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Gramian-based model order reduction methods, like balanced truncation, and interpolation-based methods, such as H2-optimal reduction, are two important types of model reduction algorithms. Although both are known for their accuracy, they are often seen as two different approaches. This paper shows that these two methods are closely related, with Gramian-based reduction being roughly an interpolation problem, and vice versa. For Galerkin projection, we find that when the reduced model has enough order to capture the significant eigenvalues of the controllability or observability Gramian, preserving these eigenvalues becomes an interpolation problem. In this case, both Gramian-based and interpolation-based model reduction methods produce the same transfer function but with different state-space realizations. When the reduced model's order is too small to capture all significant eigenvalues, the methods begin to differ, and the difference depends on the eigenvalues that were left out. In the case of Petrov-Galerkin projection, if the reduced model's order is large enough to capture the significant Hankel singular values, balanced truncation becomes the same as H2-optimal model order reduction. Again, both methods give the same transfer function but with different state-space realizations. When the order is smaller, the methods diverge, with the difference depending on the truncated Hankel singular values. Numerical examples are provided to support these findings, showing that Gramian-based and interpolation-based model reduction methods are more connected than previously thought and can be viewed as approximations of each other.

Linear quadratic output systems constitute an important class of dynamical systems with numerous practical applications. When the order of these models is exceptionally high, simulating and analyzing these systems becomes computationally prohibitive. In such instances, model order reduction offers an effective solution by approximating the original high-order system with a reduced-order model while preserving the system's essential characteristics. In frequency-limited model order reduction, the objective is to maintain the frequency response of the original system within a specified frequency range in the reduced-order model. In this paper, a mathematical expression for the frequency-limited $\mathcal{H}_2$ norm is derived, which quantifies the error within the desired frequency interval. Subsequently, the necessary conditions for a local optimum of the frequency-limited $\mathcal{H}_2$ norm of the error are derived. The inherent difficulty in satisfying these conditions within a Petrov-Galerkin projection framework is also discussed. Based on the optimality conditions and Petrov-Galerkin projection, a stationary point iteration algorithm is proposed that enforces three of the four optimality conditions upon convergence. A numerical example is provided to illustrate the algorithm's effectiveness in accurately approximating the original high-order model within the specified frequency interval.

An important class of dynamical systems with several practical applications is linear systems with quadratic outputs. These models have the same state equation as standard linear time-invariant systems but differ in their output equations, which are nonlinear quadratic functions of the system states. When dealing with models of exceptionally high order, the computational demands for simulation and analysis can become overwhelming. In such cases, model order reduction proves to be a useful technique, as it allows for constructing a reduced-order model that accurately represents the essential characteristics of the original high-order system while significantly simplifying its complexity. In time-limited model order reduction, the main goal is to maintain the output response of the original system within a specific time range in the reduced-order model. To assess the error within this time interval, a mathematical expression for the time-limited $\mathcal{H}_2$-norm is derived in this paper. This norm acts as a measure of the accuracy of the reduced-order model within the specified time range. Subsequently, the necessary conditions for achieving a local optimum of the time-limited $\mathcal{H}_2$ norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. After that, a stationary point iteration algorithm based on the optimality conditions and Petrov-Galerkin projection is proposed. Upon convergence, this algorithm fulfills three of the four optimality conditions. To demonstrate the effectiveness of the proposed algorithm, a numerical example is provided that showcases its ability to effectively approximate the original high-order model within the desired time interval.

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.