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I have always wondered about the relationship between feedback control and loop shaping mainly pertaining to the compatibility between information that can be derived from both of them.

Point in case, consider a static output feedback that stabilizes a given linear system. Now, just fooling around with the static output feedback gains, one may realize that there is a limitation to, say, speed of convergence and damping, for example, that can be achieved. While this is true for static output feedback, would this observation limit the perforamnce, in those specific terms, that can be achieved with loop shaping?

What I think:

In principal, you can cancel out all the poles and zeros of the transfer function using loop shaping (it is highly advised against since you would not be able to deal with initial condition if the original system is unstable and you, also, never know the zeros and poles of the system exactly anyways). The reason for making this point is, in principal, you can make the system behave in whatever manner you want for zero initial condition. This shows that the limitations of static output feedback do not, necessarily, apply to loop shaping techniques.

Though a trivial statement, it is worthwhile to note that the state space approach of static output feedback gives more intuitive understanding of system response in terms of eigenvalue location while the loop shaping techniques tell more about the frequency domain response. I know it is possible to convert any loop shaping design to a time domain representation (mainly minimal realization) but given the order of systems we generally end up using loop shaping, it doesnt sound like a good line of investigation.

To cut the long story short, using one technique, can comments about performance of the system be made when the other is used?

P.S. I know its a loaded question but I really would like to hear opinions.

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  • $\begingroup$ Check out the bode's sensitivity integral, en.wikipedia.org/wiki/Bode%27s_sensitivity_integral. $\endgroup$ – WG- Sep 27 '16 at 10:13
  • $\begingroup$ @WG- the Bode Integral does not apply in all cases OP is considering; you assume the system is LTI. $\endgroup$ – SZN Dec 17 '16 at 0:53
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As far as I can tell, this has less to do with a design methodology like loop shaping and more to do with controllability. If the system is controllable and you do not restrict the controller form, there's nothing (in principle) to stop you from performing a dynamic inversion and inserting the dynamics you want on top. The reason this is never done is (1) the system must be REALLY stable, or else the pole placement errors you mention can lead to the growth of unbounded modes (which you also appear to understand), and (2) most actuators don't have that sort of control authority.

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