Optimality of Internal Model Control Tuning Rules - Industrial

An adaptive IMC-PID control scheme based on neural network. Zhicheng Zhao , Jianggang Zhang , Mingdong Hou. 2009,12-16 ...
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Ind. Eng. Chem. Res. 2004, 43, 7951-7958

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Optimality of Internal Model Control Tuning Rules Jose Alvarez-Ramirez,* Rosendo Monroy-Loperena, Alejandra Velasco, and Rafael Urrea Division de Ciencias Basicas e Ingenieria, Universidad Autonoma MetropolitanasIztapalapa, Apartado Postal 55-534, Iztapalapa DF 09340, Mexico

Internal model control (IMC) tuning rules have proven to yield acceptable performance and robustness properties when used in the control of typical processes (e.g., distillation columns, chemical reactors). In general, analytical IMC tuning rules are derived for proportional-integral (PI)/proportional-integral-derivative compensators by matching an approximate process model to a low-dimensional reference model. In the case of time-delay processes, an approximate model is obtained by taking a finite-dimensional approximation to the delay operator by means of Pade or Taylor expansions. For some typical cases arising commonly in process control, including first-order plus time-delay plants, this paper studies the optimality of PI-IMC tuning rules to match the prescribed closed-loop behavior (i.e., the reference model response). To this end, optimal PI settings are computed by means of numerical optimization based on random search algorithms. Small deviations of IMC tuning from optimality are found for moderate time delays. However, significant deviations are displayed for large time delays, which motivate the use of tuning techniques based on numerical optimization to refine IMC settings. 1. Introduction Internal model control (IMC) tuning is referred to a set of tuning procedures based on the internal model principle. The underlying idea behind internal model methodologies is to compute a controller and/or to set its values relative to a prescribed response formulated as a prescribed (internal) model. In this way, IMC designs belong to the class of model-based control settings, whose origin can be traced back to the proportional-integral-derivative (PID) tuning method proposed by Dahlin.1 In fact, that tuning approach has originated a very wide research stream (see, for instance, O’Dwyer2 and references therein). In the process control field, there has been some work along these lines, including the IMC proportional-integral (PI)/PID tuning paper by Rivera et al.3 and Smith and Corripio’s direct synthesis tuning rules.4 Existing IMC tuning guidelines for typical processes have been surveyed by Chien and Fruehauf.5 Several subsequent works on IMC tuning have addressed the problem of refining the classical results of the IMC approach. For instance, Horn et al.6 proposed an improved filter design to derive low-order controllers aimed to provide effective disturbance suppression. For low-order process models, the controllers have the form of a PID controller in series with a (low-pass) filter. It should be emphasized that standard PI and PID control configurations are obtained as long as the process model is first- and second-order, respectively. Being aware of this fact, Isaksson and Graebe7 presented an analytical model-reduction-suitable PID design, which is based on either retaining the dominant pole(s) or the low-order coefficients of the plant. Essentially, the former provides an overbound on the magnitude of the plant, whereas the latter provides an underbound term crucial for robust performance at low frequencies. Cominos and Munro8 have surveyed recent tuning methods and design specifications for PI/ * To whom correspondence should be addressed: Fax: +5255-58044900. E-mail: [email protected].

PID controllers, including IMC techniques, loop shaping, etc. In the spirit of Isaksson and Graebe’s work, Skogestad9 has used the IMC framework to derive rules for model reduction and PI/PID controller tuning. Skogestad’s IMC (SIMC in short) tuning rules are analytically derived, are simple, and work well on a wide range of processes. A salient feature is that, because SIMC rules are intended for PI/PID controllers, a first- or secondorder process model of the process must be obtained. To this end, Skogestad has proposed a simple procedure based on a “half-rule” to obtain an approximate model of the process. The result is an approximate first- or second-order plus time-delay process model. Once these reduced-order models are obtained, the PI/PID controller gains are computed to adjust the closed-loop response to a first- or second-order model reference response. This procedure gives rise to a set of simple analytical PI/PID tuning rules. In addition to the approximation involved in the model-reduction procedure, another approximation is introduced during the PI/PID control parameter computation. In fact, the matching procedure of the closed-loop response to the reference model response leads to an infinite-dimensional controller with a Smith’s predictor structure. To obtain a PI/PID control structure, the time-delay operator is expanded up to firstorder (by means of a Taylor approximation). Given these two approximation sources and the fact that IMC tuning rules are oriented to match the response of a prescribed reference model, a natural question arises on this point: How far are the SIMC tuning rules from optimal PI/PID settings? By an optimal setting, we mean PI/ PID control parameters that minimize the distance from the close-loop response to the reference model response.10 This paper explores this issue by comparing SIMC settings with an optimal one obtained from numerical optimization corresponding to typical plants, including first-order and integrator plus time-delay processes. In fact, it is shown that existing analytical model-reduction and PI/PID tuning procedures, one of them being Skogestad’s, provide acceptable responses

10.1021/ie040057k CCC: $27.50 © 2004 American Chemical Society Published on Web 10/21/2004

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for most conditions. However, optimal IMC tuning provides an additional tool for refining analytical settings at the expense of modest computational complexity and effort. 2. SIMC Tuning Procedure This section provides a brief description of the IMCbased PI/PID control tuning procedure as described by Skogestad9 (for a similar procedure, see Horn et al.6 and Isaksson and Graebe7). The procedure follows the following two steps: (i) Obtain an approximate first- or second-order plus delay model. It is recognized that first- and second-order models are nothing more than a useful abstraction of the actual process dynamics and that such reducedorder models are suitable to obtain simple tuning rules for industrial PI/PID controllers.7 In this way, the first step is to design a simple model-reduction procedure that retains the main characteristics of the process. That is, given a plant described by

g0(s) )

y(s)

∏j

k )

inv (-T0,j s

+ 1)

∏i (τ0,is + 1)

u(s)

exp(-θ0s)

(1)

inv where the T0,j ’s are inverse-response time constants, the τ0,j’s are open-loop time constants ordered according to their magnitude, and θ0 is the open-loop time delay. The first step is to obtain an approximate first- or second-order model g(s) in the form of

k exp(-θs) g(s) ) (τ1s + 1)(τ2s + 1)

(2)

To this end, Skogestad’s half-rule states that the largest neglected (denominator) time constant (lag) should be distributed evenly to the effective delay and the smallest retained time constant. Then, to obtain a first-order model (i.e., τ2 ) 0), the following relationships are used:

τ1 ) τ0,1 +

τ0,2 2

τ0,2

+

2

h

inv τ0,i + ∑T0,j + ∑ 2 ig3 j

where h is the sampling period. On the other hand, to obtain a second-order model, one should use the relationships

τ1 ) τ0,1 τ2 ) τ0,2 +

(3a)

τ0,3 2

(3b)

and

θ ) θ0 +

τ0,3 2

+

ym(s) yref(s)

)

1 exp(-θs) τcs + 1

h

inv τ0,i + ∑T0,j + ∑ 2 ig4 j

(3c)

Similar to the model-reduction approach proposed by Isaksson and Graebe,7 the relationships described above

(4)

which leads to the controller

c(s) )

τ1s + 1 k[τcs + 1 - exp(-θs)]

(5)

Notice that c(s) has the structure of a Smith predictor compensator. A finite-dimensional compensator is obtained when the time-delay operator exp(-θs) is approximated by means of a Taylor expansion about θ ) 0. In the SIMC case, this is done with the first-order expansion exp(-θs) ≈ 1 - θs. In this way, one obtains

c(s) )

τ1

[

k(τc + θ)

1+

1 τ1s

]

becoming a PI controller with gain given by Kc ) τ1/k(τc + θ) and integral time τI ) τ1. To obtain fast response with good robustness, the recommended choice for the closed-loop time constant is τc ) θ. Additionally, for lagdominant processes with τ1/θ . 1, the modified integral time τI ) 4(τc + θ) is recommended. The resulting SIMC tuning rule is

Kc )

1 τ1 k 2θ

τI ) min {τ1, 8θ}

and

θ ) θ0 +

are oriented to maintain certain robustness properties of PI/PID tuning rules. That is, when the dominant pole(s) is (are) retained, the reduced model essentially provides an overbound on the magnitude of the plant. (ii) Derive the model-based controller settings. The approximate model (2) is the departing point to derive the SIMC tuning rules. A second-order model leads to a PID compensator, and the first-order model yields a PI compensator. Because derivative action is less used in process control, we constrain ourselves to the PI compensation case. By using yref(s) to denote the reference (or setpoint) command, the SIMC tuning rule is obtained by specifying a desired first-order response of the form

(6)

The key step to obtain the SIMC tuning rule (6) is the matching between the plant response and the model response (4). However, as discussed in the Introduction section, two approximations are used to arrive at the final SIMC rules: (i) the model-reduction approximation (2) and (ii) the Taylor expansion exp(-θs) ≈ 1 - θs to obtain a PI control structure. Even if the original plant g0(s) is a first-order plus time-delay process, exact model matching with a PI control structure cannot be obtained, with the time-delay operator exp(-θs) being the main obstruction. One suspects the existence of optimal PI opt control parameters {Kopt c , τI } that minimize the distance from the process y(t) to the model response ym(s). In the following section, the closeness of the SIMC setting (6) to an optimal PI setting is studied. In other words, the optimality of the SIMC tuning rule (6) will be explored below. 3. Optimal PI Tuning The aim of this section is to describe an optimal PI control formulation. Time domain is preferred over frequency domain because the optimization problem

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corresponds to finding the optimal matching between trajectories. A state-space realization of the plant g0(s) driven by a delayed PI controller c(s) ) Kc(1 + 1/τIs) can be given as follows:

x˘ t ) Axt + bKc[et-θ + τI-1zt-θ] z˘ t ) et yt ) Cxt

(7)

where A, b, and C are matrices of suitable dimensions, et ) yref - yt is the regulation error, and yt ) Cxt is the regulated output. To be consistent with the Laplaciandomain description (1), the initial conditions

xt ) 0 and zt ) 0, for all t ∈ (-θ, 0)

(8)

are considered. On the other hand, the state-space realization of the first-order response model (4) is given by / ) y˘ m,t ) -τc-1(ym,t - yref,t-θ

(9)

with initial condition

ym,t ) 0 for t ) 0

(10)

where

{

0 if t < θ / ) y yref,t ref if t g θ

(11)

The tuning problem consists of choosing the PI control parameters {Kc, τI} such that the trajectory yt is as close as possible to the desired response ym,t. Let

t ) yt - ym,t be the instantaneous matching error. Consider the (2norm) square integral error (SIE) ||||22 ) ∫∞0 t2 dt to measure the distance between trajectories. The SIE provides an absolute measurement of the closeness of the response trajectory yt to the model trajectory ym,t. A relative SIE can be defined by comparing ||||22 to the distance of the model trajectory ym,t from the constant reference yref. Because ym,0 ) 0, for yref ) 1 such a distance can be computed as ∫∞0 (yref - ym,t)2 dt. In this way, the objective function is the relative SIE (RSIE) given by

J2 )

||||22

∫0∞(yref - ym,t)2 dt

By solving eq 9, one obtains

J2 )

2||||22 τc

(12)

In this way, J2 gives a measure of the response and model trajectory matching relative to the speed convergence of the desired response reflected by the closedloop time constant τc. The optimal PI control parameters {Kc, τI} are found by solving the following unconstrained optimization problem:

min J2

{Kc,τI}

(13)

subject to the dynamics (7)-(11). It should be stressed that the PI controller is driven by the regulation error et ) yt - yref. The optimization procedure is carried out on the instantaneous error t ) yt - ym,t because the IMC control design corresponds to an approximate matching problem with respect to the desired trajectory ym,t. In fact, one should consider minimization of the SIE of the model matching error yt - ym,t because, as described in IMC tuning papers, the PI/PID control parameters are chosen to match the response of a controlled process and the response of an internal (reference) model. Finding an analytical solution to the optimization problem (13) is not an easy task. Hence, we proceed to use numerical techniques. In a first attempt, we have tried traditional optimization techniques based on gradient information. However, given that analytical gradients are not available, gradients estimated numerically (i.e., with backward differences) lead to very poor performance (attachment to local extrema) and even instabilities. We have opted to use heuristic optimization techniques based on the recursive random search algorithms reported by Zabinsky.11 To this end, the following numerical approximations for the dynamics (7)-(11) were used: (i) The trajectories yt and ym,t were calculated with a fourth/fifth-order Runge-Kutta method. The maximum integration step was set at about 0.001 times the dominant time constant of the process g0(s). (ii) The SIE ||||22 was approximated as ∫T0 t2 dt, for sufficiently large T. We have used T ) 100τc in order to capture the full control system response. In regards to the numerical-based tuning procedure described above, the following comments are in order: (a) In practice, it is frequent that the output reference yref,t be known but not constant, e.g., ramplike setpoint changes. In such cases, one can consider the instantaneous matching error as t ) yt - yref,t-θ, with t ) 0, for all t e θ. (b) Although random search algorithms are less efficient as compared with gradient-based procedures, they have the advantages that (i) their implementation is very easy, requiring only a portable random generator, and (ii) gradient information is not required. In this way, it should be clear that, in solving the optimization problem (13), we are not looking for computationally efficient optimization but rather we are looking for easy implementation with stability, which are offered by random search algorithms.11 (c) We are looking for the global solution to the optimization problem (13) subjected to the dynamics (7)-(11). However, these kinds of optimization problems are usually plagued with several local minima. To reduce the probability of stacking at a local minimum, similar to simulated annealing and genetic algorithms, the random search algorithm proposed by Zabinsky11 uses (i) random sampling for exploration, which examines the macroscopic features of the objective function and aims to identify promising areas in the parameter space, and (ii) recursive random sampling for exploitation, which focuses on the microscopic features. This is done by performing random sampling for a number of times, then moving (similar to wrong-way movements), resizing the sample space according to the previous samples, and starting another random sampling in the

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new sample region. In this way, the probability of running away from the local minimum is increased. (d) It is noted that the computational complexity of computing an optimal PI/PID tuning is significantly higher when compared with simple analytical tuning rules. In fact, the latter is quite portable as printed paper and requires maybe a standard calculator. The former requires running a program in some standard computer language, which makes it less attractive for working-on-site situations. However, with the advent of small-size portable computers, it is apparent that such a drawback of optimization-based tuning procedures can be outperformed. 4. Examples The objective of this section is to illustrate the effects of the time-delay operator approximation and the modelreduction process in the performance of PI control tuned with SIMC rules. 4.1. First-Order plus Time-Delay Process. In the following, the SIMC tuning (6) is compared with optimal tuning computed from solving (numerically) the optimization problem (13), for first-order plus time-delay processes. The idea is to show the effects of the timedelay operator approximation exp(-θs) ≈ 1 - θs on the optimality of the PI tuning. Besides that the first-order plus time-delay model is commonly used in practice to approximate high-order process dynamics and to tune PI controllers. Given that g(s) ) [k/(τ1s + 1)] exp(-θs), normalized results can be obtained by using the following dimensionless or reduced variables:

θr ) θ/τ1 Kc,r ) kKc τI,r ) τI/τ1 τc,r ) τc/τ1

(14)

so that the normalized process can be expressed as g(s) ) [1/(s + 1)] exp(-θrs). In this way, the reduced time delay θr is the only plant parameter, and the reduced closed-loop time constant τc,r is the only controller parameter. Because we are following Skogestad’s recommendation to set τc, we have that τc,r ) θr. In this way, the reduced time delay θr is the only free parameter. For θr ) 1 and yref ) 1, Figure 1 shows the evolution of the RSIE J2 for three different initial conditions Kc,r. As a consequence of the random search, the RSIE displays decrement and constant intervals. In fact, for certain iterations the tried (at random) search direction does not produce a decrement of the RSIE, so the tried direction must be rejected. Figure 1 also shows the (Kc,r, τI,r) phase portrait corresponding to the RSIE evolution described above. Notice that all trajectories converge to the same point Kc,r ) 0.725 and τI,r ) 1.623, which is taken as the optimal setting. It should be stressed that the SIMC setting (6) has been taken as the initial guess, which leads to convergence within about 200 iterations. The random search algorithm has been implemented in a PC Pentium III, and each run contained about 1 min of wall time. opt Figure 2 shows the optimal setting {Kopt c,r , τI,r } and opt the corresponding optimal RSIE J2 as a function of the reduced time delay θr. Notice that, in the graphs

Figure 1. Evolution of the RSIE J2 for three different initial conditions Kc,r.

and Kc,r, the semilog scale has corresponding to Jopt 2 been used for a better visualization. Similar behavior is found for small θr, which can correspond to either small time delay θ or large process time constant τ1. The latter case can be seen as a process behaving approximately as an integration time process.12 The following can be observed: (i) With respect to the SIMC setting, the optimal setting is obtained by increasing both the controller gain Kc and the integral time τI. That is, the optimal setting yields larger controller gain and integral time than the SIMC setting. (ii) As the reduced time delay θr is increased, the RSIE Jopt 2 decays faster than the RSIE corresponding to the SIMC tuning. This shows that certain performance enhancement is expected for large time delays. It is noticed that for relatively large values of the reduced opt time delay (θr > 0.75) the integral gain KI,r ) Kopt c,r /τI,r is quite similar for the two settings (see Figure 3). This is because, for a delay-dominant process, the optimal PI compensator is essentially a pure integrating controller. (iii) For θr > 0.125, the RSIE for both settings is smaller than 10%. For θr > 1.5, the RSIE for the optimal setting is smaller than 1% and about 7% for the SIMC setting. This means that the SIMC tuning and the optimal tuning should give similar performances. (iv) For θr < 0.125, the SIMC setting yields a large RSIE of about 40%, whereas the optimal setting gives a RSIE of about 10%. It is noticed that this behavior can correspond to integrator-like processes. In this situation, the integral time for the SIMC settings is reduced in order to improve the load disturbance

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Figure 2. Optimal and SIMC PI compensation settings and the corresponding RSIE as a function of the time delay θr, for the process g(s) ) [1/(s + 1)] exp(-θrs).

Figure 4. Process output response under the optimal and SIMC settings for g(s) ) [1/(s + 1)] exp(-θrs) and for θr ) 1 and 3.

significant degradation of the PI control performance, so that the SIMC tuning rules (6) can be considered to be acceptably close to the optimal setting. This is illustrated in Figure 4, which presents the process response under the optimal and SIMC settings for a unity setpoint change at t ) 0 and a unity load disturbance at t ) 10. In fact, marginal differences between the optimal and SIMC responses are found for θr ) 1 and 3, even in the case of external load disturbance, showing that for first-order plus time-delay processes the PI compensator tuned with SIMC rules is quite acceptable as compared to the optimal PI setting, even in the presence of external disturbances. 4.2. Higher-Order Processes. A second approximation source is the model-reduction process, which takes a high-order process into a first- or second-order model. To this end, the following seventh-order process is used:9

g0(s) ) (-0.3s + 1)(0.08s + 1) (2s + 1)(s + 1)(0.4s + 1)(0.2s + 1)(0.05s + 1)3 Figure 3. Reduced integral gain KI,r corresponding to the settings shown in Figure 2.

response, which as shown in Figure 2 deteriorates the reference tracking response measured by J2. Except for small relative time delays θr ) θ/τ1, no significant improvement in the control performance, represented by the RSIE J2, is obtained with the optimal setting. In this way, it is concluded that the time-delay approximation exp(-θs) ≈ 1 - θs does not introduce

(15)

This process is interesting because it contains fast and slow poles (from 100 to 0.5) and minimum- and nonminimum-phase zeroes. Rule T3 from Skogestad9 yields the approximation

0.08s + 1 1 ≈ 0.2s + 1 0.12s + 1 and the half-rule (2) gives a first-order plus time-delay process with k ) 1, θ ) 1.47, and τ1 ) 2.5. In this case,

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It is noted that both the optimal and SIMC tuning rules yield a closed-loop response with a significant overshooting (about 15%), which can be due to the presence of the a right-hand zero (i.e., inverse response). Acceptable settling times with reduced undershooting can be obtained by trying larger closed-loop time constants. In fact, Figure 5b shows the response of the controlled process under optimal settings for three different values of the closed-loop time constant τc, for a unity setpoint change at t ) 0, and for a unity load disturbance at t ) 10. It is observed that the larger the value of τc, the less severe the undershooting and the slower the output response. In this way, this example shows how optimal design procedures based on numerical optimization techniques can be used to refine the PI control setting obtained from simple analytical IMC rules. So far, we have considered IMC tuning rules as reported by Skogestad.9 However, analytical tuning rules based on model-reduction procedures have been also reported by Isaksson and Graebe.7 Consider the third-order process

g0(s) )

Figure 5. (a) Response of the controlled process (15) under the optimal and SIMC PI control settings. (b) Response of the controlled process (15) under optimal settings, under different values of the closed-loop time constant τc.

the SIMC rule (6) gives KSIMC ) 0.85 and τSIMC ) 2.5. c I On the other hand, the resulting optimal setting is Kopt c ) 1.009 and τopt ) 1.28. Figure 5a shows that, at least I for this higher-order process, the optimal tuning rules based on the original seventh-order model do not give significant performance improvement over the SIMC tuning rules based on the reduced first-order plus timedelay model. This could be expected from the computations made for the first-order plus time-delay process in the above subsection. In fact, for this case one has that the normalized delay θr ) 0.588, for which no significant improvement in the control performance, represented by the RSIE J2, is obtained with the optimal setting (see Figure 2). This example is representative in the sense that we have worked all of the cases reported in Table 4 of Skogestad,9 finding similar results. In this way, one can conclude that SIMC tuning rules offer easy PI/PID tuning procedures with acceptable response and robustness margins. This can be attributed to the following facts: (i) Retaining the dominant poles(s) provides an overbound on the dominant time scales of the process, which is crucial for robust performance at low frequencies. (ii) Aggregating fast poles and inverse response zeros into an effective time delay θ and selecting a desired closed-loop response time τc not larger than θ provide a protection against unstable high-frequency dynamics. That is, the PI controller is designed in such a way that the controlled process is not faster than the effective delayed dynamics of the process.

-2s + 1 (s + 1)3

(16)

This case is more severe than the process (15). In fact, the process (16) has a nonminimum-phase zero at +0.5, which is closer to the origin than the dominant pole located at -1. Such a situation imposes a serious limitation on the performance of the controlled process.13 For this process, one can obtain a first-order plus timedelay model with τ1 ) 1.5 and θ ) 3.5. The IMC tuning obtained with Isaksson and Graebe’s method is Kc ) 0.36 and τI ) 2.0. Figure 6a shows the time response, under setpoint change and disturbance, for this tuning and for the optimal one. In this case, θr ) 2.333 so that a moderate performance improvement can be obtained with the optimal PI setting. We have considered IMC tuning when the desired response satisfies a first-order internal model. Frequently, the desired response is given as an explicit reference trajectory yref(t). In this case, the optimization problem can be posed as minimizing the SIE where the instantaneous error is given by t ) yt - yref,t. To illustrate this, consider a setpoint change following the ramplike trajectory

{

0 if t < 0 yref(t) ) 15-1t if 0 e t e 15 1 if t > 15 In this case, the optimal setting is Kc ) 0.432 and τI ) 2.65. Figure 6b presents the response of the controlled system, showing that close tracking of the ramp trajectory is not possible with a (optimal) PI controller because of the presence of the unstable zero located at +0.5. However, undershooting is eliminated while convergence to the setpoint is achieved without overshooting. This example illustrates how IMC methodologies can be extended to consider more general PI/PID tuning situations by means of numerical optimization techniques. 5. Conclusions In this paper, the optimality of IMC tuning rules to match the response of a reference model has been studied. It has been shown that, for a large range

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Figure 6. (a) Response of the controlled process (16) under Isaksson and Graebe’s tuning and the optimal one with τc ) 3.5. (b) Optimal PI tuning for tracking a ramplike reference.

of time delays, the time-delay operator expansion exp(-θs) ≈ 1 - θs does not limit seriously the performance of PI controllers tuned with IMC rules. Moreover, heuristic model-reduction procedures, similar to halfrules, to obtain first- or second-order processes work well because they provide the PI/PID controller with protection against high-frequency dynamics induced by real time delays and inverse response dynamics. Although analytical IMC tuning rules can give an acceptable industrial PI/PID performance for a large variety of plants, optimization-based procedures with heuristic techniques (i.e., random search, genetic algorithms, etc.) can offer improved PI control tuning with a minimum of computational effort under model matching conditions not considered by analytical IMC tuning approaches. In fact, further exploration of optimal PI/PID tuning incorporating performance (e.g., maximum overshooting) and control input (e.g., control input saturation, etc.) should be explored. To illustrate this, consider the process g(s) ) exp(-s)/(s + 1) subject to the control input constraint (see, for instance, work by Cominos and Munro8):

|du/dt| e 0.5

(17)

That is, the rate of change of the control input cannot be larger than 0.5, which can reflect constraints imposed by actuator (e.g., valves, heat-transfer equipment, etc.) dynamics. Incorporation of these types of nonlinear constraints within an analytical IMC formulation seems to be a hard task; however, such a constraint can be easily handled within the numerical optimization frame-

Figure 7. Output response and control input dynamics for the SIMC (with and without a simple AW scheme) and optimal settings, for the process g(s) ) exp(-s)/(s + 1) under the constraint |du/dt| e 0.5.

work described above. To this end, the time derivative is approximated by a first-order backward difference, such that the constraint (17) is approximated as |ut ut-T| e 0.5T, where T can be interpreted as the used integration step or the sampling period. The SIMC setting, which does not consider the control input constraint, is Kc ) 0.5 and τI ) 1. On the other hand, the optimal setting obtained by refining this SIMC setting is Kc ) 0.56 and τI ) 1.57. Figure 7 presents the output response and the control input dynamics for the SIMC and optimal settings. Notice that the SIMC setting leads to a severe overshoot induced by the “sosmall” integral time τI ) 1. The large overshoot is windup, which can be significantly reduced if an antiwindup (AW) scheme is employed. This is shown in Figure 7 for the SIMC setting equipped with a simple AW scheme that stops the integration while the constraint is active. A similar, but slightly faster, response is obtained with the optimal setting without using an AW scheme, with about 50% savings in the RSIE, because the PI controller has been slightly detuned by increasing both the control gain and the integral time. This example shows that retuning of analytical IMC settings can lead to significant performance improvements when hard constraints are present or imposed on the control system. In particular, additional corrective actions, such as AW schemes, can also be incorporated in the optimization scheme without adding excessive computational complexity. Similar to the approach proposed by Panagopoulus et al.,10 the numerical optimization technique, either in a time or frequency domain, is a reliable framework to extend existing

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results on IMC analytical tuning rules. In fact, we have shown that existing analytical model-reduction and PI/ PID tuning procedures, one of those being Skogestad’s, provide acceptable responses for most conditions. However, optimal IMC tuning provides an additional tool for refining analytical settings at the expense of modest computational complexity and effort. Literature Cited (1) Dahlin, E. G. Designing and tuning digital controllers. Inst. Control Syst. 1968, 41, 77-81. (2) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules; World Scientific Publishing: Singapore, 2003. (3) Rivera, D. E.; Morari, M.; Skogestad, S. Internal model control. 4. PID controller design. Ind. Eng. Chem. Res. 1986, 25, 252-265. (4) Smith, C. A.; Corripio, A. B. Principles and Practice of Automatic Process Control; John Wiley & Sons: New York, 1985. (5) Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 1990, 33-41. (6) Horn, C. C.; Arulandu, C. J.; Gombas, C. J.; VanAntwerp, J. D.; Braatz, R. D. Improved filter design in internal model control. Ind. Eng. Chem. Res. 1996, 35, 3437-3441.

(7) Isaksson, A. J.; Graebe, S. F. Analytical PID parameter expression for higher order systems. Automatica 1999, 35, 11211130. (8) Cominos, P.; Munro, N. PID controllers: recent tuning methods and design to specification. IEE Proc.-D: Control Theory Appl. 2002, 149, 46-53. (9) Skogestad, S. Simple analytical rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291-309. (10) Panagopoulus, H.; Astrom, K. J.; Hagglund, T. Design of PID controllers based on constrained optimization. IEE Proc.-D: Control Theory Appl. 2002, 149, 32-40. (11) Zabinsky, Z. B. Stochastic methods for practical global optimization. J. Global Optim. 1998, 13, 433-444. (12) Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/dead time processes. Ind. Eng. Chem. Res. 1992, 31, 2628-2631. (13) Astrom, K. J.; Hagglund, T. PID controllers: Theory, design and tuning; ISA: Research Triangle Park, NC, 1995.

Received for review February 13, 2004 Revised manuscript received August 30, 2004 Accepted September 30, 2004 IE040057K