The proportional-integral-differential (PID)
controller is perhaps the most common controller in general
use. Most programmable logic controllers (PLCs)
support a variety of processes with this structure; for example, many
temperature, pressure, and force loops are implemented with PID
control.
PID is a structure that can be simplified by setting one or two of
the three gains to zero. For example, a PID controller with the
differential ("D") gain set to zero reduces to a PI controller.
This
series of six articles will explore the use of six variations of P, I,
and
D 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 must
weigh complexity against performance. PID +, the most complex of the
six controllers in this series, can accomplish anything the simpler
systems can do, but there is a cost.
More complex controllers require more capability to process, in the
form of either faster processors for digital controllers or more
components for analog controllers. Beyond that, more complex
controllers are more difficult to tune. The designer must decide how
much performance is worth paying for.
The focus in this chapter will be on digital controls, although
issues specific to analog controls are covered throughout. The basic
issues in control systems vary little between digital and analog
controllers.
For all control systems, gain and phase margins must be maintained,
and phase loss around the loop should be minimized. The significant
differences between the two controller types relate to which schemes
are easiest to implement in analog or digital components.
The controllers here are all aimed at controlling a
single-integrating plant. Note especially that the PID controller
discussed in this chapter is for a single-integrating plant, unlike a
PID position loop, which is for a double-integrating plant. A PID
position loop is fundamentally different from the classic PID loops
discussed here.
Tuning in
Throughout this series of articles, a single tuning procedure will be
applied to multiple controllers. The main goal is to provide a
side-by-side comparison of these methods. A consistent set of stability
requirements is placed on all of the controllers.
Of course, in industry, requirements for controllers vary from one
application to another. The requirements used here are representative
of industrial controllers, but designers will need to modify these
requirements for different applications.
The specific criteria for tuning will be as follows: In response to
a square wave command, the high-frequency zone (P and D) can overshoot
very little (less than 2%), and the low-frequency zone can overshoot up
to 15%.
Recognizing that few people have laboratory instrumentation that can
produce Bode plots, these tuning methods will be based on time-domain
measures of stability, chiefly overshoot in response to a square wave.
This selection was made even though it is understood that few
control systems need to respond to such a waveform. However, square
waves are the signals of choice in many cases for exposing marginal
stability; testing with gentler signals may allow marginal stability to
pass undetected.
Using zone-based tuning methods, each of the six controllers has
either one or two zones. The proportional and differential gains
combine to determine behavior in the higher zone and thus will be set
first, so the P and D gains must be tuned simultaneously. The integral
gain 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 control
law: the plant, the power converter, and the feedback filter. The lower
zone is limited primarily by the higher zone.
Note that sampling delays can be thought of as parts of these
processes; calculation delay and sample-and-hold delay can be thought
of as part of the plant and feedback delay as part of the feedback
filter.
The tuning in this series of articles will set the loop gains by
optimizing the response to the command. Higher loop gains will improve
command response and they will also improve the disturbance response.
Depending on the application, command or disturbance response may be
more important. However, command response is usually preferred for
determining stability, for a practical reason: Commands are easier to
generate in most control systems. Disturbance response is also an
important measure.
When tuning, the command should be as large as possible to maximize
the signal-to-noise ratio. This supports accurate measurements.
However, the power converter must remain out of saturation during these
tests.
For this series, the example systems are exposed only to the
relative quiet of numerical noise in the model; in real applications,
noise can be far more damaging to accurate measurements.
 |
| Figure
6-1. Experiment 6A, a P controller |
Using the Proportional Gain
Each of the six controllers in this series is based on a combination of
proportional, integral, and differential gains. Whereas the latter two
gains may be optionally zeroed, virtually all controllers have a
proportional gain.
Proportional gains set the boundaries of performance for the
controller. Differential gains can provide incremental improvements at
higher frequencies, and integral gains improve performance in the lower
frequencies. However, the proportional gain is the primary actor across
the entire range of operation.
P Control
The proportional, or "P," controller is the most basic controller. It
is simple to implement and
easy to tune. A P-control system is provided is
shown in Figure 6-1 above. The
command is provided by a square wave
feeding a digital signal analyzer (DSA). The error is formed as the
difference between command
and feedback.
That error is scaled by the single control law gain Kp to create the
command 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 feedback
must also pass through a sample-and-hold. The sample time for the
digital controller, set by the Live Constant "TSample," is 0.0005
seconds. The response vs. command is shown on the Live Scope at the
bottom left.
The chief shortcoming of the P-control law is that it allows DC
error and droops in the presence of fixed disturbances. Such
disturbances are ubiquitous in controls: Ambient temperature drains
heat, power supply loads draw DC current, and friction slows motion. DC
error cannot be tolerated in many systems, but where it can, the modest
P controller can suffice.
Next in Part 2: How to Tune a
Proportional Controller
Editor's Note:
Experiments 6A-6F
All the examples in this series of
articles were run on Visual
Mode1Q. Each of the six experiments, 6A-6F, models one of the six
methods, P, PI, PI+, PID, PID+, and PD, respectively.
These are models of digital systems,
with sample frequency defaulting to 2 kHz. If you prefer experimenting
with an analog controller, set the sample time to 0.0001 second, which
is so much faster than the power converter that the power converter
dominates the system, causing it to behave like an analog controller.
The default gains reproduce the results
shown in this series, but you can go further. Change the power
converter bandwidth and investigate the effect on the different
controllers.
Assume noise is a problem, reduce
the low-pass filter on the D gain (fD), and observe how this reduces
the benefit available from the derivative-based controllers (PID, PID+,
and PD). Adjust the power converter bandwidth and the sample time, and
observe the results.
This
series of articles was excerpted from Control
System Design Guide by George Ellis with the permission of the
publisher - Elsevier/Academic Books - and can be purchased online which
retains all copyrights.
George Ellis is senior scientist
at Danaher Motion. He has
designed and applied motin control systems for over 20 years and has
written for Machine Control Magazine, Control Engineering, Motion
Systems Design, Power Control and Intelligent Motion, EDN Magazine. In
addition to Control System Design Guide, he is also the author of
Observers in Control Systems (Academic Press).
