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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 ....
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- Spring '06