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Control invariant set for not pointwise-in-time constraints
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Given is a system of the form
$$ x_k+1 = Ax_k+Bu_k, kgeq 0 $$
with $x_0$ known, $x_k in mathbbR^n,u_k in mathbbR^m$ and $A,B$ of appropriate dimensions. Let $u = beginbmatrixu_0 \ vdots \ u_N-1endbmatrix$ with $N$ being the prediction horizon of the MPC problem that should be solved, i.e.
$$ beginaligned
u^* = &undersetu in mathbbUtextargmin sum_k=1^N (x_k-barx_k)^T (x_k-barx_k)\
&beginalignedtexts.t. &x_k in mathbbX,\
&x_k+1 = Ax_k+Bu_k,\
&u in mathbbU.
endaligned
endaligned
$$
Here, $barx_k in mathbbR^n$ and known $forall k$, $mathbbX$ is characterized by a finite number of hyperplanes, not necessarily bounded, and $mathbbU$ is also a set characterized by a finite number of hyperplanes but bounded.
The focus is not so much on the specific problem itself (e.g. the cost function), but on the fact that there is no pointwise-in-time constraint for $u_k$ but rather a "combined" constraint for $u$, i.e. $u in mathbbU$.
I want to find a control invariant set for this setup; however, all I can find is literature on how to do it if
- $mathbbX$ is bounded,
- $mathbbU$ is a pointwise-in-time constraint set, i.e. $u_kin mathbbU$.
Any hints/references appreciated.
control-theory optimal-control set-invariance
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
asked Jul 16 at 12:58
VGD
608
add a comment |Â
up vote
0
down vote
favorite
Given is a system of the form
$$ x_k+1 = Ax_k+Bu_k, kgeq 0 $$
with $x_0$ known, $x_k in mathbbR^n,u_k in mathbbR^m$ and $A,B$ of appropriate dimensions. Let $u = beginbmatrixu_0 \ vdots \ u_N-1endbmatrix$ with $N$ being the prediction horizon of the MPC problem that should be solved, i.e.
$$ beginaligned
u^* = &undersetu in mathbbUtextargmin sum_k=1^N (x_k-barx_k)^T (x_k-barx_k)\
&beginalignedtexts.t. &x_k in mathbbX,\
&x_k+1 = Ax_k+Bu_k,\
&u in mathbbU.
endaligned
endaligned
$$
Here, $barx_k in mathbbR^n$ and known $forall k$, $mathbbX$ is characterized by a finite number of hyperplanes, not necessarily bounded, and $mathbbU$ is also a set characterized by a finite number of hyperplanes but bounded.
The focus is not so much on the specific problem itself (e.g. the cost function), but on the fact that there is no pointwise-in-time constraint for $u_k$ but rather a "combined" constraint for $u$, i.e. $u in mathbbU$.
I want to find a control invariant set for this setup; however, all I can find is literature on how to do it if
- $mathbbX$ is bounded,
- $mathbbU$ is a pointwise-in-time constraint set, i.e. $u_kin mathbbU$.
Any hints/references appreciated.
control-theory optimal-control set-invariance
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
asked Jul 16 at 12:58
VGD
608
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given is a system of the form
$$ x_k+1 = Ax_k+Bu_k, kgeq 0 $$
with $x_0$ known, $x_k in mathbbR^n,u_k in mathbbR^m$ and $A,B$ of appropriate dimensions. Let $u = beginbmatrixu_0 \ vdots \ u_N-1endbmatrix$ with $N$ being the prediction horizon of the MPC problem that should be solved, i.e.
$$ beginaligned
u^* = &undersetu in mathbbUtextargmin sum_k=1^N (x_k-barx_k)^T (x_k-barx_k)\
&beginalignedtexts.t. &x_k in mathbbX,\
&x_k+1 = Ax_k+Bu_k,\
&u in mathbbU.
endaligned
endaligned
$$
Here, $barx_k in mathbbR^n$ and known $forall k$, $mathbbX$ is characterized by a finite number of hyperplanes, not necessarily bounded, and $mathbbU$ is also a set characterized by a finite number of hyperplanes but bounded.
The focus is not so much on the specific problem itself (e.g. the cost function), but on the fact that there is no pointwise-in-time constraint for $u_k$ but rather a "combined" constraint for $u$, i.e. $u in mathbbU$.
I want to find a control invariant set for this setup; however, all I can find is literature on how to do it if
- $mathbbX$ is bounded,
- $mathbbU$ is a pointwise-in-time constraint set, i.e. $u_kin mathbbU$.
Any hints/references appreciated.
control-theory optimal-control set-invariance
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
asked Jul 16 at 12:58
VGD
608
Given is a system of the form
$$ x_k+1 = Ax_k+Bu_k, kgeq 0 $$
with $x_0$ known, $x_k in mathbbR^n,u_k in mathbbR^m$ and $A,B$ of appropriate dimensions. Let $u = beginbmatrixu_0 \ vdots \ u_N-1endbmatrix$ with $N$ being the prediction horizon of the MPC problem that should be solved, i.e.
$$ beginaligned
u^* = &undersetu in mathbbUtextargmin sum_k=1^N (x_k-barx_k)^T (x_k-barx_k)\
&beginalignedtexts.t. &x_k in mathbbX,\
&x_k+1 = Ax_k+Bu_k,\
&u in mathbbU.
endaligned
endaligned
$$
Here, $barx_k in mathbbR^n$ and known $forall k$, $mathbbX$ is characterized by a finite number of hyperplanes, not necessarily bounded, and $mathbbU$ is also a set characterized by a finite number of hyperplanes but bounded.
The focus is not so much on the specific problem itself (e.g. the cost function), but on the fact that there is no pointwise-in-time constraint for $u_k$ but rather a "combined" constraint for $u$, i.e. $u in mathbbU$.
I want to find a control invariant set for this setup; however, all I can find is literature on how to do it if
- $mathbbX$ is bounded,
- $mathbbU$ is a pointwise-in-time constraint set, i.e. $u_kin mathbbU$.
Any hints/references appreciated.
control-theory optimal-control set-invariance
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
asked Jul 16 at 12:58
VGD
608
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
edited Jul 16 at 15:19
Rodrigo de Azevedo
12.5k41751
12.5k41751
asked Jul 16 at 12:58
VGD
608
asked Jul 16 at 12:58
VGD
608
asked Jul 16 at 12:58
VGD
608
608
add a comment |Â
add a comment |Â
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