CHE 185 – PROCESS CONTROL AND DYNAMICS FREQUENCY RESPONSE ANALYSIS

Frequency Response Analysis Is the response of a process to a sinusoidal input Considers the effect of the time scale of the input. Important for understanding the propagation of variability through a process. Important for terminology of the process control field. But it is NOT normally used for tuning or design of industrial controllers.

Frequency Response Analysis Process Exposed to a Sinusoidal Input.

Frequency Response Analysis Key Components – INPUT FREQUENCY, ampLitude ratio, phase angle.

Frequency Response Analysis Effect of Frequency on Ar and Φ PEAK TIME DIFFERENCE Φ= 𝜔 ∆ 𝑡 𝑝 2𝜋 360 °

Frequency graphics Bode plot of Ar and Φ versus frequency ω

BODE STABILITY PLOT BASIS OF BODE PLOT IS A MEASURE OF RELATIVE AMPLITUDE AND PHASE LAG BETWEEN A REGULAR (SINUSOIDAL) SET POINT CHANGE AND THE OUTPUT SIGNAL THIS TECHNIQUE INDICATES STABILITY OF THE SYSTEM THE ANALYSIS IS COMPLETED WITH AN OPEN LOOP

BODE GENERATION Direct excitation of process. Ways to Generate Bode Plot INCLUDE: Direct excitation of process. Combine transfer function of the process with sinusoidal input. Substitute s=i w into Gp(s) and convert into real and imaginary components which yield Ar(w) and f(w). SEE APPLICATION METHOD IN EXAMPLES 11.1 AND 11.2 Apply a pulse test.

BODE STABILITY PLOT A SINUSOIDAL SETPOINT IS SENT TO THE LOOP AFTER THE SYSTEM REACHES STEADY STATE, THERE IS A LAG, CALLED THE PHASE LAG, BETWEEN THE AMPLITUDE PEAK OF THE INLET SIGNAL AND THE AMPLITUDE PEAK OF THE OUTLET SIGNAL THE FREQUENCY IS ADJUSTED SO THE PHASE LAG OF THE OUTLET SIGNAL IS 180° BEHIND THE INPUT SIGNAL

BODE STABILITY PLOT THE RESULTS ARE THEN APPLIED TO A CLOSED LOOP THE SETPOINT IS CHANGED TO A CONSTANT VALUE SINCE THE ERROR SIGNAL IS 180° OUT OF PHASE AND IS NEGATIVE RELATIVE TO THE INPUT SIGNAL, IT REINFORCES THE PREVIOUS ERROR SIGNAL

BODE STABILITY PLOT THE AMPLITUDE OF THE ERROR SIGNAL BECOMES THE OTHER FACTOR IF THE AMPLITUDE OF THE ERROR SIGNAL TO THE AMPLITUDE OF THE ORIGINAL SINUSOID SETPOINT, CALLED THE AMPLITUDE RATIO, IS LESS THAN ONE THEN THE ERROR WILL DECAY TO ZERO OVER TIME, IF THE AMPLITUDE RATIO WAS EQUAL TO ONE, A PERMANENT STANDING WAVE WILL RESULT IF THE AMPLITUDE RATIO WAS GREATER THAN ONE, THE ERROR WILL GROW WITHOUT LIMIT.

BODE STABILITY PLOT BODE’S STABILITY CRITERION SAYS: WHEN THE AMPLITUDE RATIO IS LESS THAN ONE, THE SYSTEM IS STABLE WHEN THE AMPLITUDE RATIO IS GREATER THAN ONE, THE SYSTEM IS UNSTABLE THE AMPLITUDE RATIO IS DEFINED AS: WHERE ar REFERS TO AMPLITUDE AS SHOWN IN FIGUREs 11.2.1 and 11.2.2 IN THE TEXT AND ω IS THE FREQUENCY OF THE SINUSOID

BODE STABILITY PLOT BODE PLOTS GENERAL TECHNIQUE TO PLOT THESE ARE SHOWN FOR FOPDT PROCESSES IN FIGURES 11.2.2 AND 11.3.2 A SECOND ORDER PLOT IS FIGURE 11.3.3 GENERAL TECHNIQUE TO PLOT Write the transfer function in proper form (unit value for lowest order term in denominator) Separate the transfer function into parts based on poles and zeros Draw bode diagram for each part Sum the parts to get the final plot

Bode Plot creation example (http://lpsa. swarthmore Transfer function in proper form 𝐻 𝑠 = 100 𝑠+30 or 𝐻 𝑠 =3.3 1 𝑠 30 +1 Parts are based on pole at s=30 and constant of 3.3 Pole plot is constant 0 db up to break ω, then drops off Constant has value of 10.4 db

Bode Plot creation example (http://lpsa. swarthmore Function with real poles and zeros

Bode Plot creation example (http://lpsa. swarthmore Function with real poles and zeros

Bode Plot creation example (http://lpsa. swarthmore Function with pole at origin

Bode Plot creation example (http://lpsa. swarthmore Function with repeated real poles, negative constant

Bode Plot creation example (http://lpsa. swarthmore Function with complex conjugate poles

Bode Plot creation example (http://lpsa. swarthmore Function with multiple poles at origin, complex conjugate zeros

Bode Plot creation example (http://lpsa. swarthmore Function with multiple poles at origin, complex conjugate zeros

Bode Plot creation example (http://lpsa. swarthmore Function with time delay

BODE STABILITY PLOT Developing a Bode Plot from the Transfer Function

BODE STABILITY PLOT Derivation for a First Order Process

BODE STABILITY PLOT Properties of Bode Plots

BODE STABILITY PLOT Bode Plot of Complex Transfer Functions Break transfer function into a product of simple transfer functions. Identify Ar(ω) and Φ(ω) of each simple transfer function from Table 11.1. Combine to get Ar(ω) and Φ(ω) for complex transfer function according to properties. Plot results as a function of ω.

BODE STABILITY PLOT BODE PLOTS CAN BE PLOTTED FROM TRANSFER FUNCTIONS WE CAN SET UP THE TRANSFER FUNCTION: Y(s) – Gp(s)C(s) WHERE C(s) IS THE SINUSOIDAL INPUT AN INVERSE LaPLACE TRANSFORM OF THE RESULT THEN PROVIDES A TIME FUNCTION

BODE STABILITY PLOT TAKING THIS TO A LIMIT TO ELIMINATE TRANSIENTS THAT WILL DECAY LEAVES THE STANDING WAVE FUNCTION THIS CAN BE USED TO EVALUATE Ar AS A FUNCTION OF ω AND φ AS A FUNCTION OF ω TABLE 11.1 PROVIDES FUNCTIONS TO CALCULATE Ar AND φ FOR A NUMBER OF COMMON TRANSFER FUNCTIONS

BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN THE BODE STABILITY CRITERION IS EVALUATED AT THE POINT WHERE φ IS EQUAL TO -180°. THE FREQUENCY AT THIS POINT IS CALLED THE CRITICAL FREQUENCY

BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN THE VALUE OF Ar CALCULATED AT THE CRITICAL FREQUENCY, Ar* DETERMINES THE PROCESS STABILITY THIS IS EXPRESSED AS THE GAIN MARGIN: WHEN GM > 1, THE SYSTEM IS STABLE

BODE STABILITY PLOT GAIN MARGIN AND PHASE MARGIN THE PHASE MARGIN IS THE VALUE OF THE PHASE ANGLE AT THE POINT WHERE Ar = 1 AND IS RELATIVE TO THE PHASE ANGLE OF -180°: (EQUATION 11.3.2) THE FREQUENCY WHERE THIS CONDITION OCCURS IS CALLED THE CROSSOVER FREQUENCY

BODE STABILITY PLOT PULSE TEST THIS IS AN OPEN LOOP TEST USED TO OBTAIN THE VALUES NECESSARY TO CREATE A BODE PLOT RESULTS COMPARE THE AMPLITUDE AND THE DURATION TIMES FOR THE INPUT AND OUTPUT VALUES FOR AN OPEN LOOP. THESE ARE USED WITH EQUATIONS 11.4.1 THROUGH 11.4.8 TO OBTAIN THE BODE PLOT

BODE STABILITY PLOT PULSE TEST EXAMPLE

BODE STABILITY PLOT Developing a Pulse Test Process Transfer Function

BODE STABILITY PLOT Limitations of Transfer Functions Developed from Pulse Tests They require an open loop time constant to complete. Disturbances can corrupt the results. Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.

NYQUIST DIAGRAM PULSE TEST COMBINES THE VALUE OF Ar AND φ ON A SINGLE DIAGRAM OTHERWISE IT HAS NO ADVANTAGE OVER THE BODE PLOTS

Closed Loop Frequency Response REFERENCE FIGURE 11.6.1

Example of a Closed Loop Bode Plot REFERENCE FIGURE 11.6.2

Analysis of Closed Loop Bode Plot REFERENCE FIGURE 11.6.2 At low frequencies, the controller has time to reject the disturbances, i.e., Ar is small. At high frequencies, the process filters (averages) out the variations and Ar is small. At intermediate frequencies, the controlled system is most sensitive to disturbances. The peak frequency indicates the frequency for which a controller is most sensitive.