What is the best method to estimate the PID controller parameters? I need a reference paper or a book chapter that explain a suitable method to calculate the PID controller gains for Single Input - Single Output control system.
You need to find those parameters to satisfy your Lyapunov stability proof. Zillion papers, google "pid controller lyapunov stability". You'll find things such as https://doi.org/10.1017/S0263574700017641. Good luck.
A method for the automatic tuning of PID controllers in a closed loop, based on the estimation of a parametric ‘black-box’ transfer function model.The system is excited by generating limit cycle oscillations at two different frequencies, which are approximately the crossover frequency and the critical frequency for the feedback loop. A discrete parametric transfer function model is estimated from the experimental data. Important parameters concerning the estimation, such as the prefilter cut-off frequency and the sampling interval, are determined automatically from the experimental data. The PID parameters are determined from a constrained optimization in the frequency domain. The constraints are classical control system properties, such as the maximum amplitudes of the sensitivity and the complementary sensitivity functions. Given these constraints, the PID parameters are determined such that the low frequency amplitude characteristic for the controller is maximized. Simulation experiments show that the tuning procedure has low sensitivity to disturbances and noise during the tuning experiment.
https://www.sciencedirect.com/science/article/pii/0005109894900604
Lyapunov stability is not enough, since it will only show the region of controller gains in which the equilibrium of the closed-loop system is stable. Lyapunov stability doesn't tell anything about performance. I guess you are looking for gains that satisfies some optimality criterion or result in minimum settling time or minimum overshoot. Is your SISO system linear or nonlinear? For general PID tuning see Article PID controllers : theory, design, and tuning / Karl J. Astro...
and for a linear sytem LQI (which is kind of similar to PID) will produce optimal gains: https://www.mathworks.com/help/control/ref/lqi.htmlFor a nonlinear system finding optimal gains is not straight forward. Experimentally, you can use the Ziegler-Nichols method which uses the transient or step response of the closed-loop system: http://educypedia.karadimov.info/library/Ziegler_Nichols.pdf
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