---
title: Local Affine State-Space Models
url: https://www.emergentmind.com/topics/local-affine-state-space-models
type: topic
---

# Local Affine State-Space Models

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 [1601.02777, 1609.09789, 2507.07792]. 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 [1601.02777]. 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 [1609.09789]. 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 [2507.07792].

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 [1601.02777].

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 [1601.02777].

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 \(f:\mathcal U\times\mathcal P\to\mathcal Y\) under \(x(0)=0\) [1508.03446].

The corresponding input-output behavior admits an impulse-response expansion. In discrete time,
\[
f(u,p)(t)=\sum_{m=0}^{t}(h_m\diamond p)(t)\,u(t-m),
\]
with
\[
(h_1\diamond p)(t)=C(p(t))B(p(t-1)),
\]
\[
(h_m\diamond p)(t)=C(p(t))A(p(t-1))\cdots A(p(t-m+1))B(p(t-m)),\quad m>1.
\]
Because \(A,B,C\) are affine in \(p\), 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 [1508.03446].

## 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 \(x_o\) if and only if it is observable and span-reachable from \(x_o\); in strong form, strong minimality is equivalent to observability plus span-reachability from the zero state [1601.02777]. The corresponding rank tests are based on extended reachability and observability matrices. For \(D(\cdot)\equiv 0\),
\[
\mathcal R_0=\begin{bmatrix}x_o & B_0 & \cdots & B_{n_p}\end{bmatrix},\qquad
\mathcal R_{n+1}=\begin{bmatrix}\mathcal R_n & A_0\mathcal R_n & \cdots & A_{n_p}\mathcal R_n\end{bmatrix},
\]
\[
\mathcal O_0= \begin{bmatrix} C_0^\top & \cdots & C_{n_p}^\top \end{bmatrix}^\top,\qquad
\mathcal O_{n+1}= \begin{bmatrix} \mathcal O_n^\top & A_0^\top\mathcal O_n^\top & \cdots & A_{n_p}^\top\mathcal O_n^\top \end{bmatrix}^\top{}^\top,
\]
and
\[
\Sigma\text{ span-reachable from }x_o \iff \mathrm{rank}(\mathcal R_{n_x-1})=n_x,\qquad
\Sigma\text{ observable} \iff \mathrm{rank}(\mathcal O_{n_x-1})=n_x.
\]

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 \(T\) such that
\[
A_i' T = T A_i,\qquad B_i'=T B_i,\qquad C_i' T = C_i,\qquad D_i'=D_i
\]
for all \(i\) [1601.02777]. 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 [1601.02777].

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
\[
\eta^\Sigma_{q,q_0}(\epsilon)=C_qB_{q_0},
\]
and, for a word \(s=j_1\cdots j_m\),
\[
\eta^\Sigma_{q,q_0}(j_1\cdots j_m)=C_qA_{j_1}\cdots A_{j_m}B_{q_0},
\]
where \(q,q_0,j_k\in\{0,\dots,n_p\}\) and \(\epsilon\) is the empty word [1508.03446]. A reduced model \(\bar\Sigma\) is an \(N\)-partial realization if
\[
\eta^{\Sigma}_{q,q_0}(s)=\eta^{\bar\Sigma}_{q,q_0}(s)\quad \forall q,q_0,\ \forall s,\ |s|\le N.
\]

The finite-horizon significance is exact rather than asymptotic. If two LPV-SS models have the same sub-Markov parameters of length up to \(t-1\), then
\[
Y_{\Sigma_1}(u,p)(k)=Y_{\Sigma_2}(u,p)(k),\qquad k=0,\dots,t,
\]
for every input/scheduling pair \((u,p)\); conversely, equality of outputs for all input/scheduling sequences up to time \(t\) implies equality of those sub-Markov parameters up to length \(t-1\) [1508.03446]. Hence an \(N\)-partial realization reproduces the original input-output behavior exactly for all input and scheduling sequences of length up to \(N+1\), equivalently outputs match up to time \(N\) in the paper’s indexing.

The same construction admits a geometric interpretation as partial reachability or partial observability reduction. The \(N\)-partial reachability space is
\[
\mathscr{R}_0(\Sigma)=\operatorname{span}\bigcup_{q_0\in\mathbb I_0^{n_p}}\operatorname{im}(B_{q_0}),
\]
\[
\mathscr{R}_N(\Sigma)=\mathscr{R}_0(\Sigma)+\sum_{j\in\mathbb I_0^{n_p}}\operatorname{im}(A_j\mathscr{R}_{N-1}(\Sigma)),\quad N\ge 1,
\]
and the \(N\)-partial unobservability space is
\[
\mathscr{O}_0(\Sigma)=\bigcap_{q\in\mathbb I_0^{n_p}}\ker(C_q),
\]
\[
\mathscr{O}_N(\Sigma)=\mathscr{O}_0(\Sigma)\cap \bigcap_{j\in\mathbb I_0^{n_p}}\ker(\mathscr{O}_{N-1}(\Sigma)A_j),\quad N\ge 1.
\]
If \(V\) satisfies \(\operatorname{im}(V)=\mathscr R_N(\Sigma)\), then
\[
\bar A_i=V^{-1}A_iV,\qquad \bar B_i=V^{-1}B_i,\qquad \bar C_i=C_iV
\]
defines an \(N\)-partial realization. If \(W\) satisfies \(\ker(W)=\mathscr O_N(\Sigma)\), then
\[
\bar A_i=WA_iW^{-1},\qquad \bar B_i=WB_i,\qquad \bar C_i=C_iW^{-1}
\]
also defines an \(N\)-partial realization. If both are available and \(\operatorname{rank}(WV)=r\), then
\[
\bar A_i=WA_iV(WV)^{-1},\qquad \bar B_i=WB_i,\qquad \bar C_i=C_iV(WV)^{-1}
\]
yields a \(2N\)-partial realization [1508.03446].

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 \(N\ge n_x-1\) the partial spaces become the full reachable and unobservable spaces, so the procedure becomes a full minimization method [1508.03446].

## 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
\[
x_{t+1}=F(x_t)+Bu_t+w_t,\qquad y_t=Cx_t+\nu_t,
\]
and assumes that one state component \(\eta_t\) determines the active local model. The state is partitioned as
\[
x_t=\begin{bmatrix}\eta_t & \zeta_t & \chi_t^\top\end{bmatrix}^\top,
\]
with switching regions
\[
\mathcal R_i=\{\eta_t:\ l_i<\eta_t\le l_{i+1}\},\qquad i=1,\dots,N_r,
\]
and regionwise affine nonlinearity
\[
f(\eta_t)=f_i(\eta_t)=a_i\eta_t+b_i,\qquad \eta_t\in \mathcal R_i.
\]
In region \(\mathcal R_i\), the state equation becomes
\[
x_{t+1}=A_i x_t+Bu_t+b_i+w_t,\qquad y_t=Cx_t+\nu_t
\]
[1609.09789].

For continuous piecewise affine models, continuity at the boundaries imposes
\[
a_i l_{i+1}+b_i=a_{i+1}l_{i+1}+b_{i+1},\qquad i=1,\dots,N_r-1,
\]
which allows the intercepts to be parameterized by one base intercept and the slopes:
\[
b_i = -a_i l_i + b_1 + a_1 l_1 + \sum_{j=1}^{i-1} a_j(l_{j+1}-l_j).
\]
This reduces the free parameters from all \(\{a_i,b_i\}\) to \(\{a_i\}_{i=1}^{N_r}\) plus \(b_1\), together with the common matrices \(F,\Phi,\phi\) [1609.09789].

Identification is performed by expectation maximization. The latent variables are the continuous state trajectory \(x_{1:T}\) and the discrete submodel trajectory \(r_{1:T}\), but the paper explicitly avoids introducing a jump Markov transition density for the mode. Instead, when \(\eta_t\) is directly measured up to noise, the region probability is computed from a Gaussian CDF:
\[
p(r_t=i\mid y_t)=\int_{l_i}^{l_{i+1}} \mathcal N(\eta_t; y_t,[R]_{(1,1)})\,d\eta_t.
\]
Conditional on a sampled mode sequence, the model is linear-Gaussian and the Kalman filter together with the Rauch-Tung-Striebel smoother provides
\[
\hat x_{t\mid 1:T}^{(j)},\qquad \hat P_{t\mid 1:T}^{(j)},\qquad \hat P_{t,t-1\mid 1:T}^{(j)}.
\]
The M-step maximizes the EM surrogate
\[
Q(\theta,\hat\theta_k)=\mathbb E_{\hat\theta_k}\!\left[\log p(x_{1:T},y_{1:T},r_{1:T}\mid \theta)\mid y_{1:T}\right],
\]
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 [1609.09789].

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
\[
p(r_{1:T}\mid y_{1:T})\approx \prod_{t=1}^T p(r_t\mid y_t).
\]

The method is demonstrated on a simulated longitudinal JAS 39 Gripen aircraft model with four regions and boundaries
\[
l=[-1^\circ,\,4^\circ,\,7^\circ,\,12^\circ,\,16^\circ],
\]
over \(T=1800\) time steps with sample time \(1/60\) s, using \(M=300\) 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 [1609.09789].

## 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
\[
x(k+1)=\sum_{j=1}^{n_{\mathrm{LM},x}} \big(A_j x(k)+b_j u(k)+q_j\big)\,\omega_j^x(k),
\]
\[
\hat y(k)=\sum_{j=1}^{n_{\mathrm{LM},y}} \big(c_j x(k)+d_j u(k)+p_j\big)\,\omega_j^y(k),
\]
where \(\omega_j^x,\omega_j^y\) are normalized radial basis function weights. Each local model is valid in a region of the extended input/state space \(\tilde u=[x,u]^T\), 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 [2507.07792].

The specific contribution is a space-filling regularization for the learned state trajectory. The state trajectory is viewed as the point set
\[
\mathcal{S}=\{\tilde u(k)\mid k=1,\dots,N\},
\]
and the main regularization indicator is the minimum-distance-to-grid measure
\[
V_p = \frac{1}{n_g}\sum_{j=1}^{n_g}\min_{k\in\{1,\dots,N\}} d(\tilde u(k),g_j),
\]
with Euclidean distance
\[
d(\tilde u(k),g_j)=\sqrt{(\tilde u(k)-g_j)^T(\tilde u(k)-g_j)}.
\]
Small \(V_p\) 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 \(V_p\) [2507.07792].

Two regularization forms are proposed. The first directly penalizes poor space-filling,
\[
\min_\theta \; J(\theta)+\lambda V_p(\theta)^2 \quad \text{s.t. dynamics in (2)},
\]
and the second penalizes deviation from a desired target space-filling level,
\[
\min_\theta \; J(\theta)+\lambda\big(V_p(\theta)-V_{p,\mathrm{Target}}\big)^2 \quad \text{s.t. dynamics in (2)}.
\]
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 [2507.07792].

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 [2507.07792].

The benchmark system is the Bouc-Wen Hysteretic System. The main study focuses on the target-space-filling objective with
\[
V_{p,\mathrm{Target}}=0.25,\qquad \lambda=10^3.
\]
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
\[
0.981 \cdot 10^{-5}\,\text{m},
\]
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 [2507.07792].

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,
\[
\mathcal{M}_{\text{loc}} = \mathcal{M}_1 \cup \dots \cup \mathcal{M}_I,\qquad
\mathcal{M}_i := \{ \bar{\boldsymbol{h}}_i + \boldsymbol{V}_i \boldsymbol{\beta}_i \mid \boldsymbol{\beta}_i \in \mathbb{R}^{D_i} \}.
\]
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 [2007.01543]. 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 \(E\subset \mathbb R^p\) are defined by affine local characteristics
\[
b(x)=a^0+\sum_{i=1}^p a^i x_i,\qquad
c(x)=A^0+\sum_{i=1}^p A^i x_i,\qquad
K(x,dz)=K^0(dz)+\sum_{i=1}^p K^i(dz)\,x_i,
\]
and satisfy the affine transform formula
\[
\mathbb{E}_x\!\left[e^{u^\top X_t}\right] = \exp\big(\psi_0(t,u)+\psi(t,u)^\top x\big)
\]
when the Riccati system is well posed [1005.1099]. 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, \(1\times 1\) convolutions, and residual fusion around a dynamic state-space scan. In the encoder, the local-global fusion is summarized by
\[
z = \mathrm{Natten}(x),\qquad
g = \mathrm{SS2D}(z),\qquad
y = \mathrm{Conv}_{1\times1}(z + g),
\]
with local attention first, then SS2D, followed by shortcut fusion [2412.11890]. 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.

Source: https://www.emergentmind.com/topics/local-affine-state-space-models