A unity-feedback system with third-order plant is driven by a reference
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.

The Laplace transform description of the closed-loop system is:

Shown below are the open-loop Bode magnitude and phase plots with the
corner frequencies *a* and *b* marked on the magnitude plot and
the gain and phase crossover frequencies labeled. The left mouse button
can be used to drag corner frequencies to desired locations, thereby adjusting
the values of *a*, *b*, and *k*. The corresponding closed-loop
Bode magnitude plot is also shown, with the closed-loop bandwidth labeled.

A selection of reference inputs is available, and 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. Mean-square
steady-state error is computed on the range *15* < *t* <
*25*.)

**Sample Exercise on Closed-Loop Bandwidth:**
With no sensor noise, adjust the open-loop gain and corner frequencies
so that the step response exhibits overshoot no greater than *20*%
and a rise time no greater than* 0.2 sec*. Note the
value of mean-square steady-state error in the Table. Then select low-gain
sensor noise and adjust the open-loop parameters to achieve mean-square,
steady-state error no greater than* 0.0020* with* 20*% overshoot
bound and rise time as small as possible.

** Sample Exercise on Stability Margins: **Fix
one of the open-loop corner frequencies at *a = 2 rad/sec. *
Adjust the other corner frequency (thereby adjusting *b *and *k
*) to obtain a step response with rise time less than *0.6 sec*,
gain margin at least *6 db*, and phase margin at least *35 deg.*

**Sample Exercise on Closed-Loop Frequency Response:**
Adjust the open-loop corner frequencies and gain so that the closed-loop
frequency response has the following properties: the maximum magnitude
is *0 db*, the *3 db * bandwidth *W**bw
= 2 rad/sec*, and the closed-loop
magnitude at *8 rad/sec* is less than *- 20 db*.

Applet by Steven Crutchfield. |