TRAJECTORY TRACKING CONTROLLER DESIGN BASED ON LINEAR ALGEBRA WITH INTEGRAL ACCION: APPLICATION TO CSTR SYSTEMS.

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1 TRAJECTORY TRACKING CONTROLLER DESIGN BASED ON LINEAR ALGEBRA WITH INTEGRAL ACCION: APPLICATION TO CSTR SYSTEMS. Romina B. Suvire *1, Gustavo J. E. Scaglia 2, Mario E. Serrano 3, Jorge R.Vega 4 and Oscar A. Ortiz 5 * 1 Universidad Nacional de San Juan (UNSJ) CONICET Av. Libertador San Martín 1109 (oeste) San Juan; Argentina (Corresponding author phone: ; rsuvire@unsj.edu.ar) 2, 3 UNSJ CONICET 4 INTEC - Universidad Nacional del Litoral (UNL) - CONICET- 5 UNSJ Abstract. This work presents a new methodology based on Linear Algebra to design control algorithms for the trajectory tracking of a continuously stirred tank reactor (CSTR) system. The methodology is simple and was designed originally for non-linear multivariable mechanical systems: Mobile Robotic Systems however can be applied to the design of a large class of control systems. Particularly, a typical CSTR plant was chosen as a realistic example problem for the application of this technique. Keywords: Continuously Stirred Tank Reactor, Control System Design, Linear Algebra, Tracking Trajectory Control. 1. Introduction Continuous Stirred Tank Reactors (CSTR) are central components of many Industrial Plants in the Chemical and Biochemical Process and has been perhaps the most widely studied unit operation, from both dynamic analysis and control perspectives. These systems may exhibit highly nonlinear dynamic behavior (Vojtesek et al, 2009). In such * To whom all correspondence should be sent AAIQ Asociación Argentina de Ingenieros Químicos - CSPQ

2 cases, the use of conventional control strategies can results in poor performance and knowledge about the static and dynamic properties is a necessary condition for the design of a controller. Because of its importance, the CSTR has been the subject of numerous studies on stability (Melo et al., 2003; Perez et al., 2004; Ma et al., 2010), states estimation (Jana, 2007; Fissore, 2008; Prakash et al., 2008) and process control (Alvarez-Ramirez et al., 2001; Czeczot, 2006; Yazdi et al., 2009; Favache et al., 2010). The CSTR control has been addressed with a variety of control techniques linear and non-linear. (Velasco et al., 2011). For example, classic control and its derivations (Alvarez-Ramirez et al., 2001, Perez et al., 2004, Jana, 2007; Prakash et al., 2009), adaptive control (Czeczot, 2006, Pan et al., 2007), robust control (Fissore, 2008;), geometric control (Viel et al., 1997; Yazdi et al., 2009), among others (Alvarez- Ramirez, 1994; Favache et al., 2010, Prakash et al., 2008). As consequence of the nonlinearity of the reaction kinetics, the CSTR can deploy a variety of dynamic behavior, from multiple steady states (Bequette Wayne B., 2002) to sustained oscillations (Melo et al., 2003, Ma et al., 2010). Moreover, the literature shows that feedback control schemes in CSTR may lead to instabilities in closed loop process (Alvarez-Ramirez, 1994; Paladino et al., 2000; Perez et al., 2004 ). Often chemical reactors have significant heat effects, so it is important to be able to add or eliminate heat from them. In a Continuously Stirred Tank Reactor (CSTR) the heat is add or removed by virtue of the temperature difference between a jacked fluid and the reactor fluid (Bequette et al., 2002; Kalhoodashti, 2011; Aslam- Kaur, 2011). The heat transfer fluid often is pumped through agitation nozzle that circulates the fluid through the jacket at a high velocity. The product concentration for a CSTR can be controlled by manipulating the feed flow rate, which changes the residence time for a constant chemical reactor (Kalhoodashti, 2011; Aslam- Kaur, 2011). The heat extraction process is controlled by manipulating the speed of the cooling fluid. In many cases the jacket dynamic is faster than reactor dynamic and therefore the cooling fluid velocity binds to the jacket temperature (Ogunnaike et al., 1994).

3 One the main problems found in CSTR control are trajectory tracking. In most cases the variable of greatest interest to be monitored is the exit concentration of any component of the reactant mixture, or the reactor operation temperature control in the no isothermal case (Ogunnaike et al., 1994), while manipulated input is the temperature or cooling medium flow. Various control strategies have been developed in literature for trajectory tracking of Continuously Stirred Tank Reactor (CSTR) systems. In Lightbody et al., (1995) a reference adaptive control scheme (MRAC) is proposed and used to improve the control of a nonlinear continuously stirred tank reactor (CSTR). The performance of this technique for the control of nonlinear plants is demonstrated by comparison with Lyapunov adaptive control for a number of example plants. The adaptive tracking controller using multilayer neural networks (MNNs) proposed by Ge et al., (1998) ensures that the system output tracks a given bounded reference signal while stability of the closed-loop system is guaranteed. The effectiveness of the proposed controller is illustrated through to composition control in the CSTR system given in Lightbody et al., (1995). In Kalhoodashti, (2011) presents a control algorithm called Neural Network Approximate Generalized Predictive Control (NNAPC) for concentration tracking of a CSTR. The algorithm basically seeks to minimize the prediction error over the training data test. In order to have a good training, the data must contain sufficient information about the system dynamics. In comparison with others previous published, our controller does not present the disadvantage of the controller proposed by A. Velasco-Perez et al., 2011, Monroy- Loperena et. al., 2004; Alvarez-Ramirez et. al., 2004; where among other things, you must factorize the transfer function of the plant and solve optimization problems to obtain final control inputs to implement. In this work a trajectory-tracking controller, designed originally for robotic systems (Scaglia et al. 2008; Scaglia et al. 2009; Scaglia et al. 2010; Rosales et al. 2011), is applied to Continuously Stirred Tank Reactor (CSTR) Systems.

4 The main contribution of this control approach is the application of a novel control technique based on Linear Algebra, to design control algorithms for the trajectory tracking of a CSTR system. The control objective is described as given a desired trajectory for the reactor effluent concentration, C a,ref, find a value for the control action (coolant flow rate, qc) needed to force the system output to go from its current state to a desired one. The main advantage is that, knowing the system model only needs C a,ref to calculate the control action, and the calculation of this control actions, are obtained solving a system of linear equations. The methodology is based on the search for conditions under which a system of linear equations has an exact solution, to generate a tracking error tending to zero. The response of the CSTR system is compared with its reference model for a variety of step changes in the desired set point covering the operating range from 0.08 to 0.12 mol/l is obtained. Furthermore, the algorithm developed is easier to be implemented in a real system because the use of discrete equations allows direct adaptation to any computer system or programmable device running sequential instructions to a programmable clock speed. In this paper uses a generic model of CSTR where the irreversible, exothermic chemical reaction,, occurs and the reactor has a cooling jacket (Aris and Amundson, 1958; Uppal et al., 1974; Alvarez-Ramirez, 1994; Viel et al., 1997). Besides, it is shown how to solve a misgiven parameter s error by modifying the sample time. Moreover, using Monte Carlo method the system behavior is analyzed when it has modeling errors. By utilizing an integrator in each state variable, ensures the convergence to zero of tracking errors against modeling errors. In addition, the proposed methodology is validated and discussed through computer simulations which show the effectiveness of the proposed controller. The paper is organized as follows: Section 2, describes the methodology for the design of a control system, using Linear Algebra. Section 3 shows the CSTR model (Lightbody et al., 1995; Ge et al., 1998; Bequette Wayne B., 2002), Section 4 shows the design of the controller for the CSTR proposed, Section 5 shows the results of the

5 simulation, by applying the methodology proposed in the CSTR example. Finally, Section 6 presents the conclusions obtained in this work and some topics that will be addressed in future contributions. 2. Methodology for Controller Design 2.1. Nomenclature and Design Methodology Let us consider the first-order differential equation, (1) In Equation (1) y represents the output to the system to be controlled, u is the control action, and t is the time. The values of y(t) at discrete time t=nt 0, where T 0 is the sampling period and nϵ{0, 1, 2, }, will be denoted as y n. Thus, when computing y n+1 by knowing y n, Eq. (1) should be integrated over the time interval nt 0 t (n+1)t 0 as follows: (2) Where, u remains constant during the interval nt 0 t (n+1)t 0. Therefore, if one knows beforehand the reference trajectory (referred to as y ref (t)) to be followed by y(t), then y n+1 can be substituted by y ref(n+1) into Eq. (2), then it is possible to calculate u n that represents the control action required to go from the current state to the desired one. There are several numerical integration methods to calculate the integral in Eq. (2). For example, the Euler method approaches can be used, (3) The use of numerical methods in the simulation of the system is based mainly on the possibility to determine the state of the system at instant n+1 from the state, the control action, and other variables at instant n. So, y n+1 can be substituted by a function of reference trajectory and then the control action to make the output system evolve from the current value (y n ) to the desired one can be calculated. To accomplish this, it is necessary to solve a system of linear equations for each sampling period, as shown in

6 next Section. This represents an important advantage mainly for two reasons, first for complex systems (linear or nonlinear), the equations can be solved using iterative methods for solving systems of linear equations, which only need an initial value to start the iteration. This value may be precisely the estimate calculated in the previous sampling instant. Second, this methodology can be applied to other types of systems and the accuracy required by the numerical method is less than the one needed to simulate the behavior of the system under study. This is because, when state variables are available for feedback, at each sampling instant, the method corrects any differences caused by the cumulative error (for example, "rounding errors"). So, the approximation is used to find the best way to go from one state to the next, according to the availability of the system model. 3. CSTR Model The continuously stirred tank reactor (CSTR) system given by Lightbody et al., 1995; Ge et al.,1998; Bequette Wayne B., 2002 is shown in Figure 1. This system consists of a constant volume reactor cooled by a single coolant stream flowing in a co-current mode. The irreversible, exothermic reaction,, occurs in the tank. Because of the reaction is exothermic, the producing heat acts to slow the reaction down. By introduction of a coolant flow rate q c the temperature can be varied and hence the product concentration controlled. q C a0 T f q c T cf q c C a T a Figure 1. Continuous stirred tank reactor (CSTR) with a cooling jacket

7 The following modeling assumptions are commonly made: - Perfect mixing in reactor and jacket. - Constant volume reactor and jacket. - Reactor working at steady state therefore, their properties do not vary with time. - Because of the perfect mixing within the reactor, the properties of the reactant mixture are considered uniform anywhere within the vessel and thus are identical to the properties of the output current With all these simplifications, the process is described by the following continuoustime, nonlinear, simultaneous, differential equations: [ ] (4) Where variables C a and T a are the concentration and temperature of the tank, respectively; the coolant flow rate q c is the control input; and the parameters of the system are given in the Table 1 (see Appendix). 4. Controller Design The control objective is described as given a desired trajectory for the reactor effluent concentration, C a,ref, find a value for the control action (coolant flow rate, q c ) so that the reactor effluent concentration, C a, can follow the pre-established trajectory. The response of the CSTR system is compared with its reference model for a variety of step changes in the desired set point covering the operating range from 0.08 to 0.12 mol/l. From (2) and (4) it follows,

8 ( ) (5) * + Through the Euler s approximation of the nonlinear model of the CSTR system (5), the following set of equations is obtained: ( ) ( [ ] (6) Calling X cn as: [ ] (7) And writing the equations in matrix form: [ ] [ ] (8) Now we will consider the problem of designing a control law capable of generating the signal X cn (and therefore q cn ), with the objective that the reactor effluent concentration, C a, follows the reference trajectory (C a,ref,). To calculate X cn the system of equations (8) must have an exact solution. Then, the condition for the system (8) to have an exact solution is that the first equation is equal to zero, i.e.

9 ( ) (9) It is important to remark that the value of the difference between the reference and real trajectory will be called tracking error. It is given by: (10) Then, the following equations are defined, (11) (12) Where T adn represents the necessary reactor temperature, so that (8) has exact solution. Then, by replacing Eq. (11) and Eq. (12) in Eq.(8): [ ] [ ] (13) For the system of equations (8) has exact solution should comply that the reactor temperature (T an ): * ( ) + (14) The value obtained from Eq. (14) represents the temperature must have the reactor to follows the reference concentration (C a, ref ) and named T ad. From the second equation of (13) it is obtain:

10 (15) The value of X cn represent the control action necessary so that the reactor effluent concentration C a, follows the pre-established trajectory (C a,ref,). 5. Simulations Results In this section, we carry out computer simulations to demonstrate the performance of our tracking controller. The control approach is applied on the original time-continuous system. The CSTR configuration is obtained from Lightbody et al., 1995; Ge et al.,1998; Bequette Wayne B., It has all its parameters meeting in Table 1 (See Appendix). The reference trajectory is a linear model with a variety of step changes in the desired set point covering the operating range from 0.08 to 0.12 mol/l about the steady-state nominal concentration of 0.1 mol/l. Then, a value of 0.1 minute was chosen for the sampling time T 0 of the simulation. The values of the controller parameters are: [ ] [ ] (16) And the initial condition of the CSTR simulation is: [ ] [ ] (17) The trajectory in Ca versus time along, with their respective reference value (C a,ref ) are shown in Fig. 2 a). This demonstrates how the reactor effluent concentration tends to the reference trajectory quickly and then continues without undesirable oscillations, then the tracking error tends to zero as shown in Fig. 2 b). Figure 3 a) shows the control action required to drive this plant to follow the linear reference model for the same sequence of set points. As can be seen the coolant flowrate is well behaved, without any undesirable oscillations.

11 qc (l/min) Ta (K) Ca (mol/l) Ca-Caref (mol/l) VII CAIQ 2013 y 2das JASP In addition, Fig. 3 b) shows the temperature which adopts the reactor over time, this temperature is equal to the output current. Therefore the performance of the tracking system is satisfactory Ca Ca,ref Zoom View (a) Figure 2. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time. (b) (a) 430 Figure 3. a) Control Action vs. Time; b) Reactor Temperature vs. Time. (b) 5.1 System Response to Modeling Errors We analyze the case when the system presents modeling errors. Two methodologies are proposed and analyzed the results when there is a parametric uncertainty in the values of a 1, a 2 and a 3 (see Appendix) Modification of Sampling Time This method introduces a 10% error in the model parameters (above and below its nominal values) and tries to reduce the error that shows the system response to these modeling errors by sampling time decrease and controller gains, taking the new ranges of values:

12 Ca (mol/l) Ca - Ca,ref (mol/l)) VII CAIQ 2013 y 2das JASP [ ] [ ] (18) [ ] (19) Figure (4) shows a very favorable response of the system in C a (a) with tracking error tends to zero (b) Ca Ca,ref 3 x Zoom View (a) Figure 4. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time. (b) Application of Monte Carlo Sampling Experiment. The Monte Carlo method is usually used to find the controller parameters so as to obtain a tracking error tend to zero. However, also be applied to make an analysis of the system in case of appearing modeling errors. In this section is analyzed the controller's performance by simulation when the controller parameters vary according to the Monte Carlo experiment. We introduce a determined error in the model parameters (above and below their nominal values) and perform 100 simulations (N = 100). In each simulation the controller parameters are chosen in a random way by Monte Carlo based sampling experiment (AuatCheein and Carelli, 2012).

13 a3 a1 a2 Ca (mol/l) Ca-Caref (mol/l) VII CAIQ 2013 y 2das JASP It is observed that the performance of controller designed with the technique proposed in this paper, remains very satisfactory in the following ranges of variation: [ ] [ ] (20) [ ] For 100 iterations and whit these parametric uncertainties, the system response and tracking error can be seen in Fig. 5 a) and b), respectively. While random values that take the parameters in each of the 100 simulations are shown in Fig. 6 a), b) and c): Zoom View Ca ref x Zoom View x 1013 (a) (b) Figure 5. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time Iterations (a) Iterations (b) Iterations (C) Figure 6. Parameters Random Values in Monte Carlo Experiment a) a 1 ; b) a 2 ; c) a 3

14 Ca (mol/l) Ca-Caref ( mol/l) Ca (mol/l) Ca-Caref (mol/l) VII CAIQ 2013 y 2das JASP Application of Integral Action In this section we study the case where there is a perturbation in action control (q c ). For example: (21) In Fig. 7 can be seen that the system deviates much from the reference trajectory. The system response and tracking error can be seen in Fig. 7 a) and b), respectively. This can be solved by introducing in Eq. (11) an action integral type: (22) With this action integral can be seen in Fig. 8 a good performance of the controller because the output of the system (C a ) follows the reference trajectory (C a,ref ). The system response and tracking error can be seen in Fig. 8 a) and b), respectively Ca Ca,ref (a) (b) Figure 7. No Action Integral and a "20% perturbation in qc. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time Ca Ca Ca,ref Figure 8. Whit Action Integral and a "20% perturbation in qc. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time.

15 6. Conclusions In this paper, the trajectory tracking problem of the Continuously Stirred Tank Reactor (CSTR) Systems has been considered. The main contribution of this work is a new methodology to design control algorithms for trajectory tracking of a CSTR based on Linear Algebra. The methodology is based on the search for conditions under which a system of linear equations has an exact solution. These conditions establish the desired values of temperature and finally the control actions (coolant flow rate, q c ) for that the tracking error goes to zero. One advantage of the methodology applied is that knowing the system model only needs C a,ref to calculate the control actions. Simulation results show the effectiveness of the proposed controller. Besides, when the system s behavior is tested to modeling errors by the modification of sampling time and Monte Carlo sampling experiment it can be seen that the performance of controller designed with the technique proposed in this paper, remains very satisfactory. Similarly, when there is a disturbance of 20% in the control action and introducing an integral action can be seen a very good performance of the controller. The developed methodology for the controller design in this work has been successfully applied to several nonlinear multivariable systems, where it is experimental and simulation results, as seen in Serrano et al., 2013, Scaglia et al. 2010; Rosales et al. 2011, among others. In comparison with others previous published, our controller does not present the disadvantage of the controller proposed by A. Velasco-Perez et al., 2011; Monroy-Loperena et. al., 2004; Alvarez-Ramirez et. al., 2004; where among other things, you must factorize the transfer function of the plant and solve optimization problems to obtain final control inputs to implement. This tracking controller presents the advantages of being easy to design and to implement, which favors the implementation of algorithms on hardware dedicated to process control study. The algorithm can be implemented directly on microcontrollers without the need to implement it on an external computer, because the simple calculations by standard algebraic-numerical techniques are used to perform. The developed methodology for the controller design can be applied to other types of systems. The possibility to include in the controller design the saturation of the control

16 signals and observer-controller schemes, as shown in Wondergem et al. (2011), will be addressed in future contributions. Acknowledgments We gratefully acknowledge the Universidad Nacional de San Juan and the National Council of Scientific and Technological Research (CONICET), Argentina, by the financial support to carry out this work. References Alvarez-Ramirez J. (1994). Stability of a class of uncertain continuous stirred chemical reactors with a nonlinear feedback. Chemical Engineering Science 49, Alvarez-Ramirez, J y Puebla, H. (2001). On classical PI control of chemical reactors. Chemical Engineering Science 56, Alvarez-Ramirez, J; Velasco, A; Fernandez-Anaya, G. (2004). A note on the stability of habituating process control. Journal of Process Control 14, Aris R. y Amundson, N.R. (1958). An analysis of chemical reactor stability and control-i, II, III. Chemical Engineering Science 7, Aslam, Farhad and Kaur, Gagandeep: Comparative analysis of conventional P, PI, PID and fuzzy logic controllers for the efficient control of concentration in CSTR, International Journal of Computer Applications ( ), Vol. 17, No.6, March AuatCheein F. L. F. d. F. and Carelli R., (2012), Autonomous simultaneous localization and mapping driven by montecarlo uncertainty maps-based navigation. The Knowledge Engineering Review. Bequette Wayne B.: Behavior of a CSTR with a Recirculating Jacket Heat Transfer System, Proceedings of the American Control Conference Anchorage, AK May 8-10, Czeczot, J. (2006). Balance-based adaptive control methodology and its application to the nonisothermal CSTR. Chemical Engineering and Processing 45, Favache, A. y Dochain, D. (2010). Power-shaping control of reaction systems: The CSTR case. Automatica 46, Fissore, D. (2008). Robust control in presence of parametric uncertainties: Observer-based feedback controller design. Chemical Engineering Science 63, Ge, S.S.; Hang, C.C. and Zhang, T.: Nonlinear adaptive control using neural networks and its application to CSTR systems, Journal of Process Control 9, , Jana, A.K. (2007). Nonlinear state estimation and generic model control of a continuous stirred tank reactor. International Journal of Chemical Reactor Engineering 5, A42. Kalhoodashti, Hossein E.: Concentration Control of CSTR using NNAPC, International Journal of Computer Applications ( ), Vol. 26, No 6, July Lightbody, G. and Irwin, G.W. (1995) Direct neural model reference adaptive control, IEE Proc. Control Theory Appl., Vol. 142, No 1,

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18 Vojtesek and Dostal: Simulation of Adaptive Control of Continuous Stirred Tank Reactor, Review scientific paper, Int j simul model 8 (2009) 3, , ISSN Wondergem M., Lefeber E., Pettersen K. Y., and Nijmeijer H. (2011). Output Feedback Tracking of Ships. IEEE Transactions on Control Systems Technology, vol. 19, no. 2. Yazdi, M.B. y Jahed-Motlagh, M.R. (2009). Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chemical Engineering Journal 155, Appendix Table 1. Parameters of the mathematical model of the CSTR Parameter Description Nominal Value q Process flowrate 100 ml/min C a0 Concentration of component A 1 mol/l T f Feed temperature 350 K T cf Inlet coolant temperature 350K V Volume of tank 100 l h a Heat transfer coefficient 7x10 5 J/min K a 0 Preexponential factor 7.2x10 10 min -1 E/R Activation energy 1x10 4 K (-ΔH) Heat of reaction 2x10 4 cal/mol ρ 1, ρ c Liquid densities 1x10 3 g/l C p, C pc Heat capacities 1 cal/g K Model parameter Model parameter Model parameter Where a 1, a 2 and a 3 they are: ; ;