Getting Controller Parameters: Integral Criteria Aims to minimize - - PowerPoint PPT Presentation

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

Getting Controller Parameters: Integral Criteria

Aims to minimize area measure based on the set point response

• r load rejection response.

Based on a FOPDT model including sensor, valve, transmitter etc) Methods based on Integrals tends to be conservative Write out the error analytically in terms of the controller parameters or use an optimizer

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

Getting Controller Parameters: Integral Criteria

Start with a model of the plant and guess controller parameters Simulate the time profile of the error Get the Integral value and check for constraints Are the controller parameters

• ptimal ?

Design complete Yes No Optimizer New values of parameters

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

Getting Controller Parameters: Integral Criteria

From: Seborg et al. (1989)

Empirical parameters for a FOPDT model

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

Relative comparison of the three criteria ISE and ITSE result in short rise times but larger overshoots

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

Direct Synthesis tuning

A desired closed loop trajectory that reflects the performance requirement is specified. Such a trajectory must be chosen with care (realizability). If there is a delay in the process, the trajectory must also have a delay term. The controller is directly obtained by equating the closed loop response equation (in terms of the controller) to the desired closed loop trajectory. Synthesis could result in a non-PID type controller and therefore simplifying assumptions need to be made.

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

] 1 1 1 [ 1 ) 1 ( 1 ) ( ) ( ) ( ) ( 1 ) (         q g c g q q g c g s d y s q s y s d y c gg c gg s y

) 1 1 ( 1 1 ; 1 ) ( s K c g s q s K s g

c c

          

Example 1 : Actual closed loop transfer function Desired closed loop transfer function

Direct Synthesis Procedure and example

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

Direct Synthesis Procedure and example

) 1 1 ( ) ( ) ( 1 ; 1 ) ( s K s c g s e q s Ke s G

c c s s

c

     

 

      

 

Example 2: Presence of time delay in the loop

Notes: q(s), the desired closed loop transfer function must be specified carefully. A rough rule of thumb is to start with a time constant that is half of the process time constant.

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

) 1 )( ( ) 1 3 )( 1 2 )( 1 1 ( 1 2 2 2 1 ] 1 1 1 [ 1 ) 1 ( 1 ) ( ) ( ) ( ) ( 1 ) ( ) 1 3 )( 1 2 )( 1 1 ( g process Let the                      s s s s s c g s r r s r q q g c g q q g c g s d y s q s y s d y c gg c gg s y s s s K           

                    r r r K r r gc s s s s Kc I D            2 2 1 1 2 1 1 2 1 2 1 2 1 2 ; ( ) ( )( ) ( ) ; ( )

Direct Synthesis Procedure: Another example

Designed Achieved This is a PID controller

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

Methods based on approximate models

Generation of approximate models Method is a little approximate but is adequate for feedback controller design purposes. For more advanced control strategies, model accuracy is important.

1 ) (  



s Ke s G

s





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

Controller Kc I D P 1/K(/)

• PI

0.9/K(/) 3.3

• PID

1.2/K(/) 2.0 0.5

Approximate model tuning rules (valid for 0.1 < / < 1.0)

Cohen-Coon Direct Synthesis

Controller Kc I D PI / K(r+) 

• PID

((2+)/ (2K(r+)) +(/2) r/(2K(r+)

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

Approximate model tuning rules (Cohen- Coon) (valid for 0.1 < / < 1.0)

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

Approximate model tuning rules (ITAE) (valid for 0.1 < / < 1.0)

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

Methods based on Stability Margins

Controller parameters calculated as a back-off from stability limits The stability limit is defined based on the value of the proportional gain that sets sustained oscillations of constant amplitude in the system. Any of the methods for example, root locus or routh criteria could be

• used. Alternately, back off from the stability limits on the bode and

nyquist diagrams could also be used. Stability limits can indicate the proportional gain and the frequency at which sustained oscillations are observed in the system.

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

) 3 )( 2 )( 1 ( 2 ) (     s s s s G 2 ) 3 )( 2 )( 1 (     

c

K s s s

Example of stability limits and stability margin Plant Controller Sustained oscillation are observed at Kc=30 (call this Kcu) and the frequency of oscillations can be read from the root locus plot(or calculated by direct substitution method). This gives w=3.32 rad/s and the period is Pu=2/w.

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

Controller Kc I D P 0.5 Kcu

• PI

0.45Kcu Pu/1.2 - PID 0.6Kcu Pu/2 Pu/8

Zeigler Nichols Stability margin based tuning So, for the previous example, if a P controller were used, Kc = 15; For a PI controller, Kc=13.5 and I = 1.577; For a PID controller, Kc=18, I = 0.946, D=0.24

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

Lectures 16 : Stability& Design in the Frequency Domain

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

Plant controller yd +

• y

u

Stability limit based on sustained oscillations

 b

Consider the following simple experiments: 1) Let the setpoint be perturbed in a sinusoidal fashion with the loop open at b If at that frequency, the controller and plant effectively add a phase lag of –180o, then b will be signal that is out of phase with the set-point. 2) When the oscillations become steady, suppose the loop is closed at b and the setpoint is set to zero; the –ve sign will introduce another phase change of – 180o and the wave at b will pass through the loop over and over. If the controller and plant gain is less (greater, equal) than unity, these oscillations will die (grow,stay constant) as the signal traverses repeatedly through the loop.