State Feedback

This experiment shows how state feedback can be used to regulate the states of a system to zero. The system we consider is an electrical network - an inductor and capacitor in parallel with a current source.

The input is the voltage source *i*, the first state is the voltage across
the three components, the second state is the current through the inductor, and
the output is equal to the first state. We therefore obtain the following state
equation.

This system has both its eigenvalues and both its poles on the imaginary axis.
It is therefore Lyapunov stable but it is not BIBO stable (see the Stability web
page). However, we can use state feedback to place the eigenvalues (and
poles) anywhere that we want in the complex plane (as long as complex poles
occur in complex conjugate pairs). We do this by choosing a two-element vector *k*
such that *u = **- kx* places the closed loop poles as desired. It
turns out that if we want the poles to be at (*a + jb) *and *(a**- jb)*
then we should choose *k _{1} = *

In the simulation below, you can choose the location of the
closed loop poles. When you run the simulation you will see the output and the
control plotted. Notice that as you make *a* more negative, the output
goes to zero more quickly but the control is larger. As you make *a*
smaller (but still negative), the output goes to zero more slowly but the
control is smaller. If you make *a* positive, then you are placing the
closed loop poles in the right half plane, thus making the system unstable. In
this case you will see the output grow unbounded to infinity. If you make *a
= 0*, you are making the system marginally stable, and the output will
appear as a sustained oscillation.

Linear Systems Experiments Home Page

Last Revised: August 5, 2002