Integral Square Error Research Articles

An optimal second‐order sliding mode based inter‐area oscillation suppressor using chaotic whale optimization algorithm for doubly fed induction generator

AbstractThis paper suggests an inter‐area oscillation suppressor based on second‐order sliding mode controller (SOSMC) for doubly fed induction generator (DFIG) to suppress the inter‐area power system oscillations. SOSMC‐based suppressor targets at improving the reactive power control capability of DFIG to retrieve the stability of affected power system caused by fault occurrence into the stable condition. This controller has been initially designed for two‐area power system, after that it has been extended for multi‐area power systems. SOSMC‐based suppressor has no sensitivity to the uncertainty and variation of the parameter in all system operation conditions as compared with the conventional suppressors. Due to exploration and exploitation capability of chaotic whale optimization algorithm (CWOA), it is applied as main optimizer to optimally tune the suppressor's parameters and merge with the dynamic stability scheme. This optimization algorithm is beneficial for complex dynamic stability problem, because during the iterative procedure, it can enhance the exploration capability to find the best solution. The optimization scheme based on CWOA has been compared with the conventional whale optimization algorithm (WOA) and particle swarm optimization (PSO). The suppressor damping capability has been evaluated with consideration of some possible disturbance scenarios. Furthermore, the objective function has been carried out based on the prominent dynamic performance criteria such as integral of time multiplied absolute error (ITAE), integral of absolute error (IAE), integral of squared error (ISE), and integral of time multiplied squared error (ITSE). Considering the various operating points, the SOSMC‐based suppressor ameliorates the system robustness as compared to the conventional suppressor. At last, the simulation results extracted from both interconnected power systems confirm the enhancement of power system operability and performance in suppression of the inter‐area oscillations, and also validate the robustness of the suggested control strategy against the model uncertainties and unmodeled dynamics.

Conventional and Evolutionary Order Reduction Techniques for Complex Systems

Order reduction of the large-scale linear dynamic systems (LSLDSs) using stability equation technique mixed with the conventional and evolutionary techniques is presented in the paper. The reduced system (RS) is obtained by mixing the advantages of the two methods. For the conventional technique, the numerator of the RS is achieved by using the Pade approximations and improved Pade approximations, whereas the denominator is obtained by the stability equation technique (SET). For the evolutionary technique, numerator of the RS is obtained by minimizing the integral square error (ISE) between transient responses of the original and the RS using the genetic algorithm (GA), and the denominator is obtained by the stability equation method. The proposed RS retains almost all the essential properties of the original system (OS). The viability of the proposed technique is proved by comparing its time, frequency responses, time domain specifications, and ISE with the new and popular methods available in the literature.

Optimisation of fuzzy logic quadrotor attitude controller – particle swarm, cuckoo search and BAT algorithms

Bio-inspired optimisation algorithms have recently attracted much attention in the control community. Most of these algorithms mimic particular behaviours of some animal species in such a way that allows solving optimisation problems. The present paper aims at applying three metaheuristic methods for optimising fuzzy logic controllers used for quadrotor attitude stabilisation. The investigated methods are particle swarm optimisation (PSO), BAT algorithm and cuckoo search (CS). These methods are applied to find the best output distribution of singleton membership functions of the fuzzy controllers. The quadrotor control requires measured responses, therefore, three objective functions are considered: integral squared error, integral time-weighted absolute error and integral time-squared error. These metrics allow performance comparison of the controllers in terms of tracking errors and speed of convergence. The simulation results indicate that BAT algorithm demonstrated higher performance than both PSO and CS. Furthermore, BAT algorithm is capable of offering 50% less computation time than CS and 10% less time than PSO. In terms of fitness, BAT algorithm achieved an average of 5% better fitness than PSO and 15% better than CS. According to these results, the BAT-based fuzzy controller exhibits superior performance compared with other algorithms to stabilise the quadrotor.

Quantum deep reinforcement learning for rotor side converter control of double-fed induction generator-based wind turbines

The control performance of conventional analytic algorithms for double-fed induction generator-based wind turbines is a fixed feature, which needs to be optimized by optimization processes. To avoid the optimization processes and update control strategies online, this paper proposes an online control algorithm based on the quantum process, deep belief networks, and reinforcement learning for double-fed induction generator-based wind turbines. The proposed approach is named quantum deep reinforcement learning, consisting of deep belief networks, one reinforcement learning framework, and multiple subsidiary reinforcement learning parts with quantum processes. The quantum deep reinforcement learning can update the control strategy online with general initialization for dynamic systems, avoid optimal local solutions, and predict the next systemic states of double-fed induction generator-based wind turbines. The proportional–integral–derivative, fractional-order proportional–integral–derivative, active disturbance rejection controller, reinforcement learning, and the quantum deep reinforcement learning are compared under four cases, i.e., variable wind speed with step magnitude, variable wind speed with sine magnitude, voltage dropout of power grids, and both variable wind speed with random magnitude and voltage dropout of power grids. Consequently, the proposed quantum deep reinforcement learning can obtain better control performance for double-fed induction generator-based wind turbines with smaller integrated absolute error, integral squared error, and integrated time-weighted absolute error of the control error than other compared methods.

An enhanced feedback-feedforward control scheme for process industries

Abstract In this article, a unified control scheme is proposed for dead-time compensation and disturbance rejection via feedback and feedforward controller. The objectives of this work are suggested in two folds, first tuning of fractional order feedback controller via delayed Bode’s ideal transfer function instead of conventional Bode’s ideal transfer function with the benefits of dead time compensator and second feedforward controller for disturbance rejection. An existing method is utilized for comparison with the proposed scheme. To examine the efficacy of the proposed method robustness test is also carried out via sensitivity analysis. For quantifiable evaluation of the proposed scheme Integral Absolute Error (IAE) and Integral Square Error (ISE) are utilized. For the usefulness of the proposed scheme, two practical problems are demonstrated in this paper. The limpidity and instinctive appeal of the proposed scheme make it beautiful for industrial applications.

System identification and control of heat integrated distillation column using artificial bee colony based support vector regression

Distillation is a high-energy process widely employed in separating fluid mixtures in the oil and gas industries. Heat integration is one of the practical approaches for energy saving in the distillation columns. Proper identification or modeling of heat-integrated distillation column (HIDC) is employed to predict the composition of fluid mixtures. The nonlinear modeling of HIDC is highly challenging, and methods based on the first principles are not effective in coping with the nonlinearities. Hence, a novel, non-parametric support vector regression (SVR) approach is proposed for system identification and control of HIDC in this work. SVR parameters were optimized using artificial bee colony (ABC) algorithm, which resulted in better performance over other meta-heuristic algorithms. Moreover, the SVR model demonstrated better performance than the artificial neural network models in root mean square error (RMSE) and regression coefficient (R). RMSE and R values for ABC-SVR were found to be 0.0010 and 0.99992, respectively, with the validation dataset. The performance of the SVR and PID controllers are also compared. Integral square error (ISE), integral average error (IAE), integral time square error (ITSE), and integral time average error (ITAE) are the comparison metrics employed, which yielded minimal values of and respectively, for the SVR controller. The proposed model outperforms all other related methods, and it can be used to predict the mole fraction of Benzene in Benzene-Toluene HIDC accurately.

A Novel Control Scheme for PWM Boost Converter based on Sliding Mode Control using Particle Swarm Optimization

Background: The PWM Boost converter is a strongly nonlinear discrete system, especially when the input voltage or load varies widely, therefore, tuning the control parameters of which is a challenge work. Objective: In order to overcome the issues, particle swarm optimization (PSO) is employed for tuning the parameters of a sliding mode controller of a boost converter. Methods: Based on the analysis of the Boost converter model and its non-linear characteristics, a mathematic model of a boost converter with a sliding mode controller is built firstly. Then, the parameters of the Boost controller are adjusted based on the integrated time and absolute error (ITAE), integral square error (ISE) and integrated absolute error (IAE) indexes by PSO. Results: Simulation verification was performed, and the results show that the controllers tuned by the three indexes all have excellent robust stability. Conclusion: The controllers tuned by ITAE and ISE indexes have excellent steady-state performance, but the overshoot is large during the startup. The controller tuned by IAE index has better startup performance and slightly worse steady-state performance.

Characteristic of Fuzzy, ANN, and ANFIS for Brushless DC Motor Controller: An Evaluation by Dynamic Test

Brushless DC (BLDC) motors are the most popular motors used by the industry because they are easy to control. BLDC motors are generally controlled by artificial controls such as Fuzzy Logic Controller (FLC), Artificial Neural Network (ANN), and Adaptive Neuro-Fuzzy Inference System (ANFIS). However, the performance of the BLDC control system in previous studies was compared separately with their respective parameters, making it difficult to evaluate comprehensively. Therefore, in order to investigate the characteristic performance of Fuzzy, ANN, and ANFIS, this article provides a comparison of these artificial controls. Two scenarios of the dynamic tests are conducted to investigate control performance under constant torque-various speed and constant speed-various torque. By dynamic testing, characteristics of Fuzzy, ANN, and ANFIS can be observed as real applications. The testing parameters are: Settling Time, Overshoot and Overdamp (in the graph and average value), and then statistic performance are: Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Absolute Error (ITAE), and Mean Absolute Error (MAE). The test result in scenario 1 showed that the ANN has a better performance compared to other controllers with the MAE, IAE, ITAE, and ISE value of 31.3003; 105.6280; 208.0630; and 5,7289 e4, respectively. However, in scenario 2, ANN only has a better performance compared to other controllers on just a few parameters. In scenario 2, ANN is indeed able to maintain speed but it has a more ripple value than ANFIS. Even so, the ripple that occurs in ANN does not have too much value compared to the setpoint. Therefore, the MAE value of the ANN is smaller than the ANFIS (18.8937 of ANN and 28.4685 of ANFIS).

Two-loop controller design and implementations for an inverted pendulum system with optimal self-adaptive fuzzy-proportional–integral–derivative control

Since inverted pendulum systems (IPSs) are under-actuated systems, controlling these systems has become an important problem. Because of their irregular movements and being structurally unstable, the balancing of these systems poses a major problem. In this study, the dynamic model of a linear IPS has been obtained by using the Newton–Euler method and also conventional proportional–integral–derivative (PID), fuzzy logic (FLC), and self-adaptive fuzzy-proportional–integral–derivative (SAF-PID) control algorithms have been designed for the control of the system. The purpose of these designed controllers is to keep the arms of the IPS on the moving cart in a vertical position and bring the cart to the specified equilibrium position. In conventional control methods, unwanted oscillations have occurred during the movement of the cart. By using two-loop control methods, it is aimed to reduce these oscillations and to design a robust system. The results have been obtained using the conventional PID, FLC, and SAF-PID controllers. To increase the performance of the SAF-PID control type, the optimum value of the coefficients of the points where the legs of the membership functions touch were calculated using the firefly optimization algorithm. To conclude, the designed controllers were performed on computer simulation. The results are given graphically and numerically. Mean absolute error (MAE), mean squared error, integral absolute error IAE, and integral squared error have been compared and examined with each other and with the studies in the literature using performance criteria. As a result, the proposed control method was at least 26.31% more successful in the angular control of the pendulum and at least 9.26% better in the position control of the cart than the studies in the literature.

Nonlinear autoregressive-moving average-L2 (NARMA-L2) controller for multivariable ball mill plant

Abstract Mineral processing facilities concern an enormous amount of dynamically complex unit operations (due to nonlinearities), for instance ball mill system. Normally, these processes need multivariable controllers to smooth actions by designing for plant constraints such as deadtimes and dynamics interactions. The present work presents a comparison between a classical PI and nonlinear moving average autoregressive-linearization level 2 (NARMA-L2) controllers based on artificial neural network (ANN) for a ball mill system. The manipulated variables of this plant are the rotation velocity (Vr) and the feeding weight (Wf), while the controlled parameters are the hold up (HU) and the mass fraction under 45 μm (P45). The simulation was built in the MATLAB software (Simulink), comparing the actions of PI and NARMA-L2 controllers in the face of operational changes in specific regions (constraints). The performance of proposed controllers was verified by the integral of absolute error (IAE), integral of squared error (ISE), or the integral of time-weighted absolute error (ITAE). The results of simulation showed the validity of the model obtained and the control technique proposed in this paper, which contributes to studies of multivariate controller designs for ball mills with significant applications. Additionally, this paper brings a first hybrid approach (PI/NARMA-L2) with successful implementation described in the literature.

Regulation of Nonlinear Chemical Processes with Variable Dead Time: a Generalized Proportional Integral Controller Proposal

This work shows the development and application of a Generalized Proportional Integral (GPI) Controller based on a First Order Plus Dead Time (FOPDT) approximation for an industrial chemical process. GPI control is a relatively recent advancement in automatic control. Nowadays, several enhancements with the GPI technique come from integral reconstructions of the system states. Chemical engineering processes present numerous challenging control problems, including nonlinear dynamic behavior. GPI can become a new option to consider in industrial applications since, along with the arrival of Industry 4.0, there are many improvements in computers and automation architecture to implement controller algorithms. The new controller's functionality shows significant accessibility of mathematical and logical potential. A comparison between the GPI and a PID controller is made under the same conditions to evaluate their performance. After carrying out some tests, the GPI shows better performance and a smoother controller action when applied to the mixing tank with a variable delay than the PID. Performance indexes as Integral Square Error (ISE) for evaluating the Output Variable and Control Effort Total Variation (TVu) for evaluating the control action are used to measure performance. Finally, designing an appropriate controller like the GPI that recognizes and incorporates nonlinearities is required for chemical processes. Simulations were developed using Simulink-MATLAB.

An Improved Variable Kernel Density Estimator Based on L2 Regularization

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share a fixed bandwidth (scalar for univariate KDE and vector for multivariate KDE) in the fixed KDE (FKDE). In this paper, we propose an improved variable KDE (IVKDE) which determines the optimal bandwidth for each data point in the given dataset based on the integrated squared error (ISE) criterion with the L2 regularization term. An effective optimization algorithm is developed to solve the improved objective function. We compare the estimation performance of IVKDE with FKDE and VKDE based on ISE criterion without L2 regularization on four univariate and four multivariate probability distributions. The experimental results show that IVKDE obtains lower estimation errors and thus demonstrate the effectiveness of IVKDE.

Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems

The problem of control and stabilizing inherently non-linear and unstable magnetic levitation (Maglev) systems with uncertain equilibrium states has been studied. Accordingly, some significant works related to different control approaches have been highlighted to provide robust control and enhance the performance of the Maglev system. This work examines a method to control and stabilize the levitation system in the presence of disturbance and parameter variations to minimize the magnet gap deviation from the equilibrium position. To fulfill the stabilization and disturbance rejection for this non-linear dynamic system, the fractional order PID, fractional order sliding mode, and fractional order Fuzzy control approaches are conducted. In order to design the suitable control outlines based on fractional order controllers, a tuning hybrid method of GWO–PSO algorithms is applied by using the different performance criteria as Integrated Absolute Error (IAE), Integrated Time Weighted Absolute Error (ITAE), Integrated Squared Error (ISE), and Integrated Time Weighted Squared Error (ITSE). In general, these objectives are used by targeting the best tuning of specified control parameters. Finally, the simulation results are presented to determine which fractional controllers demonstrate better control performance, achieve fast and robust stability of the closed-loop system, and provide excellent disturbance suppression effect under nonlinear and uncertainty existing in the processing system.

Integration of genetic algorithm tuned adaptive fading memory Kalman filter with model predictive controller for active fault-tolerant control of cement kiln under sensor faults with inaccurate noise covariance

Reliable control of cement kiln under sensor faults is vital to ensure cement clinker quality. To accomplish this objective, an active fault-tolerant control (FTC) scheme is proposed for burning zone temperature control of cement kiln even under inaccurate measurements. FTC is achieved by integrating adaptive fading memory Kalman filter (AFMKF) in an already existing model predictive controller (MPC) framework. Residue from the Kalman filter is used to detect faults, and the model prediction replaces sensor measurements. Hence, there is a need for an accurate model and its associated noise covariance. Model accuracy is ensured by the online estimation of sensor noise covariance using the proposed AFMKF. Hyperparameters governing the AFMKF are tuned using the multi-objective genetic algorithm (GA), aiming to minimise the modelling and tracking error. Sensor measurement failure, bias, and stuck are considered for performance validation of the proposed FTC. Percentage recovery from fault is evaluated in terms of Integral square error (ISE) and energy utilisation compared with a conventional MPC. Experimental results illustrate the significant improvement in model accuracy and fault-tolerant behaviour using the proposed GA tuned AFMKF.

Control of electric machines using flower pollination algorithm based fractional order PID controller

In this work, fractional order proportional integral derivative controller (FOPIDC) is proposed and implemented for speed control of electric machines using nature-inspired metaheuristic optimization algorithm (MOA) named as flower pollination algorithm (FPA). Investigations are performed by considering the second-order transfer function of D.C Motor. FOPIDC constants KP, KI, KD, λ, and μ are optimized by considering FPA. The desired response of the electric machine is obtained by using tuned parameters of FOPIDC. After that, FOPIDC response using FPA is compared with PIDC response using FPA in the sense of time-domain specifications and performance indices (PI) like ‘integral of square error (ISE), integral of time-weighted square error (ITSE), integral of absolute error (IAE) and integral of time-weighted absolute error (ITAE)’. Finally, the desired response of electric machine using FPA based FOPIDC is compared with desired response of electric machine using firefly algorithm (FA) and particle swarm optimization (PSO) algorithm based FOPIDC.

Experimental validation of optimized fast terminal sliding mode control for level system

This research article presents a process control application of a single-input single-output (SISO) level control system using the combination of fast terminal sliding mode control (FTSMC) and optimization method. Non-dominated sorted genetic algorithm-ii (NSGA-ii); a modern optimization technique is used to optimized the parameters of FTSMC. Here, a comparative analysis of conventional sliding mode control (SMC), FTSMC, NSGA tuned FTSMC and NSGA-ii tuned FTSMC has being carried out through MATLAB/ Simulink. The performance indices such as integral absolute error (IAE), integral square error (ISE), and integration of weighted errors have been considered as the objective functions. The robustness of the controller is tested through real-time experimentation. The stability is obtained by using Lyapunov stability criteria. Simulation and experimental results show that NSGA-ii tuned FTSMC method outperforms the conventional methods and the error is converging to zero in a finite-time. Also, NSGA-ii tuned FTSMC provides better-estimated setpoint and disturbance rejection responses.

Optimal Fractional PID Controller for Buck Converter Using Cohort Intelligent Algorithm

The control of power converters is difficult due to their non-linear nature and, hence, the quest for smart and efficient controllers is continuous and ongoing. Fractional-order controllers have demonstrated superior performance in power electronic systems in recent years. However, it is a challenge to attain optimal parameters of the fractional-order controller for such types of systems. This article describes the optimal design of a fractional order PID (FOPID) controller for a buck converter using the cohort intelligence (CI) optimization approach. The CI is an artificial intelligence-based socio-inspired meta-heuristic algorithm, which has been inspired by the behavior of a group of candidates called a cohort. The FOPID controller parameters are designed for the minimization of various performance indices, with more emphasis on the integral squared error (ISE) performance index. The FOPID controller shows faster transient and dynamic response characteristics in comparison to the conventional PID controller. Comparison of the proposed method with different optimization techniques like the GA, PSO, ABC, and SA shows good results in lesser computational time. Hence the CI method can be effectively used for the optimal tuning of FOPID controllers, as it gives comparable results to other optimization algorithms at a much faster rate. Such controllers can be optimized for multiple objectives and used in the control of various power converters giving rise to more efficient systems catering to the Industry 4.0 standards.

Multivariate locally adaptive kernel density estimation

When the underlying density exhibits multiple modes with different scales and orientations, density estimators with locally adaptive smoothing parameters show substantial gains over those with fixed bandwidths. However, it is a concern that the local smoothing matrices may not be well parameterized, and the corresponding optimization problems will be difficult. In this paper, we build a more promising and practical algorithm. The local bandwidth factors are chosen through clustering, and the global smoothing parameter is achieved by optimizing the Asymptotic Mean Integrated Squared Error. Most importantly, our locally adaptive estimator involves optimizing a scalar rather than solving a costly multivariate optimization problem. Our method, which can also be applied to manifold density estimation, is an improvement and generalization of the binned version estimator of Sain.

Design, implementation, control and optimization of single stage pilot scale reverse osmosis process.

In this paper, a single-stage pilot-scale reverse osmosis (RO) process is considered. The process is mainly used in various chemical industries such as dye, pharmaceutical, beverage, and so on. Initially, mathematical modeling of the process is to be done followed by linearization of the system. Here a dual loop construction with a master and a slave is used. The slave uses the conventional proportional integral derivative (PID) with a reference model of the RO process and the master uses the fractional order proportional integral derivative (FOPID) with a real time RO process. The slave's output is compared with output of the real time RO process to obtain the error which is in turn used to tune the master. The slave controller is tuned using Ziegler Nichols method and the error criterion such as integral absolute error (IAE), integral squared error (ISE), integral time squared error (ITSE), integral time absolute error (ITAE) are calculated and the minimum among them was chosen as the objective function for the master loop tuning. Hence the tuning of the controller becomes a whole. Therefore two optimization techniques such as particle swarm optimization (PSO) and bacterial foraging optimization algorithm (BFO) are used for the tuning of the master loop. From the calculations, the ITSE had the minimum value among the performance indices, hence it was used as the objective function for the BFO and PSO. The best-tuned values will be obtained with the use of these techniques and the best among all can be considered for various industrial applications. Finally, the performance of the process is compared with both techniques and BFO outperforms the PSO from the simulations.

Dynamic modeling with experimental calibration for the syngas production from biomass fixed‐bed gasification

AbstractIn this article, a dynamic biomass gasification model was developed based on the hybrid peripheral fragmentation and shrinking‐core model. To improve the accuracy of syngas generation transient prediction, the chemical kinetic model was trained using global surrogate optimization techniques. The pre‐exponential corrections of kinetic reaction rates are calibrated under noncatalytic conditions, employing experimental transient data of syngas generation rate and compositions under different temperatures and gasifying agents. The Dynamic Coordinate search using Response surface model and the Gap Optimized Multi‐objective Optimization using Response Surfaces framework were employed as the numerical solvers for finding the global optimum solution of the pre‐exponential factors. The calibrated particle‐level kinetic models based on both single‐objective and multi‐objective approaches have been validated by experimental data in four different biomass gasification scenarios. The calibrated model shows an over 95% decrease in terms of integrated squared error (ISE)‐based model mismatch when compared with the original model.