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The tubular chemical reactor is a industrial equipment widely used in the chemical or biochemical industry for production of various kinds of products. The mathematical model of such system is described by partial differential equations that are solved numerically. This article presents simulation results of the mean reactant’s temperature control inside the plug-flow tubular chemical reactor. The adaptive approach here is based on the recursive identification of the external linear model as a simplified mathematical representation of the originally nonlinear system. The control synthesis is based on the polynomial theory with the Pole-placement method and the spectral factorization. These methods are easily programmable and they also offers tuning of the controller. Used two degrees-of-freedom (2DOF) control structure divides the controller into two parts – the first in the feedback part and the second one in the feedforward part of the control loop.

Ball & Plate is a well-known concept and interesting example of an unstable process. Numerous types of Ball & Plate structure can be found and this paper tries to extend its potential of moving the whole plate in space by using an industrial robotic manipulator as the most flexible way to achieve this goal. The collaborative dual-arm robot YuMi from ABB is chosen for this task as the balance between precision and safety. The purpose of the paper is to investigate restrictions and boundaries of such solution, thus the model is identified and based on this identification is designed a controller, which is tested in a simulation environment.

Many processes in industry exhibit time-delay in their dynamic behavior. Time-delay is mainly caused by the time required to transport energy, information or mass, but it can be caused by processing time as well. The typical process with time-delay is a heat exchanger that is a specialized device that assists in the transfer of heat from one fluid to the other. This paper deals with design of universal digital controller algorithms for control of a great deal of processes with time-delay. The first algorithm is realized by the digital Smith Predictor (SP) based on polynomial approach — Linear Quadratic (LQ) method. The second algorithm utilizes Model Predictive Control (MPC) approach with the possibility of measurable disturbance compensation. Both control principles were tested by a real-time control of an experimental laboratory heat exchanger.

This paper proposes the design of linear quadratic (LQ) digital controller in Ball & Plate model in experimental environment. The non-linear mathematical model of Ball & Plate structure is presented and adequately linearized. Polynomial approach to controller design for two degrees of freedom (2DOF) controller structure is introduced as the main tool for determination of unknown parameters. This method requires placing poles of characteristic polynomial, which are semi-optimally determined using minimization of linear quadratic criterion. This criterion is minimized by spectral factorization with the aid of the Polynomial Toolbox for MATLAB. Experiments have proved that this type of controller is able to stabilize the ball in desired position on the plate, reject external disturbances and follow reference path without much effort. In addition, a simple maze was created on the plate to extend possibilities of the choice of reference signal. The algorithm is able to determine correct path through the maze and navigate the ball along this path.

This paper presents extensions for educational Ball & Plate model CE151 from Humusoft. Its purpose is to offer students a complex view on the issues with designing controllers for real systems as a real engineering problem. Students can design their own controllers and directly implement them to provided Simulink structure, which can be examined by them to better understand concepts of real plant control. A graphical user interface was also designed to quickly access important data and choose the reference value. As an example a linear quadratic (LQ) digital controller is also presented. It is designed for 2DOF controller structure and accordingly tested.

Most of the processes in the industry have nonlinear behavior. Control of such processes with conventional control methods could lead to unstable, suboptimal, etc., results. On the other hand, the adaptive control is a technique widely used for controlling of nonlinear systems. The approach here is based on the recursive identification of the external linear model as a linear representation of the originally nonlinear system. The controller then reacts to the change of the working point or disturbances which could occur by the change of the parameters, structure, etc. The polynomial synthesis together with the linear quadratic (LQ) approach is employed here for the controller synthesis. These techniques satisfy basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation. Resulted controller could be tuned with the choice of weighting factors in LQ approach. This work investigates the effect of these factors on control results. Proposed methods are tested on the mathematical model of the isothermal continuous stirred-tank reactor and simulated results are also verified on the real model of the continuous stirred tank reactor.

A virtualization is one of the very progressive fields in information technologies. It is widely used especially in the server virtualization because it offers better usage of hardware resources by using one physical hardware for various virtual host servers. The history of the virtualization starts nearly before fifty years and the development runs from the virtualization of the mainframe to the virtualization on the hardware level. The second IT field which uses a virtualization is a testing and an education which uses a specialized software for virtualization of the operating system. The task of this contribution is to compare a desktop virtualization software from the performance and the hard-disk drive usage point of view. The practical part is focused mainly on the comparison of the VMware Player and the Oracle VM VirtualBox as one of the best-known free software for the desktop virtualization.

The goal of this paper is to compare two identification methods – continuous-time and discrete-time. The continuous-time identification model is more accurate but not very suitable for on-line identification. This disadvantage was overcome with the use of differential filters. On the other hand, discrete-time identification model has is more suitable for identification but less accurate. Compromise can be found in the delta model as a special type of the discrete-time model parameters of which are related to the sampling period. The adaptive approach is based on the choice of the External Linear Model, parameters of which are identified recursively which satisfies the adaptivity of this system. Proposed control strategy was applied on the mathematical model of the Continuous Stirred-Tank reactor as a typical nonlinear lumped-parameters system used in the industry.

The hybrid adaptive control with a delta External Linear Model is presented in this work. The adaptive approach is based on the choice of the External Linear Model of the originally nonlinear system parameters of which are identified recursively during the control and parameters of the controller are recomputed according to results of this estimation too. Two methods, Pole-placement and LQ approach, are compared with similar results. Advantage of these methods is that they are easily programmable and the result can be affected by tuning parameters in both methods. The polynomial approach with 2DOF control configuration satisfies basic control requirements and moreover, it could suppress overshoots of the output variable. Proposed methods were tested by the simulation on a very complex nonlinear system represented by the continuous stirred tank reactor with cooling in the jacket.

The paper deals with the adaptive control of the nonlinear process represented by the Continuous Stirred Tank Reactor (CSTR). This type of reactor belongs to the class of systems with lumped-parameters. Mathematical model of this system consist of two nonlinear Ordinary Differential Equations (ODEs). The steady-state analysis uncovers two stable and one unstable operating points. Used adaptive approach is based on recursive identification of the system's parameters during the control. A polynomial approach used for the controller synthesis has satisfied control requirements and moreover, it could be used for systems with negative properties such as nonlinearity, non-minimum phase etc. The goal of this work is to compare control for two stable steady-states.

Chemical reactors are often characterized by highly nonlinear behaviour. In such cases, the use of conventional control strategies where controllers have fixed parameters can result in poor performance. Knowledge about the static and dynamic properties is a necessary condition for the design of a controller. Unfortunately, measurements on the real system are not always feasible. The only way as to how to obtain static and dynamic behaviour of these systems is through the use of simulations, i.e. experiments on their mathematical model. Steady-state analysis shows the behaviour of the system in the steady state, which can help with the choice of the optimal working point, i.e. the appropriate combination of the input variables, which results in maximal production with minimal cost. On the other hand, the dynamic analysis results in the suitable control strategy. The adaptive approach in this work uses the recursive identification and the polynomial synthesis with the pole-placement method.

This paper deals adaptive control of the real model represented by the Continuous-stirred tank reactor (CSTR). This type of reactor belongs to the class of systems with lumped-parameters. The chemical process inside the reactor is dilution of the salt with the clean water. The resulted mixture has specific conductivity, which is about to be controlled, depends on the content of the salt inside the reactant. Used adaptive approach is based on recursive identification of the system's parameters during the control. A polynomial approach used for the controller synthesis has satisfied control requirements and moreover, it could be used for systems with negative properties such as nonlinearity, non-minimum phase etc.

This paper deals with adaptive control of the Continuous Stirred Tank Reactor (CSTR) which is a typical member of nonlinear processes with lumped parameters. Since the controller is an adaptive one, parameters of the controller are estimated recursively during the control with different recursive least squares methods. A polynomial approach used for the controller synthesis has satisfied control requirements and moreover, it could be used for systems with negative properties such as nonlinearity, non-minimum phase etc. Delta models were used as a external linear model of the nonlinear process. There were used two control configurations with one and two degrees-of-freedom for controller design.

The paper presents the cascade control design of a tubular chemical reactor with an exothermic consecutive reaction. The control is performed in primary and secondary control-loops where the primary controlled output of the reactor is the concentration of the main reaction product and the secondary output is the mean temperature of the reactant. A common control input is the coolant flow rate. The controller in the primary control-loop is a P-controller with the gain calculated using simulated or measured steady-state characteristics of the reactor. The controller in the secondary control-loop is an adaptive controller. The proposed method is verified by control simulations.

This paper shows control results of simulations on mathematical model and real measurements on the real model of the water tank. The level in the tank is controlled via input volumetric flow rate through the proportional solenoid valve. The adaptive approach with the recursive identification was employed here for controlling of this system. The controller could be tuned by the choice of the pole position in the pole-placement method. The benefit of this paper could be found also with the verification of the obtained simulation results on the real model of the water tank which shows usability of this control approach although the system has nonlinear properties. Control measurements also show hysteresis of the proportional solenoid valve which was used as an actuator in the control.

The contribution shows benefits of using mathematical software Matlab and its add-on for graphical programming Simulink. There are shown two ways of using Simulink for simulating of the dynamic behaviour of the nonlinear system. The first method is used for those who have already done simulating program in Matlab and need to implement it inside Simulink scheme. The second option is to use special S-function used for simulink. Proposed methods are tested on the simulation of the dynamic behaviour of the mathematical model of the continuous stirred-tank reactor as a typical nonlinear equipment from the industry.

Unfortunately, the major group of the systems in industry has nonlinear behavior and control of such processes with conventional control approaches with fixed parameters causes problems and suboptimal or unstable control results. An adaptive control is one way to how we can cope with nonlinearity of the system. This contribution compares classic adaptive control and its modification with Wiener system. This configuration divides nonlinear controller into the dynamic linear part and the static nonlinear part. The dynamic linear part is constructed with the use of polynomial synthesis together with the pole-placement method and the spectral factorization. The static nonlinear part uses static analysis of the controlled plant for introducing the mathematical nonlinear description of the relation between the controlled output and the change of the control input. Proposed controller is tested by the simulations on the mathematical model of the continuous stirred-tank reactor with cooling in the jacket as a typical nonlinear system. © 2015 Jiri Vojtesek and Petr Dostal.

The modelling and simulation play a very important role in the industry where it can help with the description of the system and the choice of the optimal control strategy. This contribution is focused on the modelling and simulation procedure which usually precedes the design of the controller. The mathematical model is derived with the use of material balance and produces nonlinear Ordinary Differential Equation (ODE). The static analysis provides optimal working point and the dynamic analysis gives an overview about the behavior of the system. Mentioned procedure is tested on the real model of the water tank as a part of the process control teaching system PCT40 from Armfield. Results have shown that proposed mathematical model is accurate and can be used for the design of the appropriate controller.

An adaptive nonlinear control is modification of the classic adaptive control where the controller is divided into the dynamic linear part and the static nonlinear part. The dynamic linear part is constructed with the use of polynomial synthesis together with the pole-placement method and the spectral factorization. The static nonlinear part uses static analysis of the controlled plant for introducing the mathematical nonlinear description of the relation between the controlled output and the change of the control input. In this case, the output response could be tuned by the change of the closed-loop pole. The verification of the proposed control strategy was made by simulations on the mathematical model of CSTR with cooling in the jacket as a typical nonlinear system.

The paper deals with one possible control method of a tubular chemical reactor. The procedure is based on the decomposition of the controller on a nonlinear static part and an adaptive linear dynamic part. Derivation of the nonlinear static part uses inversion and subsequent polynomial approximation of simulated or measured steady-state characteristics of the reactor. The nonlinear part of the closed-loop composed of the nonlinear static part of the controller and nonlinear model of the reactor is then approximated by an external linear model with parameters recursively estimated via a corresponding delta model. The control law is obtained using the linear quadratic control theory.

The main aim of the paper is to investigate which technology-enhanced learning tools are used in European higher education, to what specific purposes and how intensively they are employed, and what costs are associated to them. The source of presented information is based on responses of 100 universities from 27 European countries to a "Learning Tools Survey", which has been created in Vienna University of Economics and Business Administration and which was distributed under the terms of the European Union's Sixth Framework Programme project Intercultural Learning Campus (iCamp).

The paper deals with continuous-time nonlinear adaptive control of a continuous stirred tank reactor. The control strategy is based on an application of the controller consisting of a linear and nonlinear part. The static nonlinear part is derived in the way of an inversion and exponential approximation of measured or simulated input-output data. The design of the dynamic linear part is based on approximation of nonlinear elements in the control loop by a continuous-time external linear model with directly estimated parameters. In the control design procedure, the polynomial approach with the pole assignment method is used. The nonlinear adaptive control is tested by simulations on the nonlinear model of the CSTR with a consecutive exothermic reaction.

Simulation has a big importance nowadays when the speed of computers grows rapidly and the price is on the other hand decreasing. The paper is focused on the simulation of the steady-state and dynamic behaviour of the nonlinear system represented by the tubular chemical reactor with the co-current and countercurrent cooling in the jacket. These analyses are then used for the choice of the suitable control strategy. The adaptive controller used here is based on the choice of the External Linear Model as a linear representation of the originally nonlinear system parameters of which are identified recursively. The polynomial method together with spectral factorization and pole-placement method used in control synthesis satisfies the basic control requirements. The resulted hybrid controller provides good control results although the system has strongly nonlinear behaviour.

This paper deals with simulation studies of the adaptive control on the continuous stirred tank reactor (CSTR) as a typical chemical equipment with nonlinear behaviour and continuously distributed parameters. Mathematical model of this reactor is described by the set of two nonlinear ordinary differential equations (ODE). The simulation of the steady-state and dynamics results in optimal working point and external linear model (ELM) which is used in adaptive control. Adaptive approach here is based on the recursive identification of ELM and parameters of the controller are recomputed in each step too. The polynomial approach together with the poleplacement method and spectral factorization satisfies basic control requirements although the system has negative control properties. INTRODUCTION The most of the proceses in technical praxis has nonlinear behaviour and conventional control methods could lead to unoptimal or unwanted output responses. However, adaptive control (Astrom 1989) is one way how to deal with the negative control properties such as non-minimum phase behaviour, nonlinearity, time delays etc. There are several adaptive approaches. The one used in this work is based on the choice of an External Linear Model (ELM) parameters of which are recomputed recursively during the control (Bobal et al. 2005). It means that the steady-state and dynamic analyses need to be done for contructing of optimal ELM. The ELM could be chosen from the range of the continuous-time or discrete-time models. Disadvantage of the continuous-time ELM is difficult on-line recursive identification. On the other hand, external delta models (Middleton and Goodwin 2004) used here for parameter estimation belong to the range of discrete models but parameters of these models approach up to some assumptions to their continuous-time counterparts (Stericker and Sinha 1993). Well known and widely used Ordinary recursive least squares method (Fikar and Mikles 1999) was used for parameter estimation during the control. The polynomial method (Kucera 1993) used for controller synthesis together with the pole-placement method ensures basic control requirements such as stability, reference signal tracking and disturbance attenuation. Two control system configurations (Grimble 1994) were used – the first with one degreeof-freedom (1DOF) which has regulator only in feedback part and the second with two degrees-offreedom (2DOF) with feedback and feedforward parts. The resulting controllers are hybrid because polynomial synthesis is made for continuous-time but recursive identification runs on the delta-model, which belongs to the class of discrete-time models. The Continuous Stired Tank Reactor (CSTR) which is an equipment widely used in chemical industry for its good control properties. Mathematical model of this plant is described by the set of two nonlinear ordinary differential equations which are solved numerically (Ingham et al. 2000). The system has two suitable input variables – volumetric flow rates of reactant and cooling and two outputs – product’s temperature and concentration. The article provides several control simulation studies on the CSTR, all of them were done on mathematical simulation software Matlab, version 6.5.1. MODEL OF THE PLANT The controlled process here is represented by the continuous stirred tank reactor (CSTR) and its scheme can be found in Figure 1. Due to the simplification we expect that reactant is perfectly mixed and reacts to the final product with the concentration cA(t). The heat produced by the reaction is represented by the temperature of the reactant T(t). Furthermore we also expect that volume, heat capacities and densities are constant during the control. A mathematical model of this system is derived from the material and heat balances of the reactant and cooling. The resulted model is then a set of two Ordinary Differential Equations (ODEs) (Gao et al. 2002)

The paper deals with adaptive control of a tubular chemical reactor. As a part of the control design, preliminary steady-state and dynamic analysis of the process is incorporated. A nonlinear model of the process is approximated by a continuous-time external linear model with parameters estimated using a corresponding delta model. The controller design is based on the polynomial approach. The adaptive control is tested on the nonlinear model of the tubular chemical reactor with a consecutive exothermic reaction.

Abstract The goal of this paper is to propose suitable control methods for controlling of the highly nonlinear system represented by the mathematical model of the continuous stirred tank reactor (CSTR) with so called van der Vusse reaction inside. Temperature of the reactant is controlled by the heat removal of the cooling liquid in the reactor's jacket. Two control strategies were suggested – adaptive control and predictive control. The adaptive approach uses recursive identification for the optimal setting of the controller. The predictive control computes input sequence by the minimizing of the cost function constructed by the difference between output variable and reference signal. Both control strategies shows good control results and pertinence for the controlling of such type of systems.

Virtualization, VMware, VirtualBox, XEN, Virtual PC, VNC, NoMachine NX server, Paravirtualization, Operating systems, Nowadays, the virtualization technologies are more and more often used in many informatics areas. They are applied for better utilization of existing hardware or software resources, servers? consolidation and for security reasons as well. This article shows the possibility of the virtualization technology usage also in the academic environment as a support of laboratory exercises of two academic courses. The case study is based on the idea of virtualization of the physical computer for lower costs and security reasons. The VMware products are used nowadays in our university and the article want to show other, especially free, alternatives to this commercial product like VirtualBox, Microsoft Virtual PC, XEN etc. Described approaches and techniques can be applied to many others similar subjects, not only in these two courses.

This paper presents simulation experiments on the continuous stirred tank reactor (CSTR) which is widely used equipment mainly in the chemical industry. The behaviour of these types of systems is usually nonlinear with other negative properties such as a time-delay or a non-minimum phase behaviour and simulation could help us with the understanding of them without making real experiments which could be dangerous, time or cost demanding. The simple iteration method and the RungeKutta’s method were used for solving of a steady-state and dynamics of the system. Used adaptive control is based on the recursive identification of an External Linear Model (ELM) as a representation of the originally nonlinear system. The polynomial approach together with the LQ approach gives sufficient control results although the system has negative control properties. INTRODUCTION The simulation of the system on a computer enroll big boom nowadays when speed and availability of the computer technologies grows rapidly. On the contrary, the purchasing prise and running costs are relatively low. The computer simulation is usually connected with the mathematical model as a result of modeling procedure (Ingham et al. 2000). Material and heat balances are one way how to describe the system and relations between unknown quantities in the mathematical way. These balances are then represented by ordinary or partial differential equations depending on the type of systems. The Continuous Stirred Tank Reactor (CSTR) is typical equipment used in the industry for its good properties from control point of view. The CSTR belongs to the class of lumped parameters systems, a mathematical model of which is described by the set of ordinary differential equations (ODE). The simulation analysis of the system usually consists of steady-state and dynamic analyses (Ingham et al. 2000, Luyben 1989). The simple iteration method and Runge-Kutta’s method (Lyuben 1989) were used in the work for numerical solving of the steady-state and dynamic analyses. These methods are well known, simple and Runge-Kutta’s method is fully implemented in the used mathematical software Matlab. Results from simulation experiments are then used for choosing of the control strategy and designing of the controller. The nonlinearity and negative control properties of the system should be overcome with the use of Adaptive control (Astrom 1989). Adaptive approach used in this work is based the choice of an External Linear Model (ELM) parameters of which are recomputed recursively during the control (Bobal et al. 2005). The external delta models (Middleton and Goodwin 2004) were used for parameter estimation. Although delta models belong to the range of discrete models, parameters of these models approaches to their continuous-time counterparts up to some assumptions (Stericker and Sinha 1993). Ordinary recursive least squares method (Fikar and Mikles 1999) was used for parameter estimation during the control. A polynomial approach with one degree-of-freedom (1DOF) configuration used for the controller synthesis has satisfied basic control requirements and connected with the LQ control technique, it fulfills the requirements of stability, asymptotic tracking of the reference signal and compensation of disturbances (Kucera 1993). MODEL OF THE PLANT As it is written above, the chemical process under consideration is the Continuous Stirred Tank Reactor (CSTR). The schematical representation of the CSTR is in Figure 1. We supposed that reactant is perfectly mixed and react to the final product with the concentration cA(t). The heat produced by the reaction is represented by the temperature of the reactant T(t). Furthermore we expect that volume, heat capacities and densities are constant during the control due to simplification. A mathematical model of this system is derived from the material and heat balances of the reactant and cooling. The resulted model is then set of two Ordinary Differential Equations (ODEs) (Gao et al. 2002)

Simulation is very important and popular tool nowadays, when computation speed of computers increases exponentially every day. Simulations on mathematical models has several advantages over the experiment on a real model or system. It is save, cheaper and less time demanging. This paper delas with basic simulation studies on of the common used devices in chemical industry – Continuous Stirred Tank Reactor (CSTR). The mathematical model is developed from material balances. Numerical mathematics is used for steady-state analysis and dynamic analysis. Simulation results are used for choosing of an optimal working point and an external linear model of this nonlinear plant. INTRODUCTION There are several types of stirred reactors used in chemical or biochemical industry. Continuous Stirred Tank Reactors (CSTR) are common used because of their technological paramers. Reaction inside flows continuously and we can control this reaction by for example volumetric flow rate of the reactant (Ingham et al. 2000). The first step is introducing of the mathematical model which describes relations between state variables in the mathematical way. This mathematical model comes from material or heat balances inside the reactor. In our case of isothermal reactor with complex reaction (Russell and Denn 1972) is mathematical model the set of ordinary differential equations (ODE). Simulation usually consist of steady-state analysis which observes behaviour of the system in steady-state and dynamic analysis which shows dynamic behaviour after the step change of the input quantity (Ingham et al. 2000, Luyben 1989). This paper deals with simulation experiments on one type of nonlinear systems, CSTR reactor. This simulations results in optimal working point, external linear model and they will be later used for choosing of an optimal working point and mainly for control purpose. MODEL OF THE REACTOR Reactor under the consideration is Isothermal reactor with complex reaction (Ingham et al. 2000). Reactions inside the reactor can be described by following reactions

The role and importance of the simulation rises with the increasing speed of computers and simulation tools provided nowadays. Safety and less money and time demands gives a computer simulation big advantage over the experiments on a real system or its model. The paper deals with simulation of an adaptive control on a nonlinear system represented by a Continuous Stirred Tank Reactor (CSTR). This system is mathematically described by a set of Ordinary Differential Equations (ODE) which are first solved numerically to obtain steady-state and dynamic behaviour of the system. The adaptive control is based on the recursive identification of an External Linear Model (ELM) as a representation of the originally nonlinear system. The polynomial approach together with the pole-placement method gives sufficient control results although the system has negative control properties. INTRODUCTION The Continuous Stirred Tank Reactors (CSTRs) are often used in chemical or biochemical industry for their good control properties. The flow of reactant is fluent and it can be in most cases easily controlled by the volumetric flow rate (Ingham et al. 2000). This paper deals with adaptive control of the isothermal CSTR with complex reaction inside the tank (Russell and Denn 1972). This system is mathematically described by the set of five Ordinary Differential Equations (ODE) which are solved numerically via well-known Runge-Kutta’s standard method (Johnston 1982). The steady-state and dynamic analyses presented in (Zelinka et al. 2006) results in optimal working point and the choice of the External Linear Model (ELM) which represents the originally nonlinear CSTR. The polynomial approach together with a poleplacement method and spectral factorization are used for design of a controller. Parameters of the controller are periodically recomputed according to the parameters of the ELM which are estimated recursively during the control (Bobal et al. 2005). The ordinary recursive least squares method (Fikar and Mikles 1999) was used for the parameter estimation. All proposed control strategies were verified by and simulations in mathematical software MATLAB 6.5. MODEL OF THE PLANT The reactor under the consideration is an Isothermal reactor with complex reaction (Ingham et al. 2000). Reactions inside the reactor can be described by following relations

The paper shows one approach to control of a nonlinear process represented by an isothermal continuous stirredtank reactor. The hybrid adaptive controller designed with the use of a polynomial synthesis, LinearQuadratic approach and spectral factorization was used for controlling of the product‘s concentration via change of the volumetric flow rate of the reactant. An adaptivity of the system is satisfied by the on-line recursive identification of a external linear delta model of the system. The proposed control method satisfies basic control requirements such as a stability, a disturbance attenuation and a reference signal tracking. All methods are tested by the simulation on the mathematical software MATLAB. INTRODUCTION A major group of systems in the nature not only in the industry has a nonlinear behavior. The chemical rector is a typical member of nonlinear processes widely used in the chemical or the biochemical industry. The behavior of such processes could be observed by experiments on the real system or its smaller real model (Vojtesek and Dostal 2008). This method produce more realistic results but it could be dangerous or time and money demanding. The other approach uses modeling techniques for creating of a mathematical model as an abstract representation of the system. The mathematical model in the form of the set of Ordinary Differential Equations (ODE) is then subjected to simulations which show the static and the dynamic behavior of the system. The role of the simulation grows nowadays with the increasing speed and the decreasing price of computers. The control of these processes with the conventional controllers with fixed parameters could lead to the unstable, inaccurate or unwanted output response when the state of the system change or the disturbance occurs. The adaptive control (Astrom and Wittenmark 1989) is one way how we can solve these problems. This control method uses idea from the nature where plants or animals “adapt” their behavior to the actual state or environmental conditions. The adaptive controller adapts parameters or the structure to parameters of the controlled plant according to he selected criterion (Bobal et al. 2005). The adaptive approach here is based on the choice of the External Linear Model (ELM) as a linear approximation of the originally nonlinear system, parameters of which are identified recursively and parameters of the controller are recomputed according to identified ones. The choice and the order of the ELM come from the dynamic analysis. The δ-models (Middleton and Goodwin 2004) used here are special type of discrete-time (DT) models parameters of which are related to the sampling period. It was proofed, that parameters of the δ-model approach to parameters of the continuous-time (CT) model for the small sampling period (Stericker and Sinha 1993). The polynomial synthesis (Kucera 1993) together with the spectral factorization and the Linear-Quadratic (LQ) approach were used for designing of the controller. The product of this synthesis is the continuous-time controller which satisfies basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation. The resulted controller is called “hybrid” because it works in continuous-time but its parameters are recomputed in discrete time intervals together with the δ-ELM identification. The control technique was tested on the mathematical model of the isothermal Continuous Stirred-Tank Reactor (CSTR) the mathematical model of which is described by the set of five ordinary differential equations (Ingham et al. 2000). All results shown in this contribution come from the simulation on the mathematical model and they were done on the mathematical simulation software Matlab. ISOTHERMAL CHEMICAL REACTOR The nonlinear system under the consideration is an isothermal Continuous Stirred-Tank Reactor (CSTR). The schematic representation of this reactor is in Figure 1. The reactions inside the reactor could be described by the scheme

Tubular chemical reactor are units frequently used in chemical and biochemical industry. From the system theory point of view, tubular chemical reactors belong to a class of nonlinear distributed parameter systems. Their mathematical models are described by sets of nonlinear partial differential equations (PDR). The methods of modelling and simulation of such processes are described eg. in (Luyben 1989; Ingham et al. 1994; Severance 2001; Babu 2004). Relations between process behaviour and their control methods can be found in (Seborg et al. 1989; Ogunnaike and Ray 1994; Marlin 1995; Corriou 2004). It is well known that the control of chemical reactors, and, tubular reactors especially, often represents very complex problem. The control problems are due to the process nonlinearity, its distributed nature, and high sensitivity of the state and output variables to input changes. In addition, the dynamic characteristics may exhibit a varying sign of the gain in various operating points, the time delay as well as non-minimum phase behaviour. Evidently, the process with such properties is hardly controllable by conventional control methods, and, its effective control requires application some of advanced methods (e. g. Adaptive Control, Predictive Control, Robust Control or any others). However, at all events, a previous analysis of the process behaviour is obligatory. The paper presents all mathematical models used for simulations of both steady-state and dynamic charakteristics of the tubular chemical reactor together with results of some simulations. The combinations of observed variables are chosen in accordance with purposes of prospective control design. MODEL OF THE PLANT

The paper deals with simulation experiments on the nonlinear system represented by the isothermal continuous stirred tank reactor. At first, the mathematical model derived from the material balances inside the reactor will be introduced and then the steady-state and the dynamic analyses were performed on this model. As a result of these studies, the optimal working point and the choice of the external linear model for the identification will be obtained. The spectral factorization with pole-placement method and linear-quadratic approach were employed in the controller design and computation. Both types of adaptive controllers have parameters for tuning of the output response. Moreover, controllers have satisfied basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation.

The paper deals with simulation of continuous-time adaptive control of a continuous stirred tank reactor (CSTR). The control design is based on approximation of a nonlinear model of the process by a continuoustime external linear model.with parameters estimated using a corresponding delta model. The control system with two feedback controllers is considered. The controller design is based on the polynomial approach and the pole assignment method. The adaptive control is tested by simulations on the nonlinear model of the CSTR with a consecutive exothermic reaction. INTRODUCTION Continuous stirred tank reactors (CSTRs) belong to a class of nonlinear systems where both steady-state and dynamic behaviour are nonlinear. Their models are derived and described in e.g. (Ogunnaike and Ray 1994; Schmidt 2005; Corriou 2004). The process nonlinearities may cause difficulties when controlling using conventional controllers with fixed parameters. One possible method to cope with this problem is using adaptive strategies based on an appropriate choice of an external linear model (ELM) with recursively estimated parameters. These parameters are consequently used for parallel updating of controller‘s parameters. The control itself can be either continuous-time or discrete. Experiences of authors in the field of control of nonlinear technological processes indicate that the continuoustime (CT) approach gives better results when controlling processes with strong nonlinearities. For the CT ELM parameters estimation, either the direct method or application of an external delta model with the same structure as the CT model can be used (Dostal et al. 2004). In this paper, an approach based on the delta model was applied. The basics of delta models have been described in e.g. (Middleton and Goodwin 1990; Mukhopadhyay et al. 1992; Goodwin et al. 2001). Although delta models belong into discrete models, they do not have such disadvantages connected with shortening of a sampling period as discrete z-models. In addition, parameters of delta models can directly be estimated from sampled signals. Moreover, it can be easily proved that these parameters converge to parameters of CT models for a sufficiently small sampling period (compared to the dynamics of the controlled process). Complete description and experimental verification can be found in (Stericker and Sinha 1993). This contribution presents in brief a procedure of a CSTR control design and, consequently, the control simulation results. The parameters of the CT ELM are obtained via corresponding delta model parameter estimation. The control system with two feedback controllers is used according to (Ortega and Kelly 1984; Dostal et al. 2007). This set-up gives better control results for the reference tracking than the use of only a feedback controller. Input signals for the control system are considered as step functions. The resulting continuous-time controllers derived using polynomial and pole assignment methods (Kucera 1993) guarantee stability of the control system, asymptotic tracking of step references and step load disturbances attenuation. The simulations are performed on a nonlinear model of the CSTR with a consecutive exothermic reaction.

Continuous Stirred Tank Reactors (CSTR) are one of the main technological plants used in chemical and biochemical industry. These systems are quite complex with many nonlinearities and the conventional linear control with fixed parameters can be questionable or sometimes unacceptable. The solution should be found in so-called “non-traditional” control approaches like adaptive, robust, fuzzy, or artificial intelligent methods. One way is the utilization of self-tuning adaptive schemes, but computations may be quite difficult, clumsy and time-consuming. This paper brings an alternative principle called a robust approach and the comparison of the robust and adaptive control responses. Robust control considers a CSTR model as a linear system with parametric uncertainty, which covers a family of all feasible plants. Then several controllers with fix parameters are designed so that for all possible plants, the acceptable control behavior is obtained. The two-degree-of-freedom (2DOF) structure for the control law was chosen. Both robust and adaptive control is applied to an original nonlinear model of a CSTR. All calculations and simulations of mathematical models and control responses were performed in the Matlab and Simulink environment.

The paper presents simulation results of the cascade control of a continuous stirred tank reactor. The control is performed in primary and secondary control-loops where the primary controlled output of the reactor is the concentration of a desired reaction product, and, the secondary output is the reactant temperature. A common control input is the coolant flow rate. The controller in the primary control-loop is a P-controller with the gain calculated using simulated or measured steady-state characteristics of the reactor. The controller in the secondary control-loop is an adaptive controller. The proposed method is verified by control simulations INTRODUCTION The cascade control method allows the control of processes with a main and secondary controlled variable and with a single control input. The method is especially useful when a main controlled output can be measured only in longer time intervals and with an additional output measurable in shorter time periods. Principles of the cascade control are described e.g. in (Bequette 2006; Mahoney et al. 2006; Seborg et al. 1989; Smuts 2011). Chemical reactors are typical processes suitable for a use of the cascade control. In cases of non-isothermal reactions, concentrations of the reaction products mostly depend on a temperature of the reactant. Further, it is known that while the reactant temperature can be measured almost continuously, concentrations are usually measured in longer time intervals. Then, the application of the cascade control method can lead to good results. In this paper, the cascade control description of a continuous stirred tank reactor (CSTR) with results of control simulations is presented. CSTRS are apparatus widely used in chemical industry, biotechnologies, polymer manufacturing, and many others. From the system theory point of view, CSTRs belong to a class of nonlinear systems with mathematical models described by sets of nonlinear differential equations as it can be seen e.g. in (Smith 2005; Corriou 2004). Here, in the cascade control-loop, the concentration of a desired product of reactions is considered as the primary controlled variable, and, the reactant temperature as the secondary controlled variable. The coolant flow rate represents a common control input. The primary control variable is measured in discrete time intervals. The primary controller determining the set point for the secondary (inner) control-loop is a discrete nonlinear proportional controller derived on the basis of steady-state characteristics of the reactor. Since the controlled process is nonlinear, a continuous-time adaptive controller is used as the secondary controller. The procedure for the adaptive control design in the inner control-loop is based on approximation of the nonlinear model of the CSTR by a continuous-time external linear model (CT ELM) with recursively estimated parameters. In the process of parameter estimation, the direct method by (Rao and Unbehauen 2006); Garnier and Wang 2008) is used. The control loop with two feedback adaptive controllers is used, see, e.g. (Dostal et al. 2007). The resulting controllers are derived by the pole placement method, see, e.g. (Grimble 1993; Kucera 1993; Brogan 1991; Franklin et al. 2006). The cascade control is verified by simulations on the nonlinear model of the CSTR. NONLINEAR MODEL OF THE CSTR Consider a CSTR with exothermic reactions according to the scheme 1 k A B → , 2 2 k B C → and with a perfectly mixed cooling jacket. The desired product is the component B. Using usual simplifications, the model of the CSTR is described by four nonlinear differential equations ( ) A r A A f A r d c q r c c dt V = + − (1) ( ) B r B B f B r d c q r c c dt V = + − (2) ( ) ( ) ( ) ( ) h r r r r f r c r p r r r p r A U dT h q T T T T dt c V V c = + − + − ρ ρ (3) ( ) ( ) ( ) c c h cf c r c c c p c dT q A U T T T T dt V V c = − + − ρ (4) where 2 1 1 2 2 A B r k c r k c = = (5) Proceedings 29th European Conference on Modelling and Simulation ©ECMS Valeri M. Mladenov, Petia Georgieva, Grisha Spasov, Galidiya Petrova (Editors) ISBN: 978-0-9932440-0-1 / ISBN: 978-0-9932440-1-8 (CD) and, with in (0) s r r T T = an for concentra for densities volumetric fl exchange su coefficient. S mixture, c superscript s The reaction as where k0 are energies and parameters, i given in Tabl Tab Vr = 1.7 m Vc = 0.64 m ρr = 985 kg m ρc = 998 kg m k10 = 5.616 . k20 = 1.128 . h1 = 4.8 . 10 s Af c = 2.85 k s rf T = 323 K

The most of the processes not only in the industry has nonlinear behavior and control of such processes could be difficult. The controller here consists of linear and nonlinear part where the nonlinear part is derived from the static analysis and the linear part describes nonlinear elements in the loop by the External Linear Model (ELM), parameters of which are estimated recursively with the use of delta (δ-) models. The control synthesis employs polynomial approach with the pole assignment method. The proposed control method satisfies basic control requirements and it was tested by the simulations on the mathematical model of Continuous Stirred Tank Reactor (CSTR) with spiral cooling in the jacket as a typical member of the nonlinear system with lumped parameters. INTRODUCTION Chemical reactors are tools widely used not only in the chemical industry for production of various products. Although there are several types from the construction point of view such as batch, semi-bath etc. (Ingham et al. 2000), Continuous Stirred-Tank Reactors (CSTR) and tubular reactors are the most suitable for control purposes. The Continuous Stirred-Tank Reactor used in this work represents typical nonlinear plant described mathematically by the set of two nonlinear ordinary differential equations (ODE) (Gao et al. 2002). As it is described in (Vojtesek and Dostal 2010), this system has two stable and one unstable steady-state which could lead to very unstable or unoptimal output responses with the use of conventional control methods. One way how to overcome this inconvience is the use of the adaptive control (Astrom and Wittenmark 1989) which adopts parameters of the controller to the actual state of the system via recursive identification of the External Linear Model (ELM) as a linear representation of the originally nonlinear system (Bobal et al. 2005). The results of the adaptive control on this concrete mathematical model can be found for example in (Vojtesek et al. 2011). The control method used here is based on the combination of the adaptive control and nonlinear control. Theory of nonlinear control (NC) can be found for example in (Astolfi et al. 2008) and (Vincent and Grantham 1997), the factorization of nonlinear models of the plants on linear and nonlinear parts is described in (Nakamura et al. 2002) and (Sung and Lee 2004). The controller consists of a static nonlinear part (SNP) and a dynamic linear part (DLP). The static part is obtained from the steady-state characteristic of the system, its inversion, suitable approximation and its derivative. As a result of this nonlinear description, the linear part is then described by the external linear model with the use of delta (δ-) models (Middleton and Goodwin 2004) as a special type of discrete-time models which parameters approaches to the continuous ones for the small sampling period (Stericker and Sinha 1993). The polynomial approach (Kucera 1993) in the control synthesis can be used for systems with negative properties from the control point of view such as nonlinear systems, non-minimum phase systems or systems with time delays. Moreover, the pole-placed method with spectral factorization satisfies basic control requirements such as disturbance attenuation, stability and reference signal tracking. All graphs shown in this contribution come from the simulation on the mathematical model and they were done on the mathematical simulation software Matlab, version 6.5.1. CONTINUOUS STIRRED TANK REACTOR The controlled process under the consideration is the continuous stirred tank reactor (CSTR) with the spiral cooling in the jacket. The scheme of the system can be found in Figure 1. The complete mathematical description of the process is very complex and we must introduce some simplifications. At first, we expect that reactant is perfectly mixed and reacts to the final product with the concentration cA(t). The heat produced by the reaction is represented by the temperature of the reactant T(t). Furthermore we also expect that volume, heat capacities and densities are constant during the control. A mathematical model of this system is derived from the material and heat balances of the reactant and cooling. The resulted model is then a set of two Proceedings 26th European Conference on Modelling and Simulation ©ECMS Klaus G. Troitzsch, Michael Mohring, Ulf Lotzmann (Editors) ISBN: 978-0-9564944-4-3 / ISBN: 978-0-9564944-5-0 (CD) Ordinary Differential Equations (ODEs) (Gao et al. 2002)

The paper deals with continuous-time adaptive control of a tubular chemical reactor with the countercurrent cooling as a nonlinear single input - single output process. The mean reactant temperature and the output reactant temperature are chosen as the controlled outputs, and, the coolant flow rate as the control input. The nonlinear model of the reactor is approximated by an external linear model with a structure chosen on the basis of controlled outputs step responses. Its parameters are estimated via corresponding delta model. The control system structure with two feedback controllers is considered. The resulting controllers are derived using polynomial approach. The method is tested on a mathematical model of the tubular chemical reactor.

The paper presents an effective procedure for control design of multi input - multi output nonlinear processes. The procedure is based on an approximation of a nonlinear model of the process by a continuoustime external linear model in the form of the left polynomial matrix fraction. The parameters of the continuoustime external linear model are recursively estimated either by a direct method or through an external delta model. The control system structure with two feedback controllers is used. The controllers are derived using the explicit pole assignment method. The control is simulated on the nonlinear model of two conic liquid tanks in series.