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 k1 = - 2a and k2 = (k12 + 4b2- 4) / 4. As we place the closed loop poles farther to the left, the system will respond more quickly, and the states will go to zero more quickly. But this is at the expense of a large control effort. On the other hand, if we want to limit the control action we can place the closed loop poles closer to the imaginary axis. This will result in a smaller control effort, but the states will go to zero more slowly. This is the kind of tradeoff that the controls engineer faces.

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.