---
title: Discrete Zero Dynamics in Control Systems
url: https://www.emergentmind.com/topics/discrete-zero-dynamics-dzd
type: topic
---

# Discrete Zero Dynamics in Control Systems

Searching arXiv for recent 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 [2409.06125], [2211.11334], [2601.05395]. A distinct usage also appears in topological dynamics, where “Discrete Zero Dynamics” refers to zero-entropy systems whose invariant measures have discrete spectrum [1809.05617]. 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 [2409.06125]. 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 [2409.06125].

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 [2409.06125]. In these coordinates, the input acts directly on $\eta$, whereas $z$ is not directly influenced by the input, expressed by $\partial z/\partial x \cdot g(x)=0$ [2409.06125].

The zeroing relation is specified by outputs of the form
$$
y = \eta - \psi(z),
$$
so that the associated manifold is
$$
\mathcal{M}_{\psi} = \{(\eta,z): \eta = \psi(z)\}.
$$
When the closed-loop system is constrained to this manifold, the remaining discrete step-to-step evolution of $z$ is the DZD [2409.06125]. In the same spirit, for discrete-time linear systems with
$$
x_{k+1} = A x_k + B u_k, \qquad y_k = C x_k + D u_k,
$$
the zero dynamics are the internal motions consistent with $y_k \equiv 0$ under suitable $u_k$, and minimum phase corresponds to stability of these internal motions [2601.05395].

This definition extends beyond hybrid legged systems. In sampled-data nonlinear systems written in normal form, the internal coordinates $\eta$ evolve according to a sampled map
$$
\eta_{k+1}=F_z(\eta_k;T),
$$
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 [2211.11334]. In impulsive underactuated juggling, DZD arises from discrete virtual holonomic constraints and governs the passive orientation variables at impact times [2509.08085], [2508.15040].

## 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
$$
\boldsymbol{x}_{k+1} = \boldsymbol{F}(\boldsymbol{x}_k, \boldsymbol{v}_k),
$$
which in $(\eta,z)$ coordinates takes the form
$$
\boldsymbol{\eta}_{k+1} = \hat{\boldsymbol{F}}(\boldsymbol{\eta}_k, \boldsymbol{z}_k, \boldsymbol{v}_k), \qquad
\boldsymbol{z}_{k+1} = \boldsymbol{\Omega}(\boldsymbol{\eta}_k, \boldsymbol{z}_k)
$$
under the ARCHER assumption that the input weakly affects impact time [2409.06125]. This is explicitly identified as a Poincaré-type return map evaluated at impact [2409.06125].

If the manifold $\mathcal{M}_\psi$ is controlled invariant, meaning that for all $(\eta_k,z_k)\in\mathcal{M}_\psi$ there exists a discrete control parameter $v_k$ such that the next state remains on the manifold, then the restricted autonomous dynamics
$$
\boldsymbol{z}_{k+1} = \boldsymbol{\Omega}(\boldsymbol{\psi}(\boldsymbol{z}_k), \boldsymbol{z}_k)
$$
is the DZD [2409.06125]. 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 [2409.06125].

The stability interpretation is standard Poincaré theory in reduced order. If $z^*$ is a fixed point,
$$
z^* = \boldsymbol{\Omega}(\boldsymbol{\psi}(z^*), z^*),
$$
then the Jacobian
$$
D\boldsymbol{\Pi}(z^*) \equiv D_z\big[\boldsymbol{\Omega}(\boldsymbol{\psi}(z), z)\big]\big|_{z^*}
$$
provides the Floquet multipliers, and eigenvalues inside the unit circle imply local exponential orbital stability of the periodic motion on the zeroing manifold [2409.06125]. The same reduction principle appears in classical Poincaré notation
$$
P(x)=\Delta(\varphi_{T(x)}(x)),
$$
with DZD corresponding to the restriction $P|_Z$ [2409.06125].

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 $(\theta,\omega)$, and the paper states that it “provides conditions for stable juggling” [2509.08085]. In the propeller-motion variant, the DZD is a 2D autonomous map in $(\theta,\omega)$ obtained after slaving the center-of-mass coordinates to a discrete geometric constraint [2508.15040]. 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 [2409.06125]. The central object is a mapping
$$
\psi_\theta:\mathcal{Z}\to\mathcal{Y},
$$
called a Zero Dynamics Policy (ZDP), with transverse error
$$
e_k = \eta_k - \psi(z_k).
$$
Because $\psi$ depends only on the underactuated degrees of freedom, the method “achiev[es] significant dimension reduction” while preserving structure induced by underactuation [2409.06125].

The learning objective enforces one-step invariance under an optimal discrete action computed by iLQR:
$$
\mathcal{L}(\boldsymbol{\theta}) =
\mathbb{E}_{z\sim \text{UNIFORM}}
\left\|
\boldsymbol{\eta}^*_1\big(\boldsymbol{\zeta}_{\theta}(z)\big)
-
\psi_{\theta}\!\left(\boldsymbol{z}^*_1\big(\boldsymbol{\zeta}_{\theta}(z)\big)\right)
\right\|_2^2,
$$
where
$$
\boldsymbol{\zeta}_{\theta}(z) \triangleq
\begin{bmatrix}
\psi_{\theta}(z) \\ z
\end{bmatrix}.
$$
Achieving small loss implies discrete invariance under the optimal action and therefore well-defined DZD on the learned manifold [2409.06125].

The control architecture separates manifold design from transverse stabilization. During continuous evolution, $\eta(t)$ is driven to $\psi(z(t))$ using feedback linearization and RES-CLF or PD, while the reduced step-to-step behavior is governed by the DZD [2409.06125]. The paper proves a constructive stabilization lemma stating that there exists a controller such that
$$
\|\eta_{k+1}-\psi(z_{k+1})\|\le \alpha \|\eta_k-\psi(z_k)\|,\qquad \alpha\in(0,1],
$$
under assumptions including feedback-linearizable $\eta$ dynamics, an RES-CLF inequality, a Lipschitz impact map, and a lower bound on impact time [2409.06125].

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 [2409.06125]. The proof uses the composite Lyapunov function
$$
V(e_k,z_k)=\sigma V_e(e_k)+V_z(z_k),
$$
with cross-coupling terms bounded via Lipschitz constants [2409.06125]. 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 [2211.11334]. In normal form,
$$
\dot{\eta}=f_0(\eta,\xi),\qquad
\dot{\xi}_i=\xi_{i+1},\qquad
\dot{\xi}_\rho=\alpha(\xi,\eta)+\beta(\xi,\eta)u,
$$
with zero-dynamics manifold $\mathcal{Z}_0=\{(\xi,\eta):\xi=0\}$ [2211.11334]. Under zero-order hold, the sampled internal dynamics become
$$
\eta_{k+1}=F_z(\eta_k;T),
$$
which the paper explicitly calls the DZD and interprets as a Poincaré map along the limit cycle on $\mathcal{Z}_0$ [2211.11334].

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 [2211.11334]. The Lyapunov estimate
$$
V(\phi^e)-V(\xi(k))
\le
-\lambda_{\min}(Q)\,T\,\|\xi(k)\|^2
+
N_1 T^2 \|\xi(k)\|^2
+
N_2 T^2 \|\eta(k)\|^2
$$
shows that the bounded but nonvanishing limit-cycle amplitude in $\eta$ produces a persistent residual term [2211.11334]. 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 [2601.05395]. The paper defines the zero-dynamics behavior as
$$
B_{\mathrm{ZD}} = \{u:\mathbb{N}_0\to\mathbb{R}^m \mid (u,0)\in B\},
$$
and studies stability of the internal motions consistent with $y_k\equiv 0$ [2601.05395]. The approach is based on Hankel matrices, Willems’ Fundamental Lemma, the lag $l(B)$, and the most powerful unfalsified model (MPUM) [2601.05395].

A key construction uses the subspace
$$
M := \{u\in\mathbb{R}^{mL} : (u,0)\in B_{\mathrm{MPUM},[0,L-1]}\},
$$
selects a basis of zero-dynamics initializations, and forms a data-driven recursion matrix $\widetilde Q$ whose Schur stability is equivalent to stability of the zero dynamics [2601.05395]. Algorithm 2 then determines whether the data are informative for stability, instability, or ambiguity of the zero dynamics [2601.05395]. 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 $\mathcal{V}^\star$ and describes DZD as the internal state evolution restricted to $\mathcal{V}^\star$ under inputs that render the output identically zero [2201.03656]. 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 [2201.03656].

## 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 [2509.08085]. The paper’s theorem states that the reduced dynamics is period-2 if and only if
$$
\theta_{\mathrm{even}} = \pi - \theta_{\mathrm{odd}},
$$
that is, the two impulsive orientations are symmetric about the vertical axis [2509.08085]. Under this symmetry, the DZD admits infinitely many stable 2-periodic orbits parameterized by $\omega^*<0$ [2509.08085].

The propeller-motion paper gives a different but related picture. There the DZD is
$$
\theta_{k+1}=\theta_k+\Delta\theta
$$
together with a scalar nonlinear relation coupling $\omega_k$ and $\omega_{k+1}$ [2508.15040]. For $\phi=\pm \pi/2$ and $\Delta\theta=2\pi/N$, the paper proves existence of $N$-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 [2508.15040]. Orbit selection is achieved by an impulse-controlled Poincaré map, linearized as
$$
e(j+1)=\mathcal{A}e(j)+\mathcal{B}u(j),
$$
followed by discrete feedback that places the eigenvalues of $\mathcal{A}+\mathcal{B}\mathcal{K}$ strictly inside the unit circle [2508.15040].

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 [2409.06125]. 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 [2409.06125], [2509.08085], [2601.05395]. 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
$$
x_{k+1}=A x_k + B a_{u,k},\qquad y_k=C x_k,
$$
the Rosenbrock matrix
$$
P(z)=
\begin{bmatrix}
zI-A & -B\\
C & 0
\end{bmatrix}
$$
defines discrete zeros via rank loss [2505.06070]. A zero-dynamics attack injects
$$
a_{u,k}=a_0 z_0^k
$$
with nontrivial $(x_0,a_0)$ satisfying the Rosenbrock condition, so that $y_k\equiv 0$ while the state evolves as $x_k=z_0^k x_0$ [2505.06070]. If $|z_0|>1$, the hidden state may diverge; if $|z_0|<1$, it decays [2505.06070].

“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 [2101.00556]. In the discrete-time normal form, the zero-dynamics subsystem is
$$
x_z[k+1]=Sx_z[k],
$$
and a stealthy actuator attack excites this subsystem while keeping the output arbitrarily small or identically zero [2101.00556]. The same chapter states that for relative degree $r\ge 3$ and fast sampling, at least one sampling zero lies outside the unit circle, enabling disruptive discrete zero-dynamics attacks [2101.00556].

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
$$
\mathrm{dis}_t = k_i^C-k_j,
$$
comparing predicted and observed self-triggered transmission indices [2505.06070]. 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 [2505.06070].

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 $S_T(\sigma)$ and states that a nontrivial stealth attack exists over the horizon if and only if $\operatorname{rank} S_T(\sigma)<n+mT$ [2407.18440]. Designing switching schedules and weights to enforce full column rank destroys the finite-horizon null space exploited by DZD-based attacks [2407.18440]. 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 $(X,f)$ with zero topological entropy for which every invariant probability measure has discrete spectrum [1809.05617]. 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 [1809.05617].

Here the central definition is spectral, not feedback-based. A measure-preserving system has discrete spectrum if the Koopman operator $U_f$ on $L^2(X,\mu)$ is pure point, equivalently if $L^2(X,\mu)$ is spanned by eigenfunctions [1809.05617]. 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 [1809.05617].

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 [2409.06125], [2601.05395]. In one-dimensional topological dynamics, DZD refers to the zero-entropy discrete-spectrum regime [1809.05617].

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 [2409.06125], [2211.11334]. 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 [2601.05395], [2201.03656]. In cyber-physical security, it is the hidden mode exploited by stealth attacks and countered by auxiliary dynamics, timing consistency checks, or topology switching [2505.06070], [2407.18440], [2101.00556]. In topological dynamics, it names a zero-entropy discrete-spectrum class with a different formal meaning altogether [1809.05617].

Source: https://www.emergentmind.com/topics/discrete-zero-dynamics-dzd