Discrete-Time Controllability Gramian Overview

• Discrete-Time Controllability Gramian is an algebraic construct that quantifies the energy required to steer a system's state across its entire space.
• It is computed via infinite series and Lyapunov equations, with practical methods like truncation and Krylov subspace techniques to manage large-scale systems.
• Extensions to stochastic, delayed, and nonlinear systems use generalized formulations—including Koopman operator approaches—to enhance controllability assessments and control design.

The discrete-time controllability Gramian is a central algebraic construct that quantifies the energy required to steer the state of a discrete-time linear or nonlinear system across its state space using exogenous inputs. For time-invariant linear systems $x_{k+1} = A x_k + B u_k$, the Gramian encapsulates how system topology, input configuration, delay, and noise processes shape reachability, signal amplification, and network-level control phenomena. This article provides a comprehensive account, incorporating deterministic, stochastic, delayed, and nonlinear extensions.

1. Fundamental Definition and Algebraic Properties

Given the discrete-time linear time-invariant (LTI) system

$$x_{k+1} = A x_k + B u_k,\quad x_0\in\R^N,\;u_k\in\R^m,$$

with $A\in\R^{N\times N}$ and $B\in\R^{N\times m}$, the infinite-horizon controllability Gramian $W^c\in\R^{N\times N}$ is defined as

$W^c = \sum_{k=0}^\infty A^k B B^\top (A^\top)^k.$

This series converges, providing a unique positive-definite solution to the discrete Lyapunov equation

$A W^c A^\top - W^c + B B^\top = 0,$

if and only if all eigenvalues of $A$ satisfy $|\lambda| < 1$ ($\rho(A)<1$). In practical settings, $A$ is often rescaled to ensure stability. The finite-horizon Gramian is defined over $N$ steps via

$W_c(N) = \sum_{i=0}^{N-1} A^i B B^\top (A^\top)^i,$

and its invertibility is necessary and sufficient for reachability on $N$ steps (Nazerian et al., 22 Nov 2025, Diallo et al., 2015, Liu et al., 2024).

2. Gramian and H₂-Norm: Energy Gain and Input-Output Quantification

The transfer function associated to the output equation $y_k = C x_k$,

$H(z) = C (z I - A)^{-1} B,$

admits a squared $\mathsf{H}_2$-norm

$$\|H\|_2^2 = \sum_{k=0}^\infty \|C A^k B\|_F^2 = \operatorname{Tr}[C W^c C^\top].$$

Thus, $W^c$ encodes the energy gain from impulse- or white-noise-type inputs to selected outputs. The eigenvalues $\lambda_i$ of $W^c$ quantify the excitation energy per system mode, and the trace $\operatorname{Tr} W^c$ summarizes the total reachability "energy" across all modes (Nazerian et al., 22 Nov 2025).

3. Modal Structure, Network Indices, and Approximations

The modal structure is encapsulated by the spectrum of $W^c$: large eigenvalues imply easily excitable modes, while smaller ones denote "hard-to-control" directions. The worst-case amplification over a horizon $q$ is governed by the largest eigenvalue of the finite-horizon Gramian $W_q$. In large-scale networks, computing $\operatorname{Tr} W^c$ directly is impractical; first-order contributions $d_j = \sum_{i=1}^N A_{ij}^2$ for each input node allow creation of a normalized network index

$$\alpha = \frac{ \sum_{k=1}^M [d_{i_k} - d_k^{(\min)}] }{ \sum_{k=1}^M [d_{k+N-M} - d_k^{(\min)}] },$$

which scales in $O(N\log N+M)$ time. $\alpha \approx 1$ denotes optimal selection of influential nodes, while $\alpha \approx 0$ corresponds to minimal influence (Nazerian et al., 22 Nov 2025).

Analytic approximations for $\operatorname{Tr}[C W^c C^\top]$ are available in single-input, single-output, and path-dominated cases:

$$\widehat{W}_\text{out} \approx \frac{ (C A^d B)^2 }{ 1-\rho^2 },$$

where $d$ is the input-output path length and $\rho = \rho(A)$. Errors are typically below a few percent for $d\geq2$ and sufficiently dense or large systems.

4. Stochastic Systems, Delay, and Generalizations

For stochastic systems (additive or multiplicative noise), the Gramian concept generalizes but retains its core function as a reachability/covariance certificate:

• Additive Noise: In discrete-time systems $x_{k+1} = A x_k + B u_k + w_k$ ($w_k$ i.i.d.), the reachable state covariance set at time $N$ is entirely determined by the finite-horizon Gramian $G(N,0)$:

$$G(N,0) = \sum_{j=0}^{N-1} A^{N-1-j} B B^\top (A^{N-1-j})^\top.$$

Complete reachability occurs if $G(N,0)$ is positive-definite (Liu et al., 2024).

• Multiplicative Noise: For dynamics such as $$x(k+1) = [A x(k) + B u(k)] + w(k) [\bar A x(k) + \bar B u(k)]$$, the corresponding Gramian $W_N$ is generated via recursion:

$$W_{k+1} = C W_k C^\top + \bar C W_k \bar C^\top + D D^\top,$$

for constant matrices $C, \bar C, D$ derived from $A, \bar A, B, \bar B$. Exact controllability is equivalent to invertibility of $W_N$ (Xu et al., 2023, Diallo et al., 2015).

• Delay Systems: For systems with delay, $y(r+1) = A y(r) + B y(r-p) + C u(r)$, the discrete Gramian $W$ is constructed through delayed state-transition matrices $Y_p^{A,B}(r)$ involving combinatorial sums over delay-induced paths. The rank of the set $\{\mathcal Q(r,i) C\}$, with $\mathcal Q$ recursively defined, provides a necessary and sufficient condition for controllability via Gramian non-singularity (Asadzade et al., 18 Aug 2025).

5. Nonlinear Extensions: Koopman Operator Formalism

For nonlinear discrete-time systems $x_{k+1} = f(x_k, u_k)$, the Koopman operator-based lifting constructs an infinite-dimensional linearized system

$$\psi_x(x_{t+1}) = K_x \psi_x(x_t) + K_u \psi_u(u_t),$$

where observables $\psi_x$ and $\psi_u$ generate a lifted controllability Gramian

$$W_c = \sum_{j=0}^\infty K_x^j K_u K_u^\top (K_x^j)^\top.$$

Positivity and full-rank of $W_c$ (or its appropriate projection) characterize local or global controllability in the lifted space. Computationally, extended DMDc algorithms, observable dictionaries, and spectral truncation yield finite-dimensional Gramian approximations suitable for data-driven model reduction, balanced truncation, and empirical reachability assessments (Yeung et al., 2017).

6. Computational Strategies and Scalability

Direct computation of $W^c$ via Lyapunov equations requires $O(N^3)$ flops and thus is prohibitive for large $N$. Truncation at finite horizon $K$, Krylov subspace methods, exploitation of network sparsity, and analytic shortcuts are essential for large-scale applications. The network index $\alpha$ offers practical scalability to networks with $N\sim 10^6$. For stochastic and delay systems, the respective recursions and block-structured matrix forms enable Gramian assembly in $O(N n^3)$ time (Nazerian et al., 22 Nov 2025, Xu et al., 2023, Liu et al., 2024, Asadzade et al., 18 Aug 2025).

7. Illustrative Cases and Empirical Insights

• Synthetic Networks: Directed Erdős–Rényi and scale-free graphs exhibit systematic improvement in analytic Gramian approximations (errors $<2\%$ for $d\ge2$ and large $N$).
• Empirical Systems: Application to power grids, neural connectomes, gene-regulatory networks, and food webs reveals a near-linear empirical relation between $\operatorname{Tr} W^c$ and the network index $\alpha$. Networks exhibiting "blocking" behavior (low amplification) have low $\operatorname{Tr} W^c$ and $\alpha$ (e.g., IEEE118, cat connectome); those with sensory or regulatory roles exhibit "passing" character (high indices).
• Stochastic and Delay Examples: Gramian-based criteria are validated numerically, including explicit trajectory steering using minimum-energy controls in delay systems and the effect of noise structure on reachability in stochastic setups (Asadzade et al., 18 Aug 2025, Xu et al., 2023, Liu et al., 2024).

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The discrete-time controllability Gramian serves as a unifying mathematical tool for quantifying, analyzing, and designing controllability across a broad class of deterministic, stochastic, delayed, and nonlinear discrete-time systems. Its algebraic, spectral, and computational properties underpin assessment of input-output amplification, network selection, reachability, and model reduction across large-scale engineered and complex natural networks (Nazerian et al., 22 Nov 2025, Yeung et al., 2017, Diallo et al., 2015, Xu et al., 2023, Liu et al., 2024, Asadzade et al., 18 Aug 2025).