Spectral Decomposition of Discrete-Time Controllability Gramian and Its Inverse via System Eigenvalues

Abstract: This paper develops a closed-form spectral decomposition framework for the Gramian matrices of discrete-time linear dynamical systems. The main results provide explicit decompositions of the discrete-time controllability Gramian and its inverse in terms of the eigenvalues of the dynamics matrix, yielding a mode-resolved representation of these matrices. In contrast to the more common use of aggregate Gramian characteristics, such as eigenvalues, singular values, determinants, and trace-based metrics, the proposed approach describes the internal structure of the Gramian itself through contributions associated with individual modes and their pairwise combinations. The framework is extended further to the solution of the discrete-time Lyapunov difference equation, placing the obtained formulas in a broader context relevant to the analysis and computation of time-varying and nonlinear systems. In addition, the decomposition is generalized to systems whose dynamics matrix has multiple eigenvalues, enabling a closed-form estimation of the effects of resonant interactions between eigenmodes. The proposed results provide a structural tool for the analysis of controllability, observability and stability in discrete-time systems and complement existing Gramian-based methods used in model reduction, estimation, actuator and sensor selection, and energy-aware control. Beyond their theoretical interest, the derived decompositions may support the development of improved computational procedures and more informative performance criteria for a range of discrete-time control problems.

• The paper introduces closed-form spectral decompositions that express the controllability Gramian and its inverse in terms of individual eigenmodes and pairwise modal interactions.
• It leverages residue-based and derivative formulas to handle repeated and defective eigenvalues, applying the methodology to both infinite- and finite-horizon cases.
• The approach enhances practical control design by enabling precise actuator/sensor placement, model reduction, and energy-aware control through mode-resolved insights.

Spectral Decomposition of Discrete-Time Controllability Gramian and Its Inverse via System Eigenvalues

Overview

The paper "Spectral Decomposition of Discrete-Time Controllability Gramian and Its Inverse via System Eigenvalues" (2604.01149) presents a systematic framework for deriving closed-form, mode-resolved spectral decompositions of the controllability Gramian and its inverse for discrete-time linear time-invariant (LTI) systems. Unlike conventional analysis that reduces Gramian matrices to aggregated spectral quantities—such as eigenvalues, traces, and determinants—the approach detailed in this work provides explicit contributions of individual modes and pairwise modal interactions, established directly with respect to the system’s eigenvalues. The methodology is generalized to encompass initial conditions, solutions to the discrete-time Lyapunov equation (both algebraic and difference versions), and systems with nontrivial Jordan block structure, facilitating direct analysis of resonance and defective spectrums.

Problem Formulation and Novelty

The paper considers the general discrete-time LTI system $x(t + 1) = Ax(t) + Bu(t)$, with a particular focus on the spectral resolution of the solution $P$ of the discrete Lyapunov (Stein) equation $APA^T - P = -BB^T$. Unlike prior literature that typically relates only scalar summaries of $P$ to system spectral data, this work provides a direct Gramian decomposition into Hermitian components tied to the eigenstructure of $A$. Additionally, the inverse Gramian—which underpins minimum control energy analysis and is central to estimation and actuator selection—is included in the decomposition, a direction not previously explored in discrete time according to the literature review.

The development provides:

1. Closed-form spectral decompositions for both $P$ and $P^{-1}$, factoring the Gramian into modal and pairwise contributions in both the infinite-horizon and finite-horizon (difference equation) cases.
2. Generalizations for systems with arbitrary (including repeated) eigenvalues, producing residue-based and derivative-based formulas.
3. Companion form specialization: For controllability (and by duality, observability) canonical forms, the decompositions achieve particular simplicity.

Main Theoretical Results

Spectral Decomposition Formulas

• For diagonalizable $A$, the Gramian admits a decomposition $P = \sum_{i} \tilde{P}_i = \sum_{i,j} P_{ij}$, where each term is associated with a mode or a mode pair. Explicit residue formulas employing the system’s right and left eigenvectors and eigenvalues are provided.
• For the finite-horizon problem and arbitrary initial conditions, time-evolving modal contributions are derived, using geometric progression factors based on eigenvalues.
• For systems with multiple or defective eigenvalues, the decomposition is extended using derivative operations on resolvents (Jordan block treatment), with all terms computed from generalized eigenvectors.

Inverse Gramian Decomposition

• For the first time in discrete time, the same spectral viewpoint is extended to $P^{-1}$, yielding expressions in terms of left eigenvectors and functions of the spectrum.
• Orthogonality (biorthogonality) relations between the modal parts of $P$0 and $P$1 are rigorously established and employed in constructing the expansions.

Canonical Form Results

• In controllability canonical (companion) form, the sub-Gramians and their inverses become structured (e.g., Toeplitz or Hankel), and all dependence on input/output structure is removed. This is significant for model reduction and observer/controller synthesis, where canonical forms are standard.
• The explicit dependence on the coefficients and derivatives of the characteristic polynomial of $P$2 is spelled out.

Generalization for Multi-Input Systems

• For general $P$3-input systems, the modal decompositions are constructed via transformation of canonical form results, leveraging controllability matrix and Kronecker product manipulations.

Numerical Examples and Structure Properties

Illustrative examples are provided for both stable and unstable systems, and for systems with simple and multiple eigenvalues. These confirm the correctness and computational realizability of the derived expansions. Notable technical properties—such as the symmetry, Toeplitzness, or positivity of certain modal components under stability assumptions—are precisely characterized.

Implications and Applications

Practical Applications

• Actuator and Sensor Selection: Modal decompositions of the Gramian and its inverse enable precise, mode-targeted metrics for optimal selection and placement, supporting more nuanced design compared to aggregate performance measures.
• Model Reduction: Structured representation in canonical form is directly amenable to balanced truncation and error analysis, aiding efficient model order reduction.
• Energy-Aware Control: Minimum control energy analysis can now attribute energetic bottlenecks to specific subspaces or mode interactions, with potential benefit in large-scale networked systems.
• Computational Techniques: The explicit formulas facilitate structure-exploiting computation for large-scale system analysis, e.g., by targeting dominant modes or resonances.
• Time-Varying and Finite-Horizon Analysis: The finite-horizon generalizations support transient analysis and robustness considerations in non-steady-state regimes.

Theoretical Implications

• This work moves the focus in discrete-time system analysis from scalar Gramian metrics toward fine-grained, interpretable structural summaries.
• The explicit handling of repeated or nearly resonant eigenvalues elucidates the effects of modal resonance on controllability and estimation, which are essential in the analysis of complex or high-dimensional systems with clustered spectra.
• The dual applicability to observability Gramian (by transposition and duality) means results extend directly to minimum variance estimation and sensor design.

Future Directions

The analytical decompositions open several avenues:

• Development of large-scale, structure-preserving algorithms for control and estimation using modal information rather than bulk matrix decompositions.
• Extension to stochastic, networked, or time-varying systems, particularly by integrating with domains where Gramian-based approaches already play a foundational role.
• Further exploration of Gramian decomposition in the context of data-driven system identification, where inferred modal structure can enhance the interpretability and control of learned models.

Conclusion

By providing closed-form, mode-resolved spectral decompositions for both the discrete-time controllability Gramian and its inverse, this work (2604.01149) establishes a rigorous framework for granular analysis of controllability, observability, and stability properties. The approach generalizes classical scalar metrics and offers powerful new tools for both theoretical system analysis and the design of efficient, scalable computational algorithms rooted in the fundamental structure of linear dynamical systems.