Semimodule Morphic Systems: Realization, Model Matching, and Decoupling

This dissertation studies semimodule morphic systems, which are systems evolving with variables taking values in semimodules over a semiring. Intuitively, such systems are not equipped with additive inverses. The topics in this dissertation include reachability, observability, the Kalman realization diagram, fixedpoles and fixed zeros of the model matching problem, and zerosemimodules in the disturbance decoupling problem.

The concepts of reachability and observability are defined for semimodule morphic systems. This dissertation generalizes the Kalman realization theory to semimodule morphic systems. A canonical realization of a transfer function is characterized by the property that it is reachable and observable; namely, the transfer function corresponds to a canonical factorization into an onto map and a one-to-one map. The pole semimodule of output type leads to a canonical realization for a given transfer function. The pole semimodule of input type is only a reachable realization; but it is also canonical if and only if the transfer function is steady.

Solution existence conditions for the model matching problem are established. Two fixed pole structures are introduced for solutions to the model matching problem, and relationships are established between the fixed pole semimodules and the pole semimodules of the solutions. We characterize fixed pole semimodules using pole semimodules, zero semimodules, and the extended zero semimodules of the given transfer functions. The dissertation introduces two fixed zero semimodules and establishes a connection between these fixed zero semimodules and the extended zero semimodules of the solutions. The fixed zero semimodules are characterized using pole semimodules, zero semimodules, and the extended zero semimodules of the given transfer functions.

Zero semimodules are used in the study of the disturbance decoupling problem. The research generalizes several types of invariant sub-semimodules, including (A,B)-invariant sub-semimodules, (A,B)-invariant sub-semimodules of feedback type, controllability sub-semimodules, and pre-controllability sub-semimodules. A connection between zero semimodules and these invariant sub-semimodules is established. This connection provides the relationship between the frequency domain theory and the geometric control, which helps us to obtain the solvability condition for the disturbance decoupling problem of semimodule morphic systems.