Repeat the set of open loop simulations from exercise 9 but this time estimate the gain and phase angle in each case and record the results in the table below.
Period (s)
Frequency (Hz) Gain Gain ÷ by max. Gain Phase Angle (°) 0.5 2 1 1 2 0.5 3 0.333 4 0.25 5 0.2 7.5 0.133 10 0.1 15 0.067 20 0.05 100 0.01
You will need to minimise the period shown on the graph (using the slider at the top right) after the higher frequency simulations, in order to magnify the trace to estimate the phase angle. The gain can be calculated very easily in each case by obtaining the maximum and minimum flow values from the cyan trace, using the digitising cursor. The gain may then be calculated from the following equation:
Gain = (Maximum Flow Rate - Minimum Flow Rate) / 50%
Here the 50% represents the peak-to-peak value of the input, i.e. 80% - 30%. (The peak-to-peak ratio is identical to the amplitude ratio for the input and output signals of course). The only problem with this is that the input signal is quoted as a percentage and the output signal is measured in litres per minute. In order to normalise the calculated gain values in this situation we can divide all of the gain values by the maximum magnitude of the gain; which will be when the period is very large. For practical purposes it will be sufficient to estimate the gain when the period is 100 seconds or more and to divide all of the gain values by this figure. This explains the presence of the second gain column above.
The phase angle may be estimated in each case by observing how much time passes from the moment the input reaches a peak (or trough) until the output reaches the corresponding peak (or trough). If this time interval is called ∆∆T then the phase angle (in degrees) may be calculated from the following equation:
Phase Angle = (∆∆T/Period) * 360°
Estimates of the gain and phase angle will of necessity be fairly inaccurate at the highest frequencies due to the difficulty of acquiring accurate data from the traces.
Once the table above has been completed use the data in the second, forth and fifth columns to draw Bode Gain and Bode Phase plots for the flow loop. The graph below shows the sort of results you should achieve.
The open loop section within PCUSIM gives us an opportunity to introduce the subject of ‘system frequency response’ and the Bode plot method of representing it but this concept is more usually applied to servo control systems (as found in robotics, aero-engineering, vehicle suspension etc., etc.) rather than fluid process control systems. Normally Bode plots are drawn with a logarithmic frequency scale and the frequency is plotted in radians per second rather than hertz.
Glossary of Terms
Actuator Device by means of which control output effects process variable, e.g. pump, valve or heater.
Adc Analogue to digital converter.
Algorithm see control algorithm.
Amplitude For a sinusoidal waveform, half the peak-to-peak value. Analogue to digital
conversion
Operation by which analogue measurement values from real world processes are converted electronically to a digital format that can be manipulated by a microprocessor based controller.
Anti reset windup Feature available on many industrial controllers that eliminates reset windup (integral windup).
Antiphase Situation in which the maxima of an input signal occur
simultaneously with the minima of the output signal and vice versa Automatic control Regulation of a process variable by means of an algorithm
implemented by a controller that is usually microprocessor based, e.g. computer or PLC.
Batch volume control Control of supply of a specific volume of liquid at a specific temperature over a specific period of time.
Block diagram Visual method of describing a control loop based upon process, controller and feedback blocks together with signal paths and (a) summing junction(s).
Bode plots Graphs showing system response in terms of gain and phase angle plotted against frequency.
Closed loop Any control loop in which the error value is used by the controller algorithm to determine the control output.
Comma delineated file File format compatible with software packages such as Microsoft’s Excel and Mathworks’ MATLAB.
Continuous cycling method
Ziegler/Nichols tuning method that requires that a closed control loop can be made unstable by a proportional only controller. Control algorithm Rule or set of rules by which controller generates control output(s),
based upon the error signal in a closed loop system.
Control cycle Phrase sometimes used to refer to execution of a real world control experiment or a PCUSIM simulation.
Control loop System comprising process, sensor(s), error feedback and source of control output signal. If error feedback is used to calculate control output then it is a closed loop, alternatively it is an open loop.
Control output Signal generated by controller. Calculated from the error according to control algorithm in a closed loop.
Controller Device or software routine which applies a control algorithm to determine a control output from the error signal in a closed loop. Critical damping Damping such that measured value reaches setpoint as quickly as
possible without any overshoot.
(Damping ratio = 1. Usually the optimum situation is when damping ratio is about 0.7 that gives a single overshoot). D see derivative, derivative action, derivative term.
Damping Force(s) that oppose the tendency of a system or control loop to oscillate, e.g. friction, air resistance or the effect of controller. Decay Successive reduction in amplitude of control loop oscillations due
to damping force(s). Derivative,
Derivative action Derivative term
Third component in a PID or three-term control algorithm that takes account of the time rate of change of error. Derivative action is based upon the time rate of change of error multiplied by a constant called the derivative action time D.
Error Function of time defined as the difference between setpoint (SP) and measured value (MV).
Feedback Technique of using error signal to determine control output signal that causes error to be minimised in a well tuned system.
Flat Description of setpoint that remains at a certain fixed value until changed by human intervention.
Flow control Control of rate of flow of a fluid through an open or closed channel. Flowmeter Device for measuring rate of flow of fluid, the miniature turbine
type used on the PCU produces a pulsed signal which is processed electronically to determine flow rate in litres/minute.
Free integrator A concept from Laplace transform/transfer function theory that refers to the way certain devices integrate an input with respect to time thus obviating the requirement for integral action within any associated controller. E.g. a tank integrates net inward fluid flow to produce fluid level.
Frequency The rate of repetition of any periodic waveform, i.e. the number of cycles of that waveform that occur in a unit of time. Usually quoted in cycles per second, hertz (Hz) or radians per second.
Frequency response Overview of the way a system responds to input signals across the complete range of expected frequencies. Input signals include setpoint, noise and external perturbations. Often displayed as Bode graphs, i.e. gain and phase angle plotted against frequency.
Gain The ratio of the amplitude of the output signal to the amplitude of the input signal, i.e. output amplitude divided by input amplitude. I see integral, integral action, integral term.
In phase Situation in which maxima of input and output signals occur simultaneously and minima of input and output signals occur simultaneously.
Instability Characteristic of control loop that continues to oscillate with constant or increasing amplitude.
Integral, Integral action, Integral term
Second component in a PI, PID or three-term control algorithm that takes account of the recent history of the error. Integral action is calculated by dividing the net area under the error curve against time by the integral action time I.
Integral windup see reset windup
Ladder program Control program written in a PLC language known as ‘ladder logic’. Laplace transform Result of a mathematical operation that converts functions of time
e.g. input and output signals, to functions of the Laplace variable that is universally assigned the letter ‘s’. This allows more convenient mathematical manipulation for advanced work on control theory.
Level control Control of level of a fluid in a particular vessel.
Lightly damped Description of a system where the damping forces are less than they would be for critical damping. Lightly damped systems all oscillate and the lighter the damping the more cycles of oscillation which are produced.
Linearity Characteristic of a system that exhibits increases in its output that is proportional to increases in its input. E.g. sinusoidal input and output amplitudes in an open loop flow cycle.
Logic Relationships between digital (i.e. on/off) inputs and the required output signals programmed into the memory of a PLC or control computer.
Manual control Situation in which a human operator determines the control
output(s) based upon the required value and current measured value of the process variable
Measured value Function of time returned by the sensor and signal conditioning which represents the value of the process variable.
Mv See measured value
Noise Normally random non-periodic signal superimposed upon one or more of the signals in a control loop, having an amplitude which is small in comparison to the magnitude of the signals of interest. Non linearity Characteristic of a system that does not exhibit increases in its
output that is proportional to increases in its input. E.g. flow of current through a diode as voltage is increased from
-ve. To +ve.
Normalise Process of dividing all elements in a set of data by a chosen value, (sometimes the value of one particular element, perhaps the largest) in order to render the whole set more convenient for assimilation or for comparison with other data sets.
Off-line Beyond the particular time period pertaining to real-time control of a system or process.
Open loop control Any control loop in which the error value is not used by the controller to determine the control output.
Open loop response A description of the way a system responds to certain input signals such as sine waves, step changes or square waves etc. These input signals may originate from manual/human intervention or from an open loop control output. Obtaining an open loop response to a series of well-chosen sine waves may be used to determine the frequency response in the form of Bode graphs.
Optimal tuning Tuning of the PID controller to produce the best possible results for a given control system.
Oscillation, Oscillatory
Periodic pattern, almost always sinusoidal, observed in the response of a system.
Overshoot Maximum amount by which measured value exceeds final steady state value of process variable. Often expressed as percentage of final steady state value.
Pb see proportional band.
Pcu see Process Control Unit.
Pcu trend update utility Software utility for converting old DOS based PCU trend files to PCUSIM compatible format.
Period, Periodic time
Duration of one complete cycle of oscillation i.e. interval between alternate crossings of the final steady state value.
Pg see proportional gain.
Phase angle In the context of sinusoidal input and output signals, the amount by which the input leads the output expressed in degrees or radians. PID control Automatic control based upon an algorithm that includes
proportional, integral and derivative contributions to the final control output.
Plc see Programmable Logic Controller
Process control Phrase often (though not always) assigned to control of fluid flow rate, temperature, level, batch volume, acidity, pressure etc. Process control unit Bytronic hardware system for illustrating and teaching PID control
upon which all PCUSIM simulations are based. Process reaction curve
method
Ziegler/Nichols tuning method that requires that data from an open loop response to a step change be acquired.
Process variable The feature of a process loop which is actually being controlled, e.g. temperature, flow rate, level etc.
Programmable logic controller
Microprocessor based device used widely in industry to control digital and analogue processes. Essentially a specialised ‘hardened’ computer with its own (often idiosyncratic) programming language. Proportional,
Proportional term
First component in a PI, PID or three-term control algorithm that generates a component of the control output which is directly proportional to the magnitude of the error.
Proportional band The error band where the control output is between 0% and 100%. Proportional band equals 100 divided by the proportional gain. Proportional gain A constant multiplied by the error value to give a control output.
Proportional offset Steady state discrepancy between setpoint and measured value that is characteristic of all systems which use proportional only control and which do not have an intrinsic ‘free integrator’.
Ramp Description of setpoint which changes up or down gradually at a fixed rate.
Rate of flow Velocity at which fluid moves along an open or closed channel, usually expressed in litres per minute.
Real-time Description applied to computerised operation that occurs in step with some real world process. E.g. display of process reaction curves on remote SCADA computer. Used to make a distinction from off-line presentation.
Reset windup Accumulation of controller integral action ‘debt’ after a saturation episode. This offsets measured value from setpoint for longer than the physics of the process would strictly require.
Response, Response curve
Shape of the trace produced by plotting process variable measured value against time.
Rise time Time taken for the response to increase from 10% of its final steady state value to 90% of its final steady state value.
Sample time Period between successive samples of measured value digitised by ADC.
Sampling frequency Rate at which samples of measured value are digitised by ADC. Saturation Description for the situation when, even with 100% control output,
the measured value cannot be brought to the setpoint.
Sawtooth Description of setpoint which changes up and/or down periodically, gradually and at a fixed rate.
Scada see Supervisory Control and Data Acquisition
Servo control Control of motor and solenoid actuated apparatus etc. Used to make a distinction from fluid process control.
Setpoint Desired value of the process variable.
Setpoint options PCUSIM facility to change the type of setpoints available accessed via preferences menu.
Settling time Time taken for the process variable measured value to reach its final steady state value within some specified tolerance.
Sp see setpoint.
Stability Characteristic of control loop in which all oscillations ultimately decay completely after a step change.
Steady state The condition of a control loop once all transients have decayed completely, after a step change. The steady state might be a fixed measured value or an oscillation with a fixed amplitude.
Step, Step change
Description of setpoint which changes up and/or down instantaneously.
Summing junction Point in a control loop block diagram symbolised by a crossed circle where the measured value feedback is subtracted from the setpoint to provide an error signal input for the controller.
Supervisory Control and Data Acquisition
Use of (possibly remote) computer(s) to supervise one or more PLCs that are controlling a local process. The computer can acquire data from the PLC and display it as a real-time graph or store it for off-line processing. It can also send new setpoint and/or PID values to the PLC.
System frequency response
see frequency response.
Temperature control Control of temperature of a fixed body of fluid in a vessel. Three term control Automatic control by means of a PID control algorithm.
Time constant Parameter that characterises the speed of response of a control loop or an element within a control loop.
Trace Graph of process variable measured value, plotted against time. Transfer function Mathematical model (or description) of control loop block (e.g.
process or controller). Specifically the ratio of Laplace transform of output signal to Laplace transform of test input signal.
Transport delay Period during which there is no change in the process variable after a step change has been made to the set point (in closed loop) or the open loop control output.
Trend Graph of process variable measured value plotted against time in PCUSIM. This term is used to refer to complete graph once simulation has stopped, or when retrieving from hard disk. Tuning Selection of PID terms to be used within control algorithm. Waveform The shape of one cycle of a periodic signal.
Wavelength The length of one cycle of a periodic signal.
Ziegler/Nichols tuning A pair of techniques for estimating appropriate PID values for a given control loop. See continuous cycling method and process reaction curve method.