Positive realness preserving model reduction with H∞ norm error bounds

Many physical systems that occur in applications are naturally passive, for example, mechanical systems with dual sensors and actuators, and electrical circuits with passive components. Taking advantage of this property, many controller schemes have been proposed with the property that the controller is strictly positive real. Due to design and implementation considerations, the plant or the controller may need to be approximated by a lower-order system. It is highly desirable for the reduced-order system to also possess the positive realness property to guarantee that the resulting closed-loop system remains stable. Motivated by this problem, this paper considers the general model-reduction problem for a positive real system under the constraint that the reduced system is also positive real. We present a solution based on the balanced stochastic truncation. When the higher-order system is strictly positive real, we derive an H_∞ norm bound on the approximation error. We also consider alternate approaches of approximating the spectral factors with associated H_∞ norm error bounds. An example is included to show the efficacy of this method and comparison with other approaches.