Reduced-Order LTI Models

• Reduced-order LTI models are surrogate dynamic systems that approximate the input-output behavior of high-dimensional linear systems using techniques like moment-matching and Krylov subspace projections.
• They reduce computational complexity while preserving essential system properties such as stability and performance over specified time or frequency ranges.
• These models facilitate efficient simulation and real-time control in applications like discretized PDEs and networked systems, providing practical benefits for large-scale problems.

A reduced-order linear time-invariant (LTI) model is a surrogate dynamic system, typically of order $r \ll n$, constructed to approximate the input-output response of a higher-dimensional LTI system over a specified frequency range, time interval, or parameter domain. Such models enable efficient simulation, control, and optimization in high-dimensional applications by capturing the dominant dynamic features of the original system while drastically reducing computational complexity. The reduced-order LTI approximation forms a foundational paradigm in model reduction, with techniques based on moment-matching, Gramian-based projections, optimality with respect to induced norms, and data-driven methods.

1. Mathematical Formulation and Objectives

Let the full-order continuous-time LTI system be given by: $\dot{x}(t) = A x(t) + B u(t), \quad y(t) = C x(t)$ with $x \in \mathbb{R}^n$, $u \in \mathbb{R}^m$, $y \in \mathbb{R}^p$, and state-space matrices $(A,B,C)$, where $A$ is Hurwitz. The associated transfer function is $G(s) = C(sI - A)^{-1}B$.

A reduced-order LTI system has the form: $$\dot{x}_r(t) = \widehat{A} x_r(t) + \widehat{B} u(t), \quad \widehat{y}(t) = \widehat{C} x_r(t),$$ with $x_r \in \mathbb{R}^r$, $r \ll n$, and transfer function $$\widehat{G}(s) = \widehat{C}(sI_r - \widehat{A})^{-1}\widehat{B}$$.

Model reduction seeks $\widehat{G}$ to minimize a normed error $\|G - \widehat{G}\|$, according to system-theoretic metrics such as the $\mathcal{H}_2$ norm or $\mathcal{H}_\infty$ norm, or to preserve salient structural properties (e.g., energy, passivity, stability) over prescribed time/frequency/parameter ranges. In many applications, the matching is enforced only over $t \in [0, \tau]$ for some $\tau < \infty$ (the time-limited scenario) or within specific parameter regimes.

2. Classical Projection and Krylov Subspace Algorithms

Projection-based methods dominate reduced-order LTI model construction for high-dimensional systems. These approaches use Petrov–Galerkin projections to construct low-dimensional subspaces that preserve the controllable and observable directions most relevant to external input-output behavior.

For a set of interpolation points $$\{\sigma_i\}_{i=1}^r \subset \mathbb{C} \setminus \mathrm{spec}(A)$$ and tangential directions $\{b_i\}, \{c_i\}$, the rational Krylov right and left subspaces are defined as: $$V_r = \mathrm{span} \left\{ (\sigma_i I - A)^{-1} B b_i \right\}, \qquad W_r = \mathrm{span} \left\{ (\sigma_i I - A^T)^{-1} C^T c_i \right\}$$ The reduced matrices are given via Petrov–Galerkin projection: $$\widehat{A} = Z_r^T A V_r, \quad \widehat{B} = Z_r^T B, \quad \widehat{C} = C V_r, \quad Z_r^T = (W_r^T V_r)^{-1} W_r^T.$$

The Iterative Rational Krylov Algorithm (IRKA) and its variants iteratively update $\{\sigma_i, b_i, c_i\}$ to achieve first-order $\mathcal{H}_2$ optimality conditions (bitangential Hermite interpolation at the mirror images of the reduced poles) (Necoara et al., 2018, Mlinarić et al., 2023). The resulting reduced-order systems often inherit stability, and the dominant input-output behavior of the original model is retained.

For time-limited reduction, the subspaces are modified to embed the finite-horizon constraint (Das et al., 2021): $$V_r = \mathrm{span} \left\{ (\sigma_i I - A)^{-1}(I - e^{-\sigma_i \tau} e^{A\tau}) B b_i \right\}$$ yielding surrogate models optimized over $[0, \tau]$ with respect to the time-limited $\mathcal{H}_2$ norm.

3. Time-Limited Model Reduction and Optimality Conditions

Finite-horizon applications necessitate time-limited error metrics and interpolation frameworks. For $t \in [0, \tau]$, the impulse response is truncated to $g_\tau(t)$ and the time-limited $\mathcal{H}_2$ norm is

$$\|g\|_{\mathcal{H}_2, \tau} = \sqrt{\int_0^\tau \|g(t)\|_F^2 dt }$$

or, equivalently, via

$$G_\tau(s) = G(s) - e^{-s \tau} C (sI - A)^{-1} e^{A\tau} B.$$

The first-order necessary conditions for (local) $\mathcal{H}_2(\tau)$-optimality are bi-tangential interpolation constraints: $$G_\tau(-\lambda_k) \hat{b}_k = \widehat{G}_\tau(-\lambda_k) \hat{b}_k,\quad \hat{c}_k^T G_\tau(-\lambda_k) = \hat{c}_k^T \widehat{G}_\tau(-\lambda_k),\quad \hat{c}_k^T G_\tau'(-\lambda_k) \hat{b}_k = \hat{c}_k^T \widehat{G}_\tau'(-\lambda_k) \hat{b}_k,$$ for all reduced-order poles $\lambda_k$ (Das et al., 2021).

Limited Time IRKA (LT-IRKA) enforces these conditions via rational Krylov projection on the modified subspaces and iteratively updates interpolation data; the “nearness” to optimality is quantified in terms of explicit interpolation residuals.

For systems with quadratic outputs, similar time-limited $\mathcal{H}_2$ norms and necessary (but not fully achievable) optimality conditions are derived. Iterative projection-based algorithms converge to surrogates that satisfy all algebraic conditions except for one residual, which is small in practice (Zulfiqar et al., 2024).

4. Objective-Driven Optimization and Associated Algorithms

Beyond projection, direct optimization techniques pose reduced-order LTI model construction as (typically nonconvex) minimization problems over families of interpolants, seeking minimal $\mathcal{H}_2$ (or related) error:

• Full parametrization and KKT-based optimization: Moment-matching ansätze with tunable free parameters or interpolation points yield nonconvex semidefinite programs, with optimality characterized by Karush–Kuhn–Tucker conditions (Necoara et al., 2018). Gradient-type and partial-minimization algorithms are employed, and convex (semidefinite) relaxations exist that are exact under certain system structure.
• Semi-definite relaxation for SISO cases: For first- and second-order SISO models, the $\mathcal{H}_2$ cost can be minimized globally by formulating a convex SDP over interpolation point parameters, recovering the optimal shifts and enabling rational Krylov realization (Zhu et al., 24 Aug 2025).
• Data-driven and sample-optimal reductions: Gradient-based optimization on parameter-separable representations using only frequency samples allows nonintrusive construction of $\mathcal{H}_2$-optimal reduced models, and under regularity, coincides with classical projection (Mlinarić et al., 2022).

5. Extensions: Structured, Parametric, and Time-Limited Systems

The optimality landscape of reduced-order LTI modeling encompasses broad system classes:

• Structured LTI systems: Bitangential Hermite interpolation conditions generalize to second-order, port-Hamiltonian, and time-delay systems under simultaneous diagonalizability assumptions. The reduced transfer is expressed as a sum over residues and structured scalar denominator terms, with optimality enforcing interpolation at all mirror-image poles (Mlinarić et al., 2023).
• Parametric systems: For $A(p),B(p),C(p)$ with parametric dependence, the error is measured with respect to a mixed $\mathcal{H}_2 \otimes \mathcal{L}_2$ norm, and interpolatory optimality requires parameter-averaged tangential Hermite matching, or even parameter-differentiated interpolation when poles are parameter-dependent (Mlinarić et al., 2024, Hund et al., 2021).
• Balancing and structure preservation: For second-order models with inhomogeneous initial conditions, distinct projection strategies and tailored Gramians preserve second-order structure and deliver tight error bounds (Przybilla et al., 2022).

6. Practical Impact and Computational Considerations

Reduced-order LTI models are central in applications where simulation, control, and optimization must be executed on systems with very large state dimension ($10^5 - 10^6$ and beyond), such as discretized PDEs or networked systems.

• Computational scalability: Modern Krylov-based and rational interpolatory methods require only linear solves and matrix-vector operations, and admit rigorous error quantification and robust performance for large-scale problems as long as matrix exponentials and Lyapunov solutions are tractable (Das et al., 2021). Adaptive randomized SVD and block-AAA techniques further enhance scalability when only transfer function samples or impulse responses are available (Yu et al., 2023, Pelling et al., 10 Jun 2025).
• Model-based control (e.g., ROMPC): Reduced-order LTI surrogates underpin real-time model predictive control for high-dimensional plants, with explicit error bounds on output and input constraints, robust setpoint tracking, and provable stability guarantees (Lorenzetti et al., 2020, Lorenzetti et al., 2018).
• Data-driven surrogates: Non-intrusive (operator inference, Loewner) frameworks build efficient LTI (or bilinear, quadratic) surrogates from input-output data alone, bypassing the need for full-access state-space realizations (Poussot-Vassal et al., 2020).

7. Limitations, Unification, and Outlook

No globally convergent algorithm exists for general high-order multi-input or highly parameterized LTI systems; local optimality is standard, and initialization is critical in iterative methods (Das et al., 2021, Necoara et al., 2018). The accuracy of time-limited reduction, especially outside the prescribed interval, depends critically on the choice of $\tau$ and the decay of system modes. For quadratic, nonlinear, or parametric scenarios, achieving all first-order conditions via projection is typically impossible, and alternative formulations provide only near-optimality (Zulfiqar et al., 2024, Hund et al., 2021).

A unifying view emerges: bitangential Hermite interpolation—enforced at the mirror images of the reduced poles—is the fundamental mechanism underlying $\mathcal{H}_2$-optimal reduction across unstructured, structured, time-limited, and parametric LTI models (Mlinarić et al., 2023, Mlinarić et al., 2024). This insight drives the design of efficient, structure-preserving, and application-aware reduced-order modeling algorithms, with ongoing research targeting global optimality, nonlinearity, and efficient handling of parameter and uncertainty.