CHE 185 – PROCESS CONTROL AND DYNAMICS PID CONTROLLER FUNDAMENTALS

CLOSED LOOP COMPONENTS GENERAL DEFINITIONS OPEN LOOPS ARE MANUAL CONTROL FEEDBACK LOOPS ARE CLOSED EXAMPLE P&ID FOR FEEDBACK CONTROL LOOP

CLOSED LOOP COMPONENTS GENERAL BLOCK DIAGRAM FOR FEEDBACK CONTROL LOOP (FIGURE 7.2.1 FROM TEXT)

CLOSED LOOP COMPONENTS OVERALL TRANSFER FUNCTION

TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP CONSIDER THE RESPONSE TO A DISTURBANCE WITH CONSTANT S/P (Ysp(s) = 0) REGULATORY CONTROL OR DISTURBANCE REJECTION THIS REPRESENTS A PROCESS AT STEADY STATE RESPONDING TO BACKGROUND DISTURBANCES

TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP CONSIDER THE SETPOINT RESPONSE WITH NO DISTURBANCE (D(s) = 0) SETPOINT TRACKING OR SERVO CONTROL THIS MODEL REPRESENTS THE SYSTEM RESPONSE TO A S/P ADJUSTMENT

TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP GENERALIZATIONS REGARDING THE FORM OF THE TRANSFER FUNCTIONS THE NUMERATOR IS THE PRODUCT OF ALL TRANSFER FUNCTIONS BETWEEN THE INPUT AND THE OUTPUT THE DENOMINATOR IS EQUAL TO THE NUMERATOR + 1

TYPICAL TRANSFER FUNCTIONS FOR A FEEDBACK LOOP CHARACTERISTIC EQUATION oBTAINED BY SETTING THE DENOMINATOR = 0 ROOTS FOR THIS EQUATION WILL BE: OVERDAMPED LOOP COMPLEX ROOTS, FOR AN OSCILLATORY LOOP AT LEAST ONE REAL POSITIVE ROOT FOR AN UNSTABLE LOOP

Feedback Control Analysis The loop gain (KcKaKpKs) should be positive for stable feedback control. An open-loop unstable process can be made stable by applying the proper level of feedback control.

Characteristic Equation Example Consider the dynamic behavior of a P-only controller applied to a CST thermal mixer (Kp=1; τp=60 sec) where the temperature sensor has a τs=20 sec and τa is assumed small. Note that Gc(s)=Kc.

Characteristic Equation Example- closed loop poles When Kc =0, poles are -0.05 and -0.0167 which correspond to the inverse of τp and τs. As Kc is increased from zero, the values of the poles begin to approach one another. Critically damped behavior occurs when the poles are equal. Underdamped behavior results when Kc is increased further due to the imaginary components in the poles.

PID ALGORITHM - POSITION FORM ISA POSITION FORM FOR PID: FOR PROPORTIONAL ONLY

Definition of Terms e(t) - the error from setpoint [e(t)=ysp-ys]. Kc - the controller gain is a tuning parameter and largely determines the controller aggressiveness. τI - the reset time is a tuning parameter and determines the amount of integral action. τD - the derivative time is a tuning parameter and determines the amount of derivative action.

PID Controller Transfer Function

PID ALGORITHM - POSITION FORM FOR PROPORTIONAL/INTEGRAL: FOR PROPORTIONAL/DERIVATIVE

PID ALGORITHM - POSITION FORM TRANSFER FUNCTION FOR PID CONTROLLER:

PID ALGORITHM - POSITION FORM DERIVATIVE KICK: RESULTS FROM AN ERROR SPIKE (INCREASE IN 𝑑𝑒(𝑡) 𝑑𝑡 ) WHEN A SETPOINT CHANGE IS INITIATED CAN BE ELIMINATED BY REPLACING THE CHANGE IN ERROR WITH A CHANGE IN THE CONTROLLED VARIABLE −𝑑 𝑦 𝑠 (𝑡) 𝑑𝑡 IN THE PID ALGORITHM RESULTING EQUATION IS CALLED THE DERIVATIVE-ON-MEASUREMENT FORM OF THE PID ALGORITHM

DIGITAL VERSIONS OF THE PID ALGORITHM DIGITAL CONTROL SYSTEMS REQUIRE CONVERSION OF ANALOG SIGNALS TO DIGITAL SIGNALS FOR PROCESSING. DIGITAL VERSION OF THE PREVIOUS EQUATION IN DIGITAL FORMAT BASED ON A SINGLE TIME INTERVAL, Δt: YIELDS THE VELOCITY FORM OF THE PID ALGORITHM

DIGITAL VERSIONS OF THE PID ALGORITHM FOR INTEGRATION OVER A TIME PERIOD, t, WHERE n = t/Δt:

DIGITAL VERSIONS OF THE PID ALGORITHM PROPORTIONAL KICK RESULTS FROM THE INITIAL RESPONSE TO A SETPOINT CHANGE CAN BE ELIMINATED IN THE VELOCITY EQUATION BY REPLACING THE ERROR TERM IN THE ALGORITHM WITH THE SENSOR TERM

First Order Process with a PI Controller Example

PI Controller Applied to a Second Order Process Example

PROPORTIONAL ACTION USES A MULTIPLE OF THE ERROR AS A SIGNAL TO THE CONTROLLER, CONTROLLER GAIN, HAS INVERSE UNITS TO PROCESS GAIN

Proportional Action Properties Closed loop transfer function base on a P-only controller applied to a first order process. Properties of P control Does not change order of process Closed loop time constant is smaller than open loop τp Does not eliminate offset.

P-only Control Offset

Proportional Response Action with a PI Controller

PROPORTIONAL CONTROL RESPONSE OF FIRST ORDER PROCESS TO STEP FUNCTION OPEN LOOP - NO CONTROL CLOSED LOOP - PROPORTIONAL CONTROL

PROPORTIONAL CONTROL PROPORTIONAL CONTROL MEANS THE CLOSED SYSTEM RESPONDS QUICKER THAN THE OPEN SYSTEM TO A CHANGE. OFFSET IS A RESULT OF PROPORTIONAL CONTROL. AS t INCREASES, THE RESULT IS:

Integral Action The primary benefit of integral action is that it removes offset from setpoint. In addition, for a PI controller all the steady-state change in the controller output results from integral action.

INTEGRAL ACTION WHERE PROPORTIONAL MODE GOES TO A NEW STEADY-STATE VALUE WITH OFFSET, INTEGRAL DOES NOT HAVE A LIMIT IN TIME, AND PERSISTS AS LONG AS THERE IS A DIFFERENCE. INTEGRAL WORKS ON THE CONTROLLER GAIN INTEGRAL SLOWS DOWN THE RESPONSE OF THE CONTROLLER WHEN PRESENT WITH PROPORTIONAL

INTEGRAL ACTION INTEGRAL ADDS AN ORDER TO THE CONTROL FUNCTION FOR A CLOSED LOOP FOR THE FIRST ORDER PROCESS WITH PI CONTROL, THE TRANSFER FUNCTION IS: WHERE AND

Derivative Action Properties THE DERIVATIVE MODE RESPONDS TO THE SLOPE THIS MODE AMPLIFIES SUDDEN CHANGES IN THE CONTROLLER INPUT SIGNAL - INCREASES CONTROLLER SENSITIVITY

Derivative Action Properties DERIVATIVE MODE CAN COUNTERACT INTEGRAL MODE TO SPEED UP THE RESPONSE OF THE CONTROLLER. DERIVATIVE DOES NOT REMOVE OFFSET IMPROPER TUNING CAN RESULT IN HIGH-FREQUENCY VARIATION IN THE MANIPULATED VARIABLE 7.6 DOES NOT WORK WELL WITH NOISY SYSTEMS

Derivative Action Properties Properties of derivative control: Does not change the order of the process Does not eliminate offset Reduces the oscillatory nature of the feedback response Closed loop transfer function for derivative-only control applied to a second order process.

Derivative Action Response for a PID Controller

Derivative Action The primary benefit of derivative action is that it reduces the oscillatory nature of the closed-loop response.