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Application of Linearization and Approximation

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...

Linearization and Approximation

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...

Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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Residuals and Least-Squares Property

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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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The process of fitting the best-fit...

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Related Experiment Videos

Fuzzy basis functions, universal approximation, and orthogonal least-squares learning.

L X Wang¹, J M Mendel

¹Dept. of Electr. Eng. and Comput. Sci., California Univ., Berkeley, CA.

IEEE Transactions on Neural Networks

|January 1, 1992

Summary

Fuzzy systems can approximate continuous functions using fuzzy basis functions. An orthogonal least-squares algorithm designs these systems, improving control performance for complex systems like the ball and beam.

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Area of Science:

Computational intelligence
Control theory
Fuzzy logic systems

Background:

Fuzzy systems offer a powerful framework for modeling complex nonlinear systems.
Approximation capabilities of fuzzy systems are crucial for their practical applications.
Designing effective fuzzy systems often requires efficient learning algorithms.

Purpose of the Study:

To demonstrate the approximation capabilities of fuzzy basis function expansions.
To develop an orthogonal least-squares (OLS) learning algorithm for fuzzy system design.
To evaluate the performance of the OLS-designed fuzzy system in a control application.

Main Methods:

Utilizing the Stone-Weierstrass theorem to prove uniform approximation by fuzzy basis functions.
Developing an OLS learning algorithm to select significant fuzzy basis functions from input-output data.
Applying the fuzzy basis function expansion to approximate a controller for the nonlinear ball and beam system.

Main Results:

The study proves that fuzzy basis function expansions can uniformly approximate any continuous function on a compact set.
The OLS algorithm effectively designs fuzzy systems by selecting significant fuzzy basis functions.
Simulation results indicate improved control performance for the ball and beam system using the proposed fuzzy controller.

Conclusions:

Fuzzy basis function expansions provide a robust theoretical foundation for fuzzy system design.
The OLS learning algorithm offers an efficient method for constructing accurate and parsimonious fuzzy systems.
Incorporating common-sense fuzzy control rules enhances the performance of fuzzy controllers in practical applications.