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

1. ## 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

2. Originally Posted by liptak
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

3. Originally Posted by CaptainBlack
There is no transfer function because the system is not linear.
Thanks captain. I need to mathematicaly describe such system.. what do you recommend?

4. Originally Posted by liptak
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)| the actuator is linear and is symply $\text{Sat}(s)=1$...

Kind regards

$\chi$ $\sigma$

5. Originally Posted by chisigma
If $|u(t)| 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

6. Originally Posted by liptak
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

$\chi$ $\sigma$

7. Originally Posted by chisigma
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?

8. Originally Posted by liptak
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

$\chi$ $\sigma$

9. Originally Posted by chisigma
... 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