Robustness and Parametrization of the Proportional Plus Double

Apr 17, 1999 - Some robustness issues of the PI2 compensator with respect to unmodeled dynamics and nonlinearities are discussed and illustrated with ...
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Ind. Eng. Chem. Res. 1999, 38, 2013-2020

2013

Robustness and Parametrization of the Proportional Plus Double-Integral Compensator† Rosendo Monroy-Loperena,‡ Ilse Cervantes,§ America Morales,§ and Jose Alvarez-Ramirez*,§ Areas de Ana´ lisis Aplicado e Ingenierı´a Quı´mica, Universidad Auto´ noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me´ xico D.F., 09349 Me´ xico

The proportional plus double-integral (PI2) controller was proposed by Belanger and Luyben (Ind. Eng. Chem. Res. 1997, 36, 5339-5347) as a low-frequency compensator. The objective of the additional double-integral compensation is to reject the effects of ramplike disturbances. The aim of this work is to provide a parametrization of the PI2 controller gains in terms of a nominal closed-loop and disturbance estimation time constants. A reduced-order observer is constructed to estimate ramplike disturbances, which is subsequently coupled with a feedback loop to counteract the effects of the disturbance. It is shown that such control configuration is equivalent to a PI2 compensator. Some robustness issues of the PI2 compensator with respect to unmodeled dynamics and nonlinearities are discussed and illustrated with two numerical examples. Introduction

Kc ) Ku /3.04, τI1 ) 2.26Pu, τI22 ) 20.5Pu2 (2)

In a recent paper, Belanger and Luyben1 proposed the following proportional plus double-integral (PI2) controller as a low-frequency compensator to deal with ramplike disturbances:

∫

uc(t) ) Kc{(t) + (1/τI1) (t) dt +

∫∫

(1/τI22) ( (t) dt) dt} (1) where uc(t) is the computed control input, (t) ) yref y(t) is the regulation error, yref is the reference (setpoint), Kc is the controller gain, and τI1 and τI2 are the integral and double-integral time constants, respectively. It can be easily shown that a purely proportional plus integral (PI) controller (i.e., τI2 f ∞) cannot reject efficiently ramplike disturbances.1 Even if the load-transfer function time constant is so large that it resembles an integrator over a wide range of frequencies, the closedloop performance of the system will be very poor. In this way, the objective of the double-integral action

∫∫

(1/τI22) ( (t) dt) dt is to introduce compensation at low frequencies to asymptotically reject the effects of frequently occurring ramplike disturbances. Belanger and Luyben1 developed tuning rules for the PI2 compensator under the assumption that the process model can be approximated with a (first-order) integrator-deadtime model. The proposed tuning rules for the controller parameters {Kc, τI1, τI2} are given in terms of ultimate gain, Ku, and ultimate period, Pu, of the assumed integrator plus deadtime process transfer function: * Corresponding author. Phone: +52-5-724-4649. Fax: +525-724-4900. E-mail: [email protected]. † This work was supported by Conacyt and Instituto Mexicano del Petro´leo (FIES 95-93-II). ‡ Area de Ana ´ lisis Aplicado. § Area de Ingenierı ´a Quı´mica.

Adjustments of the parameters may be required for systems that behave significantly different from an integrator plus deadtime model. The settings (2) may be very conservative for plants with excellent damping properties (i.e., small open-loop time constants). However, although these settings provide a good initial estimate, it would be desirable to dispose of tuning guidelines to obtain a prescribed closed-loop behavior. The purposes of this paper are (1) to interpret the double integral as an estimator of ramplike disturbances and clarify its role into the feedback loop; (2) to provide a parametrization of the PI2 controller gain Kc, and integral times, τI1 and τI2, in terms of a nominal closedloop time constant and a disturbance estimation time constant; (3) to show that the PI2 compensator is robust against unmodeled nonlinearities. Under the assumption that the dynamics of the plant can be approximated by a first-order model, a reducedorder observer is constructed to estimate ramplike disturbances, which is subsequently coupled with a feedback loop to counteract the effects of the disturbance. It is shown that such control configuration is equivalent to the PI2 compensator. A Controller with Disturbance Estimation To derive our PI2 control configuration, let us assume that the plant can be approximated with a first-order model with an additive load disturbance d(t):

y˘ (t) ) Ry(t) + βu(t) + d(t)

(3)

where R is the open-loop pole, which can be either stable (R < 0) or unstable (R g 0), and β is the high-frequency gain. A more realistic model must consider a deadtime in the input channel (i.e., y˘ (t) ) Ry(t) + βu(t-D) + d(t), where D g 0 denotes the deadtime). As a methodological step toward our control design, in this section we will consider that D ) 0.

10.1021/ie980468z CCC: $18.00 © 1999 American Chemical Society Published on Web 04/17/1999

2014 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999

Without loosing generality, yref ) 0 will be assumed. Let a prescribed closed-loop performance be specified by means of a first-order reference model with time constant τc > 0. The closed-loop time constant τc is chosen by the designer, to satisfy the requirements (desired damping rate) of the plant-controller combination. Hence, the ideal control law to achieve the closedloop behavior y˘ ) -τ-1 c y is given by (the use of time symbol t will be omitted unless ambiguity would result)

uid(y,d) ) [-(R + τ-1 c )y - d]/β

(4)

Such a control law is ideal (i.e., nonpractical) because it requires measurements of the load disturbance d, which in most chemical processes is not possible. To construct a practical control law, an estimate, d h , of the load disturbance must be provided. In this way, the computed control law is given by

h ) ) [-(R + uc(y,d

τ-1 c )y

-d h ]/β

(5)

To obtain the estimate d h , a ramp disturbance d(t) is assumed. Ramp disturbances can be modeled by d(2) ) 0, where d(2) denotes the second time derivative of d(t). Notice that step load disturbances can also be modeled by d(2) ) 0. Let us introduce the variable z ) d(1), where d(1) denotes the first time derivative of d(t). Hence, the model in (3) can be represented as following a timeinvariant, three-dimensional system

y˘ ) Ry + βu + d d˙ ) z

(6)

z˘ ) 0 It is interesting to note the following. Given measurements of the output y(t) and the input u(t), the system in (6) is observable.2 This means that the dynamics of the disturbance d(t) and its time derivative d(1)(t) ) z(t) can be reconstructed from the dynamics of the output y(t) and the input u(t) (see Appendix 1). Given the observability property of the system in (6), a reduced order state observer2 can be used to reconstruct the “nonmeasured states” d(t) and z(t). Since all real control systems must deal with physical constraints, let us assume that the control input is subjected to a saturation nonlinearity ur ) sat(uc):

{

ulo ur ) uc uup

if if if

uc e ulo ulo < uc < uup uc g uup

where ur is the real control input, ulo and uup are the lower and upper control input limits; let us introduce the equivalent output measurement Om ) y˘ - Ry - βur. Notice that d(t) ) Om(t). We propose the following reduced order observer:

d h˙ ) zj + 2τ-1 h ) ) zj + 2τ-1 h) e (Om - d e (y˘ - Ry - βur - d h ) ) τ-2 h) zj˙ ) τ-2 e (Om - d e (y˘ - Ry - βur - d

(7)

where d h (t) and zj(t) are respectively estimates of d(t) and z(t) and τe > 0 is the estimation time constant. Define h and e2 ) z-zj. The the estimation errors as e1 ) d - d dynamics of the estimation error e ) (e1, e2)T can be

computed from (6) and (7) to give e˘ ) A0(τe)e, where

A0(τe) )

[

-2τ-1 e -τ-2 e

1 0

]

(8)

-1 The eigenvalues of the matrix A0(τe) are {-τ-1 e , -τe }, which implies that e(t) f 0 asymptotically with a damping time constant not larger than τe. Although the dynamics of the disturbance estimator (7) depend only on the known signals {y(t), u(t)}, it cannot be used as it stands because of the presence of the first time derivative y˘ in the right-hand side. To compute a realizable estimator, let us introduce the following coordinates: w1 -1 ) d h - 2τ-1 e y and w2 ) z - τe y. From the above coordinates, the estimator (7) can be realized with the following second-order filter:

-1 -1 -1 w˘ 1 ) -2τ-1 e w1 + w2 + [τe - 2τe (R + 2τe )]y -

2τ-1 e βur -2 -1 -2 w˘ 2 ) -τ-2 e w1 - τe (R + 2τe )y - τe βur

(9)

d h ) w1 + 2τ-1 e y In this way, the practical compensator is given by the feedback function uc(y,d h ) (eq 5) and the load-disturbance estimator (9). The disturbance estimator (9) can be initialized as follows. Since the disturbance d h and its time derivative z are unknown, d h (0) ) 0 and zj(0) ) 0 can be taken. On the other hand, the output y is available from measurements, which yields the initial -2 conditions w1(0) ) -2τ-1 e y(0) and w2(0) ) -τe y(0). The following comments regarding the behavior of the closed-loop system (3),(5),(9) are in order. (i) As a consequence of the fact that e(t) f 0 asymptotically, we have that d h (t) f d(t) asymptotically and h ) approaches the ideal the computed feedback uc(y, d feedback uid(y,d) with a convergence rate of the order of τ-1 e . (ii) The smaller the value τe, the faster the reconstruction of the dynamics of the load disturbance d(t). (iii) One of the major advantages of the observerbased controller is the following. Given a (fixed) desired closed-loop time constant τc > 0, the compensator has only one adjustable parameter, τe > 0, whose tuning directionality is evident. That is, faster disturbance estimation is obtained with smaller values of τe > 0. Moreover, the smaller the value τe, the closer the behavior of the closed-loop system (3),(5) to the ideal one y˘ ) -τ-1 c y. Transfer Function of the Compensator (5)/(9) In this part of the paper we compute the transfer function of the compensator (5),(9). Since the change of coordinates w1 ) d h - 2τ-1 j - τ-2 e y and w2 ) z e y, can be -1 inverted (i.e., d h ) w1 + 2τe y and zj ) w2 + τ-2 e y), the system (9) is equivalent to the system (7). We can take the system (7) into the Laplace domain to give

h (s)] sD h (s) ) Z h (s) + 2τ-1 e [(s - R)Y(s) - βUr(s) - D (10) h (s)] sZ h (s) ) τ-2 e [(s - R)Y(s) - βUr(s) - D where Z h, D h , Y, and Ur denote the signals zj, d h , y, and ur in the Laplace domain. The above system can be solved

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for D h (s) to give D h (s) ) gY(s)Y(s) - gU(s)Ur(s), where

gY(s) )

-1 τ-1 e (2s + τe )(s - R) -2 s2 + 2τ-1 e s + τe

gU(s) )

-1 βτ-1 e (2s + τe ) -2 s2 + 2τ-1 e s + τe

These expressions, together with the computed feedback function (5), yield

Uc(s) ) fY(s)[-Y(s)] + fU(s)Ur(s)

(11)

where

fY(s) )

[

β-1

]

-1 2 -1 -1 -1 -2 -1 (R + τ-1 c + 2τe )s + τe (2τc + τe )s + τc τe -2 s2 + 2τ-1 e s + τe

fU(s) ) β-1gU(s) If the control input is not saturated (i.e., Ur(s) ) Uc(s)), the following controller transfer function is obtained:

C(s) )

Uc(s) -Y(s)

(12) ) -1 2 -1 -1 -1 -1 -2 (R + τ-1 c + 2τe )s + τe (2τc + τe )s + τc τe β-1 s2

[

]

We can take the compensator (5),(9) as a time-domain realization of the PI2 compensator (13). One of the main advantages of the time-domain representation (5),(9) over the time domain representation (1) is that the first one displays antiwindup capabilities.3 If the timedomain realization (1) is used when the control input is subjected to saturations, the real plant input ur(t) will h (t)). be different from the computed control input uc(y(t),d When this happens, the computed control input does not drive the plant and, as a result, the states of the controller are wrongly updated. This effect is called controller windup.3,4 This situation does not happens in the PI2 configuration given by (5),(7). Since the disturbance estimator (7) (equivalently, the disturbance estimator (9) is driven by the real control input ur(t), the states of the controller d h and zj (equivalently, w1 and w2) are correctly updated, even if the real plant input ur(t) is different from the computed control input h (t)). This means that the estimated disturbance uc(y(t),d d h (t) will converge to the real disturbance d(t) even in the presence of control input saturations. In other words, the compensator (5),(7) has inherent observer properties with respect to the controller states d h and zj (see Walgama and Sternby5). We conclude that the controller (5),(7) (equivalently, (5),(9)) is a time-domain realization with antireset windup structure of the PI2 compensator.3,4,5 Some Robustness Issues

(13)

One important question arises with the use of the PI2 compensator: How does the presence of model/plant mismatches affect the performance of the compensator? Model/plant mismatches are caused by unmodeled dynamics and nonlinearities.6 Let us analyze the case of unmodeled dynamics. These uncertainties arise when (a) the actual plant is of higher relative degree and order and (b) deadtimes are present in the input channel. In the above section, we have shown that a deadtime-free, first-order plant is stable for all τc > 0 and τe > 0. This is not the case for systems with unmodeled dynamics. Suppose that the transfer function of the actual plant is given by

From (12) and (13), the following equivalencies are obtained:

m m-1 + ...+ b1s + b0 Y(s) Nm(s) bms + bm-1s (15) ) ) n n-1 U(s) Dm(s) ans + an-1s + ...+ a1s + a0

We will use the above transfer function to show that if the control input is not subjected to saturations, the controller (5),(7) (equivalently, the controller (5),(9)) has the structure of the PI2 compensator (1). In fact, the PI2 compensator (1) can be expressed in the Laplace domain as

CPI2(s) )

[

Uc(s)

) Kc

-Y(s)

]

τI1τI22s2 + τI22s + τI1 τI1τI22s2

-1 βKc ) R + τ-1 c + 2τe

τI1 ) τI2 )

[

2 + (R + τ-1 c )τe -1 2τ-1 c + τe

]

2 + (R + τ-1 c )τe -1 τ-1 c τe

(14)

1/2

With these equivalencies, we have obtained a (τc,τe)parametrization of the controller gain and the integral times of the PI2 compensator given by (1). Given a desired closed-loop time constant τc > 0 and an estimation time constant τe > 0, the values of the controller gain Kc and the integral times τI1 > 0 and τI2 > 0 are uniquely determined by eq 14. It is not hard to see that the converse is not true: The closed-loop time constant τc > 0 and the estimation time constant τe > 0 are not uniquely determined by a given controller gain Kc and integral time τI1 > 0 and τI2 > 0.

where n g 2 and n g m. Note that by taking a Pade` approximation6,7 for exp(-Ds), a transfer function with deadtimes can also be written as in eq 15. Let KI1 ) Kc/τI1 and KI2 ) Kc/τI2 be the first and second integral gains, respectively. Then the PI2 transfer function becomes

Kcs2 + KI1s + KI2 U(s) ) -Y(s) s2

(16)

The closed-loop system composed by the plant (15) and the compensator (16) will be stable if and only if the (n+1)-order closed-loop characteristic polynomial

s2Dn(s) + (Kcs2 + KI1s + KI2)Nm(s) ) 0

(17)

is stable6,7 (i.e., all its roots have negative real parts). From the equivalencies in eq 14, it is not hard to see that the controller gains βKc, βKI1, and βKI2 go to infinity when either τc f 0 or τe f 0. This fact, together with

2016 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999

simple root locus arguments,6,7 imply that the polynomial (17) becomes unstable for sufficiently small values of the time constants τc and τe. Consequently, m there exist minimum time constants τm c > 0 and τe > 0 m m such that, for all 0 < τc < τc and 0 < τe < τe , the closed-loop system (15),(16) is unstable. We conclude the following: The PI2 compensator cannot stabilize and estimate uncertainties faster than the dynamics of the unmodeled dynamics. For instance, if the actual plant is first-order with deadtime D > 0, the stabilization of the plant cannot be made faster than the action of the control input. Belanger and Luyben1 proposed heuristic tuning rules for the PI2 compensator in terms of the ultimate gain and ultimate period of an integrator plus deadtime plant. In most chemical processes, the ultimate gain Ku and the ultimate period Pu can be computed from a simple relay-feedback test.8 The ultimate gain is associated with a purely proportional feedback. On the other hand, the ultimate period is of the order of the dominant time constant of the unmodeled dynamics.6 With respect to the computed feedback (5), the gain for a purely proportional feedback is R + τ-1 c , which yields the minimum closed-loop time constant τm c ) (Ku R)-1. Following the ideas in Belanger and Luyben,1 the following heuristic rules to obtain initial estimates for the compensator parameters are (a) choose τc of the order of 3τm c , and (b) choose τe of the order of 2.25Pu. In the section that follows, the performance and robustness of the PI2 compensator (5),(9) with the given tuning rule are analyzed for a linear third-order plant. In the case of nonlinearities, these are created during the linearization process about a nominal operating point. Consequently, the uncertainties affecting the linear plant do not fall into the category of disturbances normally considered in the linear control design, namely (a) they are not persistent and (b) they are of a feedback nature. Although these disturbances are not of the type routinely considered, their rejection is of utmost importance.9 Disturbances induced by unmodeled nonlinearities can also be rejected by the PI2 compensator. Let us assume that the plant can be approximated with a first-order model with an additive load-disturbance d(t) and an additive (unknown) nonlinearity φ(y), which may include parametric and structural model/plant mismatches:

y˘ (t) ) Ry(t) + βu(t) + d(t) + φ(y)

(18)

Introduce the disturbance η(t) ) d(t) + φ(y(t)). Contrary to the former case where φ ) 0, in this case the disturbance is of a feedback nature because η(t) contains model/plant mismatches. Notice that parametric uncertainties can be included in φ(y). As in the case of the model (3), the model (18) can be represented as the following three-dimensional system:

y˘ ) Ry + βu + η η˘ ) z

(19)

z˘ ) Γ(y,z,u) where Γ(y,z,u) is the second-time derivative of φ(y) computed via the chain rule. The equivalent feedback function to induce the nominal closed-loop behavior y˘ ) -τ-1 c y becomes

u ) β-1[-(R + τ-1 j] c )y - η

(20)

Since the function φ(y) was assumed to be unknown, its second-time derivative Γ(y,z,u) is also unknown. Moreover, it can also be shown that the dynamics of the disturbance η(t) can be reconstructed from the dynamics of the output y(t) and the input u(t). As in the case of the estimation of d(t), the estimation of the disturbance η(t) can be made via the following reduce-order observer (see eq 7):

j) η j˙ ) zj + 2τ-1 e (y˘ - Ry - βur - η (21)

j) zj˙ ) τ-2 e (y˘ - Ry - βur - η

where η j is an estimate of η. If φ(y) ) 0, the above secondorder filter becomes the disturbance estimator given by eq 7. As in that case, the filter (19) can be realized as follows: -2 -1 -1 w˘ 1 ) -2τ-1 e w1 + w2 + [τe - 2τe (R + 2τe )]y -

2τ-1 e βur -2 -1 -2 w˘ 2 ) -τ-2 e w1 - τe (R + 2τe )y - τe βur

(22)

η j ) w1 + 2τ-1 e y h - 2τ-1 h - τ-2 where w1 ) η e y and w2 ) z e y. Notice that the compensator (20),(22) has the same structure as the compensator (5),(9). Consequently, the transfer function of the compensator (20),(22) is given by (11), which shows that a PI2 compensator can also deal with unmodeled nonlinearities. Contrary to the case where φ(y) ) 0, and because of the presence of the unknown nonlinearity Γ(y,z,u), which is of a feedback nature, the disturbance estimator (22) presents a maximum estimation time constant τM e > 0. This implies that asymptotic convergence of the disturbance estimate η j can be . It should be stressed that assured only if 0 < τe < τM e this is only a theoretical (robustness) result and is far from being realistic since arbitrarily small values of the estimation time constant render the closed-loop system highly sensitive to practically unavoidable measurement noise and unmodeled (high-frequency) dynamics, which limit the achievable closed-loop bandwidth. A complete analysis of these phenomena is beyond the objectives of this work. It can be analyzed with techniques borrowed from the theory of stabilization of nonlinear systems using output feedback.10 Examples In this section the performance and robustness of the PI2 compensator (5)/(9) are analyzed via two examples. Example 1. A Simple Transfer Function.1 Consider a system where the process transfer function1 is GM(s) ) 1/(s + 1)3. Assume that the process model can be approximated with the first-order model G′m(s) ) 1/(s + 1), and the load-transfer function is given by GL(s) ) 1/s2 (a ramp with unit slope). In terms of the notation eq 3, we have R ) -1 and β ) 1. For this system, the minimum closed-loop time constant is τm c ) 1/9 and the ultimate period Pu ) 3.65. Let us choose τc ) 3τm c ) 1/3. For this value of τc, the approximate value τm e = 3.83 was found via numerical simulations. Figure 1 illustrates the differences in response for three values

Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2017

Figure 1. Response of the controlled system (example 1) for three different values of the estimation time constant.

Figure 3. Performance of the PI2 with and without the ARW scheme.

Figure 2. Estimation for the values of the estimation time constant in Figure 1.

of the estimation time constant τe. As expected, the smaller the value τe the better the performance of the PI2 compensator. For τe ) 4.0, which is very close to the minimum estimation time constant τm e = 3.83, the closed-loop is stable with oscillatory behavior. The oscillatory behavior indicates that the unstability of the closed-loop system for τe < τm e is due to a conjugate pair of complex eigenvalues crossing the imaginary axis toward the right-half complex plane. This route to the closed-loop unstability is typical in controlled plants with unmodeled dynamics and deadtimes.6 Figure 2 presents the estimation error d(t) - d h (t), for the values of the estimation time constant considered in Figure 1. The second-order filter (9) is able to estimate the ramplike disturbance d(t), whose adverse effects are subsequently counteracted via the feedback action (5). Now suppose that the plant is under the action of a ramplike disturbance with transfer function GL(s) ) 1/(20s + 1). Assume that input is subjected to a saturation nonlinearity with -2 and +2 as lower and upper limits, respectively. Figure 3 compares the performance of the PI2 compensator with antireset windup (ARW) structure (eqs 5 and 9) and without ARW structure (eq 1), for τe ) 6.0 and the initial conditions y(0) ) 4.0, y(1)(0) ) -2.0, and y(2)(0) ) 1.0. As expected, the controller (5),(9) performs better than controller 1. This behavior is due to the fact that the PI2 configuration given by (5),(9) is driven by the real control input

Figure 4. Dynamics of the estimation error with and without the ARW scheme.

h (t)), which allows the estimator (9) ur(t) ) sat(uc(y(t),d to update correctly the disturbance estimation d h (t) (see Figure 4) despite input saturations (see Figure 3). Example 2. An Unstable CSTR with Nonlinear Dynamics. To illustrate the performance of the PI2 compensator under severe nonlinearities, we will carry out numerical simulations with a jacketed CSTR where an irreversible A f B, first-order reaction is taking place:

c˘ ) θ(cin - c) - ck0 exp(-EA/RT) (23) T˙ ) -(θ + γ)T + ∆Hrck0 exp(-EA/RT) + γu +

θ(T/in + ∆Tin)

where c and T are the reactor composition and temperature, respectively, the index in denotes inlet conditions, T/in is the nominal inlet temperature, ∆Tin is a disturbance in the inlet temperature, θ is the residence time, γ is the heat-transfer coefficient, -∆Hr is the reaction heat, and k0 and EA/R are kinetic parameters. The

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Figure 6. Estimation for the values of the estimation time constant in Figure 5.

Figure 5. Response of the controlled reactor (example 2) for three different values of the estimation time constant.

control objective is to regulate the reactor temperature T at a prescribed reference Tref, by manipulating the jacket temperature u. The following assumptions are taken: (i) The chemical kinetic term ∆Hrck0 exp(-EA/RT) is unknown, and (ii) ∆Tin(t) is a ramplike disturbance. Under the above assumptions, a nominal linear model for the reactor temperature is given by

T˙ ) -(θ + γ)T + γu + θT/in + d

(24)

which corresponds to the first-order model (3) with R ) -(θ + γ), β ) γ, d ) θ∆Tin. Moreover, the unmodeled nonlinearity corresponds with ∆Hrck0 exp(-EA/RT). In this case, the second-order filter must provide an estimate η j (t) of the dynamics of the disturbance θ∆Tin(t) + ∆Hrc(t)k0 exp(-EA/RT(t)), which is subsequently used by the feedback function (20) to counteract its effects. For the nominal input value u ) 350, and the following set of physical parameters θ ) 1, γ ) 1, cin ) 1, Tin ) 350, ∆Tin ) 0, k0 ) exp(25), EA/R ) 104, and ∆Hr ) 200, it is not hard to see that the CSTR has an unstable (saddle-type) equilibrium point at (ceq, Teq) ) (0.5, 400). We will assume that the temperature reference Tref ) 400. Notice that while the nominal model (24) is linear and open-loop stable, the real plant is nonlinear and unstable at the operating point. Moreover, since R ) - (θ + γ) ) -2, we will assume a step disturbance for the θ∆Tin transfer function GL(s) ) 10/(25s + 1). The estimated open-loop time constant is -R-1 ) 0.5, and we choose τc ) 0.5. The lower and upper input limits were chosen as 320 and 380, respectively. Figure 5 presents the dynamics of the controlled reactor for the initial conditions c(0) ) 0.3 and T(0) ) 360.0, and three values of the estimation time constant τe, which are of the order of τc. The maximum estimation time constant was found numerically as τM e = 0.415. 2 compensator is able to stabilize the For τe < τM , the PI e reactor temperature at the prescribed setpoint Tref ) 400, despite unknown nonlinearities and unmeasured

Figure 7. Performance of the PI2 with and without the ARW scheme.

disturbance. On the other hand, for τe < τM e , the reactor temperature does not converge to the reference value Tref ) 400. An explanation for this phenomenon is provided by a close examination of the compensator structure. If the disturbance estimator (22) is unable to provide a fast and reliable estimation of the real disturbance η(t) ) θ∆Tin(t) + ∆Hrc(t)k0 exp(-EA/RT(t)) (see Figure 6), the effects of such a disturbance will not be counteracted by the feedback function (20). An unstabilizing effect is then created into the feedback loop, as is evident in Figure 5. Figure 7 compares the performance of the PI2 compensator with the ARW structure (eqs 20 and 22) and without the ARW structure (eq 1), for τc ) 0.5 and τe ) 0.3. In this case, the ARW plays an important role in the stabilization of the chemical reactor. In fact, Figure 7 shows that stabilization is not attained for the given initial conditions if an ARW structure is not used. The behavior of the controlled temperature is oscillatory,

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Appendix 1. Observability Property of the System (6) The system (6) can be written in vectorial notation as follows:

x˘ ) Ax + bu, x ∈ R3

(A.1)

y ) Cx where x ) (y,d,z)T, “T” denotes transposition, and

[

R A) 0 0

1 0 0

] []

0 1 1 , b ) 0 , C ) [1 0 0] 0 0

The observability matrix associated with the threedimensional system (A.1) is2:

[ ][

C 1 O ) CA ) R CA2 R2

Figure 8. Response of the controlled reactor under a periodical disturbance in the inlet temperature.

which is induced by the input saturation. These numerical simulations illustrate the so-called controller windup.3 An interesting question is the behavior of the PI2 compensator when the load disturbances are periodical (see Appendix 2). Assume that ∆Tin ) A sin(2πωt). Figure 8 presents the dynamics of the controlled reactor for A ) 2.5, ω ) 0.5, and three different values of the estimation time constant. The maximum estimation time constant was found numerically as τM e = 0.382. For τe < τM e , the reactor temperature converges practically (i.e., the smaller the estimation time constant, the smaller the regulation error) to the reference value. On the other hand, the jacket temperature is periodical to counteract the effects of the periodical disturbance. This control action is possible thanks to the estimator scheme introduced by (7).

0 1 R

0 0 1

]

whose rank is equal to 3. Consequently, the system (A.1) is observable, which means that the dynamics of the disturbance d(t) and its time derivative d(1) ) z(t) can be reconstructed from the dynamics of the output y(t) and the input u(t). Appendix 2. Estimation of Periodical Load Disturbances In the body of the work, we assumed ramplike load disturbances satisfying d(2) ) 0. It should be noted that this class of disturbances includes steplike load disturbances. However, periodical load disturbance is an important class of disturbances that should be considered. In this appendix, we show that the PI2 compensator can estimate in an approximate manner this class of disturbances. To this end, let us assume d(t) as a disturbance of frequency ω and amplitude A that satisfies the linear differential equation:

d(2) + (2πω)2d ) 0 Notice that step- and ramplike disturbances can be interpreted as periodical disturbances of frequency zero. The model (3) can be represented as the following timeinvariant, three-dimensional system:

y˘ ) Ry + βu + d Conclusions An observer-based derivation of the proportional plus double-integral (PI2) compensator has been given in this paper. By doing this, the double-integral action has been interpreted as an estimator of ramplike disturbances. Moreover, a parametrization of the controller gain and integral times has been obtained, which clarifies the role of the PI2 components into the feedback loop. Some tuning heuristics for the controller have been presented and illustrated with two numerical examples. It has been shown that the design methodology can be extended to deal with cases with unmodeled nonlinearities. In such a case, it was also shown that the resulting controller also has the structure of a PI2 compensator, which demonstrates the robustness capabilities of this compensator in the presence of nonlinearities.

d˙ ) z

(A.2)

z˘ ) -(2πw)2d Following the ideas depicted in this paper, it can be shown that the dynamics of the disturbance d(t) can be reconstructed from the dynamics of the output y(t) and the input u(t). In this way, the reduced-order observer (7) can be taken. Notice that the unknown function (2πω)d was not considered in the construction of the observer. The dynamics of the estimation error can be computed from (7) and (A.2) to give

e˘ ) A0(τe)e - (2πω) dB1

(A.3)

where BT1 ) (0,1) and A0(τe) is given as in (8). By integrating (A.3) and using triangle and Gronwall inequalities,11 we can conclude that the estimation error converges in finite time of the order of τe to a ball B(r)

2020 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999

with radius r of the order of τe A0ω. In this way, the smaller the estimation time constant, the smaller the estimation error. On the other hand, the larger the frequency of the periodic disturbance, the larger the estimation error. That is, high-frequency disturbances are harder to be estimated and counteracted by the PI2 compensator. Literature Cited (1) Belanger, P. W.; Lyuben, W. L. Design of low-frequency compensator for improvement of plantwide regulatory performance. Ind. Eng. Chem. Res. 1997, 36, 5359-5347. (2) Kailath, T. Linear Systems; Prentice-Hall: Englewood Cliffs, NJ, 1980. (3) Kothare, M. V.; Campo, P. J.; Morari, M.; Netts, C. N. A unified study of anti-windup designs. Automatica 1994, 30, 18691883. (4) Campo, P. J.; Morari, M. Robust control of processes subjected to saturation nonlinearities. Comput. Chem. Eng. 1990, 14, 343-358.

(5) Walgama, K. S.; Sternby, J. Inherent observer property in a class of anti-windup compensators. Int. J. Control 1990, 52, 705724. (6) Maciejowski, J. M. Multivariable Feedback Design; AddisonWesley: Great Britain, 1989. (7) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: New York, 1989. (8) Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/deadtime processes. Ind. Eng. Chem. Res. 1992, 31, 2625-2628. (9) Pachter, M.; D’Azzo, J. J.; Veth, M. Proportional and integral control of nonlinear systems, Int. J. Control 1996, 64, 679-692. (10) Khalil, H. K.; Esfandiari, F. Semiglobal stabilization of a class of nonlinear systems using output feedback. IEEE Trans. Autom. Control 1993, 38, 1412-1415. (11) Khalil, H. K. Nonlinear Systems; Prentice-Hall: Englewood Cliffs, NJ, 1997.

Received for review July 17, 1998 Revised manuscript received February 25, 1999 Accepted February 26, 1999 IE980468Z