Enhanced Particle Swarm Optimization Algorithms for Multiple-Input Multiple-Output System Modelling using Convolved Gaussian Process Models
Convolved Gaussian Process (CGP) is able to capture the correlations not only between inputs and outputs but also among the outputs. This allows a superior performance of using CGP than standard Gaussian Process (GP) in the modelling of Multiple-Input Multiple-Output (MIMO) systems when observations are missing for some of outputs. Similar to standard GP, a key issue of CGP is the learning of hyperparameters from a set of input-output observations. It typically performed by maximizing the Log-Likelihood (LL) function which leads to an unconstrained nonlinear and non-convex optimization problem. Algorithms such as Conjugate Gradient (CG) or Broyden-Fletcher-Goldfarb-Shanno (BFGS) are commonly used but they often get stuck in local optima, especially for CGP where there are more hyperparameters. In addition, the LL value is not a reliable indicator for judging the quality intermediate models in the optimization process. In this paper, we propose to use enhanced Particle Swarm Optimization (PSO) algorithms to solve this problem by minimizing the model output error instead. This optimization criterion enables the quality of intermediate solutions to be directly observable during the optimization process. Two enhancements to the standard PSO algorithm which make use of gradient information and the multi- start technique are proposed. Simulation results on the modelling of both linear and nonlinear systems demonstrate the effectiveness of minimizing the model output error to learn hyperparameters and the performance of using enhanced algorithms.