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Ind. Eng. Chem. Res. 2000, 39, 432-440
A Novel Proportional-Integral-Derivative Control Configuration with Application to the Control of Batch Distillation Jose Alvarez-Ramirez,*,† Rosendo Monroy-Loperena,‡ Ilse Cervantes,§ and America Morales†
Ind. Eng. Chem. Res. 2000.39:432-440. Downloaded from pubs.acs.org by UNIV OF NEWCASTLE on 08/05/18. For personal use only.
Divisio´ n de Ciencias Ba´ sicas e Ingenierı´a, Universidad Autonoma MetropolitanasIztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09340 Me´ xico, Estrategia Sine´ rgica, S.A. de C.V., Paseo de los Pirules 124, Col. Paseos de Taxquen˜ a, Me´ xico D.F., 04250 Me´ xico, and Escuela Superior de Ingenierı´a Meca´ nica y Ele´ ctrica, SEPI, Avenida Santa Ana 1000, Col. San Francisco Culhuacan, Me´ xico D.F., 04430 Me´ xico
The aim of this paper is to propose a novel proportional-integral-derivative (PID) control configuration based on an observer structure. Batch distillation is used as the base case study where the regulated output is the distillate composition. The proposed PID control law is derived in the framework of robust nonlinear control with modeling error compensation techniques. A reduced-order observer is proposed to estimate both the derivative of the regulated output and the underlying modeling error. These observations are subsequently used in a control loop to feedback variations of distillate composition (derivative feedback) and to counteract the effects of modeling errors. It is shown that, under certain conditions, the resulting control law is equivalent to a classical PID controller with an antireset windup scheme. Moreover, the tuning of the controller is performed very easily in terms of a prescribed closed-loop time constant and an estimation time constant. Numerical results are provided for binary and multicomponent separations. Sampled/delayed measurements and several sources of uncertainties are considered in order to provide a realistic test scenario for the proposed control design procedure. 1. Introduction Proportional-integral-derivative (PID) control is one of the most common control schemes in the chemical and process industry. The main reason for this is its relatively simple structure, which is easy to understand and to implement. Different approaches have been taken to develop tuning methods. Morari and Zafiriou1 used internal model control (IMC) to derive PID tuning methods. Rotstein and Lewin2 also used the IMC approach in the context of parameter uncertainty. They compared tuning methods to different adaptive schemes for the control of an unstable chemical reactor. The design and tuning of PID control for processes with time delay have been also considered. Stahl and Hippie3 presented pole-placing PID control for unstable systems with time delay. Modified Smith predictors were considered by De Paor.4 Astro¨m et al.5 cope with unstable and integrating processes with long time delay. For years, control engineers have used PID controllers with nonlinear processes, but there was a lack of theoretical justification for such practice. In general, PID control design is intended for linear plants. However, Pachter et al.6 highlighted the crucial role of integral action in linearization-based controller design for nonlinear systems. They showed via specific examples that, when linear controllers are applied to nonlinear plants, integral action not only yields improved performance but also is, in fact, necessary. This paper presents a systematic and simple approach for the design and tuning of PID controllers for nonlin* Corresponding author. E-mail:
[email protected]. Fax: +52-5-724-4900. Tel: +52-5-724-4649. † Universidad Autonoma MetropolitanasIztapalapa. ‡ Estrategia Sine ´ rgica, S.A. de C.V. § Escuela Superior de Ingenierı ´a Mecanica y Ele´ctrica, SEPI.
ear processes. To motivate the PID control construction and to facilitate presentation, batch distillation is used as the base case study because it behaves as a relative degree two integrating process,7 which makes it suitable for PID control design. Moreover, batch distillation has become a benchmark for testing control algorithms for batch processes. The control design method is based on the estimation, via reduced-order observers, of the regulated output derivative and the modeling error induced by imperfect models. These observations are used in a globally linearizing feedback which, in the limit as the estimated signals converge to the real ones, yields a second-order linear behavior of the regulation error. The method leads to simple expressions for PID parameters in terms of the a prescribed closed-loop time constant and the estimation time constant. These expressions are used to provide tuning rules for PID control parameters. Stability conditions are also discussed. Numerical results are provided for binary and multicomponent separations. Sampled/delayed measurements and several sources of uncertainties are considered to provide a realistic test scenario for the proposed control design procedure. Summary of Batch Distillation Control. Batch distillation is a flexible process that is becoming widely used since the last decade. The main reason is that production amounts are usually small with minimum raw material inventories, which often results in an economic incentive.7,8 Batch distillation operation is designed via optimal control techniques where a prescribed profit function is maximized. This results in a time-varying reflux-ratio operation which is implemented in open-loop mode, which suffers from the drawback of lack of guaranteed robustness against model/plant mismatches. In principle, such a drawback can be reduced via closed-loop implementation of the
10.1021/ie990287c CCC: $19.00 © 2000 American Chemical Society Published on Web 12/16/1999
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optimal profiles, although this is still an open issue.9 In a recent work, Barolo and Berto7 have pointed out that the optimal profiles cannot be tracked with conventional linear controllers, because of the nonlinear and time-varying nature of the process. Relatively few papers have addressed the problem of feedback control of batch distillation. Quintero-Marmol et al.10 proposed an inferential control scheme where composition is estimated through an extended Luenberger observer. Batch distillation behaves as an integrating process with a time-varying process gain. Fileti and Pereira11 used gain-scheduled PI control for binary batch distillation. In their approach, the controller gain is increased during the operation to enable control to be maintained. Predictive and adaptive control schemes have been proposed to confront the time-varying nature of the batch distillation process. Dechechi et al.12 have shown that nonlinear model predictive control yields good results for distillate composition regulation. In addition to its lack of guaranteed robustness, this strategy requires that an extended Luenberger observer coupled with an optimization problem be solved on-line, which may be computationally prohibitive.7 In a recent interesting paper, Barolo and Berto7 have used recent developments of nonlinear control theory13 to construct controllers for distillate composition in batch distillation. Their idea is to use input/output linearizing feedback control coupled with composition estimation,10 resulting in a nonlinear inferential control structure, which can be easily extended to some cases of multicomponent distillation. A drawback of this approach is its lack of robustness guarantee. In fact, because input/linearizing feedback is based on the cancellation of nonlinearities, the presence of model/plant mismatches may lead to serious performance degradation and even to unstabilities.13 To overcome this difficulty, a classical integral action is commonly added; however, there is not a clear justification for this practice within the framework of nonlinear control theory. In this paper, we depart from a global linearizing nonlinear control design and use a modeling error compensation technique to counteract the effects of model/plant mistmatches. To this end, an observerbased estimate of the modeling error is computed, which introduces in a natural way an integral action in the control loop. This control design approach also introduces an antireset windup structure to deal with control input saturations. It is shown that this control design approach leads to a classical PID control with an antireset windup scheme. In this way, this work shows how to obtain a PID control from modern nonlinear control theory results. To the best of our knowledge, this is the first result on the PID control of uncertain nonlinear systems with guaranteed closed-loop stability. As compared to what is currently available in the literature,2,7,10,11 our contribution can be summarized as follows: (a) A PID control configuration with guaranteed stabilization capabilities for nonlinear systems is provided. The proposed PID control configuration allows one to write the closed-loop system as a nonlinear singularly perturbed system, for which stability results are available in the literature.15,16 (b) Reliable tuning guidelines are provided in terms of a desired closed-loop performance, via close-loop time constant τc and damping factor ξc, and an estimation
time constant τe. In a first stage of the tuning procedure, τc and ξc are chosen based on a prescribed closed-loop performance. In a second stage, the PID control is tuning with a parameter τe. In this way, the structure of the PID control configuration allows a tuning procedure that is in accordance with engineering intuition. (c) In the limit as τe f 0, the proposed PID control configuration recovers the performance induced by state-feedback control with perfect knowledge. In this way, the convergence of the distillate composition to the desired set point is smooth. Organization of the Paper. In section 2, the model of batch distillation is briefly discussed. In section 3, the regulation of distillation concentration under ideal (i.e., perfect knowledge) feedback control is studied for the binary system. In section 4, the batch distillation model for binary rectification is used to show how to obtain a PID control from modern control theory. Although the batch distillation is used as the base case study, the results in this paper apply to any minimumphase nonlinear system with relative degree two. The stability of the resulting closed-loop system is outlined. It is shown that an important feature of our control design approach is that the closed-loop performance induced by the ideal globally linearizing control can be approximated with the PID control. In section 5, extensions of our control design approach to multicomponent batch distillation are discussed. In section 6, numerical simulations with binary and multicomponent batch distillation are presented. Some concluding remarks are given in section 7. 2. Process Model For the sake of clarity in the presentation, as a base case for control design, we consider a batch rectifier for the separation of binary mixtures. Extensions of our results for multicomponent mixtures are discussed in section 5. The ordinary differential equations describing the process dynamics are reported in the Appendix. This model was taken from Barolo and Berto7 and retains the basic characteristics of the batch rectification process. The model takes into account the dynamics of the molar holdups on each tray, and the internal liquid rate on each stage is determined by means of the linearized version of the Francis weir formula. We have taken the batch rectifier model as the base case study because of the following reasons: (a) It is a nonlinear system with serious model uncertainties. (b) It is an integrating process with time-varying process gain. Computing a priori linear models, from either Taylor linearization or step responses, for the integrating process is not easy. Contrary to continuous processes, a linear model must describe the process dynamics along a whole trajectory. This fact makes batch processes suitable for robust nonlinear control design. (c) It is a relative degree two system, which makes it suitable for PID control design. In fact, complete control of relative degree two systems requires feedback of both the regulation error and the time derivative of the regulation error, which results in a PD-type control structure. In section 4, we will show that the integral action appears as a consequence of the modeling error compensation. (d) It is a minimum-phase system. This means that the internal dynamics (the dynamics of the noncon-
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trolled states) are stable. In terms of the linear control theory,1 this characteristic is equivalent to the nonexistence of zeros in the closed right-hand side of the complex plane. We should remark that, although our control design approach is applied to batch rectification, it is intended for any relative nonlinear with dynamical characteristics as the batch distillation, namely, relative degree two and minimum-phase behavior. 3. Binary System: Ideal Control Design We will consider the problem of regulating the distillate concentration of a desired component about any physically realizable trajectory, say ωd(t), by manipulating the reflux rate R. As a methodological step toward our robust control design in section 4, we will assume complete knowledge of the process dynamics. Consider the case of a binary system with the distillate concentration xD as the regulated variable. Let us take E(x) as the vapor-liquid equilibrium relationship (see eq A.11). Let edef ) xD - ωd be the regulation error. Then, as in Barolo and Berto,7 we can compute the time derivative (i.e., the Lie derivative) of the regulation error:
e(1) ) -ω(1) d + ξ
(1)
where e(1) denotes the first time derivative of e and ξ ) V(E(xN) - xD)/HD. Therefore, the relative degree is not one because the control input does not directly affect e(1). We can go beyond this point by computing e(2), the second time derivative of the regulation error,
e(2) ) -ω(2)d + Φ + ΨR
(2)
where
Φ
def
(V/HD)[{dE(xN)/dxN}χ - ξ] V(yN-1 - yN)
(4)
(V/HDHN)[{dE(xN)/dxN}(xD - xN)]
(5)
χ Ψ
def
def
(3)
We will see below that Ψ(t) * 0 for almost any operating condition. Therefore, the relative degree is two. The function Ψ corresponds to the high-frequency gain (HFG) of the process. Let
e(2) + 2ξcτc-2e(1) + τc-2e ) 0
(6)
be a prescribed behavior for the regulation error. In eq 6, τc and ξc are the closed-loop time constant and the damping factor, respectively. From eq 2, the control law leading to the stable closed-loop behavior (6) is -2 (1) - τc-2e]/Ψ Rid ) [ω(2) d - Φ - 2ξcτc e
(7)
In actual applications, control inputs are subjected to input saturations. Assume that the reflux ratio is subjected to the restriction 0 < Rmin e R e Rmax e V, which can be accomplished via the saturating control law
Rid s
{
Rmin if Rid < Rmin ) Sat[R ] ) Rid if Rmin e Rid e Rmax Rmax if Rid > Rmax id
(8)
Notice that the control law (7) is well-defined as long as Ψ(t) * 0, for all t g 0. In this way, some comments on the nature of the high-frequency gain Ψ are in order. We have that Ψ(t) > 0, for all t > 0. In fact, (a) V/HDHN > 0, except at startup operating conditions; (b) from standard thermodynamical arguments, it can be established that the derivative dE(x)/dx > 0, for all compositions; and (c) xD - xN > 0, for all t g 0. At steady-state operation, this condition is equivalent to the condition dE(x)/dx > 0. However, the computed control input (7) is inversely proportional to Ψ(t): the smaller the value of Ψ(t), the larger the control effort. Moreover, soon after startup, the column operation is brought from a low HFG to a high HFG composition space. This is represented by the composition gradient xD - xN, which increases as the separation advances in time. From these arguments, we conclude that composition control in batch rectifiers requires larger control efforts just after startup. Therefore, a control law for this process must have a good transient performance with fast tracking of the reference trajectory. From a control implementation viewpoint, the ideal control law (7) has the following drawbacks: (a) It requires measurements of vapor and liquid compositions {xD, xN, yN, yN-1}. In the field of batch distillation, Quintero-Marmol et al.10 have proven that a Luenberger-type observer can provide good estimates of the product composition profile from measurements of several stage temperatures. Barolo and Berto7 used the same kind of observer to implement the linearizing control law. They showed via numerical simulations that this control law leads to a fast and smooth transition from the startup phase to the production phase until the end of the operation. (b) Liquid-vapor equilibrium relationships are only an approximation to the real thermodynamics of the separation. Important structural and parametric mismatches are always present and may deteriorate the closed-loop stability and performance. In a heuristic approach, Barolo and Berto7 introduced an integral action to compensate for model/plant mismatches:
∫0te(t) dt]/Ψ
-2 (1) -2 Rid ) [ω(2) d - Φ - 2ξcτc e - τc e + kI
(9)
where kI is the integral gain. Although this approach has proven to work, to date there is not a theoretical justification for its usage within a nonlinear control design framework. Moreover, there is a lack of reliable tuning guidelines to pick the integral gain. 4. Binary System: Robust PID Control Design The aim of this section is the development of a PID control structure with guaranteed stabilization capabilities. To this end, the following assumptions are made: H.1. The distillate concentration xD is available for feedback. Estimates of xD can be obtained either from a gas cromatograph or from an extended Luenberger observer, as proposed by Quintero-Marmol et al.10 H.2. Φ h and Ψ h estimates of the functions Φ and Ψ are available. Moreover, Ψ h (t) > 0, for all t > 0. In the worst case design, the Φ h estimate can be equal to zero.
Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 435
Introduce the variable z ) e(1). Then, eq 2 can be written as a set of ordinary differential equations as follows:
Rc ) Ψ h -1[ω(2) h ] - CPID(s) F(s) e d - Φ GARW(s) (Rc - Rs) (17)
e˘ ) z
where CPID(s) is a classical PID controller whose control gain, integral, and derivative time constants are given by
z˘ ) -ω(2) h +η+Ψ hR d + Φ
(10)
def
Φ-Φ h + (Ψ - Ψ h )R denotes the modeling where η error. In eq 10 and in the sequel, we will use indistinctly the symbols e˘ and e(1) to denote the first time derivative of the regulation error. As in the case of the ideal control, suppose that eq 6 describes the desired closedloop dynamics. From eq 10, the corresponding saturating control law is
Rs )
Sat{[ω(2) d
-2
-2
-Φ h - η - 2ξcτc z - τc e]/Ψ h (11)
This control law compensates the effects of the modeling error signal η(t). However, it cannot be implemented as it stands because the modeling error η(t) is unknown and the time derivative z ) e˘ is not measured. Our idea is to use an observer to estimate simultaneously z(t) and η(t) and to use these estimates zj(t) and η j (t) in the control law (11):
h -η j - 2ξcτc-2zj - τc-2e]/Ψ h (12) Rs ) Sat{[ω(2) d - Φ We propose the following reduced-order observer:
zj˘ ) -ω(2) h +η j+Ψ h Rs + 2ξeτe-1(e˘ - zj) d + Φ η j˘ ) τe-2(e˘ - zj)
(13)
where ξe and τe are the estimation damping factor and time constant, respectively. Notice that, because e˘ ) z, the actual observer updating is proportional to the observation error z - zj. To implement the observer (13) and to avoid the use of derivators to compute e˘ , jintroduce the variables w1 ) zj - 2ξeτe-1e and w2 ) η j are computed from τe-2e. Then, the estimates zj and η the differential equations
h +η j+Ψ h Rs - 2ξeτe-1zj w˙ 1 ) -ω(2) d + Φ w˙ 2 ) -τe-2zj
(14)
and the identities
zj ) w1 + 2ξeτe-1e η j ) w2 + τe-2e
(15)
The initial conditions w1(0) and w2(0) can be chosen as follows: Because both signals z(t) and η(t) are unknown, initial estimates are z(0) ) 0 and η(t) ) 0. Then, from eq 15, we have w1(0) ) -2ξeτe-1e(0) and w2(0) ) -τe-2e(0). 4.1. Structure of the Proposed Controller. To clarify the structure of the control laws (12), (14), and (15), let us introduce the computed control input as
h -η j - 2ξcτc-2zj - τc-2e]/Ψ h Rc ) [ω(2) d - Φ
(16)
Then Rs ) Sat[Rc]. After elaborated but straightforward algebraic manipulations, it is possible to show that the computed control input can be written as
ξcτc + ξeτe h -1 Kc ) Ψ ξeτeτc2 + ξcτcτe2 τD )
τc2 + 4ξeξcτeτc + τe2
(18a)
2[ξcτc + ξeτe]
(18b)
τI ) 2[ξcτc + ξeτe]
(18c)
F(s) is a first-order lag filter (i.e., F(s) ) (τfs + 1)-1) whose time constant is given by
τf )
τcτe 2[ξeτc + ξcτe]
(19)
and GARW(s) is an antireset windup (ARW) operator acting on the “saturation error” Rc - Rs and is given by
GARW(s) )
τc - 2ξcτe2s s(τe2τcs + 2τe[ξeτc + ξcτe])
(20)
The following comments are in order: (a) The control law (17) can be seen as a PID control h ) and endowed with a law with dc-bias Ψ h (ωd2 - Φ h ) natural ARW windup scheme GARW(s) (Rc - Rs). If Φ 0 and Ψ h ) constant, then the control law (18) is a linear PID controller with an ARW scheme whose implementation can be easily made in actual inexpensive technologies (e.g., programmable logic controllers). (b) When the actuator saturates, the feedback signal GARW(s) (Rc - Rs) tries to drive the error Rc - Rs to zero by recomputing the integral action such that the controller output becomes exactly at the saturation limit. This prevents the controller from windup.14 (c) It is interesting to note that the PID control parameters are symmetric functions of the nominal closed-loop parameters (ξc and τc) and the observer parameters (ξe and τe). In other words, the PID control parameters (18) and (19) are invariant under the simultaneous substitutions (ξc and τc) f (ξe and τe) and (ξe and τe) f (ξc and τc). This means that the reference model (6) and the observers (14) and (15) have the same effects on the PID performance. Note also that this symmetry property does not hold for the ARW operator GARW(s). (d) Although the PID representation (18) and the control laws (12), (14), and (15) are input/output equivalent to each other, probably the key advantage of the proposed PID controller configuration lies in the fact that the controller states are meaningful variables as estimates of the physical plant states and the model/ plant mismatches. It follows that the estimates zj and η j can be used to monitor (on-line or off-line) the performance of the process or detect failures of actuators and sensors. 4.2. Stability Analysis. In this work, we do not cover the complete stability analysis of the resulting closedloop system. Instead, we outline the stability proof under some physically restrictive conditions, which are
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likely to be satisfied by the batch rectifier in the production phase, namely, smooth and slow dynamics. Define the observation errors as
1 ) (z - zj)/τe 2 ) η - η j
(21)
Because e˘ ) z, from eq 13 we get
˘ ) -τe-1Ae + B1γ
(22)
where ) (1, 2)T, B1 ) (0, 1)T, γ ) η˘ , and
Ae )
[
-2ξe 1 -κ(t) 0
]
(23)
where κ(t) ) Ψ(t)/Ψ h (t). Notice that κ(t) > 0. As a first approach, assume that κ(t) ) 1, such that Ae(t) is a constant and stable matrix. Then, system (22) can be seen as a stable linear system perturbed by the time derivative of the modeling error. On the other hand, let us assume that the control input is not saturated (i.e., Rs ) Rc). Then, we can compute the controlled system from eqs 10 and 12 to give
e(2) + 2ξcτc-1e(1) + τc-2e ) -2ξcτc-1τe1 + 2 (24) which corresponds to the prescribed behavior for the regulation error (see eq 6) with the estimation error disturbance -2ξcτc-1τe1 + 2. If (t) f 0, we have that e(t) f 0. This means that the regulation error converges to zero as long as the estimation error also converges to zero. Let us analyze the stability of system (22). To this end, assume that |γ(t)| e γmax. The solution to (22) is
(t) ) exp(τe-1Aet)(0) + exp(τe-1Aet)
∫0texp(-τe-1Aeσ)γ(σ) dσ
(25)
behavior closer and closer to that induced by the ideal global linearizing controller (see eq 12). Of course, the value of τe is limited by several factors, such as measurement noise, dead times, and unmodeled dynamics, which limit the crossover frequency of the closed-loop system.1 The analysis of the time-varying case is more elaborated and can be approached with tools from the stability theory of singularly perturbed nonlinear systems.15,16 The stability results for this case lead to conclusions analogous to the ones discussed above. 4.3. Estimation of the High-Frequency Gain Ψ. The implementation of the proposed PID control law and the stability of the underlying closed-loop system rely strongly on the Ψ h estimate of the high-frequency gain. Estimates of the vapor flow rate V and the holdups {HD, HN} can be obtained from the nominal operating conditions. These variables change slightly during the separation phase. On the other hand, the term [dE(xN)/ dxN](xD - xN) is time-varying. We can use the approximation
dE(xN)/dxN = dE(xD)/dxD
(26)
Moreover, if the composition gradient xD - xN is assumed to be almost constant during the separation phase, we obtain a time-varying estimate Ψ h given as
h N)[(dE(xD(t))/dxD)Π] Ψ h (t) ) (V h /H h DH
(27)
where the constant Π denotes the approximation xD xN = Π. Because linear PID controllers can be easily implemented in inespensive technologies (e.g., PLCs), it is desirable to have a constant estimate of Ψ. To this end, a further approximation dE(xD)/dxD = dE(xDref)/dx can be taken, so that the estimate Ψ h given by
h N)[(dE(xDref)/dxD)Π] Ψ h (t) ) (V h /H h DH
(28)
Because Ae is a stable matrix, there exist two positive constants δ and L such that |exp(τe-1Aet)| e L exp(τe-1δt). By taking norms and triangle inequality in eq 25, we have that
is a positive constant. Finally, the constant estimate Π of the gradient xD - xN can be taken as
|(t)| e L exp(-τe-1δt){|e(0)| +
where xNref is an underestimate of the concentration in the top tray. The parameter Π ) xDref - xNref can also be obtained from a McCabe-Thiele diagram of the column at total reflux.
∫0 exp(τe-1δσ)|γ(σ)| dσ} t
e L exp(-τe-1δt){|e(0)| + γmax
∫0texp(τe-1δσ) dσ}
) L exp(-τe-1δt)|e(0)| + -1
(τeγmaxL/δ){1 - exp(-τe δt) Then, (t) is bounded for all t > 0, and |(t)| f τeγmaxL/δ as t f ∞. The following conclusions are immediate consequences of the above analysis: (a) The larger the rate variations of the modeling error, the larger the residual error τeγmaxL/δ. This is in agreement with engineering intuition. In fact, uncertainties with large time variations are harder to counteract. (b) The smaller the estimation time constant, the smaller the residual error τeγmaxL/δ and, in accordance with eq 24, the smaller the regulation error. This conclusion establishes that small estimation time constant values must be selected to get a closed-loop
Π ) xDref - xNref
(29)
5. Multicomponent Systems As in Barolo and Berto’s paper,7 the proposed control procedure can be extended to multicomponent mixtures. In fact, if the objective of the separation is to recover the more volatile component at a constant purity and the mixture can be approximately modeled as a constant relative-volatility mixture, then the dynamical behavior of the separation can be approximately modeled as a pseudobinary system. For an easy derivation and implementation of the control law, it can be assumed that during the production phase a binary mixture is found in the top trays of the batch rectifier. This scenario is found for most industrial separations.10 Hence, the proposed PID control laws (13)-(15) are able to regulate the distillate composition of the component that is being withdrawn at the desired purity. Notice that the estimated HFG Ψ h must be obtained with respect to the
Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 437 Table 1. Model and Systems Characteristics vapor boilup, mol/h nominal tray holdup, mol reflux drum holdup, mol total feed charge, mol nominal feed composition tray hydraulic time constant, h number of ideal trays relative volatility setpoint
binary
ternary
5400 30 250 8000 0.4/0.6 0.001 8 nonideal 0.81
6000 30 300 30000 0.2/0.6/0.2 0.001 10 4/2/1 0.95/0.95
Figure 2. Time evolution of the controlled batch distillation (binary system) for two different values of the estimation time constant.
Figure 1. Time evolution of the uncontrolled batch distillation (binary system).
components of the pseudobinary mixture in the top trays of the column. This means that
Ψ h (t) ) (V h /H h DH h N)[(dE h (xDref)/dxD)Π]
(30)
where
E h (x) )
R jx R jx + 1 - x
(31)
and R j is the relative volatility of the pseudobinary mixture. 6. Simulation Results We have carried out several numerical simulations with two examples (a binary system and a ternary system) to illustrate the performance of the proposed PID control configuration. Both examples were taken from Barolo and Berto.7 The model and system characteristics are given in Table 1. 6.1. Binary System. The binary system corresponds to an ethanol/water mixture. Figure 1 presents the time evolution of the distillate composition for three different values of the internal reflux rate R/V, namely, R/V ) 0.9, 0.95, and 1.0 (total reflux). As expected, the distillate composition decreases with time. Notice that the separation phase lasts about 15-20 h. On the other hand, the maximum distillate composition (about 0.8146) is obtained with R/V ) 1.0.
We used the proposed control scheme to regulate the distillate composition under the following conditions and control parameters: (a) Track a constant trajectory with ωD ) 0.81. (b) Φ h (t) ) 0, for all t g 0. (c) From eq 18, we get the estimate Ψ h ) 0.0141. (d) As in Barolo and Berto,7 ξc ) 1 and τc ) 0.04 h. (e) The feed composition is 0.4. Figure 2 shows the plant response for three different values of the estimation time constant with ξe ) 1. Figure 2 also shows the response of the plant under a saturated ideal control law (see eq 7). As expected from the analysis above, a better closed-loop response is obtained for smaller values of the estimation time constant. Moreover, the response under the PID control laws (12), (14), and (15) approaches the response under the ideal control law. This shows that, in principle, the PID control law can achieve the closed-loop performance induced by the global linearizing control law. In this way, our control design strategy has the enormous advantage that it leads to a linear PID-like controller with easy-to-use tuning guidelines. Moreover, because η(t) f η j (t) as τe f 0, such PID control can be seen as a τe approximation to the inverse dynamics control law.7 Batch processes are run through repeated chains of operations where the actual feed composition may not been accurate. In fact, the composition usually is obtained from the blending of the main feed with recycled cuts. Figure 3 shows the time evolution of the batch distillation for the same values of the estimation time constant as those in Figure 2. In this case the feed composition is 0.45. The proposed PID control law yields a good closed-loop performance despite variations in the feed composition.
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Figure 3. Time evolution of the controlled batch distillation (binary system) for two different values of the estimation time constant and feed composition 0.45/0.55.
Finally, suppose that the control input is subjected to dead times and sampling induced by composition measurements or estimation. In this case, the control law must be detuned to avoid closed-loop instabilities.1 To select the parameters of the control law, we suggest the following rule: Choose the damping coefficients ξc and ξe on the order of 1.25 and choose the time constants τc on the order of the larger dead time. The heuristic idea behind this rule is that the modeling error cannot be estimated faster than the measurement dead time. Besides, the closedloop cannot be stabilized faster than the dead time. A source for these heuristic ideas can be found in the robust control literature.1 Figure 4 shows the response of the plant for three different values of the input dead time, namely, 0.1, 0.2, and 0.3 h. As expected, the larger the dead time, the more degraded the closed-loop performance. However, the PID control is able to give an acceptable closed-loop performance for moderate dead-time values. The above-described numerical simulations demonstrate that a PID control is capable of achieving the control objectives with output measurements only. Moreover, because such control is of a linear nature, it can be implemented in inexpensive hardware without major modifications. This is in contrast with previously reported control strategies for batch distillation,7,10 which make use of Luenberger-like composition estimators. In such a case, the control law is higher dimensionally and nonlinear, which must be implemented on a computer. 6.2. Ternary System. The ternary system corresponds to a ternary mixture with relative volatility 4/2/ 1. The objective of the separation is to recover the more
Figure 4. Time evolution of the controlled batch distillation (binary system) for three different values of the input dead time.
volatile component and the intermediate component at a constant purity. The composition setpoints were 0.95/ 0.95. The estimated HFG is Ψ h ) 0.015 for recovery of the more volatile component and Ψ h ) 0.016 for the recovery of the intermediate component. The dead time due to composition measurement/estimation was 0.2 h. The controller is switched for intermediate product recovering when R/V ) 0.98. At this point, the reflux rate is lowered following a ramp decrease, to allow the second component to reach the top of the column. When this happens, the control starts again. Figure 5 shows the response of the process. Notice that the performance of the controller is very good. Both products are kept at the desired setpoints, and the reflux profile is smooth. For the sake of comparison, the performance of the global linearizing controller is also reported in Figure 5. The proposed PID control law provides a good performance despite serious model/plant uncertainties and measurement dead times. 7. Conclusions It was developed and presented a PID control configuration that can be used to regulate the distillate composition in batch distillation. The control design departs from a global linearizing control for relative degree two systems. Model/plant mismatches were considered, and modeling error compensation is included in the control law. Estimates of the modeling error are obtained through a reduced-order observer. This control design leads to a PID control law. Probably the key advantage of this PID controller configuration lies in the fact that the controller states are meaningful variables as estimates of the physical plant states and the model/plant mismatches. It follows that the esti-
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Li ) Li,0 +
Hi - H0 τL
(A1)
where Li,0 is the reference value of the internal liquid flow rate, Hi and H0 are the actual and reference molar holdups on tray i, and τL is the tray hydraulic time constant. The energy balances are not included in the model; therefore, the vapor rate is constant inside the column. Other assumptions are ideal trays, well-mixed trays, boiling feed, total condensation with no subcooling, and negligible heat losses. At the beginning of the operation, it is assumed that the reboiler, all trays, and the reflux drum are filled with the liquid feed. In the equations below, the symbol “‚” denotes the time derivative operator d/dt and C denotes the number of components. Reboiler (subscript B)
H˙ B ) L1 - V
(A2)
x˘ B,j ) (L1x1,j - VyB,j - xB,jH˙ B)/HB
(A3)
Bottom Tray (subscript 1). For j ) 1, ..., C - 1
H˙ 1 ) L2 - L1
(A4)
x˘ 1,j ) (L2x2,j + VyB,j - L1x1,j - Vy1,j - x1,jH˙ 1)/H1 (A5)
Figure 5. Time evolution of the controlled batch distillation (ternary system).
mates zj and η j can be used to monitor (on-line or offline) the performance of the process. The simulation results for binary and ternary systems show a good performance of the PID control law with fast and smooth tracking of the reference, even in the presence of mild dead times. Moreover, our simulation results show that the PID control law can provide the performance obtained with the saturating global linearizing control law as the estimation time constant approaches zero. The proposed PID configuration can be tuned very easily; the estimation time constant needs to be adjusted for each component that is recovered as the distillate product. Moreover, the smaller the estimation time constant, the faster the stabilization of the regulation error. However, several effects induced by dead times and unmodeled high-frequency dynamics limit the achievable closed-loop performance. Tuning guidelines to select the controller parameters in the presence of the these effects were also discussed. Acknowledgment R.M.-L., I.C., and A.M. are grateful to CONACyT and Instituto Mexicano del Petroleo (FIES 95-93-II) for the financial support that made this work possible. AppendixsA Dynamical Model of Batch Distillation A basic model of the process dynamics is described here, as taken from Barolo and Berto.7 The internal flow rates of liquid are calculated by means of the Francis weir formula
Intermediate Trays (subscript i). For j ) 1, ..., C-1
H˙ 1 ) Li+1 - Li
(A6)
x˘ 1,j ) (Li+1xi+1,j + Vyi-1,j - Lixi,j - Vyi,j)/Hi - xi,jH˙ i (A7) Top Tray (subscript N). For j ) 1, ..., C - 1
H˙ N ) R - LN
(A8)
x˘ N,j ) (RxD,j+ VyN-1,j - LNxN,j - VyN,j - xN,jH˙ N)/HN (A9) Reflux Drum (subscript D). For j ) 1, ..., C - 1
x˘ D,j ) [V(yN,j - XD,j)]/HD
(A10)
The vapor and liquid compositions are related by an equilibrium relationship
yi,j ) Ei(xj)
(A11)
where xj ) (x1,j, ..., xc,j)T. This expression can be computed from any state equation or computational package. Literature Cited (1) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: Englewood, NJ, 1989. (2) Rotstein, G. E.; Lewin, D. R. Simple PI and PID tuning for open-loop unstable systems. Ind. Eng. Chem. Res. 1991, 30, 1864. (3) Stahl, H.; Hippie, P. Design of pole placing controllers for stable and unstable systems with pure time delay. Int. J. Control 1987, 45, 2173. (4) De Paor, A. M. A modified Smith predictor and controller for unstable processes with time delay. Int. J. Control 1985, 41, 1025.
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(5) Astro¨m, 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. (6) Pachter, M.; D’Azzo, J. J.; Veth, M. Proportional and integral control of nonlinear systems. Int. J. Control 1996, 63, 679. (7) Barolo, M.; Berto, F. Composition control in batch distillation: Binary and multicomponent mixtures. Ind. Eng. Chem. Res. 1998, 37, 4689. (8) Diwekar, U. M. Batch Distillation, Simulation, Design and Control; Taylor & Francis: London, 1995. (9) Edgar, T. F. Control of unconventional processes. J. Process Control 1996, 6, 99. (10) Quintero-Marmol, E.; Luyben, W. L.; Georgakis, C. Application of an extended Luenberger observer to the control of multicomponent batch distillation. Ind. Eng. Chem. Res. 1991, 30, 1870. (11) Fileti, A. M. F.; Pereira, J. A. F. R. The development and experimental testing of two adaptive control strategies for batch distillation. In Distillation and Absorption ‘97; Darton, R., Ed.; IChemE: Rugby, U.K., 1997; p 249.
(12) Dechechi, E. C.; Luz, L. F. L., Jr.; Assis, A. J.; Maciel, M. R. W.; Maciel Filho, R. Interactive supervision of batch distillation with advanced control capabilities. Computers Chem. Eng. 1998, 22, S867. (13) Henson, M. A., Seborg, D. E., Eds. Nonlinear Process Control; Prentice-Hall PTR: London, 1997. (14) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, N. N. A unified framework for the study of anti-windup designs. Automatica 1994, 30, 1869. (15) Khalil, H. K.; Esfandiari, F. Semiglobal stabilization of a class nonlinear systems using output feedback. IEEE Trans. Autom. Contr. 1993, 38, 1412. (16) Teel, A.; Praly, L. Tools for semiglobal stabilization by partial state and output feedback. SIAM J. Control Opt. 1995, 33, 1443.
Received for review April 21, 1999 Revised manuscript received October 11, 1999 Accepted October 21, 1999 IE990287C
