lecture_12 - 2.160 System Identification, Estimation, and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2.160 System Identification, Estimation, and Learning Lecture Notes No. 1 2 March 20, 2006 7 Nonlinear Models 7.1 Nonlinear Black-Box Models The predictor of a linear system: [ ] ) ( ) , ( 1 ) ( ) , ( ) , ( ) ( 1 1 t y q H t u q G q H t y + = ) ( ) ( t t y T = or ) , ( ) ( t t y T = Linear Regression or Pseudo Linear Regression Regression Space u(t-1),u(t-2), y(t-1),y(t-2) Observed/known data Z t-1 1 2 d Parameters to tune p y 1 y This linear regression or pseudo-linear regression can be extended to representation of a class of nonlinear function. To generate a nonlinear map from to y, let us consider the following function expansion: ) ( 1 = = m k k k g y (1) where ) ( k g , k = 1,, m , are basis functions and k is the corresponding coordinate. There are a number of Basis Functions that can be used for (1). They are classified into: Global basis functions } Varying over a large area in the variable space Representing global features Fourier series Volterra series 1 Local basis functions Neural networks Significant variation only in a local area Radial basis functions Wavelets Local basis functions are powerful tools for capturing local features and representing a nonlinear function with locally-tunable resolution and accuracy. Over the last few decades, local basis functions have been investigated extensively and have been applied to a number of system identification, learning, and control problems. We will focus on local basis functions for the following few lectures. 7.2 Local Basis Functions We begin with a problem to approximate a scalar nonlinear function, R x R y x g y = , ), ( , with a group of basis functions, ) , ; ( k k k x K g = , each of which covers only a local interval of axis x . See the figure below. ) , ; ( k k k x K g = y The original nonlinear function ) ( x g y = x Varying only in a local area All the basis functions ) ( k g , k = 1,, m are generated from a single mother function of a single input variable, i.e. univariate: ) , ; ( k k x K ....
View Full Document

Page1 / 8

lecture_12 - 2.160 System Identification, Estimation, and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online