vDSSM: Vector Discrete-Time State-Space Model

• vDSSM is a discrete-time state-space model featuring vector-valued states, inputs, and observations that evolve through structured transition and observation equations.
• It integrates continuous-to-discrete construction with direct discrete-time formulations, enabling exact discretization of linear-Gaussian systems and facilitating practical implementations.
• Modern vDSSMs support advanced filtering, parameter estimation, and adaptive control, while also extending to neural and augmented-state variants for complex real-world applications.

Searching arXiv for the cited state-space modeling papers to ground the article in current literature. A vector discrete-time state-space model (vDSSM) is a discrete-time dynamical representation in which the state, input, observation, and disturbance variables are vector-valued and evolve through a state-transition equation and an observation equation. In recent arXiv literature, the exact label “vDSSM” is often absent, but the underlying object appears directly as the standard discrete-time latent Markov model, the linear-Gaussian discrete-time state-space system, and a range of augmented, constrained, and neural state-space constructions (Dahdah et al., 18 May 2025, Hargreaves et al., 29 May 2025, Rojas et al., 2022). In its canonical stochastic form, a vDSSM is written with a latent vector state $x_k$, a vector input $u_k$, a vector observation $y_k$, and vector disturbances or noise terms; depending on the setting, the transition and observation laws may be linear or nonlinear, time-invariant or time-varying, exact discretizations of continuous-time systems or directly discrete-time constructions (Dahdah et al., 18 May 2025, Tomonaga et al., 22 Dec 2025).

1. Canonical form and model class

At the most general probabilistic level, a discrete-time state-space model is specified by an initialization distribution $p_\theta(x_0)$, a transition distribution $p_\theta(x_t\mid x_{t-1})$, and an observation distribution $p_\theta(y_t\mid x_t)$, with joint factorization

$$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$

This is the form adopted in probabilistic software-oriented work on state-space modeling, and it directly covers vector-valued latent states and observations as well as more structured state types (Hargreaves et al., 29 May 2025).

A more abstract realization-theoretic statement is that every deterministic, discrete-time, causal, time-invariant system admits a state-space representation of the form

$x(n+1)=f(u(n),x(n)),\qquad y(n)=g(u(n),x(n)),$

with state constructed from Nerode-type equivalence classes of input histories (Rojas et al., 2022). This establishes the conceptual legitimacy of state-space modeling in discrete time, although it does not by itself guarantee a finite-dimensional Euclidean state.

For linear stochastic models derived from continuous-time systems, the canonical vector discrete-time form is

$$x_k=A^d x_{k-1}+B^d u_{k-1}+w^d_{k-1},\qquad w^d_{k-1}\sim\mathcal N(0,Q^d),$$

$$y_k=C^d x_k+M^d v_k^d,\qquad v_k^d\sim\mathcal N(0,R^d),$$

with $u_k$0, $u_k$1, $u_k$2, and matrix dimensions interpreted accordingly (Dahdah et al., 18 May 2025).

Formulation family	Representative form	Representative sources
General probabilistic SSM	$u_k$3	(Hargreaves et al., 29 May 2025)
Abstract discrete-time realization	$u_k$4	(Rojas et al., 2022)
Linear-Gaussian vDSSM	$u_k$5	(Dahdah et al., 18 May 2025)

This range of formulations suggests that “vDSSM” is best treated as a model class rather than a single notation. The common structural requirement is first-order discrete-time evolution of a vector state together with a vector-valued observation mechanism.

2. Exact continuous-to-discrete construction

A central practical route to a vDSSM begins from the continuous-time linear stochastic system

$u_k$6

with zero-mean Gaussian white process and measurement noise having intensities $u_k$7 and $u_k$8 (Dahdah et al., 18 May 2025). For sampling period $u_k$9, the exact discrete-time matrices are

$y_k$0

$y_k$1

$y_k$2

together with

$y_k$3

under the convention used in the note (Dahdah et al., 18 May 2025).

The most implementation-relevant result is that $y_k$4, $y_k$5, and $y_k$6 can be computed simultaneously from one augmented matrix exponential. Define

$y_k$7

If $y_k$8 is partitioned conformably, then

$y_k$9

This is a modified Van Loan–style construction, and it gives an exact discrete-time vDSSM for linear time-invariant dynamics under the stated assumptions (Dahdah et al., 18 May 2025).

A practical implication is that process-noise discretization is structurally different from measurement-noise scaling. The note emphasizes that $p_\theta(x_0)$0 is generally not obtainable by trivial scaling alone because continuous-time process noise is filtered by the system dynamics, whereas $p_\theta(x_0)$1 follows the paper’s sampling convention directly (Dahdah et al., 18 May 2025). This distinction is foundational in filtering, simulation, and stochastic control.

3. Inference, identification, and control

Once a vDSSM is specified, the standard tasks are filtering, smoothing, and parameter inference. A recent software-oriented formulation within the Turing.jl ecosystem defines state-space models compositionally from latent-dynamics and observation-process objects and provides a unified predict/update interface through SSMProblems.jl and GeneralisedFilters.jl (Hargreaves et al., 29 May 2025). Linear-Gaussian models are exposed through calc_A, calc_b, and calc_Q, enabling exact Kalman filtering, while the same model abstraction also supports particle filtering, Rao-Blackwellised particle filtering, PMCMC, and GPU-accelerated batched filtering (Hargreaves et al., 29 May 2025). This suggests that the modern computational identity of a vDSSM is as much an interface contract as a set of equations.

For fully observed linear discrete-time systems with known state equality constraints, parameter estimation can be rewritten as a vectorized constrained least-squares problem. Starting from

$p_\theta(x_0)$2

and the state constraint $p_\theta(x_0)$3, the compatibility conditions $p_\theta(x_0)$4 and $p_\theta(x_0)$5 become linear equality constraints on the stacked parameter vector

$p_\theta(x_0)$6

yielding the regression

$p_\theta(x_0)$7

This converts a matrix-valued constrained vDSSM identification problem into a standard equality-constrained vector regression problem (Ricco et al., 2019).

A related fully observed identification route estimates $p_\theta(x_0)$8 and $p_\theta(x_0)$9 row by row from state trajectories using kernel ridge regression with the linear kernel $p_\theta(x_t\mid x_{t-1})$0. In that setting, the controlled plant

$p_\theta(x_t\mid x_{t-1})$1

is treated as a deterministic discrete-time state equation with direct state observation, and the resulting estimates can be used in discrete-time LQR via the Riccati equation

$p_\theta(x_t\mid x_{t-1})$2

to construct feedback $p_\theta(x_t\mid x_{t-1})$3 (Colonius et al., 2015). This is a restricted but important vDSSM regime: linear, discrete-time, finite-dimensional, and fully observed.

On the control side, adaptive state tracking for a full-state measured discrete-time plant

$p_\theta(x_t\mid x_{t-1})$4

relative to a stable reference model

$p_\theta(x_t\mid x_{t-1})$5

can be solved with gradient-based direct and indirect adaptive laws rather than classical continuous-time Lyapunov cancellation arguments (Tao, 2023). In this setting, the vDSSM is deterministic and full-state measured, but the paper is notable because it shows how filtered regressors, composite errors, and normalized gradient updates can achieve boundedness and asymptotic state tracking in discrete time (Tao, 2023).

4. Augmented, generalized, and structured variants

One important extension of the vDSSM idea augments the state with disturbances or other latent quantities. For the discrete-time multivariable plant

$p_\theta(x_t\mid x_{t-1})$6

an augmented state

$p_\theta(x_t\mid x_{t-1})$7

leads to

$p_\theta(x_t\mid x_{t-1})$8

where disturbances are treated as appended states driven by their increments (Chen et al., 1 Oct 2025). In this formulation, exact delayed disturbance reconstruction is tied to a structural condition: there must be no invariant zeros between disturbance channels and outputs (Chen et al., 1 Oct 2025). This is a precise example of a vDSSM enlarged to accommodate unknown-input estimation.

A second extension concerns generalized descriptor structure. Descriptor state-space models of the form

$p_\theta(x_t\mid x_{t-1})$9

are continuous-time rather than discrete-time, but they provide a conceptual precursor for generalized vector state-space modeling with algebraic constraints and improper transfer behavior (Li et al., 2023). The closest discrete-time analogue,

$p_\theta(y_t\mid x_t)$0

is not derived in that work, and the paper is explicit that it should be regarded as a generalized continuous-time foundation rather than a ready-made discrete-time vDSSM (Li et al., 2023).

A third structured extension introduces a static domain variable $p_\theta(y_t\mid x_t)$1 into the transition law: $p_\theta(y_t\mid x_t)$2 Here $p_\theta(y_t\mid x_t)$3 is the latent state, $p_\theta(y_t\mid x_t)$4 is the observation, and $p_\theta(y_t\mid x_t)$5 is a sequence-level latent variable that remains constant within a sequence while modulating the shared transition mechanism (Miladinović et al., 2019). From a vDSSM perspective, this can be interpreted as an augmented-state model in which the static context $p_\theta(y_t\mid x_t)$6 encodes domain-specific dynamics.

5. Neural and modern sequence-modeling interpretations

Recent sequence-modeling literature has broadened the meaning of a vDSSM beyond classical control notation. One line begins from the continuous-time controlled SSM

$p_\theta(y_t\mid x_t)$7

then discretizes it for forecasting. In Time-SSM, the discrete parameters are written as

$p_\theta(y_t\mid x_t)$8

and the practical recurrence is presented as

$p_\theta(y_t\mid x_t)$9

with input-conditioned $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$0, $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$1, and $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$2 producing a time-varying selective model (Hu et al., 2024). The paper emphasizes patch embeddings, hidden state dimension, and an additional variable Fourier operator for multivariate dependence, so the model is naturally vector-valued even when the theoretical exposition is initially SISO (Hu et al., 2024).

A complementary line derives discrete-time state recurrences directly, without passing through a continuous-time ODE. Lag Operator SSMs construct the vector state $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$3 as basis-projection coefficients of past history and obtain

$$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$4

where $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$5 is computed from a lag-operator inner product between basis functions at adjacent time steps (Tomonaga et al., 22 Dec 2025). A specific instance exactly recovers the HiPPO recurrence, which suggests that some influential structured SSMs can be understood as direct discrete-time vDSSMs rather than discretized continuous-time systems (Tomonaga et al., 22 Dec 2025).

Other modern variants depart further from the linear-Gaussian template. In variational quantization for state-space models, the latent state is discrete and finite-valued, with predictive law

$$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$6

trained through an ELBO and a two-stage alternating procedure (David et al., 2024). In D4, by contrast, the latent dynamics are retained explicitly but the generative observation model $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$7 is replaced by a discriminative decoder $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$8, producing a probabilistic state-space filter with explicit latent transition law and a neural inverse observation model (Rezaei et al., 2022). Together these models demonstrate that a vDSSM need not be continuous-valued, linear, or generative on the observation side.

6. Scope, misconceptions, and open boundaries

A common misconception is that a vDSSM must be a finite-dimensional linear-Gaussian model. The literature supports a narrower statement: linear-Gaussian systems remain the canonical and most analytically tractable form, but discrete-time state-space structure also encompasses nonlinear deterministic realizations, discrete latent HMM-like models, discriminative decoders, and domain-conditioned neural filters (Rojas et al., 2022, David et al., 2024, Rezaei et al., 2022).

A second misconception is that every useful discrete-time state-space model must arise from continuous-time discretization. Exact discretization is central in control and stochastic estimation, especially for $$p_\theta(x_{0:T}, y_{1:T}) = p_\theta(x_0)\prod_{t=1}^T p_\theta(x_t\mid x_{t-1})p_\theta(y_t\mid x_t).$$9, $x(n+1)=f(u(n),x(n)),\qquad y(n)=g(u(n),x(n)),$0, and $x(n+1)=f(u(n),x(n)),\qquad y(n)=g(u(n),x(n)),$1 (Dahdah et al., 18 May 2025), but direct discrete-time derivations also exist and can recover important structured recurrences such as HiPPO (Tomonaga et al., 22 Dec 2025). These two viewpoints are complementary rather than contradictory.

A third boundary concerns generalized models. Descriptor formulations with singular $x(n+1)=f(u(n),x(n)),\qquad y(n)=g(u(n),x(n)),$2 matrices provide modular handling of algebraic constraints and improper systems, but the cited descriptor work is explicitly continuous-time and does not itself furnish a discrete-time descriptor realization or discretization algorithm (Li et al., 2023). Likewise, software ecosystems may support controlled and time-inhomogeneous models operationally through generic arguments such as kwargs... without presenting the canonical control-theoretic equation $x(n+1)=f(u(n),x(n)),\qquad y(n)=g(u(n),x(n)),$3 in the main text (Hargreaves et al., 29 May 2025).

Finally, the abstract existence of a state-space realization does not imply a practical vector model of fixed finite dimension. The general realization theorem constructs state as an equivalence class of histories; finite-dimensional Euclidean vector states require additional structure beyond causality and time invariance alone (Rojas et al., 2022). This marks the broadest conceptual limit of the term “vDSSM”: it is most precise when it denotes a finite-dimensional vector discrete-time model, but its mathematical ancestry extends to more general state constructions.