Washington University ChE 433 Digital Process Control Laboratory PID Control Systems Lecture

Feedback Control Systems...deliberate guidance or manipulation is used to achieve a prescribed value of a variable The Process is a series of Transfer Functions Can be Active, requires power Or passive, self powered

Transfer Function Example: First Order, first order differential equation RC Circuit, Low Pass Filter In Laplase Tank Temperature Change

A Second Order System  Where  = damping factor  n = Undamped natural angular frequency = 2  f n f n is the natural frequency.

Under Damped  1.0, Critical Damped  = 1.0

In the field of servomechanisms studying the sinusoidal frequency response defines the systems’ behaviour. Differential equations can be written in a Block Diagram, a Transfer Function Steady state non-frequency dependent term K and a time variant term G(s)

Transfer Function Transfer Function Magnitude in Decibels Phase in Degrees

Delay time, Deadtime a Major Parameter in Process Control Systems The magnitude of pure dead time is 1.0 or 0.0 db. However the output is phase shifted relative to the input in degrees by

A single input single output (SISO) feedback control system

Stability SIN wave input at the set point input, results in a return signal at the summing point because of KG(s) and H(s). Return signal compared with the set point input. If the signal has arrived at this summing point, has a phase shift of 180 degrees and of sufficient magnitude, the input signal will be reinforced, a greater output and still greater signal. This process continues and the amplitude of oscillation becomes constant. If the set point reference input is removed the system will continue to oscillate. It is not even necessary to impress a sinusoid upon an unstable system to cause it to break into oscillation. Any small amplitude disturbance may bring about an oscillation. Instability, the Output/Set Point to become infinite, 1 + KG(s)H(s) = Zero, KG(s)H(s) = - 1. When this condition occurs, the gain will be infinite and the output will permit sustained oscillations.

The purpose or function of the “controller” is to adjust the process to a desired set point by modifying the total KG(s) term to insure that the closed loop system will be stable. The stability of the closed loop system can be studied and compensated for with the knowledge of the open loop transfer function. This is fortunate; not necessary to solve the entire closed loop equation to determine stability. One method; a semi log plot of the system gain and phase, Bode plot. Pair; gain and phase on the y-axis and a log frequency or angular frequency, omega, 2  f, on the x-axis.

Example:

First Order with Dead Time Step Response

PID Controller Proportional Integral Derivative In Process Control Input is the set point, SP. Output is the PV. SP - PV is the signal error, E. The signal between the controller output and the process is called output OUT, or manipulated variable, MV.

PID Controller Proportional Integral Derivative Single Input Single Output (SISO) Controller H is unity in most process controller

PID Controller Proportional Integral Derivative - The controller function block responds to the error signal. This response is called the control law. - With microprocessor circuits the digital implementation of this law is called a control algorithm.

PID Controller Proportional Integral Derivative The controller has modes of operation, define operational states. Auto - controller’s algorithm is functioning on the ERROR. Man - manual mode, directly set the controller’s output independent of the algorithm’s calculation. RSP - Remote Set Point will allow the output of one controller to set the set point of another controller. This is called cascade control. Supervisory and DDC, Direct Digital Control, are terms used where the control logic from a supervisory computer either sets the set point of the controller or directly operates the output.

PID Controller Proportional Integral Derivative Action - direct or reverse. Direct action means that the output increases with increasing error. Reverse the opposite. The action of a controller is selected based on the failure state of the final element, usually a control valve. Done to insure safe operation. This implies a sign term, + or -, to the overall controller algorithm.

PID Controller Proportional Integral Derivative Proportional only, P, control. Proportional - single static, non-dynamic gain inserted after the summer. P only controller, an offset between PV and SP. A manual bias setting can off set the error so PV = SP. Higher gain, smaller off set. Steady state value of Kc*Kc*G/(1+Kc*Kc*G*H) Kc being the controller gain. Proportional Band, PB: Gain = 100 / % PB

PID Controller Proportional Integral Derivative Integral inserts an integrator in the algorithm. Integral term called Reset; the controller output is “reset” therefore no offset between the SP and the PV. Integral only is used for constraint control and other advanced control algorithms. Reset units “repeats per minute” or “minutes” where per repeat is generally understood but frequently not written. The repeats term means that the error amount is “repeated” T times per minute. Consult controller manual. DeltaV units in “seconds” May not be possible to set the reset term to zero.

PID Controller Proportional Integral Derivative Derivative, D is called Rate, its contribution to the equation is that of inserting the derivative of the change. Implementing either - Derivative of the Error. - Derivative of the PV. Derivative of the error will cause the rate term to change due to a set point change, which can cause a large change in the output just due to set point change. The units of this rate are usually “minutes” or “seconds”.

PID Controller Proportional Integral Derivative Ideal controller algorithm, E is the error term: Real controller:

PID Controller Proportional Integral Derivative PID in DeltaV Drop and drag a PID block for the pallet. Controller algorithm selected under the Structure attribute. Check the controller type and action.

PID Controller Ziegler Nichols tuning settings Ultimate gain, Ku; Oscillation Period, Pu in minutes, where the closed loop begins to oscillate. The Ziegler Nichols, ZN controller tuning settings can be calculated: 1) For a proportional only controller:Gain = 0.5* Ku 2) For a proportional plus reset controller: Gain = 0.45*Ku Reset = 1.2/Pu 3) For a proportional plus reset plus rate controller: Gain = 0.6*KuReset = 2/Pu Rate = Pu/8 Reset units are repeats per time units. Rate units are in time units.

PID Controller Ziegler Nichols tuning settings ZN tuning method should only be used for linear non-self regulating processes. The PV response is different for set point changes than for disturbances. The ultimate period method yields better results because the latter requires finding slopes and is subject to graphical error. For three mode controller tuning settings, the controller responses' damped period is very close to the ultimate period. For three mode tuning settings the damping factor, , is 0.22 and the first peak occurs at 77.6 degrees not at 90. The so-called ¼ decay possible with a proportional only controller is more often 1/3 for PI and PID controllers.

PID Controller Ziegler Nichols tuning settings

PID Controller Proportional Integral Derivative Degree of stability of the open loop compensated network defined by two terms, phase and gain “margin” of stability. Define the amount of additional gain or phase that if added to the network would cause instability. The gain margin is that amount of gain required to product an unstable network when the phase is at –180 deg. Phase margin is 180 deg – phase Gain = 1.0

PID Controller Stable Settings Gain Margin > 8 db and Phase Margin > 30 deg

Time Domain vs. Frequency Domain Chemical or allied industries most frequently analyze systems’ transient behaviour. Therefore study the process in the time domain. Those who apply servomechanisms usually study these systems in the frequency domain.

Time Domain vs. Frequency Domain Regulators vs. Servos The primary function of a regulatory system is to maintain a constant value of the controlled variable or system output even in the event of severe load inputs. In processing industries most processes can be considered self regulating and have many first order time constants in series as well as a significant dead time between the time the manipulated variable is changed and any change is detected in the process. A servo-system is normally subjected to a continuously varying command signal or set point; its primary function, causing the output to follow the command signal.

Regulators vs. Servos Regulator Servo

Digital Filter - Used to filter hydraulic, process induced noise. - Should not be used to filter electrical noise! - The filter is an exponentially weighted moving average EWMA, of the current variable, PV current, and the past filtered signal, Pv f - Avoid using a large number, high filter time constants result in poor control performance. - Avoid exceeding 1.5 seconds filter time if possible.

Digital Filter Implementation The filtered signal, PV f, is an EWMA, exponentially weighted moving average, of the current variable and the past filtered signal: The weighting factor,, is calculated by knowing: Sampled time, t s Filter time constant, .

The universe of SISO loops can be separated into two classifications self-regulating and non self-regulating. Self regulating - These loops are such that when the controller is placed in manual, the process variable will go to some stable state Non Self regulating - These loops behave such that if the controller is placed in manual, the process variable will go to some saturated state

The PID Controller Controller Simulation Workshop Go to class Web site Download ACSL ctrl Workshop.zip Open zip file and follow instructions Extra credit-Lambda Tuning method of PI controller