---
title: Augmented Discrete-Time Stochastic Linear System
url: https://www.emergentmind.com/topics/augmented-discrete-time-stochastic-linear-system-dt-asls
type: topic
---

# Augmented Discrete-Time Stochastic Linear System

Searching arXiv for the cited papers and closely related dt-ASLS terminology.
I’ll look up the relevant arXiv records to ground the article in the cited literature.
An augmented discrete-time stochastic linear system (dt-ASLS) is a discrete-time stochastic linear model obtained by enlarging the state so that exogenous structure not directly captured by a standard state update—such as continuous-time process noise after sampling, delayed inputs, packet losses, or unknown inputs—appears within a finite-dimensional, Markovian, and typically linear-in-state representation. In the literature, dt-ASLS denotes several closely related constructions rather than a single canonical model. One line of work derives exact sampled discrete-time stochastic models from continuous-time stochastic differential equations, yielding $x_{k+1}=\Phi x_k+w_k$ with exact covariance $Q_d$ [1402.1358]. Another line introduces explicit augmentation to represent communication delays, packet drops, or latent inputs, thereby converting a non-Markovian or partially specified system into a larger stochastic linear system amenable to control, filtering, or Lyapunov analysis [2507.15031], [1309.6627], [1911.00308].

## 1. Concept and scope

The unaugmented discrete-time stochastic linear system in the networked-control setting is written as
\[
x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},
\]
where $x_k\in\mathbb{R}^n$, $u_k\in\mathbb{R}^m$, $\omega_k\in\mathbb{R}^q$ is an i.i.d. zero-mean disturbance, and $A$, $B$, and $W$ are the state, input, and disturbance-gain matrices, respectively [2507.15031]. In filtering with unknown inputs, the underlying model instead includes an additional disturbance $d_k\in\mathbb{R}^p$ entering both dynamics and measurements [1309.6627]. In stochastic-stability analysis, the basic form is often written as
\[
x_{k+1} = A_k x_k + B_k w_k,
\]
with random matrices $A_k$ and $B_k$ driven by an underlying process $\xi_k$ [1911.00308].

The defining operation in a dt-ASLS is augmentation of the state. In the unknown-input setting, one forms
\[
z_k=\begin{bmatrix}x_k\\ d_k\end{bmatrix},
\]
and, under the filtering assumption $d_{k+1}=d_k+\eta_k$, obtains an equivalent augmented stochastic linear system with block matrices $\mathcal A_k$, $\mathcal B_k$, and $\mathcal G_k$ [1309.6627]. In the communication-constrained setting, one appends a finite history of control inputs to represent a constant uplink delay of $d$ steps [2507.15031]. In the second-moment-stability setting, one augments by past disturbances, first via $X_k:=\mathrm{col}(x_k,w_{k-1})$ and then via $Z_k=\mathrm{col}(X_k,w_k)$, so that the entire evolution is rewritten in homogeneous form [1911.00308].

This suggests that dt-ASLS is best understood as a modeling pattern: augmentation is used to convert delayed, input-corrupted, unknown-input, or sampled stochastic dynamics into a finite-dimensional discrete-time stochastic linear representation on which standard tools—Kalman-type filtering, Lyapunov inequalities, and control barrier certificates—can be imposed.

## 2. Exact sampled stochastic models from continuous time

A foundational exact discretization result starts from the continuous-time stochastic differential equation
\[
dx(t) = A\,x(t)\,dt + L\,d\beta(t),\qquad E[d\beta(t)d\beta(t)^\top]=S\,dt,
\]
sampled with interval $T$ [1402.1358]. The sampled state is $x_k=x(kT)$, and the homogeneous dynamics yield the state-transition matrix
\[
\Phi \equiv e^{AT}.
\]
The exact discrete-time stochastic model is then
\[
x_{k+1} = \Phi x_k + w_k,\qquad w_k\sim \mathcal N(0,Q_d),
\]
with
\[
Q_d = \int_0^T e^{A\tau} S e^{A^\top\tau}\,d\tau
\]
after absorbing $LSL^\top$ into a single matrix $S$ [1402.1358].

Within this framework, augmentation appears in the computation of the discrete-time covariance rather than in the state definition itself. Van Loan’s method constructs the block upper-triangular matrix
\[
H=\begin{bmatrix}
A & S\\
0 & -A^\top
\end{bmatrix},
\]
whose exponential contains $Q_d$ in the $(1,2)$ block:
\[
\exp(HT)=
\begin{bmatrix}
\Phi & Q_d\\
0 & \Phi^{-\,\top}
\end{bmatrix}.
\]
A related variant uses
\[
\widetilde H=
\begin{bmatrix}
-A & S\\
0 & A^\top
\end{bmatrix},
\]
for which $Q_d=e^{AT}M_{12}$ [1402.1358].

The same source emphasizes a Lyapunov-equation formulation. Defining
\[
V \equiv S-\Phi S\Phi^\top,
\]
the exact covariance satisfies
\[
A Q_d + Q_d A^\top = -V.
\]
Hence one may compute $\Phi=e^{AT}$, form $V=S-\Phi S\Phi^\top$, and solve the Lyapunov equation $A Q_d + Q_d A^\top = -V$ using a standard stable solver [1402.1358]. The stated numerical rationale is that this method only exponentiates an $n\times n$ matrix instead of a $2n\times 2n$ one and avoids overflow or underflow issues for large $T$ or stiff $A$; if $A$ has integrators, the block-triangular structure can be exploited further [1402.1358].

Although this discretization paper does not use the exact phrase “dt-ASLS” in its title, the resulting model
\[
x_{k+1}=\Phi x_k+w_k
\]
is the discrete-time stochastic linear system to which later augmented constructions are applied. A plausible implication is that exact covariance discretization is an important precursor whenever a dt-ASLS is built from continuous-time physics and then further enlarged to encode delay or latent inputs.

## 3. Delay and packet-loss augmentation in networked control

In the communication-constrained safety-control setting, the augmented state is introduced to represent a constant uplink delay of $d$ steps together with packet losses on both uplink and downlink channels [2507.15031]. The augmented state is defined as
\[
X_k \;=\;
\begin{bmatrix}
x_k\\
u_{k-1}\\
u_{k-2}\\
\vdots\\
u_{k-d}
\end{bmatrix}
\in\mathbb{R}^{\,n+md},
\]
while the fresh control computation serves as the external input:
\[
U_k=u_k\in\mathbb{R}^m.
\]
Packet losses are modeled by independent Bernoulli-distributed random processes: $\theta_k\sim \mathrm{Bernoulli}(p_\theta)$ for the uplink and $\phi_k\sim \mathrm{Bernoulli}(q_\phi)$ for the downlink [2507.15031].

The paper also introduces the zero-mean auxiliary variable $\zeta_{\phi,k}$ through
\[
\phi_k = q_\phi\bigl(1-\zeta_{\phi,k}\bigr),\qquad
E[\zeta_{\phi,k}]=0,\qquad
\mathsf{Var}[\zeta_{\phi,k}]=\tfrac{1-q_\phi}{q_\phi},
\]
in order to decompose the dynamics into mean plus zero-mean multiplicative noise [2507.15031].

A more extensive augmented variable is then formed:
\[
\mathbf{x}_k := \begin{bmatrix}x_k & x_{k-1} & \cdots & x_{k-d}\end{bmatrix},\qquad
\mathbf{u}_k := \begin{bmatrix}u_k & u_{k-1} & \cdots & u_{k-d}\end{bmatrix},
\]
and
\[
\mathcal Z_k=
\begin{bmatrix}
\mathbf{x}_k\\
\mathbf{u}_k
\end{bmatrix}
\in \mathbb R^{\,n(d+1)+m(d+1)}.
\]
The resulting dt-ASLS takes the standard form
\[
\mathcal Z_{k+1}
= A_{\rm aug}\,\mathcal Z_k
+ B_{\rm aug}\,U_k
+ W_{\rm aug}\,\Omega_k,
\]
where $\Omega_k$ collects the last $d+1$ disturbances [2507.15031].

The block structure is central. The $(1,2)$ block $B\phi_k$ switches on or off according to the downlink loss $\phi_k$; the shift rows are identity matrices that push the input history down the register; and $W_{\rm aug}$ collects the current and past $d$ disturbances so that $\mathbf{x}_{k+1}$ accumulates $w_k,\dots,w_{k-d}$ via powers of $A$ [2507.15031]. The total dimension is finite and fixed once the maximum delay $d$ is chosen:
\[
n(d+1)+m(d+1).
\]

The stated reason augmentation works is that carrying along the last $d$ inputs allows the state update for $x_{k+1}$ to reconstruct the effect of a constant sensor-to-controller delay, while the Bernoulli variables enter as multiplicative switches on the appropriate blocks of $A_{\rm aug}$ or $B_{\rm aug}$ without any additional memory beyond $\mathcal Z_k$ [2507.15031]. Once augmented, $\mathcal Z_{k+1}$ depends only on $\mathcal Z_k$, the new disturbance $\omega_k$, and the current Bernoulli draws $\theta_k,\phi_k$; the process is therefore time-homogeneous and Markov [2507.15031].

## 4. Unknown-input augmentation and unbiased minimum-variance filtering

A distinct use of dt-ASLS appears in simultaneous input and state estimation, where the true system contains an unknown input $d_k$:
\[
\begin{aligned}
x_{k+1} &= A_k x_k + B_k u_k + G_k d_k + w_k,\\
y_k &= C_k x_k + D_k u_k + H_k d_k + v_k.
\end{aligned}
\]
Here $w_k$ and $v_k$ are zero-mean white noises with $\E[w_k w_k^\top]=Q_k$, $\E[v_k v_k^\top]=R_k$, and $\E[w_k v_j^\top]=0$ [1309.6627].

For filtering purposes, one assumes a trivial random walk
\[
d_{k+1}=d_k+\eta_k,
\]
with $\eta_k$ white and of small covariance. The augmented state
\[
z_k=\begin{bmatrix}x_k\\ d_k\end{bmatrix}\in\mathbb R^{\,n+p}
\]
then yields the equivalent dt-ASLS
\[
z_{k+1}
=
\mathcal A_k z_k
+
\mathcal B_k u_k
+
\mathcal G_k \bar w_k,
\]
with
\[
\mathcal A_k=
\begin{bmatrix}
A_k & G_k\\
0 & I_p
\end{bmatrix},\qquad
\mathcal B_k=
\begin{bmatrix}
B_k\\
0
\end{bmatrix},\qquad
\mathcal G_k=
\begin{bmatrix}
I_n & 0\\
0 & I_p
\end{bmatrix},
\]
and measurement model
\[
y_k = \mathcal C_k z_k + D_k u_k + v_k,\qquad
\mathcal C_k=\begin{bmatrix}C_k & H_k\end{bmatrix}
\]
[1309.6627].

The estimation objective is to produce $\hat z_{k|k}=\E[z_k\mid y_0,\dots,y_k]$ with covariance
\[
P_{k|k}
=\E\bigl[(z_k-\hat z_{k|k})(z_k-\hat z_{k|k})^\top\bigr]
\]
such that the estimator is unbiased and, among all linear unbiased filters, has minimum trace of $P_{k|k}$ [1309.6627]. The prediction step is
\[
\hat z_{k|k-1}
=
\mathcal A_{k-1}\hat z_{k-1|k-1}
+
\mathcal B_{k-1}u_{k-1},
\]
\[
P_{k|k-1}
=
\mathcal A_{k-1}P_{k-1|k-1}\mathcal A_{k-1}^\top
+
\mathcal G_{k-1}\bar Q_{k-1}\mathcal G_{k-1}^\top,
\]
and the update step is
\[
\hat z_{k|k}
=
\hat z_{k|k-1}
+
L_k\bigl(y_k-\mathcal C_k\hat z_{k|k-1}-D_k u_k\bigr)
\]
[1309.6627].

A specific structural constraint arises from unbiasedness: $\E[L_k H_k]=0$ [1309.6627]. The paper then derives the minimum-variance gain and covariance update, and distinguishes two variants, ULISE and PLISE, with ULISE denoting “Updated Linear Input {data} State Estimator” [1309.6627].

The observability and detectability theory is likewise formulated for the augmented setting. In the time-invariant case, strong observability holds iff
\[
\rank\!\bigl[\;zI-A\;-G;\;C\;H\;\bigr]=n+p
\quad \text{for all } z\in\mathbb C,
\]
while strong detectability holds iff the same rank condition is satisfied for all $|z|\ge 1$ [1309.6627]. The paper further states that when $H_k\equiv 0$ the unified filter reduces to the Gillijns-De Moor filter, when $H_k$ has full column rank one recovers the Darouach-Hou or Fang–Simon–Yong filters, and when also $G_k\equiv 0$ the filter collapses to the standard Kalman filter [1309.6627].

A common misconception is that augmentation is merely a bookkeeping device. In this filtering context it is instead the mechanism by which unknown inputs are converted into explicit state components so that unbiased minimum-variance estimation, observability analysis, and stability statements can be expressed within a single linear stochastic framework.

## 5. Second-moment stability and Lyapunov conditions

For stochastic systems with general random dynamics, augmentation is used to absorb additive disturbances into a homogeneous random-coefficient model [1911.00308]. Starting from
\[
x_{k+1}=A_k x_k+B_k w_k,
\]
one first forms
\[
X_k:=\mathrm{col}(x_k,w_{k-1})\in\mathbb R^{n+p},
\]
which leads to an intermediate model with an additive term in $w_k$. A further augmentation
\[
Z_k=\mathrm{col}(X_k,w_k)
\]
then yields the homogeneous form
\[
Z_{k+1}=\bar A(\xi_k)Z_k,
\]
where the system matrix has the block-upper-triangular form
\[
\bar A(\xi_k)=
\begin{bmatrix}
A_k & B_k & 0\\
0 & 0 & I_p\\
0 & 0 & 0
\end{bmatrix}
\]
[1911.00308].

The associated stability notions are mean-square stable, uniformly mean-square stable, mean-square asymptotically stable, uniformly mean-square asymptotically stable, and exponentially mean-square stable, with the last defined by the existence of $a>0$ and $\lambda\in(0,1)$ such that
\[
E_0[\|x_k\|^2]\le a\|x_{k_0}\|^2 \lambda^{2(k-k_0)}
\]
for all $k\ge k_0$ [1911.00308]. Under a minimal bounded-moment assumption on $A(\xi_k)$, the paper states that these five notions collapse into just two classes, and that uniform asymptotic stability is equivalent to exponential stability [1911.00308].

The principal criterion is a general Lyapunov condition. Under second-moment boundedness of the entries of $A(\xi_k)$, exponential mean-square stability holds iff there exist $\lambda_1\in(0,1)$ and a matrix-valued function $P$ satisfying conditional-expectation inequalities of the form
\[
E_0\!\left[\lambda_1^2 P(\cdot)-A(\xi_{k_0})^\top E_0[P(\cdot)\mid \mathcal F_{k_0}]A(\xi_{k_0})\right]\ge 0
\]
together with lower and upper positive-definite bounds [1911.00308]. Under essential boundedness, one may equivalently use a related matrix function $R(\cdot)$ [1911.00308].

When specialized to the dt-ASLS augmented form, one sets $P(S_k \xi^{k+})\equiv P(\xi_k)$ or $R(S_k \xi^{k+})\equiv R(\xi_k)$ and uses the Lyapunov function
\[
V(Z_k)=Z_k^\top P(\xi_k) Z_k.
\]
If one seeks a constant Lyapunov matrix, the sufficient condition becomes
\[
E\!\left[\lambda^2 P_0-\bar A(\xi_k)^\top P_0 \bar A(\xi_k)\right]\ge 0
\]
for all $k$ [1911.00308].

The practical verification procedure given in the same source proceeds by identifying the augmented matrix $\bar A(\xi)$, deciding whether to seek a constant or $\xi$-dependent Lyapunov matrix, formulating the corresponding LMI, and using a standard semidefinite-program solver such as SeDuMi or Mosek to check feasibility [1911.00308]. If $\xi_k$ takes finitely many values, the expectation condition becomes a finite family of LMIs [1911.00308].

## 6. Safety certification and the Markovian role of augmentation

In the communication-constrained control problem, augmentation is explicitly motivated by safety synthesis [2507.15031]. Once the delayed, packet-lossy loop is rewritten as a dt-ASLS, the resulting model has a purely augmented linear-in-state form with both additive and multiplicative zero-mean noise terms, which permits the use of a quadratic Control Barrier Certificate
\[
\mathbb B(\mathcal Z)=\mathcal Z^\top P\mathcal Z,\qquad P\succ 0.
\]
The safety constraints are translated into matrix inequalities on the blocks of $A_{\rm aug}$ and $W_{\rm aug}$ so as to certify probabilistic safety [2507.15031].

The stated guarantee is that all trajectories of the original dt-SLS remain within safe regions with a quantified probabilistic bound, despite constant delay and Bernoulli packet loss in the communication channels [2507.15031]. The validation example is an RLC circuit subject to both constant delay and probabilistic data loss [2507.15031].

This use of dt-ASLS clarifies a broader methodological point. Delay-free linear stochastic tools often presuppose a Markov state, whereas delayed and lossy networked loops are not naturally represented in that form unless the state is enlarged. The augmentation step therefore does not remove stochastic complexity; rather, it relocates that complexity into larger matrices and multiplicative switching variables while recovering a finite-dimensional Markov process [2507.15031].

## 7. Relations among formulations and recurring structural themes

Across the cited literature, dt-ASLS has no single universally fixed state vector, but several structural motifs recur.

| Setting | Augmented variable | Purpose |
|---|---|---|
| Exact discretization | Block matrix $H$ or $\widetilde H$ in matrix exponential | Compute exact $Q_d$ |
| Networked control | $X_k$ or $\mathcal Z_k$ with input/state history | Encode delay and packet loss |
| Unknown-input filtering | $z_k=[x_k^\top\ d_k^\top]^\top$ | Estimate states and unknown inputs |
| Second-moment stability | $X_k=\mathrm{col}(x_k,w_{k-1})$, then $Z_k=\mathrm{col}(X_k,w_k)$ | Convert additive-noise model to homogeneous form |

One recurring theme is dimensional enlargement in exchange for structural regularity. In the delay setting, the dimension becomes $n(d+1)+m(d+1)$ once the maximum delay is chosen [2507.15031]. In the unknown-input setting, the dimension becomes $n+p$ [1309.6627]. In the homogeneous stability reformulation, the augmented dimension is $n+p+p$ [1911.00308]. This suggests a common trade-off: augmentation preserves finite dimensionality while moving temporal dependence, hidden inputs, or additive disturbances into explicit coordinates.

A second recurring theme is the centrality of Lyapunov structure. In exact discretization, $Q_d$ satisfies a continuous-time Lyapunov equation [1402.1358]. In second-moment stability, exponential mean-square stability is characterized through conditional-expectation Lyapunov inequalities [1911.00308]. In safety synthesis, quadratic control barrier certificates induce matrix inequalities on augmented blocks [2507.15031]. Although these are distinct problems—covariance computation, stability analysis, and safety certification—they all exploit the fact that augmentation yields a linear representation for which matrix-inequality machinery is available.

A third theme is that augmentation often restores standard theory rather than creating a fundamentally new dynamics class. Once unknown inputs are appended, Kalman-type recursion, detectability, and steady-state convergence can be discussed within a unified minimum-variance framework [1309.6627]. Once delayed packet-loss dynamics are embedded in $\mathcal Z_k$, the system becomes a time-homogeneous Markov process suitable for control barrier certificates [2507.15031]. Once additive stochastic terms are absorbed into a homogeneous random-coefficient matrix, general second-moment Lyapunov theorems apply directly [1911.00308].

Taken together, these formulations establish dt-ASLS as a general augmented modeling device for discrete-time stochastic linear analysis. Its significance lies less in a single canonical equation than in a repeatable transformation principle: enlarge the state until the relevant stochastic, delayed, or latent structure becomes linear, finite-dimensional, and compatible with the desired analytical toolchain [1402.1358], [2507.15031], [1309.6627], [1911.00308].

Source: https://www.emergentmind.com/topics/augmented-discrete-time-stochastic-linear-system-dt-asls