Generalized Reachability & Observability Gramians

Updated 9 August 2025

Generalized reachability and observability Gramians are operator-theoretic tools that quantify energy transfer and capture how inputs affect state dynamics across diverse system classes.
They extend classical Gramian frameworks to nonuniform, bilinear, stochastic, switched, and nonlinear systems, enabling robust model reduction and optimal sensor/actuator placement.
Their formulation using Lyapunov equations, Koopman analysis, and empirical methods offers a unified approach for system analysis, network design, and error quantification.

Generalized reachability and observability Gramians are algebraic and operator-theoretic constructs that extend the classical Gramian formalism from linear deterministic systems to a broad spectrum of system classes, including nonuniformly sampled discrete systems, bilinear and quadratic-bilinear systems, systems with stochastic excitation, switched and descriptor systems, continuum (PDE) systems, and nonlinear systems leveraging data-driven or operator-theoretic perspectives such as Koopman analysis. These objects quantify input-to-state and state-to-output energy transfer, encode system directions that are difficult to reach or observe, and underpin model reduction, system analysis, sensor/actuator placement, and error quantification even in challenging non-classical settings.

1. Extending Gramian Concepts Beyond Linear Time-Invariant Systems

The classical reachability (controllability) Gramian $P$ and observability Gramian $Q$ for an LTI system are defined via Lyapunov equations, capturing how state space directions respond to input excitation and how they manifest in output observation. Modern developments generalize these notions in several ways:

Sampling Generalization: In nonuniformly sampled discrete systems, reachability and observability are characterized by the fundamental solution matrix evaluated at general sampling sequences, replacing uniform-sampling-based criteria with determinant conditions involving characteristic modes and the actual sampling intervals (Fúster-Sabater, 2010). This enables flexible digital control design robust to delays and asynchrony.
Robustification and Generalization to Networks: Pseudo Gramians for semistable or weakly connected systems subtract nondecaying components, yielding integrals that converge even for systems without full asymptotic stability (Cheng et al., 2019). These pseudo Gramians underpin network aggregation (clustering) and Petrov-Galerkin-based reduction in large-scale networked dynamical systems.
Bilinear and Nonlinear Systems: For bilinear systems, the reachability Gramian $\mathcal{W}$ is used to bound input energies needed to reach target states, with metrics such as the minimum eigenvalue, trace, and determinant capturing the worst-case, average, and volumetric controllability, respectively (Zhao et al., 2015). For quadratic-bilinear and more general nonlinear systems, algebraic Gramians are defined via generalized quadratic Lyapunov or Hamilton–Jacobi equations or via contraction mappings in Volterra or observable subspaces, with further extensions to empirical and variational gramians derived from first-order trajectory sensitivity or tangent linearization (Benner et al., 2017, Kawano et al., 2019, Kazma et al., 22 Feb 2024).
Stochastic and Switched Systems: For systems with stochastic noise or switching, the Gramians satisfy generalized Lyapunov inequalities or equations which couple Lyapunov operators with stochastic or mode-dependent terms (Benner et al., 2015, Duff et al., 2018, Manucci et al., 29 Jul 2024). In stochastic settings, reachability and observability Gramians coincide with Fisher information matrices bounding state estimation error via the Cramér–Rao inequality (Boyacıoğlu et al., 25 Oct 2024).
Operator-Theoretic Lifting and Decomposition: The Koopman operator permits the definition and balancing of Gramians in the space of observables for nonlinear dynamics, where linear methods can then be used for analysis and model reduction (Yeung et al., 2017, Brown et al., 5 Jul 2025).

2. Algebraic and Analytical Formulations

The formulations of generalized Gramians depend on the system class:

Nonuniformly Sampled Discrete Systems: Given a minimal realization $(A, b, c)$ , system is $n$ -reachable and $n$ -observable if for $n$ successive sampling instants $t_i$ , the determinant $\det[\varphi_i(\alpha_m)] \neq 0$ for all $i=1,\ldots,n$ and $m=0,\ldots,n-1$ , where $\varphi_i(\cdot)$ are system modes and $\alpha_m = t_{n-1} - t_{n-m-1}$ (Fúster-Sabater, 2010).
Quadratic-Bilinear Systems: The reachability Gramian $P$ satisfies a generalized Lyapunov equation:

$A P + P A + H(P \otimes P) H^\top + \sum_{k=1}^m N_k P N_k^\top + BB^\top = 0$

with similar structure for observability via the adjoint and output matrices (Benner et al., 2017).

Nonlinear/Stochastic Systems: For fully nonlinear stochastic systems, Gramians are defined via operator inequalities:

$L V(X^{-1})(x, y; \delta) \leq -S_{X^{-1}}(x, y)^\top U_{X^{-1}}^{-1} S_{X^{-1}}(x, y)$

for a candidate Lyapunov function $V_X(x) = x^\top X x$ , where $L$ is a second-order operator incorporating differences of drift, diffusion, and noise coefficients (Redmann, 4 Aug 2025). Sufficient conditions for the existence of such Gramians are framed in terms of incremental stability (one-sided Lipschitz and linear growth) and mean square (or exponential) stability criteria.

Balanced Truncation in Stochastic and Switched Systems: Generalized Lyapunov equations for mode-dependent or stochastic systems take forms:

$A P + P A^\top + \sum_{j=1}^M [D_j P D_j^\top + B_j B_j^\top] = 0$

and

$A^\top Q + Q A + \sum_{j=1}^M [D_j^\top Q D_j + C_j^\top C_j] = 0$

which couple nominal (reference) dynamics with deviations induced by stochasticity or switching (Duff et al., 2018, Manucci et al., 29 Jul 2024).

Empirical and Variational Gramians: For data-driven and high-dimensional systems, empirical Gramians are constructed using central differences of output trajectories under small perturbations, formally:

$W_o^\epsilon(t) = \frac{1}{4\epsilon^2} \int_0^t \Delta Y(\tau)^\top \Delta Y(\tau) d\tau$

with $\Delta Y(\tau)$ collecting output differences across perturbed initial conditions (Andrews et al., 3 Jan 2025, Kawano et al., 2019). The variational Gramian computes the same sensitivity using tangent linear propagation of state/output Jacobians, yielding computational efficiency and a direct link to Lyapunov exponents (Kazma et al., 22 Feb 2024).

3. Applications: Model Reduction, Sensor/Actuator Placement, and Network Design

Generalized reachability and observability Gramians provide the algebraic backbone for a spectrum of applications:

Balanced Truncation and Model Reduction: Projection-based reduction schemes, especially balanced truncation, generalize to nonlinear, stochastic, quadratic-bilinear, and switched systems, using generalized Gramians to identify low-energy (difficult to reach and/or observe) modes to truncate. Typical algorithms compute a transformation that diagonalizes both Gramians, with diagonal entries (editor's term: "Hankel singular values") quantifying the importance of each state direction (Benner et al., 2017, Redmann, 4 Aug 2025, Duff et al., 2018). For stochastic systems, error bounds for the reduced-order model are obtained in the $\mathcal{H}_\infty$ or $\mathcal{L}_2$ norm, often in terms of the sum of neglected singular values (Benner et al., 2015, Redmann, 4 Aug 2025).
Sensor and Actuator Placement: Empirical and energy-based Gramians inform optimal sensor selection by prioritizing configurations that maximize observability metrics (e.g., minimum eigenvalue or log-determinant of the observability Gramian) (Andrews et al., 3 Jan 2025, Brace et al., 3 Jan 2025). Submodularity properties enable scalable, near-optimal greedy algorithms for large networks (Kazma et al., 22 Feb 2024).
Network Clustering and Approximation: In directed or semistable networks, pseudo Gramians are used to define input/output effort-based dissimilarity measures for vertices. Graph clustering based on such measures preserves essential input-output properties in reduced-order models and guarantees approximation accuracy (Cheng et al., 2019).
Error and Information Theoretic Analysis: In stochastic settings, the (generalized) Gramian coincides with the Fisher information matrix (FIM) for state estimation problems—its smallest eigenvalue bounding the best achievable estimation uncertainty via the Cramér–Rao inequality. Duality between stochastic observability and constructability allows efficient and robust recursive computation of the FIM for long-horizon state estimation and smoother design (Boyacıoğlu et al., 25 Oct 2024).

4. Theoretical Properties and Generalization Criteria

Much of the recent literature characterizes when generalized Gramians exist and the stability or theoretical guarantees they provide:

Existence: Existence is often tied to Lyapunov-type (operator) inequalities, one-sided Lipschitz or linear growth conditions, or mean square exponential stability for stochastic systems (Redmann, 4 Aug 2025, Duff et al., 2018).
Containment of Reachable/Observable Sets: For switched systems, the image (range) of the Gramian computed from generalized Lyapunov equations encloses the reachable or observable set of the underlying DAE or ODE system, not merely the span of states from a particular input (Manucci et al., 29 Jul 2024).
Duality and Symmetries: Rigorous duality between reachability and observability Gramians persists across system classes; in stochastic discrete-time systems, observability and constructability Gramians are related by a duality transformation (time-reversal, parameter mapping), enabling transfer of analysis and computation tools between the two (Boyacıoğlu et al., 25 Oct 2024).
Supermodularity and Greedy Construction: The mapping from actuator/sensor sets to Gramian-based metrics (e.g., minimum eigenvalue) exhibits an "increasing returns" property (supermodularity) in several networked and bilinear settings, which guarantees the efficacy of greedy optimization for combinatorial sensor/actuator placement (Zhao et al., 2015, Kazma et al., 22 Feb 2024).

5. Computational Methods and Numerical Considerations

Generalized Gramians often require nonstandard, computationally efficient algorithms:

Recursive and Block Algorithms: Dual formulation enables the use of efficient recursive computation for stochastic observability Gramians, with a sequence of $n\times n$ matrix recursions rather than large block matrix inversions, greatly improving memory and numerical stability (Boyacıoğlu et al., 25 Oct 2024).
Frequency-Domain/Lifted Approaches: For time-periodic systems, Gramians may be constructed in the frequency domain (editor's term: "frequential Gramians") via harmonic resolvent operators and Fourier expansions. This approach permits efficient computation via parallelizable algebraic solves and extends applicability to unstable or slow-decaying systems (Padovan et al., 2022).
Data-Driven and Empirical Methods: In high-dimensional, nonlinear, or black-box settings, empirical Gramians are built from finite-difference or Monte Carlo-based output perturbation experiments (Andrews et al., 3 Jan 2025, Kawano et al., 2019, Powel et al., 2020). For systems with noise, expected values and covariance structures are computed to reveal the impact of process noise in "exciting" otherwise unobservable directions (Powel et al., 2020).

6. Impact on System Design, Estimation, and Future Research

Generalized Gramians fundamentally alter the analysis and design space for modern dynamical systems:

Sensor/Actuator Design: Observability-driven sensor placement for nonlinear spacecraft, continuum systems (beams), and chemical networks leads to substantiated performance gains in estimation, as measured by reduced error covariances and improved condition numbers (Andrews et al., 3 Jan 2025, Brace et al., 3 Jan 2025, Kazma et al., 22 Feb 2024).
System Identification and Validation: Subspace identification and state-space model consistency checks leverage criteria relating reachability and observability matrices to feasible system realizations (Ferrante et al., 2013).
Stochastic and Uncertain Systems: Incorporating process and measurement noise via generalized Gramians impacts the assessment of system observability and constructability, with empirical and Fisher information-based Gramians directly quantifying estimation limits under stochastic dynamics (Boyacıoğlu et al., 25 Oct 2024, Powel et al., 2020).
Nonlinear and Non-affine Extensions: Operator-theoretic and SVD-like decompositions, norm conditions, and balancing/truncation procedures extend rigorous model reduction—with $\mathcal{H}_\infty$ error bounds—to classes of nonlinear, non-affine control systems, providing explicit tractable criteria for ensuring preservation of dynamic features post-reduction (Brown et al., 5 Jul 2025, Redmann, 4 Aug 2025).
Open Problems and Future Directions: Ongoing research focuses on scaling these frameworks to high-dimensional and distributed systems, synthesizing data-driven Gramian construction with machine learning for operator approximation, and extending Lyapunov-type analysis to non-Euclidean state spaces and nonclassical noise structures, while further quantifying the role of stochasticity in enabling and limiting reachability and observability.

This comprehensive synthesis delineates the mathematical formulations, properties, computational methods, and system-theoretic implications of generalized reachability and observability Gramians across modern system classes, aligning classical energy-based analysis with contemporary requirements in system modeling, reduction, and estimation in complex, noisy, and high-dimensional settings.