LQR with derivative-dependent performance? The Next CEO of Stack OverflowOptimal control problemConditions for optimal controlHow to determine optimal control lawHow to determine the optimal control law?Integrate all estimated states from a LQG controller?Solution set of all LQR problems smaller than all stabilizing controllers?Using Jacobian insted of Lie Derivativeminimum control frequency for closed loopRole of the weight matrix $M$ in $x^T M u$ in the LQR cost functionExosystem for reference generation
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LQR with derivative-dependent performance?
The Next CEO of Stack OverflowOptimal control problemConditions for optimal controlHow to determine optimal control lawHow to determine the optimal control law?Integrate all estimated states from a LQG controller?Solution set of all LQR problems smaller than all stabilizing controllers?Using Jacobian insted of Lie Derivativeminimum control frequency for closed loopRole of the weight matrix $M$ in $x^T M u$ in the LQR cost functionExosystem for reference generation
$begingroup$
Given a standard LTI system with
$$
dotx = A x + B u
$$
The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that
$$
J_1 = int_0^infty big( x^T Q x + u^T R u big) dt tag1
$$
is minimized, given some positive (semi) definite $Q$ and $R$.
However, what if one is also interested in penalizing the derivatives of the controlled system? With that I mean to extend $(1)$ like
$$
J_2 = int_0^infty big( x^T Q x + u^T R u + dotx^T S dotx big) dt tag2
$$
with some positive (semi) definite $S$.
Questions:
- Has something like $(2)$ been considered before? If so, I am very interested in references.
- Is it possible to minimize $(2)$ in a similar way like $(1)$? Will the optimal controller be linear after all?
optimization dynamical-systems control-theory optimal-control linear-control
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
$endgroup$
add a comment |
$begingroup$
Given a standard LTI system with
$$
dotx = A x + B u
$$
The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that
$$
J_1 = int_0^infty big( x^T Q x + u^T R u big) dt tag1
$$
is minimized, given some positive (semi) definite $Q$ and $R$.
However, what if one is also interested in penalizing the derivatives of the controlled system? With that I mean to extend $(1)$ like
$$
J_2 = int_0^infty big( x^T Q x + u^T R u + dotx^T S dotx big) dt tag2
$$
with some positive (semi) definite $S$.
Questions:
- Has something like $(2)$ been considered before? If so, I am very interested in references.
- Is it possible to minimize $(2)$ in a similar way like $(1)$? Will the optimal controller be linear after all?
optimization dynamical-systems control-theory optimal-control linear-control
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
$endgroup$
$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31
add a comment |
$begingroup$
Given a standard LTI system with
$$
dotx = A x + B u
$$
The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that
$$
J_1 = int_0^infty big( x^T Q x + u^T R u big) dt tag1
$$
is minimized, given some positive (semi) definite $Q$ and $R$.
However, what if one is also interested in penalizing the derivatives of the controlled system? With that I mean to extend $(1)$ like
$$
J_2 = int_0^infty big( x^T Q x + u^T R u + dotx^T S dotx big) dt tag2
$$
with some positive (semi) definite $S$.
Questions:
- Has something like $(2)$ been considered before? If so, I am very interested in references.
- Is it possible to minimize $(2)$ in a similar way like $(1)$? Will the optimal controller be linear after all?
optimization dynamical-systems control-theory optimal-control linear-control
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
$endgroup$
Given a standard LTI system with
$$
dotx = A x + B u
$$
The standard LQR finds the control gain $K$ of the state feedback $u = -Kx$ such that
$$
J_1 = int_0^infty big( x^T Q x + u^T R u big) dt tag1
$$
is minimized, given some positive (semi) definite $Q$ and $R$.
However, what if one is also interested in penalizing the derivatives of the controlled system? With that I mean to extend $(1)$ like
$$
J_2 = int_0^infty big( x^T Q x + u^T R u + dotx^T S dotx big) dt tag2
$$
with some positive (semi) definite $S$.
Questions:
- Has something like $(2)$ been considered before? If so, I am very interested in references.
- Is it possible to minimize $(2)$ in a similar way like $(1)$? Will the optimal controller be linear after all?
optimization dynamical-systems control-theory optimal-control linear-control
optimization dynamical-systems control-theory optimal-control linear-control
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
asked Mar 19 at 21:26
SampleTimeSampleTime
55639
55639
$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31
add a comment |
$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31
$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31
$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You have $dotx^TSdotx = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
$endgroup$
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
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
$begingroup$
You have $dotx^TSdotx = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
$endgroup$
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
add a comment |
$begingroup$
You have $dotx^TSdotx = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
$endgroup$
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
add a comment |
$begingroup$
You have $dotx^TSdotx = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
$endgroup$
You have $dotx^TSdotx = (Ax+Bu)^TS(Ax+Bu)$ so it's just a standard LQ problem with a mixed term (the general LQ penalty is $x^TQx+u^TRu + x^TNu$)
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
answered Mar 19 at 21:33
Johan LöfbergJohan Löfberg
5,5301811
5,5301811
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
add a comment |
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
$begingroup$
Right, forgot that mixed term, thanks.
$endgroup$
– SampleTime
Mar 19 at 21:44
add a comment |
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$begingroup$
LTI, LQR... DWA (down with abbreviations) EWTATFTYUT (explain what they mean the first time you use them)
$endgroup$
– Jean Marie
Mar 19 at 21:31