The proportional-integral-differential (PID)controller is perhaps the most common controller in generaluse. Most programmable logic controllers (PLCs)support a variety of processes with this structure; for example, manytemperature, pressure, and force loops are implemented with PIDcontrol.
PID is a structure that can be simplified by setting one or two ofthe three gains to zero. For example, a PID controller with thedifferential (“D”) gain set to zero reduces to a PI controller.
Thisseries of six articles will explore the use of six variations of P, I,andD gains: Proportional Control, Proportional-Integral Control,Psueodo-derivative feedback with feed-forwared (PDFF), PID control,PID+ control, and Proportional Derivative Control.
When choosing the controller for an application, the designer mustweigh complexity against performance. PID +, the most complex of thesix controllers in this series, can accomplish anything the simplersystems can do, but there is a cost.
More complex controllers require more capability to process, in theform of either faster processors for digital controllers or morecomponents for analog controllers. Beyond that, more complexcontrollers are more difficult to tune. The designer must decide howmuch performance is worth paying for.
The focus in this chapter will be on digital controls, althoughissues specific to analog controls are covered throughout. The basicissues in control systems vary little between digital and analogcontrollers.
For all control systems, gain and phase margins must be maintained,and phase loss around the loop should be minimized. The significantdifferences between the two controller types relate to which schemesare easiest to implement in analog or digital components.
The controllers here are all aimed at controlling asingle-integrating plant. Note especially that the PID controllerdiscussed in this chapter is for a single-integrating plant, unlike aPID position loop, which is for a double-integrating plant. A PIDposition loop is fundamentally different from the classic PID loopsdiscussed here.
Throughout this series of articles, a single tuning procedure will beapplied to multiple controllers. The main goal is to provide aside-by-side comparison of these methods. A consistent set of stabilityrequirements is placed on all of the controllers.
Of course, in industry, requirements for controllers vary from oneapplication to another. The requirements used here are representativeof industrial controllers, but designers will need to modify theserequirements for different applications.
The specific criteria for tuning will be as follows: In response toa square wave command, the high-frequency zone (P and D) can overshootvery little (less than 2%), and the low-frequency zone can overshoot upto 15%.
Recognizing that few people have laboratory instrumentation that canproduce Bode plots, these tuning methods will be based on time-domainmeasures of stability, chiefly overshoot in response to a square wave.
This selection was made even though it is understood that fewcontrol systems need to respond to such a waveform. However, squarewaves are the signals of choice in many cases for exposing marginalstability; testing with gentler signals may allow marginal stability topass undetected.
Using zone-based tuning methods, each of the six controllers haseither one or two zones. The proportional and differential gainscombine to determine behavior in the higher zone and thus will be setfirst, so the P and D gains must be tuned simultaneously. The integralgain and a command filter, which will be presented in due course,determine behavior in the lower zone.
The higher zone is limited by the control loop outside the controllaw: the plant, the power converter, and the feedback filter. The lowerzone is limited primarily by the higher zone.
Note that sampling delays can be thought of as parts of theseprocesses; calculation delay and sample-and-hold delay can be thoughtof as part of the plant and feedback delay as part of the feedbackfilter.
The tuning in this series of articles will set the loop gains byoptimizing the response to the command. Higher loop gains will improvecommand response and they will also improve the disturbance response.
Depending on the application, command or disturbance response may bemore important. However, command response is usually preferred fordetermining stability, for a practical reason: Commands are easier togenerate in most control systems. Disturbance response is also animportant measure.
When tuning, the command should be as large as possible to maximizethe signal-to-noise ratio. This supports accurate measurements.However, the power converter must remain out of saturation during thesetests.
For this series, the example systems are exposed only to therelative quiet of numerical noise in the model; in real applications,noise can be far more damaging to accurate measurements.
|Figure6-1. Experiment 6A, a P controller|
Using the Proportional Gain
Each of the six controllers in this series is based on a combination ofproportional, integral, and differential gains. Whereas the latter twogains may be optionally zeroed, virtually all controllers have aproportional gain.
Proportional gains set the boundaries of performance for thecontroller. Differential gains can provide incremental improvements athigher frequencies, and integral gains improve performance in the lowerfrequencies. However, the proportional gain is the primary actor acrossthe entire range of operation.
The proportional, or “P,” controller is the most basic controller. Itis simple to implement andeasy to tune. A P-control system is provided isshown in Figure 6-1 above . Thecommand is provided by a square wavefeeding a digital signal analyzer (DSA). The error is formed as thedifference between commandand feedback.
That error is scaled by the single control law gain Kp to create thecommand to the power converter. The command is clamped (here, to±20) and then fed to a power converter modeled by a 500-Hz,two-pole low-pass filter with a damping ratio of 0.7.
The plant is a single integrator with a gain of 500. The feedbackmust also pass through a sample-and-hold. The sample time for thedigital controller, set by the Live Constant “TSample,” is 0.0005seconds. The response vs. command is shown on the Live Scope at thebottom left.
The chief shortcoming of the P-control law is that it allows DCerror and droops in the presence of fixed disturbances. Suchdisturbances are ubiquitous in controls: Ambient temperature drainsheat, power supply loads draw DC current, and friction slows motion. DCerror cannot be tolerated in many systems, but where it can, the modestP controller can suffice.
Next in Part 2: How to Tune aProportional Controller
Editor's Note:Experiments 6A-6F
All the examples in this series ofarticles were run on VisualMode1Q. Each of the six experiments, 6A-6F, models one of the sixmethods, P, PI, PI+, PID, PID+, and PD, respectively.
These are models of digital systems,with sample frequency defaulting to 2 kHz. If you prefer experimentingwith an analog controller, set the sample time to 0.0001 second, whichis so much faster than the power converter that the power converterdominates the system, causing it to behave like an analog controller.
The default gains reproduce the resultsshown in this series, but you can go further. Change the powerconverter bandwidth and investigate the effect on the differentcontrollers.
Assume noise is a problem, reducethe low-pass filter on the D gain (fD), and observe how this reducesthe benefit available from the derivative-based controllers (PID, PID+,and PD). Adjust the power converter bandwidth and the sample time, andobserve the results.
Thisseries of articles was excerpted from ControlSystem Design Guide by George Ellis with the permission of thepublisher – Elsevier/Academic Books – and can be purchased online whichretains all copyrights.
George Ellis is senior scientistat Danaher Motion. He hasdesigned and applied motin control systems for over 20 years and haswritten for Machine Control Magazine, Control Engineering, MotionSystems Design, Power Control and Intelligent Motion, EDN Magazine. Inaddition to Control System Design Guide, he is also the author ofObservers in Control Systems (Academic Press).