Online trained support vector machines-based generalized predictive control of non-linear systems

Abstract

In this work, an online support vector machines (SVM) training method (Neural Comput. 2003; 15: 2683-2703, referred to as the accurate online support vector regression (AOSVR) algorithm, is embedded in the previously proposed support vector machines-based generalized predictive control (SVM-Based GPC) architecture (Support vector machines based generalized predictive control, under review), thereby obtaining a powerful scheme for controlling non-linear systems adaptively. Starting with an initially empty SVM model of the unknown plant, the proposed online SVM-based GPC method performs the modelling and control tasks simultaneously. At each iteration, if the SVM model is not accurate enough to represent the plant dynamics at the current operating point, it is updated with the training data formed by persistently exciting random input signal applied to the plant, otherwise, if the model is accepted as accurate, a generalized predictive control signal based on the obtained SVM model is applied to the plant. After a short transient time, the model can satisfactorily reflect the behaviour of the plant in the whole phase space or operation region. The incremental algorithm of AOSVR enables the SVM model to learn the new training data pair, while the decremental algorithm allows the SVM model to forget the oldest training point. Thus, the SVM model can adapt the changes in the plant and also in the operating conditions. The simulation results on non-linear systems have revealed that the proposed method provides an excellent control quality. Furthermore, it maintains its performance when a measurement noise is added to the output of the underlying system.

Generalized self-tuning regulator based on online support vector regression

“…where e is the upper value of tolerable error, n's and n H 's denote the deviation from e tube and called as slack variables [13,39]. The primal form has non-convex objective function and the solution may get stuck at local minima.…”

“…When new sample P mc is introduced, the coefficient a mc corresponding this the new sample should be changed in a finite number of discrete steps until it meets the KKT conditions while ensuring the existing samples in T continue to satify the KKT conditions at each step [40]. The KKT conditions [38][39][40] that are fundamental in convergence and migration of the closed-loop data are given as: …”

“…where ''w'' represents the weights of the SVR network in feature space (F), ''Uðx i Þ'' is the projection of the input samples to feature space, ''b'' typifies the bias of regressor and h:; :i is inner product in F [39]. The essence of the optimization problem for support vector machines (SVM) is based on finding the optimal separator or regressor.…”

“…The bioreactor system is frequently used in technical literature as a benchmark nonlinear system so as to appraise and compare the performances of proposed control methodologies [19,39,49,50]. A biorector is a vessel in which water, cells (e.g., yeast or bacteria) and nutrients (substrate) to be consumed by cells are mixed.…”

“…where x 1 ðtÞ symbolizes the cell concentration , x 2 ðtÞ indicates the amount of nutrients in tank, u(t) is the flow rate by which the bioreactor is controlled, cðtÞ is nutrient inhibition parameter, bðtÞ is grow rate parameter [19,39,[49][50][51]. In …”