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Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system
Indranil Pan, Saptarshi Das
TL;DR
AVR control must balance contradictory objectives such as tracking, disturbance rejection, robustness, and control effort. The paper compares fractional-order PIλDµ and PID controllers using chaotic multi-objective optimization and Pareto fronts, finding that neither controller dominates across all specifications. The preferred controller depends on the objective combination and control-cost considerations.
Problem
AVR controller design must address contradictory objectives, while single-objective formulations cannot represent the trade-offs among tracking, disturbance attenuation, robustness, and control effort.
Method
The paper compares PID and fractional-order PIλDµ controllers using a multi-objective optimization framework with Pareto-front analysis and fractional-order implementation by rational approximation.
Results
Neither controller is superior for all specifications: FOPID is better for set-point tracking and load disturbance rejection, whereas PID is better for set-point tracking with lower control signal.
Takeaways & Limitations
Controller selection should depend on the design trade-off, with PID preferred when control cost is expensive and different controllers favored in other objective combinations.
Takeaways & Limitations
The fractional-order operators require band-limited rational approximation, creating a trade-off between implementation complexity and achievable accuracy.
Abstract
from arXiv · showhide
In this paper, a fractional order (FO) PIλDμcontroller 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 PIλDμand PID controller. A comparative analysis is done with respect to the standard PID controller to demonstrate the merits and demerits of the fractional order PIλDμcontroller.
1. Introduction
AVR control must address disturbances, instability, and competing design objectives. The paper motivates fractional-order controllers and multi-objective optimization as ways to represent these trade-offs.
- AVRs regulate generator primary voltage, but disturbances can cause prolonged electrical oscillations and instability.
- Traditional, expert-system, stochastic, particle-swarm, genetic, bacterial-foraging, and chaotic-ant optimization methods have been applied to AVR controller tuning.
- Most prior AVR studies optimize a single objective, although practical design requires trade-offs among time-domain performance, robustness, and other criteria.
- Fractional-order PIλDµ controllers provide extra flexibility, and prior work reports advantages over PID for selected robustness or time-domain objectives.
- The paper organizes AVR modeling, fractional calculus, controller structure, contradictory objectives, chaotic multi-objective NSGA II, results, and conclusions into successive sections.
2. Description of the AVR system
The AVR model comprises amplifier, exciter, generator, and sensor subsystems represented by transfer functions. Their parameter values and ranges vary with the load.
- The AVR schematic contains amplifier, exciter, generator, and sensor models represented by subsystem transfer functions.
- Amplifier gain varies between 10 and 400, with a time constant between 0.02 and 0.1.
- Exciter gain varies between 10 and 400, with a time constant between 0.5 and 1.
- Generator gain varies between 0.7 and 1, with a time constant between 1 and 2; these constants depend on load.
- The selected nominal parameters are amplifier gain 10 and time constant 0.1, exciter gain 1 and time constant 0.4, generator gain 1 and time constant 1, and sensor gain 1 and time constant 0.01.
3. Basics of Fractional Calculus
The paper introduces Grunwald-Letnikov, Riemann-Liouville, and Caputo definitions for fractional differ-integration. It uses Caputo-based operators to realize the fractional controller.
- Fractional differentiation and integration are presented through the Grunwald-Letnikov, Riemann-Liouville, and Caputo definitions.
- 3.1. Grunwald-Letnikov (G-L) definition: The Grunwald-Letnikov definition extends backward finite differences and is widely used for numerical fractional differentiation or integration.
- 3.2. Riemann-Liouville (R-L) definition: The Riemann-Liouville definition extends successive integration and defines fractional differentiation through integer-order differentiation of a fractional integral.
- 3.3. Caputo definition: The paper identifies Caputo differentiation as the typical choice for deriving fractional-order transfer-function models from fractional-order differential equations with zero initial conditions.
- 3.3. Caputo definition: Caputo-based operators are used in the paper to realize the fractional integro-differential operators of the FOPID controller.
4. Fractional order PIλDµ controller structure and its time domain optimization
The PIλDµ controller extends PID with independently tunable fractional orders and gains, while frequency-domain approximation enables implementation. The design balances flexibility, approximation accuracy, and realization complexity.
- Under zero initial conditions, the fractional operator D^α is represented in the frequency domain as s^α, giving PIλDµ additional degrees of freedom.
- The FOPID structure has five independent tuning parameters: gains K_p, K_i, K_d and fractional orders λ, µ.
- The paper uses Oustaloup band-limited frequency-domain rational approximation to implement the fractional control law.
- A filter approximates the fractionally differentiated or integrated signal, with order (2N + 1) and a specified fitting frequency range.
- The study uses a 5th-order Oustaloup approximation over a band-limited frequency range, reflecting a trade-off between realization complexity and achievable accuracy.
5. Need for multi-objective optimisation and contradictory objective functions
Multi-objective optimization is needed because controller specifications such as tracking, disturbance attenuation, robustness, and control effort conflict and cannot be represented adequately by a single norm. The paper therefore evaluates contradictory objective combinations for fractional- and integer-order PID controllers.
- Why multi-objective optimization is required: Different design specifications, including disturbance attenuation, robust stability, and tracking, cannot be represented by a single norm, motivating multi-objective optimization.The cited discussion contrasts H2, H∞, and L1 criteria and emphasizes their distinct design domains and characteristics.
- Contradictory objective functions: The first objective case combines ITSE for set-point tracking with ISDCO for controller-output variation.ITSE targets fast tracking and settling, while ISDCO penalizes control-signal changes and large actuator effort.
- Contradictory objective functions: Reducing tracking error or obtaining faster tracking increases controller effort, so the tracking and control-signal objectives are contradictory.The paper states that minimizing J1 requires more controller effort, which increases J2, and vice versa.
- Contradictory objective functions: The second objective case simultaneously minimizes ITSE for set-point tracking and ITSE for load-disturbance rejection.J1 emphasizes tracking error and faster settling, whereas J3 minimizes deviation from the set-point during disturbances.
- Contradictory objective functions: The third case imposes the tracking, control-effort, and load-disturbance objectives together as three contradictory objectives.The paper combines J1, J2, and J3 to assess trade-offs among all three performance requirements.
6. Multi-objective chaotic Non-dominated Sorting Genetic Algorithm-II (chaotic NSGA-II)
The paper formulates controller design as a Pareto-based multi-objective problem and uses NSGA-II to rank and preserve diverse non-dominated solutions. Its chaotic variant replaces uniform random-number generation with a logistic map while optimizing controller parameters.
- Problem formulation: The generalized problem minimizes a vector of m objective functions over a decision space Ω mapped into an m-dimensional objective space.F: Ω→R^m contains the real-valued objectives being optimized.
- Pareto optimization: A Pareto-optimal solution cannot be improved in one objective without worsening at least one other objective.The associated Pareto set contains Pareto-optimal decision points, and the Pareto front contains their objective vectors.
- NSGA-II: NSGA-II addresses the complexity and non-elitism of earlier multi-objective evolutionary algorithms while improving solution spread and convergence near the Pareto front.It uses non-dominated sorting and crowding-distance assignment to rank and diversify solutions.
- NSGA-II: NSGA-II selects lower-front solutions first and uses higher crowding distance to retain sparsely populated regions when solutions share a front.This tournament rule combines convergence toward non-dominated fronts with niching across the front.
- Optimization variables: The fractional-order controller optimizes Kp, Ki, Kd, λ, and μ, whereas the integer-order PID controller optimizes only Kp, Ki, and Kd.The stated bounds are [0,100] for the gains and [0,2] for the differ-integral orders.
- Chaotic NSGA-II: The chaotic implementation replaces uniform random generation in NSGA-II with a logistic map constrained to produce values between 0 and 1.The selected initial condition is x0 = 0.2027 and the parameter is a = 4; unsuitable initial values can become constant after iterations.
7. Results and Discussions
The results compare PID and FOPID controllers across contradictory objective sets using Pareto fronts and representative simulations. Neither controller dominates universally: PID is preferable for J1–J2, whereas FOPID is superior for J1–J3 and the three-objective case depends on designer-assigned weights.
- Case I: J1 and J2: For Case I, the FOPID Pareto front lies inside the PID front, so PID outperforms FOPID across the considered J1–J2 cases.The fronts nearly merge at higher J1 values, where controller choice makes little difference.
- Case I: J1 and J2: FOPID representative solutions have higher overshoot and longer settling time than PID solutions, while control signals vary with J1.PID control signals are higher at lower J1 values, but PID performs better at higher J1 values.
- Multi-objective trade-offs: The Pareto front defines performance limits: improving one objective necessarily worsens another under the selected controller structure and conflicting objectives.Single-objective tracking could improve set-point response, but the resulting load-disturbance response would be worse than the multi-objective solutions.
- Case II: J1 and J3: For Case II, the FOPID Pareto frontier encloses the PID frontier and therefore outperforms PID for all considered J1–J3 cases.Representative solutions include extreme-end and median points, with FOPID solutions showing faster settling in the cited comparisons.
- Case III: J1, J2, and J3: In Case III, the PID and FOPID Pareto fronts intersect, so controller preference depends on the weights assigned to the three contradictory objectives.An interpolated surface is constructed from discrete Pareto solutions for visualization, with representative solutions reported in Table 3.
8. Robustness analysis of the obtained solutions
Robustness is assessed by varying generator parameters associated with load changes. Both controllers remain sufficiently robust, while FOPID provides more consistent time-domain performance at greater implementation complexity.
- Generator gain and time constant are varied to assess robustness under load-related parameter changes.The study restricts robustness analysis to these generator-transfer-function variations.
- Both PID and FOPID controllers offer sufficient robustness despite changes in system parameters.
- FOPID provides more consistent time-domain performance than PID under parameter changes.The comparison uses the respective median Pareto-front solutions.
- FOPID’s greater performance consistency comes with a more complex and potentially more expensive hardware implementation.The designer must weigh the additional performance against implementation cost and complexity.
- Median Pareto-front solutions provide average performance across contradictory objectives, while other solutions can be selected when one objective is prioritized.
9. Conclusions
The paper compares PID and FOPID controllers through multi-objective optimization for AVR systems. Neither controller dominates across all specifications; preference depends on the objectives and their importance to the designer.
- The proposed multi-objective framework compares PID and FOPID controllers for AVR systems.
- FOPID is better for the contradictory objectives of set-point tracking and load-disturbance rejection.
- PID outperforms FOPID for set-point tracking with lower control signal, favoring PID when control cost is expensive.
- When all three objectives are considered, the Pareto fronts intersect, so controller preference depends on the designer’s objective priorities.
- Future work is directed toward frequency-domain robust fractional-order controller design for AVR uncertainty.
Appendix
The appendix contains additional representative solutions for the three Pareto-front cases.
- Table 4 lists additional representative solutions on the Pareto front for Case I.
- Table 5 lists additional representative solutions on the Pareto front for Case II.
- Table 6 lists additional representative solutions on the Pareto front for Case III.