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 $t<t_0$?
control-theory linear-control
edited Nov 27 at 3:35
SZN
2,713720
asked Nov 21 at 23:08
Ali G
113
add a comment |
up vote
1
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 $t<t_0$?
control-theory linear-control
edited Nov 27 at 3:35
SZN
2,713720
asked Nov 21 at 23:08
Ali G
113
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
Nov 23 at 14:39
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37
add a comment |
up vote
1
down vote
favorite
up vote
1
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 $t<t_0$?
control-theory linear-control
edited Nov 27 at 3:35
SZN
2,713720
asked Nov 21 at 23:08
Ali G
113
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 $t<t_0$?
control-theory linear-control
control-theory linear-control
edited Nov 27 at 3:35
SZN
2,713720
asked Nov 21 at 23:08
Ali G
113
edited Nov 27 at 3:35
SZN
2,713720
asked Nov 21 at 23:08
Ali G
113
edited Nov 27 at 3:35
SZN
2,713720
edited Nov 27 at 3:35
SZN
2,713720
edited Nov 27 at 3:35
SZN
2,713720
2,713720
asked Nov 21 at 23:08
Ali G
113
asked Nov 21 at 23:08
Ali G
113
asked Nov 21 at 23:08
Ali G
113
113
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
Nov 23 at 14:39
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37
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
Nov 23 at 14:39
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37
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
Nov 23 at 14:39
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
Nov 23 at 14:39
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37
add a comment |
1 Answer
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Recall that $dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$
where
$$W_c(t) = int_0^te^{Atau}BB^{T}e^{A^{T}tau}dtau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
add a comment |
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1 Answer
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up vote
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down vote
Recall that $dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$
where
$$W_c(t) = int_0^te^{Atau}BB^{T}e^{A^{T}tau}dtau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
add a comment |
up vote
0
down vote
Recall that $dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$
where
$$W_c(t) = int_0^te^{Atau}BB^{T}e^{A^{T}tau}dtau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
add a comment |
up vote
0
down vote
up vote
0
down vote
Recall that $dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$
where
$$W_c(t) = int_0^te^{Atau}BB^{T}e^{A^{T}tau}dtau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
Recall that $dot{x} = Ax + Bu$, then $(A,B)$ is controllable with control law (with minimum energy) given by
$$u(t) = -B^{T}e^{A^{T}(t_1-t)}W_c^{-1}(t_1)(e^{At_1}x_0 -x_1), $$
where
$$W_c(t) = int_0^te^{Atau}BB^{T}e^{A^{T}tau}dtau.$$
For $W_c$ nonsingular, then equivalence is $(A,B)$ is controllable.
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
answered Nov 27 at 3:57
Euler....IS_ALIVE
2,64311338
2,64311338
add a comment |
add a comment |
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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
Nov 23 at 14:39
What have you done? Perhaps it would be useful to find the solution of the system for $x$. Can you write down this formula?
– SZN
Nov 27 at 3:37