Mixing
Help undersanding system order reduction method
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I'm trying to understand the method exposed in Dorf's "Modern Control Systems", for approximating a high order system with a lower order one. The method goes as follows (using the exact same notation): Let $G_H(s)$ and $G_L(s)$, be the transfer functions of the original system and the approximate one, respectively. Take $G_H(s)/G_L(s)$, and call $M(s)$ and $Delta(s)$ the numerator and denominator polynomials of the resulting quotient, respectively. Define
$$M_2q = sum_k=0^2q frac(-1)^k+qM^(k)(0)M^(2q-k)(0)k!(2q-k)!$$
and
$$Delta_2q = sum_k=0^2q frac(-1)^k+qDelta^(k)(0)Delta^(2q-k)(0)k!(2q-k)!$$
and make $M_2q=Delta_2q$, for $q=0,1,2dots$, as needed to solve for all the unknowns in $G_L(s)$.
The questions is, why this works, how'd they derived it. I tried to undersand it by looking at the book's references and found a paper from the 60s by E. Davison but couldn't access it (paywall). Any help to see how it works or another source to delve in.
dynamical-systems control-theory
asked Jul 27 at 1:58
Felipe Vega
163
add a comment |Â
up vote
2
down vote
favorite
I'm trying to understand the method exposed in Dorf's "Modern Control Systems", for approximating a high order system with a lower order one. The method goes as follows (using the exact same notation): Let $G_H(s)$ and $G_L(s)$, be the transfer functions of the original system and the approximate one, respectively. Take $G_H(s)/G_L(s)$, and call $M(s)$ and $Delta(s)$ the numerator and denominator polynomials of the resulting quotient, respectively. Define
$$M_2q = sum_k=0^2q frac(-1)^k+qM^(k)(0)M^(2q-k)(0)k!(2q-k)!$$
and
$$Delta_2q = sum_k=0^2q frac(-1)^k+qDelta^(k)(0)Delta^(2q-k)(0)k!(2q-k)!$$
and make $M_2q=Delta_2q$, for $q=0,1,2dots$, as needed to solve for all the unknowns in $G_L(s)$.
The questions is, why this works, how'd they derived it. I tried to undersand it by looking at the book's references and found a paper from the 60s by E. Davison but couldn't access it (paywall). Any help to see how it works or another source to delve in.
dynamical-systems control-theory
asked Jul 27 at 1:58
Felipe Vega
163
Why the downvote?
– Chandler Watson
Jul 27 at 2:31
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm trying to understand the method exposed in Dorf's "Modern Control Systems", for approximating a high order system with a lower order one. The method goes as follows (using the exact same notation): Let $G_H(s)$ and $G_L(s)$, be the transfer functions of the original system and the approximate one, respectively. Take $G_H(s)/G_L(s)$, and call $M(s)$ and $Delta(s)$ the numerator and denominator polynomials of the resulting quotient, respectively. Define
$$M_2q = sum_k=0^2q frac(-1)^k+qM^(k)(0)M^(2q-k)(0)k!(2q-k)!$$
and
$$Delta_2q = sum_k=0^2q frac(-1)^k+qDelta^(k)(0)Delta^(2q-k)(0)k!(2q-k)!$$
and make $M_2q=Delta_2q$, for $q=0,1,2dots$, as needed to solve for all the unknowns in $G_L(s)$.
The questions is, why this works, how'd they derived it. I tried to undersand it by looking at the book's references and found a paper from the 60s by E. Davison but couldn't access it (paywall). Any help to see how it works or another source to delve in.
dynamical-systems control-theory
asked Jul 27 at 1:58
Felipe Vega
163
I'm trying to understand the method exposed in Dorf's "Modern Control Systems", for approximating a high order system with a lower order one. The method goes as follows (using the exact same notation): Let $G_H(s)$ and $G_L(s)$, be the transfer functions of the original system and the approximate one, respectively. Take $G_H(s)/G_L(s)$, and call $M(s)$ and $Delta(s)$ the numerator and denominator polynomials of the resulting quotient, respectively. Define
$$M_2q = sum_k=0^2q frac(-1)^k+qM^(k)(0)M^(2q-k)(0)k!(2q-k)!$$
and
$$Delta_2q = sum_k=0^2q frac(-1)^k+qDelta^(k)(0)Delta^(2q-k)(0)k!(2q-k)!$$
and make $M_2q=Delta_2q$, for $q=0,1,2dots$, as needed to solve for all the unknowns in $G_L(s)$.
The questions is, why this works, how'd they derived it. I tried to undersand it by looking at the book's references and found a paper from the 60s by E. Davison but couldn't access it (paywall). Any help to see how it works or another source to delve in.
dynamical-systems control-theory
asked Jul 27 at 1:58
Felipe Vega
163
asked Jul 27 at 1:58
Felipe Vega
163
asked Jul 27 at 1:58
Felipe Vega
163
asked Jul 27 at 1:58
Felipe Vega
163
163
Why the downvote?
– Chandler Watson
Jul 27 at 2:31
add a comment |Â
Why the downvote?
– Chandler Watson
Jul 27 at 2:31
Why the downvote?
– Chandler Watson
Jul 27 at 2:31
Why the downvote?
– Chandler Watson
Jul 27 at 2:31
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863980%2fhelp-undersanding-system-order-reduction-method%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Why the downvote?
– Chandler Watson
Jul 27 at 2:31