Transforming linear dynamical system to reduce magnitude of eigen values

Question

Consider a linear dynamical system $\dot{x} = Ax + Bu$ with eigen values of high magnitude (can be both positive or negative), say in 900's. Can the system be transformed to reduce the magnitude of eigen values, say 900 reduces to 9 and vice versa.

My approach to was to consider a transformation matrix $T$ such that $z = Tx$, where $T$ is transformation matrices. Then the new dynamical system is

$\dot{z} = T\dot{x} = TAx + TBu = TAT^{-1}z + TBu$.

I was however not sure what should be the structure of $T$ be to minimizes the magnitude of eigen values of the system. Any lead in solving the problem will be appreciated.

Answer 1

A similarity transform can only change the eigenvectors and not the eigenvalues. The only way to achieve what you want would be to change the time scale, so $z(\tau)=x(\alpha\,t)$.

Answer 2

You can do it, although the change of coordinates is generally just continuous (the term topological equivalence may come in handy).

Let me exemplify it for a simple scalar problem. Consider $\dot x=ax$, the solution is $x(t)=\exp(at)x(0)$.

Now let $z=x^{b/a}$. Then the solution is transformed to $z(t)=[\exp(at)x(0)]^{b/a}=(\exp(at))^{b/a}x(0)^{b/a}=\exp(bt)z(0)$, which is the solution of $\dot z = bz$.

Of course a problem is the exponent $b/a$...

As commented previously, if you want a differentiable change of coordinates then the answer is negative, you cannot change the eigenvalues.