Show that if a linear dynamical equation is controllable at $t_0$, then it is controllable at any $t<t_0$.
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Consider a $n$-dimentional $p$-input equation:
$$dot{x}=Ax+Bu$$
where $A$ and $B$ are constant $ntimes n$ and $ntimes p$ real matrices.
By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.
Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?.
I hope that you can help me.
control-theory linear-control
edited Nov 22 at 1:45
GuadalupeAnimation
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asked Nov 21 at 23:08
Ali G
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Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
up vote
0
down vote
favorite
Consider a $n$-dimentional $p$-input equation:
$$dot{x}=Ax+Bu$$
where $A$ and $B$ are constant $ntimes n$ and $ntimes p$ real matrices.
By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.
Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?.
I hope that you can help me.
control-theory linear-control
edited Nov 22 at 1:45
GuadalupeAnimation
1659
asked Nov 21 at 23:08
Ali G
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider a $n$-dimentional $p$-input equation:
$$dot{x}=Ax+Bu$$
where $A$ and $B$ are constant $ntimes n$ and $ntimes p$ real matrices.
By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.
Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?.
I hope that you can help me.
control-theory linear-control
edited Nov 22 at 1:45
GuadalupeAnimation
1659
asked Nov 21 at 23:08
Ali G
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Consider a $n$-dimentional $p$-input equation:
$$dot{x}=Ax+Bu$$
where $A$ and $B$ are constant $ntimes n$ and $ntimes p$ real matrices.
By definition, the latter state equation is said to be controllable if for any initial state $x(0)=x_0$ and any final state $x_1$, there exists an input that transfers $x_0$ to $x_1$ in a finite time.
Then, how can I show that if a linear dynamical equation is controllable at $t_0$ then it is controllable at any $t<t_0$?.
I hope that you can help me.
control-theory linear-control
control-theory linear-control
edited Nov 22 at 1:45
GuadalupeAnimation
1659
asked Nov 21 at 23:08
Ali G
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 22 at 1:45
GuadalupeAnimation
1659
asked Nov 21 at 23:08
Ali G
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Nov 22 at 1:45
GuadalupeAnimation
1659
edited Nov 22 at 1:45
GuadalupeAnimation
1659
edited Nov 22 at 1:45
GuadalupeAnimation
1659
1659
asked Nov 21 at 23:08
Ali G
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 21 at 23:08
Ali G
62
asked Nov 21 at 23:08
Ali G
62
62
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ali G is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago
add a comment |
Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago
Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago
add a comment |
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Ali G is a new contributor. Be nice, and check out our Code of Conduct.
Ali G is a new contributor. Be nice, and check out our Code of Conduct.
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Do you mean a linear time invariant system, because general linear systems also include linear time variant systems?
– Kwin van der Veen
Nov 22 at 0:20
yeah linear time invariant system
– Ali G
Nov 22 at 3:54
Without loss of gbenerality, you can show if the terminal state $x_1=0$ at time $t_0$, then $x(t)=0$ for all $t>t_0$ .
– Gustave
2 days ago