Scaled Controllability Gramian

• Scaled controllability Gramian is a modified matrix that quantifies system reachability while incorporating adjustments for time horizon, physical scaling, and unstable behaviors.
• It employs similarity, energy-weighted, and operator-norm scaling techniques to achieve unit consistency and bounded behavior in both linear and nonlinear settings.
• Practical algorithms, such as scaling-and-squaring and iterative normalization, leverage these Gramians for robust actuator placement, statistical estimation, and control in complex systems.

A scaled controllability Gramian is a generalization or modification of the classical controllability Gramian to handle issues of time horizon, input/output magnitude, physical scaling, nonlinearity, marginal/unstable dynamics, or problem-specific normalization. Scaled Gramians are central to modern controllability analysis and design in both finite- and infinite-dimensional, linear and nonlinear, classical and statistical-mechanical frameworks.

1. Definitions and Core Constructions

The classical controllability Gramian for a linear time-invariant system,

$\dot{x}(t) = Ax(t) + Bu(t),\quad x(0)=0,$

with $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, is

$W(T) = \int_0^T e^{A t} B B^T e^{A^T t} \, dt$

over $[0,T]$, or, for stable $A$,

$$W_\infty = \int_0^\infty e^{A t} B B^T e^{A^T t} \, dt.$$

This object quantifies the reachability of the state-space by unit $L^2$-norm controls.

A scaled controllability Gramian modifies this core definition in several distinct but overlapping ways:

• Similarity and diagonal scaling: Used to obtain bounded objects as $T\to\infty$ when $A$ has nonnegative real eigenvalues, as in

$$W_\mathrm{scaled}(T) = D(T)^{-1} Q^{-1} W_\text{unscaled}(T) Q^{-T} D(T)^{-T},$$

where $D(T)$ appropriately normalizes block contributions according to their dynamical growth rates (Umezu et al., 15 Jan 2026).

• Energy-weighted scaling: If the physical energy is $x^T Q x$, the Gramian is typically recomputed in appropriately weighted coordinates, e.g. $A\to F A F^{-1}$, $B\to F B$ for $F^T F=Q$ (Herrmann et al., 2023).
• Rescaling for robustness or operator-norm bounds: In 1D convolutional neural networks, the discrete-time Gramian $X_i$ is rescaled by a weight $Q_{i-1}$ recursively propagated throughout the network to ensure the global mapping has a prescribed Lipschitz bound (Pauli et al., 2023).
• Path or domain scaling in PDEs: In the multitime PDE case, the $\gamma$-Gramian weights the contributions from each time axis by integrating along paths $\gamma$ in the time domain (Ghiu et al., 2011).

The scaled Gramian often underpins more sophisticated control objectives (e.g. volumetric or average energy scores) and explicit optimization/inference procedures in non-classical settings (Umezu et al., 15 Jan 2026).

2. Motivations for Scaling

Scaling Gramians is essential for several reasons:

• Infinite Horizon/Lyapunov Divergence: For unstable or marginally-stable systems, the usual integral diverges as $T\rightarrow\infty$. Carefully constructed similarity and diagonal scaling (often following Jordan or real-Schur decompositions) yields a bounded, well-posed limit even in the presence of unstable modes (Umezu et al., 15 Jan 2026).
• Unit and Energy Consistency: Physical units of state and control vary widely across applications. Scaling enforces unit invariance and ensures dynamic metrics (e.g., controllability indices, input-output pairings) are not dominated by arbitrary choices of units or signal range. Sinkhorn–Knopp scaling achieves full unit-invariance (Bengtsson et al., 2019).
• Operator-norm and Robustness Guarantees: In machine learning applications, particularly 1D CNN parameterizations, scaling the Gramian via layerwise recursive weights constrains the induced operator norm or global Lipschitz constant. This enables precise robustness control via linear matrix inequalities (LMIs) (Pauli et al., 2023).
• Statistical and Information-Theoretic Connections: Scaling directly governs the tradeoff between energetic controllability and statistical uncertainty, as revealed in the explicit relations between scaled Gramians, Fisher information matrices, and (thermodynamic/Shannon) entropy—scaling up the Gramian lowers control energy but increases estimation entropy (Silva, 8 Jul 2025).

3. Theoretical Properties and Scaling Formulas

Infinite-Horizon and Block Scaling

For general $A$ (possibly unstable), the scaled Gramian is defined via a change of coordinates and block-diagonal scaling: $$W(B;T) = D(T)^{-1} Q^{-1} \widetilde{W}(B;T) Q^{-T} D(T)^{-T}$$ where $Q$ diagonalizes $A$ into Jordan blocks, and $D(T)$ is block-diagonal, normalizing each block according to its growth (e.g., identity for stable blocks, polynomial or exponential for marginal/unstable blocks, respectively).

As $T\to\infty$, this construction guarantees convergence: $W(B;T) \rightarrow W_\infty(B)$ which is block-diagonal with each block corresponding to stable, marginal, or unstable dynamics—with explicit formulas for the limiting cases (Umezu et al., 15 Jan 2026).

Information and Duality Scaling

Suppose $W_c$ is rescaled by $\alpha$: $\widetilde{W}_c = \alpha W_c$. This induces scaling laws (Silva, 8 Jul 2025):

Quantity	Scaling Law
Gramian $\widetilde{W}_c$	$\alpha W_c$
Determinant $\det \widetilde{W}_c$	$\alpha^n \det W_c$
Minimum energy $E_{\min}$	$1/\alpha\,E_{\min}$
Fisher Information $I$	$I/\alpha$
Entropy $H$	$H + \frac{1}{2} \ln \alpha$
Thermodynamic Entropy $S$	$S + k_B \ln \alpha$

Thus, increasing the Gramian scaling increases reachable set volume (controllability), decreases minimum control energy, but reduces estimation precision and increases entropy.

PDEs, Multitime, and Nonuniform Scaling

In systems with multiple time variables or nonuniform time growth, scaling enters via path-integral or explicit time-dependent exponential factors: $$W(t, t+\sigma) = \int_t^{t+\sigma} \Phi_A(t, s) B(s) B^T(s) \Phi_A^T(t, s) ds$$ with time-dependent bounds: $$e^{-2\mu_0 t}\alpha_0(\sigma) I \leq W(t, t+\sigma) \leq e^{2\mu_1 t}\alpha_1(\sigma) I$$ capturing nonuniform growth or decay in the dynamics and subsuming both Kalman's UCC and weaker notions (Huerta et al., 2024). In the multitime PDE setting, the $\gamma$-Gramian flexibly reflects the integration path and coordinate scaling (Ghiu et al., 2011).

Nonlinear and Stochastic Extensions

For nonlinear systems, Gibbs-type density scaling is introduced, leveraging minimum-energy functions and stochastic path distributions: $$W = \frac{\int x x^T e^{-\beta E(x)} dx}{\int e^{-\beta E(y)} dy}$$ with $\beta$ (inverse temperature) setting the control-noise tradeoff (Kashima, 2016). Monte Carlo or noise-excited trajectory ensembles can thus estimate effective controllability.

4. Numerical Algorithms and Practical Computation

Contemporary scalable algorithms exploit scaling for conditioning and stability.

• Scaling-and-Squaring and Doubling: High-accuracy computation of $e^{A}$ and $G(T)$ proceeds by repeated squaring after initial (Padé/Legendre) scaling. For the Gramian,

$G(2T) = G(T) + e^{A T} G(T) e^{A^T T}$

enables recursive Cholesky/QR-based schemes that preserve low-rank factors and numerical stability. Scaling (in $A$ and $B$) is an essential ingredient for backwards error control and subspace preservation (Stillfjord et al., 2023).

• Iterative Scaling for Input-Output Matching: In process control, Gramian-based interaction matrices $\Gamma$ are rescaled by (i) column-sum, (ii) row-sum, or (iii) Sinkhorn-Knopp (bi-stochastic) normalization. Sinkhorn-Knopp is unit-invariant (independent of original units/range) and yields statistically superior pairing performance in benchmark studies (Bengtsson et al., 2019).
• Monte Carlo Estimation in Nonlinear Settings: For nonlinear or noise-excited dynamics, the covariance of end-point distributions from noise-driven trajectories estimates the scaled Gramian, enabling data-driven extraction of controllable directions (Kashima, 2016).
• Layerwise Scaling in Deep Networks: In 1D CNNs, recursive scaling by prescribed gains ensures that the product of layerwise Gramians satisfies strict LMI-induced operator-norm (Lipschitz) constraints (Pauli et al., 2023).

5. Applications and Interpretive Frameworks

Scaled controllability Gramians underpin key methods and analysis in diverse areas:

• Controllability Centrality Scores: Scaled Gramians allow for dynamics-aware optimization of actuator placement and quantification of nodal influence, notably via the volumetric (VCS) and average-energy (AECS) controllability scores—these remain well defined for unstable systems and have unique, numerically stable minimizers on the infinite horizon (Umezu et al., 15 Jan 2026).
• Nonlinear Statistical Mechanics: The Gibbs-type scaled Gramian reduces nonlinear controllability quantification to covariance estimation for thermalized or noise-resolved systems, connecting control theory to statistical mechanics (Kashima, 2016).
• Estimation–Control Duality: The explicit relationship of Gramian scaling to Fisher information and entropy establishes a quantitative energetic–informational duality; improved controllability (lower energy) corresponds to increased uncertainty (entropy), while estimation accuracy is proportional to Gramian inverse (Silva, 8 Jul 2025).
• Process Systems Engineering: Input-output pairing and structure selection in MIMO systems require robust scaling of Gramian-based indices to avoid errors due to range/unit disparity, addressed systematically via Sinkhorn-Knopp (Bengtsson et al., 2019).

6. Sufficient Conditions, Limitations, and Open Problems

Scaled Gramians play a decisive role in modern sufficient controllability conditions:

• In general nonlinear or control-affine systems, invertibility of appropriately scaled (e.g., frozen-reference) Gramians yields constructive control laws and sharp $L^2$-energy bounds (Tamekue et al., 7 Mar 2025).
• For nonuniform growth or time-varying systems, explicit time-dependent exponential scalings in the Gramian inequalities guarantee stabilization and controllability, providing intermediate notions between classical complete and uniform complete controllability (Huerta et al., 2024).
• In infinite-dimensional or large-network contexts, maintaining bounded and well-conditioned scaled Gramians prevents controllability degradation as system size increases; pathological topologies (e.g., stars) cannot be stabilized by scaling alone without unfavorably increasing leader fractions (Enyioha et al., 2014).

Nonetheless, scaling cannot recover controllability when essential structural rank loss or unreachable subspaces are present. Choice of scaling may also interact subtly with physical interpretability, and over-scaling can increase estimation uncertainty as quantified by entropy metrics (Silva, 8 Jul 2025). In marginally stable or highly-inhomogeneous systems, further research is ongoing to fully characterize optimal scaling strategies and their effect on robust control synthesis.

7. Illustrative Examples

A selection of scenarios highlights the qualitative behavior of the scaled controllability Gramian:

• Nonlinear Oscillator Networks: In the FitzHugh–Nagumo oscillator network, the eigenvectors of the noise-driven scaled Gramian distinguish synchronization modes under varying noise scales, revealing controllable directions (Kashima, 2016).
• Large-Scale Networks: In directed chain/ring networks, scaling by maximum degree ensures that Gramian condition numbers remain bounded as the network size increases; in contrast, stars diverge under any scaling if the center is not actuated (Enyioha et al., 2014).
• Damped Oscillators: For a two-dimensional damped oscillator, scaling the Gramian by $\alpha$ reduces the minimum energy by $1/\alpha$ and increases the entropy by $\log \alpha$ (or $k_B \log \alpha$ in thermodynamic units), highlighting the energy–information tradeoff (Silva, 8 Jul 2025).
• Multitime and PDE Control: The $\gamma$-Gramian in PDEs encodes controllability along paths in multitime, and path-dependent scaling enables fine-grained reachability analysis absent in classical ODEs (Ghiu et al., 2011).

These examples underscore the flexibility and power of scaling in extending classical Gramian-based methodology to contemporary control, inference, and machine learning domains.