Algorithmic analysis of control systems with affine input and polynomial state dynamics

Abstract: We provide simple algorithms for the formal analysis of deterministic continuous-time control systems whose dynamics is affine in the input and polynomial in the state (in short, polynomial systems). We consider the following semantic properties: input-output equivalence, input independence, linearity, and analyticity. Our approach is based on Chen-Fliess series, which provide a unique representation of the dynamics of such systems via their generating series (in noncommuting indeterminates). Our starting point is Fliess' seminal work showing how the semantic properties above are mirrored by corresponding combinatorial properties on generating series. Next, we observe that the generating series of polynomial systems coincide with the class of shuffle-finite series, a nonlinear generalisation of Schützenberger's rational series which we have recently studied in the context of automata theory and enumerative combinatorics. We exploit and extend recent results in the algorithmic analysis of shufflef-finite series to show that the semantic properties above are decidable for polynomial systems.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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