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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 ) ( ) , ( ) , ( ) ( ˆ 1 1 t y q H t u q G q H t y θ θ θ θ − − − + = θ ϕ θ ) ( ) ( ˆ t t y T = or θ θ ϕ θ ) , ( ) ( ˆ t t y T = Linear Regression or Pseudo Linear Regression θ Regression Space u(t-1),u(t-2),… y(t-1),y(t-2)… ϕ Observed/known data Z t-1 1 ϕ 2 ϕ d ϕ Parameters to tune p y ˆ 1 ˆ 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 ϕ to y, let us consider the following function expansion: ) ( ˆ 1 ϕ α ∑ = = m k k k g y (1) where ) ( ϕ k g , k = 1,…, m , are basis functions and k α 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 γ β = , each of which covers only a local interval of axis x . See the figure below. ) , ; ( k k k x K g γ β = y The original nonlinear function ) ( x g y = x Varying only in a local area All the basis functions ) ( ϕ k g , k = 1,…, m are generated from a single mother function of a single input variable, i.e. univariate: ) , ; ( k k x K γ β ....
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- Spring '06
- Nonlinear Models, Nonlinear system, neural network, basis functions, Radial basis function network