A Note on High-Order Controllers for Time-Delay Processes

Ind. Eng. Chem. Res. 2003, 42, 4993-5002

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RESEARCH NOTES A Note on High-Order Controllers for Time-Delay Processes Basilio del-Muro-Cue´ llar* and Jose´ Alvarez-Ramı´rez† Programa de Investigacio´ n en Matema´ ticas Aplicadas y Computacio´ n, Instituto Mexicano del Petro´ leo, Eje Central Lazaro Cardenas 152, Col. San Bartolo Atepehuacan, Mexico D.F., 07730 Mexico

Processes with significant transport delays in the input channel are commonly found in practice. The control of these processes is a challenging problem because of the difficulties in compensating the adverse effects induced by delayed control inputs. The aim of this paper is to show that high-order controllers can be required to obtain fast convergence rates in the face of significant transport delays. To this end, the control design is based on a sampled (discrete-time) version of the process dynamics where the sampling of the transport delay can be made in a straigthforward way. This model is then used to compute an output-feedback controller based on pole-placement polynomial methodologies. The resulting controller has the structure of a state-feedback controller equipped with integral action and a reduced-order observer to estimate unmeasured internal states. Numerical simulations are used to illustrate advantages and drawbacks of the use of high-order controllers to compensate transport delays. 1. Introduction Control input time-delay dynamics are often found in, e.g., chemical, manufacturing, metallurgical, and biotechnological processes. In several cases, time delays are induced by transport (convective and diffusive) phenomena. It is well-known in practice that the achievable performance (e.g., settling times, overshoots, etc.) of a control system can be seriously degraded if a process has a relatively large time delay as compared to the dominant time constant.1,2 In this case, transport-delay compensation may be necessary in order to enhance the control performance. Compared with a process without time delay, the presence of time delay increases the difficulty to obtain a controller. The Smith control scheme is one of the most popular,3 with successful extensions to multivariable processes.4 Improvements of the basic Smith control structure have been subsequently reported. Watanabe et al.5 proposed a modified plant model to be used in the inner feedback loop to improve regulation capabilities. It has been pointed out that Smith predictor structures are sensitive to model mismatch and have poor disturbance rejection capabilities.6 To obtain more robust control structures, an approximate inverse of the time-delay operator has been proposed by Huang et al.6 The idea is to improve the disturbance predicting capability of the Smith control structure by “inverting” time delays. Observer-based predictors aimed at enhancing the stability of the control loop,7 autotuning configuration,8 and robustified Smith control structures have also been explored. Recently, Tan et al.9 have used a generalized predictive control approach to obtain an * To whom correspondence should be addressed. Tel.: +525-30037571. Fax: +52-5-30036277. E-mail: [email protected]. † Also at Departamento de Ingenieria de Procesos e Hidraulica, Universidad Autonoma Metropolitana-Iztapalapa, Mexico D.F., 09340 Mexico. E-mail: [email protected].

optimum control performance in terms of a specified cost function. An interesting result is that when a general second-order model is used, the primary controller reduces to a proportional-integral-derivative (PID) control. In this way, traditional PID control can be seen as a controller endowed with a simple predicting structure induced by the derivative part. This property of derivative compensation has been exploited to propose advanced autotuning strategies.10 Luyben11 has highlighted the advantages of using derivative compensation to reduce the adverse effects of transport-delayed control inputs. Luyben has shown, via numerical simulations, that PID control can yield improved performance, as compared with proportional-integral (PI) control, with acceptable measurement noise sensitivity. From our viewpoint, the main conclusion drawn from the results discussed above is that high-order controllers can be required in order to obtain a good control performance (e.g., high convergence rates and fast disturbance rejections) when transport delays are present. For instance, Luyben11 has shown that the use of derivative control enhances the control performance. In a strict sense, derivative action increases the control order by 1. In this way, the order of the process is increased by 1, and a proportional-derivative (PD) control configuration can be easily obtained for relative degree 1 processes affected by a control input time delay. Subsequently, the PD controller is equipped with an integral control to improve performance in the face of external loads and process parameter uncertainty. In this way, as discussed before, the derivative feedback can be seen as an approximate inverse of the transportdelay operator.6 In principle, better time-delay compensation can be obtained by using higher-order lag approximationsWn(0)/Wn(s), where Wn(s) is the product of n lags τθ,i/(τθ,is + 1). The result is a higher-order and relative degree process for which a higher-order (i.e., generalized PD) controller with an enhanced time-delay

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4994 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

compensation capability can be obtained. Following this procedure, one recovers the nature of time delay in the sense that it introduces high order (in fact, infinitedimensional) dynamics into feedback control loops. The main drawback of this approach is that an accurate nthorder lag approximation of time delays is not available. In principle, such an approximation procedure should involve complex nonlinear programming problems in the frequency domain. Because the time-delay dynamics is a purely transport operator [i.e., a signal r(t) is transported to r(t + θ) without shape modification], a more reliable framework to approximate exp(-θs) is by means of delay operators in the discrete-time domain. The aim of this paper is to show that high-order controllers can be required to obtain fast convergence rates in the face of significant transport delays. To this end, the control design is based on a sampled (discretetime) version of the process dynamics where the sampling of the transport delay can be made in a straightforward way. The underlying idea is that time delays, when expressed in discrete time, provide fewer difficulties in controller design. The resulting model is then used to compute an output-feedback controller based on pole-placement polynomial methodologies. The resulting controller has the structure of a state-feedback controller equipped with integral action and a reduced-order observer to estimate unmeasured internal states. Numerical simulations are used to illustrate advantages and drawbacks of the use of high-order controllers to compensate transport delays. It must be remarked that we do not intend to provide a better control design methodology with tight tuning procedures. Rather, our intention is to show the following: (a) High-order controllers can be required to obtain an enhanced performance for the process with large transport delay. (b) Discrete-time approaches offer a framework for a systematic design of high-order compensation. Besides, high-order control designs can be easily made with standard pole-placement techniques. 2. Class of Transport-Delay Processes

GT(z) ) z-kθTHT(z) ≡

In this section, a control design methodology based on a discrete-time version of the process will be provided. 2.1. Process Studied. Consider the following singleinput single-output process:

Y(s) ) G(s) [U(s) + Q(s)] G(s) )

Let yref be a prescribed output setpoint. The control problem consists of designing a feedback controller C(s) such that y(t) f yref asymptotically. Because the time delay exp(-θs) imposes serious limitations in control performance, the feedback controller C(s) should provide acceptable convergence and disturbance rejection rate. By far the most commonly used model in process control13 is G(s) ) [Kp/(τs + 1)] exp(-θs). Several process control design and tuning studies are based on this model. For instance, Luyben11 has given tight PI/PID tuning guidelines that yield a good performance in the face of external disturbances. However, higher order models can be required to accurately describe the dynamics of paper-making processes. In fact, Astrom and Hagglund13 have pointed out that a second-order model better describes the paper-making process dynamics. This case, among others, justifies the development of control design methodologies for the class of processes given by eq 1. On the other hand, step load disturbances are used because they are usually the most important disturbance in continuous processes.11 2.2. Process Discretization. To obtain a discretetime version of the process (1), an exact sampling procedure with sampling period T is used. It should be emphasized that we are not focusing on the control design problem for sampled processes. Rather, we are taking the sampling period T as an adjustable parameter in order to vary the controller order. That is, the sampling period T can be chosen freely, which affects the order of the feedback controller. To simplify notation and presentation, in the sequel Q(s) ) 0 will be taken. Let v(t) ) u(t - θ). Then, the process (1) can be written as Y(s) ) G(s) V(s). Let HT(z) ) R(z)/P(z) be a sampling version, with sampling period T, of the transfer function G(s). Here, R(z) and P(z) are polynomials with deg(R(z)) ) m and deg(P(z)) ) n. To obtain a discrete-time representation of exp(-θs), let kθT be a positive integer number such that θ/T ) kθT. Because v(t + θ) ) u(t), one has that zkθTV(z) ) U(z) + Q(z). Therefore, the transfer function GT(z) corresponding to the map U(z) f Y(z) is

(1)

N(s) exp(-θs) D(s)

where θ > 0 is the transport or time delay, deg(N(s)) ) m, deg(D(s)) ) n, r ) n - m g 0 is the relative degree of the delay-free process, and Q(s) is a matched step disturbance corresponding to either a setpoint change or an external load. In general, for chemical,12 metallurgical, and pulp- and paper-making processes,10 the processes (1) is stable. That is, the polynomial has D(s) and all of its roots in the left-hand side of the complex plane. In the sequel, it will be assumed that (a) only the output y(t) and the input u(t) are available for measurements, and (b) r g 1. Assumption b is not restrictive because if r ) 0, one can add a lag to increase the relative degree.

R(z) W(z)

(2)

where W(z) ) zkθTP(z) and deg(W(z)) ) n + kθT, so that GT(z) corresponds to a (n + kθT)th-order process with relative degree rT ) n + kθT - m g r. One has that deg(R(z)) ) m and deg(W(z)) ) n + kθT. Notice that the process GT(z) has kθT poles at zero induced by the approximation exp(-θs) = zkθT. In this way, the discretetime approximation of the delay operator increases the relative degree of the process. Besides, the smaller the sampling period T, the higher the order of the discretetime process GT(z). In the limit as T f 0, one obtains an infinite-dimensional process with rT f ∞. This fact reflects clearly the high-order nature of feedback processes affected by transport delays in the control input channel. 3. Control Design Once the transfer function GT(z) ) R(z)/W(z) has been computed, a feedback control design based on a quite standard polynomial approach will be proposed below. The controller will be represented as follows:

U(z) ) C(z) [Yref(z) - Y(z)]

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4995

Figure 1. Simulations for the controlled output and the control input with kθT ) 1, for θ/τ0 ) 0.5, and second-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controller is C(z) ) (3.5z2 - 1.9z)/(z2 + 0.2z - 1.2).

where Yref(z) is the reference command and the controller transfer function C(z) is given as a rational function given by

C(z) ) M(z)/P(z) where deg(M(z)) ) deg(P(z)) ) n + kθT. To reject external loads Q(z), the controller should be equipped with an integrator 1/(z - 1). Notice that the controller has 2(n + kθT) parameters represented in the vector

π ) {mn+kθT, mn+kθT - 1, ..., m0, pn+kθT - 2, ..., p0} The characteristic polynomial induced by the controller C(z) is

W(z) P(z) + R(z) M(z)

(3)

The polynomial (3) has 2(n + kθT) free parameters for vector π that can be assigned arbitrarily in order to obtain a desired closed-loop behavior. An easy way to assign π is by means of pole-placement techniques. This is made by constructing a Schur-stable, 2(n + kθT)thorder polynomial E(z), where its roots are selected according to a given criterion (e.g., convergence rate, etc.). When the actual closed-loop polynomial and the prescribed one are matched, the following polynomial equation is obtained:

W(z) P(z) + R(z) M(z) ) E(z)

(4)

In this way, the controller parameters for vector π can be computed by solving a set of linear algebraic equations obtained by matching the coefficients of both polynomials in the left- and right-hand sides of eq 4. The following comments are in order: (a) The order nc of the controller C(z) is equal to nc ) n + kθT. In this way, the order nc can be adjusted by increasing or decreasing the sampling parameter T. (b) The main part of the controller computation is the selection of the prescribed closed-loop poles or, equivalently, the roots of the polynomial E(z). One observes

that, after the kθTth-order discrete-time approximation z-kθT of the time-delay operator exp(-θs) is given, the effective nonmodeled time delay is θ/kθT. Such a delay imposes a limitation in the performance (e.g., convergence rate) of the discrete-time controller C(z) when it is acting on the actual continuous-time process (1). (c) Noisy measured signals can seriously limit the practical usage of high-order controllers. In fact, noisy measurements can lead to undesirable effects, such as control input saturation due to noise amplification. The use of low-pass filters can reduce the noise power in measured signals. However, because low-pass filtering is based on lag operations, some care should be taken in order to avoid severe degradation of the computed feedback controller. Generally, if a low-pass filter is added after the control design, performance degradation of the control loop can be expected. A more systematic way to design a feedback controller when a filter is required to reduce noise effects is to design a compensator that takes into account the filter dynamics. In fact, if F(z) is a filter transfer function with F(1) ) 1 (unitary gain) and YF(z) corresponds to the filtered signal [i.e., YF(z) ) F(z) Y(z)], the idea is to regulate the filtered signal YF(z) rather than the measured one Y(z). In this way, the process to be considered for control design is YF(z) ) GT(z) F(z) [U(z) + Q(z)], so that the full transfer function is GT(z)F(z). If F(z) is an fth-order filter, the resulting controller is of (n + kθT + f)th order. Hence, by considering that the filter F(z) is a component of the controller C(z), the order of the controller is increased by 2f. This is illustrated with a simulation example. (d) For a first-order plus time-delay process G(s) ) [Kp/(τ0s + 1)] exp(-θs), the controller C(z) has the structure of a discrete-time PI and PID compensator for kθT ) 0 and 1, respectively. That is, a PI control structure is obtained if the time delay is not accounted for in control design. On the other hand, if T ) θ (i.e., kθT ) 1), the simplest approximation exp(-θs) = z-1 is used for control design. In this case, the derivative action induced by the delay z-1 plays the role of a firstorder (i.e., linear) Smith predictor for the regulated


Figure 2. Simulations for the controlled output and the control input with kθT ) 2, for θ/τ0 ) 0.5, and a third-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controller is C(z) ) (8.9z3 - 6z2)/(z3 + 0.4z2 + 0.3z - 1.7).

Figure 3. Simulations for the controlled output and the control input with kθT ) 5, for θ/τ0 ) 0.5, and a sixth-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controler is C(z) ) (37.4z6 - 30.7z5)/(z6 + 0.5z5 + 0.5z4 + 0.5z3 + 0.4z2 + 0.4z 3.23).

output y(t + θ). In this way, one can say that the approximation exp(-θs) = z-kθT leads to a generalized PIDkθT controller with a kθTth-order Smith prediction of the regulated output y(t + θ). In the limit as T f 0 (i.e., kθT f ∞), one obtains an infinite-dimensional controller. 3.1. Closed-Loop Polynomial Specification. As mentioned above, the key part of the controller construction is the specification of the characteristic polynomial E(z) via the selection of the controlled process poles. As in work by Lin and Hwang,14 we propose to follow a IMC-type algorithm. To this end, let us consider the following poles classification: (i) Fixed Poles Set. This set contains all of the poles that are chosen to remain fixed under the action of the

feedback controller. These poles are as follows: (a) Welldamped process poles, which do not limit the closedloop process dynamics. It is noted that this subset contains kθT zero poles associated with the transportdelay operation z-kθT. (b) kθT additional poles to be placed at zero, which corresponds to the estimation of internal (unmeasured) states of the delay state-space representation. In fact, because z-kθT induces only transportdelay dynamics, the estimation of such dynamics can be made just by signal shifting. It can be shown that such a procedure is equivalent to placing kθT poles at zero. (c) Low-pass filter poles, which should be maintained in order to ensure acceptable reduction of measurement noise effects.


Figure 4. Simulations for the controlled output and the control input with kθT ) 1, for θ/τ0 ) 5, and a second-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controller is C(z) ) 0.65z2/(z2 + 0.4z - 0.61).

Figure 5. Simulations for the controlled output and the control input with kθT ) 2, for θ/τ0 ) 5, and a third-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controller is C(z) ) (0.76z3 - 0.06z2)/(z3 - 0.32z2 - 0.7).

(ii) Assigned Poles Set. These poles correspond to open-loop unstable and poorly damped poles, which must be relocated in order to obtain closed-loop stability and acceptable convergence rates. Besides, such a set also contains n poles corresponding to (implicit) state observer dynamics. Fast state estimation is desirable in order to attain the performance induced by state feedback. On the basis of this classification, let us factor E(z) as

E(z) ) Eas(z) Efx(z) where Efx(z) and Eas(z) are polynomials corresponding respectively to fixed and assigned. In this way, the pole

placement design is achieved by assigning deg(Eas(z)) poles. Noted that deg(Efx(z)) g 2kθT + f, so that deg(Eas(z)) e 2n + f. According to the comments made in item c of the above subsection, assigned poles should be located in the region D Sch ) D unit/DT. To ensure T fast convergence, we suggest to locate the assigned poles close to the boundary of DT. Besides, to avoid undesirable oscillatory behavior, such assigned poles should also be located around the positive real axis. In conclusion, as a heuristic tuning guideline we propose to choose the assigned poles around the value exp(-kθT/θ). The effectiveness of this tuning guideline is illustrated in the following section by means of numerical simulations.


Figure 6. Simulations for the controlled output and the control input with kθT ) 5, for θ/τ0 ) 5, and a sixth-order discrete-time controller, compared with the PID-CM, considering R ) 0.1. The controler is C(z) ) (1.6z6 - 0.6z5)/(z6 - 1).

Figure 7. Simulations for the controlled output and the control input with kθT ) 1, 2, and 5, for θ/τ0 ) 10 (kθT ) 1, 2, and 5 correspond with sampling time T ) 10, 5, and 2, respectively). The controllers are C(z) ) 0.64z2/(z2 + 0.4z - 0.61), 0.65z3/(z3 - 0.39z2 - 0.64), and (0.86z6 - 0.13z5)/(z6 - 0.3z5 - 0.74), respectively.

4. Example

controller of the form

Consider the first-order plus time-delay process

Kp G(s) ) exp(-θs) τ0s + 1

(

C(s) ) Kc 1 + (5)

where Kp and τ0, the process open-loop gain and time constant, respectively, are set equal to unity, so that all results can be scaled in terms of ratios (e.g., time t/τ0). The load disturbance is a unit step at time zero, and it affects the process output y(t) through a firstorder lag with time constant τ0. A continuous-time PID

τDs 1 + τIs RτDs + 1

)

has been used for comparisonwith our digital controller (DC). Ciancone-Marlin (CM) tuning, as described in Table 1 in Luyben’s paper,11 has been used. In that paper, CM tuning presents the best performance when compared with the IMC tuning developed by Morari and Zafiriou and with the classical ZieglerNichols tuning. It should be stressed that we are not intending to underscore the performance of tradi-


Figure 8. Simulations for the controlled output and the control input of the second-order system, with kθT ) 1, 2, and 5, for θ/τ0 ) 5 (kθT ) 1, 2, and 5 correspond with sampling time T ) 5, 2.5, and 1, respectively). The controllers are C(z) ) 4.3z3/(z3 - 0.4z2 - 0.6z), (5.1z4 0.45z3)/(z4 - 0.3z3 - 0.7z2), and (10.8z7 - 4z6)/(z7 - 0.9z - 0.14), respectively.

Figure 9. Simulations for the controlled output and the control input of the second-order system, with kθT ) 5, for θ/τ0 ) 5 (kθT ) 2 corresponds with sampling time T ) 2.5), for time delay disturbances of (15%.

tional PID control. Instead, we are using PID control to confront the performance (convergence rate) of a high-order controller as compared with an industrial PID one. In Figures 1-3, the simulations for the controlled output and the control input with three different values of kθT, namely, 1, 2, and 5, respectively, for θ/τ0 ) 0.5, are compared with the PID-CM, considering R ) 0.1. The order of the associated controllers is 4, 6, and 12, respectively. We can see in Figures 4-6 the simulations for θ/τ0 ) 5, with the same three different values of kθT (1, 2, and 5 and controller order 4, 6, and 12, respectively), also compared with the

PID-CM but considering R ) 2. In Figure 7, the simulations for θ/τ0 ) 10, with the same three different values of kθT (1, 2, and 5) are shown (in Figure 7, kθT ) 1, 2, and 5 correspond to sampled time T ) 10, 5, and 2, respectively). In all of these simulations, two poles are relocated at {0.2, 0.2} and the rest are set at the origin. Note that this criterion follows the guideline proposed in subsection 3.1 and also corresponds with the spirit of the methodology suggested by Astrom and Wittenmark.15 In fact, they suggested specifying only two dominant poles and requiring the remaining poles to be close to the origin. As expected, the larger


Figure 10. Simulations for the controlled output and the control input of the second-order system, with kθT ) 5, for θ/τ0 ) 2 (kθT ) 2 corresponds with sampling time T ) 2.5), for parameter disturbances of (15% on Kp and τ0.

Figure 11. Simulations for the controlled output and the control input with kθT ) 2, for θ/τ0 ) 0.5, and a sixth-order discrete-time controller. White noise has been added in the controlled output.

the value of kθT (and larger the order of the associated controller), the faster the disturbance rejection. Compared to the PID-CM tuning, the discrete-time controller can offer a similar performance but with a slightly large overshoot. That is normal because the discrete-time controller takes at least one time period to start acting. In practice, the first-order process model (5) is commonly used as an approximation for a higher-order process where τ0 can be interpreted as a dominant process pole. As in the case of paper basis weight control,10 assume that the actual plant is the second-

order one

G(s) )

0.15Kp (τ0s + 1)(0.2s + 1)

exp(-θs)

Figure 8 shows the dynamics of the controlled process for θ/τ0 ) 5. Because the control design was based on the dominant pole model (5), the controller is able to provide a good performance. In fact, the process pole located at 5 is a well-damped pole, which can be seen as belonging to the fixed poles polynomial Efx(z).


Figure 12. Simulations for the controlled output and the control input with kθT ) 2, for θ/τ0 ) 0.5, and a sixth-order discrete-time controller. White noise has been added in the controlled output, and a filter H(S) ) 1/(s + 1) is used. The filter is consider in the controller design.

Figure 13. Performance of the proposed control strategy as compared to the performance of a third-order (unconstrained) MPC.

Time-delay uncertainties can lead to closed-loop instabilities with Smith predictors4 and PID controllers.11 For θ/τ0 ) 5 and kθT ) 2, Figure 9 presents the behavior of the controlled process for +15% and -15% disturbances in the time delay. Because the control design is not based on an exact prediction of future states, the controller is still able to yield acceptable disturbance rejection. Similar results are found when uncertainties are present in the process parameters Kp and τ0. The performance of the controller for θ/τ0 ) 5 and kθT ) 2 and (20% is shown in Figure 10. These results show that, for moderated controller orders (i.e., moderated values of kθT), the discrete-time controller is able to

provide an acceptable performance with good robustness capabilities. High-order controllers can display high sensitivity to measurement noise, leading to excessive control efforts. This effect is shown in Figure 11, which shows that a measurement noise added at the mesured signal can be amplified into the input channel. Of course, such noisy behavior of the control input cannot be accepted, in general, in practice. In fact, noisy control inputs can lead to premature degradation of control actuators, such as valves and motors. To reduce the adverse effect of measurement noise, a second-order low-pass filter was added at the output channel. Besides, the low-pass filter was taken into account during the control design (see remark d in section 3). Figure 12 shows the simulation results, where it is observed that the effects of measurement noise have been reduced significantly. These numerical simulations show that it is possible to use high-order controllers in the presence of measurement noise. The key hint is that one has to consider the filters in the control design process. Finally, because model predictive control (MPC) is commonly formulated within a discrete-time version, it would be interesting to compare the proposed control architecture, tuned with the tuning rules described before, with an unconstrained MPC control of the same order. To this end, we have chosen the following index function:

J)

1N

(Yref - Yj)2 ∑ 2j)1

It is well-known that, for the corresponding exact discrete-time process, this index function can lead to a dead-beat controller, such that Yj converges to Yref in N steps. However, when the MPC is applied to the continuous-time process, only asymptotic stability is obtained. For N ) 3, Figure 13 compares the performance of the controller, showing that the performance of the resulting dead-beat controller (from the MPC


approach) is not significantly better than the performance of the proposed control approach. This is not surprising because the proposed control strategy can also be seen as a MPC with a nonexplicitly constructed index function. In fact, departing from inverse-optimality results,16 a stabilizing feedback function is also an optimal feedback controller. In this way, Figure 13 shows that, by a suitable tuning of the controller parameters, one can find a performance similar to that induced by MPC. In summary, the above numerical simulations have shown that a discrete-time approach for controlling the process with significant transport delays is a promising methodology for practical applications. 5. Conclusions As in the case of Luyben’s contribution,11 it is hoped that our results can help to understand the role and effects of high-order compensation for a process with large transport delay. If time delays are small, a good closed-loop performance can be obtained with controllers on the order of the process order or relative degree. In situations where time delays are large, high-order controllers can be necessary. This is because time delays introduce high-order dynamics in feedback control loops. In this way, the use of derivative control increases the order of a simple PI controller, hence yielding improved convergence rates and disturbance rejection capabilities. However, a simple derivative action cannot suffice in processes where faster convergence is required. In such cases, generalized derivative (e.g., state feedback plus state observer) control can be required. Our results have shown that simple high-order controllers can be obtained by means of a discrete-time approach where the time delay has a straightforward equivalent representation. Literature Cited (1) Marshall, J. E. Control of Delay Systems; Stevenage, Peter Peregrimus Ltd.: London, U.K., 1979.

(2) Malek-Zavarei, M.; Jamshidi, M. Time-delay System Analysis, Optimization and Applications; North-Holland Systems & Control Series; North-Holland: New York, 1987; Vol. 9. (3) Smith, O. J. M. Closer control of loops with dead-time. Chem. Eng. Prog. 1957, 53, 217-223. (4) Palmor, Z. J.; Halevi, Y. On the design and properties of multivariable dead time compensation. Automatica 1983, 19, 255263. (5) Watanabe, K.; Ishiyama, Y.; Ito, M. Modified Smith predictor control for multivariable systems with delays and unmeasurable step disturbances. Int. J. Control 1983, 37, 959-971. (6) Huang, H.-P.; Chen, Ch.-L.; Chao, Y.-Ch.; Chen, P.-L. A modified Smith predictor with an approximate inverse of dead time. AIChE J. 1990, 36, 1025-1031. (7) Furakawa, T.; Shimemura, E. Predictive control for systems with time delay. Int. J. Control 1983, 37, 399-411. (8) Lee, T. H.; Wang, Q. G.; Tan, K. K. Automatic tuning of the Smith predictor controller. J. Syst. Eng. 1995, 5, 102-114. (9) Tan, K. K.; Lee, T. H.; Leu, F. M. Optimal Smith predictor design based on GPC approach. Ind. Eng. Chem. Res. 2002, 41, 1242-1248. (10) Tang, W.; Shi, S. Autotuning PID control for large timedelay processes and its application to paper basis weight control. Ind. Eng. Chem. Res. 2002, 41, 4318-4327. (11) Luyben, W. L. Effect of derivative algorithm and tuning selection on the PID control of dead-time processes. Ind. Eng. Chem. Res. 2001, 40, 3605-3611. (12) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New York, 1989. (13) Astrom, K. J.; Hagglund, T. PID Controllers: Theory, Design and Tuning, 2nd ed.; Instrument Society of America: Research Triangle Park, NC, 1995. (14) Lin, M.-L.; Hwang, S. H. Robust design method for discrete-time controllers with simple structure. Ind. Eng. Chem. Res. 2002, 41, 2705-2715. (15) Astrom, K. J.; Wittermark, T. Automatic tuning of simple regulators with specifications of phase and amplitude margins. Automatica 1984, 20, 645-654. (16) Krstic M.; Li, Z. H. Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Trans. Autom. Control 1998, 4, 336-350.

Received for review February 24, 2003 Revised manuscript received July 1, 2003 Accepted August 13, 2003 IE030175S

A Note on High-Order Controllers for Time-Delay Processes

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