The basics of control system design: Part 1 - Moving beyond PIDThe 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.
series of six articles will explore the use of six variations of P, I,
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.
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.
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
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).