For what values does this system show BIBO stability?

Question

I got this system state representation:

$$\begin{align} \frac{dx}{dt} &= \begin{bmatrix}-3&0&0\\0&\alpha-2&0\\\alpha&0&2+\alpha\end{bmatrix}x+\begin{bmatrix}6\\2\\1\end{bmatrix}u\\ y &= \begin{bmatrix}0&0&1\end{bmatrix}x \end{align} $$

I shall find the biggest possible range of values for the system to be BIBO(Bounded Input Bounded Output) stable.

My eigenvalues are $-3, \alpha-2, \alpha +2$.
So my system is asymptotically stable for all values of $\alpha < -2$.

The eigenvalues of the system matrix A are the poles of the transfer function of the system. And if the poles have a negative real part, the system is BIBO stable.
So my system is also BIBO stable for all values of $\alpha < -2$.

Is my guessing here correct?

Answer 1

As the name BIBO (Bounded Input-Bounded Output) suggests, pure oscillations or constant trajectories are allowed. Consider the following simpler cases for the autonomous systems (with initial condition $x(0) = \pmatrix{6&2&1}^T$ if you like):

1. $$\dot x = \pmatrix{0&&\\&0&1\\&-1&0}x$$
2. $$\dot x = \pmatrix{0&&\\&-1&1\\&0&-1}x$$
3. $$\dot x = \pmatrix{-1&&\\&-1&1\\&0&-1}x$$

A simple $e^{At}$ calculation reveals that the trajectories are all bounded for all initial conditions in $\mathbb{R}^3$ but only 3. is asymptotically stable.

Hence, the answer for the system given in the question is : If $\alpha\leq-2$ then BIBO stability. If further, $\alpha<-2$ then asymmptotical stability. In the case where $\alpha=-2$ you get poles on the imaginary axis, hence oscillations. Here you have the so-called marginal stability.