Algorithmic analysis of systems with affine input and polynomial state
Lorenzo Clemente
TL;DR
This work provides exact, terminating algorithms for semantic analysis of deterministic continuous-time control systems with affine input and polynomial state dynamics by representing the dynamics with Chen-Fliess generating series. It identifies polynomial systems with the class of shuffle-finite series, enabling decidability of zeroness, equality, input independence, linearity, and analyticity by reducing these properties to decision problems on shuffle-finite series and their commutative regular-language restrictions. A novel technical contribution shows shuffle-finite series are effectively closed under support restrictions by commutative regular languages, which underpins the regular-support inclusion and commutativity procedures. The results yield a complete, decidable framework for equivalence, input independence, linearity, and analyticity of polynomial systems (including MISO/MIMO), advancing formal verification for this important class of nonlinear systems and offering tools for exact analysis without convergence concerns.
Abstract
The goal of this paper is to provide exact and terminating algorithms for the formal analysis of deterministic continuous-time control systems with affine input and polynomial state dynamics (in short, polynomial systems). We consider the following semantic properties: zeroness and 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 formal generating series. 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 has recently been studied in the context of automata theory and enumerative combinatorics. We exploit and extend recent results in the algorithmic analysis of shuffle-finite series (such as zeroness, equivalence, and commutativity) to show that the semantic properties above can be decided exactly and in finite time for polynomial systems. Some of our analyses rely on a novel technical contribution, namely that shuffle-finite series are closed under support restrictions with commutative regular languages, a result of independent interest.