Discrete-Time Controllability Gramian

• Discrete-Time Controllability Gramian is a matrix that aggregates the effect of input signals over time, defined by a convergent sum that solves a discrete-time Lyapunov equation.
• It determines system controllability by ensuring that a positive definite Gramian signifies full access to the state space, facilitating energy-based input-state analysis.
• The Gramian concept extends to stochastic, delay, and nonlinear systems, offering scalable computational methods for optimal control input placement and network design.

A discrete-time controllability Gramian is a fundamental matrix object that quantifies how input signals propagate through a discrete-time linear system (and various classes of nonlinear or stochastic systems) to influence the dynamics and outputs. The Gramian enables rigorous characterization of input–state and input–output energy, formalizes controllability criteria, and provides computational means to analyze, compare, and design networks and systems, including those with delays, stochasticity, or complex nonlinear dynamics.

1. Definition and Algebraic Characterizations

For a discrete-time linear time-invariant (LTI) system:

$x_{k+1} = A x_k + B u_k,\quad x_0 = 0,$

the infinite-horizon controllability Gramian $W_d$ is defined by the convergent sum:

$W_d = \sum_{k=0}^{\infty} A^{k} B B^T (A^T)^k,$

which measures input energy delivered to the state-space over all future time. $W_d$ equivalently solves the discrete-time Lyapunov equation:

$$A W_d A^T - W_d + B B^T = 0 \quad \Leftrightarrow \quad W_d = A W_d A^T + B B^T.$$

Convergence of $W_d$ requires $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$; practical computations often rescale $A$ to satisfy this condition (Nazerian et al., 22 Nov 2025).

Generalizations appear across system types:

• In delay systems (Asadzade et al., 18 Aug 2025), $W_d$ is a finite sum involving recursively defined delayed matrix exponentials.
• In linear systems with multiplicative noise (Xu et al., 2023, Diallo et al., 2015), the deterministic Gramian is replaced by random-coefficient Gramians or by a limiting process of backward stochastic Riccati schemes.
• In nonlinear systems, a Koopman-lifted Gramian can be defined in the space of observables (Yeung et al., 2017).

2. Role in Controllability and Network Signal Transmission

The Gramian encodes system controllability: $W_d \succ 0$ (positive definite) if and only if the system is controllable. For stochastic or delay systems, positivity/non-singularity is established for appropriately defined Gramians:

• In stochastic systems, invertibility of the expectation-defined Gramian is necessary and sufficient for exact controllability (Xu et al., 2023).
• For discrete-time delay systems, the constructed Gramian's non-singularity is equivalent to relative controllability (Asadzade et al., 18 Aug 2025).

In networked systems, $W_d$ quantifies the capacity of external inputs at specified nodes to propagate through the topology, distinguishing "passing" (amplifying) from "blocking" (attenuating) networks (Nazerian et al., 22 Nov 2025).

3. Computational Strategies and Approximations

Explicit computation of $W_d$ can be performed via direct solution of the Lyapunov equation, incurring $O(N^3)$ cost for size $N$ systems (Nazerian et al., 22 Nov 2025).

Approximation techniques include:

• Path-based geometric series: For input-output node pairs at distance $d$ in graph-based systems, $$W_d^{\mathrm{out}} \approx \frac{(C A^d B)^2}{1-\rho(A)^2}$$ yields accurate large-scale estimates for long paths or dense networks (Nazerian et al., 22 Nov 2025).
• Structural proxies: The efficient index $\alpha$ based on squared column sums of $A$ (per-node scores) correlates linearly with $\operatorname{Tr}(W_d)$ and $\lambda_{\max}(W_d)$ in networks with small $\rho(A)$, affording $O(\|A\|_0)$ scaling for rapid signal-passing estimates (Nazerian et al., 22 Nov 2025).
• In finite impulse-response (FIR) systems (e.g., state representations of 1D convolutional layers), $A$ is nilpotent and $W_d$ admits a finite exact sum (Pauli et al., 2023).

Optimization for control and design leverages $W_d$:

• Input selection for maximum or minimum transmission (e.g., maximizing $\operatorname{Tr}(W_d)$ or $\lambda_{\max}(W_d)$) can be formulated as mixed-integer linear or semidefinite programs (Nazerian et al., 22 Nov 2025).

4. Relationship to System Norms and Indices

The Gramian is directly related to the $\mathcal H_2$-norm of the system's transfer function:

$$\|G\|_2^2 = \sum_{k=0}^\infty \|C A^{k-1} B\|_F^2 = \operatorname{Tr}(C W_d C^T),$$

where $Y_k = C A^{k-1} B$ is the $k$-step impulse response and $\|\cdot\|_F$ is the Frobenius norm (Nazerian et al., 22 Nov 2025).

Principal scalar indices derived from $W_d$ include:

• $\operatorname{Tr}(W_d)$: total energy transferred from input to state/output, used to classify network "passing" or "blocking" behavior.
• $\lambda_{\max}(W_d)$: amplitude of maximal response direction.
• Proxy $\alpha$: input-structure-based index for rapid assessment of network signal transmission capacity, with empirically established linearity to Gramian-derived measures (Nazerian et al., 22 Nov 2025).

In FIR-based CNN architectures, embedding the Gramian within Lyapunov-related linear matrix inequalities ensures layerwise and end-to-end Lipschitz bounds, guaranteeing network robustness by design (Pauli et al., 2023).

5. Generalizations: Stochastic, Delay, and Nonlinear Systems

In stochastic and delay settings, the classical Gramian requires adaptation.

Stochastic linear systems: For $$x_{k+1} = (A x_k + B u_k) + w_k (\bar A x_k + \bar B u_k)$$, a Gramian $G_N$ is defined as a sum of expected-state backward products involving random matrices; invertibility enta ils both exact and null controllability (Xu et al., 2023). Parallel constructs appear via backward stochastic Riccati-difference schemes in Markov-trend and multiplicative noise models, where the limit $$\liminf_{\varepsilon\rightarrow0} P_0^{\varepsilon}$$ functions as a Gramian-equivalent controllability metric (Diallo et al., 2015).

Discrete-delay systems: In controlled difference equations with delays, the Gramian $G_d$ is constructed as a finite sum involving recursively generated "delayed matrix exponentials," and its rank conditions provide necessary and sufficient controllability criteria (Asadzade et al., 18 Aug 2025).

Nonlinear systems: Koopman operator methods embed the dynamics in a lifted space of observables, where a lifted Gramian $W_c^\psi$ is defined by:

$$W_c^\psi = \sum_{j=0}^{\infty} K_x^j K_u K_u^T (K_x^T)^j,$$

and is projected back to the original state variables via a projection matrix $P$. Non-singularity of $W_c^\psi$ (or its projection) relates to local controllability via the rank of the accessible Lie derivative directions (Yeung et al., 2017).

6. Empirical, Algorithmic, and Network-Theoretic Insights

Large-scale empirical studies in (Nazerian et al., 22 Nov 2025) reveal systematic differences across network families (e.g., power grids vs. brain connectomes) in $\operatorname{Tr}(W_d)$ and $\lambda_{\max}(W_d)$, correlating with their passing/blocking characteristics. This justifies the use of scalar Gramian metrics and proxy indices for network comparison and classification.

Approximately optimal input placements (as determined via MILP/MISDP) reveal that most real-world networks deviate systematically from random choices, highlighting the structural significance of node selection in amplification or attenuation properties.

Path-based approximations provide near-exact evaluation of $W_d^{\mathrm{out}}$ for large, sparse, or otherwise computationally intensive networks, validating context-driven simplifications in applied and theoretical settings.

7. Extensions and Applications

The discrete-time controllability Gramian, together with its generalizations, underpins a wide class of results across system theory, including:

• Robustness certification for convolutional neural networks via layerwise Gramian-embedded LMIs (Pauli et al., 2023).
• Balancing and model reduction of nonlinear systems in Koopman frameworks, enabling balanced truncation and reduced-order representation (Yeung et al., 2017).
• Analysis and optimal intervention in stochastic or Markov-networked systems (e.g., gene networks), based on the positive-definite stochastic analogues of $W_d$ (Diallo et al., 2015).
• Design and synthesis of optimal control inputs, particularly for delay and stochastic systems, leveraging explicit solutions derived from the Gramian inverse (Asadzade et al., 18 Aug 2025, Xu et al., 2023).

The discrete-time controllability Gramian thus serves as a unifying analytical and computational construct across deterministic, stochastic, delay, and nonlinear domains in discrete-time systems and networks.