Approach for the Assessment of Stability and Performance in the S- and Z – Complex Domains

This paper presents a systematic approach for rapid assessment of the performance and robustness of linear control systems through geometric analysis in the complex plane. By combining indirect performance indices within a defined zone of desired performance in the complex s-plane, a connection is established with direct performance indices, forming a foundation for the synthesis of control algorithms that ensure root placement within this zone. Analytical relationships between the complex variables s and z are derived, thereby defining an equivalent zone of desired performance for discrete-time systems in the complex z-plane. Methods for verifying digital algorithms with respect to the desired performance zone in the z-plane are presented, along with a visual assessment of robustness through radii describing robust stability and robust performance—representing performance margins under parameter variations. Through parametric modeling of controlled processes and their projections in the complex s- and z- domains, the influence of the discretization method and sampling period - as forms of a priori uncertainty is analyzed. This paper offers original derivations for MISO systems, facilitating the analysis, explanation, and understanding of the dynamic behavior of real-world controlled processes in both the continuous and discrete-time domains, and is aimed at integration into expert systems supporting control strategy selection.