Robust Proportional−Integral Control - American Chemical Society

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Ind. Eng. Chem. Res. 1998, 37, 4740-4747

Robust Proportional-Integral Control Jose Alvarez-Ramirez,* America Morales, and Ilse Cervantes Departamento de Ingenieria de Procesos e Hidraulica, Universidad Autonoma MetropolitanasIztapalapa, Apartado Postal 55-534, Mexico D.F., 09340 Mexico

In this paper, the classical proportional-integral controller is derived by using an on-line uncertainty estimation approach. The crucial role of integral action in linearization-based controller design for uncertain and externally disturbed systems is highlighted. Specifically, the integral action is interpreted as a modeling error compensator, which induces certain robustness capabilities against external disturbances, uncertain parameters, and nonlinearities. Our approach leads to a parametrization of the controller gain and the integral time in terms of estimation and nominal closed-loop time constants. As a consequence of our results, the role of the integral time in the robustness of the closed-loop system is clarified. 1. Introduction Proportional-integral (PI) control is a very popular control method. The rationale behind the derivation of the classical PI controller is as follows.1 Proportional action is used to either stabilize or enhance the convergence of the plant. However, proportional control alone cannot remove output steady-state offset when load disturbances affect the plant. From a simple Laplace domain analysis of the resulting closed-loop system, it is then concluded that the integral action is sufficient to remove steady-state offset. A wide variety of tuning rules have been reported. For instance, IMC2,3 and relay methodologies4 are very popular and effective. In a lot of practical cases, PI controllers tuned with such methodologies provide acceptable closed-loop performance and good robustness margins against uncertain plant parameters, nonconstant bounded external disturbances, and nonlinearities. It is well-known that a system driven by a PI controller is robust because the integral action asymptotically estimates the load disturbance and the feedback loop counteracts its effects.5 By continuity arguments, it can be concluded that the integral error becomes an estimate of low-frequency disturbances. However, as Pratcher et al.6 have remarked, the role of the integral action in the robustness capabilities of the PI controller when nonlinearities are present has not been completely clarified. The aim of this work is to derive the classical PI controller by using a modeling error compensation approach. The crucial role of integral action in linearization-based controller design for uncertain and externally disturbed systems is highlighted. Specifically, the integral action is interpreted as a modeling error compensator, which induces certain robustness capabilities against external disturbances, uncertain parameters, and nonlinearities into the controlled system. Our approach yields a parametrization of the controller gain and the integral time in terms of estimation and nominal closed-loop time constants. In this way, the role of the integral time in the robustness of the closedloop system is clarified. The paper is organized as follows. In section 2, a controller with uncertainty estimation is derived. In * Corresponding author. Phone: +52-5-7244649. Fax: +525-7244900. E-mail: [email protected].

section 3, it is shown that this controller is the wellknown PI controller with a special parametrization. The role of the integral action in nonlinear control is discussed in section 4. In section 5, the discussed concepts are illustrated in the context of two chemical process control examples. Conclusions are made in section 6. 2. A Simple Controller with Uncertainty Estimation It is considered that a linear model of the plant is given as

G(s) ) Y(s)/U(s) )

b h e-Dh S s+a j

where a j, b h , and D h are estimates of the actual parameters of the plant. This model can be obtained either from linearization of a nonlinear model or from plant data. This type of model is able to represent the dynamics of many processes over the frequency of interest for feedback controller design.1 One-point distillation columns2 and heat exchangers7 are examples of processes whose dynamics can be represented by this model. If a > 0, the plant is stable. If a ) 0, the plant is an integrator. If a < 0, the plant is unstable. The following assumptions are considered. Assumption 1. For simplicity, D ) 0 will be assumed in a first stage. Subsequently, ideas from IMC2 will be used to account for delayed input. It interesting to note that the delay term in the process model can be seen as a multiplicative model uncertainty, which will limit the design bandwidth of the feedback control loop.2 Assumption 2. The sign of the high-frequency gain b is known. Assumption 3. The plant is affected by nonmodeled nonlinearities F(y) and external disturbances π(t) with bounded variation (i.e., |π(1)| e δ1, where π(1) denotes the variation of π(t)). This is a realistic assumption. In fact, nonlinearities are always present and cannot be removed during the linearization process. Under the assumptions above, the plant under consideration can be represented as

y˘ ) ay + F(y) + π(t) + bu

10.1021/ie980180+ CCC: $15.00 © 1998 American Chemical Society Published on Web 12/07/1998

(1)

Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4741

The control objective is to have the output of the plant track a reference yref. Define the regulation error x(t) ) yref - y(t). If the term -ayref is taken as a constant disturbance that can be included in d(t), the plant (1) can be written as

x˘ ) ax + σ(x) + d(t) - bu

(2)

where σ(x) ) -F(yref - x) and d(t) ) -π(t) - ayref. Note that the nominal linearization of this plant is x˘ ) a jx b h u. If a PI-controller design C(s) ) U(s)/X(s) ) KP(1 + jx - b h u and is 1/TIs) is based on the nominal plant x˘ ) a used to control the actual plant (2), the closed-loop system is

x˘ ) (a - bKP)x - bKIz + σ(x) + d(t) z˘ ) x

(3)

where KI ) KP/τI is the integral gain. Suppose that a control objective is to reduce the effects of the external disturbance d(t) on the regulation error x(t), while maintaining closed-loop stability. A common practice to achieve this objective is to induce a faster transient by increasing the magnitude of the PI-controller gains KP and KI. This practice does not have a formal basis and may lead to adverse effects. In fact, Kokotovic and Marino8 have shown that if the nonlinearity is severe (for instance, σ(x) ) x3), the stability region for systems of the form (3) vanishes as the magnitudes of KP and KI become excessively large. Since all practical systems are subjected to unmodeled nonlinearities, a linearization-based control design, as in the case of PI controllers, must be designed to deal with this class of uncertainties. We will design a linear controller with capabilities to deal with the unmodeled nonlinearity σ(y) and the unmeasured disturbance d(t). Let us describe the system (2) as

x˘ ) a j x + φ(x,u,d) - b hu

(4)

where φ(x,u,d) ) ∆ax + σ(x) + d(t) - ∆bu, ∆a ) a - a j, and ∆b ) b - b h . The function φ(x,u,d) contains all of the uncertain terms of the system (2). In fact, φ(x,u,d) is the modeling error induced by external disturbances and unmodeled nonlinearities. If φ(x,u,d) ) 0, the nominal model x˘ ) a jx - b h u is obtained. The uncertainty φ(x,u,d) affecting the linear plant x˘ ) a jx - b h u is persistent and of feedback nature. Although these plant perturbations are not of the type routinely considered to be handled by standard linear controllers, their rejection is of utmost importance to assure stable closedloop operation. Let us define an augmented state η ∈ R. The system (4) is equivalent to the following5

x˘ ) a jx + η - b hu η˘ ) ψ(x,η,U,D)

(5)

where U ) (u,u(1)), D ) (d,d(1)), and u(1) and d(1) denote the time derivatives of u and d, respectively. The function ψ(x,η,U,D) is the time derivative of φ(x,u,d), which is computed via the chain rule for derivation (see the appendix). If a controller stabilizes the extended system (5), then such a controller also stabilizes the actual plant (2). The augmented state representation (5) is the departing point for our control design.

Let ac < 0 be a prescribed closed-loop pole or, equivalently, let τc ) -ac-1 be a prescribed closed-loop time constant. A feedback controller for the system (5) is then given by

u ) -[(ac - a j )x - η]/b h

(6)

so the closed-loop behavior is governed by x˘ ) acx ) -τc-1x. Since the augmented state η (equivalently the modeling error φ(x,u,d)) is unknown, the feedback control (6) cannot be implemented just as it is. In fact, a procedure to estimate η is required. To this end, let us remark the following important property of the system (5): “The dynamics of the augmented state η(t) can be reconstructed from the dynamics of the output y(t) and input u(t) signals. In fact, note that η(t) ) x˘ (t) - a j x(t) + b h u(t). Thus, η(t) can be reconstructed by using measurements of x(t) and u(t) and a derivator to realize x˘ (t).” Since perfect derivators cannot be realized in practice, the most common alternative is to use observers9 to estimate unmeasured states. Let η j (t) be an estimate of η(t). When it is recalled that η ) x˘ - a˘ x + b h u, the following reduced-order observer is proposed as an estimator for the modeling error η(t)

η j˙ ) (η - η j )/τe ) (x˘ - a j x + bu - η j )/τe

(7)

where τe > 0 is the estimation time constant, which is an adjustable parameter to be specified in the sections below. Since φ(x,u,d) is unknown, ψ(x,η,U,D) is also unknown, so that this term was not considered in the construction of the estimator (7). To motivate the choice of the estimator (7), note that because of the asymptotic convergence of η j˙ to zero (7) implies that η j asymptotically reconstructs the modeling error signal η. Since x˘ is on the right-hand side of (7), the variable j + x is introduced. Consequently, the estimaw ) -τeη tor (7) can be rewritten as

w˘ ) a jx - b hu + η j η j ) (x - w)/τe

(8)

Given the regulation error x(t) and the input u(t) signals, the first-order filter (8) provides an estimate η j of the modeling error η ) φ(x,u,d) ) ∆ax - ∆bu + σ(x) + d. From (6), the corresponding control signal is

j )x - η j ]/b h ) [(τc-1 + a j )x + η j ]/b h u ) -[(ac - a

(9)

The estimator (8) together with the feedback (9) comprises our controller design. The estimator (8) provides the estimate η j , which is subsequently used in (9) to counteract the effects of the modeling error η ) ∆ax ∆bu + σ(x) + d. In this way, if η j asymptotically reconstructs the modeling error signal η, the plant (2) under the controller consisting of (8) and (9) will behave asymptotically as the asymptotically stable system x˘ ) acx ) -τc-1x. 3. Proportional and Integral Control The concepts discussed in section 2 will be discussed in the context of proportional and integral control actions.

4742 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998

The feedback function (9) yields a jx - b hu + η j ) -τc-1x, -1 which can be used in (8) to give ω˘ ) -τc x. Define ) ν ) -τcw. Hence, the estimator (8) can be written as ν˘ ) x and η j ) (x + τc-1ν)/τe. The controller consisting of (8) and (9) can be equivalently written as

u ) [(τc-1 + τe-1 + a j )x + (τcτe)-1ν]/b h ν˘ ) x

(10)

which is a PI controller with proportional and integral gains given by

b h KP ) τc-1 + τe-1 + a j bKI ) KP/τI ) (τcτe)-1

(11)

Since τI ) KP/KI, the integral time is given by

τI ) τcτe(τc-1 + τe-1 + a j)

(12)

Consequently, our control design with modeling error estimation is equivalent to a classical PI controller, as long as the feedback control is not affected by actuator nonlinearities (e.g., input saturation). Comments. (a) The control gain and integral time parametrization consisting of (11) and (12) has the enormous advantage that the tuning up of τc and τe is particularly easy in view of the fact that, up to the point where the influence of nonmodeled plant behavior is no longer negligible, the velocity of convergence of the closed-loop system increases monotonically with τc, while the sensitivity of the feedback loop increases monotonically with τe. (b) It should be noted that an interesting interconnection exists between the proposed PI control configuration and previous developments on self-tuning PI controllers with zero-gain predictors10 (see also the de Santis’ paper11 for PI and PID configurations in the context of speed and position control). These developments typically involve the designs of feedback controllers with estimation of zero-mean disturbances, which yields integral action in the controller and offset removal in the tuning algorithm. When the zero-mean predictor is realized in a discrete time system, it becomes equivalent to a one-step-ahead estimator. In the case of the proposed PI configuration (10), the disturbance estimation via a reduced-order observer (7) plays the role of the zero-mean estimator in the work of Gawthrop.10 However, our approach leads to a different parametrization of the control gain and integral time (see section IV.A of Gawthrop’s paper), which we believe is easy to tune up. (c) From (11), it is concluded that a control parameter {τc, τe} setting defines a unique proportional and integral gains {KP, KI} setting. The converse is not true. Given a PI controller {KP, KI} setting, {τc, τe} setting corresponding to the control structure consisting of (8) and (9) is given as the positive solutions Rc and Rf of the equations

h KP - a j Rc + Rf ) b

(13a)

RcRf ) b h KI

(13b)

where Rc ) τc-1 and Rf ) τf-1. On the one hand, since

b h KI > 0, (13b) corresponds to a hyperbole with a branch into the first quadrant R+. On the other hand, (13a) corresponds to a straight line with a slope equal to -1 and intersection b h KP - a j . It is not hard to show that the nonlinear algebraic system (13) has positive solutions if and only if b h KP - a j g 2(b h KI)1/2. Moreover, (i) if 1/2 b h KP - a j ) 2(b h KI) (the critical case), Rc ) Rf ) (b h KP a j )/2 is the unique positive solution, and (ii) if b h KP - a j h kI)1/2 (the regular case), there exist two > 2(b h KI) × 2(b positive solutions {Rc,1, Rf,1} and {Rc,2, Rf,2} satisfying Rc,1 ) Rf,2 and Rc,2 ) Rf,1. This is a very interesting finding since it implies that a PI controller with gains {KP, KI} can be realized with the controller consisting of (8) and (9) via two different pairs, {τc,1, τf,1} and {τc,2, τf,2}. (d) If the nominal system is open-loop stable (a j < 0), the value -a j -1 corresponds to the nominal open-loop time constant τj, and k h)b h τj corresponds to the nominal steady-state gain of the plant. In this case, the proportional gain and integral time can be written as

k h KP ) τj(τc-1 + τe-1 - τj-1) τI ) τcτe(τc-1 + τe-1 - τj-1) h KP ) τjτc-1 and τI ) τj, which If τe ) τj is chosen, then k corresponds to a PI controller tuned with IMC rules for first-order plants with no input delay.2 Note that with IMC tuning rules the correspondence between the settings {KP, KI} and {τc, τe} is one-to-one. Thus, a PI controller tuned with IMC rules corresponds to a controller consisting of (8) and (9) with an estimation time constant equal to the nominal open-loop time constant τj. That is, estimation of the modeling error is made as fast as the open-loop dynamics. (e) So far, we have assumed that the control input is never saturated. When a sudden change in the reference input or disturbance affects the plant, the control input can meet its saturation limits. In this situation, the feedback is effectively broken and the plant behaves as an open-loop, yielding deterioration of the performance of a PI controller. Hence, because of the integral action, the control saturation gives rise to undesirable side effects such as large overshoots and large settling times.12 This phenomenon is known to as reset windup. From the viewpoint of implementation, reset windup appears because the integral action ν˘ ) x has no knowledge of the control input saturation, so that it continues integrating even if the feedback is broken. If the control input returns to admissible values and the feedback is effective, there exists a mismatch between the computed and the current integral error. Our control design displays a natural model-based antiwindup structure for PI controllers. Given a PI controller with gains {KP, KI} (equivalently {KP, τI}), it can be realized as

jx + b h us + η j) ν˘ ) τc (a η j ) (x + ν/τc)/τe

(14)

where the {τc, τe} setting is obtained as a positive solution of (13), us is the computed control input

j )x + η j ]/b h} us ) Sat{[(τc-1 + a

(15)

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Sat( ) is the saturation function

of completeness in presentation, a sketch of the proof of the results below is given in the appendix. Case 1. Either d(t) ) constant, for all t g 0, or d(t) f constant as t f ∞. In general, this corresponds to the case of step disturbances. Take

{

umin if ϑ e umin if umin < ϑ < umax Sat(ϑ) ) ϑ umax if ϑ g umax and umin and umax are estimated minimum and maximum control input limits, respectively. If umin and umax correspond to the exact values of the control max input limits, the variables ν(t) and η j (t) are the integral error and the estimate of the modeling error, respectively. Moreover, when the control input is unsaturated, us ) j )x + η j ]/b h and consequently the compensator [(τc-1 + a consisting of (14) and (15) is equivalent to a PI controller. It is worth noting that the antireset windup structure discussed above cannot reduce completely the adverse effects of integrator windup. In fact, suppose that the controller is realized in a discrete-time system and that at each sampling time the antiwindup controller is a discrete-time version of (14) and (15). If saturation occurs and the actual control input us(t) may be measured, this signal can be used for computing the next control state ν(t+1). In this way, a mismatch between the actual η(t) and the estimated η j (t) modeling errors still exists. Of course, the higher the sampling frequency, the smaller the modeling error mismatch. 4. Stability Analysis The equivalence between our control design and the classical PI controller was stated in section 3. The stability of the system (2) under the controller consisting of (8) and (9) will be studied in this section. Let e ) η - η j be the estimation error. The closedloop system becomes

x˘ ) -τc-1x + e e˘ ) - τe-1µe + Ξ(x,e) + d(1)

(16)

where µ ) b/b h and Ξ(x,e) is given by (A.4) in the appendix. For simplicity in the analysis, let us take the following assumption. Assumption 4. |σ′(x)| e γ1 < ∞. That is, the gradient of the nonlinearity σ(x) is bounded. In this way, the class of unmodeled nonlinearities includes functions in coniclike sections (i.e., |σ(x)| e γ2|x| + γ3). From assumptions 3 and 4 and (A.4) (see the appendix), we get

|Ξ(x,e)| e λ1|x| + λ2|e|

(17)

where

h )(τc-1 + a)|} λ1 ) τ-1{|∆a| + γ1 + |(∆b/b λ2 ) τcλ1

(18)

Note that both λ1 and λ2 are independent of the estimation time constant τe. By prescribing a nominal closed-loop performance through the closed-loop time constant τc, the controller consisting of (8) and (9) has a single adjustable parameter τe, which is used to achieve stability of the closedloop system. Depending on the dynamics of the disturbance d(t), two cases can be distinguished. For the sake

t/e )

µ λ2 + 0.25τc(1 + λ1)2

(19)

Then, y(t) f 0 and e(t) f 0 as t f ∞, for all 0 < τe < τ/e. The limiting behavior (y(t), e(t)) as t f ∞ shows that y(t) approaches its nominal closed-loop trajectory x(t) ) x(0) exp(-t/τc) as τe f 0 uniformly in t for all t g t1 and any fixed t1 > 0. In other words, the closed-loop behavior with perfect knowledge x˘ ) acx ) -τc-1x is achieved asymptotically as the estimation time constant τe f 0. Case 2. d(t) is a persistent external disturbance with bounded variation. This corresponds to cases where the plant is persistently excited by the external disturbance d(t). For all 0 < τe < τ/e, y(t) and e(t) are bounded and the mean-square value of y(t) is of the order O(τe). Comments. (a) When persistent disturbances affect the plant, perfect regulation cannot be achieved. However, the regulation error can be made as small as desired by taking small values of the estimation time constant τe. (b) Stability results show that τ/e is a maximum allowable estimation time constant. For values of τe > τ/e, closed-loop stability cannot be assured. In this way, τ/e can be taken as a measure of the robustness of the e controlled plant. The larger the value of τ/e, the better the robustness capabilities of the controlled plant. (c) From (18) and (19) and after some algebraic manipulations, one can see that the smaller the value of the closed-loop time constant τc, the smaller the value of the critical estimation time constant τ/e. This conclusion establishes a well-known fact in robust control theory:2 In order to obtain larger stability margins, the controller must be detuned, thus leading to slower closedloop dynamics. (d) From the discussion above, it is concluded that very small values of the estimation time constant τe are desirable. However, measurement noise and unmodeled high-frequency dynamics (e.g., actuator dynamics) impose limitations on the estimator bandwidth. Excessively small values of τe must be avoided in practice. For open-loop stable systems, IMC tuning guidelines can be used to choose τe. Specifically, τe must be chosen on the order of the open-loop time constant τj (see Chien and Fruehauf3). (e) Delayed control inputs (i.e., D * 0) impose additional limitations in the achievable closed-loop performance. For instance, persistent disturbance cannot be attenuated arbitrarily (the mean-square tracking error is no longer of the order O(τe)). Following ideas from IMC tuning guidelines (see Morari and Zafiriou2), the prescribed closed-loop time constant τc is lowerlimited by the time delay D. In this way and as can be concluded from robust control theory results,2 the performance of the controller consisting of (8) and (9), and consequently of PI controllers, is strongly limited by nonminimum-phase components, neglected highfrequency dynamics, and measurement noise.

4744 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998

Figure 1. Performance of the controlled system (example 1) for two different values of the estimation time constant τe.

Figure 2. Effects of the antireset windup structure on the performance of the controlled system (example 1).

5. Examples We illustrate the controller consisting of (8) and (9) of the previous sections via two simulation examples: a first-order example and a control of temperature in a CSTR example. Example 1. Consider a first-order plant described as in (2) whose estimated linear model is represented by the transfer function

behavior of the reference system x˘ ) -τc-1x. The approximate value τ/e = 1.35 was found by means of numerical simulations. For τe > τ/e, the regulation error does not converge to zero. To demonstrate the antiwindup capabilities of our control design, suppose the input limits umin ) -1 and umax ) +1. Figure 2 presents the behavior of the closedloop system when the plant is driven by PI controller (21) and by the PI controller with antiwindup structure

G(s) ) Y(s)/U(s) ) b h e-Dh S/s

w˘ ) -b hu + η j

(20)

If the input is not subjected to saturation nonlinearities, the controller consisting of (8) and (9) is written as

w˘ ) τc-1x η j ) (x - w)/τe u ) [τc-1x + η j ]/b h

(21)

which corresponds to a PI controller with b h KP ) τc-1 + h KI ) KP/τI ) (τcτe)-1. Suppose that the plant τe-1 and b is affected by the nonlinearity σ(x) ) -2x3 and the step disturbance d(t) ) 1, where t g 0. Moreover, let yref ) 1, b ) 2, b h ) 3, and τc ) 1. Initial conditions are taken as yO ) 0 and wO ) 0. Figure 1 presents numerical simulations for two different values of the estimation time constant τe. It can be observed that as the estimation time constant τe takes smaller values, the behavior of the closed-loop system approaches the

η j ) (x - w)/τe u ) Sat{[τc-1x + η j ]/b h}

(22)

Note that, since the PI-controller structure (22) has information about input saturation, faster convergence is obtained when the controller (22) is used. Moreover, the control input remains saturated for a larger period when the classical PI-controller configuration is used. Let us assume that the plant is subjected to the persistent disturbance d(t) ) 1.0 + 0.5 sin(2t). Figure 3 presents the behavior of the controlled closed-loop system. Note that the smaller regulation error is obtained when the estimation time constant τe takes smaller values. Theoretically, the regulation error x(t) f 0 as τe f 0; however, excessively large overshoot can be induced. Suppose now that D h ) 1.0. Following ideas from IMC h ) is chosen. Figure 4 presents tuning rules,3 τc ) O(D the behavior of the controlled plant for τc ) 0.5 and

Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4745

Figure 3. Behavior of the controlled system (example 1) under the action of a persistent disturbance.

Figure 4. Behavior of the controlled system (example 1) in the presence of a delayed control input (D ) 1.0).

different values of τe. As before, faster convergence is obtained for small values of the estimation time constant τe. In fact, the achievable closed-loop performance is obtained on the limit as τe f 0. Example 2. Consider the model of a CSTR with a single exothermical reaction

where x ) Tref - T, u ) T h j - Tj, and T h j is the nominal jacket temperature. If a first-order reaction rate R(c,T) ) ckO exp(-EA/RT) is taken with kO ) exp(25), EA/R ) hj 104, ∆Hr ) 200, θ ) 1, γ ) 1, cin ) 1, Tin ) 350, and T ) 350, the CSTR has an unstable equilibrium point at (ceq, Teq) ) (0.5, 400). The performance of the controller (25) is illustrated by using the following set of perturbations: The reactor is initially stabilized at the unstable point (0.5, 400) (i.e., Tref ) 400), a -10% step disturbance in Tin is introduced at t ) 10, and a setpoint change from Tref ) 400 to 410 is made at t ) 20. Figure 5 presents the behavior of controlled CSTR for initial conditions (cO, TO) ) (1, 350), the control input bounds Tj,min ) 300 and Tj,max ) 400, the closed-loop time j ) 1.25, and several values of the constant τc ) 1, γ estimation time constant τe. For reference, the ideal closed-loop behavior T˙ ) -τc-1(T - Tref) is also presented. As in example 1, as τe takes smaller values, the closed-loop behavior approaches the ideal one. To illustrate the effects of input saturation on the closed-loop performance, Figure 6 presents numerical simulations for τc ) 1 and τe ) 0.2. Note that when the control input is saturated, the performance of the PI controller without antiwindup structure is seriously degraded. Finally, to demonstrate the capabilities of the controller (25) to reconstruct the modeling error, Figure 7 presents the behavior of the actual η(t) and the estimated η j (t) modeling error for different values of the estimation time constant τe. Note that the smaller the

c˘ ) θ(cin - c) - R(c,T) T˙ ) θ(Tin - T) + ∆HrR(c,T) + γ(Tj - T)

(23)

Assume that the control objective is to regulate the reactor temperature T via manipulations of the jacket temperature Tj. Let γ j be an estimate of the heattransfer coefficient γ. If the reaction rate R(c,T) and the reaction enthalpy ∆Hr are unknown, a very crude input-output model is

j e-DS/(s + (γ j + θ)) ∆T(s)/∆Tj(s) ) γ

(24)

which represents a stable plant with open-loop pole a j ) -(γ j + θ) (or open-loop time constant τj ) 1/(γ j + θ)) and high-frequency gain b h ) γ j . In this case, the controller consisting of (8) and (9) can be written as

w˘ ) -(γ j + θ)x - γ ju + η j η j ) (x - w)/τe Tj ) T h j + Sat{[(τc-1 - γ j - θ)x + η j ]/γ j}

(25)

4746 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998

Figure 5. Behavior of the controlled CSTR under the proposed PI configuration.

Figure 6. Effects of the antireset windup structure on the performance of the controlled CSTR.

value of τe, the faster the reconstruction of the modeling error η(t). 6. Conclusions The classical PI controller was derived by using a modeling error estimation approach. The role of integral action in linearization-based controller design for uncertain and externally disturbed systems was clarified. In fact, the integral action was interpreted as a modeling error compensator, which induces certain robustness capabilities against external disturbances, uncertain parameters, and unmodeled nonlinearities. Moreover, our control design provides a model-based antireset windup structure for PI controllers. Numerical examples were used to illustrate the performance of the resulting closed-loop system. Our main conclusion is that even though integral control might not be mandated by the linear analysis, integral action is necessary for the control of plants with unmodeled nonlinearities.

Appendix A (I) Computation of ψ(x,η,U,D). Since φ(x,u,d) ) ∆ax + σ(x) - ∆bu + d, we have that

η˘ ) d(φ(x,u,d))/dt ) (∆a + σ′(x))(a jx + η - b h u) - ∆bu(1) + d(1)

Figure 7. Behavior of the actual η(t) and estimated η j (t) modeling errors.

Consequently

ψ(x,η,U,D) ) (∆a + σ′(x))(a jx + η - b h u) ∆bu(l) + d(1) (A.1) (II) Computation of Ψ(x,e,D). To get the function Ψ(x,e,D), the function ψ(x,η,U,D) must be written in (y, e) coordinates, where e ) η - η j is the estimation error. hu ) To this end, the derivative u(1) is required. Since b

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(τc-1 + a j )x + η j ),

Z ) (|x|, |e|)T. From (A.5), we have that the derivative of V(x,e) can be written as V˙ ) -ZTM(τe)Z, where

b h u(1) ) (τc-1 + a j )(a jx + η - b h u) + (η - η j )/τe ) (τc-1 + a j )(-τc-1x + e) + e/τe

(A.2)

From (A.1),

Ψ(x,e,D) ) (∆a + σ′(x))(a jx + η - b h u) - ∆bu(1) + d(1) ) (∆a + σ′(x))(-τc-1x - e) - (∆b/b h )[(τc-1 + a j )(-τc-1x + e) - e/τe] + d(1) ) -(∆b/b h )e/τe + Ξ(y,e) + d(1)

(A.3)

where

j )](-τc-1x + e) Ξ(x,e) ) [∆a + σ′(x) - (∆b/b h )(τc-1 + a (A.4) The term -(∆b/b h )e/τe reflects the feedback effects induced by the input-related uncertainty ∆bu. Note that -(∆b/b h )e/τe is the only term in Ψ(y,e,D) which is on the order of τe-1. (III) Computation of the Dynamics of the Reconstruction Error. Recall that η˘ ) ψ(x,η,U,D) and j , then η j ) (η - η j )/τe. If e ) η - η

e˘ ) -e/τe + ψ(x,η,U,D) ) -e/τe - (∆b/b h )e/τe + Ξ(x,e) + d(1)

(from (A.3))

(1)

) -(b/b h )e/τe + Ξ(x,e) + d

Note that sign(b) ) sign(b h ) by assumption. Consequently, the nominal dynamics e˘ ) -(b/b h )e/τe is exponentially stable. Appendix B Stability Analysis. Consider the Lyapunov function candidate V(x,e) ) (x2 + e2)/2. The time derivative of V(x,e) along the trajectories of the closed-loop system (16) is

V˙ ) -τc-1x - τe-1µe2 + xe + e(Ξ(x,e) + d(1)) which, by virtue of inequality (17) and assumption 3, yields

V˙ e -τc-1|x|2 - (τe-1µ - λ2)|e|2 + (1 + λ1)|x||e| + δ2|e| (A.5) The idea of the proof is to fix the nominal closed-loop performance via a prescribed closed-loop time constant τc > 0 and to tune the estimation time constant τe > 0 to achieve closed-loop stability. Case 1. Either d(t) ) constant, for all t g 0, or d(t) f constant as t f ∞. It suffices to study the subcase d(t) ) constant. Thus, d(1) ) 0 and δ1 ) 0. Define the two-dimensional vector with nonnegative components

M(τe) )

[

-(1 + λ1)/2 τc-1 -(1 + λ1)/2 τe-1µ - λ2

]

Therefore, V˙ < 0, for all (x, e)T ∈ R, if M(τe) is a positivedefinite matrix. Since λ1 and λ2 are independent of the estimation time constant τe, it is not hard to see that M(τe) > 0 for all 0 < τe < τ/e, where τ/e is given as in (17). This shows that, if d(t) ) constant, for all t g 0, both signals x(t) f 0 and e(t) f 0 as t f ∞, for all 0 < τe < τ/e. The subcase d(t) f constant as t f ∞ follows from the Barbalat’s lemma.13 Case 2. d(t) is a persistent disturbance with bounded variation. Let us rewrite the closed-loop system (16) as

x˘ ) acx + e ) - τc-1x + e e˘ ) -µe + (Ξ(x,e) + d(1))

(A.6)

where  ≡ τe is taken as a small parameter. System (6) is written into the singular perturbation form,13 with an exponentially stable boundary-layer model, and its reduced model is the closed-loop system with perfect knowledge (i.e., x˘ ) -τc-1x). Using standard results from singularly perturbed systems,13 we can conclude that, for all 0 < τe < τ/e, x(t) and e(t) are bounded and the mean-square value of x(t) is of the order O(τe). Literature Cited (1) Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers; McGraw-Hill: New York, 1990. (2) Morari, M.; Zafiriou, E. Robust Process Control; PrenticeHall: Englewood Cliffs, NJ, 1989. (3) Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 1990, 86 (Oct), 33. (4) Astro¨m, K. J.; Hagglund, T. Automatic Tuning of PID Controllers; Instrument Society of America: Research Triangle Park, NC, 1988. (5) Alvarez-Ramirez, J.; Femat, R.; Barreiro, A. A PI controller with disturbance estimation. Ind. Eng. Chem. Res. 1997, 36, 3668. (6) Pratcher, M.; D’Azzo, J. J.; Veth, M. Proportional and integral control of nonlinear systems. Int. J. Control 1996, 64, 679. (7) Alvarez-Ramirez, J.; Cervantes, I.; Femat, R. Robust controllers for a heat exchanger. Ind. Eng. Chem. Res. 1997, 36, 382. (8) Kokotovic, P. V.; Marino, R. On vanishing stability regions in nonlinear systems with high-gain feedback. IEEE Trans. Autom. Control 1986, 31, 967. (9) Kailath, T. Linear Systems; Prentice-Hall: Englewood Cliffs, NJ, 1980. (10) Gawthrop, P. J. Self-tuning PID controllers: Algorithms and implementation. IEEE Trans. Automatic Contr. 1986, AC31, 201. (11) de Santis, R. M. A novel PID configuration for speed and position control. J. Dyn. Syst., Meas., Control 1994, 116, 542. (12) Kothare, M. V.; Campo, P. J.; Morari, M.; Nett, C. L. A unified framework for the study of anti-windup designs. Automatica 1994, 30, 1869. (13) Khalil, H. K. Nonlinear Systems; MacMillan: New York, 1992.

Received for review March 24, 1998 Revised manuscript received September 10, 1998 Accepted September 11, 1998 IE980180+