Robust Controllers for a Heat Exchanger - Industrial & Engineering

Modeling and Control of the In-Situ Thermoplastic Composite Tape-Laying Process. Wei-Ching Sun , Susan C. Mantell , Kim A. Stelson. Journal of Dynamic...
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Ind. Eng. Chem. Res. 1997, 36, 382-388

Robust Controllers for a Heat Exchanger Jose Alvarez-Ramirez,* Ilse Cervantes, and Ricardo Femat Departamento de Ingenierı´a de Procesos, Universidad Autonoma MetropolitanasIztapalapa, Apartado Postal 55-534, D.F. 09000, Mexico

A robust controller with uncertainty estimation is proposed for temperature control of process fluid in a fluid-fluid heat exchanger. The controller comprises an approximate input-output linearizing feedback and an observer-based uncertainty estimator. The performance of the proposed controller is evaluated by simulation for both regulatory and servo problems. The simulation results show the effectiveness of the present controller and its robustness to modeling errors and disturbances. 1. Introduction An important advance in nonlinear control theory in the last decade has been the characterization of linearizable systems, that is, systems that can be linearized by a change of coordinates and state feedback (Isidori, 1989). Linearization of nonlinear systems is based on cancellation of nonlinearities and assumes perfect knowledge of the system nonlinearities. A feedback control designed by this approach guarantees closed-loop stability and output tracking. However, it is well known that linearizing controllers may have performance degradation and even instability under nonlinear uncertainties (Khalil, 1996). Although there have been attempts at improving robustness of controlled feedback linearizable systems (Kanellakopolous et al., 1992; Lin and Saberi, 1995), there is still a need to find robust control laws. In this paper, we develop a robust output feedback controller for heat exchangers that belongs to the class of fully linearizable nonlinear systems. The process is simple, but it has enough structure to capture many of the important features of both feedback linearization and compensation of uncertainties in nonlinear processes. Feedback linearization of heat exchangers was studied by Malleswararao and Chidambaram (1992), and some tuning rules for model reference control were provided. However, robustness properties of the linearizing feedback controller were not studied. Our main contribution to the robust control problem of this class of processes is to interpret transformed coordinates and lumped uncertainties as observable states that can be reconstructed from output measurements. In this way, the robust stabilization problem is posed in the framework of recently developed methodologies (Esfandiari and Khalil, 1992) for output feedback stabilization of nonlinear systems. The controller proposed in this work comprises an approximate linearizing feedback and an observer-based uncertainty estimator. Such a controller attempts to estimate a function representing the effect of uncertainties via a high-gain observer. The gathered information is subsequently used to cancel the unknown dynamics and the unexpected disturbances simultaneously. In this way, when the estimated uncertainty approaches the actual one, the cancellation of nonlinearities is enhanced, and the dynamics of the closed-loop system approaches the closed-loop dynamics under an ideal linearizing feedback control. Although in this work we restrict ourselves to the case of heat exchangers, the proposed control design meth* Corresponding author. E-mail: [email protected]. Fax: +52-5-7244900. S0888-5885(96)00496-4 CCC: $14.00

odology can be easily extended to general SISO linearizable systems, such as distillation columns and chemical reactors. The work is organized as follows. Section 2 presents the model equations of a fluid-fluid heat exchanger. Section 3 describes the control design with exact knowledge of system dynamics. To address the uncertainty compensation problem, section 4 presents a implicit state representation of open-loop dynamics. In section 5, a robust control which accounts for uncertainties is derived. In section 6, a numerical example is presented to illustrate the performance of the resulting controller. The work is closed with some concluding remarks. 2. Model Equations A simplified but realistic model of a fluid-fluid heat exchanger is given by the following set of nonlinear equations (Malleswararao and Chidambaram, 1992):

T˙ po ) 2[Fp(t)(Tpi - Tpo) - (U(t)A∆T(u )/Cpp)]/Mp (1) T˙ co ) 2[Fc(Tci - Tco) + (U(t)A∆T(u )/Cpc)]/Mc (2) where u ) (Tci,Tco,Tpi,Tpo) ∈ R4 is a vector of input and output fluid temperatures, ∆T(u ) is the effective mean temperature difference, which can be the arithmetic mean temperature difference (AMTD),

∆T(u ) ) [(Tpo - Tci) + (Tpi - Tco)]/2

(3a)

or, as in most practical cases, the log mean temperature difference (LMTD),

∆T(u ) )

(Tpo - Tci) - (Tpi - Tco) log(Tpo - Tci) - log(Tpi - Tco)

(3b)

Equations 1 and 2 can be written in the following statevariable form:

x˘ ) f(x,d) + g(x)u

(4)

y ) Cx where x ) (Tpo,Tco)T, u ) Fc, d ) (Fp,Tpi,U)T, C ) (1,0),

f(x,d) )

[ [

]

f1(x,d) ) f2(x,d) 2[Fp(t)(Tpi - Tpo) - (U(t)A∆T(u )/Cpp)]/Mp 2U(t)A∆T(u )/(CpcMc)

© 1997 American Chemical Society

]

g(x) )

[ ] [

g1(x) 0 ) 2(Tci - Tco)/Mc g2(x)

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 383

]

u ) uI(z,D) ) [-γ1(z,D) - K1z1 - K2z2]/γ2(z,D) (7)

The controlled variable is the outlet process fluid temperature (Tpo), and the manipulated variable is the heating fluid flow rate (Fc). The disturbance variables are the inlet temperature (Tpi) and flow rate (Fp) of process fluid. The time dependence of the heat transfer coefficient accounts for variations in the heat transfer surface. We assume (a) U(t) * 0, t g 0 and (b) Tci > Tpi (respectively Tci < Tpi). The above assumptions imply that, under normal operation conditions, Tci < Tco (respectively Tci > Tco), so that the control system (4) is well defined for all t > 0 (i.e., g(x) * 0 for all t > 0). 3. Nonlinear Feedback Control Design The control objective is to regulate the system output y ) Tpo about a prescribed reference value yr ) T* po. If we recall that the relative degree of a control system is the lowest order time derivative of the output y that is directly related to the control input u (Isidori, 1989), it is clear that the relative degree of system (2) is 2. That is, the dynamics from the input u to the input y can be expressed through a second-order differential equation. Consider the change of coordinates x ) Φ(z), given as

z1 ) y - yr ) Tpo - yr

(5)

z2 ) y(1) ) f1(x,d) where we used the notation y(k) to designate the time derivative dky/dtk. The Jacobian matrix of (5) is JΦ-1(x) )

[

1 0 -2Fp(t)/Mp - U(t)A∂1∆T(u )/Cpp -U(t)A∂2∆T(u )/Cpp

]

where ∂1∆T(u ) ) ∂∆T(u )/∂Tpo and ∂2∆T(u ) ) ∂∆T(u )/ ∂Tco. Assume that ∂2∆T(u ) is bounded away from zero. Then |J(x)| ) U(t)A|∂2∆T(u )|/Cpp * 0, so that the change of coordinates (5) is globally invertible. Define the following functions (1) R1(x,D) ) 2FpTpi /Mp + 2F(1) p (Tpi - Tpo)/Mp -

2Fpf1(x,d)/Mp -2U(1)A∆T(u )/MpCpp 2UA(∂1∆T(u )f1(x,d) + ∂2∆T(u )f2(x,d) + (1) ∂3∆T(u )Tpi + ∂4∆T(u )T(1) ci )/CppMp

R2(x,D) ) -4U(t)A(Tci - Tco)∂2∆T(u )/Mc where ∂3∆T(u ) ) ∂∆T(u )/∂Tpi, ∂4∆T(u ) ) ∂∆T(u )/∂Tci, and D is a vector containing the disturbance d and its time derivatives. In coordinates z, system (4) is written as follows:

z˘ 1 ) z2

(6)

z˘ 2 ) γ1(z,D) + γ2(z,D)u where γi(z,D) ) Ri(Φ(z),D), i ) 1, 2. The system (6) is (1) equivalent to z(2) 1 - γ1(z,z1 ,D) ) γ2(z1,z2,D)u, which is a nonlinear second-order equation in the variable z1 ) Tpo - yr. From (6) we obtain the following input-output linearizing feedback control:

where the gains Ki’s are chosen such that the polynomial Pc(s) ) s2 + K2s + K1 is Hurwitz. The feedback control (7) is well defined since, by assumption, ∂2∆T(u ) * 0 and Tci - Tco * 0, so that γ2(z,D) is bounded away from zero. The control input (7) stabilizes system (6) about the origin, and since x ) Φ(z) is globally invertible, (7) stabilizes the heat exchanger system (1) about the reference value yr. The feedback control (7) will be referred to as ideal feedback control because it yields a linear input-output behavior of the closed-loop system through exact cancellation of the nonlinearities γi’s. As mentioned earlier, the change of coordinates bringing system (1) into system (6) is invertible, and the control input (7) is well defined if ∂2∆T(u ) is bounded away from zero. Consequently, it must be proven that such condition is satisfied by system (1). Suppose that ∆T(u ) is given by the AMTD (3a). Then it is clear that ∂2∆T(u ) ) -1/2. On the other hand, if ∆T(u ) is taken as the LMTD (3b), we have that

∂2∆T(u ) ) [ln(a) + 1/a - 1]/(ln(a))2 where a ) (Tpo - Tci)/(Tpi - Tco). It can be shown that for 0 < a < ∞, |∂2∆T(u )| > σ/2, where σ is a certain positive number. It is interesting to note that, in the case of LMTD, ∂2∆T(u ) f -1/2 as a f 1, so around the value a ) 1, the LMTD behaves very likely the AMTD. 4. An Implicit State Representation To confront the problem of uncertainties compensation, in this section we introduce an alternative representation of the system (6). Let us assume the following. Assumption 1: The temperatures Tpo, Tco, and Tci are available from measurements. This assumption is physically realizable since temperatures can be continuously monitored at high sampling rates of the order of 1000 Hz. Assumption 2: Process disturbance Fp(t), Tpi(t) are bounded and not measured. Only a nominal value of them, e.g., F* p and T* pi, are known. Assumption 3: The heat transfer coefficient U(t) is uncertain. This is a reasonable assumption since U(t) is hardly known. We assume that an upper bound estimate U* of U(t) is known. Due to the above assumptions, the change of coordinates (5) depends on uncertainties and unmeasured disturbances. Hence, we will refer to (5) as an uncertain change of coordinates. Consequently, the state z2 and the functions γi’s in (6) are also uncertain. Notice that the heat exchanger (1) is transformable to the system (6) with unmeasured state (z2) and uncertainties acting only on the equation containing the control input u. In other words, uncertainties in (6) satisfy the so-called matching condition (Gutman, 1979). Note that the state z2 in the state representation (6) is observable from measurements of the output y ) z1. Thus, a state reconstructor can be provided to obtain an estimate zj2. We will look for a control strategy to compensate for the uncertain functions γ1(z,D) and γ2(z,D). Let γe1(z) and γe2(z) be known estimates of γ1(z,D) and γ2(z,D), respectively. Define the uncertainties δi(z,D) ) γi(z,D) - γei (z) (i ) 1, 2). Then system (6) can be rewritten in the following manner:

384 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997

z˘ 1 ) z2

j , such that the practical control the estimates zj2 and η input is

z˘ 2 ) γe1(z) + γe2(z)u + η(t)

u ) up(zj,η j ) ) [-γe1(zj) - η j - K1zj1 - K2zj2]/γe2(zj) (10)

where η(t) ) δ1(z(t),D(t)) + δ2(z(t),D(t))u(t). The term η(t) can be interpreted as a unknown disturbance. Of course, the dynamics of η(t) depend on the state z(t) and the control input u(t), such that the disturbance η(t) includes internal feedback effects, which may lead to unstabilization of the controlled system. One can note that η(t) is accessible from y ) z1. That is, the dynamics of η(t) can be reconstructed from measurements of y ) z1 in the following manner: η(t) ) y(2) - γe1(y,y(1)) - γe2 (y,y(1))u. Thus, in principle, we can develop an observer to reconstruct the dynamics of z2 together with the dynamics of η. To this end, η will be interpreted as an additional state through the extended state space representation:

z˘ 1 ) z2 z˘ 2 ) γe1(z) + γe2(z)u + η

(8a)

η˘ ) Γ(z,η,U,D) where is comprised of the control input u and its time derivative u(1),

Γ(z,η,U,D) ) ∂1δ1(z,D)z2 + ∂2δ1(z,D)(γe1(z) + γe2(z)u + η) + ∂Dδ1(z,D)D(1) + [∂1δ2(z,D)z2 + ∂2δ2(z,D)(γe1(z) + γe2(z)u + η) + ∂Dδ2(z,D)D(1)]u δ2(z,D)u(1)

Following the ideas of Esfandiari and Khalil (1992) on output feedback stabilization of nonlinear system, we propose the following state observer:

zj˘1 ) zj2 + Lβ1(z1 - zj1) j + L2β2(z1 - zj1) zj˘2 ) γe1(zj) + γe2(zj)u + η

(11)

η j˘ ) L3β3(z1 - zj1) where L is a positive parameter and the βj’s are chosen such that the polynomial s3 + β1s2 + β2s + β3 ) 0 is Hurwitz. The parameter L is adjusted to large positive values. In general, this adjustment of L causes an impulsive-like behavior known as the peaking phenomenon (Sussman and Kokotovic, 1989), which may lead to unstabilization of the resulting closed-loop system. To reduce the effects of peaking, Esfandiari and Khalil (1992) proposed bounding the action of the control input by introducing a saturating function in the following form:

u ) sat[up(zj,η j )]

(12)

where the saturation function sat:R f [umin,umax] is defined as

+ (8b)

∂iδj(z,D) ) ∂δj(z,D)/∂zi (i,j ) 1, 2), and D(1) ) dD/dt. It is clear that, since the δi’s are uncertain functions, Γ(z,η,U,D) is also an uncertain function. Despite system (8) involving three states (z1,z2,η), it is actually a twodimensional system. In fact, the function Ψ(z,η,u,D) ) η - δ1(z,D) + δ2(z,D)u is invariant under the dynamics of the system (8) (i.e., dΨ/dt ) 0 along the trajectories of system (8)). In this way, we can see system (8) as a three-dimensional system evolving on a manifold Ψ(z,η,u,D) ) ψ, for a given constant ψ ∈ R (implicit state representation). Based on the above arguments, the extended system (8) is equivalent to the original system (6) as long as initial conditions satisfy Ψ(z0,η0,u0,D0) ) 0. From (8), it is now clear that η is a new state whose dynamics can be reconstructed from measurements of the output

y ) z1 ) Tpo - yr 5. A Robust Control Scheme In this section, we will use the state representation (8) to derive a robust controller for the heat exchanger. An ideal linearizing control law can be written as

u ) uI(z,η) ) [-γe1(z) - η - K1z1 - K2z2]/γe2(z) (9) which leads to exact cancellation of uncertainties η(t) and nonlinearities γe1(z) and γe2(z). The control law (9) cannot be applied just as it is. The uncertain states z2 and η are not available for measurements. However, we take advantage of the property that the dynamics of z2 and η can be reconstructed from measurements of y ) z1. The idea is to construct a state observer to obtain

{

umin sat(u) ) u umax

if u e umin if umin < u < umax if u g umax

In some sense, the saturation of up(zj,η j ,D) can be seen as an anti-wind-up structure to contend with peaking phenomenon. To establish the stabilization properties of the output feedback controller (10)-(12), the following scaled observation errors are defined:

e1 ) L2(z1 - zj1), e2 ) L(z2 - zj2), e3 ) η - η j In (z,e)-coordinates, the right-hand side of system (11) can be written as

zj˘1 ) z2 + L-1(e2 + β1e1) zj˘2 ) γe1(z - M(L)) + γe2(z - M(L))sat(up(z,e)) + η - e3 + β2e1 η j˘ ) Lβ3e1 Using the equality η(t) ) δ1(z(t),D(t)) + δ2(z(t),D(t))u(t), the unsaturated control input (10) can be written as

up(z,e) ) [(-γ1(z,D) - γe1(z - M(L)) + γe1(z)) + e3 - K1(z1 - L-2e1) - K2(z2 - L-1e2)]/(γ2(z,D) γe2(z) + γe2(z - M(L))) where  ) (e1,e2) and M(L) ) diag(L-2,L-1). Note that only the dynamics of η j are of the order of L. The righthand side of the equations for zj are of the order of Lr, r e 0. Using the identity η ) δ1(z,D) + δ2(z,D)u, and through elaborate but straightforward algebraic ma-

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 385

nipulations, it is possible to obtain the following expressions:

e˘ ) LA0(κ)e + φ1(z,N(L)e,D)

(13)

where e ) (e1,e2,e3)T, φ1(z,N(L)e,D) is a nonlinear function of its arguments and depends on the function Γ(z,η,U,D), κ ) [γe2(z) + S ′ ( )(γ2(z,D) - γe2(z))]/γe2(z), S ′( ) is the derivative of the saturating function sat( ), N(L) ) diag[L-2,L-1,1], and

[

-β1 1 0 A0(κ) ) -β2 0 1 -κβ3 0 0

]

The term κ in the above matrix arises when the time derivative of the saturated control input (12) is taken to compute the expression (8b) (recall that the time derivative u(1) is required to compute the dynamics of e3 ) η - η j ). In fact, straightforward algebraic manipulation leads to an expression of the form

u(1) ) Lβ3S ′( )δ2(z,D)/γe2(z)e1 + ϑ(z,N(L)e,D) where ϑ(z,N(L)e,D) is a nonlinear function of its arguments, which is of the order of Lr, r e 0, with respect to the observation error. That is, only the first term in the right-hand side of the above expression is magnified as L is adjusted to large values. Notice that, if κ ) 1 (i.e., γ2(z,D) ) γe2(z)), then the matrix A0(1) is Hurwitz. A result on stability margins in Su (1994) implies that, if 0 < κ < 2, the system e˘ ) LA0(κ)e is globally exponentially stable about the origin. We know that -1 < S ′( ) < +1, so that a sufficient condition to satisfy 0 < κ < 2 is that γe2(z) > γ2(z,D) and sign(γe2(z)) ) sign(γ2(z,D)). That is, the sign of the “high-frequency gain” γ2(z,D) must be known. In process control, the sign of high-frequency gains can be determined by means of step perturbations of the uncontrolled plant. In the case of heat exchangers, it is evident that the sign of γ2(z,D) is positive (respectively negative) if the process fluid is the cooler (respectively warmer) one. On the other hand, the controlled system can be written as

z˘ 1 ) z2

(14)

z˘ 2 ) γ1(z,D) + γ2(z,D)sat[uI(z,D)] + γ2(z,D){sat[up(z,e)] - sat[uI(z,D)]} Notice that up(z,0) ) uI(z,D), and we recover the closedloop system under a saturating version of the ideal feedback control (7). Stability of the closed-loop system (13),(14) can now be established. In this case, Theorem 4 in Esfandiari and Khalil (1992) leads to the conclusion that the closed-loop system is semiglobally, practically stable about the origin. That is, given an arbitrary region of initial conditions R ⊂ R5 containing the origin, there exists a set of parameters (umin,umax,L*), such that, for all L > L*, all trajectories starting into R remain there and converge to a neighborhood R0 of the origin (semiglobal stability). Besides, the size of R0 can be made as smaller as desired (practical stability) by adjusting the parameter L beyond the threshold value L*. Suppose that γe2(z) does not satisfy globally the condition 0 < κ < 2. In this case, one can conclude that the closed-loop system (13),(14) can be stabilized only

in regions R where the condition 0 < [γe2(z) + S ′ ( )(γ2(z,D) - γe2(z))]/γ2(z,D) < 2 is satisfied. Computation of γe1(z) and γe2(z). According to the feedback control law (10)-(12), estimates of γ1(z,D) and γ2(z,D) are required. Since the estimate γe2(z) determines the stability of the observation dynamics e˘ ) LA0(κ), the more demanding estimate is the “high-frequency gain” γe2(z). If the temperatures Tco and Tci were available from measurements, we can choose γe2(z) ) Re2(Φ(z),D), where Re2(x,D) ) -4U*A*(Tci - Tco)∂2∆T(u )/Mc and U*A* is an upper bound for U(t)A (assumption 3). However, the computation of ∂2∆T(u ) is not an easy task. Although most heat exchanger models use LMTD to represent ∆T(u ), the effective temperature difference is actually a very complex function of boundary and internal temperatures. With the temperatures Tci and Tco, and if ∆T(u ) is approximated by the AMTD, then an estimate of γ2(z,D) is given as

γe2(z) ) γ2(Φ-1(x)) ) -2U*A*(Tci - Tco)/Mc On the other hand, the design of the controller (10)(12) can be based on a linear model. In this approach, the estimates γe1(z) and γe2(z) are computed from openloop experiments in actual equipment or linearization around certain nominal operating conditions. From (6), we note that the heat exchanger (1) is represented as a second-order plant with relative degree 2. Furthermore, for given constant input and disturbances (F* c,D*), the uncontrolled heat exchanger is globally asymptotically stable about a certain equilibrium point. If we assume that the system (1) behaves linearly, then the inputoutput dynamical behavior of the plant can be represented as ∆y ) G(s)∆u, where G(s) ) Kp/(s2 + a1s + a1) is a transfer function with poles in the left half plane. Simulation and experimental results (Malleswararao and Chidambaram, 1992; Xuan and Roetzel, 1993) showed that heat exchangers present overdamped responses for most operating conditions. Consequently, we can choose the following transfer function:

G(s) )

Kp

, τ1,τ2 > 0

(τ1s + 1)(τ2s + 1)

(15)

where τ1 and τ2 are the characteristic time constants of the process. If we identify ∆y and ∆u as being equal to z1 ) Tpo - yr and u - ur, respectively, then the plant (15) can be realized in state space form as

z˘ 1 ) z2 z˘ 2 ) -[z1 + (τ1 + τ2)z2 + Kp(u - u*)]/τ1τ2 where z2 ) d(∆y)/dt. A direct comparison of the above system with (6) yields

γe1(z) ≡ -[z1 + (τ1 + τ2)z2]/τ1τ2

(16)

γe2(z) ≡ Kp/τ1τ2 According to the stability analysis of the closed-loop system, |γe2(z)| must be an upper bound of |γ2(z,D)| in a certain region of interest. Hence, in the identification stage of the parameters (Kp,τ1,τ2), the magnitude of the gain Kp must be overestimated, and the magnitude of the time constant τ1,τ2 must be underestimated. That

386 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 Table 1. Model Parameters Used in Numerical Simulations U A Cpp Mp F* p F*c T*pi Mc Tci

0.6 Btu min-1 F-1 ft-2 20 ft2 0.38 Btu lbm-1 °F 15 lbm 25 lbm min-1 40 lbm min-1 0.456 Btu lbm-1 °F-1 40 lbm 70 °F

is, the model (15) must be constructed on the basis of the faster dynamics of the process. The identification of the parameters Kp, τ1, and τ2 can be carried out with standard identification methods (Shinskey, 1994). Since the heat exchanger is open-loop stable, such identification is easier than in the general case, and very standard identification procedures can be used. A direct approach is to use output system response to step changes in the control input (Luyben, 1990). We will use this identification approach in the numerical simulations of the next section.

Figure 1. Performance of the output feedback controller (10)(12). A step change in Fp from 25 to 30 lbm/min occurs at t ) 20 min. The controller adjusts the heating fluid flow rate Fc to keep the process fluid temperature Tpo at the reference value of 130 °F.

6. Numerical Simulations In this section, we illustrate the features of the output feedback (10)-(12) controlling the heat exchanger system (1). The system parameters for numerical simulations have been taken from Malleswararao and Chidambaram (1992). For the sake of completeness, the values are reported here in Table 1. In that table, F* c, F* po, and T* pi designate nominal values used to identify the parameters in the plant model (15). The parameters in (15) were estimated in the following manner. First, a +0.25% step perturbation in the nominal value of the control input u ) Fc was applied to the heat exchanger. Then, the parameters of the second-order model (15) were adjusted to fit the dynamic response of the output y ) Tpo. For the nominal values in Table 1, we get the approximate values Kp ) -0.23 °F min lbm-1, τ1 ) 0.4 min, and τ2 ) 0.3 min, so that

γe1(z) ) -8.3z1 - 5.8z2 γe2(z) = -1.9 The control gains K1,K2 were chosen such that the time constants of the ideal closed-loop system z(2) 1 + K2z(1) 1 + K1z1 ) 0 are τc,1 ) τc,2 ) 0.25 min (i.e., the roots of the characteristic polynomial s2 + K2s + K1 ) 0 are placed at -4.0). Consequently, K1 ) 16.0, and K2 ) 8.0. As in internal model control approach, the idea of this tuning of the control loop is to induce a slightly faster closed-loop response than the open-loop response. In this way, most control effort will be devoted to reject disturbances and to track set-point changes. Regarding the observer, parameters βj’s are chosen as follows: β1 ) 3, β2 ) 3, and β1 ) 1, such that all the eigenvalues of matrix A0(1) in (13) are located at -1. All numerical simulations were carried out in MATLAB. Robustness against External Disturbances. To evaluate the disturbance rejection capacity of controller (10)-(12), the following disturbance was taken:

Tpi(t) ) T* pi + 5 sin(0.5t) The desired set value of Tpo was 130 °F. Initial conditions were taken as Tpo(0) ) 150 °F and Tco(0) )

Figure 2. Dynamics of the control input Fc corresponding to the dynamical behavior in Figure 1.

115 °F. Analogous results as described below were obtained for different initial conditions. The controller states were initialized at zj1(0) ) 145 °F, zj2(0) ) ηj (0) ) 0. The performance of the control system is shown in Figure 1. A good regulatory response is obtained for L ) 5, such that the nominal dynamics of the observer (11) are 20 times faster than the nominal dynamics of the controlled system. Tuning of the parameter L was performed on the basis of pole placement. This tuning method is valid under the assumption of very small measurement noise. In actual applications, an optimal tuning method (i.e., optimal Kalman filtering) can be used. In some cases, prefiltering of measured signals together with sliding mode observation can be used to reduce the effects of noise in the estimated states (Elmali and Olgac, 1996; Lee and Tomizuka, 1996). The process fluid temperature Tpo is taken very close to the reference value yr ) 130 °F, despite uncertainties and disturbances acting on the system. In fact, the error in the regulated temperature is not larger than 0.2 °F. The performance of the control system for the step change in Fp from 25 to 30 lbm min-1 at t ) 20 min is also shown in Figure 1. For larger values of process flow rate Fp, the process fluid temperature Tpo is more sensitive to disturbances. This behavior can be observed in Figure 1, where the oscillations amplitude in Tpo becomes larger after the perturbation in Fp. Under these operating conditions, the regulation error |Tpo - yr| can be reduced if the parameter L is adjusted. The corresponding values of the manipulated variable are shown in Figure 2.

Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997 387

Figure 3. Response of the controlled system for +20% perturbations in the heat transfer coefficient U at t ) 20 min.

Figure 6. Control input Fc used to change the process fluid temperature Tpo from 130 to 125 °F.

Figure 4. Dynamics of the control input Fc corresponding to the dynamical behavior in Figure 1.

Figure 7. Performance of the controlled system under first-order, unmodeled actuator dynamics. Notice the oscillations in Tpo induced by the unmodeled dynamics.

Robustness against Unmodeled Dynamics. Although in the above section the problem of robustness against unmodeled dynamics was not addressed, we show with simulations that the system control (10)(12) yields closed-loop stability despite a certain class of unmodeled dynamics. A theoretical framework to study this class of problems is under development and will be reported in a future work. Unmodeled dynamics arise when, for instance, the dynamics of actuators are not considered in the control design stage. In general, the dynamics of actuators such as valves are faster than dynamics of processes. Suppose that the valve actuator is subjected to the following unitary gain, first-order dynamics: Figure 5. Performance of the closed-loop system under servo operating conditions when the reference value yr is changed from 130 to 125 °F.

The robustness of the control system for +20% perturbations in the heat transfer coefficient U at t ) 20 min is shown in Figure 3. The control system (10)(12) adjusts the input Fc to regulate the process fluid temperature Tpo in the new operating conditions. The dynamics of the control input Fc are shown in Figure 4. Steady-state oscillations in Fc are due to the external disturbance Tpi(t). The performance of the control system for servo problem (from yr ) 130 to yr ) 125 °F) is considered. The performance of the present control system is shown in Figure 5. Under the control action shown in Figure 6, the process fluid temperature Tpo converges to the new reference value without overshooting.

Fc )

Fc τas + 1

where τa is the time constant of the actuator, Fc is the actual heat fluid flow rate, and Fc is the command input from the control system (10)-(12). Notice that, if τa ) 0 (infinitely fast actuator dynamics), then Fc ) Fc, and stability of the closed-loop system follows from the analysis in the above sections. For the sake of numerical simulations, we choose τa ) 0.1 min and the same control parameters as in the regulation case. Besides, it was assumed that Tpo ) T*po (no external perturbations). Figure 7 shows the performance of the controller for a step change in Tci from 70 to 75 °F. Due to the delay induced by the actuator dynamics, the closed-loop dynamics are slightly slower than in the case with no actuator dynamics. However, the performance of the

388 Ind. Eng. Chem. Res., Vol. 36, No. 2, 1997

Acknowledgment This work was partially supported by Industrias Negromex, S.A. de C.V. Nomenclature e1, e2, e3 ) scaled observation errors Fc ) heating fluid flow rate (manipulated variable) Fp ) process fluid flow rate J(x) ) Jacobian matrix K1, K2 ) control gains L ) observer adjustable parameter Tci, Tco ) inlet and outlet heating flow temperature, respectively Greek Letters Figure 8. Heating fluid flow rate Fc induced by the controller (10)-(12). Notice that, due to the delay induced by actuator dynamics, Fc oscillates before it attains a steady-state value.

controller is still very good. Contrary to the case in the above section, the regulated temperature Tpo does not present significant overshooting. This behavior is caused by the dynamics of the actuator, which acts as a smoother of the control actions (low-pass filter) (Morari and Zafiriou, 1989). Figure 8 presents the heating fluid flow rate Fc induced by the controller (10)-(12). Notice that, due to the delay induced by actuator dynamics, Fc oscillates before it attains a steady-state value. The above-described dynamical behavior is obtained thanks to the uncertainty estimator included in the control system (10)-(12). In all the above cases, one can note the impulsive-like behavior (peaking phenomenon) in the controlled temperature Tpo, which is caused by the transient behavior in the uncertainty estimator (11). That is, the high-gain observer (11) acts as an integral action. A possible remedy to reduce overshooting is to use a control strategy as in neural network control. In a first stage (learning stage), the observer (11) evolves under open-loop conditions. In a second stage, the controller (10) is connected to system. In this way, overshooting induced by initial conditions could be reduced. A second approach is to use low-pass filtering to smooth control actions (Morari and Zafiriou, 1989). 7. Conclusions An scheme for the robust control of a fluid-fluid heat exchanger was presented. It is composed of two controller components: an input-output linearizing feedback and an observer-based uncertainty estimator. The performance of the robust controller has been examined through numerical simulations under various uncertainties and external disturbances. Tuning of the controller was based on standard methods of linear control systems. Even though better results can be obtained by using more sophisticated tuning, the effectiveness of the proposed method is proven by a numerical example. The proposed structure has been shown to maintain good properties, even when confronting significant modeling errors, such as parameter uncertainties and unmodeled actuator dynamics. It must be pointed out that the applicability of the proposed controller (10)-(12) is not restricted to heat exchangers. In fact, our design methodology can be easily extended to higher dimensional, linearizable (partially or fully) systems, such as chemical reactors and distillation columns. Results in this direction will be reported in a future work.

β1, β2, β3 ) coefficients of the characteristic polynomial of the observation error companion matrix A0(κ) γe1, γe2 ) estimates of the uncertain transformed functions ∆T(u ) ) effective mean difference of temperature η ) lumped uncertainties and external disturbances

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Received for review August 8, 1996 Revised manuscript received October 23, 1996 Accepted October 23, 1996X IE960496M

X Abstract published in Advance ACS Abstracts, December 15, 1996.