Optimal zeros for model reduction of continuous-time systems

Abstract

Closed-form expressions of transfer-function responses are applied to model-reduction of n-th order continuous-time systems. The contribution of each eigenvalue to the impulse response over an infinite time horizon is evaluated. The reduced-order or truncated model is set up truncating the eigenvalues that contribute the least, maintaining the same DC gain as the original system, but leaving other numerator coefficients to be determined. Then, a cost function is formed measuring the impulse-response deviation between the original and the reduced-order model. The cost function is subsequently minimized, rendering new numerator coefficients for the reduced model. Essentially, the proposed method results in reduced-order models that are easily computed, maintain stability for an originally stable system, and render impulse responses very close to the original system's. In addition, the eigenvalues are a subset of the original eigenvalues, which may be important, e.g., in cases where the original system states are directly measured or correspond to physical quantities. Finally, the method allows freedom in the relative degree of the reduced system, for example, maintaining the same relative degree as in the original system may be beneficial in some cases.