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Frequency Domain Design of Fractional Order PID Controller for AVR System Using Chaotic Multi-objective Optimization
Indranil Pan, Saptarshi Das
TL;DR
The paper addresses AVR controller design with contradictory performance objectives and compares fractional-order and conventional PID designs. It applies chaotic multi-objective NSGA-II in the frequency domain, finding that Henon-map augmentation produces better FOPID trade-offs than the classical and Logistic-map-assisted versions. The study also identifies a frequency-domain robustness advantage for FOPID designs.
Problem
AVR design must address contradictory performance objectives, while prior approaches often optimized time-domain criteria rather than frequency-domain trade-offs.
Method
The paper uses a Henon-map-augmented NSGA-II to design FOPID and PID controllers through frequency-domain multi-objective optimization.
Results
The Henon-map-augmented NSGA-II outperformed classical NSGA-II and Logistic-map-assisted NSGA-II, while FOPID produced better designs than PID for the considered objectives.
Takeaways & Limitations
Frequency-domain design exposes phase-margin and gain-crossover-frequency trade-offs and shows that FOPID can tolerate more gain variation than PID at the same phase margin.
Takeaways & Limitations
The AVR model omits nonlinearities, which the paper identifies as an important avenue for more realistic real-world modeling.
Abstract
from arXiv · showhide
A fractional order (FO) PID or FOPID controller is designed for an Automatic Voltage Regulator (AVR) system with the consideration of contradictory performance objectives. An improved evolutionary Non-dominated Sorting Genetic Algorithm (NSGA-II), augmented with a chaotic Henon map is used for the multi-objective optimization based design procedure. The Henon map as the random number generator outperforms the original NSGA-II algorithm and its Logistic map assisted version for obtaining a better design trade-off with an FOPID controller. The Pareto fronts showing the trade-offs between the different design objectives have also been shown for both the FOPID controller and the conventional PID controller to enunciate the relative merits and demerits of each. The design is done in frequency domain and hence stability and robustness of the design is automatically guaranteed unlike the other time domain optimization based controller design methods.
1. Introduction
AVR systems regulate generator terminal voltage to support reliable power-system operation, while controller design must balance multiple performance objectives. This paper motivates FOPID control and frequency-domain multi-objective optimization, using fractional-order operators realized through rational approximation.
- AVRs maintain alternator terminal voltage and support reactive-power distribution among parallel generators in power plants.
- Single-objective AVR controller tuning has used time-domain criteria, whereas practical designs must address multiple objectives simultaneously.Prior studies considered measures including integral absolute error, overshoot, rise time, settling time, steady-state error, and controller effort.
- The paper designs an FOPID controller with a chaotic Henon-map-augmented NSGA-II for contradictory frequency-domain objectives.The study compares FOPID and PID trade-offs rather than optimizing a single performance criterion.
- Fractional calculus and FOPID controller: Fractional calculus extends differentiation and integration to arbitrary real orders through the operator a_tD^α.Here, α is the differentiation or integration order, while a and t define the operation bounds.
- Fractional calculus and FOPID controller: FOPID controllers add fractional integration and differentiation orders as tuning parameters alongside proportional, integral, and derivative gains.These additional orders provide two extra tuning knobs beyond the conventional PID gains.
3. Multi-objective optimization framework for FOPID controller design
The framework motivates frequency-domain multi-objective design for the AVR FOPID controller, balancing speed, damping, robustness, and stability through Pareto optimization. It models the AVR with linear component transfer functions and uses effective unity-feedback frequency analysis.
- Requirement for frequency domain multi-objective controller design: Frequency-domain design is preferred because time-domain methods cannot explicitly quantify parametric robustness, while multiple norms capture different performance properties.H2 emphasizes disturbance stabilization, whereas H∞ emphasizes robust stability; neither alone addresses both properties.
- Description of the AVR system: The AVR model comprises amplifier, exciter, generator, and sensor dynamics represented by first-order transfer functions with parameter ranges.The linear model captures major time constants but neglects saturation and other nonlinearities.
- Description of the AVR system: The linear frequency-domain formulation omits nonlinearities because they complicate Bode analysis, especially for fractional-order controllers.The paper presents this formulation as a first step toward analytically tractable fractional-order frequency-domain results.
- Description of the AVR system: The controller uses an effective open-loop transfer function equivalent to the sensor-feedback AVR system with unity feedback for Bode, gain-margin, and phase-margin analysis.This transformation enables the reported frequency-domain stability analysis while preserving the represented closed-loop system.
- Conflicting objectives: Trade-off between gain crossover frequency and phase margin: Gain crossover frequency and phase margin are conflicting objectives: higher gain crossover frequency improves speed, whereas higher phase margin improves damping and robustness but can make the response sluggish.The paper therefore treats their trade-off as a multi-objective frequency-domain design problem.
- Multi-objective optimization framework: The optimization retains solutions with positive gain and phase margins, using penalties for violations because positive margins imply asymptotic stability for the linear system.NSGA-II ranks solutions by non-domination and crowding distance to approximate a diverse Pareto front.
Algorithm 1
Algorithm 1 begins by randomly generating N chromosomes within the feasible search space.
- Algorithm 1: N chromosomes are randomly generated within the feasible search space.
Step 1. [Start]
Step 1 initializes the optimization procedure by evaluating the multiple fitness values of each population solution.
- Step 1. [Start]: The algorithm calculates the multiple fitness values of each solution in the population.
Step 3. [Rank]
The algorithm ranks individuals using domination rank and crowding distance, then creates a new population.
- Step 3. [Rank]: Individuals are ranked using domination rank and crowding distance before a new population is created.
Step 4. [New Population]
The new population is generated by selecting parents, applying chaotic-map-guided crossover and mutation, and replacing the old population before testing termination.
- Step 4. [New Population]: Parents are selected from the previous population using the crowding selection operator.
- Step 4. [New Population]: A specified probability and chaotic map guide crossover to form new offspring.
- Step 4. [New Population]: Chromosomes are mutated using a chaotic map and specified probability to form additional offspring.
- Step 4. [New Population]: The new offspring are placed in the new population.
- Step 4. [New Population]: The old population is replaced by the new population, and the algorithm returns the Pareto set when termination criteria are satisfied.
- Step 4. [New Population]: Domination-rank assignment is required as part of the NSGA-II procedure.
Algorithm 2
Algorithm 2 assigns domination ranks by repeatedly identifying non-dominated individuals, assigning them a rank, removing them, and continuing until no individuals remain.
- Algorithm 2: The procedure finds non-dominated individuals in the current population and assigns them a rank.
- Algorithm 2: After ranking, the identified individuals are removed from the population and the procedure continues.
- Algorithm 2: The procedure stops when the working population is empty; otherwise, it returns to the next iteration.
- Algorithm 2: Crowding distance measures the relative density of solutions near a particular solution across multiple objective functions.
- Algorithm 2: The crowding-distance assignment is specified as an algorithm for the non-dominated solution set.
Algorithm 3
The crowding-distance procedure sorts solutions for each objective, uses boundary values, and accumulates normalized neighbor distances for interior solutions.
- Algorithm 3: For each objective function, the solution set is sorted in ascending order.
- Algorithm 3: Boundary solutions are identified using the maximum objective values.
- Algorithm 3: Interior solutions receive accumulated crowding-distance contributions from neighboring objective values.
- Algorithm 3: Crowding selection compares chromosomes using domination rank and crowding distance.
Algorithm 4
The chaotic Henon map supplies normalized values for the NSGA-II design process, which compares chaotic and classical variants while configuring the AVR controller optimization.
- Algorithm 4: The chaotic Henon-map approach is motivated by prior reports that chaotic sequences can improve evolutionary optimization results and convergence speed.
- Algorithm 4: The Henon map is used as a chaotic random-number generator whose output is scaled to [0,1].The first 200 scaled samples are shown in Figure 2.
- Algorithm 4: The Henon map is chaotic for a = 1.4 and b = 0.3 and is described as a simplified model of the Poincare section of the Lorenz system.
- Algorithm 4: The study compares frequency-domain FOPID tuning using classical NSGA-II, Henon-map augmentation, and Logistic-map augmentation.
- Algorithm 4: The chaotic Henon-map NSGA-II uses a population size of 200, 150 generations, and an elite count of 30.
- Algorithm 4: Controller gains are selected in [0,10], fractional orders in [0,2], and MATLAB code is coupled with fractional-order AVR transfer functions.
4. Results and discussions
The frequency-domain results compare PID and FOPID controllers across gain crossover frequency, phase margin, robustness, and optimization variants. FOPID provides superior trade-offs and robustness, while the chaotic Henon map yields the strongest optimization performance among the tested NSGA-II variants.
- Trade-off between frequency-domain objectives: At a 70° phase margin, the FOPID reaches approximately 70.8 radians/sec versus 8.4 radians/sec for the PID.This illustrates the FOPID’s higher achievable gain crossover frequency at the same phase-margin level.
- Trade-off between frequency-domain objectives: The FOPID Pareto front completely dominates the PID front for gain crossover frequency and phase margin.The comparison targets high control speed and accuracy through simultaneous maximization of both objectives.
- Trade-off between frequency-domain objectives: For comparable speed increases, the FOPID maintains higher phase margin, damping, and lower overshoot than the PID.Both controllers exhibit the trade-off that increasing gain crossover frequency reduces phase margin.
- Chaotic-map optimization comparison: The chaotic Henon-map NSGA-II produces the best FOPID Pareto front, with wide spread and high phase margin and gain crossover frequency.It outperforms both classical NSGA-II and the Logistic-map variant; changing crossover probability mainly improves low-speed, high-phase-margin regions, which are less desirable practically.
- Contributions: The study introduces frequency-domain multi-objective AVR tuning and compares FOPID with PID using gain crossover frequency and phase margin.The paper identifies these comparisons and the coupling of multi-objective evolutionary optimization with fractional-order frequency-domain design as novel contributions.
- Robustness analysis: The FOPID tolerates more system-gain variation than the PID, although derivative-kick oscillations increase as system gain rises.The PID develops high oscillations at an exciter gain of 120 and eventually becomes unstable, whereas the FOPID has faster settling and quickly decaying initial oscillations.
5. Conclusions
The paper compares frequency-domain PID and FOPID design trade-offs for AVR systems and reports advantages for FOPID. It also finds benefits from chaotic Henon-map assistance in NSGA-II.
- The study optimizes the trade-off between phase margin and gain crossover frequency using a multi-objective frequency-domain formulation.
- FOPID produces better designs than PID and can provide faster time response while tolerating greater system-gain variation at the same phase margin.
- The chaotic Henon map improves NSGA-II performance over both original NSGA-II and Logistic-map-assisted NSGA-II for FOPID design trade-offs.
Appendix
The appendix lists additional representative solutions for the frequency-domain multi-objective designs of both PID and FOPID controllers.
- Table 3 presents additional representative solutions for the frequency-domain multi-objective PID controller design.
- The appendix includes representative-solution tables for both conventional PID and fractional-order FOPID designs.
- Table 4 presents additional representative solutions for the frequency-domain multi-objective FOPID controller design.