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I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two of the three original eigenvalues.

What I want is to try to reconstruct the original $\Bbb{R}^3$ matrix. I do not know which operation or algebraic manipulation should I do to get the $\Bbb{R}^3$ system from two of the three systems obtained.

To show a numerical example we have the following:

$A=\begin{pmatrix}05099 & 0.2649 & 0.01\\ -0.99 & -0.485 & 0.01\\ -1 & -1.5 & 0\end{pmatrix}$

It has the following eigenvalues: $-0.0875+0.2592i$, $-0.0875-0.2592i$ and $0.1998$. Through some process I have obtained three different systems:

$A_{11}=\begin{pmatrix}0.5825-0.1494i & 0.2796-0.1092i\\ -0.9170-0.1502i & -0.4702-0.1097i\end{pmatrix}$

Its eigenvalues are: $0.1998$ and $-0.0875-0.2592i$

$A_{22}=\begin{pmatrix}0.5825+0.1494i & 0.2796+0.1092i\\ -0.9170+0.1502i & -0.4702+0.1097i\end{pmatrix}$

Its eigenvalues are: $0.1998$ and $-0.0875+0.2592i$

$A_{33}=\begin{pmatrix}0.3502 & 0.2249\\ -1.1505 & -0.5252\end{pmatrix}$

Its eigenvalues are: $-0.0875-0.2592i$ and $-0.0875+0.2592i$

Thanks for your help. I hope it is clear, but if not please ask.

Gina Torres

Answers to the question about the process that i have done.

Yes i have missed the i's, sorry for that, the numbers are complex. The process what i have done is the following, this matrix A, named extended in the language that i am using contains n+1 eigenvalues (3 eigenvalues in the case that i am talking about). Is possible to obtain these three new models through this $A_{11}=SAMP^kS^T(SMP^kS^T)^{-1}$ which is the same $A_{11}=SMEP^kS^T(SMP^kS')^{-1}$ where S is a selection matrix, in this case is the following $S=\begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\end{pmatrix}$ and P is a permutation Matrix $P=\begin{pmatrix}0 & 0 & 1\\ 1 & 0 & 0\\0 & 1 & 0\end{pmatrix}$ for eigenvectors. E is a diagonal Matrix which contains the eigenvalues and M is the eigenvectors matrix. Another way to express that solutions is $A=(SMP^{k}S')S(MP^k)^{-1}A(MP^k)S'(SMP^kS')^{-1}$. What i have done is like a projection from $R^3$ to $R^2$ or $C^2$ eliminating one eigenvalue in each case to obtain A11, A22, A33.

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    The complete explanation is in the question that$i$have edited some minutes ago.2012-12-10

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