** **

**Introduction**

** **

Continuous-time,
dynamical systems that are linear and time-invariant, usually called *LTI systems*, comprise an important
subclass of all systems. Several different mathematical representations are
used to describe the input-output behavior of a causal *LTI* system, including

·
unit
impulse response and convolution integral,

·
linear
differential equation

·
Fourier
frequency response function (stable system)

·
Laplace
transfer function

*LTI* systems exhibit a rich
variety of input-output behaviors. Characteristics of input-output behavior
become important in a *control systems*
setting, where the objective is to achieve desired output signal behaviors by
applying an appropriate input signal.

**The Applet**

The
applet permits exploration of the input-output behavior of several basic *LTI*
systems. By selecting targets for the output signal, the applet becomes a video
game. To make the game easier, the autonomous behavior of time is suspended,
and you define the time scale as you sketch the input signal!

All
the systems are causal, and the mathematical descriptions listed above are
displayed using the notation _{}for the unit impulse and *u(t)* for the unit step
function. Included are plots of the unit impulse response, unit step response,
and a Bode magnitude diagram for the transfer function.

All
signals are assumed to be zero for *t <
0*, a condition that often is made explicit by appending *u(t)*.

Select
a system and a number of targets. Then sketch an input signal *x(t)* so that the response *y(t)*
minimizes the sum-squared miss distance at the targets. To make the task
more challenging, impose on youself an upperbound on the elapsed time. An
equally important objective is to correlate the mathematical description of the
system with the characteristics of input-output behavior.