Ball and Plate Sytem i read some documents about PSO PID tuning and i recognize that they make inner and outer loop but i do not know why and how , i mean where to set PSO algorithm and if i run it while system is running it will cause delay so how can i solve this problem and how can i implement PSO PID tuning for the ball and plate system.
i expect solution for this problem and how can i implement PSO algorithm for the ball and plate system.
Given a linear system $\ddot{x} + a_1 \dot{x} + a_0 x = u$, we design a PID controller $u = k_2 \int\limits_0^\infinity (x - x_r) dt + k_1 (x - x_r) + k_0 (\dot{x} - \dot{x}_r)$ such that the objective function $J$ given by $\int\limits_0^\infinity (\dot{x} - \dot{x}_r)^2 dt$. We now can use find adequate values for variables $(k_0, k_1, k_2)$ which minimize objective function $J(k_0, k_1, k_2)$. We may even add control boundary constraints, like $\abs{u} < u_r$
Among optimization methods, we may cite Particle swarm optimization method: It consists on a heuristic based on multiple agents evaluated on different points of parameter space. After perform the optimization procedure, we have the (sub-)optimal values for desired control performance.