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Math Help - Control theory - Closed-loop transfer function with actuator saturation

  1. #1
    Newbie liptak's Avatar
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    Question Control theory - Closed-loop transfer function with actuator saturation

    Hey guys,

    consider a closed-loop with actuator saturation (see the image). What is the transfer function of this saturation block? And what is the closed-loop transfer function of the system?

    I came up with this
    Closed-loop transfer function:
    H(s)= K(s)SAT(s)G(s) / 1+K(s)SAT(s)G(s)
    where
    SAT(u)=sign(u).min(|u|, u_lim) (u is bounded from -u_lim to u_lim)

    but I don't think it's correct...

    Any help would be greatly appreciated. Thanks A LOT
    Have a great day

    Liptak
    Attached Thumbnails Attached Thumbnails Control theory - Closed-loop transfer function with actuator saturation-closed-loop-sat.png  
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by liptak View Post
    Hey guys,

    consider a closed-loop with actuator saturation (see the image). What is the transfer function of this saturation block? And what is the closed-loop transfer function of the system?

    I came up with this
    Closed-loop transfer function:
    H(s)= K(s)SAT(s)G(s) / 1+K(s)SAT(s)G(s)
    where
    SAT(u)=sign(u).min(|u|, u_lim) (u is bounded from -u_lim to u_lim)

    but I don't think it's correct...

    Any help would be greatly appreciated. Thanks A LOT
    Have a great day

    Liptak
    There is no transfer function because the system is not linear.

    CB
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  3. #3
    Newbie liptak's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    There is no transfer function because the system is not linear.
    Thanks captain. I need to mathematicaly describe such system.. what do you recommend?
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by liptak View Post
    Hey guys,

    consider a closed-loop with actuator saturation (see the image). What is the transfer function of this saturation block? And what is the closed-loop transfer function of the system?

    I came up with this
    Closed-loop transfer function:
    H(s)= K(s)SAT(s)G(s) / 1+K(s)SAT(s)G(s)
    where
    SAT(u)=sign(u).min(|u|, u_lim) (u is bounded from -u_lim to u_lim)

    but I don't think it's correct...

    Any help would be greatly appreciated. Thanks A LOT
    Have a great day

    Liptak
    If |u(t)|<u_{\text{lim}} the actuator is linear and is symply \text{Sat}(s)=1...

    Kind regards

    \chi \sigma
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  5. #5
    Newbie liptak's Avatar
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    Quote Originally Posted by chisigma View Post
    If |u(t)|<u_{\text{lim}} the actuator is linear and is simply \text{Sat}(s)=1...
    Thank you for the answer. Yeah, but if |u(t)| >u_{\text{lim}}, then saturation is non-linear, right? Saturation is non-linear, so we can't build a closed-loop transfer function, which is for linear time-invariant systems only (thanks captain, i should know that already ). How to describe it? Through non-linear differential equations or state space equations? Oh man, I'm screwed
    Last edited by liptak; March 26th 2011 at 07:13 AM.
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  6. #6
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by liptak View Post
    Thank you for the answer. Yeah, but if |u(t)| >u_{\text{lim}}, then saturation is non-linear, right? Saturation is non-linear, so we can't build a closed-loop transfer function, which is for linear time-invariant systems only (thanks captain, i should know that already ). How to describe it? Through non-linear differential equations or state space equations? Oh man, I'm screwed
    Pratically, from the point of view of control theory, what is important is to extablish if a control system is stable with sufficient margin, if it has low sensitivity to disturbances and 'reasonable' transient time. All these parameters are of pratical importance if the system operates in linearity and that means that the actuator can be considered as a unity gain block...

    Kind regards

    \chi \sigma
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  7. #7
    Newbie liptak's Avatar
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    Quote Originally Posted by chisigma View Post
    Pratically, from the point of view of control theory, what is important is to extablish if a control system is stable with sufficient margin, if it has low sensitivity to disturbances and 'reasonable' transient time. All these parameters are of pratical importance if the system operates in linearity and that means that the actuator can be considered as a unity gain block...
    So you're saying that actuator saturation doesn't affect stability of the control system?
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  8. #8
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by liptak View Post
    So you're saying that actuator saturation doesn't affect stability of the control system?
    ... more precisely the saturation of the actuator prevents the 'explosion' of the system in case of instability...

    Kind regards

    \chi \sigma
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  9. #9
    Newbie liptak's Avatar
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    Quote Originally Posted by chisigma View Post
    ... more precisely the saturation of the actuator prevents the 'explosion' of the system in case of instability...
    Ok, thank you, you've helped me tremendously.

    Have a nice day

    Liptak
    Last edited by liptak; March 27th 2011 at 11:34 PM.
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