Determination of Optimal Q and R Matrices for the LQR Control of Nonlinear BOS System Using Fish Swarm Algorithms

Q (state weighting) and R (input weighting) matrices are the essential components of the linear quadratic regulator (LQR) optimization process. Trial and error (TE) and Bryson’s methods are conventionally used for tuning them. However, these methods are time-consuming and tedious. Additionally, they do not give optimal values. Therefore, this paper uses two fish swarm algorithms, namely: artificial fish swarm algorithm (AFSA) and weighted AFSA (wAFSA), to determine the optimal Q and R matrices for the control of the ball-on-sphere (BOS) system. wAFSA was developed to solve the problem associated with AFSA of going into local minima because visual distance and step size are constant. System parameters were derived, controllability and observability were checked and optimal Q and R matrices were obtained using AFSA and wAFSA. Using these matrices, the system was controlled. Settling times of 7.0 sec, 5.70 sec, and 3.35 sec were obtained for TE, AFSA, and wAFSA, respectively. The steady-state error for the three algorithms was found to be zero. wAFSA was found to be more efficient because the objective of the control is to bring the system back to equilibrium.