Local Affine State-Space Models

• Local affine state-space models are structured nonlinear paradigms that use locally valid affine laws, driven by scheduling variables, state partitions, or soft interpolations.
• They employ methods such as LPV-SSA, PWASS, and LMSSN to decompose and rebuild global dynamics from regional affine behaviors.
• These models offer practical benefits in control and system identification by enabling finite-horizon equivalence, exact model reduction, and stable local dynamics.

Searching arXiv for the cited LPV/local affine state-space papers to ground the article. Local affine state space models are state-space descriptions in which the dynamics and output relations are constrained to be affine in a locally valid variable or representation. In the cited literature, closely related constructions include linear parameter-varying state-space representations with affine static scheduling dependence, piecewise affine state-space models whose active affine law is determined by a region of the state or input domain, and local model state-space networks formed by smoothly blended local affine models (Petreczky et al., 2016, Rui et al., 2016, Klein et al., 10 Jul 2025). This suggests that the term denotes a family of structured nonlinear modeling paradigms rather than a single canonical formalism.

1. Taxonomy of local affine state-space constructions

A recurring source of ambiguity is that “local affine” refers to different mechanisms of locality in different communities. The cited papers separate at least three principal mechanisms: external scheduling, state-dependent region selection, and soft interpolation across local models. Closely adjacent literatures also use affine structure in parameter space or in local stochastic characteristics rather than directly in a deterministic state equation.

Family	Locality mechanism	Affine structure
LPV-SSA / LPV-SS	Scheduling signal $p(t)$	Matrices $A(p),B(p),C(p),D(p)$ affine in $p$
PWASS	Region of a measured state component or input variable	One affine state equation per region
LMSSN	NRBF-weighted local regions in extended input/state space	Global model is a superposition of local affine models

For affine LPV models, the system is linear in the state and input, but its matrices vary affinely with the instantaneous scheduling variable; the realization-theoretic literature denotes this class by LPV-SSA and emphasizes static affine dependence (Petreczky et al., 2016). In PWASS models, one measured state component determines which regional affine law is active, so the global model is obtained by stitching together finitely many affine submodels (Rui et al., 2016). In LMSSN, the local affine laws are blended smoothly by normalized radial basis function weights, yielding a soft partition-of-unity representation over extended input/state space (Klein et al., 10 Jul 2025).

A common misconception is to identify all affine state-space models with a single construction. The available literature instead distinguishes externally scheduled affine models from state-dependent switching models and from soft local-model networks. Another misconception is that “local affine” necessarily implies piecewise discontinuous switching. In LMSSN the interpolation is smooth, whereas in PWASS the regional assignment is explicit and the continuous piecewise affine case is enforced by boundary constraints.

2. Affine LPV state-space representations

The realization-theoretic LPV formulation considered in the cited work has the form

$$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$

with

$$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$

Here $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$ is the scheduling variable and $\xi$ denotes either differentiation in continuous time or the forward shift in discrete time (Petreczky et al., 2016).

This formulation restricts attention to matrices that depend affinely and statically on the instantaneous scheduling signal. The restriction is deliberate: the realization theory emphasizes that this class is close to the models used in identification and control, while retaining a finite-dimensional structure. The assumption $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$ is highlighted because it ensures that the scheduling set contains enough variation to identify the affine coefficients uniquely (Petreczky et al., 2016).

The model-reduction paper treats the discrete-time, zero-initial-state case

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$

with the same affine dependence

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$

and studies the induced input-output map $A(p),B(p),C(p),D(p)$0 under $A(p),B(p),C(p),D(p)$1 (Bastug et al., 2015).

The corresponding input-output behavior admits an impulse-response expansion. In discrete time,

$A(p),B(p),C(p),D(p)$2

with

$A(p),B(p),C(p),D(p)$3

$A(p),B(p),C(p),D(p)$4

Because $A(p),B(p),C(p),D(p)$5 are affine in $A(p),B(p),C(p),D(p)$6, these coefficients expand into sums of products of scheduling components and constant matrix products. This is the basis for the sub-Markov-parameter formalism used throughout LPV realization and reduction theory (Bastug et al., 2015).

3. Realization theory, minimality, and moment matching

A central structural result for affine LPV-SSA is the Kalman-style minimality theorem: an LPV-SSA realization is minimal with respect to an initial state $A(p),B(p),C(p),D(p)$7 if and only if it is observable and span-reachable from $A(p),B(p),C(p),D(p)$8; in strong form, strong minimality is equivalent to observability plus span-reachability from the zero state (Petreczky et al., 2016). The corresponding rank tests are based on extended reachability and observability matrices. For $A(p),B(p),C(p),D(p)$9,

$p$0

$p$1

and

$p$2

Minimal realizations are unique up to a scheduling-independent linear isomorphism. If two minimal LPV-SSA realizations generate the same input-output map, then there exists a nonsingular matrix $p$3 such that

$p$4

for all $p$5 (Petreczky et al., 2016). The same paper proves a Hankel-rank criterion: an input-output function has an LPV-SSA realization if and only if it has an impulse response representation and its Hankel matrix has finite rank; the rank equals the dimension of a minimal LPV-SSA realization. It also proves correctness conditions for a Ho-Kalman-style partial-realization algorithm applied to finite Hankel blocks (Petreczky et al., 2016).

The model-reduction theory specializes these ideas to moment matching for discrete-time LPV-SS models with affine dependence. It defines sub-Markov parameters by

$p$6

and, for a word $p$7,

$p$8

where $p$9 and $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$0 is the empty word (Bastug et al., 2015). A reduced model $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$1 is an $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$2-partial realization if

$$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$3

The finite-horizon significance is exact rather than asymptotic. If two LPV-SS models have the same sub-Markov parameters of length up to $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$4, then

$$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$5

for every input/scheduling pair $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$6; conversely, equality of outputs for all input/scheduling sequences up to time $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$7 implies equality of those sub-Markov parameters up to length $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$8 (Bastug et al., 2015). Hence an $$\Sigma:\quad \begin{cases} \xi x(t)=A(p(t))x(t)+B(p(t))u(t),\ y(t)=C(p(t))x(t)+D(p(t))u(t), \end{cases}$$9-partial realization reproduces the original input-output behavior exactly for all input and scheduling sequences of length up to $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$0, equivalently outputs match up to time $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$1 in the paper’s indexing.

The same construction admits a geometric interpretation as partial reachability or partial observability reduction. The $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$2-partial reachability space is

$$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$3

$$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$4

and the $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$5-partial unobservability space is

$$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$6

$$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$7

If $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$8 satisfies $$A(p)=A_0+\sum_{i=1}^{n_p}A_i p_i,\quad B(p)=B_0+\sum_{i=1}^{n_p}B_i p_i,\quad C(p)=C_0+\sum_{i=1}^{n_p}C_i p_i,\quad D(p)=D_0+\sum_{i=1}^{n_p}D_i p_i.$$9, then

$p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$0

defines an $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$1-partial realization. If $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$2 satisfies $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$3, then

$p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$4

also defines an $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$5-partial realization. If both are available and $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$6, then

$p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$7

yields a $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$8-partial realization (Bastug et al., 2015).

A further misconception is that LPV moment matching is merely a numerical approximation heuristic. In the formulation above, the preservation of sub-Markov parameters gives an exact finite-horizon equivalence result, and for $p(t)\in\mathbb P\subseteq\mathbb R^{n_p}$9 the partial spaces become the full reachable and unobservable spaces, so the procedure becomes a full minimization method (Bastug et al., 2015).

4. State-dependent and piecewise affine state-space models

A different local-affine construction partitions the state or input domain into finitely many regions and assigns an affine state-space law to each region. The PWASS literature begins from

$\xi$0

and assumes that one state component $\xi$1 determines the active local model. The state is partitioned as

$\xi$2

with switching regions

$\xi$3

and regionwise affine nonlinearity

$\xi$4

In region $\xi$5, the state equation becomes

$\xi$6

(Rui et al., 2016).

For continuous piecewise affine models, continuity at the boundaries imposes

$\xi$7

which allows the intercepts to be parameterized by one base intercept and the slopes: $\xi$8 This reduces the free parameters from all $\xi$9 to $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$0 plus $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$1, together with the common matrices $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$2 (Rui et al., 2016).

Identification is performed by expectation maximization. The latent variables are the continuous state trajectory $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$3 and the discrete submodel trajectory $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$4, but the paper explicitly avoids introducing a jump Markov transition density for the mode. Instead, when $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$5 is directly measured up to noise, the region probability is computed from a Gaussian CDF: $\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$6 Conditional on a sampled mode sequence, the model is linear-Gaussian and the Kalman filter together with the Rauch-Tung-Striebel smoother provides

$\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$7

The M-step maximizes the EM surrogate

$\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$8

which is approximated by Monte Carlo over sampled mode trajectories, and the paper notes that the gradient and Hessian can be computed exactly, so Newton’s method can be used (Rui et al., 2016).

The distinction from HMM-style switching models is substantive. In the PWASS formulation, the active region is a deterministic function of the current state in the underlying model, not an independent Markov chain. The CDF-based probabilities therefore encode measurement uncertainty about a state-dependent region label rather than stochastic mode dynamics. The paper also states explicit assumptions and limitations: known region boundaries, direct measurement of the switching variable up to noise, Gaussian independent noises, Gaussian posterior approximation for RTS smoothing, and the factorized approximation

$\mathrm{Aff}\,\mathbb P=\mathbb R^{n_p}$9

The method is demonstrated on a simulated longitudinal JAS 39 Gripen aircraft model with four regions and boundaries

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$0

over $$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$1 time steps with sample time $$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$2 s, using $$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$3 sampled trajectories per EM iteration. The reported outcome is convergence over 100 EM iterations, good recovery of the piecewise affine function shape, and small residual bias in some boundary values (Rui et al., 2016).

5. Local-model state-space networks and space-filling regularization

In nonlinear system identification, local affine state-space modeling also appears in the Local Model State Space Network. For a single-input single-output system, the LMSSN representation is

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$4

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$5

where $$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$6 are normalized radial basis function weights. Each local model is valid in a region of the extended input/state space $$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$7, and the global model is formed by soft interpolation across these regions. The paper relates this structure to Takagi-Sugeno, piecewise affine, and qLPV models, and emphasizes the LOLIMOT partitioning strategy with axis-orthogonal splits and automatically parametrized validity functions (Klein et al., 10 Jul 2025).

The specific contribution is a space-filling regularization for the learned state trajectory. The state trajectory is viewed as the point set

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$8

and the main regularization indicator is the minimum-distance-to-grid measure

$$\Sigma\left\{ \begin{aligned} x(t+1) &= A(p(t))x(t)+B(p(t))u(t),\ y(t) &= C(p(t))x(t), \end{aligned}\right.$$9

with Euclidean distance

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$0

Small $$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$1 corresponds to good grid coverage. The paper also discusses the convex hull volume and a Kullback-Leibler-divergence-based density diagnostic, but the regularization term itself is built primarily on $$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$2 (Klein et al., 10 Jul 2025).

Two regularization forms are proposed. The first directly penalizes poor space-filling,

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$3

and the second penalizes deviation from a desired target space-filling level,

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$4

The motivation is geometric rather than merely aesthetic: the paper argues that state-trajectory compression degrades coverage of local regions, reduces interpretability, and can lead to unstable local dynamics, including local models with unstable poles (Klein et al., 10 Jul 2025).

The training procedure begins from a deterministic initialization based on a global linear state-space model, obtained from the Best Linear Approximation and subspace identification, then transformed into a balanced realization. LOLIMOT is applied split-by-split, each split being optimized by nonlinear minimization with Quasi-Newton and a BFGS Hessian approximation. The paper notes an early-stopping effect under regularization because the space-filling term conflicts with pure output-error reduction, so the optimizer terminates much earlier than in the unregularized case (Klein et al., 10 Jul 2025).

The benchmark system is the Bouc-Wen Hysteretic System. The main study focuses on the target-space-filling objective with

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$5

Selection is based on validation error, with a normalized RMSE threshold tied to 40 dB SNR. The reported findings are that the best regularized model used seven splits, the corresponding unregularized best model stopped at four splits, the regularized model achieved a test RMSE of

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$6

and the regularized model ensemble showed no detected instability on test data. In the illustrative example, regularized training ended after 324 iterations versus 3232 for the unregularized case (Klein et al., 10 Jul 2025).

These results support a more specific interpretation of locality: in LMSSN the issue is not only whether each local affine law can approximate the dynamics, but whether training preserves a state-space geometry that actually excites and uses the local regions in a balanced manner.

6. Adjacent affine-state-space notions and modern reinterpretations

Several neighboring literatures use local affine structure in ways that are related but not identical to the classical control-identification meaning of a local affine state-space model. One example is online supervised acoustic system identification, where the unknown room impulse responses are assumed to lie near a low-dimensional manifold approximated by a union of affine subspaces,

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$7

The offsets and bases are learned offline from training data using unsupervised K-means followed by PCA, and online evidence maximization selects the active affine subspace. The denoising step projects the adaptive-filter update onto the selected local affine subspace (Haubner et al., 2020). This is an affine local model in parameter space, not a local affine state equation for the plant, but it shares the same geometric intuition: locally Euclidean structure is exploited to constrain estimation.

Another adjacent notion appears in stochastic-process theory. Affine jump-diffusions on a general closed convex state space $$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$8 are defined by affine local characteristics

$$A(p(t))=A_0+\sum_{i=1}^{n_p}A_i\,p_i(t),\quad B(p(t))=B_0+\sum_{i=1}^{n_p}B_i\,p_i(t),\quad C(p(t))=C_0+\sum_{i=1}^{n_p}C_i\,p_i(t),$$9

and satisfy the affine transform formula

$A(p),B(p),C(p),D(p)$00

when the Riccati system is well posed (Spreij et al., 2010). The connection is conceptual rather than structural: the affine dependence is imposed on local stochastic characteristics and generator coefficients on a convex state space, not on a bank of local deterministic subsystems.

A further contemporary reinterpretation appears in deep state-space architectures for vision. SegMAN is built around state space models and local attention rather than pure attention or pure convolution. The paper does not introduce a separately named affine-SSM module; the relevant pattern is the repeated use of linear projections, $A(p),B(p),C(p),D(p)$01 convolutions, and residual fusion around a dynamic state-space scan. In the encoder, the local-global fusion is summarized by

$A(p),B(p),C(p),D(p)$02

with local attention first, then SS2D, followed by shortcut fusion (Fu et al., 2024). This suggests a modern analogical use of “local affine/state-space” behavior rather than the classical realization-theoretic meaning.

Taken together, these adjacent developments show that affine locality can refer to local scheduling dependence, local state-space partitions, smoothly blended local affine laws, local affine manifolds in parameter space, or affine local characteristics on a state domain. The technical commonality is not a single syntax but a structural restriction: the model is made nonlinear globally by composing pieces that are affine in an appropriately local variable.