Study on Setpoint Tracking Performance of the PID SISO and MIMO Under Noise and Disturbance for Nonlinear Time-Delay Dynamic Systems

Abstract

This paper presents a case study of the setpoint tracking performance of the proportional integral derivative (PID) controller on the Single-Input Single-Output (SISO) and Multi-Input Multi-Output (MIMO) nonlinear digital plants under Gaussian white noise and constant load disturbance for the nonlinear time-delay dynamic system. With the objective of getting a better understanding of the nonlinear discrete-time PID controller, we proposed a case study using two SISO and two MIMO digital plants, and then do the numerical simulations along with the addition of Gaussian white noise and load disturbance to simulate the real environment. In this paper, we compare the results of the system working with and without noise and load disturbance. The study result of this paper shows that on the discrete-time digital nonlinear plant, the PID controller is working well to follow the nonlinear setpoint even under heavy noise and load disturbance. The study compared the performance indexes of the controllers in terms of the maximum error, the Root mean square error (RMSE), the Integral square error (ISE), the Integral absolute error (IAE), and the Integral of time-weighted absolute error (ITAE).

Output Recurrent Fuzzy Broad Learning Systems for Adaptive MIMO PID Control: Theory, Simulations, and Application

“…By considering the predictive cost function (41), the increment of the updating parameter vector ∆𝑷 𝐶 (𝑘 + 𝑗) is obtained from the following formula 58) into (53) and then one rewrites it to obtain ( )…”

“…The first evaluative example deploys the two-by-two MIMO system model characterized by time-varying dynamics, uncertainty, and exhibiting strong nonlinearity and coupling, akin to the model described in [41], [42]. The complex model is written by the following; where the slowly time-varying factors 𝑎 1 (𝑘) = 1.2 (1 − 0.8𝑒 −0.5𝑘 ) and 𝑎 2 (𝑘) = 1.2 (1 − 0.8𝑒 −0.1𝑘 ) evolve the dynamics of the system over time.…”

“…Expanding on the preliminary simulations, another challenging test system adapted from [41], [42] is taken as the second multivariate simulation in order to exemplify augmented complexity through a customized nonlinear timevarying dynamics system model exhibiting exacerbated coupling effects. This modified system model is described by…”