Parametric Analysis of Root Location for a Fourth-Order Characteristic Equation

Abstract. This paper develops a root-based method for analyzing dynamic systems with fourth-order characteristic equations based on a spatial analog of the Vyshnegradsky diagram. Such an approach allows visualizing system stability and, moreover, determining key performance indices of transients without directly computing the roots. Within the proposed methodology, a parametrization of complex surfaces is employed to construct a three-dimensional (3D) root location diagram for a fourth-order characteristic equation. On this diagram, the solutions are divided into stable and unstable ones by the stability boundary. The surface of zero discriminant is used to classify the roots into real and complex. An additional surface defines the boundary of different root locations relative to the imaginary axis. A method for deriving these parametrically defined surfaces is presented. Parametric expressions are obtained for families of surfaces corresponding to particular system performance requirements (level surfaces for the stability degree, the oscillatory index, and the maximum absolute value among the real parts of all roots). The research results can be conveniently applied in practice to design automatic control systems. The 3D diagrams constructed in MATLAB visualize the parameter regions simultaneously ensuring the desired values of transient characteristics, the response speed, and the oscillatory index.

Keywords: Vyshnegradsky stability diagram, fourth-order system, characteristic equation, stability, discriminant, root location, parametrization, oscillation.

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Sorokvashin, A.V. and Ignatov, A.I., Parametric Analysis of Root Location for a Fourth-Order Characteristic Equation. Control Sciences 4, 24–34 (2026).

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