Closed-Form Gramians and Riccati Solutions for Reduced Second-Order Energy-System Benchmarks

Abstract

Reduced circuit and electromechanical models often lead to the scalar second-order system ay″+by′+cy=u(t). General controllability, minimum-input-energy, and LQR results for such systems are standard; the contribution here is an explicit specialization that exploits the companion-matrix decomposition A=μI+B, B2=ηI. This structure yields damping-regime-independent state-transition formulas, closed forms for the finite-horizon controllability Gramian and its critical-damping limit, and an explicit stabilizing CARE solution for diagonal state weights. To separate analytic continuity from numerical stability, a near-critical evaluation rule is supplied: when |η|T2≤10−4, the Gramian coefficients are evaluated by convergent Taylor series rather than differences of nearly equal root-based terms. A two-sided sweep down to |η|=10−14 shows that the stable evaluation remains at floating-point accuracy against adaptive quadrature, whereas direct root-difference evaluation loses accuracy as cancellation grows. A fixed-seed 1000-case CARE experiment provides deterministic algebraic verification against SciPy’s Schur/QZ solver. Normalized RLC, averaged LC-filter, and small-signal swing examples are included only as reduced-model benchmarks; they do not constitute validation on a switching converter, full electrical network, or experimental platform. An explicit LCL-filter connection is also derived for the ideal capacitor-voltage resonant coordinate, while full practical LCL inverter control is retained as a higher-order problem outside the scope of the scalar model.