Compact analysis of the necessity of padé approximation for delayed continuous-time models in lqr, h-infinity and root locus control strategies

This paper presents a comprehensive analysis of the need for the Padé approximation for continuous-time models with delays, focusing on its critical role in addressing the control challenges posed by time delays. Time delays, often referred to as dead times, transport delays or time lags, are inherent in a wide range of industrial and engineering processes. These delays introduce phase shifts that degrade control performance by reducing control bandwidth and threatening the stability of closed-loop systems. Accurate modelling and compensation of these delays is essential to maintain system stability and ensure effective control. This paper highlights the difficulties that arise when using advanced control techniques such as root locus (RL), linear quadratic regulator (LQR) and H-infinity (H_∞) control in systems with delays. Representing delays in exponential form leads to an infinite number of state problems, complicating the design and analysis of controllers in such systems. To address these challenges, the Padé approximation is proposed as an effective method for approximating time delays with rational polynomials of appropriate order. This approach allows for more accurate simulation, system analysis and controller design, thereby mitigating the problems caused by delays. The paper also provides a detailed comparative analysis between the Padé approximation and Taylor polynomials, demonstrating the superiority of the former in achieving accurate delay modelling and control performance. The results show that the use of Padé approximation not only improves the accuracy of system models, but also improves the robustness and stability of control strategies such as RL, LQR, and H_∞. These results highlight the importance of the Padé approximation as a valuable tool in the design of delay-affected control systems, offering significant advantages for both theoretical and practical applications.