Discrete Zero Dynamics in Control Systems

• Discrete Zero Dynamics (DZD) is a reduced-order model capturing internal dynamics that remain when outputs are driven to zero.
• It applies to hybrid underactuated, sampled-data, and linear systems, enabling stability certification, orbital analysis, and controller synthesis.
• DZD extends to cyber-physical security and topological dynamics, offering insights into stealth attacks and discrete spectrum behavior.

Searching arXiv for papers on Discrete Zero Dynamics and closely related usages. Discrete Zero Dynamics (DZD) denotes a reduced-order description of internal or passive dynamics under output-zeroing constraints in discrete time. In control-theoretic usage, especially for hybrid underactuated systems, sampled-data nonlinear systems, and discrete-time linear systems, DZD is the impact-to-impact or sample-to-sample evolution that remains when actuated coordinates or measured outputs are constrained by feedback, virtual constraints, or zero-output conditions (Csomay-Shanklin et al., 2024, Shenoy et al., 2022, Schaa et al., 8 Jan 2026). A distinct usage also appears in topological dynamics, where “Discrete Zero Dynamics” refers to zero-entropy systems whose invariant measures have discrete spectrum (Li et al., 2018). Across these settings, the common thread is dimensional reduction: DZD isolates the internal dynamics consistent with an imposed output relation, making it central to stability certification, orbital analysis, controller synthesis, data-driven inference, and attack analysis.

1. Control-theoretic definition and formal setting

In hybrid underactuated robotics, the starting point is a constrained Lagrangian model

$$D(q)\ddot{q} + H(q,\dot{q}) = B u + J(q)^\top \lambda,$$

with underactuation expressed by $\operatorname{rank}(B)=m<n$, and a hybrid structure given by continuous flow and impact resets (Csomay-Shanklin et al., 2024). The unconstrained dynamics are written in control-affine form

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$

with impacts

$$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$

and a guard $\mathcal{S}$ marking touchdown or liftoff events (Csomay-Shanklin et al., 2024).

A diffeomorphic change of coordinates separates actuated and underactuated components:

$$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$

where $\eta$ collects the actuated coordinates and $z$ the underactuated coordinates (Csomay-Shanklin et al., 2024). In these coordinates, the input acts directly on $\eta$, whereas $z$ is not directly influenced by the input, expressed by $\operatorname{rank}(B)=m<n$0 (Csomay-Shanklin et al., 2024).

The zeroing relation is specified by outputs of the form

$\operatorname{rank}(B)=m<n$1

so that the associated manifold is

$\operatorname{rank}(B)=m<n$2

When the closed-loop system is constrained to this manifold, the remaining discrete step-to-step evolution of $\operatorname{rank}(B)=m<n$3 is the DZD (Csomay-Shanklin et al., 2024). In the same spirit, for discrete-time linear systems with

$\operatorname{rank}(B)=m<n$4

the zero dynamics are the internal motions consistent with $\operatorname{rank}(B)=m<n$5 under suitable $\operatorname{rank}(B)=m<n$6, and minimum phase corresponds to stability of these internal motions (Schaa et al., 8 Jan 2026).

This definition extends beyond hybrid legged systems. In sampled-data nonlinear systems written in normal form, the internal coordinates $\operatorname{rank}(B)=m<n$7 evolve according to a sampled map

$\operatorname{rank}(B)=m<n$8

which the paper explicitly interprets as the sampled zero dynamics, with fixed points and periodic points corresponding to intersections of a continuous-time limit cycle with a Poincaré section (Shenoy et al., 2022). In impulsive underactuated juggling, DZD arises from discrete virtual holonomic constraints and governs the passive orientation variables at impact times (Khandelwal et al., 9 Sep 2025, Khandelwal et al., 20 Aug 2025).

2. Hybrid locomotion, Poincaré maps, and invariant manifolds

For hybrid locomotion, DZD is obtained by composing flow-to-impact with reset, producing an impact-to-impact map

$\operatorname{rank}(B)=m<n$9

which in $$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$0 coordinates takes the form

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$1

under the ARCHER assumption that the input weakly affects impact time (Csomay-Shanklin et al., 2024). This is explicitly identified as a Poincaré-type return map evaluated at impact (Csomay-Shanklin et al., 2024).

If the manifold $$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$2 is controlled invariant, meaning that for all $$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$3 there exists a discrete control parameter $$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$4 such that the next state remains on the manifold, then the restricted autonomous dynamics

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$5

is the DZD (Csomay-Shanklin et al., 2024). This controlled invariance is weaker than classical hybrid invariance, since the paper learns and enforces discrete invariance under the optimal action rather than assuming a priori hybrid invariance of the zeroing manifold (Csomay-Shanklin et al., 2024).

The stability interpretation is standard Poincaré theory in reduced order. If $$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$6 is a fixed point,

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$7

then the Jacobian

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$8

provides the Floquet multipliers, and eigenvalues inside the unit circle imply local exponential orbital stability of the periodic motion on the zeroing manifold (Csomay-Shanklin et al., 2024). The same reduction principle appears in classical Poincaré notation

$$\dot{\boldsymbol{x}} = \boldsymbol{f}(\boldsymbol{x}) + \boldsymbol{g}(\boldsymbol{x})\,\boldsymbol{u},$$9

with DZD corresponding to the restriction $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$0 (Csomay-Shanklin et al., 2024).

A closely related construction appears in devil-stick juggling with discrete virtual holonomic constraints. There the DZD is an implicit reduced-order map in the passive variables $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$1, and the paper states that it “provides conditions for stable juggling” (Khandelwal et al., 9 Sep 2025). In the propeller-motion variant, the DZD is a 2D autonomous map in $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$2 obtained after slaving the center-of-mass coordinates to a discrete geometric constraint (Khandelwal et al., 20 Aug 2025). These cases show that DZD is not restricted to legged locomotion; it is a general reduced model for impulsive underactuated orbital tasks.

3. Zero Dynamics Policies and learned DZD manifolds

“Robust Agility via Learned Zero Dynamics Policies” develops a learned realization of DZD for hybrid underactuated systems (Csomay-Shanklin et al., 2024). The central object is a mapping

$$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$3

called a Zero Dynamics Policy (ZDP), with transverse error

$$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$4

Because $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$5 depends only on the underactuated degrees of freedom, the method “achiev[es] significant dimension reduction” while preserving structure induced by underactuation (Csomay-Shanklin et al., 2024).

The learning objective enforces one-step invariance under an optimal discrete action computed by iLQR:

$$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$6

where

$$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$7

Achieving small loss implies discrete invariance under the optimal action and therefore well-defined DZD on the learned manifold (Csomay-Shanklin et al., 2024).

The control architecture separates manifold design from transverse stabilization. During continuous evolution, $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$8 is driven to $$\boldsymbol{x}^+ = \boldsymbol{\Delta}(\boldsymbol{x}^-),$$9 using feedback linearization and RES-CLF or PD, while the reduced step-to-step behavior is governed by the DZD (Csomay-Shanklin et al., 2024). The paper proves a constructive stabilization lemma stating that there exists a controller such that

$\mathcal{S}$0

under assumptions including feedback-linearizable $\mathcal{S}$1 dynamics, an RES-CLF inequality, a Lipschitz impact map, and a lower bound on impact time (Csomay-Shanklin et al., 2024).

This leads to a composite theorem: if the manifold is controlled invariant and the DZD is exponentially stable, then any controller that exponentially stabilizes the transverse error also exponentially stabilizes the full discrete system (Csomay-Shanklin et al., 2024). The proof uses the composite Lyapunov function

$\mathcal{S}$2

with cross-coupling terms bounded via Lipschitz constants (Csomay-Shanklin et al., 2024). This makes DZD not only a reduced model, but the core stability certificate for the full hybrid closed loop.

4. Sampled-data and data-driven formulations

In sampled-data nonlinear systems, discretization changes the role of zero dynamics. “Data-Driven Feedback Linearization of Nonlinear Systems with Periodic Orbits in the Zero-Dynamics” studies systems whose zero dynamics possess a stable periodic orbit (Shenoy et al., 2022). In normal form,

$\mathcal{S}$3

with zero-dynamics manifold $\mathcal{S}$4 (Shenoy et al., 2022). Under zero-order hold, the sampled internal dynamics become

$\mathcal{S}$5

which the paper explicitly calls the DZD and interprets as a Poincaré map along the limit cycle on $\mathcal{S}$6 (Shenoy et al., 2022).

A central conclusion is that higher-order internal-dynamics terms enter the sampled controllable subsystem as disturbance-like terms, and that coupling from the internal periodic orbit prevents asymptotic convergence of the controllable states unless the coupling vanishes (Shenoy et al., 2022). The Lyapunov estimate

$\mathcal{S}$7

shows that the bounded but nonvanishing limit-cycle amplitude in $\mathcal{S}$8 produces a persistent residual term (Shenoy et al., 2022). This suggests a broader lesson: in sampled-data settings, DZD is not only a reduced model for stability, but also the mechanism by which discretization-induced coupling constrains achievable convergence rates.

For discrete-time linear systems, “Data-Based Analysis of Relative Degree and Zero Dynamics in Linear Systems” gives an input-output, data-driven characterization of zero dynamics without explicit model identification (Schaa et al., 8 Jan 2026). The paper defines the zero-dynamics behavior as

$\mathcal{S}$9

and studies stability of the internal motions consistent with $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$0 (Schaa et al., 8 Jan 2026). The approach is based on Hankel matrices, Willems’ Fundamental Lemma, the lag $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$1, and the most powerful unfalsified model (MPUM) (Schaa et al., 8 Jan 2026).

A key construction uses the subspace

$$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$2

selects a basis of zero-dynamics initializations, and forms a data-driven recursion matrix $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$3 whose Schur stability is equivalent to stability of the zero dynamics (Schaa et al., 8 Jan 2026). Algorithm 2 then determines whether the data are informative for stability, instability, or ambiguity of the zero dynamics (Schaa et al., 8 Jan 2026). In this formulation, DZD is an internal behavior inferred directly from trajectories rather than a model-derived state-space reduction.

The same discrete-time viewpoint underlies data-driven geometric control. “Data-driven Meets Geometric Control: Zero Dynamics, Subspace Stabilization, and Malicious Attacks” defines the output-nulling controlled invariant subspace $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$4 and describes DZD as the internal state evolution restricted to $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$5 under inputs that render the output identically zero (Celi et al., 2022). That paper also gives a data-driven invariant-zero test and shows how undetectable attacks can be synthesized by driving the state along zero dynamics using only measured trajectories (Celi et al., 2022).

5. Stability, periodic orbits, and orbital control

DZD is especially important when the target behavior is periodic rather than equilibrium stabilization. In the devil-stick juggling problem, the DZD is the reduced map governing the passive orientation and angular velocity at impulse times once the discrete virtual holonomic constraint has been enforced (Khandelwal et al., 9 Sep 2025). The paper’s theorem states that the reduced dynamics is period-2 if and only if

$$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$6

that is, the two impulsive orientations are symmetric about the vertical axis (Khandelwal et al., 9 Sep 2025). Under this symmetry, the DZD admits infinitely many stable 2-periodic orbits parameterized by $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$7 (Khandelwal et al., 9 Sep 2025).

The propeller-motion paper gives a different but related picture. There the DZD is

$$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$8

together with a scalar nonlinear relation coupling $$\boldsymbol{\eta} = \boldsymbol{\Phi}_{\eta}(\boldsymbol{x}), \qquad \boldsymbol{z} = \boldsymbol{\Phi}_{z}(\boldsymbol{x}),$$9 and $\eta$0 (Khandelwal et al., 20 Aug 2025). For $\eta$1 and $\eta$2, the paper proves existence of $\eta$3-periodic solutions under the stated conditions, and then shows that these periodic DZD orbits are “stable but not attractive,” with Floquet matrix eigenvalues of unit modulus verified numerically (Khandelwal et al., 20 Aug 2025). Orbit selection is achieved by an impulse-controlled Poincaré map, linearized as

$\eta$4

followed by discrete feedback that places the eigenvalues of $\eta$5 strictly inside the unit circle (Khandelwal et al., 20 Aug 2025).

This use of DZD for orbital analysis is conceptually aligned with the ARCHER hopping results. There, stability of the full closed-loop hybrid system follows from exponentially stable DZD combined with fast transverse stabilization (Csomay-Shanklin et al., 2024). In both cases, DZD acts as the decisive low-dimensional model on which periodic motion is designed, classified, and stabilized.

A common misconception is that DZD is merely a notational shorthand for any Poincaré map. The literature summarized here is narrower. The reduced map must be the autonomous internal evolution that remains after enforcing a zeroing relation, discrete virtual holonomic constraint, or output-nulling condition (Csomay-Shanklin et al., 2024, Khandelwal et al., 9 Sep 2025, Schaa et al., 8 Jan 2026). A plausible implication is that not every return map is a DZD; the reduction must arise from output elimination or manifold restriction.

6. Security, invariant zeros, and stealth attacks

In cyber-physical security, DZD appears through invariant zeros and output-nulling trajectories. For a discrete-time system

$\eta$6

the Rosenbrock matrix

$\eta$7

defines discrete zeros via rank loss (Eslami et al., 9 May 2025). A zero-dynamics attack injects

$\eta$8

with nontrivial $\eta$9 satisfying the Rosenbrock condition, so that $z$0 while the state evolves as $z$1 (Eslami et al., 9 May 2025). If $z$2, the hidden state may diverge; if $z$3, it decays (Eslami et al., 9 May 2025).

“Zero-dynamics Attack, Variations, and Countermeasures” emphasizes that sampled-data implementation can create “sampling zeros,” so that even a continuous-time minimum-phase plant may become vulnerable after discretization (Shim et al., 2021). In the discrete-time normal form, the zero-dynamics subsystem is

$z$4

and a stealthy actuator attack excites this subsystem while keeping the output arbitrarily small or identically zero (Shim et al., 2021). The same chapter states that for relative degree $z$5 and fast sampling, at least one sampling zero lies outside the unit circle, enabling disruptive discrete zero-dynamics attacks (Shim et al., 2021).

Recent work addresses detection by invalidating the stealth conditions. “Zero Dynamics Attack Detection and Isolation in Cyber-Physical Systems with Event-triggered Communication” augments the plant with an auxiliary system without zero dynamics and introduces a timing discrepancy residual

$z$6

comparing predicted and observed self-triggered transmission indices (Eslami et al., 9 May 2025). The key point is that an attacker can mask auxiliary output residuals, but cannot simultaneously preserve the equality of plant-side and command-side self-triggered schedules while maintaining a genuine DZD attack on the plant input (Eslami et al., 9 May 2025).

A complementary mitigation strategy is topology switching in networked systems. “Robust Optimal Network Topology Switching for Zero Dynamics Attacks” defines a finite-horizon stealth matrix $z$7 and states that a nontrivial stealth attack exists over the horizon if and only if $z$8 (Tsukamoto et al., 2024). Designing switching schedules and weights to enforce full column rank destroys the finite-horizon null space exploited by DZD-based attacks (Tsukamoto et al., 2024). In this security literature, DZD is therefore both the vulnerability mechanism and the algebraic object targeted by detection and mitigation.

7. Distinct usage in topological dynamics

A separate research tradition uses “Discrete Zero Dynamics” in a measure-theoretic and topological sense rather than a control-theoretic one. In “Quasi-graphs, zero entropy and measures with discrete spectrum,” DZD is interpreted as the class of dynamical systems $z$9 with zero topological entropy for which every invariant probability measure has discrete spectrum (Li et al., 2018). The paper establishes this property for quasi-graphs and, with orbit-closure qualifications, for dendrites whose endpoint sets are closed and have only finitely many accumulation points (Li et al., 2018).

Here the central definition is spectral, not feedback-based. A measure-preserving system has discrete spectrum if the Koopman operator $\eta$0 on $\eta$1 is pure point, equivalently if $\eta$2 is spanned by eigenfunctions (Li et al., 2018). The paper proves that every invariant measure of a quasi-graph map with zero topological entropy has discrete spectrum, and also proves a quasi-graph analogue of the Llibre–Misiurewicz horseshoe theorem for positive entropy (Li et al., 2018).

This meaning of DZD is conceptually unrelated to zeroing manifolds, input-output linearization, or invariant zeros in control systems. The shared terminology comes from “discrete” and “zero,” but the objects are different: discrete spectrum versus zero-output internal dynamics. A common source of confusion is therefore terminological rather than mathematical. In control and robotics, DZD refers to reduced internal dynamics under output-zeroing constraints (Csomay-Shanklin et al., 2024, Schaa et al., 8 Jan 2026). In one-dimensional topological dynamics, DZD refers to the zero-entropy discrete-spectrum regime (Li et al., 2018).

Taken together, these literatures show that DZD is best treated as a family of related but nonidentical notions. In hybrid robotics and sampled-data control, it is a reduced-order dynamical model essential for stability and orbit design (Csomay-Shanklin et al., 2024, Shenoy et al., 2022). In linear systems and data-driven control, it is the internal zero-output behavior whose stability determines minimum-phase structure and admissibility of inversion-based control (Schaa et al., 8 Jan 2026, Celi et al., 2022). In cyber-physical security, it is the hidden mode exploited by stealth attacks and countered by auxiliary dynamics, timing consistency checks, or topology switching (Eslami et al., 9 May 2025, Tsukamoto et al., 2024, Shim et al., 2021). In topological dynamics, it names a zero-entropy discrete-spectrum class with a different formal meaning altogether (Li et al., 2018).