digital control of a continuous stirred tank reactor · in this paper we deal with a novel digital...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2011, Article ID 439785, 18 pagesdoi:10.1155/2011/439785

Research ArticleDigital Control of a Continuous StirredTank Reactor

M. P. Di Ciccio, M. Bottini, and P. Pepe

Department of Electrical and Information Engineering, University of L’Aquila,67040 Poggio di Roio (AQ), Italy

Correspondence should be addressed to P. Pepe, [email protected]

Received 4 August 2010; Revised 24 January 2011; Accepted 18 February 2011

Academic Editor: Maria do Rosario de Pinho

Copyright q 2011 M. P. Di Ciccio et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We project a novel digital control law for continuous stirred tank reactors, based on sampledmeasures of temperatures and reactant concentration, as it happens in practice. The methodologyof relative degree preservation under sampling is used. It is proved that a suitably approximatedsampled system, obtained by Taylor series expansion and truncation, in closed loop with theprojected control law, is asymptotically stable, provided that a condition on the sampling periodis verified. Such condition allows for values of the sampling period larger than necessary inpractical implementation with usual technology. Many simulations show the high performanceof the proposed digital control law.

1. Introduction

The control of the operation of chemical reactors has attracted the attention of researchersfor a long time. The underlying motivation relies on the fact that industrial chemical reactorsare frequently operated at unstable operating conditions, which often corresponds to optimalprocess performance [1–19]. Polymerization processes [5] and bioreactors fermentors [18] areimportant examples of large-scale chemical reactors operated at unstable conditions. As wellknown, the measures of the reactant concentration cannot be achieved on continuous time. Inpractice, thesemeasures are available at certain time intervals. It is therefore very important toproject control laws which are piecewise constant and are updated at each measures update.That is, for practical use, a digital control law has to be projected. Applications of nonlineardigital control theory to some chemical reactors can be found in [20, 21]. In [11] a discrete-time controller is obtained by linear approximations successively executed at each samplingtime, and an application to a continuous stirred tank reactor is shown.

2 Mathematical Problems in Engineering

In this paper we deal with a novel digital controller design for a continuous stirredtank reactor, where an irreversible, exothermic, liquid-phase reaction A → B evolves. Thecontrolled output is the reactor temperature. The jacket temperature dynamics is considered.The control law is actuated by means of a pneumatic valve which regulates the coolingwater flow rate. This actuation process is easier to implement than the ones based on theuse of the inlet reactor concentration or of the jacket temperature as manipulated inputs.The model consists of three nonlinear ordinary differential equations. We develop a digitalcontrol law by means of a suitably approximated sampled model, obtained by Taylor seriesexpansions (see [21, 22]). The preservation of relative degree methodology shown in [22]is here used. For the underlying continuous time model, the relative degree is not full. Therelative degree is preserved for the approximated sampled model by introducing a dummyoutput. Then a feedback control law is projected, by which the approximated sampledsystem becomes asymptotically stable. Many performed computer simulations show thehigh performance of the proposed digital controller. Saturation effects have been taken intoaccount in simulations. The convergence of the state variables to the arbitrarily chosenoperating point is always obtained with the many considered system initial conditions (evenstart-up initial conditions).

2. The Model of the CSTR

Here we study a CSTR with jacket cooling in which a first-order irreversible exothermicreaction takes place [23]: A → B (Figure 1).

The liquids in the reactor and in the jacket are assumed perfectly mixed, that is, withno radial, axial, or angular gradients in properties (temperature, concentrations). This allowsto consider reactor and jacket temperatures and reactant concentrations as space invariantvariables. If physical properties are assumed constant (densities and heat capacities), thereactor volume is constant and the jacket volume is constant, the mathematical model of theCSTR is given by the following set of nonlinear functional differential equations:

x1(t) =F(Ca0 − x1(t))

Vr− x1(t)K0e−E/Rx2(t),

x2(t) =F( T0 − x2(t))

Vr− λx1(t)K0e−E/Rx2(t)ρ−1Cp

−1 − UAj(x2(t) − x3(t))VrρCp

,

x3(t) =u(t)(Tcin − x3(t))

Vj+UAj(x2(t) − x3(t))

VjρjCj,

(2.1)

where x1 = Ca = reactant concentration (kmol/m3), x2 = TR = reactor temperature (K),x3 = TJ = jacket temperature (K), u = flow rate of coolant (control input) (m3/s), K0 =pre-exponential factor (s−1), E = activation energy (J/kmol), R = universal gas constant,8314 Jkmol−1K−1, F = flow rate of feed and product (m3/s), ρ = density of product stream(kg/m3), Ca0 = concentration of reactant A in feed (kmol/m3), VR = volumetric holdup ofliquid in reactor (m3), T0 = temperature of feed (K), Cp = heat capacity of product (Jkg−1K−1),λ = heat of reaction (J/kmol), U = overall heat transfer coefficient (WK−1m−2), Aj = jacketheat transfer area (m2), ρj = density of coolant (kg/m3), Cj = heat capacity of coolant(Jkg−1K−1), and Tcin = supply temperature of cooling medium (K).

Mathematical Problems in Engineering 3

Fresh feed

FJ , TJ (t)

Coolant flowFJ , Tcin

TR(t), Ca(t), F

Effluent

TR(t)

VRCa(t)

A ⇀ B

T0, Ca0, F

Figure 1: Schematic reactor type CSTR.

Let � ∈ [0, 1] be the reaction conversion

� =Ca0 − Ca

Ca0. (2.2)

Let �eq be the chosen operating point reaction conversion. By (2.2) it follows that theoperating point concentration of reactant in the reactor Caeq is given by

Caeq = Ca0(1 − �eq

). (2.3)

From equations in (2.1), setting zero the left-hand side, it follows that the operating pointreactor temperature (TReq), the operating point jacket temperature (Tjeq), the necessary coolantwater flow rate (ueq) for the chosen operating point are given by

TReq = −E(

ln

(

−F(−Ca0 +Caeq

)

CaeqK0Vr

))−1R−1,

TJeq =λCa0F�eq

UAj− FρCpT0

UAj− E

(

ln

(F�eq

(1 − �eq

)K0Vr

))−1R−1

− FρCpE

(

ln

(F�eq

(1 − �eq

)K0Vr

))−1R−1U−1Aj

−1,

ueq = UAjEρj−1Cj

−1(Tcin − TJeq

)−1(

ln

(F�eq

(1 − �eq

)K0Vr

))−1R−1

+UAjTJeq

ρjCj

(Tcin − TJeq

) .

(2.4)


3. Sampled System and Digital Control Law

The system (2.1) is in the form

x(t) = f(x(t)) + g(x(t)) · u(t), (3.1)

where the functions f , g can be easily defined by (2.1) as

f(x(t)) =

⎡

⎢⎢⎣

f1(x(t))

f2(x(t))

f3(x(t))

⎤

⎥⎥⎦ =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

F(Ca0 − x1(t))Vr

− x1(t)K0e−E/Rx2(t)

F(T0 − x2(t))Vr

− λx1(t)K0e−E/Rx2(t)ρ−1Cp−1 − UAj(x2(t) − x3(t))

VrρCp

UAj(x2(t) − x3(t))VjρjCj

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

g(x(t)) =

⎡

⎢⎢⎣

g1(x(t))

g2(x(t))

g3(x(t))

⎤

⎥⎥⎦ =

⎡

⎢⎢⎢⎢⎢⎣

0

0

Tcin − x3(t)Vj

⎤

⎥⎥⎥⎥⎥⎦.

(3.2)

In order to rewrite (3.1) in normal form, as required in [22], let us now consider the followingvariables:

ξ(t) =

⎡

⎢⎢⎣

ξ1(t)

ξ2(t)

ξ3(t)

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

x2(t) − TReq

f2(x(t))

x1(t) − Caeq

⎤

⎥⎥⎦ = φ(x(t)), (3.3)

from which the easy inverse relation holds

x(t) = Φ−1(ξ(t)) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ξ3(t) + Caeq

ξ1(t) + TReq

−FρCp

UAj

(T0 − ξ1(t) − TReq

)

+λVr

(ξ3(t) + Caeq

)K0e−E/R(ξ1(t)+TReq )

UAj+ξ2(t)VrρCp

UAj+ ξ1(t) + TReq

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (3.4)

Let the control input be equal to

u(t) = ueq + v(t). (3.5)


Let the controlled output be

y(t) = h(ξ(t)) = ξ1(t). (3.6)

We want to control to zero this output, which corresponds to drive the reactor temperatureto the reactor temperature operating point TReq .

We can write the system equations by using the new variables ξ(t) as follows:

ξ(t) = p1(ξ(t)) + p2(ξ(t))v(t), (3.7)

where p1, p2 are given by

p1(ξ(t)) =

⎡

⎢⎢⎣

p11(ξ(t))

p21(ξ(t))

p31(ξ(t))

⎤

⎥⎥⎦,

p11(ξ(t)) = ξ2(t),

p21(ξ(t)) = −λK0FCa0Vr−1ρ−1Cp

−1(eE/R(ξ1(t)+TReq )

)−1 − ueqUAjξ1(t)VjVrρ Cp

+ueqFT0

VjVr

+ λξ3(t)K0FVr−1ρ−1Cp


)−1 −ueqUAjTReq

VjVrρ Cp− ueqFξ1(t)

VjVr

+ λK0CaeqFVr−1ρ−1Cp


)−1− ueqξ2(t)

Vj+ueqUAjTcin

VjVrρ Cp

+ λK02ξ3(t)

(eE/R(ξ1(t)+TReq )

)−2ρ−1Cp

−1 −ueqFTReq

VjVr− UAjξ2(t)

VjρjCj

+ λK02Caeq

(eE/R(ξ1(t)+TReq )

)−2ρ−1Cp

−1 − ξ2(t)FVr

− ξ2(t)λK0Eξ3(t)ρ−1Cp−1R−1

(ξ1(t) + TReq

)−2(eE/R(ξ1(t)+TReq )

)−1

− ξ2(t)λK0ECaeqρ−1Cp

−1R−1(ξ1(t) + TReq

)−2(eE/R(ξ1(t)+TReq )

)−1

− UAjξ2(t)VrρCp

+UAjFT0

VjρjCjVr− UAjFξ1(t)

VjρjCjVr−UAjFTReq

VjρjCjVr

−UAjλK0ξ3(t)Vj−1ρj−1Cj

−1ρ−1Cp−1(eE/R(ξ1(t)+TReq )

)−1

−UAjλK0CaeqVj−1ρj−1Cj

−1ρ−1Cp−1(eE/R(ξ1(t)+TReq )

)−1


− ueqλK0ξ3(t)Vj−1ρ−1Cp


)−1

− ueqλK0CaeqVj−1ρ−1Cp


)−1,

p31(ξ(t)) =F(Ca0 − ξ3(t) − Caeq

)

Vr− (ξ3(t) +Caeq

)K0e−E/R(ξ1(t)+TReq),

p2(ξ(t)) =

⎡

⎢⎢⎣

p12(ξ(t))

p22(ξ(t))

p32(ξ(t))

⎤

⎥⎥⎦,

p12(ξ(t)) = 0,

p22(ξ(t)) =TcinUAj

VjVrρCp+

FT0VjVr

− Fξ1(t)VjVr

−FTReq

VjVr− UAjξ1(t)

VjVrρCp−TReqUAj

VjVrρCp− ξ2(t)

Vj

− λK0ξ3(t)Vj−1ρ−1Cp


)−1

− λK0CaeqVj−1ρ−1Cp


)−1,

p32(ξ(t)) = 0.

(3.8)

Since ξ = 0 corresponds to x =[ CaeqTReqTJeq

], it follows that p1(0) = 0. The introduction of

the new variables allows to obtain the CSTR equations in normal form [22], and the origin isan equilibrium point. Therefore, the stability of the origin for the system described by (3.7) isequivalent to the stability of the chosen operating point for the system described by (2.1). Letus now sample and approximate, by Taylor series expansion and truncation, the system (3.7)as follows: setting v(t) = v(kδ) = vk , kδ ≤ t < (k + 1)δ, k = 0, 1, 2, . . .,

⎡

⎣ξ1

ξ2

⎤

⎦((k + 1)δ) =

[1 δ

0 1

]⎡

⎣ξ1

ξ2

⎤

⎦(kδ) +

⎡

⎢⎣

δ2

2

δ

⎤

⎥⎦R0(kδ),

ξ3((k + 1)δ) = ξ3(kδ) + δp31(ξ(kδ)

)+δ2

2q(ξ(kδ)

),

(3.9)

where ξ(kδ) is the approximation of ξ(kδ); R0 is given by (Lie derivatives are used),

R0(kδ) = l2p1h(ξ(kδ)

)+ lp2 lp1h

(ξ(kδ)

)v(kδ);

l2p1h(ξ(kδ)

)=

∂l1p1h(ξ(kδ)

)

∂(ξ(kδ)

) p1(ξ(kδ)

); l1p1h

(ξ(kδ)

)=∂h(ξ(kδ)

)

∂(ξ(kδ)

) p1(ξ(kδ)

);

lp2 lp1h(ξ(kδ)

)=

∂lp1h(ξ(kδ)

)

∂(ξ(kδ)

) p2(ξ(kδ)

);

(3.10)


q(ξ(kδ)) is given by

q(ξ(kδ)

)=∂p31

(ξ(kδ)

)

∂ξ(kδ)p1(ξ(kδ)

),

q(ξ(kδ)

)= −K0Eξ2(kδ)ξ3R−1

(ξ1(kδ) + TReq

)−2(eE/R(ξ1(kδ)+TReq )

)−1

−K0Eξ2(kδ)CaeqR−1(ξ1(kδ) + TReq

)−2(eE/R(ξ1(kδ)+TReq )

)−1

− F2Ca0

Vr2

+F2ξ3

Vr2+F2Caeq

Vr2

+ 2FK0ξ3Vr−1(eE/R(ξ1(kδ)+TReq )

)−1

+ 2FK0CaeqVr−1(eE/R(ξ1(kδ)+TReq )

)−1+K0

2Caeq

(eE/R(ξ1(kδ)+TReq )

)−2

− FK0Ca0Vr−1(eE/R(ξ1(kδ)+TReq )

)−1+ ξ3(kδ)K0

2(eE/R(ξ1(kδ)+TReq )

)−2.

(3.11)

The approximated sampled system (3.9) is obtained by exploiting the normal formof (3.7) (see [22]) and by neglecting, in the Taylor series expansion, the terms which are0(δ3) in the expression of ξ1((k + 1)δ) and in the expression of ξ3((k + 1)δ), and the termswhich are 0(δ2) in the expression of ξ2((k + 1)δ) (see terms N2

r,δin [22, Proposition 3.2]).

Notice that, since (∂p31(ξ(kδ))/∂ξ(kδ))p2(ξ(kδ)) = 0, the inclusion of a term which is 0(δ2)in the approximation of ξ3((k + 1)δ) does not cause a new presence (besides the one in R0)of the control input vk . As well, neglecting terms which are 0(δ2) in the expression of ξ2((k +1)δ) avoids to have further terms (besides R0) which involve the control input. This fact isinstrumental for building the digital control law here proposed, as will be shown below.

The approximated sampled system (3.9) admits relative degree equal to 1, thus notpreserving the relative degree equal to 2 of the original continuous time system (2.1) withrespect to the output x2(t) (reactor temperature). Thus, a digital input/output linearizingfeedback control law built up on the basis of the given output (3.6) yields a further internaldynamics. In order to preserve the relative degree, let us introduce the dummy output asproposed in [22],

yδ(kδ) =[

1 −δ2

]⎡

⎣ξ1(kδ)

ξ2(kδ)

⎤

⎦, (3.12)

and let us define the variables

χ(k) =

⎡

⎢⎢⎣

χ1(k)

χ2(k)

χ3(k)

⎤

⎥⎥⎦ =

⎡

⎢⎢⎣

yδ(kδ)

yδ((k + 1)δ)

ξ3(kδ)

⎤

⎥⎥⎦ = Mξ(kδ), (3.13)


where the nonsingular matrixM is given by

M =

⎡

⎢⎢⎢⎢⎢⎢⎣

1 −δ2

0

1δ

20

0 0 1

⎤

⎥⎥⎥⎥⎥⎥⎦

. (3.14)

Let M1 be the upleft submatrix of M

M1 =

⎡

⎢⎢⎣

1 −δ2

1δ

2

⎤

⎥⎥⎦. (3.15)

We obtain, from (3.9),

[χ1(k + 1)

χ2(k + 1)

]

= M1

[1 δ

0 1

]

M−11

[χ1(k)

χ2(k)

]

+M1

⎡

⎢⎣

δ2

2

δ

⎤

⎥⎦R0(kδ),

χ3(k + 1) = χ3(k) + δp31(M−1χ(k)

)+δ2

2q(M−1χ(k)

),

(3.16)

where R0, rewritten with the variables χ, becomes

R0(kδ) = l2p1h(M−1χ(k)

)+ lp2 lp1h

(M−1χ(k)

)vk. (3.17)

Theorem 3.1. Let

Γ =[Γ1 Γ2

] ∈ R1x2 (3.18)

be chosen such that the matrix[ 0 1Γ1 Γ2

]has eigenvalues inside the open complex unitary circle. Let the

sampling period δ > 0 satisfy the following inequality:

δ < δmax =2

(F/Vr) +K0e−E/RTReq. (3.19)

Then, the following discrete time feedback control law

vk =−l2p1h

(M−1χ(k)

)

lp2 lp1h(M−1χ(k)

) +[ 1 −2 ]

[χ1(k)χ2(k)

]+ Γ[χ1(k)χ2(k)

]

δ2lp2 lp1h(M−1χ(k)

) (3.20)


is such that the trivial solution of the closed-loop system (3.16)–(3.20) is asymptotically stable.Moreover, the eigenvalues of the Jacobian (evaluated at 0) of the right-hand function describingthe closed-loop system (3.16)–(3.20) are given by the eigenvalues of the matrix

[ 0 1Γ1 Γ2

]and by the

eigenvalue

1 − δ

(F

Vr+K0e−E/RTReq

)+δ2

2

(F

Vr+K0e−E/RTReq

)2

, (3.21)

which takes its minimum, equal to 1/2, at δ = δ(1/2) = 1/((F/Vr) +K0e−E/RTReq ).

Proof. By the form of the matrixM1, the following equation holds for χ1 and for χ2, in (3.16)

[χ1(k + 1)

χ2(k + 1)

]

=

[0 1

−1 2

][χ1(k)

χ2(k)

]

+

[0

δ2

]

R0(kδ). (3.22)

Notice that, by construction, such subsystem (and thus the overall system (3.16)) admitsrelative degree equal to 2, in a neighborhood of the origin, with respect to the dummy outputχ1(k). Indeed, taking into account of the form of R0(kδ), we have

χ1(k + 1) = χ2(k + 1),

χ1(k + 2) = χ2(k + 1) =[−1 2

]χ(k)

+ δ2(l2p1h(M−1χ(k)

)+ lp2 lp1h

(M−1χ(k)

)v(kδ)

),

(3.23)

and the term multiplying v(kδ) is nonzero in a neighborhood of the origin, sincelp2 lp1h(M

−1χ(k)) is smooth in a neighborhood of the origin and lp2 lp1h(0)/= 0. Therefore,taking again into account the form of R0, the proposed discrete-time feedback control lawyields the following equations for the closed-loop-system (3.16), (3.20)

[χ1(k + 1)

χ2(k + 1)

]

=

[0 1

Γ1 Γ2

][χ1(k)

χ2(k)

]

,

χ3(k + 1) = χ3(k) + δp31(M−1χ(k)

)+δ2

2q(M−1χ(k)

).

(3.24)

In order to show the asymptotic stability of the nonlinear system (3.24), it is sufficient to showthat the linear system obtained by linearizing (3.24) is asymptotically stable. The Jacobian, atzero, of the function in the right-hand side of (3.24) is given, taking into account of p31 and ofq, by

J =

⎡

⎢⎢⎣

0 1 0

Γ1 Γ2 0

α β γ

⎤

⎥⎥⎦, (3.25)


where

γ = 1 − δ

(F

Vr+K0e−E/RTReq

)+δ2

2

(F

Vr+K0e−E/RTReq

)2(3.26)

and α, β are suitable reals. Given the triangular structure of the Jacobian J , the eigenvalues aregiven by the eigenvalues of the matrix

[ 0 1Γ1 Γ2

]and by γ . The eigenvalues of the matrix

[ 0 1Γ1 Γ2

]

are inside the open unitary circle. The eigenvalue γ is always positive (for positive samplingperiod), and is inside the open unitary circle if the inequality (3.19) is satisfied. Moreover, byusing the derivative of γ with respect to δ, we obtain that γ takes the minimum value, 0.5, atδ(1/2).

Now, going back to initial variables x, by means of the (approximated) relation χ(k) �Mφ−1(x(kδ)), the digital feedback control law here proposed is given, for kδ ≤ t < (k + 1)δ,k = 0, 1, 2, . . ., by

u(t) = 4VjλK0FCa0(Tcin − x3(kδ))−1U−1Aj−1(eE/Rx2(kδ)

)−1+ 2

VjVrρ CpTReqΓ1δ2(Tcin − x3(kδ))UAj

− 2VjFx2(kδ)

(Tcin − x3(kδ))Vr+

VjρCpF2T0

(Tcin − x3(kδ))UAjVr− 2

VjVrρCpx2(kδ)δ2(Tcin − x3(kδ))UAj

+ 2Vjx2(kδ)

δ(Tcin − x3(kδ))− 2VjλK0Fx1(kδ)(Tcin − x3(kδ))−1U−1Aj

−1(eE/Rx2(kδ)

)−1

− VrVjλK02x1(kδ)(Tcin − x3(kδ))−1U−1Aj

−1(eE/Rx2(kδ)

)−2

+ VjFT0λx1(kδ)K0E(Tcin − x3(kδ))−1U−1Aj

−1R−1x2(kδ)−2(eE/Rx2(kδ)

)−1

− VjλK0Fx1(kδ)E(Tcin − x3(kδ))−1U−1Aj−1x2(kδ)−1R−1

(eE/Rx2(kδ)

)−1

− VrVjλ2x1(kδ)2K0

2E(Tcin − x3(kδ))−1U−1Aj

−1ρ−1Cp−1(eE/Rx2(kδ)

)−2 1Rx2(kδ)

− Vjλx1(kδ)K0(Tcin − x3(kδ))−1ρ−1Cp−1(eE/Rx2(kδ)

)−1− 2

VjVrρCpΓ1x2(kδ)δ2(Tcin − x3(kδ))UAj

− Vjλx1(kδ)K0E(Tcin − x3(kδ))−1ρ−1Cp−1x2(kδ)−1R−1

(eE/Rx2(kδ)

)−1

− VjUAjx2(kδ)(Tcin − x3(kδ))Vrρ Cp

+Vjx3(kδ)F

(Tcin − x3(kδ))Vr

+VjFT0

(Tcin − x3(kδ))Vr− VjρCpF2x2(kδ)(Tcin − x3(kδ))UAjVr

+ Vjx3(kδ)λx1(kδ)K0E(Tcin − x3(kδ))−1ρ−1Cp−1R−1x2(kδ)−2

(eE/Rx2(kδ)

)−1


+VjUAjx3(kδ)

(Tcin − x3(kδ))VrρCp− UAjx2(kδ)(Tcin − x3(kδ))ρjCj

+UAjx3(kδ)

(Tcin − x3(kδ))ρjCj

− 2Vjx3(kδ)

δ(Tcin − x3(kδ))+ 2

VjVrρCpTReq

δ2(Tcin − x3(kδ))UAj

− 2VjρCpFT0

δ(Tcin − x3(kδ))UAj+ 2

VjρCpFx2(kδ)δ(Tcin − x3(kδ))UAj

+ 2VrVjλx1(kδ)K0δ−1(Tcin − x3(kδ))−1U−1Aj

−1(eE/Rx2(kδ)

)−1.

(3.27)

The digital control law (3.27) is applied to the CSTR system, described by thedifferential equations (2.1), by means of a zero-order hold device. Such control law canbe easily implemented on a microprocessor. It is a static feedback control law, that is, nodynamics are involved. The control law has to be computed at every sampling period whichis very much larger than the computation time (many minutes versus some seconds atmaximum). The great advantage of this digital control law is also given by the fact that onlysampled-data measures, provided by a suitable device, are necessary. Take into account that,in practice, continuous timemeasures are difficult to obtain, or even impossible. The feedbackcontrol law here presented is therefore easily realizable. The closed-loop system (2.1), (3.27)(with the zero-order hold) is simulated in the following section.

4. Simulation Results

Simulations, using MatLab (The MathWorks, Inc.) Software Package, have been carried outto verify the effectiveness of the proposed method. The values of the model parameters usedin simulation are given in Table 1 (taken from [23]). In all the carried out simulations, aminimum and a maximum value for the input u, specifically the volumetric flow rate of therefrigerant, is imposed, to take into account of the actuator saturation. It is considered:

FJmin ≤ u(t) ≤ FJmax (4.1)

with the following acceptable physical values for a control valve

FJmin = 0.002m3/s = 7.2m3/h,

FJmax = 0.08m3/s = 288m3/h.(4.2)

With these parameters, δmax in condition (3.19) is equal to 0.57 h, which corresponds to about34 min. The value of δ(1/2) is equal to about 0.28 h which corresponds to about 16 min.


Table 1: Reactor and controller parameters values.

K0 = 7.47 × 1010 h−1 Ca0 = 8.01 kmol ·m−3

E = 69.71 × 106 J · kmol−1 F = 15.76m3 · h−1

ρ = 801 kg ·m−3 Tcin = 294K

R = 8314 J · kmol−1 · K−1 T0 = 294K

ρj = 1000 kg ·m−3 Aj = 45.2m2

Cp = 3137 J · kg−1 · K−1 Vr = 30.3m3

Cj = 4183 J · kg−1 · K−1 Vj = 9m3

λ = −69.71 × 106 J · kmol−1 Γ1 = −6 · 10−6U = 3.0636 × 106 J · h−1 ·m−2 · K−1 Γ2 = −5 · 10−3

Table 2: Process parameters of Simulation 1.

Reactor operating point Initial stateχeq = 85%

⎧⎪⎨

⎪⎩

Ca(t = 0) = Ca0

TR(t = 0) = T0

TJ(t = 0) = Tcin

δ = 0.1h = 6 minCaeq = 1.2015 kmol ·m−3

TReq = 350KTJeq = 312Kueq = 69.86m3 · h−1

Parameters mismatch: absent

We performed several sets of simulation runs and all showed the high performance of thedigital control law here proposed. We choose the reactor conversion �eq = 0.85, to whichthe following unstable operating point corresponds: Caeq = 1.2015 kmol/m3, TReq = 350K,TJeq = 312K, ueq = 69.86m3/h. In the simulations here presented, the process is initiallyassumed to be at the start up, that is x1(0) = Ca0 (concentration of reactant in feed) =8.01 kmol/m3, x2(0) = T0 (temperature of feed) = 294K, x3(0) = Tcin (supply temperatureof cooling medium) = 294K. This means that the reactor is very far from the operatingpoint.

In Simulation 1 (see Table 2, Figure 2) we consider the reactor at nominal parametersvalues (see Table 1), without any uncertainty. Figure 2 shows the evolution of the reactantconcentration, of the reactor temperature, of the jacket temperature, and of the piecewiseconstant control signal, respectively, with sampling period equal to 10−1 h (6min).

As can be seen, all the state variables converge to the operating point ones in avery good fashion, without dangerous oscillations. The reactor temperature settling time isestimated from plots to be about 6 h, the steady-state error is equal to 0.

In Simulation 2 (see Table 3, Figure 3) we consider again the reactor at nominalparameters values (see Table 1), without any parameters mismatch, and a sampling periodequal to 0.25 h (15min) is considered. The same desirable convergence to the operatingpoint is achieved, with a reactor temperature settling time of about 8 h. Again no dangerousoscillations appear and the reactor temperature is driven in a very good fashion to theoperating point, with steady-state error equal to 0.

In Simulations 3, 4 (see Table 4, Figure 4, and Table 5, Figure 5, resp.) we consider thereactor whose parameters are uncertain, with sampling time again equal to 0.1 h and 0.25 h,


8

7

6

5

4

3

2

1

0

0 4 8 12 16 20 24

Time (h)

Ca(kmol

·m−3)

Caeq = 1.2015 kmol · m−3

(a)

TR(K

)

TReq = 350K

0 4 8 12 16 20 24

Time (h)

500

450

400

350

300

(b)

300

200

100

00 4 8 12 16 20 24

Time (h)

8580757065605550

4 4.5 5 5.5 6 6.5 7 7.5 8

u(m

3·h

−1)

ueq = 69.86m3 · h−1

(c)

0 4 8 12 16 20 24

Time (h)

312

308

304

300

3 3.2 3.4 3.6 3.8 4TJeq = 312K

TJ(K

)

360

340

320

300

(d)

Figure 2: Results of Simulation 1.



⎧⎪⎨

⎪⎩

Ca(t = 0) = Ca0

TR(t = 0) = T0TJ(t = 0) = Tcin

δ = 0.25h = 15 minCaeq = 1.2015 kmol ·m−3


Parameters mismatch: absent

respectively. We apply the controller obtained using the nominal values of the parameters(see Table 1), to a reactor whose parameter values are different with respect to the nominalones as reported in Table 5. In this case we observe a reactor temperature steady-state errorof about 5K. Take into account that the parameters mismatch has been deliberately chosenvery critical, in order to show the robustness of the proposed controller.

The performed simulations show the high performance of the proposed controller,as evidenced, in the case the reactor parameters are exactly known, by zero steady stateerror, no dangerous oscillations, good settling time. When parameters mismatch occurs, theproposed controller yields a very good behavior for the reactor, evidenced by small steady-state errors, no dangerous oscillations, and good settling time. We stress the fact that the


8

7

6

5

4

3

2

1

0

0 4 8 12 16 20 24

Time (h)

Ca(kmol

·m−3)

Caeq = 1.2015 kmol · m−3

(a)

TR(K

)

TReq = 350K

0 4 8 12 16 20 24

Time (h)

500

450

400

350

300

(b)

300

200

100

00 4 8 12 16 20 24

Time (h)

u(m

3·h

−1)

ueq = 69.86m3 · h−1

70

60

50

40

30

203 4 5 6 7 8 9

(c)

310

305

300

3 3.25 3.5 3.75 4

0 4 8 12 16 20 24

Time (h)

TJeq = 312K

TJ(K

)

380

360

340

320

300

(d)




⎧⎪⎨

⎪⎩

Ca(t = 0) = Ca0

TR(t = 0) = T0TJ(t = 0) = Tcin

δ = 0.1h = 6 minCaeq = 1.2015 kmol ·m−3


Parameters mismatch:F = F/1.3, T0 = T0 · 1.05;

Ca0 = Ca0/1.1, Tcin = Tcin · 1.03U = 0.8 ·U

proposed controller allows for sampled-data measures of the state variables, which is veryuseful in practical applications.

5. Conclusion

In this paper we have presented a digital control law for a continuous stirred tank reactor,constructed on the basis of an approximated sampled model and of the preservation of the




⎧⎪⎨

⎪⎩

Ca(t = 0) = Ca0

TR(t = 0) = T0TJ(t = 0) = Tcin

δ = 0.25h = 15 minCaeq = 1.2015 kmol ·m−3


Parameters mismatch:F = F/1.3, T0 = T0 · 1.05,

Ca0 = Ca0/1.1, Tcin = Tcin · 1.03,U = 0.8 ·U

8

6

4

2

00 4 8 12 16 20 24

Time (h)

Ca(kmol

·m−3)

Ca(t = 24 h) = 0.61 kmol · m−3

(a)

TR(K

)

0 4 8 12 16 20 24

Time (h)

500

450

400

350

300

TR(t = 24 h) = 355.9K

(b)

50

45

40

35

305 5.5 6 6.5 7 7.5 8

300

240

180

120

60

00 4 8 12 16 20 24

Time (h)

u(m

3·h

−1)

u(t = 24 h) = 34.54m3 · h−1

(c)

330

325

320

315

310

3054 4.5 5 5.5 6 6.5 7

0 4 8 12 16 20 24

Time (h)

TJ(K

)

330

310

320

300

290

TJ (t = 24 h) = 325.8K

(d)


relative degree methodology. The proposed feedback control law is easy to implement andsimulations show high performance. A condition on the sampling period is found, by whichthe asymptotic stability of the closed-loop approximated sampled system is proved. Suchcondition is verified by sampling periods belonging to feasible sets, allowing for values largerthan necessary in practical implementation. Future work will concern the construction of adiscrete-time observer, in order to avoid reactant concentration measures, which are difficult


8

6

4

2

00 4 8 12 16 20 24

Time (h)

Ca(kmol

·m−3)

Ca(t = 24 h) = 0.64 kmol · m−3

(a)

TR(K

)

0 4 8 12 16 20 24

Time (h)

500

450

400

350

300

TR(t = 24 h) = 355.1K

(b)

50454035

3520

30

5 5.5 6 6.5 7 7.5 8

300

240

180

120

60

00 4 8 12 16 20 24

Time (h)

u(m

3·h

−1)

u(t = 24 h) = 35.71m3 · h−1

(c)

330

325

320

315

310

3054 4.5 5 5.5 6 6.5 7

0 4 8 12 16 20

Time (h)

TJ(K

)

330

310

320

300

290

TJ (t = 24 h) = 325.1K

(d)


to do and expensive (see [24–27]). Another topic which will be investigated is the problemarised by the recycle time-delay (see [21, 28–31]), on the basis of the results in [32].

Acknowledgments

This paper is supported by the University of L’Aquila Atheneum RIA Project and by theCenter of Excellence for Research DEWS, L’Aquila, Italy.

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