In this paper, an efficient technique is developed to examine the bounded-input bounded-output (BIBO) stability of linear time-invariant (LTI) systems. Our method is based on eigenvalue separation, traditionally relying on the matrix sign function. To simplify the intricate calculations, we propose a theorem linking the matrix sign function to the matrix hyperbolic tangent function. This reformulation reduces complexity, requiring at most two matrix exponentials and a matrix inversion. Additionally, we enhance the computation of matrix exponentials using a scaling and squaring technique that features pre-adjusted accuracy up to any desired level. Unlike conventional approaches, our method does not require the characteristic polynomial of the system, avoiding numerical errors that can lead to a misrepresentation of the system’s stability. The proposed criterion approximately achieves a 77-fold reduction in CPU time for a system of order 5, with this improvement increasing progressively, reaching a 1465-fold reduction at order 200, as confirmed by our analysis. Based on an uncertain parameter test conducted on 1000 random systems of order 10 subject to ±20% perturbation, our criterion achieved a Matthews correlation coefficient of unity, whereas the Routh–Hurwitz method attained a value of 0.5827. Several case studies, including a binary distillation column from chemical engineering and an enhanced Sallen-Key filter system from electronics engineering, are presented to illustrate how the criterion can be applied in practice.

ABSTRACTThis paper presents a novel design framework termed differential Extreme Learning Machine (DELM) for addressing nonlinear process dynamics in time series modelling. DELM is constructed via a single‐layer feed‐forward ELM network featuring a skip net topology. This innovative network is engineered to accurately assess nonlinear time series patterns utilizing an nth order Legendre polynomial activation and imposing constraints at the output layer. The DELM persistently monitors trends in streaming process data and adjusts dynamic model predictive control (DMPC) settings inside the feedback loop. The Adaptive Distributed Model Predictive Control (ADMPC) is engineered to provide optimal control responses that meet both local and global stability requirements. The efficacy of DELM‐driven DMPC is evaluated for reference tracking and disturbance rejection goals and compared with RELM‐based DMPC and model‐based adaptive MPC (AMPC). The DELM‐DMPC surpasses alternative methods by providing superior generalization, stability, and computational efficiency. Average performance accuracy of 95% is attained across the operational range, exhibiting superior computing speed relative to its controller counterparts.

The most common control philosophy followed in the chemical process industries is the SISO system using the conventional PID controller algorithms. One drawback is relying on models for both control and design work in Chemical process industries (CPI) is that many problems are very complex and accurate models are difficult, if not impossible to obtain. To overcome these problems, it will be helpful to apply techniques that use human judgment and experience rather than precise mathematical models, which in the major cases deduced from the linearization of the system and simplification hypothesis. The fuzzy logic systems are capable of handling complex, nonlinear systems using simple solutions. However, obtaining an optimal set of fuzzy membership functions is not an easy task. In this chapter a solution based on artificial intelligence is proposed to improve the control of a binary distillation column. The solution is based on the use of the advantages of both fuzzy logic and genetic algorithms. The fuzzy logic is used as a supervisory PI controller that is a simple PI controller that generally used in controlling distillation columns with parameters deduced from the fuzzy supervisor. The membership functions shape is deduced by using research algorithms based on hierarchical genetic algorithms. The results show that the Fuzzy supervisory PI controller provide an excellent tracking toward set point change.

Identification, Analysis and Implementation of Model Predictive Controller (MPC) for Binary Distillation Column

In this paper, a new multidimensional Takagi–Sugeno (T-S) identification technique is proposed for multivariable nonlinear systems. In this technique, multidimensional membership functions are designed using concepts from solid mechanics. The design of membership functions is carried out in multidimensional space, defining the principal axes from the eigenvectors of the inertia matrix, and it has the characteristic of dividing the space into regions with the same inertia. These regions are analyzed to define the center of gravity for each rule. Illustrative examples of multivariable nonlinear systems, such as a thermal mixing process and a binary distillation column, are selected to evaluate the effectiveness of the proposed method. The proposed method is compared with traditional T-S identification that uses one-dimensional membership functions and shows a reduction in the relative identification error and the algorithm execution time. Additionally, the proposed method prevents rules from being positioned outside the system’s range, thereby avoiding the generation of unnecessary rules.

The closed-loop system’sperformance in synthesizing modelpredictive control (MPC) heavily relies on the model used for prediction.In continuously operating plants, a linear model-based MPC is designedbased on the operating point’s linear model during the commissioningstage. However, if the plant requires significant transitions fromits normal operating point, the linear model-based MPC may not beeffective. Therefore, to maintain the MPC performance under changingnominal operating conditions, the model (deterministic and stochasticcomponents) needs to be updated to predict every sampling instant.This study focuses on designing an adaptive MPC (AMPC) scheme basedon the linear model estimated from the input–output perturbationdata under nominal operating conditions. The OBF–ARX (generalizedorthonormal basis filters with ARX structure) parametrizes the observer’sdynamic components. The proposed fixed and variable pole AMPC schemes’efficacy is demonstrated using a simulation study on a binary distillationcolumn and experimental evaluation studies on a benchmark two-tankheater setup. The efficacy of the proposed AMPC schemes in addressingboth servo and regulator problems has been demonstrated through simulationand experimental results. Specifically, these schemes have been shownto effectively track set points while simultaneously rejecting disturbances.These findings suggest that the AMPC schemes hold promise for usein a variety of applications in which precise control is required.

This paper proposes a 2-DOF preview feedforward sliding mode controller (PFFSMC), which has a combination of feedback and feedforward terms in the sliding surface, for controlling the multivariable process with dead time, for both set point tracking and disturbance rejection. The proposed controller is applied for the control of the Wood and Berry binary distillation column having the composition of top and bottom products. The performance of the 2-DOF PFFSMC is compared with that of the 1-DOF preview sliding mode controller. The proposed controller delivered better responses for set point tracking and for disturbance rejection. The performance of the proposed 2-DOF PFFSMC is compared with the other controllers reported in the literature and, it was found that there was a significant reduction in the settling time by more than 76%, without any overshoot in the closed loop response. The proposed controller was robust against ±5% parametric uncertainties in gain, time constant, and dead time when these uncertainties were considered alone and together. The stability of the proposed controller was verified using the Lyapunov stability criterion.

A liquefied petroleum gas plant facility (LPGPF) is a series of binary distillation columns used to separate natural gas into four alkanes: ethane, propane, butane, and pentane. The conventional distillation column design consists of three binary distillation columns and six heat exchangers to perform the process. Each heat exchanger consumes immense energy to heat up the reboiler and condense the distillate. There are several process technologies that can minimize distillation column energy consumption. In this research, a fully thermally coupled distillation column (FTCDC) was proposed to minimize energy consumption by reducing the number of heat exchangers and tray columns. An FTCDC has the capability to reduce capital expenditure, operational expenditure, and total annual cost (TAC). The complexity of the FTCDC arises from its process integration. In each column, the intersection composition depends on complex mass and energy balances at the column inlet and outlet and each tray. Process integration, including material recycling and heat recovery, increases the complexity significantly. Moreover, the decision variables are multi-intersection composition for each column to achieve optimum objective function, increasing the number and complexity of the computational load such that effective stochastic optimization algorithms are required. The proposed method was designed using a rigorous vapor liquid equilibrium (VLE) FTCDC model and incorporated with recent stochastic optimization algorithms, such as a genetic algorithm, particle swarm optimization (PSO), an imperialist competitive algorithm, and a duelist algorithm, to determine hydrocarbon composition in the FTCDC intersection. To increase the efficiency and effectiveness of the FTCDC optimization design, cloud computing was utilized. The result was compared with conventional methods such as Fenske-Underwood-Gilliland, a Fenske-Underwood-Gilliland modification, and VLE. The optimization objective function is to minimize TAC with hydrocarbon composition in the FTCDC intersection as decision variables. The optimization using the VLE-PSO method reduces TAC up to 26.28%. All designs were validated using a rigorous model with Aspen HYSYS commercial software. This study's primary goal is to improve the performance of FTCDCs using stochastic algorithms and cloud-based computing capacity. The large amount of computation is handled by cloud-based computing resources, enabling reliability and durability.

Abstract In this article, we address the integration of design and nonlinear model‐based control under uncertainty and structural decisions for naturally ordered structures. We propose an algorithmic framework to determine the optimal location of process units or streams over an ordered discrete set that operates in closed‐loop with a model‐based controller. The formulation corresponds to a mixed‐integer bilevel problem (MIBLP) that is transformed into a single‐level mixed‐integer nonlinear problem (MINLP) using a KKT transformation strategy. In our methodology, the integer decisions are partitioned into subsets called external variables, such that the MINLP is decomposed into an integer‐based master problem and primal subproblems with fixed discrete variables. The master and primal problems are solved using a Discrete‐Steepest Descent Algorithm (D‐SDA). We illustrate the discrete‐based methodology in a case study for a binary distillation column. The D‐SDA showed an improved performance compared to a benchmark continuous‐based formulation using differentiable distribution functions (DDFs).

PurposeThe purpose of this paper is to propose and validate a robust industrial control system. The aim is to design a Multivariable Proportional Integral controller that accommodates multiple responses while considering the process's control and noise parameters. In addition, this paper intended to develop a multidisciplinary approach by combining computational science, control engineering and statistical methodologies to ensure a resilient process with the best use of available resources.Design/methodology/approachTaguchi's robust design methodology and multi-response optimisation approaches are adopted to meet the research aims. Two-Input-Two-Output transfer function model of the distillation column system is investigated. In designing the control system, the Steady State Gain Matrix and process factors such as time constant (t) and time delay (?) are also used. The unique methodology is implemented and validated using the pilot plant's distillation column. To determine the robustness of the proposed control system, a simulation study, statistical analysis and real-time experimentation are conducted. In addition, the outcomes are compared to different control algorithms.FindingsResearch indicates that integral control parameters (Ki) affect outputs substantially more than proportional control parameters (Kp). The results of this paper show that control and noise parameters must be considered to make the control system robust. In addition, Taguchi's approach, in conjunction with multi-response optimisation, ensures robust controller design with optimal use of resources. Eventually, this research shows that the best outcomes for all the performance indices are achieved when Kp11 = 1.6859, Kp12 = −2.061, Kp21 = 3.1846, Kp22 = −1.2176, Ki11 = 1.0628, Ki12 = −1.2989, Ki21 = 2.454 and Ki22 = −0.7676.Originality/valueThis paper provides a step-by-step strategy for designing and validating a multi-response control system that accommodates controllable and uncontrollable parameters (noise parameters). The methodology can be used in any industrial Multi-Input-Multi-Output system to ensure process robustness. In addition, this paper proposes a multidisciplinary approach to industrial controller design that academics and industry can refine and improve.

Solution of the maximum distillate on a batch distillation column by a Pseudo-Newton method

Artificial neural network based identification of process dynamics and neural network controller design for continuous distillation column

A convex optimization approach to multi-objective design of two-stage compensators for MIMO linear systems

Mathematical programming can express competency concepts in a well-defined mathematical model for a particular Any system that runs is always be expected to experience faults in different ways. Any change in the physical state of numerous components, control machinery, as well as environmental factors, might result in these problems. In process industries, where prompt detection is crucial in maintaining high product quality, dependability, and safety under various operating situations, finding these flaws is one of the most difficult tasks. The goal of this project is to implement several machine learning techniques for fault identification and classification in a binary distillation column. A pilot binary distillation unit (UOP3CC) is utilized for this purpose. The set up is run under normal operating conditions and the real time data is collected. Three common faults namely reboiler fault, feed pump fault and sensor fault are introduced one at a time and the faulty data is collected. These data are then introduced in to different machine learning algorithms like Logistic Regression, KNN, Naive Bayes, Decision Tree, Gradient Boosting, X Gradient Boosting, SVC and Light Gradient Boosting for model development. 70% of the data samples used for training and 30% of data samples are used for testing. It is found the Decision tree algorithm gives the best accuracy possible with 99.9%. Using decision tree algorithm, fault classification is performed for different datasets and is found that the algorithm was able to classify accurately even for new untrained datasets.

The effectiveness of evolutionary algorithms (EAs) such as differential search algorithm (DSA), Real‐Coded genetic algorithm with simulated binary crossover (RGA‐SBX), particle swarm optimization (PSO), and chaotic gravitational search algorithm (CGSA) on the optimal design of cross‐coupled nonlinear PID controllers is compared in this paper. A cross‐coupled multivariable PID controller structure for the binary distillation column was developed with two inputs and two outputs. EA simulations are run to lower IAE using two stopping criteria, namely, maximum number of functional evaluations (Fevalmax) and Fevalmax plus PID parameter and IAE tolerance. Over 20 separate trials were used to compare the performances of several EAs using statistical measures such as best, mean, standard deviation of outcomes, and average calculation time. This article presents the design of a cross‐coupled nonlinear PID controller using single‐objective evolutionary algorithms. Using evolutionary algorithms (EAs) with a multicrossover strategy, the results achieved by various EAs are compared to previously reported results. The results of a multivariable cross‐coupled system clearly show that a single‐objective nonlinear PID controller performs better. Simulations further show that all four techniques evaluated are suitable for PID controller tweaking off‐line. However, only the single‐objective evolutionary algorithms are acceptable for online PID tuning due to their higher consistency and shorter computation time.

Separation technique with distillation column is an energy intensive process even though it is one of the most extensively used separation process in the petrochemical, chemical and agro-allied industries. Studies have proved that some other schemes of distillation columns other than the conventional columns could be more energy efficient. This has led to a number of different configurations of the columns. This study presents a review of second law of thermodynamics in determining and enhancing the effectiveness of energy usage in the distillation columns. The review covers binary, multicomponent and crude distillation columns and it is not limited to the conventional columns. It can be concluded that exergy analysis of the columns while providing a true analysis of the column efficiency can be used to improve the column’s energy efficiency. It is therefore imperative that process engineers should be armed with this tool to design and operate energy efficient columns. Keywords: Distillation, Exergy analysis, Efficiency, Thermodynamics.

Convex reformulations for self-optimizing control optimization problem: Linear Matrix Inequality approach

The paper considers the problem of maximum efficiency for the system of distillation columns. Columns in such systems are connected in parallel or sequential way. The mixture being separated is assumed to be close to ideal one. Authors parameterize the relationship between feed flow rate and heat duties of a steady‐state binary distillation column using two parameters: the reversible efficiency and the irreversibility coefficient. This relationship is later being used to solve the problems of optimal distribution of heat and feed flows within the system. The results obtained allow one to estimate minimum heat energy demand for distillation of the given feed flow, maximum performance, and efficiency of the system.

This paper deals with the fractional order multi-loop controller design for a distillation column. The idea is a generalization of the IMC-PID-FOF controller design method developed for monovariable systems to the distillation column model which is multivariable. The principle is based on the IMC paradigm and the choice of appropriate control configuration with minimum of interactions. The proposed method is illustrated with an example of a distillation column model taken from the literature.

The paper considers the limitation problem of the distillation column systems separating multicomponent mixtures with serial and parallel structures. The solution takes into account the irreversibility of processes. Using entropic balance conditions, the dependence of load on heat consumption is obtained for a binary distillation column. This dependence is parameterized through two characteristic coefficients–reversible efficiency and irreversibility factor. This dependence was used to solve problems of distribution of heat and raw material fluxes in parallel column structure and selection of optimal separation order in serial structure. The obtained results make it possible to estimate the minimum heat consumption for the separation of a given flow of raw materials, the maximum productivity, and efficiency of the system.