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Register a new userA Novel Approach for Complete Identification of Dynamic Fractional Order Systems Using Stochastic Optimization Algorithms and Fractional Calculus
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Added by
Deepyaman Maiti
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The said simultaneous equations are generated and updated using particle swarm optimization (PSO) technique, the fitness function being the sum of squared deviations from the actual set of observations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.
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This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.
The Proportional-Integral-Derivative Controller is widely used in industries for process control applications. Fractional-order PID controllers are known to outperform their integer-order counterparts. In this paper, we propose a new technique of fractional-order PID controller synthesis based on peak overshoot and rise-time specifications. Our approach is to construct an objective function, the optimization of which yields a possible solution to the design problem. This objective function is optimized using two popular bio-inspired stochastic search algorithms, namely Particle Swarm Optimization and Differential Evolution. With the help of a suitable example, the superiority of the designed fractional-order PID controller to an integer-order PID controller is affirmed and a comparative study of the efficacy of the two above algorithms in solving the optimization problem is also presented.
System identification refers to estimation of process parameters and is a necessity in control theory. Physical systems usually have varying parameters. For such processes, accurate identification is particularly important. Online identification schemes are also needed for designing adaptive controllers. Real processes are usually of fractional order as opposed to the ideal integral order models. In this paper, we propose a simple and elegant scheme of estimating the parameters for such a fractional order process. A population of process models is generated and updated by particle swarm optimization (PSO) technique, the fitness function being the sum of squared deviations from the actual set of observations. Results show that the proposed scheme offers a high degree of accuracy even when the observations are corrupted to a significant degree. Additional schemes to improve the accuracy still further are also proposed and analyzed.
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations.
Several approaches to the formulation of a fractional theory of calculus of variable order have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpis ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.
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