Learning Systems

A learning system can be defined as a system which can adapt its behaviour to become more effective at a particular task or set of tasks. It consists of an architecture with a set of variable parameters and an algorithm. Learning systems are useful in many fields, one of the major areas being in control and system identification. This work covers major aspects of learning systems: system architecture, choice of performance index and methods measuring error. Major learning algorithms are explained, including proofs of convergence. Artificial neural networks, which are an important class of learning systems and have been subject to rapidly increasing popularity, are discussed. Where appropriate, examples have been given to demonstrate the practical use of techniques developed in the text. System identification and control using multi-layer networks and CMAC (Cerebellar Model Articulation Controller) are also presented.

1 Introduction to Learning Systems.- 1.1 Systems, Memory.- 1.2 Performance Index.- 1.2.1 Random Input.- 1.2.2 Deterministic Input.- 1.3 Learning Algorithms.- 1.4 Some Examples of Learning Systems.- 1.4.1 The Learning Linear Combiner.- 1.4.2 Neurons of Higher Order.- 1.4.3 The Learning Non-Linear Transformer.- 1.4.4 Linear and Non-Linear Learning Filters.- 1.4.5 Learning control systems.- 1.4.6 The Learning Multilayer Neural Network.- 1.4.7 Learning CMAC.- References.- 2 Deterministic Algorithms.- 2.1 Simple Projection Algorithms in Spaces With Different Norms (Structure, Convergence, Properties).- 2.1.1 Construction Of Kaczmarz Algorithm.- 2.1.2 Convergence.- 2.2 Modified Projection Algorithms With a High Rate of Convergence.- 2.2.1 Construction.- 2.2.2 Transient Mode.- 2.2.3 Properties Of The Estimates.- 2.2.4 “Bad” Measurements.- References.- 3 Deterministic and Stochastic Algorithms of Optimisation.- 3.1 Deterministic Methods for Unconstrained Minimisation.- 3.1.1 The Gradient Algorithm.- 3.1.2 Convergence.- 3.1.3 Newton’s Algorithm.- 3.1.4 Example Application.- 3.2 Stochastic Approximation and Recurrent Estimation.- 3.2.1 Random Input.- 3.2.2 Example System.- References.- 4 Stochastic Algorithms: The Least Squares Method in the Non-Recurrent and Recurrent Forms and the Gauss-Markov Theorem.- 4.1 The Least Squares Method in Recursive and Non-Recursive Forms (The White Noise Case).- 4.1.1 The Least Squares Method in Non-Recursive Form.- 4.1.2 A Priori Information.- 4.2 The Gauss-Markov Theorem.- 4.2.1 Optimal Estimates.- 4.2.2 Connection With the Maximum Likelihood Estimates [2].- 4.3 Example System.- References.- 5 Stochastic Algorithms.- 5.1 Algorithms With Forgetting Factor.- 5.2 The Least Squares Method by Correlated Noise in the Non-Recursive and Recursive Forms: Connection With Decorrelation Procedures.- 5.2.1 The Arbitrary Case.- 5.2.2 Special Case.- 5.3 Introduction to the Kaiman filter [1], [3].- References.- 6 Multilayer Neural Networks.- 6.1 Multilayer Neural Network as a Non-Linear Transformer. The Kolmogorov and Cybenko Theorems.- 6.1.1 Introduction.- 6.1.2 MNN Architecture.- 6.1.3 The Input-Output Mapping of a Multilayer Network of the First Order.- 6.1.4 Approximation.- 6.2 Learning Algorithms for Single Elements of Multilayer Neural Networks.- 6.2.1 Differentiable Activation Functions.- 6.2.2 Non-Differentiable Activation Functions.- References.- 7 Learning Algorithms for Neural Networks.- 7.1 The Back-Propagation Algorithm for MNN Learning.- 7.1.1 Main Equations.- 7.1.2 Fully Connected MNNs.- 7.1.3 The Transient Sensitivity Matrix D(i+1) for Fully Connected MNNs of the First Order.- 7.1.4 The Transient Sensitivity Matrix D(i+1) for Fully Connected MNNs of the Second Order.- 7.1.5 Non-Fully Connected MNNs.- 7.2 Autonomous Algorithms for Adjusting MNNs.- 7.2.1 Introduction.- 7.2.2 Neural Networks With a Single Output.- 7.2.3 Neural Network With Many Outputs.- References.- 8 Identification and Control of Dynamic Systems Using Multilayer Neural Networks.- 8.1 Identification of Dynamic Systems Using Multilayer Neural Networks.- 8.1.1. Linear Systems.- 8.1.2 Non-Linear Systems.- 8.1.3 The Structure of MNNs for Identification.- 8.2 Control of Dynamic Systems Using Multilayer Neural Networks.- 8.2.1 Linear Systems.- 8.3 Control of Non-Linear Dynamic Systems Using Multilayer Neural Networks.- 8.3.1 Example use of Neural Networks for MRAC.- References.- 9 The Cerebellar Model Articulation Controller (CMAC).- 9.1 Introduction to CMAC.- 9.1.1 Introduction.- 9.2 Data Storage and Learning Process in the CMAC.- 9.2.1 Introduction to the CMAC.- 9.2.2 The CMAC system.- 9.3 Albus’ Learning Algorithm.- 9.4 Modified Albus’ Algorithm.- 9.4.1 Modification.- 9.4.2 Special Features of the Modified Albus Algorithm.- 9.5 CMAC for Identification and Adaptive Control.- References.