Module 4: PID Control
This article is contained in Scilab Control Engineering Basics study module, which is used as course material for International Undergraduate Program in Electrical-Mechanical Manufacturing Engineering, Department of Mechanical Engineering, Kasetsart University.
Module Key Study Points
- understand the basics of PID control
- learn the effects of 3 parameters to system response
- learn how response is degraded by integrator windup
- How to tune PID parameters by Ziegler-Nichols Frequency Domain method
PID stands for Proportional, Integral, and Derivative. It is a
standard control structure used successfully in many industrial
applications. Commercial PID controllers can be bought off the shelf and
installed in the system. After some setup and tuning the three PID
gains, and perhaps some additional parameters, the system is up and
running in no time. Of course, this means the control engineer is
certain the plant can be handled by the PID controller and understand
how to adjust the control parameters. Without such knowledge, it could
be a frustrating experience to her.
So, in this module we discuss some basics of PID control and focus on
how to analyze a feedback sysem, with Scilab, build an Xcos diagram
and simulate. For a detailed study, we recommend [1]. For those
interested in digital PID implementation, document [3] on this site
might be helpful. The e-book [4] provides more information on
independent robot joint control using PID.
Different forms of PID
The so-called “textbook” form of PID controller is described as follows
(1) 
This algorithm computes the control variable
as output, given the input
,
the error between the command and plant output. We see that the
control variable is a function of 3 terms: P (proportional to error), I
(time integral of error), and D (derivative of error), with
corresponding control parameters
(proportional gain),
(integral time), and
(derivative time), respectively. Taking Laplace transform of (1), we have
(2) 
with the transfer function of PID controller
(3) 
Another common structure of PID algorithm is represented by
(4) 
In this form, the controller gains are distributed to each of the PID terms separately, with its transfer function
(5) 
For our discussion of PID controller, we focus on the form (4) and
(5). Nevertheless, it is easy to convert between parameters of (1) and
(4) by the relations
(6) 
Ex. 1: To get started with Xcos simulation of a PID feedback system, download file
pid_feedback.zcos .
This launches an Xcos diagram in Figure 1. This diagram can easily be
constructed using Xcos standard palettes. It requires only a
step_function, clock_c (from Sources palette), PID and CLR blocks (from
Continuous time systems palette), SUMMATION (from Mathematical
operations palette), MUX (from Signal Routing palette), and CSCOPE (from
Sinks palette). The plant transfer function is a robot joint driven by
DC motor model from module 1, with
and
.
Figure 1 pid_feedback.zcos basic PID simulation diagram
Click on the Simulation Start button to see the step response. Try
adjusting the PID gains and plant parameters to see their effects on the
closed-loop system.
PID parameter tuning is well studied in control engineering. Here we give a brief guideline. The proportional gain
is a dominant quantity that normally has some nonzero value. The integral gain
helps eliminate steady state error but too high value could introduce overshoot and oscillation. The derivative gain
could help the response to reach steady state faster but could amplify
high frequency noise, and could affect stability if set too high.
When the plant model is not known, adjusting these 3 gains to achieve
good response could be problematic for an inexperienced user. Later we
discuss a tuning procedure. Commercial PID controller products usually
have auto-tuning functions for user convenience.
Ex. 2: To experiment with the tracking and disturbance attenuation performance of PID control, we setup a feedback diagram in Figure 2.
Figure 2 PID feedback diagram
which can be constructed as an Xcos model
pid_dist.zcos in Figure 3. The joint inertia and friction are set to
and
, respectively. The step disturbance at the plant input is active at t = 2 sec, with step value of 80.
Figure 3 pid_dist.zcos Xcos model for PID feedback with disturbance
Note that in this example, we have the luxury of knowing the exact
plant model. So it is possible to solve for a good set of PID gains. An
approach used in [2] turns off the
gain first and solve for
and
that yields a critical-damped system at chosen natural frequency. Then
is adjusted to eliminate steady-state error.
To elaborate, it is shown in [2] that when
is set to 0, the resulting second order closed-loop characteristic polynomial is described by
(7) 
which gives
(8) 
For example, suppose we select a critical-damped closed loop system
, with natural frequency
. From (8), this yields
and
. Table 1 below gives calculated P and D gain values for 3 selected natural frequencies.
 |
|
|
| 4 |
160 |
80 |
| 8 |
640 |
160 |
| 12 |
1440 |
240 |
Table 1 and gains for 3 different natural frequencies
Using these 3 sets of gain values, we simulate the step responses.
The results are shown in Figure 4. As expected, the higher the natural
frequency, the faster the response. We also see that the steady-state
error from step disturbance cannot be eliminated with the PD controller.
Figure 4 step responses of PD control with disturbance d = 80
To fix this problem, we need to turn on the
gain of PID controller. With the controller transfer function described by (5), the closed-loop system is now of third order
(9) 
with the characteristic polynomial
(10) 
Applying Routh-Hurwitz criteria to (10), it can be concluded that,
given all positive PID gains, the feedback system is stable if
(11) 
The same Xcos diagram in Figure 3 can be used by putting some positive integral gain in the PID block. Let us fix
and
, corresponding to natural frequency
rad/s. Figure 5 shows the responses from setting some values of
. We see that the steady-state error to command and disturbance is eliminated by increasing
, though higher value could introduce more overshoot.
Figure 5 step responses of PID control
Applying the bound (11) to this example yields
This can be verified by setting
. The oscillating response in Figure 6 shows that the closed-loop system is at the brink of stability.
Figure 6 closed-loop response at controller gains 
Effect from Saturation
Analysis of PID feedback control is often performed under the
assumption that the system is purely linear. In real devices, nonlinear
elements emerge at certain points in the feedback loop. Saturation is
one of them that could cause undesirable system response. This nonlinear
effect normally results from physical limits of signals and system
parameters, say, maximum motor torque and current. Another common
saturation is the input limit of servo amplifier. An industrial servo
amp might allow input voltage command in the range
volts, for example.
Effect from saturation is more pronounced when integral term is used
in PID. The response could have excessive overshoot due to error
accumulation in the integral term. This is known as
integrator windup. See [1] for more detail.
Ex. 3: To see an effect from input saturation, construct an Xcos model as in Figure 7, or download
pid_ilim.zcos.
It represents a comparison of two closed-loop systems, which are
basically the same except that the lower one has its input bounded
between
units.
Figure 7 pid_ilim.zcos Xcos diagram for Example 3
The controller gains are set at
. Running the simulation yields the step responses in Figure 8, and the
control variables (controller output) in Figure 9. We see that input
saturation causes significantly larger overshoot and also worsens
disturbance response.
Figure 8 effect of integrator windup on output response
Figure 9 controller output with plant input limit
Tuning the PID Parameters
As stated before, adjusting the PID gains from scratch to achieve a
good response may not be trivial. As a result, most commercial PID
controllers have functions to tune the 3 parameters automatically. This
is normally called “autotuning” feature. One of autotuning methods
suggested in the literature makes use of some relay feedback mechanisms,
which is closely related to a manual tuning scheme known as
Ziegler-Nichols Frequency Domain (ZNFD) method. Hence we discuss this ZNFD scheme in the last part of this module.
Note: The original ZNFD method
applies the “textbook” PID equation (1). But since (1) and (4) are
closely-related by (6), here we make a conversion so that standard PID
block in Xcos, which assumes the form (4), can be conveniently used.
To tune a PID controller manually by ZNFD method, follow this procedure
- turn off both the integral and derivative terms; i.e., setting
. So now the PID is left only with the proportional gain . We crank
up to the point that the closed-loop system starts to oscillate. At
this point, the plant output will swing in a constant sinusoid motion,
not growing and not dying out. Write this value down on a paper as 
.
- Measure the period of oscillation. Write it down as
.
- Use Table 2 to tune the controller parameters
| Controller Form |
|
|
|
| P |
 |
– |
– |
| PI |
 |
 |
– |
| PID |
 |
 |
 |
Table 2: suggested PID parameters from ZNFD method
From the above procedure, we could observe a drawback of the basic
ZNFD method: the system must oscillate at some point of gain setting
before it becomes unstable. This implies the Nyquist plot must cross the
negative real axis at some point. For the DC motor model we are using,
the gain margin is infinity. Hence it is difficult to find the
oscillating point and the
value.
There is some trick to get around such limitation, but this is beyond
the scope of this module. For our basic study, we simply modify the DC
motor robot joint by adding some dynamics to it, so that the Nyquis plot
of
crosses the negative real axis.
Ex. 4: Let us assume that the
PID output is passed through a LPF with cutoff frequency 500 Hz before
feeding the plant input. Using the filter design guide from module 1, we
have the LPF transfer function as
(12) 
To simulate, construct an Xcos model in Figure 10, or download
pid_znfd.zcos.
With this model, we are ready to perform the ZNFD tuning procedure
described above. Click on the PID block to set the I and D gains to
zero. Then start increasing the P gain until the output oscillates. This
is what you must do in a real application when the plant model is not
known.
Figure 10 pid_znfd.zcos Xcos diagram for ZNFD tuning method
Well, in this example we do know the plant model, don’t we? So, to save us some time, the
value that causes output oscillation can be found, for example, using
the method from Module 2. First from the loop transfer function
containing the plant, LPF, and PID controller with
.
-->kp = 1;
-->ki = 0;
-->kd = 0;
-->s=poly(0,'s');
-->P=syslin('c',1/(10*s^2+0.1*s)); // plant
-->H = syslin('c',1/(1+0.0003*s)); // filter
-->C=syslin('c',kp + ki/s + kd*s); // controller
-->L = C*H*P; // loop t.f
|
Then find the gain where the closed-loop pole touches the
axis.
->[kmax,s]=kpure(L)
s =
5.7735027i
kmax =
333.33433
|
resulting in
.
Verify this by setting the P gain to 333 in PID block. This yields
the desired oscillatory response as shown in Figure 11. The oscillation
period can be roughly measured from the plot to yield
sec.
Figure 11 oscillatory response with 
So from the last row of Table 2, ZNFD method suggests the three PID parameters as
Setting these gains gives the response in Figure 12.
Figure 12 response from PID gains suggested by ZNFD method
Note that the overshoot is quite excessive (70%). In a sense, ZNFD
just gives us some good values to start with. We may want to fine-tune
the PID gains to improve the response further. For example, decreasing
the Ki gain would bring the overshoot down. Figure 13 shows the response
from original PID gains (green), compared with the case when Ki is
reduced to 300 (blue), and 200 (magenta).
Figure 13 reduce overshoot in the response by decreasing
The ZNFD method could be explained using a Nyquist diagram in Figure 14. The diagram shows how a point
on the curve is moved related to the P, I , and D terms. Using the P
term alone, x could be moved in radial direction only. The I and D terms
help provide more freedom to move perpendicular to the radius. It can
be shown that by using ZNFD method, the critical point
is moved to the point
.
The distance of this point to the critical point is 0.5. So the
sensitivity peak is at least 2. This explains the high overshoot in the
step response.
Figure 14 How a point on Nyquist curve is moved with PID control
Summary
PID is a simple control structure that is still used widely in
various industrial applications. It is a close relative to the lead-lag
compensator explained in module 2, except that its functionality may be
more user-friendly. With some knowledge and practice, an engineer or
technician would be able to tune and operate a plant equipped with PID
control.
In this module we discuss the basics of PID feedback systems, with
emphasis on Xcos simulations to show how the responses are related to
three control parameters, as well as effect from input saturation that
could worsen the response. Without some good starting values, tuning the
PID gains can be cumbersome for a novice. So at the end, we mention a
manual tuning procedure known as the Ziegler-Nichols frequency domain
method. Some auto-tuning scheme of a commercial PID controller, such as
the relay feedback method, is based on the ZNFD manual tuning.
References
- K.J. Astrom and T.Hagglund. PID Controllers, 2nd ed., Instrument Society of America, 1995.
- M.W.Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control. John Wiley & Sons. 2006.
- V. Toochinda. Digital PID Controllers, a previous online document available at controlsystemslab.com. 2009.
- V.Toochinda. Robot Analysis and Control with Scilab and RTSX. Mushin Dynamics, 2014.
