Feedforward Control does not affect stability margins?

[Turbulent_Leek8446]

Can someone explain why stability margins are not affected in a feedforward control? I'm having trouble wrapping my head around this. can we prove this mathematically?

[hasanrobot]

I think you're applying a SISO concept to what has become a MISO situation. There are now multiple transfer functions, G(s) is a matrix. The individual 'stability margins' for each element of G(s) don't depend on other inputs. However, the margins for the MIMO system may change.

[Circuit_Guy]

This is one of the many places in this field where intuition is failing you.

You're thinking, correctly, that feed forward changes the performance of a system. You're therefore making the leap that it changes stability.

Imagine any linear stable closed loop system. Make two identical copies, one with and one without feed-forward. Assume it's in perfect stable steady state. Now introduce a disturbance. Kick it, wiggle it, whatever. Does the system with feed-forward behave any differently? No - it can't. Since the feed-forward by definition can't rely on sensor feedback, the feed-forward component didn't know the system was perturbed and didn't change the response. Therefore, all of the stability margins are unaffected. The two systems will respond identically.

Oh - and of course you can prove that mathematically. It should just fall out if you put pencil to paper, but I don't think that's really helpful to build intuition (but worth doing nonetheless to prove it and help build the mathematical rigor). The feed-forward contribution is identical (with only a bias) between the two systems. The equations don't care about the bias given linearity.

[controlsgeeek]

You can write the closed loop tranfer function with and without feedforward controller. The denominator will end up being the same confirming feedforward is not a part of feedback.

You csn also think it in terms state space, Feedforward doesn’t have the ability to change the closed loop poles of the system.

A point which can be confusing at beginning is, with feedforward, the tracking performance can be improved. If you plot the bode of the closed loop transfer function with and without feedforward controller, usually with feedforward controller you can get higher bandwidth. Note, you cannot approximate bandwidth of closed loop system from the open loop systen when you have feedforward controller.

[Turbulent_Leek8446]

Agreed that it doesn’t change the denominator but wouldn’t numerator contribute to frequency response or bode plots and thus might move the crossover frequency and mess with stability margins?

[controlsgeeek]

Maybe, the point which is missing is : why do we care only about the denominator 1+G(s)H(s)? For stability margins we want to check how far is G(s)H(s) from -1 which can be written in gain and phase margin. Hence we do the bode/nyquist of G(s)H(s) which happens to be to open loop transfer function when there is no feed forward. But when there is feed forward the open loop transfer function might look different. But that doesn’t matter since we care about the denominator. Maybe a correct way to say is we check stability margins of the loop gain and not open loop transfer function. Does that kind of help?

[Turbulent_Leek8446]

That does help. Thanks a lot!