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Hope that you will find this post in good health. I am Mr. Adnan from Pakistan with research background in Control systems. I am working on one problem in which Hadamard weights are using.

During solving that problem, i am facing one problem to calculate inverse vectorization/stacking operator i.e $\text{vec}^{-1}$. I would be highly thankful if could you please guide me that how to calculate for example $$\text{vec}^{-1}(Wq)$$ where $W$ is non singular matrix and $q$ is controller having state space representation \begin{align} X'&=Ax+Bu, \\ Y&=Cx+Du \end{align}

I just have two queries to calculate above inverse

  1. what should be order of $q$ to calculate inverse vector operator $\text{vec}^{-1}$

  2. if order of controller $q$ is know then Please tell me how to solve $\text{vec}^{-1}$, any formula,relation or command.

I Knew these two commands $\text{vec}$ and $\text{vec}^{-1}$ but that is not working in my problem.

-Thanks in Advance

Rodrigo de Azevedo
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adnan jaffar
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    Hi Adnan. I edited your post to put the formulas in LaTeX/Mathjax format, I hope you don't mind. I'm still a bit confused about what you are trying to do though - are you following a procedure that exists in a reference like a book or paper? If so, posting a link to the reference would help – Nick Alger Apr 13 '16 at 17:47

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I do not really understand what your problem is?

Let $A$ be defined as: $$A = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}$$ Then $\operatorname{vec}(A)$ is defined as $$\operatorname{vec}(A) = \begin{bmatrix} a_{11} \\ a_{21}\\ a_{12} \\ a_{22} \end{bmatrix}$$ As a result the inverse is defined as $$\operatorname{vec}^{-1}(\operatorname{vec}(A)) = A$$

Now, when I typed this down, I think I understand your problem. You have the matrix $B = Wq$ which is vectorized and want to inverse it. This will only work if you know the size of $W$ and $q$ before they were vectorized. Then you know size which the result should be after the inverse vectorization.

Furthermore, what is $q$ exactly? You say that it is a state space system? What do you imagine the result of $Wq$ to be then?

WG-
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  • Yes exactly. Suppose W is the 22 matrix and q is the controller having state space representation. I know the order of q(suppose 6 states,2 inputs and 2 outputs). Now please tell me that how can i calculate inverse vectorization of Wq ? – adnan jaffar Apr 15 '16 at 10:46
  • @adnanjaffar again please tell me what you imagine the result of $Wq$ to be because what you are saying does not make sense to me. When $q$ is a state space system. $Wq = \begin{align} WX'&=WAx+WBu, \ WY&=WCx+WDu \end{align}$? Please give a concrete example. – WG- Apr 18 '16 at 09:11