A feedback system with a third-order plant transfer function

where both *a* and *b* are positive, and both forward-path
and feedback-path time delays (transport lags) is driven by a reference input
signal *r(t)* and a sensor noise signal *n(t)*. Typical performance
objectives involve tracking the reference signal while rejecting the influence
of the sensor noise.

Time delays are commonly found in chemical process control systems, biological control systems, and control systems that involve long-distance communications. Typically, time delays have adverse effects on closed-loop stability properties.

The presence of time delays greatly complicates the analysis of control systems because the closed-loop transfer functions

are not rational functions of *s*. Therefore many
standard techniques, such as the Routh-Hurwitz test for stability,
and partial-fraction expansion methods for computing responses,
do not apply. However the Nyquist stability criterion applies,
and furthermore systems of the class considered here are stable for
sufficiently low gain and become unstable as gain increases. Therefore Bode
plots can be used to obtain meaningful values for phase margin and
gain margin.

Shown below are the open-loop Bode magnitude and phase plots with the
corner frequencies *a* and *b* marked on the magnitude plot.
The gain and phase crossover frequencies are labeled. For this
form of plant transfer function, it can be shown that closed-loop
stability corresponds to positive
phase margin. The left mouse button
can be used to drag corner frequencies to desired locations, thereby adjusting
the values of *a*, *b*, and *k*. The time delays
are set by slider bars.
The corresponding closed-loop
Bode magnitude plot is also shown, with the closed-loop bandwidth labeled.

Two reference inputs can be selected, or click the *Custom Input*
button and draw a reference input signal
with the mouse. Also, high-frequency
sensor noise can be selected at two amplitude levels. Plots of *r(t)*
and *y(t)* are shown, and the *Table* button provides numerical values
of important system parameters and response characteristics. (Rise time is the
*10% - 90%* rise time, and thus ignores an initial delay in
the response. Mean-square
steady-state error is computed on the range *15* < *t* <
*25*.)

The red-green lights show when computation is in process - it can take a bit of time on leisurely computers. Also note that the stability/instability border is imprecise: The instability sign is based on Phase Margin, though the numerical indication might be slightly different.

**Exercises:** Set the plant parameters to approximately *k=100, a=5, b=10*,
and sketch the Custom Input *r(t) = 2u(t) - 3u(t-1) + u(t-2)*, where
*u(t)* is the unit step function.

1. What are the effects of *T _{1}* and

2. What are the effects of

3. What is the value of

4. What is the value of

5. Why should the answers to questions 3 and 4 be the same?

6. What is the effect of

7. What is the effect of

8. Why should the answers to questions 6 and 7 be different?

Applet by Seth Kahn |