Testing Backward-Flatness of Nonlinear Discrete-Time Systems

Despite ongoing research, testing the flatness of discrete-time systems remains a challenging problem. To date, only the property of forward-flatness - a special case of difference-flatness - can be checked in a computationally efficient manner. In this paper, we propose a systematic approach for testing backward-flatness, which is another special case of difference-flatness, and for deriving a corresponding backward-flat output. Additionally, we discuss the relationship between the Jacobian matrices associated with the flat parameterization of backward- and forward-flat systems and illustrate our results by an academic example.

💡 Research Summary

The paper addresses a notable gap in the theory of discrete‑time flatness: while efficient geometric tests exist for forward‑flat (difference‑flat) systems, no comparable method has been available for backward‑flat systems, a special subclass where the flat output depends only on past (backward) shifts. The authors introduce a systematic approach that leverages the existing forward‑flat test by constructing an “associated system” whose trajectories are in a one‑to‑one correspondence with those of the original system but with time reversed.

Starting from a general nonlinear time‑invariant discrete‑time system
(x^{+}=f(x,u)) with full‑rank input Jacobian and submersivity, they define the inverse diffeomorphism ((\psi_x,\psi_u)) such that ((x^{+},\zeta)= (f(x,u),g(x,u))) can be locally inverted. Using this inverse, they build the associated system
(z^{+}= \psi_x(z,v),\quad \eta = \psi_u(z,v)).
A key theorem proves that the trajectories of the original and associated systems satisfy
(x(k+l)=z(k-l+1),; u(k+l)=\eta(k-l),; \zeta(k+l)=v(k-l)) for all integers (l). Consequently, flatness of one system implies flatness of the other.

The authors then specialize to the forward‑flat and backward‑flat subclasses. Corollary 9 shows a duality: a system is backward‑flat iff its associated system is forward‑flat, and vice‑versa. This duality enables an indirect test for backward‑flatness: construct the associated system, apply the existing geometric forward‑flat algorithm (Algorithm 1 from Kolar et al. 2023), and interpret a positive result as proof of backward‑flatness for the original system.

Algorithm 1 iteratively builds a sequence of distributions (E_k) on the extended state‑input manifold. Starting with the input distribution (E_0=\text{span}{\partial_{u_i}}), it extracts the largest projectable subdistribution, pushes it forward through the system map, pulls it back via the projection, and repeats until the dimension stabilizes. The system is forward‑flat precisely when the final dimension equals the total number of states plus inputs ((n+m)).

Beyond the algorithmic contribution, the paper analyses the Jacobian matrices of the flat parameterizations. For forward‑flat systems, certain sub‑blocks of the Jacobians (\partial_xF_x) and (\partial_uF_u) must be full rank. In the backward‑flat case these sub‑blocks appear with reversed shift indices, and the authors demonstrate that the Jacobians of the original and associated systems are related by transposition of the relevant blocks. This algebraic insight clarifies why the forward‑flat test can be reused for backward‑flat verification.

An academic example with a two‑dimensional nonlinear system illustrates the theory. The original system is shown to be backward‑flat with a flat output involving only past states and inputs. After constructing its associated system, the forward‑flat algorithm confirms flatness, and the backward‑flat output is recovered by time‑reversing the forward‑flat output of the associated system. Rank checks on the Jacobian sub‑matrices corroborate the theoretical predictions.

In summary, the paper provides a novel, computationally tractable method for testing backward‑flatness of nonlinear discrete‑time systems by exploiting a time‑reversed associated system and the existing forward‑flat geometric test. It also clarifies the structural relationship between the Jacobians of forward‑ and backward‑flat parameterizations, thereby enriching the theoretical foundation of difference‑flatness and opening avenues for practical control design, trajectory planning, and system identification in discrete‑time nonlinear settings.