A PI Controller with Disturbance Estimation - American Chemical Society

Aug 15, 1997 - Many chemical processes can be modeled for the purpose of feedback controller tuning by a first-order process. In this case, PI feedbac...
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Ind. Eng. Chem. Res. 1997, 36, 3668-3675

PROCESS DESIGN AND CONTROL A PI Controller with Disturbance Estimation† Jose´ Alvarez-Ramirez,* Ricardo Femat, and Arturo Barreiro Departamento de Ingenieria Quı´mica, Divisio´ n de Ciencias Ba´ sicas e Ingenierı´a, Universidad Auto´ noma MetropolitanasIztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09000 Me´ xico

Many chemical processes can be modeled for the purpose of feedback controller tuning by a first-order process. In this case, PI feedback controllers are mostly used to regulate the process. Due to the capabilities of the integral action to estimate load disturbance, a certain stability margin in the closed-loop system is obtained. That is, the integral variable provides an estimate of the load disturbance, which is counteracted in the feedback loop. The aim of the work is to derive a model-based PI feedback controller with extended capabilities to estimate dynamic disturbances (PII2 feedback controller). It is shown that, under certain gain choices, the proposed controller reduces to a classical PI one. The performance of the controller is illustrated via numerical simulations of two chemical processes. 1. Introduction Many chemical processes can be modeled for the purpose of PI feedback controller tuning by a first-order process containing only a gain and a time constant.

Kp′e-TS G(s) ) τs + 1

(1)

This type of model is able to represent the dynamics of many processes over the frequency range of interest for feedback controller design (Luyben, 1990). Examples of processes whose dynamics can be represented by (1) are distillation columns (Skogestad and Morari, 1988), heat exchangers (Alverez-Ramirez et al., 1997), and continuously stirred chemical reactors (Luyben, 1990). Additionally, the simple model (1) is very convenient for process identification because it contains only three parameters: Kp′, τ, and T. Relay feedback tests can be used to estimate such a set of model parameters (Wang et al., 1997). IMC approach (Chien and Fruehauf, 1990) and relay methodologies (Astro¨m and Hagglund, 1988) can be used to derive tuning rules for PI controllers. In most practical cases, the PI controllers tuned with such rules provide acceptable closed-loop performances and robustness margins against uncertainties and load disturbances. Robustness of closed-loop systems under PI feedback control can be explained due to the capabilities of the integral action to estimate load disturbances. That is, the integral variable becomes an estimate of uncertainties and load disturbances at low frequencies. In this way, steady-state offset is eliminated. The aim of this work is to derive a PI controller with enhanced capabilities to estimate disturbances. The main idea is to interpret disturbances as an external state whose dynamics can be reconstructed from output measurements. The information obtained in this way is subsequently used in a PI-like (which we named PII2) feedback controller to counteract the effects of disturbances. It

is shown that, under certain choices of controller parameters, the proposed controller becomes a classical PI controller. Tuning rules for the proposed controller are discussed, and it is shown that the number of controller parameters can be reduced to two, which can be tuned independently. The paper is organized as follows. In section 2, derivation of a PI controller with enhanced disturbance estimation is proposed. In section 3, the stability properties of the closed-loop system when the model (1) matches the structure of the process are studied. Numerical simulations are provided in section 4. The conclusions are in section 5. For the sake of completeness, some stability concepts are provided in appendices. 2. PI Controller with Enhanced Disturbance Estimation Consider the plant (1) with no input delay (i.e., T ) 0) and an input-channel unknown disturbance d(t)

Y(s) )

Kp′ (U(s) + D(s)) τs + 1

In time domain, the plant (2) can be written as

y˘ ) -y/τ + Kpu + d

S0888-5885(97)00230-3 CCC: $14.00

(3)

where Kp ) Kp′/τ and d(t) ) Kp′d(t)/τ. Let us write a PI feedback controller in the following form:

u ) [(1/τ - 1/τc)y]/Kp - z z˘ ) Kiy

(4a)

where τc is a prescribed closed-loop time constant and Ki is the integral gain. In standard notation, the above PI controller is

(

C(s) ) Kc 1 + * Corresponding author. E-mail: [email protected]. † This work was supported by Instituto Mexicano del Petroleo (FIES-95-93-II).

(2)

1 τIs

)

Kc ) (1/τ - 1/τc)/Kp © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3669

τ1 ) -(1/τ - 1/τc)/KpKi

(4b)

The closed-loop system is given by

y˘ ) -y/τc - Kpz + d z˘ ) Kiy Assume that the closed-loop system attains an asymptotic equilibrium point, which is given as yeq ) 0 and zeq ) d∞ ) limtf∞ d(t). That is, as t f ∞, the integral variable z estimates the asymptotic value of the load disturbance d. In this way, d∞ is counteracted by zeq and steady-state offset is eliminated. From continuity arguments, it is expected that the integral variable z(t) provides an estimate of d(t) at low frequencies. In fact, the value of the integral gain Ki can be interpreted as a frequency cutoff for a disturbance whose dynamics could be estimated by the integral variable z(t). As discussed above, integral action introduces a certain closed-loop stability margin because z(t) (interpreted as an estimate of d(t)) counteracts the effects of d(t). In what follows we will derive a model-based PI feedback controller with enhanced disturbance estimation capabilities. Notice that, in principle, the dynamics of the disturbance d(t) can be reconstructed from measurements of the output y(t) and the input u(t). That is, d(t) ) y(t)/τ ) Kpu(t). In order to systematically derive an estimator for d(t), assume that d(t) is continuous and rewrite eq 3 as an extended state-space system

y˘ ) -y/τ + Kpu + d d˙ ) f(t) where f(t) is a bounded, unknown time function. Then the system

h + g1(y - yj) yj˘ ) -y/τ + Kpu + d d h˙ ) g2(y - yj)

e˘ ) M(g) e + F(t) where F(t) ) (0

(6a)

and

[

-g M(g) ) -g1 2

u ) ((1/τ - 1/τc)y - d h )/Kp

1 0

]

In order that M(g) be Hurwitz, g1 > 0 and g2 > 0. It is easy to arrive at the following conclusions: (i) if limtf∞ F(t) < ∞, the estimator (5) is asymptotically stable; (ii) more generally, if F(t) is bounded, then the estimation error is bounded. It is possible to integrate (6) to obtain the following error estimate:

(7)

where d h is obtained from (5). Equations 5 and 7 become a feedback controller with second-order compensation. The underlying idea of the feedback (8) is similar to some extent to the idea of state feedback control with disturbance decoupling (see Isidori, 1989). That is, if the disturbance d(t) is exactly known, then the feedback control (7) decouples the effects of the disturbance d(t) and the reference signal. The difference between the state feedback and the proposed scheme in this work is the following: while in the former case a feedback controller is provided such that disturbances are contained in the output null space, in the later case an estimate of the disturbance is used to counteract disturbance effects in the output dynamics. Physically, the idea behind the controller structure is that the compensator detect disturbances quickly and take corrective action to prevent the controlled output from going far off the desired operation point. Transfer Function C(s) of the Controller. The feedback controller given by eqs 5 and 7 is linear with respect to the states y, yj, and d h . It is not hard to obtain the following transfer function U(s)/Y(s) ) C(s):

[

C(s) ) Kc 1 +

(5)

represents a stable estimator system: yj and d h are estimated values for y and d, whereas g1 and g2 are estimator parameters which must be positive to guarantee the stability. The dynamics of d h are driven by the estimation error (y - yj), which is supposed to reflect the deviation between d and d h. Define the estimation errors e1 ) y - yj and e2 ) d d h . The dynamics of e(t) are governed by

f(t))T

||e(tf∞)|| ) βκ2/λ*. An upper bound for λ* is given by min{Re(λ1),Re(λ2)}, where λ1 and λ2 are the eigenvalues of M(g). The constant κ is known as the peaking factor (Hinrichsen and Pritchard, 1993) and is an estimate of the overshoot in the estimation error due to initial conditions. In principle, the asymptotic error ||e(tf∞)|| can be made as small as desired by choosing g1 and g2 such that the eigenvalues of M(g) are located deeper in the left-half complex plane; however, large overshooting can be present. If the estimator system (5) provides an estimate of d, it can be used to counteract the effects of the disturbance in the feedback loop. We propose the following feedback controller:

Ke 1 + τIs s(s + g1)

]

(8)

where

kc ) -(1/τ - 1/τc)/Kp τI-1 ) -g2/(1/τ - 1/τc) Ke ) -g2(g1 - 1/τc)/(1/τ - 1/τc) The controller (8) is composed of three parts: (i) a proportional feedback, (ii) an integral action 1/τIs, and (iii) a “second-order” integral action Ke/s(s + g1). The third part enhances the disturbance estimation capabilities of the PI controller. The controller (8) will be called the PII2 feedback controller. The PII2 controller becomes a PI controller by choosing Ke ) 0, or equivalently g1 ) 1/τc. In the time domain, making use of (7), we have that

yj˘ ) -y/τc + g1(y - yj) ) -g1yj d h˙ ) g2(y - yj)

(9)

2

||e(t)|| e κ||e(0)|| exp(-λ*t) + βκ (1 - exp(-λ*t))/λ* (6b) where |f(t)| e β and κ and λ* are positive constants depending only on the matrix M(g). One can see that

By integrating the first equation in (9) and substituting its solution into the second one, we obtain

d h˙ ) g2y - exp(-g2t)yj(0)

3670 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

Hence, when g1 ) 1/τc, the controller (5), (7) is reduced to the PI controller (4) with a “perturbing” decaying term exp(-g2t)yj(0) (recall that g2 > 0). A useful parameterization of the estimator gains g1 and g2 is as follows: g1 ) 2L and g2 ) L2 where L > 0. With such parametrization, the eigenvalues of M(g) are {-L, -L}, and the number of controller parameters is reduced to two, namely, τc and L. The corresponding PII2 controller becomes a PI one if L ) 1/2τc. The parameter L can be interpreted as the rate of disturbance estimation, which can be tuned independently from τc. In this sense, there is a Separation Principle for designing the proposed PI controller with disturbance estimation. Then the larger the value of L is, the faster the convergence of d(t) to the actual disturbance d(t). In fact, let us define the following estimation errors: e1 ) L(y - yj) and e2 ) d - d h . Then, the dynamics of such estimation errors are given by e˘ ) LM*e + F(t), where F(t) ) [0 d] ) [0 f(t)] and the Hurwitz matrix M* is given by

M* )

[

-2 -1

1 0

]

Assume that d h (t) ) sin(2πωt), so that f(t) ) 2πω cos(2πωt). In this way, |F(t)| e 2πω. It is not hard to integrate the system e˘ ) LM*e + F(t) and obtain the following estimate:

|e(t)| e κ||e(0)|| exp(-Lλ*t) + ω′κ(1 - exp(-Lλ*t))/λ* where the positive numbers κ and λ* depend only on the Hurwitz matrix M*, and ω′ ) 2πω/L. Notice that |e(t)| converge with a decaying rate of the order O(L). On the other hand, |e(∞)| e ω′κ/λ*. This inequality makes clear the role of the gain L in the dynamics of the estimation error. In this way, for a desired asymptotic estimation error bound e∞, L must be chosen to be on the order of the dominant frequency of the disturbance d h (t). However, it can be seen in eq 6b that for very large values of L the overshoot is also very large, so that the transient closed-loop performance could be unacceptable. Heuristically, the parameter L can be also interpreted as a cutoff frequency of estimated disturbances. That is, the estimator (5) is able to reconstruct the dynamics of disturbances whose frequencies are not beyond L. This fact will be illustrated in the numerical simulations section. 3. Disturbance Estimation in the Presence of Parametric Uncertainties In most cases, only an estimate of the actual plant parameters is available. In this case it is also possible to use a control strategy like the one described in the above section. Let τj and K h p be estimates of τ and Kp, respectively. Equation 3 can be rewritten as follows:

y˘ ) -y/τj + K h pu + w

(10)

where w ) d - (1/τ - 1/τj)y + (Kp - K h p)u. If the signal w is seen as a disturbance, the plant (10) can be described as an extended state-space system where w is interpreted as a new state:

y˘ ) -y/τj + K h pu + w w˘ ) ψ(y,u,u˘ ,d,d˙ )

(11)

where ψ(y,u,u˘ ,d,d˙ ) is an unknown function. As in the

case of the section above, the following second-order compensator can be proposed:

j )/K hp u ) ((1/τj - 1/τc)y - w yj˘ ) - y/τj + K h pu + w j + g1(y - yj) w j˘ ) g2(y - yj)

(12)

where w j is an estimate of the uncertainties term w ) d - (1/τ - 1/τj)y + (Kp - K h p)u. Notice that system (12) has the structure of a (Luenberger) linear state observer, except that the unknown function ψ( ) was not included in the construction of the estimator. In this way, system (12) makes use only of measured signals y(t) and u(t) and internal states yj(t) and w j (t). Different from the case where only d(t) is unknown, in this case the uncertainty w(t) has a feedback effect. Consequently, the controller (12) can lead to unstabilization of the closed-loop system. The dynamics of the controlled system and the estimation error are governed by

y˘ ) -y/τc + e2 e˘ 1 ) -g1e1 + e2 e˘ 2 ) -g2e1 + ψ(y,u,u˘ ,d,d˙ )

(13)

Notice that, if e2(t) f 0, then y(t) f 0. From the definition of w, one can obtain the following expressions:

ψ(y,u,u˘ ,d,d˙ ) ) d˙ - (1/τ - 1/τj)(-y/τc + e2) + (Kp - K h p)u˘ hp u˘ ) [(1/τj - 1/τc)(-y/τc + e2) - g2e1]/K Therefore, the dynamics of the estimation error e2 ) w -w j are governed by e˘ 2 ) -γg2e1 + Φ(y,u,u˘ ,e,d,d˙ ), where γ ) Kp/K h p and Φ(y,u,u˘ ,e,d,d˙ ) ) ψ(y,u,u˘ ,d,d˙ ) + (Kp - K h p)g2e2/K h p. Then, in this case the nominal part of the estimation error dynamics is e˘ ) M(g,γ) e, where

[

-g1 M(g,γ) ) -γg 2

1 0

]

If g1 and g2 > 0, a necessary condition for the closedloop system to be internally stable is that γ > 0. This implies that the sign of the process gain Kp must be known. This is not a serious restriction since in most SISO chemical processes, sign(Kp) is known. In order to study the effects of γ in the dynamics of the reconstruction error e, the parametrization g1 ) 2L and g2 ) L2 is used. The matrix M(g,γ) becomes

M(g,γ) )

[

-2L -γL2

1 0

]

The eigenvalues of M(g,γ) are ) L(-1 - (1 - γ)1/2). up Suppose that upper Kp and lower Klo p bounds of the actual plant gain Kp are available (i.e., Kp ∈ [Klo p, Kup ]). Here, the problem is how to choose the estimate p K h p to obtain good performance of the error dynamics. If K h p ) Klo p , then γ > 1 and the eigenvalues of the matrix M(g,γ) are complex: the larger the value of γ is, the larger the imaginary part is. Then, when K h p ) Klo p , one can expect underdamped convergence of the closed-loop system. On the other hand, if K h p ) Kup p , we have that γ < 1, λ+ > -L, and λ- < -L. That is, one mode of the λ(

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3671

estimation error is slower than the other. Then, for small values of γ slow convergency (due to the eigenvalue λ+) of the estimation error can be expected. Based up on the above discussion, a (Klo p , Kp ) average could give better performance. For instance, one can use K hp ) up (Klo + K )/2. p p The stability properties of the closed-loop system (13) can be obtained by considering it as a controlled system coupled with a state observer. In fact, by using results from Esfandiari and Khalil (1992), we can conclude that if γ > 0, there exists a positive constant L* (minimum gain) such that for all L > L* the closed-loop system (13) is asymptotically stable. In this way, L* is a minimum gain (Luyben, 1990) for the closed-loop system. In principle, the closed-loop system (13) is asymptotically stable for any arbitrarily large gain L > L*. However, in practice there is a restriction in the maximum value of L that can be used. In fact, measurement noise, sampled measurements, and nonmodeled dynamics (e.g., neglected actuator and highfrequency dynamics) limit the maximum achievable closed-loop performance. In the case of noisy measurement, noise is amplified with a gain on the order of O(L2). Prefiltering of measured signals and scaling of estimated uncertainties can be used to reduce the effects of noise into the feedback loop. The cases of sampled measurements and neglected actuator and filter dynamics are not easy to deal with. Such cases can be seen as an estimation with delayed input where the sampling period and the dominant characteristic time of the nonmodeled dynamics are interpreted as the delay time. Since the parameter L can be seen as the rate of uncertainty/disturbance estimation, L-1 can be interpreted as the characteristic time of estimation. From a practical viewpoint, it is clear that it is not possible to estimate delayed signals faster than the delay time. So, a restriction in the maximum allowable value of L must be considered in the sense that the minimum value of L-1 should be on the order of the delay time. The case of delayed input will be analyzed in the following subsection. Uncertainty Estimation under Input Delay. In the cases studied in the sections above, the disturbance estimator matches the structure of the process. So, the results on closed-loop stability and estimation are precise. When a delay T is present in the input channel, the results cannot be precise because disturbance estimation is not causal. That is, estimation of future (at time t + T) uncertainties is required. Of course, this estimation requirement is not physically possible, so we will propose an approximation. For T > 0, the dynamics of the plant are described by

w˘ t ) (Kp - K h p)u˘ t-T ) (Kp - K h p)((1/τj - 1/τc)(-yt/τj + Kput-T) hp g2e2,t-T)/K From this equality, the nominal parts of the estimation error dynamics are given as follows:

e˘ 1,t ) -g1e1,t + e2,t e˘ 2,t ) -g2e1,t + (1 + γ)g2e1,t-T

(15)

Asymptotic stability of system (13) is necessary for the asymptotic stability of the closed-loop system. If either T ) 0 or γ ) 1 (no input related to uncertainty), positivity of g1, g2, and γ is a necessary and sufficient condition to obtain asymptotic stability. However, if γ * 1 and T * 0, stability of system (15) is not a straightforward issue. From continuity arguments we can conclude that when (1 - γ)g2 is sufficiently small, the delayed system (15) is asymptotically stable (Hale, 1976). On the other hand, the controlled system becomes

y˘ t ) -yt/τ + (1/τ - 1/τc)yt-T + (wt - wt-T) (16) Thus, the asymptotic stability of the controlled nominal plant y˘ t ) -yt/τ + (1/τ - 1/τc)yt-T is also necessary for closed-loop stability. From continuity arguments (Hale, 1976), we conclude that, for values of τc on the order of the open-loop characteristic time τ, the nominal system (16) is asymptotically stable. From internal model control methodologies (Chien and Fruehauf, 1990), we suggest to select τc around 0.75τ. This means that it is desired that the nominal closed-loop system be faster than the open-loop system, but not so fast to avoid closed-loop instability. As mentioned above, our results for the input-delay case are not precise. Derivation of asymptotic stability results for this case is quite complex and is beyond the aim of this work. However, in the next section we will show through extensive simulations that our procedure is robust and yields good results when the model does not match the process. Some comments regarding possible extensions of the proposed scheme to different process models other than simple first-order ones are in the following order: (i) Suppose that the plant is minimum-phase and is given by Y(s)/U(s) ) KpPm(s)/Pn(s), where Kp is the plant gain,

Pn(s) ) sn + a1sn-1 + ... + an-1s + an

y˘ t ) -yt/τj + K h put-T + wt where wt ) dt - (1/τ - 1/τj)yt + (Kp - K h p)ut-T′ and the subindex t is used to denote that the signal is taken at time t. Analogous to the delay-free case, the following controller is proposed:

ut ) [(1/τj - 1/τc)yt - w j t]/K hp yj˘ t ) -yt/τj + K h put-T + w j t + g1(yt - yjt) j˘ t ) g2(yt - yjt) w

cannot be canceled exactly. For the sake of simplicity, let us assume that τ ) τj and d ) 0. Then

(14)

Notice that due to the delayed input ut-T′ uncertainties

Pm(s) ) sm + b1sm-1 + ... + bm-1s + bm The relative degree of such a plant is n - m (Isidori, 1989). So, the plant can be stabilized with a (n - m)order compensator. If the plant parameters ai’s, bj’s, and Kp are uncertain, the uncertainties can be lumped into an augmented state so that an uncertainty estimator (of order n - m) like that given by eq 14 can be constructed to obtain an estimate of the uncertainties. (ii) The general condition to reconstruct disturbances and uncertainties from output measurements is that they can be lumped into augmented states which are observable (in the usual rank sense). To check observ-

3672 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

Figure 1. Actual d(t) ) 0.5 sin(2πωt) and estimated d h (t) disturbance for several values of the estimator parameter L and ω ) 0.1. The selected closed-loop characteristic time was 0.666. As expected, d h (t) approaches d(t) as L becomes larger.

Figure 3. Dynamics of the controlled system under different choices of the estimated plant gain K h p. The parameters of the controller were L ) 1.0 and τc ) 0.75. The better closed-loop lo behavior was obtained for the choice K h p ) (Kup p + Kp )/2.

Figure 2. Actual d(t) ) 0.5 sin(2πωt) and estimated d h (t) disturbance for several values of the estimator parameter L and ω ) 0.2. Since ω is larger than in Figure 1, larger values of L are required to obtain comparable estimation errors.

ability conditions, a state-space representation is more natural and general. 4. Numerical Simulations In this section, we will use two simulation examples to illustrate the performance of the controller proposed in the above sections. 4.1. Example 1. The first numerical case considered is a first-order system with Kp ) 1 and τ ) 1. Suppose that the system is subjected to the unitary amplitude disturbance d(t) ) 0.5 sin(2πωt). The selected closedloop time constant is 2/3 times the open-loop time constant (it is asked that the nominal closed-loop system be slightly faster than the open-loop one). This heuristic rule is a common choice for chemical processes (Luyben, 1990). The estimator gains are parametrized as g1 ) 2L and g2 ) L2. The objective of the simulations below is to show that if the frequency ω is large, the estimator (5) requires large values of L in order to reconstruct the dynamics of d(t). For ω ) 0.1, Figure 1 presents the estimated disturbances d h (t) for several values of L. Notice that, for L on the order of 10ω, the estimate d h (t) is quite acceptable. This fact is corroborated by Figure 2 where, for ω ) 0.2, the estimated disturbance d h (t) is presented for several values of L. We consider parametric uncertainties. Assume that τ is exactly known, Kp ∈ [0.25, 2], and a constant load

disturbance d ) -1 is acting in the process at t ) 10. Again we assume that Kp ) 1. Uncertainties are associated with parametric uncertainties, so that their dynamics have a characteristic time on the order of the plant characteristic time τ. Since L was interpreted as a rate of uncertainties estimation, we tuned L at values on the order of τ-1. For L ) 1.0 and τc ) 0.75, Figure 3 shows the dynamics of the control input u and the controlled output y(t) for the choices K h p ) 0.25, K hp ) 2.0, and K h p ) (0.25 + 2.0)/2 ) 1.125. As expected, the slower closed-loop response is present for K h p ) Kup p . The faster convergence of y(t) is found for K h p ) k1o p ; however, a serious overshoot is present, and an excessive control effort is also required. Better closed-loop up performance is found for the choice K h p ) (K1o p + Kp )/ 2: acceptable convergence of y(t) with moderate control effort. Suppose now that the actual plant is nonlinear:

y˘ ) -y/τ + Kpu + d + f (y)

(17)

with f (y) ) 0.3 cos (2y). As before, assume that a constant load disturbance d ) -1 is acting in the process. In addition, a change of setpoint is made at t ) 12.0. Figure 4 presents the dynamics of the control input u(t) and the controlled output y(t) for several values of the estimator parameter L. It can be seen that convergence of the controlled output is faster for large values of the estimator parameter L. As before, large values of L induce severe overshoots in the control input. Notice that L has frequency units (i.e., L [)] t-1). Notice that, for L small (L ) 0.01), the controlled output y(t) does not converge to the setpoint. In this way, Figure 4 illustrates the fact that the closed-loop system has a minimum gain L* > 0. In this case, and with the aid of numerical simulations, we found L* = 0.2. For values L < L*, the output y does not converge to the desired setpoint. As in classical PI control, L can be interpreted as an inverse reset time associated with the disturbance (and uncertainty) estimator. Following the IMC-based

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3673

Figure 4. Dynamics of the controlled system (example 1) in the presence of a nonmodeled nonlinearity f (y) ) 0.3 cos(2y). This figure illustrates the fact that the closed-loop system has a minimum gain L* (=0.2).

Figure 5. Dynamics of the controlled system (example 1) when the PII2 compensator is tuned with IMC ideas (see Chien and Fruehauf, 1990). The performance of the controller is acceptable.

tuning strategies for PI controllers derived by Chien and Fruehauf (1990), we propose using values of L of around τ. Figure 5 presents the dynamics of the control input u(t) and the controlled output y(t). With L ) 1.33 (=τ-1), the performance of the controlled system is acceptable. Finally, let us consider the case of delayed control input. Assume that the delay time T is 5 times the open-loop characteristic time τ (i.e., T ) 5). That is, we have a delay-time-dominant process. In addition, assume that τ is exactly known and the actual and the estimated values of Kp are 1.0 and 0.75, respectively. The plant is affected by a constant load disturbance at

Figure 6. Closed-loop behavior of the controlled system (example 1) for several values of the compensator parameter L. In this case, an input delay T ) 5 was considered.

t ) 60 and by a setpoint change at t ) 200. Figure 6 presents the behavior of the controlled output and the manipulated input for several values of L. It can be seen that, for large values of L, the convergence is underdamped. As mentioned in the sections above, L can be seen as a rate of disturbance (uncertainty) estimation. In this way, underdamped (oscillatory) behavior is induced because the estimator tries to reconstruct the dynamics of the uncertainty faster than the underlying delay dynamics induced by the delayed control input. Thus, for delayed systems we propose L to be tuned at values of around T-1. We compared the performance of the proposed controller with a classical PI one tuned with IMC methodologies (Chien and Fruehauf, 1992). Figure 7 presents the closed-loop performance for both controllers. It can be seen that the PI controller with enhanced estimation is superior to the PI-IMC controller. 4.2. Example 2. As a second numerical example, we considered a rectifying column modeled as in Luyben (1990) (see Appendix B). Contrary to the above example, the model of the rectifying column does not match the system structure (1). However, we will see that the proposed PII2 is able to regulate the distillation process. The parameters and nominal flowrate specifications are given in Table 1. The control objective is to regulate the purity of distillate xD by manipulating the reflux flowrate R. For distillate concentrations of about 99% and nominal reflux flowrate R* ) 22 mol/min, a +2% step change was induced in the reflux flowrate R. The time response was fitted with a first-order model G(s) ) Kp′/(τs + 1). The estimated parameters were Kp′ ) 0.002 min-1/mol and τ ) 14.5 min. The next step is to go with closed-loop composition control. A conservative selection of τc ) 10 min was used following the guidelines outlined in the previous example. The controller tuning constant L was selected on the order of τc. That is, it is asked that the uncertainty estimation rate be on the order of the nominal closed-loop convergence rate.

3674 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

Figure 7. Closed-loop performance of the PII2 controller compared with the closed-loop performance of a PI controller tuned with IMC methodologies (Chien and Fruehauf, 1990).

Figure 8. Dynamics of the distillate composition xd and the reflux flowrate R under the action of a PII2 controller with L ) 1 min-1 and τc ) 10 min.

Table 1. Parameter and Nominal Values for the Rectifying Column relative volatility, R ) 1.5 hydraulic time constant, β ) 0.5 min feed flowrate, Fo ) 50 (lb mol)/min feed composition, zo ) 0.5 vapor flowrate, V ) 35 (lb mol)/min reflux flowrate, R ) 27 (lb mol)/min tray holdup, M h j ) 2 lb mol condenser holdup, Mc ) 100 lb mol reboiler holdup, Mr ) 100 lb mol total number of rectifying trays, N ) 13

In order to obtain better converge of the estimation error, we used K h p′ ) 0.004 min-1/mol as an estimate of Kp′. From certain initial conditions, the column was established at the setpoint xd,ref ) 0.99. Then the following set of perturbations to the nominal values were applied: change of setpoint from xd,ref ) 0.99 to xd,ref ) 0.97 at t ) 100 min, -5% in the feed composition at t ) 180 min; change of setpoint from xd,ref ) 0.97 to xd,ref ) 0.99 at t ) 260 min. Figure 8 presents the results for the controlled column. The behavior of the distillate composition is smooth, while the manipulated variable R is saturated only at the initial time and in setpoint step changes. This behavior of R(t) is induced by the disturbance (uncertainty) estimator (5), which, in a first stage must converge to the actual disturbance (uncertainty). If L is excessively large, this learning stage induces severe overshoot in w j (t) and consequently in the manipulated input. As a remedy to overshooting effects, we included input saturation in numerical simulation (R ∈ [20, 35]). In this way, if |w j (t)| takes excessively large values, the control input R(t) saturates and the overshooting is not injected into the feedback loop. In the simulations above, we assumed that the distillate composition was available without delay. This is not a realistic assumption. A 4-min delay in the control input due to the composition analyzer was assumed. Numerical results for the controlled column are presented in Figure 9. Compared with the delay-

Figure 9. Same as in Figure 8, except that an input delay T ) 5 min was considered. Compared with the delay-free case, the performance is still acceptable.

free case, the performance of the controller is not degraded excessively, so it leads to acceptable closedloop performance. Contrary to the delay-free case, the disturbance in the feed composition (at t ) 200 min) induces more effects in the controlled output xD. Of course, this effect is caused by the delayed input Rt-T, which cannot detect the disturbance and correct it faster than the delay time T. 5. Conclusions In this paper we presented a PI controller with enhanced uncertainty estimation capabilities. As a

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3675

departing point, we interpreted the integral action of a classical PI feedback controller as an estimator of load disturbances. Then, by interpreting uncertainties as generated by an extended state, we proposed an uncertainty estimator, which resembles the structure of linear state observers. The transfer function of the underlying feedback controller was derived, and it was shown that, under a certain selection of controller parameters, the controller becomes a classical PI controller. Due to its structure, the proposed controller was called PII2. Physically, the idea behind the controller structure is that the compensator detect disturbances quickly and take corrective action to prevent the controlled output from going far off the desired operation point. Guidelines for tuning of controller parameters were derived from analysis of the resulting closed-loop system and extensive numerical simulations of two examples. For instance, in the case of delayed manipulated inputs, it was observed that the rate of uncertainty estimation L cannot be excessively larger than the rate of control action (T-1). That is, uncertainties cannot be estimated faster than the action of the control input. If L is excessively large, severe underdamped closed-loop behavior may appear. The tuning guidelines obtained in this way were used to simulate the control of a highpurity rectifying column. Numerical results showed that our controller is able to control the distillate composition despite disturbance actions, setpoint changes, and dead time in the composition analyzer. Notation d ) load disturbance g1, g2 ) uncertainty estimator gains Kc ) PI controller gain Ke ) PII2 controller gain Kp ) process gain R ) reflux flowrate T ) time delay u ) manipulated input w ) lumped uncertainties xd ) distillate composition y ) controlled output Greek Letters τ ) process characteristic time τc ) nominal closed-loop characteristic time τI ) PI integral reset time

Appendix A. Some Useful Basic Concepts For the sake of clarity in presentation, this appendix presents some basic stability concepts used within the work. (i) A matrix A is said to be Hurwitz if all its eigenvalues have negative real parts. (ii) A linear system x˘ ) Ax + b is said to be asymptotically stable about the equilibrium point xeq ) -A-1b if ||x(t) - xeq|| f 0 as t f ∞. If the matrix A is Hurwitz, the linear system x˘ ) Ax + b is asymptotically stable. (iii) Consider the linear system x˘ ) Ax + F(t). The following expression is a solution for the system above:

x(t) ) exp(At) x(0) +

∫0texp(A(t - s))F(s) ds

If the matrix A is Hurwitz, there exist positive numbers κ and Λ* such that ||exp(At) x(0)|| e κ||x(0)|| exp(-Λ*t). Appe ndix B. Model of a Binary Rectifying Column (Luyben, 1990) Equimolar overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODE per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship).

dMj/dt ) Lj+1 - Lj d(Mjxj)/dt ) Lj+1xj+1 + Vyj-1 - Ljxj - Vyj yj ) Rxj/(1 + (R - 1)xj) Lj ) L h j + (Mj - M h j)/β where xj and yj are the liquid and vapor compositions in the jth plate, respectively. Lj and Mj are the liquid flowrate and the holdup in the jth plate, and L h j and M hj designate nominal values of Lj and Mj. The parameter β is the hydraulic time constant. Literature Cited Alvarez-Ramirez, J.; Cervantes, I.; Femat, R. Robust controllers for a heat exchanger. Ind. Eng. Chem. Res. 1997, 36, 382-388. Astro¨m, K. J.; Hagglund, T. Automatic Tuning of PID Controllers; Instrument Society of America: Research Triangle Park, NC, 1988. Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 1990, 86 (Oct), 3341. Esfandiari, F.; Khalil, H. K. Output feedback stabilization of fully linearizable systems. Int. J. Control 1992, 56, 1007-1037. Hale, J. Theory of Functional Differential Equations; SpringerVerlag: New York, 1976. Hinrichsen, D.; Pritchard, A. J. Robust exponential stability of time-varying systems. Int. J. Robust Nonlinear Control 1993, 3, 63-83. Isidori, A. Nonlinear Control Systems; Springer-Verlag: Berlin, 1989. Luyben, W. L. Process Modelling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1990. Skogestad, S.; Morari, M. LV-control of high-purity distillation columns. Chem. Eng. Sci. 1988, 43, 33-48. Wang, Q.-G.; Hang, Ch.-Ch.; Zou, B. Low order modeling from relay feedback. Ind. Eng. Chem. Res. 1997, 36, 375-385.

Received for review March 18, 1997 Revised manuscript received June 5, 1997 Accepted June 6, 1997X IE970230V

X Abstract published in Advance ACS Abstracts, August 15, 1997.