PID_ct.sq
Continuous-time PID controller
PID controllers, or proportional-integral-derivative controllers, are
probably the most popular kind of linear single-input single-output
controllers. This is justified by their simplicity and their effectiveness
for a large class of systems. Taking as input the difference between the
desired set-point r(t) and the measured system output
y(t) ("error"
e(t)=r(t)-y(t)), they have three terms
with easy-to-understand effects which are added up, and three parameters to
adjust their weights:
- a proportional term (the larger the error, the larger the control signal
to reduce it);
- an integral term (if a nonzero control signal is required to cancel out
the error, the control signal is increased until the error vanishes);
- a derivative term (the evolution of the error is anticipated to increase
damping).
Weights can be specified either separately for the three terms, or as
a global gain kp and two time values
Ti and Td which do not
depend on the gain of the
system. PID_ct.sq uses the latter parameterization. The control signal
u(t) is

The transfer function of the controller
K(s) = U(s)/E(s), where
U(s) and E(s) are the Laplace
transforms of y(t) and e(t),
respectively, is

Translating the conceptual simplicity of the PID into an effective design
is not always straightforward. PID_ct.sq displays the graphics where common
specifications can be checked;
you can manipulate the PID parameters, the
controller gain kp in the Bode, Nyquist, or root locus
diagram, or the
time values of the integrator and the derivator in the Bode, root locus,
or open-loop poles diagram.

For set-point tracking, filtering the same way the measured output and
the set-point by considering only the error e(t)=y(t)-r(t)
does not give a good transient behavior when the set-point is discontinuous.
The set-point is usually not differentiated. In addition, the proportional
term of the controller kp applied to the set-point
can be reduced by a factor b smaller than 1. A third
common improvement is to filter the derivative term to limit the amplification
of noise at high frequencies (this is actually required to have a causal
controller); the filter is parameterized with a number
N, typically between 10 and 20, which is the bandwidth
of the effect of the derivator term. In the Laplace domain, the control signal is

Figures
The figures are the same as those defined for RST_ct.sq, except for
the Open-Loop Zeros and Poles and the Closed-Loop Poles which are not defined.
Settings
The System, Sampling Period, method for converting to digital controller,
and Damping Specification have the same effect as the corresponding menu
entries defined in RST_ct.sq. Two new entries are defined.
PID Coefficients
The three parameters of the PID (kp,
Ti and ) can be edited
in a dialog box. For P, PI, or PD controllers, set the parameter of the
missing component to the empty matrix [].
No Derivator On Reference
When the input of the PID controller is the error between the set-point
and the measured output, discontinuities of the set-point are differentiated by
the derivator component of the PID and yield infinite values for the
control signal (see above).
When No Derivator On Reference is checked, the set-point is not differentiated.
Display Frequency Line
When selected, moving the mouse above a frequency response (Bode or
sensitivity) will display a corresponding line in other frequency
responses, Nyquist diagrams, and zero/pole diagrams.
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