=

⎛

_⎜

⎞

_⎟

⎝

⎠

;

† 1

2

ζ

= (21) It should be noted that the optimal damping coefficient

†

ζ

is independent of the process dynamics and weighting factor.

Case B (

σ ϖ

₁

=

₂

=

ϖ

₃

=

0

): The global optimum, denoted by

(

ζ τ

∗h

,

_H∗h

)

, is located on the constraint

τ

_H2

=

γ

_h

h

( )

ζ

. This case occurs when the following conditions are satisfied:

† † †

and

max( ( ), ( ))

H < hh f f

τ

γ ζ

γ

ζ

ζ∗h >max(ζ ζhf, _min)

Then, from (15)-(17), the following equation is given.

2 2 5

2 '

2 3 4

4 8

( ) ( )

3 (1 4( )

( )( )

2 ( ) ( ) 2 ( )

h

h h

h

h h

h h h

h h

h

h h

∗

∗ ∗

∗ ∗

∗ ∗ ∗

− −

+

− =

β

αζ

γ

ζ

ζ

β

α

ζ

ζ

γ

ζ

ζ

ζ

) 0

(22)

h

ζ

∗

can be calculated by using any simple root finding method from (22).

Finally,

τ

Hh

∗

can be accordingly found by using (17) as:

(

)

1

2

(

)

h

H h

h

τ

∗

₌

γ ζ

∗h

(23) Case C (

ϖ

₁=

σ

₂ =

ϖ

₃ =0): The global optimum, denoted by

(

ζ τ

*g

,

H*g

)

, is on the constraintg( )

ζ

=

γ

g. This case

happens when the conditions [

ζ

_min

>

ζ

† ] and [ Hg max( Hgh, gf)

∗ _>

H

τ

τ τ

] are satisfied.

The minimum allowable damping factor,

ζ

_min , in the feasible region is calculated from (18).

ζ

∗gis equivalent to

min

ζ

.

τ

_H∗gis obtained from (15) as follows:

ζ

∗g

=

ζ

_min;

1 4 2 4 min min 3

( )

(1 4 )

g H

β

τ

α

ζ

ζ

∗ ₌

+ (24)

Case D (

σ

₁ =

σ

₂ =

ϖ

₃ =0 ): The global optimum, denoted by(

ζ τ

gh

,

_Hgh), is located on the vertex point formed by

τ

_H2 =

γ

_hh( )

ζ

and g( )

ζ

=

γ

_g . This case occurs either when [τ_H† ≥max( γ_hh(ζ†) ,γ_ff(ζ†)) ], [ _min †], and [τ_Hgh >max(τ τ_H∗g, _Hgf)] or when [ζgh >max(ζ ζhf, ∗h)] and [τ_H† <max( γ_hh(ζ†) ,γ_f f(ζ†))].

ζ

>

ζ

min gh

ζ

=

ζ

;

(

)

1 2 h ( min) gh

H

h

τ

=

γ ζ

(25)

Case E( ): The global optimum, denoted

by

1 2 3 0

ϖ

=

ϖ

=

σ

=

* *

(

ζ

f

,

τ

Hf), is on the constraint

τ

H =

γ

f f( )

ζ

. This case

happens when [ τ_H† <max( γ_hh(ζ†) ,γ_f f(ζ†)) ] and [

ζ

_min

<

ζ

*f

<

ζ

hf ].

Once the global optimum is obtained in terms of

ζ

and

τ

_H , the corresponding optimal PI parameters can be directly calculated using (5) and (6):

1

c opt

p H K

k

τ

−

= ; 4( opt)2 opt

I H

τ

=

ζ

τ

(30)

The PI controller designed by the proposed method gives the optimal responses whereas strictly satisfying all the given three constraint specifications.

From (15), (16) and (19)

ζ

∗fis evaluated by using (26):

The overall procedure for a quick finding of the global optimum and PI parameters is presented in Fig. 3.

3 3 5

2

4 4 4

4

8 ( )

( )( )

3

( (1 4( ) ) '( ))

( )( )

f f

f f f

f

f f

f

f f

f

f

f

f f

∗ ∗

∗ ∗

∗ ∗

∗ ∗

− −

− + =

β

αγ ζ ζ

γ ζ ζ

β

α ζ γ ζ

γ ζ ζ 0

(26)

IV. CASE STUDY

The most common integrating process is the liquid level system. In this section, a servo problem for the liquid level control system shown in Fig.4 is considered to demonstrate the performance of the proposed method. In this particular integrating process, k_D , , , , and Y s correspond to

p

k D s( ) U s( ) ( )

1

A , 1

A

−

, Q s_i( ) , Q s_o( ) , and H s( ) , respectively.

f H

τ

∗

is then found by using (19) as follows:

(

f

H

)

f f

f

τ

∗

₌

γ

ζ

∗

(27) Case F ( ): The global optimum, denoted by

1 2 3 0

ϖ

=

σ

=

σ

=

(

gf

,

gf

)

H

ζ τ

, is located on the vertex point formed by

τ

_H =

γ

_f f( )

ζ

andg( )

ζ

=

γ

_g. This case happens either when [τ_H† ≥max( γ_hh(ζ†) ,γ_f f(ζ†)) ], [ _min †], and [

τ

_Hgf >max(

τ τ

_H∗g, _Hgh ] or when [

ζ

*f

<

ζ

gf

<

ζ

hf ] and [τ_H† <max( γ_hh(ζ†) ,γ_f f(ζ†))].

ζ

>

ζ

)

where Q is maximum allowable rate of change of outlet flow rate, m

max

'_o

3

/min2; is maximum allowable of outlet flow rate, m

max

o Q

3

/min; is maximum allowable level deviation from set-point, m.

max

H

gf

ζ

is equivalent to

ζ

_min.

τ

_Hgf is calculated as follows:

In the level control, the modified PI controller is expressed in (31):

ζ

gf

=

ζ

_min;

τ

_Hgf =

γ

_f f(

ζ

_min) (28)

)

Case G ( ): The global optimum, denoted by

1 2 3 0

σ

=

ϖ

=

σ

=

(

ζ τ

hf

,

_Hhf , is located on the vertex point formed by

τ

_H =

γ

_f f( )

ζ

and

τ

_H2

=

γ

_h

h

( )

ζ

. This case occurs when [ H† <max( hh( †) , ff( ))

†

τ γ ζ γ ζ ], [

γ

h

<

γ

f ], and

[ *

min

max(ζ ζh, )<ζhf <ζ∗f].

( ) ( ) L

(

( ) set( )

o L

I

K

)

K H s H s H s

s τ

= + −

Q s (31)

where o max

c L

span

Q

K

K

H

ν

=

∆

The closed-loop transfer functions for the level control system are:

2 H

1

(s)= ( )

1

set

I I

H

s s

τ τ

+

τ

+ H s (32)

hf

ζ

and

τ

_Hhf are calculated as follows:

H 2

H 1

( )

L set

( )

o

I I

K

s

Q s

H

s

s

s

= −

+

+

τ

τ τ

τ

(33)

γ

_f f(

ζ

hf)=

γ

_hh(

ζ

hf);

τ

_Hhf =

γ

_f f(

ζ

hf) (29) Note that when

γ

h

<

γ

f, the vertex point formed by the two constraints

τ

H2

=

γ

h

h

( )

ζ

and

τ

H =

γ

f f( )

ζ

exists. The derivative of the constraint

τ

_H2

=

γ

_h

h

( )

ζ

and the constraint are always negative with any value of

ζ

. Furthermore, at the vertex point, the derivative of the constraint

τ

_H2

=

γ

_h

h

( )

ζ

is always greater than the derivative of the constraint

τ

_H =

γ

_f f( )

ζ

, which means the constraint

τ

H =

γ

f f( )

ζ

are always upper than the constraint

τ

_H2

=

γ

_h

h

( )

ζ

for

ζ

_min

< <

ζ ζ

hf .

where V

H

L c

A

K

K

τ

τ

=

=

;

V

max max

( span) T

o o

H A V

Q_ν Q_ν

τ = ∆ =

( )

H f f

τ

=

γ

ζ

† † †

H h f

if τ ≥max( γ h(ζ ),γf (ζ )) if ζmin≤ ζ†

† † H (ζ τ, )

g *g H (ζ τ∗ , )

Global optimum

h f

if γ < γ hf

min

if ζ ≤ ζ if ζmin < ζ < ζ*f hf

f *f H (ζ τ∗ , )

gf gf H (ζ τ, ) f

min if ζ ≥ ζ∗

h hf

if ζ ≤ ζ∗ hf hf

H (ζ τ, )

*h *h H (ζ τ, ) h

min

if ζ ≤ ζ∗ gh gh

H (ζ τ, )

*h *h H (ζ τ, )

N

Y Y

Y

Y Y

N N

N

Y Y

Y

N

opt opt H (ζ ,τ ) c opt

p H 1 K

k

− =

τ ;τ = ζI 4( opt 2 opt) τH

gh g gf

H H H

if τ >max(τ τ∗ , ) g gh gf

H H H

if τ >∗ max(τ τ, ) gf g gh

H H H

if τ >max(τ τ∗ , )

gh gh H (ζ τ, )

gf gf H (ζ τ, )

N

Y Y Y N

N

† † †

H h f

if τ ≥max( γ h(ζ ),γf (ζ )) if ζmin≤ ζ†

† † H (ζ τ, )

g *g H (ζ τ∗ , )

Global optimum

h f

if γ < γ hf

min

if ζ ≤ ζ if ζmin < ζ < ζ*f hf

f *f H (ζ τ∗ , )

gf gf H (ζ τ, ) f

min if ζ ≥ ζ∗

h hf

if ζ ≤ ζ∗ hf hf

H (ζ τ, )

*h *h H (ζ τ, ) h

min

if ζ ≤ ζ∗ gh gh

H (ζ τ, )

*h *h H (ζ τ, )

N

Y Y

Y

Y Y

N N

N

Y Y

Y

N

opt opt H (ζ ,τ ) c opt

p H 1 K

k

− =

τ ;τ = ζI 4( opt 2 opt) τH

gh g gf

H H H

if τ >max(τ τ∗ , ) g gh gf

H H H

if τ >∗ max(τ τ, ) gf g gh

H H H

if τ >max(τ τ∗ , )

gh gh H (ζ τ, )

gf gf H (ζ τ, )

N

Y Y Y N

N

Fig.3. Procedure to find a global optimum and PI parameters The PI parameters were found based on the minimum of

the sum of the rate of change of the outlet flow rate and the level deviation.

2 '

0

2 ' 2

max max

(1

min ( )] ( )] (34)

( o )

t dt Q t dt

H Q

∞ ∞

0 0

ω − ω)

Φ =

_∫

[ Η

δ

+

_∫

[ 2

where

2 max

y

H

ω = ω ,

' 2 max

(1 ( )

u o

Q

ω = − ω) , ( ) ( ) set

( )

.

t H t H

t

δ

Η = −

The controller should satisfy the following conditions: (1) The maximum allowable level deviation;

(2) The maximum allowable rate of change of the outlet flow (3) The maximum allowable outlet flow

Therefore, the three constraints corresponding to (7b)-(7d) are

H t

( )

≤

H

_max (35)

' ( )

o

'

Q

t

≤

Q

omax (36)

Q t_o( ) ≤Q_omax (37) The parameters

α

,

β

,

γ

_h,

γ

_g, and

γ

_f given in (9) and (10) are then:

2

max

( )

2

set H H

α

=ω ∆ ;

2

max 1

32 '

set

o A H

Q

ω

β

=⎛_⎜ − ⎞_⎟⎛_⎜ ∆

⎝ ⎠⎝ ⎠

⎞

⎟ (38)

max

'

h

set o

A H

Q

γ

=

∆

; _g

H

max_set

H

γ

=

∆

; max

f

set o

A H

Q

γ

=

∆

( 39)

Suppose that the liquid level of a drum with a cross section area of 1m2and a working volume A H∆ _span of 2 is controlled by the PI controller. The initial steady- state level is 50% and the nominal flow rates of the inlet and outlet are both 1 /min. The maximum outflow is 5m

3

m

3

m

Q

ovmax

3 /min. The expected maximum set point change in the level∆H_sp is 0.6m. In Table.1, seven examples are considered for illustrating seven corresponding to cases A-G.

Table.1. Constraints for illustrative examples example case

max

'

_o

Q

(m3/min2)

max o

Q

(m3/min)

max

H

(m)

ω

1 A 3.0 4.0 0.72 0.5

2 B 1.5 1.5 0.72 0.7

3 C 3 4.0 0.612 0.3

4 D 1.5 4.0 0.609 0.5

5 E 5 0.5 0.72 0.5

6 F 3 0.5 0.603 0.4

7 G 3 0.6 0.72 0.7

Fig.5. Response of level deviation, rate of change of outlet flow, and outlet flow rate for Examples (1-4).

Fig.6. Response of level deviation, rate of change of outlet flow, and outlet flow rate for Examples (5-7).

Figs. 5 and 6 compare the responses of the level deviation, the rate of change of the outlet flow rate, and the outlet flow rate for several cases.

A set point step change of 0.6m was introduced at t=1. As seen in the figures, the PI controller designed by the proposed

method gives the optimal responses whereas strictly satisfying the givenH_max,Q'_omax, and Q_omax specifications.

V. CONCLUSION

A novel PI controller design method for constrained optimal servo control of the integrating process system is proposed by solving the constrained optimization problem to deal with the important three specifications in the optimal control as well as minimize the optimal performance measure. A simple constrained problem with two independent variables is obtained from a proper variable transformation. The Lagrangian multiplier method is applied to find the analytical optimal solution. The performance of the proposed method has been demonstrated through the example of typical liquid level system.

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