Continuous Stirred Tank Reactor Problem Research Articles

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.

Optimized Fuzzy-Based Wavelet Neural Network Controller for a Non-Linear Process Control System

One of the common non-linear process control problems is the continuous stirred tank reactor problem and the reactor is applied widely in chemical process industries. It is of high importance to develop a suitable controller scheme for the concentration and temperature control of the considered non-linear continuous stirred tank reactor (CSTR) model. In this paper, a novel wavelet neural network (WNN) controller model is developed to carry out the control action and to meet the performance requirements of the system model. The developed wavelet neural network model is tuned for its weights employing the proposed deterministic grey wolf optimizer algorithm and its input is fed from the Mamdani fuzzy model. The optimal number of rules to be formulated for the fuzzy rule base model is computed using the deterministic grey wolf optimizer (DGWO) which is the variant of the classic grey wolf optimizer (GWO). WNN controller devise, based on the inputs from the fuzzy model and weights from DGWO, is designed to perform control action on the non-linear CSTR model. The concentration and temperature control is carried out by the proposed controller and the responses are obtained. Also, the respective time response specifications are evaluated for the developed WNN controller. Simulation experiments prove the effectiveness of the proposed controller for the considered non-linear CSTR model in comparison with that of the existing controllers from the literature.

A stable class of modified Newton-like methods for multiple roots and their dynamics

Abstract There have appeared in the literature a lot of optimal eighth-order iterative methods for approximating simple zeros of nonlinear functions. Although, the similar ideas can be extended for the case of multiple zeros but the main drawback is that the order of convergence and computational efficiency reduce dramatically. Therefore, in order to retain the accuracy and convergence order, several optimal and non-optimal modifications have been proposed in the literature. But, as far as we know, there are limited number of optimal eighth-order methods that can handle the case of multiple zeros. With this aim, a wide general class of optimal eighth-order methods for multiple zeros with known multiplicity is brought forward, which is based on weight function technique involving function-to-function ratio. An extensive convergence analysis is demonstrated to establish the eighth-order of the developed methods. The numerical experiments considered the superiority of the new methods for solving concrete variety of real life problems coming from different disciplines such as trajectory of an electron in the air gap between two parallel plates, the fractional conversion in a chemical reactor, continuous stirred tank reactor problem, Planck’s radiation law problem, which calculates the energy density within an isothermal blackbody and the problem arising from global carbon dioxide model in ocean chemistry, in comparison with methods of similar characteristics appeared in the literature.

Modified Optimal Class of Newton-Like Fourth-Order Methods for Multiple Roots

Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergence analysis that demonstrated the establishment of fourth-order convergence of the developed methods. It is interesting to note that some existing schemes are found to be the special cases of our proposed scheme. Numerical experiments have been performed on a good number of problems arising from different disciplines such as the fractional conversion problem of a chemical reactor, continuous stirred tank reactor problem, and Planck’s radiation law problem. Computational results demonstrates that suggested methods are better and efficient than their existing counterparts.

Discrete-Time Nonlinear Stochastic Optimal Control Problem Based on Stochastic Approximation Approach

In this paper, a computational approach is proposed for solving the discrete-time nonlinear optimal control problem, which is disturbed by a sequence of random noises. Because of the exact solution of such optimal control problem is impossible to be obtained, estimating the state dynamics is currently required. Here, it is assumed that the output can be measured from the real plant process. In our approach, the state mean propagation is applied in order to construct a linear model-based optimal control problem, where the model output is measureable. On this basis, an output error, which takes into account the differences between the real output and the model output, is defined. Then, this output error is minimized by applying the stochastic approximation approach. During the computation procedure, the stochastic gradient is established, so as the optimal solution of the model used can be updated iteratively. Once the convergence is achieved, the iterative solution approximates to the true optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, an example on a continuous stirred-tank reactor problem is studied, and the result obtained shows the applicability of the approach proposed. Hence, the efficiency of the approach proposed is highly recommended.

Design of an H∞ PI controller with given relative stability and its application to the CSTR problem

H? control theory has achieved a very high level, which is not followed by appropriate applications in industry. Some reasons are: synthesis of H? controllers is complex and based on the numerical Nevanlinna-Pick procedure, high order of controllers (higher than the order of the process) and sensitivity to deviation of controller coefficients (fragile controller). On the other hand, PI (PID) controllers are still dominant in industry, which places the problem of design of fixed structure controllers at the forefront. The mentioned problem is very difficult. This paper proposes a simple interactive procedure for design of H PI controllers with the presence of constraints (given the relative stability) based on D-decomposition. The catalogue of responses to references, suppression of disturbances and minimum of H? criteria of control is created. Selection of controllers, based on the catalogue, is left to the user. The application of H? PI controllers to a CSTR (continuous stirred-tank reactor) is demonstrated.

Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences

In this paper, we propose an output regulation approach, which is based on principle of model-reality differences, to obtain the optimal output measurement of a discrete-time nonlinear stochastic optimal control problem. In our approach, a model-based optimal control problem with adding the adjustable parameters is considered. We aim to regulate the optimal output trajectory of the model used as closely as possible to the output measurement of the original optimal control problem. In doing so, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated. During the computation procedure, the differences between the real plant and the model used are measured repeatedly. In such a way, the optimal solution of the model is updated. At the end of iteration, the converged solution approaches closely to the true optimal solution of the original optimal control problem in spite of model-reality differences. It is important to notice that the resulting algorithm could give the output residual that is superior to those obtained from Kalman filtering theory. The accuracy of the output regulation is therefore highly recommended. For illustration, a continuous stirred-tank reactor problem is studied. The results obtained show the efficiency of the approach proposed.

An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems

An iterative algorithm, which is called the integrated optimal control and parameter estimation algorithm, is developed for solving a discrete time nonlinear stochastic control problem. It is based on the integration of the principle of model-reality differences and Kalman filtering theory, where the dynamic integrated system optimization and parameter estimation algorithm are used interactively. In this approach, the weighted least-square output residual is included in the cost function by appropriately monitoring the weighted matrix. An improved linear quadratic Gaussian optimal control model, rather than the original optimal control problem, is solved. Subsequently, the model optimum is updated using the adjusted parameters induced by the differences between the real plant and the model used. These updated solutions converge to the true optimum, despite model-reality differences. For illustration, the optimal control of a nonlinear continuous stirred tank reactor problem is considered and solved by using the method proposed.

Structured Singular Valued based robust nonlinear model predictive controller using Volterra series models

A methodology is proposed for designing a robust nonlinear model predictive controller based on a Volterra series model with uncertain coefficients. A key benefit of using the Volterra series model is that it can be split into a nominal and an uncertainty model thus permitting the application of robust analysis tools. The controller is based on the on-line solution of a robust performance test based on a Structured Singular Value norm. The cost function of the controller can be formulated to account for manipulated variable movement weighting, manipulated variable constraints and a terminal condition. Finally, the proposed methodology is applied to a single-input–single-output continuous stirred tank reactor problem and to a multiple-input–multiple-output pH neutralization process.

LMI-based robust model predictive control and its application to an industrial CSTR problem

In this paper, robust model predictive control (MPC) is studied for a class of uncertain linear systems with structured time-varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the “worst-case” objective function over infinite moving horizon, subject to input and output constraints. A sufficient state-feedback synthesis condition is provided in the form of linear matrix inequality (LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor (CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed.

Application of Taylor transformation to nonlinear predictive control problem

A Taylor transformation technique for nonlinear process predictive control is presented. In this approach, a prediction model for nonlinear predictive control (NPC) is introduced, using the Taylor transformation technique, to make the NPC design problem similar to the linear predictive control problem. It is then applied to the continuous stirred tank reactor (CSTR) control problem. The stability of the proposed nonlinear predictive control scheme to the CSTR problem is also studied. The model mismatch can also be made to cope with the proposed scheme. The results are compared with those obtained using nonlinear model predictive control and a fuzzy adaptive controller. The comparison shows that the proposed method leads to feasible results for NPC research.