State-Space Construction of Continuous-Time Orthogonal Systems with Applications to System Identification and Control

Continuous-time orthogonal basis functions play a fundamental role in system identification, signal approximation, model reduction, and control, where compact and numerically efficient representations of dynamical systems are required. Most existing constructions are based on predefined frequency-domain basis functions and their associated pole configurations. This paper introduces a novel state-space framework for the construction of continuous-time orthogonal and biorthogonal systems by characterizing orthogonality directly through the dynamics of stable linear systems. The central result shows that orthogonality can be enforced by a Lyapunov-type matrix condition linking the system matrix and the initial state, thereby enabling the systematic generation of orthogonal basis functions as state trajectories. The proposed framework naturally encompasses the classical Laguerre and Kautz systems as special cases while providing substantially greater design flexibility through the independent parametrization of system matrices and initial conditions. It is further extended to output-orthogonal, state-biorthogonal, and output-biorthogonal systems, yielding a unified state-space formulation of orthogonal and dual functional representations. In addition, new algebraic formulations of convolution and signal decomposition are derived, leading to explicit coefficient relations expressed through Lyapunov and Sylvester matrix equations. The applicability of the proposed framework is demonstrated through numerical examples in system identification, model reduction, and optimal control. The results show that the additional parametrization freedom enables orthogonal representations that are better adapted to oscillatory and weakly damped dynamics while preserving the computational advantages of orthogonal-function-based approaches.