This chapter recalls the nonlinear polynomial neurons and their incremental and batch learning algorithms for both plant identification and neuro-controller adaptation. Authors explain and demonstrate the use of feed-forward as well as recurrent polynomial neurons for system approximation and control via fundamental, though for practice efficient machine learning algorithms such as Ridge Regression, Levenberg-Marquardt, and Conjugate Gradients, authors also discuss the use of novel optimizers such as ADAM and BFGS. Incremental gradient descent and RLS algorithms for plant identification and control are explained and demonstrated. Also, novel BIBS stability for recurrent HONUs and for closed control loops with linear plant and nonlinear (HONU) controller is discussed and demonstrated.
TopIntroduction
Machine learning and control algorithms have been amazingly, theoretically developing in recent decades. The data-driven trends of system approximation based on neural networks and this way-based adaptive control is of significant interest to the research community; however, we may observe that these trends are not correspondingly spreading into industrial practice. The main reasons are the relatively high demands on operator qualification, i.e. education in mathematics and dynamical systems and operators’ technological competences and ability to understand the concepts and proper applications of machine learning related to neural networks (there are many more parameters and aspects than there are with PID controllers, e.g.). For sure, the stability of nonlinear control loops with neural networks, i.e. the stability of such nonlinear time invariant systems, together with the risk of heavy costs resulting from unstable development within factory lines (or power plant units, for example) is also a crucial consideration, so the control algorithms in the industry remain conservative, preserving comprehensible and analyzable techniques, such as PID control and other linear-based approaches. Thus, we believe that the transition from conservative linear techniques of control toward nonlinear and machine learning–based ones can be successful, if such control principles are well comprehensible both to the academic and industrial community and if the stability analysis is also not acceptable and comprehensible for both sides.
The recent trends of Deep Networks and Deep Learning (Goodfellow, Bengio, & Courville, 2016; LeCun, Bengio, & Hinton, 2015) are not always achievable in industrial practice for control, especially when dynamical systems should be approximated from data, and these neural networks require large training datasets. The networks with Long Short-Term Memory (LSTM) neurons (Hochreiter & Schmidhuber, 1997) are very popular today as well; however, their application to control might be practically limited due to the need for heavier computations for training and due to a relatively complex architecture for analysis. Furthermore, many industrial control tasks would not need to implement a too complex and too nonlinear plant model and controller. When a conventional control (like PID) is doing a less or more sufficient job in practice, then there is a high chance that the control performance can be significantly optimized with reasonably nonlinear neural networks
Thus, the neural computation presented in this chapter relates to polynomial neural networks that can be ranked among shallow neural networks and that, contrary to other shallow networks, feature a linear optimization problem, while the input-output mapping is customable nonlinear. Other shallow neural architectures that shall be mentioned are random vector functional link (RVFL) networks (Zhang & Suganthan, 2016) and recently published their alternatives known as Extreme Learning Machines (ELMs) (Huang, Zhu, & Siew, 2006a, 2006b; Zhang & Suganthan, 2016) and of course also the multilayer perceptrons (MLPs) (Hornik, Stinchcombe, & White, 1989) with a very few hidden layers. Regarding MLPs, it shall be mentioned that novel types of hidden neurons with non-sigmoid somatic operations, i.e. ELU or RELU (Glorot, Bordes, & Bengio, 2011), shall be also considered for their applications due suppressing the issue of vanishing gradients and improving the convergence.
Higher order neural units (Bukovsky, Hou, Bila, & Gupta, 2008; Gupta, Homma, Hou, Solo, & Bukovsky, 2010; Gupta, Bukovsky, Homma, Solo, & Hou, 2013) are standalone architectures stemming from the branch of polynomial neural computation originating from the works, or higher order neural networks (Ivakhnenko, 1971; Kosmatopoulos, Polycarpou, Christodoulou, & Ioannou, 1995; Nikolaev & Iba, 2006; Taylor & Coombes, 1993; Tripathi, 2015).
The adaptive control that is presented in this chapter belongs among the control that utilizes a data-driven model, which is also approximated as a polynomial neural architecture.
Nowadays, the neural network–based control approaches can be classified according to the main principle as follows: