Empirical Observability Gramian

• Empirical observability Gramian is a data-driven measure that quantifies how small state perturbations affect output trajectories in nonlinear, time-varying, and stochastic systems.
• It is computed by simulating perturbed initial conditions and using finite-difference approximations to integrate output differences over time, applicable to ODEs and PDEs.
• Spectral metrics such as eigenvalues and condition numbers derived from the Gramian guide practical applications like sensor placement, model reduction, and system analysis.

The empirical observability Gramian is a simulation-based, data-driven generalization of the classical observability Gramian for nonlinear, parametric, time-varying, and even stochastic systems. It quantifies the local sensitivity of system output trajectories to small perturbations in initial state and is central to observability analysis, sensor placement, and model reduction in nonlinear dynamical systems. Empirical Gramians unify theory and computation for both finite-dimensional systems and distributed parameter (PDE) models, with efficient algorithms and well-understood interpretative properties.

1. Mathematical Definition and Construction

The empirical observability Gramian, denoted $W_o^\varepsilon$, encodes the effect of infinitesimal state perturbations on output trajectories. For a general nonlinear system

$\dot{x} = f(x, u),\qquad y = h(x),$

and a nominal initial state $x_0$, outputs are simulated for initial states $x_0^{+i} = x_0 + \varepsilon e_i$ and $x_0^{-i} = x_0 - \varepsilon e_i$ along each standard basis $e_i$ in $\mathbb{R}^n$, with $\varepsilon > 0$ a small perturbation. The empirical Gramian over $[0, T]$ is computed as

$$W_o^\varepsilon(x_0, u; T) = \frac{1}{4\varepsilon^2} \int_{0}^{T} \Phi^\varepsilon(t)^\top \Phi^\varepsilon(t)\,dt,$$

where $\dot{x} = f(x, u),\qquad y = h(x),$0, with $\dot{x} = f(x, u),\qquad y = h(x),$1 the outputs under the perturbed initial states. In the limit $\dot{x} = f(x, u),\qquad y = h(x),$2, $\dot{x} = f(x, u),\qquad y = h(x),$3 reduces to the linear observability Gramian if $\dot{x} = f(x, u),\qquad y = h(x),$4 and $\dot{x} = f(x, u),\qquad y = h(x),$5 are linear (Boyacıoğlu et al., 2022, Qi et al., 2014, Alaeddini et al., 2016, Kazma et al., 2024, Himpe, 2016, Powel et al., 2020).

For discrete-time systems, the sum replaces the integral. The construction generalizes straightforwardly to PDEs by perturbing low-dimensional subspaces of the initial condition, and to nonlinear or data-driven measurement operators (Kang, 2011, Kang et al., 2014, Brace et al., 3 Jan 2025).

2. Computational Algorithms and Practical Choices

Algorithmic computation involves the following steps:

1. Select a nominal $\dot{x} = f(x, u),\qquad y = h(x),$6, perturbation magnitude $\dot{x} = f(x, u),\qquad y = h(x),$7, and simulation horizon $\dot{x} = f(x, u),\qquad y = h(x),$8.
2. For each state direction $\dot{x} = f(x, u),\qquad y = h(x),$9, simulate outputs from $x_0$0.
3. At each time $x_0$1, compute output differences and form $x_0$2.
4. Approximate the integral by a quadrature rule (e.g., Riemann, trapezoidal).
5. Accumulate $x_0$3.

Essential parameter considerations:

• $x_0$4 must be small enough for local linearity, but robust to numerical noise (empirically $x_0$5–$x_0$6 times typical state magnitude).
• $x_0$7 must capture all relevant system dynamics.
• Sampling interval $x_0$8 must resolve system time scales (Boyacıoğlu et al., 2022, Himpe, 2016, Brace et al., 3 Jan 2025, Qi et al., 2014).

For high-dimensional systems (e.g., PDEs, large-scale networks), the method generalizes by restricting perturbations to a subspace of the initial state, often corresponding to modal decomposition or a physically meaningful basis (Brace et al., 3 Jan 2025, Kang, 2011, Kang et al., 2014).

Averaging over multiple perturbation scales can enhance robustness in the presence of nonlinearities (Himpe, 2016). Centering options (arithmetic mean, steady-state, etc.) are available for conditioning and interpretability.

3. Interpretation: Spectral Measures and Observability Indices

$x_0$9 is a symmetric positive semidefinite matrix. Its eigenvalues $x_0^{+i} = x_0 + \varepsilon e_i$0 and corresponding eigenvectors $x_0^{+i} = x_0 + \varepsilon e_i$1 illuminate observability structure:

• Large $x_0^{+i} = x_0 + \varepsilon e_i$2: Perturbations along $x_0^{+i} = x_0 + \varepsilon e_i$3 cause strong output variations—those state components are highly observable.
• Small $x_0^{+i} = x_0 + \varepsilon e_i$4: Weak sensitivity—corresponding directions are nearly unobservable.

Key scalarizations used in sensor placement and system analysis include:

• Minimum eigenvalue $x_0^{+i} = x_0 + \varepsilon e_i$5: Worst-case or weakest observable direction.
• Condition number $x_0^{+i} = x_0 + \varepsilon e_i$6: Observability "balance" or numerical stability.
• Determinant $x_0^{+i} = x_0 + \varepsilon e_i$7 or $x_0^{+i} = x_0 + \varepsilon e_i$8: Overall "volume" of the output-perturbation ellipsoid, global observability measure.
• Trace $x_0^{+i} = x_0 + \varepsilon e_i$9: Sum of observable variances, but insensitive to unobservable modes (Boyacıoğlu et al., 2022, Qi et al., 2014, Qi et al., 2014, Brace et al., 2021).

Interpretation underpins sensor selection, control design (via feedback optimization), and model reduction. For example, maximizing the minimum eigenvalue ensures all state directions are sufficiently sensed, whereas maximizing determinant improves overall system identifiability (Qi et al., 2014, Kazma et al., 2024, Brace et al., 3 Jan 2025).

4. Extensions: Stochastic, Delayed, and PDE Systems

Stochastic Systems

The empirical observability Gramian extends to stochastic systems with process noise: $x_0^{-i} = x_0 - \varepsilon e_i$0 Here, $x_0^{-i} = x_0 - \varepsilon e_i$1 becomes a random matrix due to process noise, and observability properties are characterized by the statistics (mean, variance) of eigenvalues computed via Monte Carlo runs. The "2n-simulation" methodology ensures positive semidefiniteness, and process noise can actually render previously unobservable modes observable (stochastic observability) (Powel et al., 2020, Boyacıoğlu et al., 2023).

Delay, Composite, and Nonlinear Output Operators

Nonlinear delayed outputs, as in neural encoding models, are treated by simulating the full composite measurement chain—no analytic derivative of the nonlinearity or delay is needed. The Gramian is still constructed by finite-difference output trajectories (Boyacıoğlu et al., 2022).

PDE/Continuum Systems

For infinite-dimensional systems (PDEs), the Gramian quantifies the observability of finite-dimensional subspaces (e.g., modal amplitudes or physical modes). Convergence and consistency results guarantee that discretized/empirical Gramians recover the true unobservability index in the limit of fine discretization (Kang, 2011, Kang et al., 2014, Brace et al., 3 Jan 2025).

5. Sensor Placement and Optimal Configuration

Sensor/actuator placement is a canonical application. Given candidate outputs (sensors), individual empirical Gramians $x_0^{-i} = x_0 - \varepsilon e_i$2 are precomputed. The sensor placement problem is formulated as

$x_0^{-i} = x_0 - \varepsilon e_i$3

where $x_0^{-i} = x_0 - \varepsilon e_i$4 is the desired number of sensors. The objective is typically to minimize an unobservability metric such as $x_0^{-i} = x_0 - \varepsilon e_i$5 or maximize $x_0^{-i} = x_0 - \varepsilon e_i$6 or $x_0^{-i} = x_0 - \varepsilon e_i$7. Convex relaxations (e.g., $x_0^{-i} = x_0 - \varepsilon e_i$8) and combinatorial solvers (e.g., NOMAD/MADS, greedy submodular maximization) are employed for tractability (Qi et al., 2014, Qi et al., 2014, Kazma et al., 2024, Brace et al., 3 Jan 2025, Boyacıoğlu et al., 2023).

Sensor selection criteria adapt to regime: when observability is low (few sensors), prioritizing $x_0^{-i} = x_0 - \varepsilon e_i$9 or $e_i$0 avoids nearly-unobservable configurations. In large systems, maximizing determinant improves global performance. These approaches are robust to system perturbations and parameter changes, as demonstrated empirically on power networks and bioinspired sensing models (Qi et al., 2014, Boyacıoğlu et al., 2022, Brace et al., 3 Jan 2025, 2411.7016).

Table: Scalar Observability Measures (abbreviated) | Metric | Interpretation | Limitations/Use Context | |------------------|-------------------------------------|-------------------------------------| | $e_i$1 | Volume, overall observability | Insensitive to weakest direction | | $e_i$2 | Worst-case, guarantees no blind spot| Neglects high-energy directions | | $e_i$3 | Balance, numerical stability | May prefer flat but small ellipsoids| | $e_i$4 | Sum of output energy | Ignores completely unobservable directions |

6. Connections to Variational Analysis and Model Reduction

The empirical Gramian is closely related to the variational (linearized) observability Gramian, and formal equivalence can be shown for small perturbations. Taylor expansion of nonlinear outputs around $e_i$5 yields that the finite-difference-based empirical Gramian approximates the variational Gramian to $e_i$6. This connection extends to Lyapunov exponents, which can be used to jointly characterize observability and system stability. Scalar measures such as log-determinant of the Gramian have been shown to relate to sums of Lyapunov exponents (Kazma et al., 2024, Kawano et al., 2019).

Empirical Gramians underpin practical model reduction via (empirical) balanced truncation for nonlinear and parametric systems, especially where explicit linearization is infeasible or unavailable (Himpe, 2016, Kawano et al., 2019). The balanced empirical Gramian identifies and retains the most observable states, supporting projection-based model order reduction.

7. Limitations, Consistency, and Theoretical Guarantees

Practical and theoretical limitations include:

• Locality: $e_i$7 reflects observability only in the neighborhood of $e_i$8; global observability cannot be inferred.
• Computational scaling: For high-dimensional or PDE systems, cost is $e_i$9 (with $\mathbb{R}^n$0 the number of time steps), but parallel and GPU-based algorithms can mitigate this.
• Choice of perturbation $\mathbb{R}^n$1: Must balance linearization validity against numerical noise.
• Unobservable subspaces: If $\mathbb{R}^n$2 is singular (zero minimum eigenvalue), the system is locally unobservable in some directions.
• PDE systems: Finite-dimensional empirical Gramians converge to the true unobservability index under well-posed discretizations, with explicit convergence theorems (Kang, 2011, Kang et al., 2014).

Empirical Gramians are consistent estimators of true linearized observability properties under standard regularity and discretization assumptions, both for ODE and PDE settings. Theoretical results guarantee convergence and robustness to discretization choices, ensuring reliability of the measure in both applied and theoretical contexts (Kang, 2011, Kang et al., 2014, Himpe, 2016, Brace et al., 3 Jan 2025).

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References:

• (Boyacıoğlu et al., 2022) Neural-inspired Measurement Observability
• (Qi et al., 2014) Adaptive Optimal PMU Placement Based on Empirical Observability Gramian
• (Kazma et al., 2024) Observability for Nonlinear Systems: Connecting Variational Dynamics, Lyapunov Exponents, and Empirical Gramians
• (Himpe, 2016) emgr - The Empirical Gramian Framework
• (Brace et al., 3 Jan 2025) Sensor Placement on a Cantilever Beam Using Observability Gramians
• (Kang, 2011) The Consistency of Partial Observability for PDEs
• (Qi et al., 2014) Optimal PMU Placement for Power System Dynamic State Estimation by Using Empirical Observability Gramian
• (Boyacıoğlu et al., 2023) Sensor Placement for Flapping Wing Model Using Stochastic Observability Gramians
• (Powel et al., 2020) Empirical Observability Gramian for Stochastic Observability of Nonlinear Systems
• (Kawano et al., 2019) Empirical Differential Gramians for Nonlinear Model Reduction