Inverse of State-space representation (control)












2












$begingroup$


Ask two questions from a paper (2012 ACC):



Consider the plant:



enter image description here



Let X be the stabilizing solution of the Riccati equation:



enter image description here



where enter image description here.

Define the LQR gain by enter image description here.



The transfer matrix enter image description here has a left spectral factorization enter image description here,

where WL is given by



enter image description here



Questions:




  1. If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)


  2. (Basic question) Why does the Riccati equation is $(A, B_2, C_1)$ not $(A, B_1, C_1)$ or
    $(A, B_2, C_2)$?







control-theory optimal-control







share|cite|improve this question











$endgroup$












  • $begingroup$
    Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
    $endgroup$
    – JimmyK4542
    Aug 15 '14 at 3:35










  • $begingroup$
    I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
    $endgroup$
    – sleeve chen
    Aug 15 '14 at 3:42










  • $begingroup$
    What is $X$? I believe it is a solution to some matrix equation.
    $endgroup$
    – obareey
    Aug 17 '14 at 14:36










  • $begingroup$
    More complete. Thanks!
    $endgroup$
    – sleeve chen
    Aug 17 '14 at 16:00
















2












$begingroup$


Ask two questions from a paper (2012 ACC):



Consider the plant:



enter image description here



Let X be the stabilizing solution of the Riccati equation:



enter image description here



where enter image description here.

Define the LQR gain by enter image description here.



The transfer matrix enter image description here has a left spectral factorization enter image description here,

where WL is given by



enter image description here



Questions:




  1. If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)


  2. (Basic question) Why does the Riccati equation is $(A, B_2, C_1)$ not $(A, B_1, C_1)$ or
    $(A, B_2, C_2)$?







control-theory optimal-control







share|cite|improve this question











$endgroup$












  • $begingroup$
    Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
    $endgroup$
    – JimmyK4542
    Aug 15 '14 at 3:35










  • $begingroup$
    I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
    $endgroup$
    – sleeve chen
    Aug 15 '14 at 3:42










  • $begingroup$
    What is $X$? I believe it is a solution to some matrix equation.
    $endgroup$
    – obareey
    Aug 17 '14 at 14:36










  • $begingroup$
    More complete. Thanks!
    $endgroup$
    – sleeve chen
    Aug 17 '14 at 16:00














2












2








2


1



$begingroup$


Ask two questions from a paper (2012 ACC):



Consider the plant:



enter image description here



Let X be the stabilizing solution of the Riccati equation:



enter image description here



where enter image description here.

Define the LQR gain by enter image description here.



The transfer matrix enter image description here has a left spectral factorization enter image description here,

where WL is given by



enter image description here



Questions:




  1. If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)


  2. (Basic question) Why does the Riccati equation is $(A, B_2, C_1)$ not $(A, B_1, C_1)$ or
    $(A, B_2, C_2)$?







control-theory optimal-control







share|cite|improve this question











$endgroup$




Ask two questions from a paper (2012 ACC):



Consider the plant:



enter image description here



Let X be the stabilizing solution of the Riccati equation:



enter image description here



where enter image description here.

Define the LQR gain by enter image description here.



The transfer matrix enter image description here has a left spectral factorization enter image description here,

where WL is given by



enter image description here



Questions:




  1. If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)


  2. (Basic question) Why does the Riccati equation is $(A, B_2, C_1)$ not $(A, B_1, C_1)$ or
    $(A, B_2, C_2)$?







control-theory optimal-control




control-theory optimal-control






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 15 '17 at 20:38







sleeve chen

















asked Aug 15 '14 at 3:23









sleeve chensleeve chen

3,08641852




3,08641852












  • $begingroup$
    Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
    $endgroup$
    – JimmyK4542
    Aug 15 '14 at 3:35










  • $begingroup$
    I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
    $endgroup$
    – sleeve chen
    Aug 15 '14 at 3:42










  • $begingroup$
    What is $X$? I believe it is a solution to some matrix equation.
    $endgroup$
    – obareey
    Aug 17 '14 at 14:36










  • $begingroup$
    More complete. Thanks!
    $endgroup$
    – sleeve chen
    Aug 17 '14 at 16:00


















  • $begingroup$
    Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
    $endgroup$
    – JimmyK4542
    Aug 15 '14 at 3:35










  • $begingroup$
    I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
    $endgroup$
    – sleeve chen
    Aug 15 '14 at 3:42










  • $begingroup$
    What is $X$? I believe it is a solution to some matrix equation.
    $endgroup$
    – obareey
    Aug 17 '14 at 14:36










  • $begingroup$
    More complete. Thanks!
    $endgroup$
    – sleeve chen
    Aug 17 '14 at 16:00
















$begingroup$
Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
$endgroup$
– JimmyK4542
Aug 15 '14 at 3:35




$begingroup$
Multiplying $W_L W_L^{-1}$ doesn't seem to yield the identity matrix, unless we know stuff about the matrices $A, B_2, K$. Can you provide more context?
$endgroup$
– JimmyK4542
Aug 15 '14 at 3:35












$begingroup$
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
$endgroup$
– sleeve chen
Aug 15 '14 at 3:42




$begingroup$
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
$endgroup$
– sleeve chen
Aug 15 '14 at 3:42












$begingroup$
What is $X$? I believe it is a solution to some matrix equation.
$endgroup$
– obareey
Aug 17 '14 at 14:36




$begingroup$
What is $X$? I believe it is a solution to some matrix equation.
$endgroup$
– obareey
Aug 17 '14 at 14:36












$begingroup$
More complete. Thanks!
$endgroup$
– sleeve chen
Aug 17 '14 at 16:00




$begingroup$
More complete. Thanks!
$endgroup$
– sleeve chen
Aug 17 '14 at 16:00










1 Answer
1






active

oldest

votes


















2












$begingroup$

First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.



If you write down the equations explicitly:



$$
begin{align}
dot x &= Ax+Bu\
y &= Cx+Du
end{align}
$$
Now assuming $D$ is invertible, we get



$$
begin{align}
dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\
u &= -D^{-1}Cx + D^{-1}y
end{align}
$$
So the state space from $y$ to $u$ is given by the realization
$$
G^{-1}(s) = begin{bmatrix}A-BD^{-1}C &BD^{-1}\-CD^{-1}&D^{-1}end{bmatrix}
$$



Then you can see the similarity between this inverse and $W_L$.



For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.



Hence you want the second input appearing on the performance channel.






share|cite|improve this answer











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    active

    oldest

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    active

    oldest

    votes









    2












    $begingroup$

    First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.



    If you write down the equations explicitly:



    $$
    begin{align}
    dot x &= Ax+Bu\
    y &= Cx+Du
    end{align}
    $$
    Now assuming $D$ is invertible, we get



    $$
    begin{align}
    dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\
    u &= -D^{-1}Cx + D^{-1}y
    end{align}
    $$
    So the state space from $y$ to $u$ is given by the realization
    $$
    G^{-1}(s) = begin{bmatrix}A-BD^{-1}C &BD^{-1}\-CD^{-1}&D^{-1}end{bmatrix}
    $$



    Then you can see the similarity between this inverse and $W_L$.



    For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.



    Hence you want the second input appearing on the performance channel.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.



      If you write down the equations explicitly:



      $$
      begin{align}
      dot x &= Ax+Bu\
      y &= Cx+Du
      end{align}
      $$
      Now assuming $D$ is invertible, we get



      $$
      begin{align}
      dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\
      u &= -D^{-1}Cx + D^{-1}y
      end{align}
      $$
      So the state space from $y$ to $u$ is given by the realization
      $$
      G^{-1}(s) = begin{bmatrix}A-BD^{-1}C &BD^{-1}\-CD^{-1}&D^{-1}end{bmatrix}
      $$



      Then you can see the similarity between this inverse and $W_L$.



      For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.



      Hence you want the second input appearing on the performance channel.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.



        If you write down the equations explicitly:



        $$
        begin{align}
        dot x &= Ax+Bu\
        y &= Cx+Du
        end{align}
        $$
        Now assuming $D$ is invertible, we get



        $$
        begin{align}
        dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\
        u &= -D^{-1}Cx + D^{-1}y
        end{align}
        $$
        So the state space from $y$ to $u$ is given by the realization
        $$
        G^{-1}(s) = begin{bmatrix}A-BD^{-1}C &BD^{-1}\-CD^{-1}&D^{-1}end{bmatrix}
        $$



        Then you can see the similarity between this inverse and $W_L$.



        For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.



        Hence you want the second input appearing on the performance channel.






        share|cite|improve this answer











        $endgroup$



        First question is about inverting a state space system that represents the dynamics from $u$ to $y$ such that the representation is from $y$ to $u$.



        If you write down the equations explicitly:



        $$
        begin{align}
        dot x &= Ax+Bu\
        y &= Cx+Du
        end{align}
        $$
        Now assuming $D$ is invertible, we get



        $$
        begin{align}
        dot x &= Ax+B(-D^{-1}Cx + D^{-1}y)\
        u &= -D^{-1}Cx + D^{-1}y
        end{align}
        $$
        So the state space from $y$ to $u$ is given by the realization
        $$
        G^{-1}(s) = begin{bmatrix}A-BD^{-1}C &BD^{-1}\-CD^{-1}&D^{-1}end{bmatrix}
        $$



        Then you can see the similarity between this inverse and $W_L$.



        For the second question, in Linear Quadratic Cost function you would like to add the cost of the input to your cost function. In this formulation, often $B_1$ is the disturbance input to the system and $B_2$ models the input matrix.



        Hence you want the second input appearing on the performance channel.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 27 '18 at 13:38









        amWhy

        1




        1










        answered Apr 3 '15 at 9:57









        percussepercusse

        21214




        21214






























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