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Tuning PID and FOPID Controllers using the Integral Time Absolute Error Criterion
Deepyaman Maiti, Ayan Acharya, Mithun Chakraborty, Amit Konar, Ramadoss Janarthanan
TL;DR
Designing a fractional-order PID controller requires optimizing five parameters rather than the conventional PID set. The paper applies PSO to minimize ITAE and realizes the resulting controller with Tustin-based continued fraction expansion, reporting effective tuning and superiority of the fractional controller over the integer-order counterpart.
Problem
Fractional-order PID design requires selecting five parameters, making optimization more challenging than conventional PID controller design.
Method
The paper uses particle swarm optimization to minimize ITAE and digitally realizes the controller with a Tustin operator-based continued fraction expansion.
Results
The proposed scheme effectively tunes both integer-order PID and fractional-order PIλDδ controllers, with the fractional-order controller reported as superior.
Takeaways & Limitations
Fractional-order control provides a more flexible controller design than integer-order PID while retaining the classical controller as a special case.
Abstract
from arXiv · showhide
Particle swarm optimization (PSO) is extensively used for real parameter optimization in diverse fields of study. This paper describes an application of PSO to the problem of designing a fractional-order proportional-integral-derivative (FOPID) controller whose parameters comprise proportionality constant, integral constant, derivative constant, integral order (lambda) and derivative order (delta). The presence of five optimizable parameters makes the task of designing a FOPID controller more challenging than conventional PID controller design. Our design method focuses on minimizing the Integral Time Absolute Error (ITAE) criterion. The digital realization of the deigned system utilizes the Tustin operator-based continued fraction expansion scheme. We carry out a simulation that illustrates the effectiveness of the proposed approach especially for realizing fractional-order plants. This paper also attempts to study the behavior of fractional PID controller vis-a-vis that of its integer order counterpart and demonstrates the superiority of the former to the latter.
I. INTRODUCTION
Fractional-order PID controllers extend classical PID design with two additional order parameters, creating a five-dimensional optimization problem. The paper uses PSO to minimize ITAE while situating the approach within fractional-calculus theory and prior controller-design methods.
- Related work: Prior work includes digital and hardware realizations, frequency-domain and pole-distribution designs, and state-space feedback pole placement for fractional controllers.These approaches form the related-work context for the proposed optimization-based design.
- Motivation: Fractional-order PIλDδ controllers add integral and derivative orders λ and δ to the conventional PID parameters.The five adjustable parameters are Kp, Ti, Td, λ, and δ.
- Motivation: Selecting the five controller parameters for a given plant requires real-parameter optimization over a five-dimensional space using the ITAE criterion.The design objective is to minimize the Integral Time Absolute Error.
- Optimization approach: The paper uses derivative-free particle swarm optimization because classical optimization techniques are unsuitable for the rough multidimensional objective surface.PSO is presented as simple to implement, computationally inexpensive, and effective.
- Paper organization: The paper introduces fractional calculus and controller fundamentals before reviewing PSO and presenting the controller-synthesis procedure.The paper concludes with the reported results and conclusions.
B. Basic Concept of PID and PIλDδ Controllers
PID control combines proportional, integral, and derivative actions in a feedback loop, while PIλDδ control generalizes the integral and derivative actions to non-integer orders. This expansion preserves classical PID as a special case while increasing design flexibility and complexity.
- PID control: A PID controller is a generic closed-loop feedback mechanism that computes corrective action from the error between a measured variable and its set point.The corrective signal is fed back to adjust the process.
- PID control: The PID algorithm weights present error, accumulated error, and error-rate information through proportional, integral, and derivative actions.Their superposition adjusts plant performance.
- Fractional-order PID: The PIλDδ controller replaces integer integral and derivative orders with adjustable powers λ and δ in the corresponding actions.Its controller equation retains the proportional, integral, and derivative structure while generalizing the orders.
- Fractional-order PID: Classical PID, PI, and PD controllers are special cases of the general PIλDδ controller.The fractional formulation expands integer-order design from a point to a plane.
C. Overview of Discretization Methods for Control Systems
Digital realization maps fractional-order operators from the s-domain to the z-domain using generating functions. The paper focuses on the Tustin operator followed by continued fraction expansion to obtain a discretized controller.
- s-to-z transformation: Fractional-order differentiators are transformed from the s-domain to the z-domain using a generating function.The realization must account for the retained history of past inputs.
- s-to-z transformation: Euler and Tustin are identified as established s-to-z schemes for obtaining fractional-calculus approximation coefficients.The paper uses the Tustin operator as its generating function.
- Continued fraction expansion: The Tustin-based approximation is processed through continued fraction expansion to produce the discretized result.This sequence supplies the digital realization used for the fractional controller.
r Z CFE
The continued-fraction-expansion realization represents fractional operators with numerator and denominator polynomials in z^-1. Approximation orders and sampling period determine the resulting discrete controller, while integer PID is recovered by setting λ = δ = 1.
- CFE representation: In the CFE realization, p and q specify approximation orders, while Pp and Qq are numerator and denominator polynomials.The polynomials are defined in the variable z^-1.
- CFE representation: The approximation commonly sets p = q = n, with T defining the sampling period.The denominator representation is summarized for selected approximation orders.
- Discrete controller: The discretized PIλDδ controller combines proportional action with discrete approximations of its fractional integral and derivative terms.The controller is expressed as Kp plus Ti wi(z) and Td wd(z).
- Discrete controller: The integer-order controller is obtained from the fractional formulation by setting λ = δ = 1.This provides a direct link between the fractional and classical discrete controllers.
III. REVIEW ON PARTICLE SWARM OPTIMIZATION
PSO searches a multidimensional parameter space with particles whose positions represent candidate solutions, updating positions and velocities iteratively until a stopping condition is met. The fittest final particle is taken as a possible solution.
- PSO uses a swarm of particles that move through a multidimensional search space over discrete time-steps.The population size remains constant, and each particle represents a candidate solution through its position vector.
- Each particle stores its current position, personal best, and global best, with lower objective-function values defining better solutions.
- Particle positions and velocities are updated recursively for each particle and search-space dimension.The update equations use algorithmic parameters including inertia, self confidence, swarm confidence, and random factors.
- Iterations continue until a preset maximum number of time-steps or a fitness tolerance is reached.
- The fittest vector in the final population is treated as a possible solution near the global optimum.
IV. DESIGN OF THE CONTROLLER
The controller-design procedure defines the optimization objective and encodes controller parameters for PSO, whose best particle supplies the optimized parameters.
- PSO controller tuning designs a fitness function, encodes the parameters to be searched, and uses the best particle’s position as the optimized parameter set.
A. The Parameters to be Optimized
The fractional-order controller requires five tuned parameters, whereas the integer-order PID requires three, so PSO searches spaces of corresponding dimensions. The experiments use specified particle, inertia, coefficient, and initialization settings.
- The PIλDδ controller is tuned over five parameters: {Kp, Ti, Td, λ, δ}.Each PSO particle therefore has five-dimensional position and velocity vectors.
- The integer-order PID controller is tuned over three parameters: {Kp, Ti, Td}.
- The PSO population contains 10 particles, with inertia decreasing linearly from 0.9 to 0.4 and c1 = c2 = 1.4.
- The experiments use the stated PSO dynamics, while parameter initialization ranges and position and velocity limits are summarized separately.
- The parameter-range tables distinguish the integral-order controller from the fractional-order controller.Table II covers the integral-order controller, while Table III covers the fractional-order controller.
C. Fitness Function
The controller-tuning fitness function is the ITAE criterion, evaluated over a finite horizon chosen so that later error is negligible. ITAE is presented as more selective and less oscillatory than IAE or ISE.
- The fitness function minimized by the design method is the ITAE performance criterion.
- ITAE is described as producing smaller overshoots and oscillations than the IAE and ISE indices.
- ITAE is identified as the most sensitive and selective among ITAE, IAE, and ISE, while ITSE is less sensitive and computationally inconvenient.
- The finite integration horizon is set to T = 10 seconds because integrating to infinity is impracticable and later error is assumed negligible.
D. Stop Criterion
The study stops PSO after 100 generations and uses the best particle to obtain optimized fractional- and integer-order controller parameters, then compares their responses and fitness histories.
- D. Stop Criterion: PSO was stopped after 100 generations, with the best particle providing the optimized controller parameters.The procedure specifies a maximum-generation stop criterion.
- D. Stop Criterion: The optimized fractional-order controller has Kp = 325.9739, Ti = 303.3286, Td = 389.4627, λ = 0.6022, and δ = 1.6188.
- D. Stop Criterion: The fractional-order controller’s best-particle fitness is 4.9094 × 10^-4, identified as the ITAE index.
- D. Stop Criterion: The optimized integer-order controller has Kp = 47.9222, Ti = 29.6641, and Td = 449.1112, with best-particle fitness 0.0979.
- D. Stop Criterion: The evaluation includes the plant’s open-loop step response, the controlled closed-loop response, error waveforms, and best-fitness variation across iterations.
V. CONCLUSIONS
The proposed scheme effectively tunes both integer-order PID and fractional-order PIλDδ controllers, with the reported responses displaying superiority for the fractional-order controller.
- V. CONCLUSIONS: The proposed scheme is reported to be highly effective for tuning both integer-order PID and fractional-order PIλDδ controllers.
- V. CONCLUSIONS: The reported time responses and best-fitness variations display superiority for the fractional-order controller.
- V. CONCLUSIONS: Future work proposes more stringent performance-index fitness functions and other stochastic optimization algorithms, including differential evolution and bacterial foraging optimization.