Register to reply

Linear System Transformations

by X89codered89X
Tags: control systems, linear algebra, odes
Share this thread:
X89codered89X
#1
Nov21-12, 08:02 PM
P: 141
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.
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100

Register to reply

Related Discussions
Linear Algebra question regarding Matrices of Linear Transformations Calculus & Beyond Homework 2
Linear Functionals, Dual Spaces & Linear Transformations Between Them Linear & Abstract Algebra 8
Another linear algebra problem, basis and linear transformations. Calculus & Beyond Homework 7
Linear Algebra - Linear Transformations, Change of Basis Calculus & Beyond Homework 3
Linear Algebra (Vector spaces, linear independent subsets, transformations) Calculus & Beyond Homework 12