Assume you have a cantilever beam connected to a rotating hub . Such a system has an infinite number of degrees of freedom. for this system , you can only measure the angular displacement and velocity of the hub as well as the deflection of the tip of the beam. If I consider 6 shape modes of the beam , the system become 7 DOF. How can design a controller for this system knowing that the 6 modes are not observable?
If the system is controllable, you need to design a controller in such a way that it is robust enough to account for uncertainty in the plant dynamics. Evidently, you will need certain assumptions about this uncertainty, for example, that it is bounded and Lipschitz.
If the system is non-observable, its transfer function has zero-pole cancellations which are zeros of a certain polynomial of degree greater than or equal to one ( the number of cancelled modes) . It is always possible to choose a state-space realization which is controllable, taht is, a transfer function with cancellations can be internally represented by state space realizations being either controllable or either observable ( this fails from the hypothesis here) but not both ( this fails too here). So , the problem can be solved with a feedback controller of transfer function S/R ( S, R polynomials) calculated from the diophantine equation of polynomials:
(AC)R+(BC)S=MC
where G = (BC)/(AC) is the transfer function of the controlled plant, C is the polynomial of cancelled poles and zeros ( it exists as a polynomial of degree being non less than one since the system is not observable). M is the set of suited ( stable) closed -loop poles . The closed -loop poles contain the cancelled pole-zero open-loop poles so that C has to be stable (i.e., the cancellations have to be stable). the above methos is useful for continuous-time systems ( polynomials in s, argument of the laplace transform) or in the derivative operator D=d/dt) and for discrete-time ones ( polynomials in z of the z- transform or q, or q(-1) , the discrete one -step time advance or delay operator).
The above controller does not need the state measurement for implementation, it needs only output measurement, but it needs the knowledge of the location of the stable unobservable modes ( the zeros of M) .
As explained by Manuel, yo can follow the mathematical procedure for the design of a controller.
Another approach would be to design a non-linear controller based on fuzzy logic with your experience on how to control the plant. Afterwards you could also optimize this controller with Genetic Algorithms for best performance. Ideally you would require the system's model in order to be able to perform simulations before its real-time implementations.
Good luck.
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