[Paper Review] Response to a Setpoint Change in PID Controlled Time-Delay Feedback Systems

Gianpasquale Martelli|ArXiv.org|Feb 22, 2007

Advanced Control Systems Design5 references4 citations

TL;DR

This paper presents an explicit analytical solution for the response to a setpoint change in PID-controlled time-delay systems using the method of steps, avoiding Padé approximations. It derives closed-form coefficient formulas for a second-order system, enabling accurate, low-time response evaluation without numerical approximation or contour integration, crucial for precise controller tuning in delay-prone processes.

ABSTRACT

The response to a setpoint change in PID controlled feedback systems plays an important role for the tuning methods. This response may be easily evaluated in linear systems without delay by solving the related ordinary differential equations. These equations are also obtained for system with delay, if the delay exponential term is approximated to a polynomial by one of the Pade expressions. If this simplification is avoided, one must deal with differential difference equations. In this paper their explicit solution, obtained with the method of steps, is presented. Moreover a second order equation is investigated as example and the complete set of the coefficients formulas is given.

Motivation & Objective

• To develop an exact analytical solution for the response to a setpoint change in PID-controlled time-delay feedback systems.
• To overcome the limitations of Padé approximation in solving differential-difference equations arising from time delays.
• To provide a systematic method for computing controller coefficients without relying on numerical integration or contour integrals.
• To enable accurate and safe tuning of PID controllers in systems with time delays by offering explicit formulas for the response.
• To extend the applicability of classical tuning methods to systems with time delays by providing a rigorous analytical framework.

Proposed method

• The method of steps is applied to solve the differential-difference equation derived from the closed-loop transfer function of a PID-controlled time-delay system.
• The solution is expressed as a sum of exponential and polynomial terms: $ y_{n,k}(t_n) = \sum_{p=1}^{m_a} e^{r_p t_n} \sum_{i=0}^{v_{k,p}+n} G_{n,k,p,i} t_n^i $, where $ r_p $ are the characteristic roots of the delay-free system.
• Coefficients $ G_{n,k,p,i} $ are determined recursively using continuity conditions and matching of terms involving $ e^{r_p t_n} t_n^i $ across intervals.
• The approach assumes initial conditions and setpoint changes are compatible with the solution form, particularly leveraging the integral action of the PID controller to ensure a zero characteristic root.
• A general procedure is derived for computing coefficients, valid for systems with real, simple characteristic roots, and extendable to multiple or complex roots.
• The derivation uses series product identities and Vandermonde systems to solve for unknown coefficients, with Cramer's rule applied to the resulting linear systems.

Experimental results

Research questions

• RQ1 Can an exact analytical solution be derived for the setpoint response in PID-controlled time-delay systems without Padé approximation?
• RQ2 How can the method of steps be systematically applied to solve differential-difference equations arising from time-delay feedback systems?
• RQ3 What are the closed-form expressions for the coefficients in the solution series when the system is second-order and time-delayed?
• RQ4 How do the initial conditions and setpoint step changes affect the structure of the solution in terms of exponential-polynomial terms?
• RQ5 Can the solution be expressed in a form that enables precise, low-time evaluation for controller tuning purposes?

Key findings

• The paper derives explicit, closed-form coefficient formulas for the solution of a second-order PID-controlled time-delay system using the method of steps.
• The solution is expressed as a finite sum of exponential and polynomial terms, $ y_{n,k}(t_n) = \sum_{p=1}^{m_a} e^{r_p t_n} \sum_{i=0}^{v_{k,p}+n} G_{n,k,p,i} t_n^i $, which is analytically tractable and avoids numerical approximation.
• The method ensures continuity across time intervals and allows recursive computation of coefficients $ G_{n,k,p,i} $ using linear systems derived from matching powers of $ t_n $.
• The solution is valid for systems with real, simple characteristic roots, and the presence of an integrator in the PID controller ensures a zero root, enabling compatibility with step changes and steady initial conditions.
• The use of Vandermonde systems and Cramer's rule allows for the systematic solution of coefficient equations, with determinants expressed via products of differences of roots.
• The approach avoids contour integration and infinite series, making it suitable for accurate evaluation at low time values where Padé approximations fail.