Augmented Discrete-Time Stochastic Linear System

• Augmented dt-ASLS is a discrete-time stochastic model that enlarges the state to explicitly capture delays, packet losses, and latent disturbances.
• It transforms non-Markovian dynamics into finite-dimensional, linear representations, enabling the use of Kalman filtering, Lyapunov analysis, and robust control techniques.
• This modeling approach bridges continuous-time processes and networked control systems by exact covariance discretization and unbiased filtering of unknown inputs.

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 $Q_d$ (Wahlström et al., 2014). 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 (Akbarzadeh et al., 20 Jul 2025, Yong et al., 2013, Hosoe et al., 2019).

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 (Akbarzadeh et al., 20 Jul 2025). In filtering with unknown inputs, the underlying model instead includes an additional disturbance $d_k\in\mathbb{R}^p$ entering both dynamics and measurements (Yong et al., 2013). In stochastic-stability analysis, the basic form is often written as

$Q_d$0

with random matrices $Q_d$1 and $Q_d$2 driven by an underlying process $Q_d$3 (Hosoe et al., 2019).

The defining operation in a dt-ASLS is augmentation of the state. In the unknown-input setting, one forms

$Q_d$4

and, under the filtering assumption $Q_d$5, obtains an equivalent augmented stochastic linear system with block matrices $Q_d$6, $Q_d$7, and $Q_d$8 (Yong et al., 2013). In the communication-constrained setting, one appends a finite history of control inputs to represent a constant uplink delay of $Q_d$9 steps (Akbarzadeh et al., 20 Jul 2025). In the second-moment-stability setting, one augments by past disturbances, first via $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$0 and then via $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$1, so that the entire evolution is rewritten in homogeneous form (Hosoe et al., 2019).

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

$$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$2

sampled with interval $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$3 (Wahlström et al., 2014). The sampled state is $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$4, and the homogeneous dynamics yield the state-transition matrix

$$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$5

The exact discrete-time stochastic model is then

$$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$6

with

$$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$7

after absorbing $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$8 into a single matrix $$x_{k+1} \;=\; A\,x_k \;+\; B\,u_k \;+\; W\,\omega_k,\qquad k\in\mathbb{N},$$9 (Wahlström et al., 2014).

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

$x_k\in\mathbb{R}^n$0

whose exponential contains $x_k\in\mathbb{R}^n$1 in the $x_k\in\mathbb{R}^n$2 block: $x_k\in\mathbb{R}^n$3 A related variant uses

$x_k\in\mathbb{R}^n$4

for which $x_k\in\mathbb{R}^n$5 (Wahlström et al., 2014).

The same source emphasizes a Lyapunov-equation formulation. Defining

$x_k\in\mathbb{R}^n$6

the exact covariance satisfies

$x_k\in\mathbb{R}^n$7

Hence one may compute $x_k\in\mathbb{R}^n$8, form $x_k\in\mathbb{R}^n$9, and solve the Lyapunov equation $u_k\in\mathbb{R}^m$0 using a standard stable solver (Wahlström et al., 2014). The stated numerical rationale is that this method only exponentiates an $u_k\in\mathbb{R}^m$1 matrix instead of a $u_k\in\mathbb{R}^m$2 one and avoids overflow or underflow issues for large $u_k\in\mathbb{R}^m$3 or stiff $u_k\in\mathbb{R}^m$4; if $u_k\in\mathbb{R}^m$5 has integrators, the block-triangular structure can be exploited further (Wahlström et al., 2014).

Although this discretization paper does not use the exact phrase “dt-ASLS” in its title, the resulting model

$u_k\in\mathbb{R}^m$6

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 $u_k\in\mathbb{R}^m$7 steps together with packet losses on both uplink and downlink channels (Akbarzadeh et al., 20 Jul 2025). The augmented state is defined as

$u_k\in\mathbb{R}^m$8

while the fresh control computation serves as the external input: $u_k\in\mathbb{R}^m$9 Packet losses are modeled by independent Bernoulli-distributed random processes: $\omega_k\in\mathbb{R}^q$0 for the uplink and $\omega_k\in\mathbb{R}^q$1 for the downlink (Akbarzadeh et al., 20 Jul 2025).

The paper also introduces the zero-mean auxiliary variable $\omega_k\in\mathbb{R}^q$2 through

$\omega_k\in\mathbb{R}^q$3

in order to decompose the dynamics into mean plus zero-mean multiplicative noise (Akbarzadeh et al., 20 Jul 2025).

A more extensive augmented variable is then formed: $\omega_k\in\mathbb{R}^q$4 and

$\omega_k\in\mathbb{R}^q$5

The resulting dt-ASLS takes the standard form

$\omega_k\in\mathbb{R}^q$6

where $\omega_k\in\mathbb{R}^q$7 collects the last $\omega_k\in\mathbb{R}^q$8 disturbances (Akbarzadeh et al., 20 Jul 2025).

The block structure is central. The $\omega_k\in\mathbb{R}^q$9 block $A$0 switches on or off according to the downlink loss $A$1; the shift rows are identity matrices that push the input history down the register; and $A$2 collects the current and past $A$3 disturbances so that $A$4 accumulates $A$5 via powers of $A$6 (Akbarzadeh et al., 20 Jul 2025). The total dimension is finite and fixed once the maximum delay $A$7 is chosen: $A$8

The stated reason augmentation works is that carrying along the last $A$9 inputs allows the state update for $B$0 to reconstruct the effect of a constant sensor-to-controller delay, while the Bernoulli variables enter as multiplicative switches on the appropriate blocks of $B$1 or $B$2 without any additional memory beyond $B$3 (Akbarzadeh et al., 20 Jul 2025). Once augmented, $B$4 depends only on $B$5, the new disturbance $B$6, and the current Bernoulli draws $B$7; the process is therefore time-homogeneous and Markov (Akbarzadeh et al., 20 Jul 2025).

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 $B$8: $B$9 Here $W$0 and $W$1 are zero-mean white noises with $W$2, $W$3, and $W$4 (Yong et al., 2013).

For filtering purposes, one assumes a trivial random walk

$W$5

with $W$6 white and of small covariance. The augmented state

$W$7

then yields the equivalent dt-ASLS

$W$8

with

$W$9

and measurement model

$d_k\in\mathbb{R}^p$0

(Yong et al., 2013).

The estimation objective is to produce $d_k\in\mathbb{R}^p$1 with covariance

$d_k\in\mathbb{R}^p$2

such that the estimator is unbiased and, among all linear unbiased filters, has minimum trace of $d_k\in\mathbb{R}^p$3 (Yong et al., 2013). The prediction step is

$d_k\in\mathbb{R}^p$4

$d_k\in\mathbb{R}^p$5

and the update step is

$d_k\in\mathbb{R}^p$6

(Yong et al., 2013).

A specific structural constraint arises from unbiasedness: $d_k\in\mathbb{R}^p$7 (Yong et al., 2013). 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” (Yong et al., 2013).

The observability and detectability theory is likewise formulated for the augmented setting. In the time-invariant case, strong observability holds iff

$d_k\in\mathbb{R}^p$8

while strong detectability holds iff the same rank condition is satisfied for all $d_k\in\mathbb{R}^p$9 (Yong et al., 2013). The paper further states that when $Q_d$00 the unified filter reduces to the Gillijns-De Moor filter, when $Q_d$01 has full column rank one recovers the Darouach-Hou or Fang–Simon–Yong filters, and when also $Q_d$02 the filter collapses to the standard Kalman filter (Yong et al., 2013).

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 (Hosoe et al., 2019). Starting from

$Q_d$03

one first forms

$Q_d$04

which leads to an intermediate model with an additive term in $Q_d$05. A further augmentation

$Q_d$06

then yields the homogeneous form

$Q_d$07

where the system matrix has the block-upper-triangular form

$Q_d$08

(Hosoe et al., 2019).

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 $Q_d$09 and $Q_d$10 such that

$Q_d$11

for all $Q_d$12 (Hosoe et al., 2019). Under a minimal bounded-moment assumption on $Q_d$13, the paper states that these five notions collapse into just two classes, and that uniform asymptotic stability is equivalent to exponential stability (Hosoe et al., 2019).

The principal criterion is a general Lyapunov condition. Under second-moment boundedness of the entries of $Q_d$14, exponential mean-square stability holds iff there exist $Q_d$15 and a matrix-valued function $Q_d$16 satisfying conditional-expectation inequalities of the form

$Q_d$17

together with lower and upper positive-definite bounds (Hosoe et al., 2019). Under essential boundedness, one may equivalently use a related matrix function $Q_d$18 (Hosoe et al., 2019).

When specialized to the dt-ASLS augmented form, one sets $Q_d$19 or $Q_d$20 and uses the Lyapunov function

$Q_d$21

If one seeks a constant Lyapunov matrix, the sufficient condition becomes

$Q_d$22

for all $Q_d$23 (Hosoe et al., 2019).

The practical verification procedure given in the same source proceeds by identifying the augmented matrix $Q_d$24, deciding whether to seek a constant or $Q_d$25-dependent Lyapunov matrix, formulating the corresponding LMI, and using a standard semidefinite-program solver such as SeDuMi or Mosek to check feasibility (Hosoe et al., 2019). If $Q_d$26 takes finitely many values, the expectation condition becomes a finite family of LMIs (Hosoe et al., 2019).

6. Safety certification and the Markovian role of augmentation

In the communication-constrained control problem, augmentation is explicitly motivated by safety synthesis (Akbarzadeh et al., 20 Jul 2025). 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

$Q_d$27

The safety constraints are translated into matrix inequalities on the blocks of $Q_d$28 and $Q_d$29 so as to certify probabilistic safety (Akbarzadeh et al., 20 Jul 2025).

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 (Akbarzadeh et al., 20 Jul 2025). The validation example is an RLC circuit subject to both constant delay and probabilistic data loss (Akbarzadeh et al., 20 Jul 2025).

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 (Akbarzadeh et al., 20 Jul 2025).

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 $Q_d$30 or $Q_d$31 in matrix exponential	Compute exact $Q_d$32
Networked control	$Q_d$33 or $Q_d$34 with input/state history	Encode delay and packet loss
Unknown-input filtering	$Q_d$35	Estimate states and unknown inputs
Second-moment stability	$Q_d$36, then $Q_d$37	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 $Q_d$38 once the maximum delay is chosen (Akbarzadeh et al., 20 Jul 2025). In the unknown-input setting, the dimension becomes $Q_d$39 (Yong et al., 2013). In the homogeneous stability reformulation, the augmented dimension is $Q_d$40 (Hosoe et al., 2019). 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$41 satisfies a continuous-time Lyapunov equation (Wahlström et al., 2014). In second-moment stability, exponential mean-square stability is characterized through conditional-expectation Lyapunov inequalities (Hosoe et al., 2019). In safety synthesis, quadratic control barrier certificates induce matrix inequalities on augmented blocks (Akbarzadeh et al., 20 Jul 2025). 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 (Yong et al., 2013). Once delayed packet-loss dynamics are embedded in $Q_d$42, the system becomes a time-homogeneous Markov process suitable for control barrier certificates (Akbarzadeh et al., 20 Jul 2025). Once additive stochastic terms are absorbed into a homogeneous random-coefficient matrix, general second-moment Lyapunov theorems apply directly (Hosoe et al., 2019).

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 (Wahlström et al., 2014, Akbarzadeh et al., 20 Jul 2025, Yong et al., 2013, Hosoe et al., 2019).