Convergence of fractional adaptive systems using gradient approach
Javier A Gallegos1, Manuel A Duarte-Mermoud1
1Department of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago, Chile; Advanced Mining Technology Center, University of Chile, Av. Tupper 2007, Santiago, Chile.
ISA Transactions
|May 10, 2017
Summary
This study derives conditions for stable fractional adaptive control systems using steepest descent. Fractional order adjustment offers advantages in adaptive control, demonstrated through experimental results in pole placement.
Area of Science:
- Control Engineering
- Applied Mathematics
- Systems Theory
Background:
- Fractional order systems offer advanced modeling capabilities.
- Adaptive control schemes require robust parameter and output error convergence.
- Steepest descent is a common optimization method in adaptive control.
Purpose of the Study:
- Derive conditions for boundedness and convergence in fractional order adaptive schemes.
- Introduce and characterize sufficiently exciting signals for fractional systems.
- Apply fractional order adaptive control to integer order systems for parameter adjustment.
Main Methods:
- Utilizing Caputo's fractional derivative in adaptive schemes.
- Applying the steepest descent method for parameter adaptation.
- Introducing and analyzing 'sufficiently exciting signals' for fractional systems.
Main Results:
- Established conditions for output and parameter error convergence in fractional adaptive schemes.
- Demonstrated the relationship between sufficiently exciting and persistently exciting signals.
- Successfully applied fractional order adaptive control for pole placement in an integer order system.
Conclusions:
- Fractional order adaptive schemes can achieve stable parameter and output error convergence.
- Sufficiently exciting signals are key for fractional adaptive system performance.
- Fractional adjustment provides experimental advantages in adaptive control applications.
Related Concept Videos
01:26Linear Approximation in Frequency Domain
409
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.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
409
01:24Approximate Integration
76
In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
76
01:21Convergence of Fourier Series
474
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
474
01:30Newton’s Method
84
Newton’s Method is a powerful iterative technique for approximating the roots of real-valued, differentiable functions, particularly when analytical solutions are impractical. This approach is widely used in scientific computing, engineering, and finance, where equations may be too complex for traditional algebraic methods to handle. The method relies on an iterative process that refines an initial estimate using the function’s derivative to approach the true solution progressively.
84
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
375
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
375
01:29Integration of Rational Functions Using Partial Fractions
169
Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
169