Bode of Controller and Sensitivity Functions in a State Feedback Controller System
I have a system:
$A$, $B$ and $C$ forms the state space representation of the system. The system has a state feed back controller with an integral controller. I want to draw bode plot of controller, sensitivity function and complementary sensitivity function. How can I create these plots from a system like this? I know that sensitivity function is defined as $1/(1+PC)$ but I can not define $P$ and $C$ in this system; it is same in other two plots.
control-theory
add a comment |
I have a system:
$A$, $B$ and $C$ forms the state space representation of the system. The system has a state feed back controller with an integral controller. I want to draw bode plot of controller, sensitivity function and complementary sensitivity function. How can I create these plots from a system like this? I know that sensitivity function is defined as $1/(1+PC)$ but I can not define $P$ and $C$ in this system; it is same in other two plots.
control-theory
add a comment |
I have a system:
$A$, $B$ and $C$ forms the state space representation of the system. The system has a state feed back controller with an integral controller. I want to draw bode plot of controller, sensitivity function and complementary sensitivity function. How can I create these plots from a system like this? I know that sensitivity function is defined as $1/(1+PC)$ but I can not define $P$ and $C$ in this system; it is same in other two plots.
control-theory
I have a system:
$A$, $B$ and $C$ forms the state space representation of the system. The system has a state feed back controller with an integral controller. I want to draw bode plot of controller, sensitivity function and complementary sensitivity function. How can I create these plots from a system like this? I know that sensitivity function is defined as $1/(1+PC)$ but I can not define $P$ and $C$ in this system; it is same in other two plots.
control-theory
control-theory
edited Nov 26 at 2:31
Kwin van der Veen
5,2752826
5,2752826
asked Nov 25 at 14:11
Arda
102
102
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The open loop $P,C$ can not be defined when you are using cascading control and you want the open loop to be a SISO system. You could define a MIMO open loop using
begin{align}
begin{bmatrix}
dot{x} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K_i \ 0 & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix} +
begin{bmatrix}
B,K & 0 \ 0 & I
end{bmatrix}v \
y &= begin{bmatrix}
I & 0 \ C & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix},
end{align}
with in closed loop $v = vec{r} - y$ where $vec{r} = begin{bmatrix}r_x^top & r^top end{bmatrix}^top$, so in order to get the same closed loop you would have to set $r_x=0$. I do have to note that the order of multiplication matters when calculating the transfer functions that you are interested, since you are dealing with a MIMO open loop which is a (transfer function) matrix and does not commute.
Another approach is to add an observer and use the error $e=r-y$ instead of the output to estimate the state. In that case only the output of the system will be used in the feedback loop and is $e$ the only input to the controller. Normally the observer dynamics is defined as
$$
dot{hat{x}} = A,hat{x} + B,u + L(y - C,hat{x}),
$$
such that the combined closed loop dynamics can be written as
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ 0 \ I
end{bmatrix} r.
$$
But if $-e=y-r$ is used instead of $y$ in $dot{hat{x}}$ then the closed loop dynamics becomes
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} r,
$$
so the same system matrix but a different input matrix, therefore the closed loop poles will be the same, but the zeros might be different. The open loop then becomes
begin{align}
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K & -B,K_i \
0 & A-B,K-L,C & -B,K_i \
0 & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} e \
y &= begin{bmatrix}
C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix}.
end{align}
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012886%2fbode-of-controller-and-sensitivity-functions-in-a-state-feedback-controller-syst%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The open loop $P,C$ can not be defined when you are using cascading control and you want the open loop to be a SISO system. You could define a MIMO open loop using
begin{align}
begin{bmatrix}
dot{x} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K_i \ 0 & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix} +
begin{bmatrix}
B,K & 0 \ 0 & I
end{bmatrix}v \
y &= begin{bmatrix}
I & 0 \ C & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix},
end{align}
with in closed loop $v = vec{r} - y$ where $vec{r} = begin{bmatrix}r_x^top & r^top end{bmatrix}^top$, so in order to get the same closed loop you would have to set $r_x=0$. I do have to note that the order of multiplication matters when calculating the transfer functions that you are interested, since you are dealing with a MIMO open loop which is a (transfer function) matrix and does not commute.
Another approach is to add an observer and use the error $e=r-y$ instead of the output to estimate the state. In that case only the output of the system will be used in the feedback loop and is $e$ the only input to the controller. Normally the observer dynamics is defined as
$$
dot{hat{x}} = A,hat{x} + B,u + L(y - C,hat{x}),
$$
such that the combined closed loop dynamics can be written as
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ 0 \ I
end{bmatrix} r.
$$
But if $-e=y-r$ is used instead of $y$ in $dot{hat{x}}$ then the closed loop dynamics becomes
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} r,
$$
so the same system matrix but a different input matrix, therefore the closed loop poles will be the same, but the zeros might be different. The open loop then becomes
begin{align}
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K & -B,K_i \
0 & A-B,K-L,C & -B,K_i \
0 & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} e \
y &= begin{bmatrix}
C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix}.
end{align}
add a comment |
The open loop $P,C$ can not be defined when you are using cascading control and you want the open loop to be a SISO system. You could define a MIMO open loop using
begin{align}
begin{bmatrix}
dot{x} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K_i \ 0 & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix} +
begin{bmatrix}
B,K & 0 \ 0 & I
end{bmatrix}v \
y &= begin{bmatrix}
I & 0 \ C & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix},
end{align}
with in closed loop $v = vec{r} - y$ where $vec{r} = begin{bmatrix}r_x^top & r^top end{bmatrix}^top$, so in order to get the same closed loop you would have to set $r_x=0$. I do have to note that the order of multiplication matters when calculating the transfer functions that you are interested, since you are dealing with a MIMO open loop which is a (transfer function) matrix and does not commute.
Another approach is to add an observer and use the error $e=r-y$ instead of the output to estimate the state. In that case only the output of the system will be used in the feedback loop and is $e$ the only input to the controller. Normally the observer dynamics is defined as
$$
dot{hat{x}} = A,hat{x} + B,u + L(y - C,hat{x}),
$$
such that the combined closed loop dynamics can be written as
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ 0 \ I
end{bmatrix} r.
$$
But if $-e=y-r$ is used instead of $y$ in $dot{hat{x}}$ then the closed loop dynamics becomes
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} r,
$$
so the same system matrix but a different input matrix, therefore the closed loop poles will be the same, but the zeros might be different. The open loop then becomes
begin{align}
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K & -B,K_i \
0 & A-B,K-L,C & -B,K_i \
0 & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} e \
y &= begin{bmatrix}
C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix}.
end{align}
add a comment |
The open loop $P,C$ can not be defined when you are using cascading control and you want the open loop to be a SISO system. You could define a MIMO open loop using
begin{align}
begin{bmatrix}
dot{x} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K_i \ 0 & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix} +
begin{bmatrix}
B,K & 0 \ 0 & I
end{bmatrix}v \
y &= begin{bmatrix}
I & 0 \ C & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix},
end{align}
with in closed loop $v = vec{r} - y$ where $vec{r} = begin{bmatrix}r_x^top & r^top end{bmatrix}^top$, so in order to get the same closed loop you would have to set $r_x=0$. I do have to note that the order of multiplication matters when calculating the transfer functions that you are interested, since you are dealing with a MIMO open loop which is a (transfer function) matrix and does not commute.
Another approach is to add an observer and use the error $e=r-y$ instead of the output to estimate the state. In that case only the output of the system will be used in the feedback loop and is $e$ the only input to the controller. Normally the observer dynamics is defined as
$$
dot{hat{x}} = A,hat{x} + B,u + L(y - C,hat{x}),
$$
such that the combined closed loop dynamics can be written as
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ 0 \ I
end{bmatrix} r.
$$
But if $-e=y-r$ is used instead of $y$ in $dot{hat{x}}$ then the closed loop dynamics becomes
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} r,
$$
so the same system matrix but a different input matrix, therefore the closed loop poles will be the same, but the zeros might be different. The open loop then becomes
begin{align}
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K & -B,K_i \
0 & A-B,K-L,C & -B,K_i \
0 & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} e \
y &= begin{bmatrix}
C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix}.
end{align}
The open loop $P,C$ can not be defined when you are using cascading control and you want the open loop to be a SISO system. You could define a MIMO open loop using
begin{align}
begin{bmatrix}
dot{x} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K_i \ 0 & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix} +
begin{bmatrix}
B,K & 0 \ 0 & I
end{bmatrix}v \
y &= begin{bmatrix}
I & 0 \ C & 0
end{bmatrix}
begin{bmatrix}
x \ x_i
end{bmatrix},
end{align}
with in closed loop $v = vec{r} - y$ where $vec{r} = begin{bmatrix}r_x^top & r^top end{bmatrix}^top$, so in order to get the same closed loop you would have to set $r_x=0$. I do have to note that the order of multiplication matters when calculating the transfer functions that you are interested, since you are dealing with a MIMO open loop which is a (transfer function) matrix and does not commute.
Another approach is to add an observer and use the error $e=r-y$ instead of the output to estimate the state. In that case only the output of the system will be used in the feedback loop and is $e$ the only input to the controller. Normally the observer dynamics is defined as
$$
dot{hat{x}} = A,hat{x} + B,u + L(y - C,hat{x}),
$$
such that the combined closed loop dynamics can be written as
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ 0 \ I
end{bmatrix} r.
$$
But if $-e=y-r$ is used instead of $y$ in $dot{hat{x}}$ then the closed loop dynamics becomes
$$
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} =
begin{bmatrix}
A & -B,K & -B,K_i \
L,C & A-B,K-L,C & -B,K_i \
-C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} r,
$$
so the same system matrix but a different input matrix, therefore the closed loop poles will be the same, but the zeros might be different. The open loop then becomes
begin{align}
begin{bmatrix}
dot{x} \ dot{hat{x}} \ dot{x}_i
end{bmatrix} &=
begin{bmatrix}
A & -B,K & -B,K_i \
0 & A-B,K-L,C & -B,K_i \
0 & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix} +
begin{bmatrix}
0 \ -L \ I
end{bmatrix} e \
y &= begin{bmatrix}
C & 0 & 0
end{bmatrix}
begin{bmatrix}
x \ hat{x} \ x_i
end{bmatrix}.
end{align}
answered Nov 26 at 4:28
Kwin van der Veen
5,2752826
5,2752826
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012886%2fbode-of-controller-and-sensitivity-functions-in-a-state-feedback-controller-syst%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown