A Novel Approach to Modelling Overdamped Second and Higher-Order Systems using Linearized Derived Equations

A second-order system is widely recognized in control systems, as many practical systems are modeled using it. The system's response to a step input is well-defined, with mathematical expressions available for key parameters like rise time, settling time, and others. However, most of these equations apply specifically to an underdamped second-order system, where an explicit solution is relatively straightforward, except for the delay time equation, which is derived from a linear equation involving the damping factor, ϛ. This paper develops mathematical equations for both delay time and rise time-based on linear equations, allowing the extraction of a mathematical model from the system's output response for both low and high damping factor values. Additionally, the proposed equations can be applied to model higher-order systems by using an equivalent second-order system, with results showing that this model accurately represents the higherorder system. Further analysis investigates the effect of the damping factor on natural frequency ratio (ϛ/wn) at high ϛ values, demonstrating that the system's response depends on ϛ/wn rather than the individual values of ϛ or wn. This implies that the system response remains consistent for a fixed ϛ/wn ratio