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Hankel singular values of Discrete time system and its inverse.

0

$begingroup$

An $n$ dimensional discrete-time system $Sigma$ is describe by the ewuations
$$x^{+}=Ax+Bu, quad y=Cx+Du $$

and has the property that $D$ is invertib;e. Its inverse system $Sigma^{-1}$ is described by the state space equations
$$xi^+=(A-BD^{-1}C)xi+BD^{-1}y, quad u=-D^{-1}Cxi+D^{-1}y$$

(Here, $x^+$ denotes the time shifted signal $x^+(t)=x(t+1)$)

a) $quad$ Show that the transfer function of $Sigma^{-1}$ is the inverse of the transfer function of $Sigma$.

b) $quad$ Show that $Sigma$ is reachable if and only if $Sigma^{-1}$ is reachable.

c) $quad$ Assuming that both $Sigma$ and $Sigma^{-1}$ are stable, is it true that the Hankel singular values of $Sigma^{-1}$ are $quad quad$the inverses of the Hankel singular values of $Sigma$?

mathematical-modeling

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asked Mar 22 at 13:07

user463102user463102

7913

$endgroup$

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0

$begingroup$

An $n$ dimensional discrete-time system $Sigma$ is describe by the ewuations
$$x^{+}=Ax+Bu, quad y=Cx+Du $$

and has the property that $D$ is invertib;e. Its inverse system $Sigma^{-1}$ is described by the state space equations
$$xi^+=(A-BD^{-1}C)xi+BD^{-1}y, quad u=-D^{-1}Cxi+D^{-1}y$$

(Here, $x^+$ denotes the time shifted signal $x^+(t)=x(t+1)$)

a) $quad$ Show that the transfer function of $Sigma^{-1}$ is the inverse of the transfer function of $Sigma$.

b) $quad$ Show that $Sigma$ is reachable if and only if $Sigma^{-1}$ is reachable.

c) $quad$ Assuming that both $Sigma$ and $Sigma^{-1}$ are stable, is it true that the Hankel singular values of $Sigma^{-1}$ are $quad quad$the inverses of the Hankel singular values of $Sigma$?

mathematical-modeling

share|cite|improve this question

asked Mar 22 at 13:07

user463102user463102

7913

$endgroup$

add a comment | 

$begingroup$

An $n$ dimensional discrete-time system $Sigma$ is describe by the ewuations
$$x^{+}=Ax+Bu, quad y=Cx+Du $$

and has the property that $D$ is invertib;e. Its inverse system $Sigma^{-1}$ is described by the state space equations
$$xi^+=(A-BD^{-1}C)xi+BD^{-1}y, quad u=-D^{-1}Cxi+D^{-1}y$$

(Here, $x^+$ denotes the time shifted signal $x^+(t)=x(t+1)$)

a) $quad$ Show that the transfer function of $Sigma^{-1}$ is the inverse of the transfer function of $Sigma$.

b) $quad$ Show that $Sigma$ is reachable if and only if $Sigma^{-1}$ is reachable.

c) $quad$ Assuming that both $Sigma$ and $Sigma^{-1}$ are stable, is it true that the Hankel singular values of $Sigma^{-1}$ are $quad quad$the inverses of the Hankel singular values of $Sigma$?

mathematical-modeling

share|cite|improve this question

asked Mar 22 at 13:07

user463102user463102

7913

$endgroup$

share|cite|improve this question

asked Mar 22 at 13:07

user463102user463102

7913

share|cite|improve this question

asked Mar 22 at 13:07

user463102user463102

7913

asked Mar 22 at 13:07

user463102user463102

7913

asked Mar 22 at 13:07

user463102user463102

7913