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 or load rejection response. Based on a FOPDT model including sensor, valve, transmitter etc) Write out the error analytically in terms of the controller parameters or use an optimizer Methods based on Integrals tends to be conservative
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 New values of parameters Simulate the time profile of the error Optimizer Get the Integral value and check for constraints Are the controller parameters Design optimal ? complete No Yes
Department of Chemical Engineering I.I.T. Bombay, India Getting Controller Parameters: Integral Criteria Empirical parameters for a FOPDT model From: Seborg et al. (1989)
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 Direct Synthesis Procedure and example Actual closed loop gg c y ( s ) y ( s ) transfer function d 1 gg c y ( s ) q ( s ) y ( s ) Desired closed loop d transfer function 1 1 1 q g ( ) g [ ] c c 1 g 1 q g 1 q Example 1 : 1 K g ( s ) ; q s 1 s 1 c 1 g ( 1 ) c K s c
Department of Chemical Engineering I.I.T. Bombay, India Direct Synthesis Procedure and example Example 2: Presence of time delay in the loop s s Ke e c G ( s ) ; q s 1 s 1 c 1 ( ) ( 1 ) g s c K ( ) s c 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 Direct Synthesis Procedure: Another example Designed Achieved K Let the process g r ( s 1 )( s 1 )( s 1 ) ; 2 K r r 1 2 3 r 2 r gg c ( ) ( ) y s y s d 1 gg ( s 1 )( s 1 ) c 1 2 y ( s ) q ( s ) y ( s ) gc s ( ) d s 1 q 1 1 g ( ) g [ ] c c 1 ( ) g 1 q g 1 1 2 q Kc ; ( ) 1 2 I 1 q 2 2 s 2 s 1 r r r 1 2 ( s 1 )( s 1 )( s 1 ) 1 2 3 D g c ( )( 1 ) s s 1 2 This is a PID controller
Department of Chemical Engineering I.I.T. Bombay, India Methods based on approximate models Generation of approximate models s Ke G ( s ) s 1 Method is a little approximate but is adequate for feedback controller design purposes. For more advanced control strategies, model accuracy is important.
Department of Chemical Engineering I.I.T. Bombay, India Approximate model tuning rules (valid for 0.1 < / < 1.0) Cohen-Coon I D Controller K c 1/K( / ) P - - 0.9/K( / ) 3.3 PI - 1.2/K( / ) 2.0 0.5 PID Direct Synthesis I D Controller K c / K( r + ) PI - ((2 + )/ +( /2) r /(2K( r + ) PID (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 Example of stability limits and stability margin Plant 2 G ( s ) ( s 1 )( s 2 )( s 3 ) Controller ( s 1 )( s 2 )( s 3 ) 2 K 0 c Sustained oscillation are observed at Kc=30 (call this K cu ) 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 P u =2 /w.
Department of Chemical Engineering I.I.T. Bombay, India Zeigler Nichols Stability margin based tuning I D Controller K c P 0.5 K cu - - PI 0.45K cu P u /1.2 - PID 0.6Kcu P u /2 P u /8 So, for the previous example, if a P controller were used, K c = 15; For a PI controller, K c =13.5 and I = 1.577; For a PID controller, K c =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 Stability limit based on sustained oscillations + y d y Plant controller u - 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 – 180 o , 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 – 180 o 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.
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