# What is the relation between eigenvalues and state-space response in control systems?

I understand the mathematics behind it but I want to know what happens physically in a real-life system. How do the eigenvalues come into the picture from a non-mathematical (physical) point of view? In linear algebra, a matrix is basically a linear transformation and eigenvectors are those which, on being applied this transformation, just stretches themselves (that is without any rotation, I'm limiting only to 2D space). So when I consider a physical system (I'm only considering a homogeneous or unforced system to make it simpler) what do these eigenvalues or eigenvectors physically represent and how do they relate to their mathematical interpretations?

• I meant the natural response, considering the basic homogeneous equation – Sagnik Mar 18 at 8:07

## 1 Answer

Just because eigenvalues apply in linear algebra doesn't mean that the same meaning applies everywhere. Think of the Fourier transform: does it only apply to the study of heat transfer? Same here. The A matrix is the companion matrix, which means its eigenvalues represent the roots of the system. For example:

\begin{align} H(s)&=\dfrac{3}{s^2+2s+3}\tag{1} \\ \mathbf{A}&=\begin{bmatrix}{{0}\quad{-3} \\ {1}\quad{-2}}\end{bmatrix}\tag{2} \\ s_{1,2}&=-1\pm j\sqrt{2}\tag{3} \\ \end{align}

I'm afraid I don't quite understand what you mean by "from a non-mathematical (physical) point of view". The eigenvalues are the mathematical roots of the system, they tell about the stability, the time and frequency response. There's nothing non-physical about it. The system itself is physical -- as in it has a meaning -- even if only described on paper, such as the example above. It's just that you don't have to think about them from a single perspective, only.

• So you mean to say that we consider eigenvalues here just for the sake of computing the solutions and they have very different interpretations here from that in linear algebra? But i actually thought that since the basic definition of eigenvalues are rooted in linear algebra they can be interpreted along those lines for a physical system – Sagnik Mar 18 at 8:15
• @Sagnik You are thinking about generic matrices, but this is a particular case, the companion matrix. And you got it backwards: e.v. are not for "the sake of computing the solutions", they are just inherent to the state-space representation. A system can be represented by its states in matrix form (provided it's LTI) based on the differential equations governing it. And the matrices make it so much easier to describe systems with mutiple I/O. Just because the origins are in the linear algebra, it doesn't mean that that's exactly how they are supposed to behave, or be treated (the FFT example). – a concerned citizen Mar 18 at 9:14