---
title: Discrete-Time Controllability Gramian Overview
url: https://www.emergentmind.com/topics/discrete-time-controllability-gramian
type: topic
---

# Discrete-Time Controllability Gramian Overview

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 [2511.18119], [1510.03157], [2406.14740].

## 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 [2511.18119].

## 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 [2511.18119].

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 [2406.14740].

- **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 $W_N$ [2312.17577], [1510.03157].

- **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 [2508.13336].

## 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 [1709.08712].

## 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 [2511.18119], [2312.17577], [2406.14740], [2508.13336].

## 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 [2508.13336], [2312.17577], [2406.14740].

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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 [2511.18119], [1709.08712], [1510.03157], [2312.17577], [2406.14740], [2508.13336].

Source: https://www.emergentmind.com/topics/discrete-time-controllability-gramian