Unverified 2026

Latent Reference Governor for Safe SSMs

Implementation & benchmark of arXiv:2607.12178 — Dynamically Feasible Planning and Control in Complex Environments: a Scalable Systematic Approach

Usefulness6/10

Difficulty6/10

Novelty7/10

Dynamics Stability Accuracy Ssm Rnn Inference speedup Training dynamics Control theory Dynamical systems Linear algebra Geometry Optimization

Source paper: Dynamically Feasible Planning and Control in Complex Environments: a Scalable Systematic Approach arXiv:2607.12178 ⓘ · analyzed Aug 30, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Insert a reference governor between a neural model's raw latent command and a linear state-space update, so that hidden states and outputs remain inside a prescribed union of polytopes. At every step, choose the largest interpolation toward the desired command whose predicted trajectory remains in the offline safe set. This can prevent hidden-state explosions and invalid latent trajectories without globally shrinking the model's weights.

Formulas

$$x_{k+1}=Ax_k+Bv_k,\qquad y_k=Cx_k+Dv_k.$$

$$\mathcal{Y}=\bigcup_{j=1}^{M}\{y\in\mathbb{R}^{n_y}:H_jy\leq h_j\},\qquad \mathcal{O}_{\infty}=\{(x,v):y_t\in\mathcal{Y}\ \forall t\geq 0\ \text{under }v_t=v\}.$$

$$v_k=v_{k-1}+\alpha_k(r_k-v_{k-1}),\qquad \alpha_k=\max\{\alpha\in[0,1]:(x_k,v_{k-1}+\alpha(r_k-v_{k-1}))\in\mathcal{O}_{\infty}\}.$$

$$\rho(A)<1\quad\Longrightarrow\quad A^t\to 0,\qquad y_{k+t}=CA^tx_k+\left(C\sum_{i=0}^{t-1}A^iB+D\right)v_k.$$

Mathematical statement

Use the discrete-time latent model $x_{k+1}=Ax_k+Bv_k$ and observable output $y_k=Cx_k+Dv_k$, where $x_k\in\mathbb{R}^{n_x}$ is the SSM hidden state, $v_k\in\mathbb{R}^{n_v}$ is a controllable latent command, and $y_k\in\mathbb{R}^{n_y}$ is the constrained latent output. Let the non-convex admissible region be $\mathcal{Y}=\bigcup_{j=1}^M\{y:H_jy\leq h_j\}$. The safe-command set is $\mathcal{O}_\infty=\{(x,v):y_t\in\mathcal{Y}\ \forall t\geq 0\}$ under constant command $v$. Given desired command $r_k$, use $v_k=v_{k-1}+\alpha_k(r_k-v_{k-1})$, with the largest $\alpha_k\in[0,1]$ for which $(x_k,v_k)\in\mathcal{O}_\infty$. If $A$ is Schur stable, $\rho(A)<1$, and $r_k$ is strictly admissible, the reference-governor construction guarantees constraint satisfaction and convergence of $v_k$ to $r_k$. In a neural implementation, the guarantee is exact only for the identified linear model, so model mismatch must be measured and compensated with constraint margins.

Implementation notes

1. Integration point: use a diagonal or low-dimensional linear SSM block inside an RNN, sequence model, or latent world model. Let its hidden state be $x_k$ and expose a trainable command $r_k$ produced by the neural network. Keep the learned state matrices $A,B,C,D$ fixed during the first experiment, or update their estimates slowly during training. Define axis-aligned or general polytopes for allowable hidden coordinates, output magnitudes, or task-specific latent regions. 2. Pseudocode: precompute an approximate safe set by sampling candidate constant commands $v$, propagating $x_{t+1}=Ax_t+Bv$ for $T$ steps, and retaining candidates satisfying every polytope constraint. Optionally fit a polyhedral union to the retained samples. At inference, compute $r_k$, search $\alpha$ on a grid such as $\{0,0.01,\ldots,1\}$, predict the trajectory for each candidate using the displayed flow equation, reject candidates violating any polytope, and apply the largest surviving $\alpha$; then update the SSM state. 3. Computed versus estimated: $A,B,C,D$ and the finite-horizon safe set are computed or fitted offline; the infinite-horizon condition is approximated by checking until $T$ such that $\lVert A^T\rVert$ is below a tolerance. Estimate model mismatch by comparing predicted and actual hidden states, and shrink each polytope by a margin proportional to a high quantile of this error. 4. First cheap experiment: use a 2D linear SSM driven by an MLP on a synthetic Adding or Copy task, comparing an unconstrained SSM, a spectral-normalized SSM, and the governor-SSM. Sweep the spectral radius and command jump size. The predicted signature is zero constraint violations for exact-model trajectories initialized in the computed safe set, a sharp failure boundary near $\rho(A)=1$, and monotone convergence of $\lVert r_k-v_k\rVert$ toward zero for strictly admissible targets. With finite-horizon approximation, violation probability should decrease as $T$ increases and $\lVert A^T\rVert$ decreases.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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