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| We study analytic control systems on time scales, with output. Under certain conditions such a system may be immersed into another system with simpler structure: affine, polynomial or rational with respect to the state variables. Though the state space of the simpler system is usually higher dimensional than the state space of the original system, but the simpler structure may have some advantage. We construct the observation space, observation algebra and observation field of the original analytic system and show that these objects play fundamental role in simplification of the structure of the system. We show that the system may be immersed into an affine system if and only if the observation space of the system is finitely dimensional. Similarly, the system may be immersed into a polynomial system if and only if the observation algebra of the system is contained in some finitely generated algebra, and the system may be immersed into a rational system if and only if the observation field of the system is a finitely generated extension of the field of reals. We also show that the observation algebra and the observation field are a skew differential algebra and a skew differential field, respectively, with respect to some skew derivation. |
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