TRAJECTORY TRACKING CONTROLLER DESIGN BASED … · trajectory tracking controller design based on linear algebra with integral accion: application to cstr systems

VII CAIQ 2013 y 2das JASP

AAIQ Asociación Argentina de Ingenieros Químicos - CSPQ

TRAJECTORY TRACKING CONTROLLER DESIGN BASED ON

LINEAR ALGEBRA WITH INTEGRAL ACCION: APPLICATION

TO CSTR SYSTEMS.

Romina B. Suvire*1,

Gustavo J. E. Scaglia2, Mario E. Serrano

3,

Jorge R.Vega4 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: +54 264 4211700; e-mail: [email protected])

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

mailto:[email protected]


AAIQ, Asociación Argentina de Ingenieros Químicos - CSPQ

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).



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.



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, Ca,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 Ca,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



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=nT0, where T0 is the

sampling period and nϵ{0, 1, 2, …}, will be denoted as yn. Thus, when computing yn+1

by knowing yn, Eq. (1) should be integrated over the time interval nT0≤ t ≤(n+1)T0 as

follows:

∫ ( ) ( )

(2)

Where, u remains constant during the interval nT0≤ t ≤(n+1)T0. Therefore, if one

knows beforehand the reference trajectory (referred to as yref(t)) to be followed by y(t),

then yn+1 can be substituted by yref(n+1) into Eq. (2), then it is possible to calculate un 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, yn+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 (yn) 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



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 qc the temperature can be varied

and hence the product concentration controlled.

Figure 1. Continuous stirred tank reactor (CSTR) with a cooling jacket

q

Ca0

Tf

qc

Tcf

qc

Ca

Ta



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 continuous-

time, nonlinear, simultaneous, differential equations:

( )

(4)

( )

[

] ( )

Where variables Ca and Ta are the concentration and temperature of the tank,

respectively; the coolant flow rate qc 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, Ca,ref, find a value for the control action (coolant flow rate, qc) so

that the reactor effluent concentration, Ca, 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,



∫ (

( )

( )

)

(5)

∫

( )

( )

*

+ ( )

Through the Euler‟s approximation of the nonlinear model of the CSTR system (5),

the following set of equations is obtained:

(

( )

)

(6)

(

( )

[

] ( )

Calling Xcn 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 Xcn (and therefore qcn), with the objective that the reactor effluent

concentration, Ca, follows the reference trajectory (Ca,ref,). To calculate Xcn 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)

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 Tadn 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 (Tan):

* ( ( )

)

( )+ ( )

(14)

The value obtained from Eq. (14) represents the temperature must have the reactor to

follows the reference concentration (Ca,ref) and named Tad.

From the second equation of (13) it is obtain:



( )

( )

( )

(15)

The value of Xcn represent the control action necessary so that the reactor effluent

concentration Ca, follows the pre-established trajectory (Ca,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., 2002. 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 T0 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 (Ca,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.



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.

(a) (b)

Figure 2. a) Tracking Trajectory Ca vs Time; b) Tracking Error vs Time.

(a) (b)

Figure 3. a) Control Action vs. Time; b) Reactor Temperature vs. Time.

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 a1, a2 and a3 (see Appendix).

5.1.1 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:

0 10 20 30 40 50 60

0.08

0.09

0.1

0.11

0.12

Time (min)

Ca (

mol/

l)

Ca

Ca,ref

10 12 14 16 18

0.08

0.085

0.09

0.095

Ca

Ca,ref

Zoom View

0 10 20 30 40 50 60-0.015

-0.01

-0.005

0

0.005

0.01

0.015Tracking Error

Time (min)

Ca

-Ca

ref

(mo

l/l)

0 10 20 30 40 50 600

20

40

60

80

100

120

140

Time (min)

qc (

l/m

in)

0 10 20 30 40 50 60430

435

440

445

Time (min)

Ta

(K

)



[ ]

[ ] (18)

[ ]

(19)

Figure (4) shows a very favorable response of the system in Ca (a) with tracking error

tends to zero (b).

(a) (b)


5.1.2 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).

0 10 20 30 40 50 60

0.09

0.1

0.11

0.12

0.13System Response to Modeling Errors

Time (min)

Ca

(m

ol/

l)

Ca

Ca,ref

11 12 130.08

0.085

Zoom View

0 10 20 30 40 50 60-2

-1

0

1

2

3x 10

-3

Time (min)

Ca -

Ca,r

ef

(mol/

l))



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):

(a) (b)


(a) (b)

(C)

Figure 6. Parameters Random Values in Monte Carlo Experiment a) a1; b) a2; c) a3

0 10 20 30 40 50 60

0.08

0.09

0.1

0.11

0.12

0.13

Time (min)

Ca

(m

ol/l)

22 23 24

0.116

0.118

0.12

Ca ref

Zoom View

0 10 20 30 40 50 600

0.002

0.004

0.006

0.008

0.01

0.012

Time (min)

Ca-C

aref (m

ol/l)

3 3.5 4 4.5 5 5.5

0

2

4

6

x 10-4

Zoom View

0 10 20 30 40 50 60 70 80 90 1001.44

1.44

1.44

1.44

1.44

1.44

1.44x 10

13

Iterations

a1

0 10 20 30 40 50 60 70 80 90 100650

700

750

Iterations

a2

0 10 20 30 40 50 60 70 80 90 1000.01

0.01

0.01

0.01

0.01

Iterations

a3



5.1.3 Application of Integral Action

In this section we study the case where there is a perturbation in action control (qc).

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 (Ca) follows the reference trajectory (Ca,ref). The

system response and tracking error can be seen in Fig. 8 a) and b), respectively.

(a) (b)

Figure 7. No Action Integral and a "20% perturbation in qc. a) Tracking Trajectory Ca vs Time; b) Tracking Error

vs Time.

Figure 8. Whit Action Integral and a "20% perturbation in qc. a) Tracking Trajectory Ca vs Time; b) Tracking

Error vs Time.

0 10 20 30 40 50 60

0.08

0.1

0.12

0.14

0.16

Time (min)

Ca (

mol/

l)

Ca

Ca,ref

0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

Time (min)

Ca-C

are

f (m

ol/

l)

0 10 20 30 40 50 60

0.08

0.09

0.1

0.11

0.12

Ca

Time (min)

Ca

(m

ol/

l)

Ca

Ca,ref

0 10 20 30 40 50 60-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time (min)

Ca-C

are

f (

mol/

l)



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, qc) for that the

tracking error goes to zero. One advantage of the methodology applied is that knowing

the system model only needs Ca,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



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.

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Appendix

Table 1. Parameters of the mathematical model of the CSTR

Parameter Description Nominal Value

q Process flowrate 100 ml/min

Ca0 Concentration of component A 1 mol/l

Tf Feed temperature 350 K

Tcf Inlet coolant temperature 350K

V Volume of tank 100 l

ha Heat transfer coefficient 7x105 J/min K

a0 Preexponential factor 7.2x1010

min-1

E/R Activation energy 1x104 K

(-ΔH) Heat of reaction 2x104 cal/mol

ρ1, ρc Liquid densities 1x103 g/l

Cp, Cpc Heat capacities 1 cal/g K

Model parameter

Model parameter

Model parameter

Where a1, a2 and a3 they are:

( )

;

;

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