What is a good, non-trivial example of a control system that is unstable without feedback and stable with feedback for certain parameter values?

Question

Consider the input/state/output (i/s/o) feedback control system

$ \left\{ \begin{array}{lll} \frac{d}{dt}x = Ax + Bu \\ y = Cx \\ u = Ky \end{array} \right.$

Here, $A \in \mathbb{R}^{3 \times 3}$, $B \in \mathbb{R}^{3 \times 1}$, $C \in \mathbb{R}^{1 \times 3}$ and $K \in \mathbb{R}$. I am interested in an example where we first have:

$B = (0,0,0)^{\intercal}$ (so there is no feedback at all) and the system resulting system $\frac{d}{dt}x = Ax$ is unstable for some $(3\times3)$ matrix $A$ (so all eigenvalues of $A$ have positive real part).

And then (second), with the same matrix $A$ as previously chosen, we choose $B$ and $C$ in such a way that there are certain values for $K$ for which the whole system now becomes stable. If you can give an example with the appropriate $A$, $B$ and $C$, then I can calculate the range of $K$ by means of the Routh-Hurwitz algorithm to actually stabilise the system.

Do you have an example of this? And does it allow a physical interpretation? Any help would be very much appreciated.

Answer 1

An inverted pendulum, to give an example that is studied to exhaustion in controls teaching material and demos, and also present in real life - the Segway, or, well, us bipeds.

Answer 2

Assuming that the constraints on the sizes of $A$, $B$, $C$ (and $K$) also implies that the state space model has to be minimal, otherwise a smaller representation could be used. Since the state space model can be assumed to be minimal, then I will make it myself easy and write it in the controllable canonical form (since otherwise there would always be a transformation which would bring this system into this form).

As states in my comment, the uncontrolled system will be unstable if any of the eigenvalues of $A$ has a real part greater than zero (or when the Jordan canonical form of $A$ contains a Jordan block bigger than one by one for an eigenvalue with a zero real part).

Since you are only asking for an example, I will just give you such a system for which you can find a $K$. But if you are interested in how I came up with such system (so maybe you can come up with different systems yourself) feel free to ask.

$$ \left[\begin{array}{c|c} A & B \\ \hline C & \end{array}\right] = \left[\begin{array}{ccc|c} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 7 & 1 & \llap{-}7 & 1\\ \hline 2 & 1 & \llap{-}1 & \end{array}\right] $$