Integral Squared Error Criterion Research Articles

Enhancing the Pitch-Rate Control Performance of an F-16 Aircraft Using Fractional-Order Direct-MRAC Adaptive Control

This study presents a comparative analysis of classical model reference adaptive control (IO-DMRAC) and its fractional-order counterpart (FO-DMRAC), which are applied to the pitch-rate control of an F-16 aircraft longitudinal model. The research demonstrates a significant enhancement in control performance with fractional-order adaptive control. Notably, the FO-DMRAC achieves lower performance indices such as the Integral Square-Error criterion (ISE) and Integral Square-Input criterion (ISU) and eliminates system output oscillations during transient periods. This study marks the pioneering application of FO-DMRAC in aircraft pitch-rate control within the literature. Through simulations on an F-16 short-period model with a relative degree of 1, the FO-DMRAC design is assessed under specific flight conditions and compared with its IO-DMRAC counterpart. Furthermore, the study ensures the boundedness of all signals, including internal ones such as ω(t).

A Method for Estimating the State of Charge and Identifying the Type of a Lithium-Ion Cell Based on the Transfer Function of the Cell

This paper proposes a new method for assessing the state of charge (SoC) and identifying the types of different lithium-ion cells used in the battery systems of light electric vehicles. A particular challenge in the development of this method was the SoC estimation time, as the method is intended for implementation in the control system of a bicycle charging station, where the state of charge must be determined immediately after the bicycle is plugged in in order to start the charging process as quickly as possible according to the appropriate charging algorithm. The method is based on the identification of the transfer function, i.e., the dynamic response of the battery voltage to the current pulse. In the learning phase of this method, a database of reference transfer functions and corresponding SoCs for a specific type of battery cell is created. The transfer functions are described by coefficients determined through the optimization procedure. The algorithm for estimating the unknown battery cell SoCs is based on the comparison of the measured voltage response with the responses of the reference transfer functions from the database created during the learning process to the same current signal. The comparison is made by calculating the integral of the square error (ISE) between the response of the specific reference transfer function and the measured voltage response of the battery cell. Each transfer function corresponds to a specific SoC and cell type. The specific SoC of the unknown battery is determined by quadratic interpolation of the SoC near the reference point with the smallest ISE for each battery type. The cell type detection algorithm is based on the fact that the integral squared error criterion near the actual SoC for the actual cell type changes less than the squared error criterion for any other battery cell type with the same SoC. An algorithm for estimating the SoC and cell type is described and tested on several different cell types. The relative error between the estimated SoC and the actual SoC was used as a measure of the accuracy of the algorithm, where the actual SoC was calculated using the Coulomb counting method.

Heat flow modeling in the alkaline activation process of fly ash

Cement production plays an important role in strengthening the infrastructure of growing countries such as Colombia. However, the production of this material has a high energy cost and contributes to the emission of large amounts of CO2. To address these environmental concerns, it is essential to explore alternative materials that can partially or completely replace traditional cement. Alkaline activated cement (AAC) has emerged as a promising candidate in this regard. Due to this, it is necessary to understand the process of alkaline activation and the variables that influence it. This research proposes a phenomenological-based semi-physical model, which predicts the performance of some variables that control alkaline activation: activator concentration (NaOH), heat flow, and degree of reaction. The model results show that with the increment of the activator concentration, the degree of reaction also increases. Furthermore, the model has an accurate response compared with the Freisleben-Hansen model. The integral square error criterion (ISE) was used in this comparison.

System identification based PSO approach for networked DC servo motor

Networked control system (NCS) suffers from inherent time delay and packet loss associated with any communication network. To build a controller that overcomes these issues, we need to know the accurate model of the system a priori. But, unfortunately most of the practical systems faced with lack in available and complete specifications of the physical system that helps in developing accurate mathematical model. This paper provides an experimentally identification method for any system based on Particle Swarm Optimization approach. The objective function to be minimized is based on the integral squared error criterion between the experimental and modeled trajectories. An armature-controlled DC motor with gearbox in lab is examined. The practical system trajectory represents the angular rotation of the output shaft. The DC motor system is modeled and controlled using fuzzy PID controller by the presented approach. Experimental results show that model equation has been successfully found with 94% accuracy. Also, a methodology for modeling the time delay in WNCS is presented.

Equilibrium optimizer for parameter extraction of a fuel cell dynamic model

The representative model of fuel cells (FCs) has a vital role in investigating the steady-state and dynamic analysis of these cells. The FC model has a significant impact on simulation studies of such systems and this is appeared in several applications like nano grids, microgrids and smart grids. This manuscript attempts a novel and simplified precise model of the proton electrolyte membrane FCs (PEMFCs). The proposed model deeply reduces the number of unknown parameters of such models to achieve a simplified one. In this regard, it reveals only four design variables within the model. Another salient feature of this study is the novel application of a meta-heuristic algorithm called equilibrium optimizer (EO) to obtain these unknown four parameters. The integral squared error criteria is used to formulate the fitness function. The effectiveness of EO-PEMFC model is tested by comparing the simulation with experimental results under various operating situations. The validity of proposed model is also verified by benchmarking its results with that obtained using other optimization-based models. The high efficacy of this model is checked under both steady-state and dynamic operating conditions. With the application of the proposed model, a very precise PEMFC model can be implemented.

A new optimized fuzzy FOPI-FOPD controller for automatic generation control of electric power systems

In the rapidly expanding size and complexity of the electricity network, automatic generation control (AGC) is contemplated to be the most remarkable option for offering good quality electric power supply to the end users. An AGC system entails highly vigorous, competent and intelligent control technique to deliver a healthy power under stochastic nature of consumers’ power demand. Hence, in this paper, a hybrid fuzzy fractional order proportional integral-fractional order proportional derivative (FFOPI-FOPD) controller is proposed as a new expert control technique to tackle AGC profitably in isolated and interconnected multi-area power systems. A recently developed imperialist competitive algorithm (ICA) is utilized for the optimization of the output gains (KP/KP1/KI/KD) and other parameters such as order of integrator (λ) and differentiator (μ) of FFOPI-FOPD controller exercising integral of squared error criterion. The proposed technique is firstly implemented on 1-area thermal system, then to express its potential and extensibility, the work is extended to 2-area hydro-thermal and 3-area thermal power systems widespread in the literature. The eminence of the method is betokened by comparing the results with the various newly published control methodologies and FPI/FFOPI controller designed in the study via ICA in terms of minimum values of various error criteria and undershoots/overshoots/settling times of frequency and tie-line power deviations following a sudden load demand in an area. The sensitivity analysis substantiates that the suggested controller is robust and performs staunchly under the wide variations in the system parameters, random load pattern and in the company of physical constraints to produce more clean electricity.

Application of model-based design tool X2C in induction machine vector control

Open source software tools for model-based design of embedded control systems represent a new method for rapid development and testing of advanced control structures in modern electric drives. The new concept of one of those tools, X2C, is presented in this paper, and in order to evaluate its performance in the development of complex control systems, it is applied to implement induction machine vector control. Based on the simulation model of the drive developed in PLECS, the parameters of the controllers were determined. The algorithm for the vector control of the induction machine was developed in X2C, implemented on the digital signal processor and applied to the real system. The system was tested during the no-load acceleration, deceleration, and reversing of the motor, while measuring the electrical and mechanical variables. Finally, the quantitative comparison of the experimental results and the results obtained by the equivalent simulations, based on the Integral Squared Error criterion, revealed that these results were well matched. This finding suggested that the control system was successfully implemented, thereby confirming the effectiveness of X2C tool in this particular type of application. With this work it has been shown that a transition from simulation to actual environment is rapidly achieved, with a simple verification of implemented methods accompanying the process. Realization of this work is a step forward in the utilization of open source software packages for implementation of induction machine vector control with the purpose of rapid verification of simulation models.

Automatic generation control of two-area electrical power systems via optimal fuzzy classical controller

The interconnected large-scale power systems are liable to performance degradation under the presence of sudden small load demands, parameter ambiguity and structural changes. Due to this, to supply reliable electric power with good quality, robust and intelligent control strategies are extremely requisite in automatic generation control (AGC) of power systems. Hence, this paper presents an output scaling factor (SF) based fuzzy classical controller to enrich AGC conduct of two-area electrical power systems. An implementation of imperialist competitive algorithm (ICA) is made to optimize the output SF of fuzzy proportional integral (FPI) controller employing integral of squared error criterion. Initially the study is conducted on a well accepted two-area non-reheat thermal system with and without considering the appropriate generation rate constraint (GRC). The advantage of the proposed controller is illustrated by comparing the results with fuzzy controller and bacterial foraging optimization algorithm (BFOA)/genetic algorithm (GA)/particle swarm optimization (PSO)/hybrid BFOA-PSO algorithm/firefly algorithm (FA)/hybrid FA-pattern search (hFA-PS) optimized PI/PID controller prevalent in the literature. The proposed approach is further extended to a newly emerged two-area reheat thermal-PV system. The superiority of the method is depicted by contrasting the results of GA/FA tuned PI controller. The proposed control approach is also implemented on a multi-unit multi-source hydrothermal power system and its advantage is established by Correlating its results with GA/hFA-PS tuned PI, hFA-PS/grey wolf optimization (GWO) tuned PID and BFOA tuned FPI controllers. Finally, a sensitivity analysis is performed to demonstrate the robustness of the proposed method to broad changes in the system parameters and size and/or location of step load perturbation.

Integrating a superconducting magnetic energy storage system for intelligent control of LFC characteristic in multi-area power system

This work exhibits use of adaptive neural control configured genetically tuned PI controller based superconducting magnetic energy storage (SMES) system for enhanced load frequency control (LFC) operation for a three area power system. The suggested scheme for the SMES control is a distinct one, as it overcomes the shortcomings in the contemporary SMES control schemes with automatic adjustment of converter duty ratio. Control in each area is accomplished via a neural estimator and a neural controller online simultaneously. The neural estimator selects the control area dynamics over a functioning zone, this feature leads to pure adaptive control, with modified gain settings obtained by integral square error criterion (ISE). Simulation studies carried out on MATLAB are presented for the power system with steam reheat constraint and governor dead band nonlinearity, after carrying out necessary modelling exercise, which portrays a reduction of 65% and 50% overshoots in frequency and tie-power oscillations respectively.

Multiobjective robust PID controller design for various industrial processes

In this paper, Non-dominated Sorting Particle Swarm Optimization (NSPSO) is applied to design multiobjective ro- bust Proportional-Integral-Derivative (PID) controllers for various industrial processes such as distillation column plant, liquid level control system and DC servo motor control system. Multiobjective robust PID controller design problem is formulated by minimizing Integral Squared Error (ISE) and balanced robust performance criteria. The performance of various optimal PID controllers is compared in terms of ISE and robust criteria. For the purpose of comparison, NSPSO and Non-dominated Sorting Genetic Algorithm-II (NSGA-II) methods are considered. The performance of PID controllers obtained using NSPSO is better than that of NSGA-II. The results reveal that NSPSO yields better robustness and consistency in terms of integral squared error (ISE) and balanced robust performance criteria than NSGA-II for various industrial processes.

Dynamic stability study of an isolated wind-diesel hybrid power system with wind power generation using IG, PMIG and PMSG: A comparison

This paper presents a comparative study of reactive power control for isolated wind-diesel hybrid power system in three different cases with wind power generation by induction generator (IG), permanent-magnet induction generator (PMIG) and permanent-magnet synchronous generator. The synchronous generator (SG) is used with diesel engine set. A mathematical model of the system based on small signal analysis, is developed considering reactive power flow balance equations. The variable reactive power needed by the system is provided by a static synchronous compensator (STATCOM) when wind power generation is done by IG and PMIG. When permanent-magnet synchronous generator (PMSG) is used for wind power generation, the variable reactive power demand is fulfilled by a voltage source converter (VSC) which is on the load side. A new mathematical approximation model for VSC connected with PMSG is proposed such that the voltage source converter fulfills the increased reactive power requirement of load and also increases its active power equal to the increased input wind power. Proportional and integral (PI) gains of the STATCOM and VSC controllers are optimized using integral square error criterion (ISE). The dynamic responses of the system for small (1%) step increase in load reactive power with and without 1% step increase in input wind power are shown. The paper also shows the dynamic responses of the system for random step change in load reactive power plus random step change in input wind power. The MATLAB/SIMULINK environment is used for simulation.

Multi-objective robust PID controller tuning using two lbests multi-objective particle swarm optimization

In this paper, two lbests multi-objective particle swarm optimization (2LB-MOPSO) is applied to design multi-objective robust Proportional-integral-derivative (PID) controllers for two MIMO systems, namely, distillation column plant and longitudinal control system of the super maneuverable F18/HARV fighter aircraft. Multi-objective robust PID controller design problem is formulated by minimizing integral squared error (ISE) and balanced robust performance criteria. During the search, 2LB-MOPSO can focus on small regions in the parameter space in the vicinity of the best existing fronts. As the lbests are chosen from the top fronts in a non-domination sorted external archive of reasonably large size, the offspring obtained can be more diverse with good fitness. The performance of various optimal PID controllers is compared in terms of the sum of ISE and balanced robust performance criteria. For the purpose of comparison, 2LB-MOPSO, NSGA-II as well as earlier reported Riccati, IGA and OSA methods are considered. The performance of PID controllers obtained using 2LB-MOPSO is better than that of others. In addition, Hypervolume-based comparisons are carried out to show the superior performance of 2LB-MOPSO over NSGA-II. The results reveal that 2LB-MOPSO yields better robustness and consistency in terms of the sum of ISE and balanced robust performance criteria than various optimal PID controllers.

OPTIMALITY CHARACTERISTICS OF PI/PID CONTROLLERS: A COMBINED MIN-MAX/ISE INTERPRETATION

This communication addresses the tuning of PI and PID controllers on the basis of the IMC approach. The tuning is based upon a first order plus time delay (FOPTD) model and aims to achieve a step response specification. Through analysis it has been found that by using the IMC approach we get a PI or a PID depending on the rational approximation used for the time delay term. This article raises the question that the use of a PID instead of a PI controller should be based on another reason more related to the control objectives rather than the use of a better approximation for the time delay. An alternative tuning is presented here, from within the IMC formulation, based on a min-max optimization. From the tuning rule provided by this approach the optimum settings from an integral squared error criterion point of view are derived. The optimal controller results in being a PI controller. From this optimal controller as the starting point, the introduction of the derivative action can be seen as a detuning procedure that can increase the robustness of the controller. This approach provides further insight into the tuning of PI and PID controllers giving the (alternative) parameters a precise engineering meaning.

PID controller tuning for integrating processes

Minimizing the integral squared error (ISE) criterion to get the optimal controller parameters results in a PD controller for integrating processes. The PD controller gives good servo response but fails to reject the load disturbances. In this paper, it is shown that satisfactory closed loop performances for a class of integrating processes are obtained if the ISE criterion is minimized with the constraint that the slope of the Nyquist curve has a specified value at the gain crossover frequency. Guidelines are provided for selecting the gain crossover frequency and the slope of the Nyquist curve. The proposed method is compared with some of the existing methods to control integrating plant transfer functions and in the examples taken it always gave better results for the load disturbance rejection whilst maintaining satisfactory setpoint response. For ease of use, analytical expressions correlating the controller parameters to plant model parameters are also given.

Control quality enhancement using fractional PIλDμ controller

The proportional–integral–derivative (PID) controllers have remained, by far, the most commonly and practically used in all industrial feedback control applications; therefore, there is a continuous effort to improve the system control quality performances. More recently Podlubny has proposed the fractional PIλDμ controller, a generalisation of the classical PID controller, involving an integration action of order λ and differentiation action of order μ. Since then, many researchers have been interested in the use and tuning of this type of controller. In this article, a new conception method of this fractional PIλDμ controller is considered. The basic ideas of this new tuning method are based, in the first place, on the classical Ziegler–Nichols tuning method for setting the parameters of the fractional PIλDμ controller for λ = μ = 1, which means setting the parameters of the classical PID controller, and on the minimum integral squared error criterion by using the Hall–Sartorius method for setting the fractional integration action order λ and the fractional differentiation action order μ. Illustrative examples and simulation results are presented to show the control quality enhancement of this proposed fractional PIλDμ controller conception method compared to the PID controller conception using Ziegler–Nichols tuning method.

A double two-degree-of-freedom control scheme for improved control of unstable delay processes

A double two-degree-of-freedom control scheme is proposed for enhanced control of unstable delay processes. The scheme is motivated by the modified Smith predictor control in [IEE Proc. Control Theory Appl. 16(5) (1999) 359] and devised to improve in the following ways: (i) one more freedom of control is introduced in our scheme to enable manipulation of disturbance transient response, and is tuned based on minimization of the integral squared error criterion; (ii) four controllers in the scheme are well placed to separately tune the denominators and numerators of closed-loop transfer functions from the set-point and disturbance. This allows easy design of each controller and good control performance for both set-point and disturbance responses. Especially, improvement of disturbance response is extremely great, compared with [IEE Proc. Control Theory Appl. 16(5) (1999) 359; Automatica 36 (2000) 1651]. Internal stability of the proposed structure is analyzed, which has not been reported in the literature on modified Smith predictor control before. Examples are provided for illustration.

パレート最適性を考慮したロバストＩ‐ＰＤ制御系設計法

In this paper, a new design method for robust I-PD controller by using Pareto-optimality is proposed. The proposed design method is formulated as a two-objective optimization problem in which optimization of the settling time and damping ratio is translated into a pole assignment problem. The uncertainties of the plant are represented as a polytope of polynomials, and the computational cost is reduced by using the edge theorem. The genetic algorithm is applied to optimize this problem because of its multiple search property. In order to demonstrate the effectiveness of the proposed design method, we applied the proposed design method to a magnetic levitation system and compared our method with the design method based on the generalized ISE (Integral of Squared Error) criterion.

Load frequency control of isolated wind diesel hybrid power systems

In this paper, a load frequency controller is designed for isolated wind Diesel hybrid power systems, and its effect on the transient performance of the system is evaluated. The modelling of the system, consisting of a wind turbine induction generator unit and a Diesel engine synchronous alternator unit, is presented along with the controller, both for continuous and discrete control cases. Optimum parameter values are obtained for different hybrid power system examples with continuous and discrete load frequency control and with and without blade pitch controllers, using the integral square error criterion (ISE). It is shown that wind Diesel or multi-wind Diesel hybrid power systems, with blade pitch control mechanism and with continuous load frequency control, have better dynamic performance than any other configuration of the isolated wind Diesel hybrid systems. Finally, some of the transient responses of the systems are shown for optimum gain settings.

Model reduction using new optimal Routh approximant technique

An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.

Computer-aided discretization of continuous data control systems

A computer-aided method is presented in this paper for optimum discretization of continuous data control systems using a time domain optimization technique. Digital controllers with different orders can be assumed and search for optimum values for their parameters is carried out by minimizing the value of the ISE (integral squared error criterion) between the output step responses of the continuous control system and that of the discretized control system. The minimization procedure adopted makes use of both Rosenbrock's automatic hill climbing technique (which optimizes the direction of search using Gram-Schmidt procedure) and the Fibanacci search method for step size optimization. The method is illustrated with numerical examples. Results show that step responses obtained using this method of discretization were very close to the desired responses as compared to other methods when one uses as the basis for comparison the integral squared error between step responses.