Abstract
Maintaining dynamic stability of the system is not an easy task especially for the systems with higher-order characteristic polynomials. For closed-loop control systems implementation of proper PID gains can be a tool to keep dynamic system within its stability range. However, selection of proper controller gains is not an easy task either. The paper describes new and effective method how to identify the stability boundaries of higher-order dynamic systems in terms of boundaries of closed-loop characteristic polynomial coefficients. Provided the precise boundaries of that coefficients are defined they can be easily manipulated by means of appropriate PID gain values to keep those coefficients within the stability range. The new intuitively discovered algorithm is an effective yet universal technique to identify stability boundaries of any higher-order dynamic system. The paper describes the derivation of a single analytical expression separately for the systems with order from three to five. It becomes a necessary and sufficient condition for such systems to remain at its stability boundaries, i.e., provides marginal stability. Once adopted these expressions can be used as a guide for selection of PID controller gains and thus keep the system within its stability range.