Algebraic Approach to Nonlinear Systems Theory

We pursue a new framework of nonlinear systems theory utilizing algebra and algebraic geometry. For example, commutative algebra is often useful to extend linear systems theory on the basis of linear algebra. We aim to establish new methodologies to solve outstanding problems for a reasonably wide class of systems by exploiting such special structures in system models as rational functions or polynomial functions. Although this approach is in contrast to real-time optimization by numerical computation, they still share the same goal: practical construction of a solution in a wide variety of problems.

One of our goals is to extend all important results in linear systems theory, i.e., controllability, observability, stability, state feedback, and optimal regulator and observer. Moreover, we also try to generalize such important notions in linear systems theory as transfer functions and eigenvalues by exploiting noncommutative rings of differential operators.


■ Keywords

Nonlinear systems, Polynomial systems, Commutative algebra, Algebraic geometry, Noncommutative algebra, Symbolic computation.

References

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