Inverse of State-space representation (control)
up vote
2
down vote
favorite
Ask two questions from a paper (2012 ACC):
Consider the plant:
Let X be the stabilizing solution of the Riccati equation:
where 
.
Define the LQR gain by 
.
The transfer matrix 
has a left spectral factorization 
,
where WL is given by
Questions:
If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)
(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
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
add a comment |
up vote
2
down vote
favorite
Ask two questions from a paper (2012 ACC):
Consider the plant:
Let X be the stabilizing solution of the Riccati equation:
where .
Define the LQR gain by .
The transfer matrix has a left spectral factorization ,
where WL is given by
Questions:
If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)
(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
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
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?
– JimmyK4542
Aug 15 '14 at 3:35
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Ask two questions from a paper (2012 ACC):
Consider the plant:
Let X be the stabilizing solution of the Riccati equation:
where .
Define the LQR gain by .
The transfer matrix has a left spectral factorization ,
where WL is given by
Questions:
If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)
(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
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
Ask two questions from a paper (2012 ACC):
Consider the plant:
Let X be the stabilizing solution of the Riccati equation:
where .
Define the LQR gain by .
The transfer matrix has a left spectral factorization ,
where WL is given by
Questions:
If I know the $W_L$, how to derive the $W_L^{-1}$, (bottom one)
(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
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
edited Feb 15 '17 at 20:38
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
asked Aug 15 '14 at 3:23
sleeve chen
3,01041852
3,01041852
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?
– JimmyK4542
Aug 15 '14 at 3:35
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00
add a comment |
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?
– JimmyK4542
Aug 15 '14 at 3:35
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00
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?
– JimmyK4542
Aug 15 '14 at 3:35
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?
– JimmyK4542
Aug 15 '14 at 3:35
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
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.
edited Nov 27 at 13:38
amWhy
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
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.
edited Nov 27 at 13:38
amWhy
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
add a comment |
up vote
2
down vote
accepted
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.
edited Nov 27 at 13:38
amWhy
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
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.
edited Nov 27 at 13:38
amWhy
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
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.
edited Nov 27 at 13:38
amWhy
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
edited Nov 27 at 13:38
amWhy
191k28224439
edited Nov 27 at 13:38
amWhy
191k28224439
edited Nov 27 at 13:38
amWhy
191k28224439
191k28224439
answered Apr 3 '15 at 9:57
percusse
21214
answered Apr 3 '15 at 9:57
percusse
21214
answered Apr 3 '15 at 9:57
percusse
21214
21214
add a comment |
add a comment |
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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?
– JimmyK4542
Aug 15 '14 at 3:35
I think the representation is not really a matrix; it is like C*inv(sI-A)B + D
– sleeve chen
Aug 15 '14 at 3:42
What is $X$? I believe it is a solution to some matrix equation.
– obareey
Aug 17 '14 at 14:36
More complete. Thanks!
– sleeve chen
Aug 17 '14 at 16:00