---
title: 'vDSSM: Vector Discrete-Time State-Space Model'
url: https://www.emergentmind.com/topics/vector-discrete-time-state-space-model-vdssm
type: topic
---

# vDSSM: Vector Discrete-Time State-Space Model

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 [2505.18187][2505.23302][2205.03366]. 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 [2505.18187][2512.18965].

## 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 [2505.23302].

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 [2205.03366]. 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 \(x(t)\in\mathbb R^n\), \(u(t)\in\mathbb R^m\), \(y(t)\in\mathbb R^p\), and matrix dimensions interpreted accordingly [2505.18187].

| Formulation family | Representative form | Representative sources |
|---|---|---|
| General probabilistic SSM | \(p(x_0)\prod_t p(x_t\mid x_{t-1})p(y_t\mid x_t)\) | [2505.23302] |
| Abstract discrete-time realization | \(x_{k+1}=f(x_k,u_k),\ y_k=h(x_k,u_k)\) | [2205.03366] |
| Linear-Gaussian vDSSM | \(x_{k+1}=A_dx_k+B_du_k+w_k,\ y_k=C_dx_k+M_d v_k\) | [2505.18187] |

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
\[
\dot{x}(t)=A\,x(t)+B\,u(t)+L\,w(t),\qquad y(t)=C\,x(t)+M\,v(t),
\]
with zero-mean Gaussian white process and measurement noise having intensities \(Q\) and \(R\) [2505.18187]. For sampling period \(\Delta t\), the exact discrete-time matrices are
\[
A^d=\exp(A\Delta t),
\]
\[
B^d=\int_0^{\Delta t}\exp\!\big(A(\Delta t-s)\big)\,B\,ds,
\]
\[
Q^d=\int_0^{\Delta t}\exp\!\big(A(\Delta t-s)\big)\,LQL^\top\,\exp\!\big(A(\Delta t-s)\big)^\top ds,
\]
together with
\[
C^d=C,\qquad M^d=M,\qquad R^d=\frac{1}{\Delta t}R
\]
under the convention used in the note [2505.18187].

The most implementation-relevant result is that \(A^d\), \(B^d\), and \(Q^d\) can be computed simultaneously from one augmented matrix exponential. Define
\[
\Xi=
\begin{bmatrix}
A & LQL^\top & 0 & 0 \\
0 & -A^\top & 0 & 0 \\
0 & 0 & A & B \\
0 & 0 & 0 & 0
\end{bmatrix},\qquad
\Upsilon=\exp(\Xi\Delta t).
\]
If \(\Upsilon\) is partitioned conformably, then
\[
A_d=\Upsilon_{11},\qquad B_d=\Upsilon_{34},\qquad Q_d=\Upsilon_{12}\Upsilon_{11}^\top.
\]
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 [2505.18187].

A practical implication is that process-noise discretization is structurally different from measurement-noise scaling. The note emphasizes that \(Q_d\) is generally not obtainable by trivial scaling alone because continuous-time process noise is filtered by the system dynamics, whereas \(R_d\) follows the paper’s sampling convention directly [2505.18187]. 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` [2505.23302]. 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 [2505.23302]. 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
\[
x_{k+1}=Ax_k+Bu_k+Gw_k,\qquad y_k=x_k+v_k,
\]
and the state constraint \(Sx_k=s\), the compatibility conditions \(SA=S\) and \(SB=0\) become linear equality constraints on the stacked parameter vector
\[
\theta=\begin{bmatrix}\operatorname{vec}(A)\\ \operatorname{vec}(B)\end{bmatrix},
\]
yielding the regression
\[
Z=\Psi\theta+\Xi,\qquad D\theta=d.
\]
This converts a matrix-valued constrained vDSSM identification problem into a standard equality-constrained vector regression problem [1904.05178].

A related fully observed identification route estimates \(A\) and \(B\) row by row from state trajectories using kernel ridge regression with the linear kernel \(K(x,y)=x^\top y\). In that setting, the controlled plant
\[
x(k+1)=Ax(k)+Bu(k)
\]
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
\[
A^\top\Bigl(P-PB(R+B^\top P B)^{-1}B^\top P\Bigr)A+Q=P
\]
to construct feedback \(u(k)=-Fx(k)\) [1507.03111]. 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
\[
x(t+1)=Ax(t)+Bu(t)
\]
relative to a stable reference model
\[
x_m(t+1)=A_mx_m(t)+B_mr(t)
\]
can be solved with gradient-based direct and indirect adaptive laws rather than classical continuous-time Lyapunov cancellation arguments [2308.02484]. 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 [2308.02484].

## 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
\[
x(k+1)=A_0x(k)+B_0u(k)+E_0f(k),\qquad y(k)=C_0x(k),
\]
an augmented state
\[
X(k)=\begin{bmatrix}x(k)\\ f(k)\end{bmatrix}
\]
leads to
\[
X(k+1)=AX(k)+Bu(k)+E\Delta f(k),\qquad y(k)=CX(k),
\]
where disturbances are treated as appended states driven by their increments [2510.01007]. In this formulation, exact delayed disturbance reconstruction is tied to a structural condition: there must be no invariant zeros between disturbance channels and outputs [2510.01007]. 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
\[
E\dot{x}=Ax+Bu,\qquad y=Cx+Du
\]
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 [2303.01701]. The closest discrete-time analogue,
\[
E x_{k+1}=A_d x_k+B_d u_k,\qquad y_k=C_d x_k+D_d u_k,
\]
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 [2303.01701].

A third structured extension introduces a static domain variable \(D\) into the transition law:
\[
S_{i+1}=f(S_i,D)+\beta_i,\qquad X_i=g(S_i)+\omega_i.
\]
Here \(S_i\in\mathbb R^L\) is the latent state, \(X_i\in\mathbb R^O\) is the observation, and \(D\) is a sequence-level latent variable that remains constant within a sequence while modulating the shared transition mechanism [1906.03255]. From a vDSSM perspective, this can be interpreted as an augmented-state model in which the static context \(D\) 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
\[
x'(t)=\mathbf A x(t)+\mathbf B u(t),\qquad y(t)=\mathbf C x(t),
\]
then discretizes it for forecasting. In Time-SSM, the discrete parameters are written as
\[
\bar{\mathbf A}=\exp(\Delta \mathbf A),\qquad \bar{\mathbf B}=\Delta \mathbf B,
\]
and the practical recurrence is presented as
\[
x_{t+1}=\bar{\mathbf A}_t x_t+\bar{\mathbf B}_t u_t,\qquad y_t=\mathbf C_t x_t,
\]
with input-conditioned \(\Delta_t\), \(\mathbf B_t\), and \(\mathbf C_t\) producing a time-varying selective model [2405.16312]. 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 [2405.16312].

A complementary line derives discrete-time state recurrences directly, without passing through a continuous-time ODE. Lag Operator SSMs construct the vector state \(\bm c_t\in\mathbb R^N\) as basis-projection coefficients of past history and obtain
\[
\bm c_{t+1}=\bm A_t\bm c_t+\bm B_t u_{t+1},
\qquad
\hat u_{t+1}=\bm\Psi_{t+1}^\mathsf T \bm c_{t+1},
\]
where \(\bm A_t\) is computed from a lag-operator inner product between basis functions at adjacent time steps [2512.18965]. 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 [2512.18965].

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
\[
p_{\theta}(y^i_{t+1:t+h}\mid \text{context})
=
\sum_{x^i_{t+1:t+h}\in \mathsf X^h}
\prod_{s=1}^h
p_{\theta_y}(y^i_{t+s}\mid x^i_{t+s},\text{context})
p_{\theta_x}(x^i_{t+s}\mid x^i_{t+s-1},\text{context}),
\]
trained through an ELBO and a two-stage alternating procedure [2404.11117]. In D4, by contrast, the latent dynamics are retained explicitly but the generative observation model \(p(y_k\mid x_k)\) is replaced by a discriminative decoder \(p(x_k\mid y_k,h_k)\), producing a probabilistic state-space filter with explicit latent transition law and a neural inverse observation model [2205.10947]. 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 [2205.03366][2404.11117][2205.10947].

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 \(A_d\), \(B_d\), and \(Q_d\) [2505.18187], but direct discrete-time derivations also exist and can recover important structured recurrences such as HiPPO [2512.18965]. These two viewpoints are complementary rather than contradictory.

A third boundary concerns generalized models. Descriptor formulations with singular \(E\) 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 [2303.01701]. Likewise, software ecosystems may support controlled and time-inhomogeneous models operationally through generic arguments such as `kwargs...` without presenting the canonical control-theoretic equation \(x_t=A_t x_{t-1}+B_tu_t+w_t\) in the main text [2505.23302].

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 [2205.03366]. 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.

Source: https://www.emergentmind.com/topics/vector-discrete-time-state-space-model-vdssm