A New Error Bound for Nonzero Initial Conditions in Linear Dynamical Systems

Balanced truncation is one of the most widely used techniques for reducing the order of asymptotically stable linear time-invariant systems because it preserves stability and provides computable a priori error bounds. However, the classical theory is formulated essentially for homogeneous initial conditions and therefore does not directly quantify the approximation error when the system starts from a nonzero state. In many applications, such as electrical circuits with pre-charged capacitors, mechanical systems released from displaced configurations, and diffusion processes initialized away from equilibrium, the initial condition contributes substantially to the output and must be accounted for in model reduction. In this paper we derive a new L 2 -error bound for balanced truncation of continuous-time linear time-invariant systems with nonzero initial conditions. The key idea is to reinterpret the initial state as an auxiliary impulsive input acting through an additional channel of an extended realization. This reformulation preserves the input–output behavior of the original initial-value problem while allowing the use of the standard balanced-truncation framework. We show that the observability Gramian remains unchanged, whereas the controllability Gramian acquires a transparent additive correction induced by the initial condition. Based on this structure, we derive an explicit a priori error estimate in which the contribution of the initial state and the contribution of the external input are separated. The resulting bound is independent of regularization parameters and reduces to the classical balancedtruncation bound in the homogeneous case. Numerical experiments for a SLICOT benchmark model, an RC-circuit example, and an additional illustrative test problem show that the proposed estimate is reliable and substantially sharper than existing bounds for systems with inhomogeneous initial conditions.