In general, the lead block of the transfer function speeds up the - - PowerPoint PPT Presentation

Department of Chemical Engineering I.I.T. Bombay, India

Lead Lag Systems If a transfer function is of the form,

1 1 ) (    s s K s G  

In general, the lead block of the transfer function speeds up the process. A pure lead block as in the case of a PD controller is not realizable. Approximations of these have to be done to implement lead or lead/lag blocks (for example in feedforward control)

Department of Chemical Engineering I.I.T. Bombay, India

Dynamic Response of a Lead Lag System

1 1 ) (    s s K s G  

Using partial fraction expansion,

] 1 [ 1 1 ) (

1

      s A A K s s K s G   

where A0=/ and A1=1-A0 = 1-

) ( 1 ) 1 ( ) ( ) ( ) ( s u s K K s u s g s y              

Department of Chemical Engineering I.I.T. Bombay, India

Step Response of a Lead Lag System

Depending on the value of , an effective lead or lag action is seen

Department of Chemical Engineering I.I.T. Bombay, India

Lectures 7: Dynamics of higher order systems

Department of Chemical Engineering I.I.T. Bombay, India

Dynamic behaviour of second order systems

Number of dynamic elements are 2. F

h1

h2

Second order systems can arise because of two first order systems in interacting or noninteracting setups Inherent dynamics in the system is of higher order U-tube manometer. Presence of a controller in a closed loop even if the process is first order (such as PI or PID controllers in a closed loop) can give a second order system.

Interaction between the holdups could give an

• scillatory response.

Department of Chemical Engineering I.I.T. Bombay, India

F0

h1 h2

IIlustrative example: Interacting system of tanks in series First Principles Model

2 2 2 1 1 2 2 2 1 1 1 1

) ( ); ( h c h h c dt dh A h h c F dt dh A

• 

    

Department of Chemical Engineering I.I.T. Bombay, India

) ( 1 ) ( ) ( ; 1 ; c A ; c A If

1 2 2 1 2 2 1 2 1 2 2 1 2 1 1 2 2 2 1 1 1

s u s K s K K s y c c K c K                

First Principles Model The roots of the denominator (poles of the transfer function) could be complex. They could give rise to oscillatory behaviour for different u.

Department of Chemical Engineering I.I.T. Bombay, India

Second Illustrative example: Control of a first order process

Ku y dt dy             



t d I d c

dt y y y y K t u ) ( 1 ) ( ) ( 

and

          



t d I d c

dt y y y y KK y dt dy ) ( 1 ) (  

Therefore,

d I c c

y y KK dt dy KK dt y d       ) 1 (

2 2 1

This is a second order system and will give rise to a second

• rder transfer function.

Plant controller yd +

• y

u

Department of Chemical Engineering I.I.T. Bombay, India

plane of initial rest when DP = 0 h h P1 P2 DP = P1 - P2

U-Tube Manometer

P g h dt dh R g L dt h d g L D       2 1 4 2

2 2 2

L = length of fluid in the manometer tube ,  = density and viscosity

• f manometer fluid

R = radius of manometer tube g = gravitation constant