How Does the Transformation Matrix P Convert a Linear System to Canonical Form?

[X89codered89X]

Hi there,

I have a linear algebra question relating actually to control systems (applied differential equations)

for the linear system

[itex]

{\dot{\vec{{x}}} = {\bf{A}}{\vec{{x}}} + {\bf{B}}}{\vec{{u}}}\\
\\

A \in \mathbb{R}^{ nxn }\\
B \in \mathbb{R}^{ nx1 }\\
[/itex]

In class, we formed a transformation matrix P using the controllability matrix [itex] M_c [/itex] as a basis (assuming it is full rank).
[itex]

M_c = [ {\bf{B \;AB \;A^2B\;...\;A^{n-1}B}}]
[/itex]

and there is a second matrix with a less established name. Given that the characteristic equation of the system is [itex] |I\lambda -A| = \lambda^n + \alpha_1 \lambda^{n-1} +... + \alpha_{n-1}\lambda + \alpha_n= 0 [/itex], we then construct a second matrix, call it M_2, which is given below.

[itex]
{\bf{M}}_2 =
\begin{bmatrix}
\alpha_{n-1} & \alpha_{n-2} & \cdots & \alpha_1 & 1 \\
\alpha_{n-2} & \cdots & \alpha_1 & 1 & 0 \\
\vdots & \alpha_1 & 1 & 0 & 0\\
\alpha_1 & 1 & 0 & \cdots & 0\\
1 & 0 & 0& \cdots & 0 \\
\end{bmatrix}

[/itex]

then the transformation matrix is then given by

[itex]

P^{-1} = M_c M_2

[/itex]


and then applying the transformation always gives.. and this is what I don't understand...

[itex]
{\overline{\bf{A}}} = {\bf{PAP}}^{-1} =

\begin{bmatrix}
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & \vdots \\
\vdots & \vdots & 0 & 1 & 0\\
0 & 0 & \cdots &0& 1\\
-\alpha_{1} & -\alpha_{2} & \cdots & -\alpha_{n-1}& -\alpha_{n}\\
\end{bmatrix}

[/itex]

Now I'm just looking for intuition is to why this is true. I know that this only works if the controllability matrix is full rank, which can the be used as a basis for the new transformation, but I don't get how exactly the M_2 matrix is using it to transform into the canonical form... Can someone explain this to me? thanks...

Disclaimer: I posted this in another PF subforum, but I think I might do better in this section.

 

[jvicens]

Hi there,

Thank you for your question. I can definitely provide some insight into the intuition behind the transformation matrix P and the resulting canonical form of the system.

First, let's start with the controllability matrix M_c. This matrix is formed using the control inputs B and the system matrix A, and its purpose is to determine whether or not the system is controllable. If M_c is full rank, it means that every state of the system can be reached from every other state using the available control inputs. This is important because it allows us to manipulate the system into a simpler form.

Next, let's look at the second matrix M_2. This matrix is formed using the coefficients of the characteristic equation of the system. The characteristic equation is derived from the eigenvalues of the system matrix A, and it tells us about the stability and behavior of the system. By constructing M_2 in this way, we are essentially using the information from the characteristic equation to transform the system into a canonical form.

Now, onto P. The transformation matrix P is formed by multiplying M_c and M_2 together. This matrix essentially combines the information from the controllability matrix and the characteristic equation to transform the system into a canonical form. This form, as you have noticed, has a specific structure with 1's along the diagonal and the coefficients of the characteristic equation in the last row. This structure is important because it allows us to easily analyze and control the system.

In summary, the transformation matrix P is using the information from both the controllability matrix and the characteristic equation to transform the system into a canonical form. This form simplifies the analysis and control of the system, making it easier to understand and manipulate. I hope this helps to clarify the intuition behind this transformation. If you have any further questions, please don't hesitate to ask.