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Model Approximation for Dead-Time Recycling Systems Basilio del-Muro-Cue´ llar,*,† Martı´n Velasco-Villa,‡ He´ ctor Puebla,† and Jose´ A Ä lvarez-Ramı´rez†,§ Instituto Mexicano del Petro´ leo, Eje Central La´ zaro Ca´ rdenas 152, Col. San Bartolo Atepehuacan, Apartado Postal 07730, Me´ xico D.F., Me´ xico, and CINVESTAV-IPN, Departamento de Ingenierı´a Ele´ ctrica, Seccio´ n de Mecatro´ nica, AP 14-740, 07000 Me´ xico D.F., Me´ xico
Chemical processes with recycling commonly contain delays in both the forward and the backward paths. The characteristic equation of such systems is a quasipolynomial function, so that the corresponding transfer function contains an infinite number of poles. This feature precludes the use of classical stability analysis and control design techniques. In this work, a simple and effective methodology to derive an approximate discrete-time model for continuous-time recycling processes with delay is proposed. The method is based on the discretization, via a fictitious sampler and hold device, of the internal delayed signal, resulting in a finite-dimensional discrete-time version of the original continuous model. In this way, standard analysis methods, such as root locus and stability margin techniques, can be easily applied to the approximate model to obtain some conclusions on the stability of the original recycling process. Illustrative examples are used to show that some stability measures (e.g., stability margin) obtained with the approximate discrete-time model closely describe the behavior of the original recycling process. 1. Introduction Recycling processes are commonly found in chemical processes. Recycling systems enable the energy and matter to be recovered in an industrial process. For instance, recycle loops encountered in reactive processes are based on recycling the nonreacted feed after withdrawal of the product, recycling the heat generated in the system (through external or internal heat exchangers), or the simultaneous recycling of both heat and mass.1 On the other hand, significant transport delays are present in most chemical plants. Thus, a realistic dynamic model of a recycling chemical process should contain delays in both the forward and the backward paths.2 This is the case, for instance, with interconnected reactor and separation units where mass recycling is incorporated to recover reactive material and delays are due to the mass transport between equipment. The design of the control system for processes with recycling and dead time presents some specific difficulties because neglecting the effect of recycling and dead time leads to unsatisfactory performance and, in some cases, instabilities may appear in the closed-loop response. Recycle and dead-time systems lead, in general, to transfer functions with quasipolynomial functions including transcendental exponential terms in both the denominator and the numerator, and consequently, the factorization into rational transfer functions and pure delay is not possible. In particular, the denominator dead-time terms preclude the use of standard stability analysis (e.g., root locus method) and control design techniques because the mathematical methods em* To whom correspondence should be addressed. Tel.: +52-55-91757571. Fax: +52-55-91756277. E-mail: bdelmuro@ imp.mx. † Instituto Mexicano del Petro´leo. ‡ CINVESTAV-IPN, Departamento de Ingenierı´a Ele´ctrica. § Also at Universidad Autonoma Metropolitana-Iztapalapa.
ployed by these techniques require transfer functions with rational denominators. In fact, the stability analysis of recycle and dead-time systems involves finding an infinite number of roots of a transcendental equation. Model approximation has been proposed to remove the exponential term from the denominator, such as the method of moments3 and Pade´ approximations.4 Other techniques, such as Taylor series expansion5 and the seasonal time-series model,6 have been proposed to obtain an approximate model to represent recycle systems. In turn, such approximate models can be used for stability analysis or control design.7-15 In this work, a simple and effective methodology to derive an approximate discrete-time model for continuous-time recycling processes with a delay is proposed. The method is based on the discretization, via a fictitious sampler and hold device, of the internal delayed signal, resulting in a finite-dimensional discrete-time model that captures the dominant dynamics of the original continuous model. In this way, standard analysis methods, such as root locus and stability margin techniques, can be easily applied to the approximate model to obtain some conclusions on the stability of the original recycling process. Two illustrative examples are used to show that some stability measures (e.g., stability margin) obtained with the approximate discrete-time model closely describe the behavior of the original recycling process. 2. A Class of Recycle Systems with Time Delays Consider recycling processes as described in Figure 1, which can be described as follows:
[ ]
Y(s) ) [Gf(s) Gf(s)Gr(s)] where
10.1021/ie0402113 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/10/2005
U(s) Y(s)
(1)
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Figure 1. General recycle configuration.
Gf(s) ) Gr(s) )
Nf(s) e-hfs Df(s) Nr(s) e-hrs Dr(s)
Figure 2. General scheme of approximation.
with Gf(s) and Gr(s) being, respectively, the transfer functions of the forward and backward (or recycle) paths, hr g 0 is the recycle-path time delay, and hf g 0 is the forward-path time delay. In addition, Nf(s), Nr(s), Df(s), and Dr(s) are polynomials, and U(s) and Y(s) are the process input and output signals, respectively. Simple block diagram manipulations give the following Gt(s)
transfer function for the overall path U(s) 98 Y(s): -hfs
Gt(s) )
Dr(s) Nf(s) e
Df(s) Dr(s) - Nf(s) Nr(s) e-(hf+hr)s
(2)
Notice that exponential terms appear explicitly in the denominator and numerator of Gt(s). As usual, the stability of the process in eq 2 is governed by the Hurwitz stability of characteristic equation -(hf+hr)s
Q(s) ) Df(s) Dr(s) - Nf(s) Nr(s) e
(3)
That is, the overall path U(s) f Y(s) is stable if and only if all of the roots of Q(s) are contained in the open lefthalf complex plane. In the absence of time delays, Q(s) is reduced to a polynomial whose stability conditions can be easily stated by means of, for example, a Routh array. However, if hf + hr > 0, the characteristic equation Q(s) has a transcendental term e-(hf+hr)s that induces an infinite number of roots. In this way, classical control design techniques and stability analysis methods, such as root locus representations, cannot be used. In this form, model-reduction techniques to obtain finite-dimensional systems are desirable to obtain useful information on the stability of the process in eq 2. A feature of processes containing exponential terms in the denominator is that they correspond to delays in internal states. In fact, the class of recycling processes with delays given by eq 2 belongs to a general class of systems whose state representation is given by
Y(s) ) C[sI - (A + A1 e-hs)]-1B e-τs U(s)
(5)
The characteristic equation of the above equation is described by a quasipolynomial of the form
R(s) ) det(sI - A - A1 e-hs)
(6)
Notice that the quasipolynomial Q(s) given by eq 3 is, in fact, a particular case of the general quasipolynomial form R(s). The time delay associated with the state leads to a characteristic equation (eq 6) with an infinite number of solutions. It is noticed that the class of systems described by eq 5 can be represented as differentialdifference equations.4 From a theoretical point of view, the stability of linear time invariant systems with time delays at the input signal and the state has been widely studied, producing several methodologies to evaluate their stability properties. In particular, in Mori and Kokame16 and in Wang,17 the stability properties of systems described by eq 4 were analyzed, leading to sufficient conditions dependent on the time-delay value. However, the stability conditions can be so conservative that their application for practical cases yields limited results. In the following section, we will provide a methodology to obtain a finite-dimensional (reducedorder) model that captures the main dynamics of the system in eq 4. 3. Approximate Discrete-Time Model
x˘ (t) ) Ax(t) + A1x(t - h) + Bu(t - τ) y(t) ) Cx(t) x(φ) ) φ(φ), φ ∈[-h, 0]
For simplicity in the presentation, and without losing generality, we will consider single-input-single-output systems, taking into account time delays in the state and input. However, the extension of this method to the multivariable case is straightforward. The Laplace transform of the system in eq 4 leads to the following expression:
(4)
where x ∈ Rn is the state vector, u ∈ R is the input, y ∈ R is the output, h g 0 is the time delay associated with the state, τ g 0 is the time delay associated with the input, and φ(φ) is a continuous function of initial conditions with - h e φ e 0. Finally, A, A1 ∈ Rn×n, B ∈ Rn×1, and C ∈ R1×n are matrices and vectors of system parameters.
Consider the delay associated with the state as a second (virtual) input to the system. When a zero-order hold (ZOH) is introduced to this virtual input, as shown in Figure 2, the following partially sampled system is obtained:
x˘ r(t) ) Axr(t) + A1ξ(tk - h) + Bu(t - τ) y(t) ) Cxr(t) x(t0 - h) ) φ(φ)
(7)
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where ξ(tk - h) is a piecewise constant function that corresponds to the virtual (sampled) input for system 4 (see Figure 2). The following result states an approximating connection between the system (eq 4) and the partially sampled system (eq 7). Theorem 1. Assume that A is a Hurwitz matrix. The solution xr(t) of system 7 is a uniform O(T) approximation of the solution x(t) of system 4 with respect to the sampling period T. That is, xr(t) f x(t) uniformly as T f 0. Proof. Consider the solution of system 4 for t0 e t < h and the initial condition ξ(t - h). Notice that for t0 e t < h, the delayed state x(t - h) is determined by the initial condition function. Then, it is possible to consider the following systems for t0 e t < h:
x˘ (t) ) Ax(t) + A1ξ(t - h) + Bu(t - τ)
period T. Therefore, increasing the frequency T f 0 will also improve the approximation. On the basis of Theorem 1, a finite-dimensional discrete-time model of the system (eq 4) can be derived. Consider the solution xr(t) of the (partially sampled) system (eq 7), over the interval t0 e t < h, with the sampled initial condition ξ(tk - h):
xr(t) ) Φ(t, t0) x(t0) +
The signals ξ(tk - h) and u(t - τ) can be seen as external inputs. By considering a ZOH over the input u(t - τ), it is possible to write
xr(tk+1) ) Φ(tk+1, tk) x(tk) +
∫tt
When the error signal is defined as ex(t) ) x(t) - xr(t), the following dynamic system is obtained:
(8)
where γ(t) is a function defined as
where Φ(tk+1, tk) ) e(tk+1-tk)A ) eTA. The following result describes the structure of the approximate model for the original delayed system (eq 7). Proposition 1. An approximate discrete-time model for the continuous-time system (eq 7) is given by
h xk + A h 1x(tk - h) + B h u(tk - τ) xk+1 ) A
γ(t) ) ξ(t - h) - ξ(tk - h) γ(t) corresponds to the error induced by the consideration of the ZOH on the delayed state feedback loop of system 4. To explicitly show the order of approximation of system 7 with respect to system 4, also consider the solution of system 8 on the segment t0 e t < h, which is given by
∫ttΦ(t, s) A1γ(s) ds 0
where Φ correspond to the transition matrix of system 8. Considering the effect of the ZOH and the fact that the initial condition is the same for systems 4 and 7, it follows that
ex(t0) ) ξ(t0) - ξr(t0) ) 0 then,
ex(t) )
∫ttΦ(t, s) A1γ(s) ds 0
Φ(tk+1, s) ds [A1ξ(tk - h) + Bu(tk - τ)]
k+1
k
ex(t) ) Φ(t, t0) ex(t0) +
0
0
x˘ r(t) ) Axr(t) + A1ξ(tk - h) + Bu(t - τ)
e˘ x(t) ) Aex(t) + A1γ(t)
∫ttΦ(t, s) A1ξ(tk - h) ds + ∫thΦ(t, s) Bu(s - τ) ds
(9)
Because γ(t) ) ξ(t - h) - ξ(tk - h) is a consequence of the ZOH, then γ(t) is a function of order 1 (with respect to the sampling period T), that is, γ(t) ) O(T). The equation (eq 9) can be seen as an integration error between functions ξ(t - h) and ξ(tk - h); therefore, when a rectangular (by the effect of the ZOH) numerical integration method (quadrature formula) is considered,18 it follows that the integral (eq 9) also represents a function of order 1. To conclude the proof, it would suffice to consider the step method19 for the solution of system 8 over the segments tk e t < tk + h. The proof of the above result states that the error ex(t) can be minimized by considering a sampling and hold device of higher order of a different type, for example, a polygonal hold (PH) that corresponds in terms of numerical integration to the use of a trapezoidal method. On the other hand, it is important to note that the error of approximation ex(t) depends directly on the sampling
with
h 1) A h ) Φ(tk+1, tk), A B h )
∫tt
∫tt
k+1
k
Φ(tk+1, s) ds A1
k+1
k
Φ(tk+1, s) ds B
Proof. From the above developments and from Theorem 1, we have determined that xr(t) is an approximation for x(t); therefore, for t0 e t < h, the following approximate discrete-time system can be obtained
h xk + A h 1ξ(tk - h) + B h u(tk - τ) xk+1 ) A
(10)
When the above developments are applied over the segments tk e t < tk + h, it follows that the initial condition ξ(tk - h) is equal to the delayed state x(tk-1 h) obtained on the precedent segment of time. This concludes the proof. It should be noted that when the time delays h and τ of system 7 are multiples of the sampling period T, that is, τ ) m1T and h ) m2T (commensurable delays), system 10 can be depicted as in Figure 3. If the time delays are both rational numbers, it is always possible to find sampling times that partition the time delays in an exact way. If one or both time delays are irrational numbers, it is always possible to approximate them with a rational number. In fact, for a given irrational number, there is a rational number arbitrarily close. On the other hand, finding irrational time delays is a rare case because even numerical processing is made with rational numbers. 3.1. Discrete-Time Approximate Model for Recycle Systems. As stated above, although the transfer function for a recycle system may be constructed from the individual linear transfer functions for each of the subunits comprising the plant, the resultant overall plant transfer function will often contain denominator dead-time terms. It is, therefore, necessary to simplify
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standard polynomial approaches, or more sophisticated model-based controllers. 4. Illustrative Examples
Figure 3. Approximate discrete time system.
these models before they can be used for stability analysis and control design purposes in a standard sense. By exploiting the transfer function of the overall recycle systems (eq 2), we have the following result. Proposition 2. An approximate discrete-time model for the continuous-time recycle system (eq 1) is given by
Y(z) ) [Gf(z) Gfr(z)]
[ ] U(z) Y(z)
(11)
where Gf(z) ) (1 - z-1)Z(Gf(s)/s) and Gfr(z) ) (1 - z-1)Z(Gfr(s)/s); that is, Gf(z) and Gfr(z) represent the z transforms of Gf(s) and Gf(s)Gr(s), respectively, considering a ZOH device. Proof. The proof can be obtained directly from the results obtained in the previous section. In fact, this is done by considering the fictitious (two inputs) system
Y(s) ) [Gf(s) Gf(s)Gr(s)]
The aim of this section is to use some typical examples from recycling processes, all with dead time in both the forward and recycle paths, to illustrate the goodness of the discrete-time approximation. To this end, some stability measures, such as stability margins in a root locus scheme, will be used. The first case study consists of a second-order system with zeros of nonminimun phase in the direct path and a first-order system in the recycle path, which can be used to represent chemical engineering prototype recycle systems. The second case consists of a transfer function representing an industrial quench column with data collected after an open-loop pulse test.20 It should be stressed that the extension of the approximation model methodology to the multiple-input-multiple-output case is straighforward from the results presented in Section 3. However, because we are concerned only with recycling processes with time delays, the worked examples are used to illustrate the ability of the finite-dimensional approximate model to describe some dynamic features of recycling processes. 4.1. Example 1. It is a well-accepted fact that the dynamics of many chemical processes can be modeled by input-output first- or second-order models with delay (see, for instance, Morari and Zafiriou12). We consider a chemical recycle system with a transfer function of a second-order system with zeros of nonminimun phase in the direct path and a first-order system in the recycle path:
[ ]
Gf(s) )
U(s) U1(s)
[ ] U(z) U1(z)
where Gf(z) and Gfr(z) represent the z transforms of Gf(z) and Gf(s)Gr(s), respectively, considering a ZOH at the inputs. In this form, the global transfer function for the approximate discrete-time model of recycle systems with delays can be obtained as
Gt(z) )
Dfr(z) Nf(z) Df(z) Dfr(z) - Nf(z) Nfr(z)
where Nf(z), Nfr(z), Df(z), and Dfr(z) are polynomials in z such that Gf(z) ) Nf(z)/Df(z) and Gfr(z) ) Nfr(z)/Dfr(z). Once the transfer function of the approximate discretetime model Gt(z) ) R(z)/W(z) has been computed, a feedback control design can be carried out on the basis of the stability analysis of the approximate model. By virtue of Theorem 1, one knows that conclusions drawn from the approximate (finite-dimensional) discrete-time model reflect the behavior of the original continuoustime (infinite-dimensional) system. For instance, a feedback control design based on the approximate discrete-time model can be addressed with different compensator designs, such as simple PI controllers,
(s + 1)2
Gr(s) )
and the corresponding discrete-time system
Y(z) ) [Gf(z) Gfr(z)]
(s - 3) e-1.2s e-0.8s s+2
This case can represent (i) interconnected reactor and separation units where mass recycling is incorporated to recover reactive material and delays are due to the mass transport between equipment or (ii) FCC units with coupled riser and regenerator units. The overall transfer function is given by
Y(s) (s + 2)(s - 3) e-1.2s ) U(s) (s + 1)2(s + 2) - (s - 3) e-2s
(12)
To obtain the approximate discrete-time representation, let us consider the fictitious (two inputs) system
Y(s) )
[
(s - 3) e-1.2s (s - 3) e-2s 2
(s + 1)
][ ]
U(s) (s + 1) (s + 2) U1(s) 2
where signal U1(s) corresponds to the fictitious input. The corresponding discrete-time system with a ZOH at the inputs and a sampling period T ) 0.4 is given by
Y(z) )
[
0.08347z - 0.4095 z - 1.351z3 + 0.4493z2 4
][ ]
0.02551z2 - 0.00792z - 0.03737 U(z) z8 - 1.79z7 + 1.052z6 - 0.2019z5 U1(z)
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Figure 4. Original system versus ZOH, PH, and Pade´ approximations.
By considering U1(z) ) Y(z), the approximate discretetime model, based on a ZOH, is obtained as
Y(z) )
[
0.08347z - 0.4095 z4 - 1.341z3 + 0.4493z2
][ ]
0.02551z2 - 0.00792z - 0.03737 U(z) z8 - 1.79z7 + 1.052z6 - 0.2019z5 Y(z) that can be rewritten as
Y(z) B(z) ) U(z) A(z) with
B(z) ) 0.08347z9 - 0.559z8 + 0.8209z7 0.4476z6 + 0.08268z5 A(z) ) z12 - 3.131z11 + 3.901z10 - 2.416z9 + 0.7432z8 - 0.09072z7 - 0.02551z6 + 0.1121z5 0.07855z4 - 0.01509z3 + 0.01679z2 In Figure 4, the transient behavior of the approximate model is shown when compared with the original continuous system under a unit step input. The response of the approximate model obtained when a Pade´ approximation is used to transform the time delay in a rational function is also shown. Simulations with a firstorder Taylor expansion (not shown) for time delays resulted in an excessively poor, even unstable, ap-
Figure 5. Root locus for approximate system 12.
proximation. Also in Figure 4, it is possible to appreciate how the accuracy of the approximation is increased when the proposed methodology is used together with a PH instead of a ZOH. To validate the accuracy of the discrete model, the root locus of the discrete-time model is shown in Figure 5. It can be seen that two open-loop (discrete-time) poles are located on the unitary circle; that is, the discretetime model indicates that the open-loop system is critically stable. Consider the discrete time controller u(kT) ) k[v(kT) - y(kT)], with v(kT) as a step signal. It is possible to
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Figure 6. Closed-loop stability properties for system 12.
Figure 7. Stability gain for controlled system 12.
see in Figure 6 that the original continuous closedloop system is marginally stable in an open loop, underdamped for k ) 0.4, and overdamped for k ) 0.77. From Figure 5, it can be appreciated that the system is marginally stable for k ≈ 0.83. This can be validated by simulation, see Figure 7, where the system result is stable with k ) 0.820 and unstable with k ) 0.833. 4.2. Example 2. The quench column is a tray steel column in which the cracking furnace exit gases are quenched and fractionated. Coke is removed from the
quench column by circulating the quench bottoms through a coke filter. The quenching overheads from both columns are condensed in two stages; the condensed liquor from the first stage is primarily used as reflux to the quench column, whereas the uncondensed vapors are fed to the second stage where the remaining vapors are condensed. The bottom cooled water is recycled to the quench column as upper reflux and a lower reflux. A rigorous model of the process consists of several balance equations. Emoto et al.20 conducted an open-loop step test on the process. When these data
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Figure 8. Output of the continuous original model vs output of the discrete approximate model with a ZOH for the industrial quench column.
Figure 9. Output system following a unit step input and rejecting a unit step input at t ) 150s for the industrial quench column.
were used, the following models were obtained for the forward and recycle paths, respectively,
Gl(s) ) Gr(s) )
0.0777 e-s 1.25s + 1
6.2173 e-11s 1.44s + 1
It can be seen from the above transfer functions that
there is a considerable time-delay separation between the direct and recycle paths. However, large time-delays are not an obstruction for the application of the model approximation methodology. Indeed, large time delays can lead only to high-order approximations. A description of the system is given by
Y(s) ) [Gl(s) Gl(s)Gr(s)]
[ ] U(s) Y(s)
(13)
where Y(s) is the output and U(s) the input. Let us
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consider the two-inputs system
Y(s) )
[
][ ]
U(s) 0.0777 e-s 0.4831 e-12s 2 1.25s + 1 1.8s + 2.69s + 1 U1(s)
Therefore, when the described methodology for T ) 1 is applied, it is possible to get the approximate discretetime model of the system (eq 13) on the basis of ZOH
Y(z) )
[
][ ]
U(z) 0.04279 0.08287z + 0.05031 14 13 12 Y(z) z - 0.4493z z - 0.9487z + 0.2244z 2
or, equivalently,
Y(z) 0.04279z14 - 0.04059z13 + 0.009601z12 ) 16 U(z) z - 1.398z15 + 0.6507z14 - 0.1008z13 0.08287z3 - 0.01308z2 + 0.0226z Figure 8 shows the transient behavior of the approximate model when compared with the original continuous system under an unitary step input. It can be seen that despite the complexity of the original system, the approximate model can follow the original system trajectory with reasonable accuracy. It must be pointed out that the mismatch between the original continuous model and the approximate discrete-time model can be improved by considering a high-order or polygonal hold in the model approximation methodology, as in Example 1. Finally, from the root locus diagram (not shown) built from the approximate discrete-time model, the following simple discrete PID-type (PID ) proportional-integralderivative) controller has been designed:
HPID(z) )
2z2 z2 - 1
In the quench column, the control input is the bypass flow rate of the reflux cooler via a bottom temperature controller. The response of the discrete PID-type controller to both a step reference and a step disturbance applied at t ) 150s can be seen in Figure 9. The controller provides an acceptable closed-loop behavior; however, it has to deal with a high-order dead time in the recycle path, which degrades its performance. Nevertheless, it can be seen from Figure 9 that a simple PID-type controller designed from the properties obtained from the approximate model can follow step inputs and reject step disturbances. Thus, this example has illustrated the accuracy of approximation of the dominant poles of the original transfer function with the approximate model. 5. Conclusions We have proposed a model approximation methodology with applications to dead-time and recycle systems. Recycling processes with a time delay in both forward and recycle paths can be considered within a general class of invariant systems involving time delays at the input signal and at the state. By means of either a ZOH or a PH on the time delay at the state, we have derived a sampled time-delay model, which is used to obtain an approximate discrete-time model of an original timedelay continuous model. Using the presented method, we can arbitrarily choose a suitable approximation of a
delay element for a desirable design. Although the suggested approach is restricted to the class of recycle systems described here, it can be applied to a wide class of linear systems including nonminimum phase systems. Numerical simulations for two recycle systems with dead time in both the forward and direct paths show both the effectiveness of the proposed model approximation methodology and the closed-loop behavior of control designs based on approximate discrete-time plant models. Acknowledgment This work was partially supported by CONACyTMe´xico under Grant 42093. Literature Cited (1) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999. (2) Shinskey, F. G. Process control: as taught vs as practiced. Ind. Eng. Chem. Res. 2002, 41, 3745-3750. (3) Papadourakis, A.; Doherty, M. F.; Douglas, J. M. Approximate dynamic models for chemical process systems. Ind. Eng. Chem. Res. 1989, 28, 546-552. (4) Malek-Zavarei, M.; Jamshidi, M. Time-Delay Systems. Analysis, Optimization and Applications; North-Holland: Amsterdam, The Netherlands, 1987. (5) Hugo, A. J.; Taylor, P. A.; Wright, J. D. Approximate dynamic models for recycle systems. Ind. Eng. Chem. Res. 1996, 35, 485-87. (6) Kwok, K. E.; Chong-Ping, M.; Dumont, G. A. Seasonal model based control of processes with recycle dynamics. Ind. Eng. Chem. Res. 2001, 40, 1633. (7) Samyudia, Y. K.; Kadiman, K.; Lee, P. L.; Cameron, I. T. Gap metric based control of processes with recycle systems. Proceedings of ADChEM, Pisa, Italy, 2000; pp 497-502. (8) Taiwo, O. The design of robust control system for plants with recycle. Int. J. Control 1986, 43 (2), 671-678. (9) Scali, C.; Ferrari, F. Performance of control systems based on recycle compensators in integrated plants. J. Process Control 1999, 9, 425-437. (10) Smith, O. J. Closer control of loops with dead time. Chem. Eng. Prog. 1957, 53 (5), 217-219. (11) Wellons, M. C.; Edgar, T. F. The generalized analytical predictor. Ind. Eng. Chem. Res. 1987, 26 (8), 1523-1536. (12) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: Englewood Cliffs, New Jersey, 1989. (13) Astrom, K. J.; Hang, C. C.; Lim, B. C. A new smith predictor for controlling a process with an integrator and long dead time. IEEE Trans. Autom. Control 1994, 39, 343-345. (14) Watanabe, K.; Ito, M. Process-model control for linear systems with delay. IEEE Trans. Autom. Control 1981, AC-26, 1261-1269. (15) Maza-Casas, L.; Velasco-Villa, M.; Alvarez-Gallegos, J. On the state prediction of linear systems with time-delays in the input and the state. In Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona, December 1999; pp 239244. (16) Mori, T.; Kokame, H. Stability of x˘ (t) ) Ax(t) + Bx(t - τ). IEEE Trans. Autom. Control 1989, 34, 460-462. (17) Wang, S. S. Further results on stability of x˘ (t) ) Ax(t) + Bx(t - τ). Syst. Control Lett. 1992, 19, 165-168. (18) Mathews, J. H. Numerical Methods for Mathematics Science and Engineering; Prentice Hall: New York, 1992. (19) Driver, R. D. Ordinary and Delay Differential Equations; Springer-Verlag: New York, 1977. (20) Emoto, G.; Miller, R. M.; Ebara, S. Modeling and identification for quench column bottom temperature control. In Proceedings of the 11th IFAC Symposium on System Identification, Fukuoka, Japan, July 1997; Elsevier Science: New York 1997; p 553.
Received for review August 4, 2004 Revised manuscript received March 14, 2005 Accepted April 8, 2005 IE0402113
