Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system (1205.1765v2)

Abstract: In this paper, a fractional order (FO) PI{\lambda}D\mu controller is designed to take care of various contradictory objective functions for an Automatic Voltage Regulator (AVR) system. An improved evolutionary Non-dominated Sorting Genetic Algorithm II (NSGA II), which is augmented with a chaotic map for greater effectiveness, is used for the multi-objective optimization problem. The Pareto fronts showing the trade-off between different design criteria are obtained for the PI{\lambda}D\mu and PID controller. A comparative analysis is done with respect to the standard PID controller to demonstrate the merits and demerits of the fractional order PI{\lambda}D\mu controller.

• The paper demonstrates that fractional order PI5D5 controllers outperform standard PID in certain multi-objective scenarios, such as combined tracking and disturbance rejection.
• Multi-objective analysis reveals trade-offs, showing standard PID performs better for objectives like minimizing control effort alongside tracking due to its simplicity.
• Incorporating chaotic logistic maps into NSGA-II significantly improves the optimization algorithm's performance and convergence efficiency towards the Pareto front.

Chaotic Multi-Objective Optimization in the Design of Fractional Order PI^D" Controllers for AVR Systems

The paper explores the design of a fractional order proportional-integral-derivative (FOPID) controller, denoted as PI^D", for an Automatic Voltage Regulator (AVR) system, focusing on a multi-objective optimization framework to balance contradictory performance objectives. The paper employs a modified Non-dominated Sorting Genetic Algorithm II (NSGA-II), enhanced with a chaotic map, to improve solution variety and convergence efficiency. This approach allows the evaluation and comparison of the fractional order PI^D" controller against traditional PID controllers, highlighting both strengths and weaknesses of the fractional order approach.

Methodology and Approach

The AVR system is evaluated using a fractional calculus-based approach for control design, specifically leveraging three prominent definitions of fractional differentiation: Grunwald-Letnikov, Riemann-Liouville, and Caputo. The FOPID controller introduces new parameters through the fractional orders, offering increased flexibility over classical PID controllers, which is particularly beneficial for systems requiring nuanced control adjustments.

The paper outlines the application of NSGA-II for multi-objective optimization, where the sophistication of control systems requires a compromise between competing objectives, such as fast settling time and actuator size constraints. The authors introduce a chaotic logistic map to improve the NSGA-II algorithm's ability to avoid local minima and improve convergence over a set of Pareto-optimal solutions.

Key Findings and Implications

1. Superior Performance of FOPID in Certain Objective Sets: The paper identifies cases where the PI^D" controller outperforms standard PID controllers, especially in scenarios demanding the simultaneous optimization of set point tracking and load disturbance rejection. The results are evidenced by Pareto front analysis, which demonstrates that the FOPID controller offers better solutions when these objectives are considered.
2. Trade-offs Illustrated by Multi-Objective Optimization: Contrastingly, when minimizing control effort alongside set point tracking, the classical PID controller shows better performance due to its simplicity and ease of implementation. This underscores the necessity of multi-objective frameworks to properly evaluate trade-offs between different design criteria.
3. Effectiveness of Chaotic Maps in Optimization: Incorporating chaotic logistic maps in NSGA-II improves the algorithm's performance, offering denser and more efficient convergence to the Pareto front compared to the conventional algorithm. This enhancement demonstrates an innovative application of chaos theory to refine evolutionary algorithms, a point of interest for optimization in complex systems.

Practical and Theoretical Implications

The paper’s findings emphasize the importance of considering fractional order controllers in modern control systems, especially where multiple competing objectives exist. The paper suggests that while fractional order controllers provide significant advantages under certain conditions, their complexity and implementation cost should be weighed against these benefits. The successful integration of chaotic maps within NSGA-II also paves the way for further research into chaos-enhanced optimization techniques in other multidimensional and unstructured optimization problems.

Future Research Directions

The research opens up several potential avenues for further exploration. A prospective direction could be the extension of the proposed optimization framework to incorporate robustness against system uncertainties and modeling inaccuracies in a frequency domain context. Another intriguing area is the application of chaos theory in other evolutionary optimization techniques, which can be a considerable enhancement to computational intelligence methodologies. Such developments could lead to breakthroughs in the design and implementation of both fractional and integer order controllers in various complex systems.

In conclusion, this paper solidifies the role of fractional order controllers and chaotic optimization in expanding the boundaries of efficient control in automatic voltage regulation systems, fostering advancements in both theoretical research and industrial applications.