New Approach for Design and Automatic Tuning of the Smith Predictor

3438

Ind. Eng. Chem. Res. 1999, 38, 3438-3445

New Approach for Design and Automatic Tuning of the Smith Predictor Controller K. K. Tan,* T. H. Lee, and R. Ferdous Department of Electrical Engineering, The National University of Singapore, 10 Kent Ridge Crescent (0511), 119260 Singapore

In this paper, we present a new Smith control design using a deliberate mismatch between the actual process and process model. With this particular representation, every Smith control system has an associated “Smith compensated” (SC) process that is an interactive function of the actual process and process model. Under this alternative Smith control structure, it is both intuitive and demonstrative by simulation study that the “best” Smith control design does not necessarily arise from the use of the perfect process model but rather the use of an adequately chosen model that results in the SC process possessing good dynamical properties from the viewpoint of the controller. Thus, the process modeling phase of the Smith control design may be viewed as a process precompensation design, and the final primary controller design will be carried out with respect to the SC process. For ease of practical applications, the entire procedure from modeling to control design may be automated and carried out online, using a new online relay tuning method. Robustness analysis of the new Smith control design is further carried out in the paper by drawing on existing results for single-input-single-output feedback systems. Simulations are provided to illustrate the applicability and effectiveness of both the online autotuning approach and the new Smith control design technique. 1. Introduction More than five decades since its inaugural application, PID (proportional-integral-derivative) is still the undisputed predominant controller in industrial practice (Astrom, 1984; Tan et al., 1996). The evergreen controller, however, is faced with a well-known dilemma when applied to processes with a long time delay or deadtime. Under these circumstances, the derivative action is often disabled to reduce excessive overshoot and oscillations due largely to misinterpretation of nonresponsiveness of the process (Hagglund and Astrom, 1991). On the other hand, with no derivative action, there is no predictive capability in the remaining PI control action, which is precisely needed for these processes. Thus, when a better control performance is necessary, some forms of deadtime compensation become necessary. One of the more popular deadtime compensation schemes is the Smith predictor controller (Smith, 1957). The scheme is only applicable to stable processes as the closed-loop poles will also include the open-loop ones. The major obstacle restricting the application of the Smith predictor is the requirement of a mathematical model for the process. It is the general notion that ideally the perfect model should be used, which unfortunately does not exist in practice. In the presence of modeling errors, the controller must be detuned to retain closed-loop stability. These necessary adjustments and requirements become strong deterents to the practitioner, since the performance from an imperfect Smith controller may be worse than the PI control, despite the higher costs in terms of implementation complexity and requirements. In Tan et al. (1996), it has been shown that a “calculated” model mismatch does not necessarily result in worse performance compared to the perfect model, but no proper guidelines were developed on how to determine the imperfect model. In this paper, a new

Smith control design method is developed that views the Smith control system as a single-loop control of an equivalent “Smith compensated” (SC) process that is an interactive function of the actual process and the process model. Santacesaria and Scattolini (1993) have used a similar representation of the Smith control system. Under a perfect modeling condition, it can be easily verified that this equivalent SC process becomes the delay-free portion of the process, so that the controller may be designed directly with respect to the delay-free part without considering the time delay. Under this alternative Smith control structure, it is both intuitive and demonstrated by simulation study that the “best” Smith control design does not necessarily arise from the use of the perfect process model but rather the use of an adequately chosen model that results in the SC process possessing good controllability characteristics. Thus, the process modeling phase of the Smith control design may be viewed as a process precompensation design and the final primary controller design will be carried out with respect to the SC process. An explicit modeling method and a new controller design based on the normalized deadtime of the process are developed and provided in this paper. For ease of application of this approach, a new online relay tuning method is applied to the Smith predictor, which automates the entire procedure, from modeling to control design, without having to disturb closed-loop operations. The relay tuning experiment is carried out on the closed Smith control system. The frequency response of the proces may be derived from the frequency response of the closed-loop system thus obtained, and it may be used for both process modeling and primary controller design for the Smith control system. Robustness analysis of the new Smith control design is carried out in this paper by drawing on existing results for single-input-single-output (SISO) feedback

10.1021/ie990059i CCC: $18.00 © 1999 American Chemical Society Published on Web 08/11/1999

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3439

Figure 2. Equivalent representation of the Smith predictor controller.

Figure 1. Smith predictor controller.

systems. The controller may also be designed based on these results for robust performance. Simulations are provided to illustrate the applicability and effectiveness of both the online relay autotuning approach and the new Smith control design. 2. Smith Predictor Controller: A Review The Smith predictor controller was proposed by Smith (1957) for deadtime compensation and is shown in Figure 1, where gp(s) ) gr(s)e-Ls and gpo(s) ) gro(s)e-Los are the process and model, respectively. The closed-loop transfer function between the setpoint and output can be shown to be

gyr(s) )

gc(s)gp(s) -sLo

1 + gc(s)[gro(s) - gro(s)e

+ gp(s)]

(1)

In the case of perfect modeling, i.e., gpo(s) ) gp(s), the closed-loop transfer function becomes

gyr(s) )

gc(s)gr(s)

e-Ls

1 + gc(s)gr(s)

This implies that the characteristic equation is free of the delay so that the primary controller gc(s) can be designed with respect to gr(s). The achievable performance can thus be improved greatly over a conventional single-loop system without the delay-free output prediction. Traditionally, it would appear that the perfect matching would give the best control results. However, this may not be true, as we will illustrate in the subsequent sections. 3. Equivalent Representation The Smith predictor structure is in many aspects similar to the internal model control (IMC) structure (Morari and Zafiriou, 1989), which has similar predictor capabilities. In Lee et al. (1996a), stability properties of the Smith predictor are inferred from an IMC equivalent representation of the Smith predictor. However, consider an equivalent representation of the Smith predictor controller as shown in Figure 2. The distinction of the Smith system from a single-loop control system is the additional compensator c(s) in the feedback path having the transfer function

c(s) )

gro(s)(1 - e-sLo) -sL

gr(s)e

c(s) ) esL

(3)

gs(s) ) gr(s)

(4)

This indicates that under a perfectly matched condition, the compensator c(s) gives a considerable phase lead in the feedback loop, in the form of a deadtime inverse. The total phase advance will increase with increasing L. It is this compensator c(s) that neutralize the deadtime of the process. The SC process gs(s) given in eq 3 is then the delay-free part of the process, gr(s), and the controller may thus be designed directly with gr(s). The concept of inherent phase lead compensation is not new. Astrom (1977) has conjectured that there exists some inherent phase lead compensation in the Smith predictor structure, and he verified the point through a specific example. With this particular representation, every Smith control system thus has an associated SC process gs(s), and all the uncertainty is contained in this process. The properties of gs(s) will thus directly affect the achievable closed-loop performance of the Smith system. For example, if the frequency response gs(jw) shows first-order process characteristics, then high gains in the controller gc(s) are permitted to yield the desirable transfer function gyr for good control performance in terms of setpoint tracking and load disturbance rejection. On the other hand, if the frequency response of gs(s) consists of resonant peaks, then the controller gc(s) has to be considerably detuned for closed-loop stability and poor performance will be expected. The “compensated” process gs(s) can thus act as a unifying element in the analysis and design of the Smith predictor controller. Furthermore, it provides an indication of when to use the Smith system and how to make the best use of it. 4. New Smith Predictor Controller Design

+1

The entire feedback element is described by

gs(s) ) c(s)gp(s)

In the case of Lo ) 0, it follows that c(s) ) 1 and gs(s) ) gr(s)e-sL. With this particular model, Figure 2 thus reduces to the single-loop control structure with the controller gc and the actual process gp in the loop. The SC process is now the actual process itself. Thus, we can actually view the single-loop control system as a particular case of a mismatched Smith system with Lo ) 0. In general, gs(s) can thus be regarded as the SC process. Under a perfectly matched condition, i.e., gp(s) ) gpo(s), we have

(2)

The proposed new Smith predictor controller design essentially consists of two phases to be described in the following subsections. 4.1. Modeling as a Process Precompensation Tool. From eq 2, the SC process may be expressed as

3440 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

gs(s) ) gro(s) - gpo(s) + gp(s) Departing from the traditional approach of deriving the process model such that ideally gpo(s) ) gp(s), the proposed approach here is to select gpo(s) such that gs(s) exhibits certain specified desired characteristics. In this paper, it is the desired and selected objective for gs(jw) to exhibit the characteristics of a first-order delayfree transfer function for the frequency range of concern. Denoting the desired SC process as g/s (s), we thus have

g/s (s) )

Ks Tss + 1

Figure 3. Relation between Θ and λ.

where Ks and Ts are the user-specified parameters. It is well-known that it is relatively easy to control a process with a first-order dynamics compared to one with a high-order dynamics. For this kind of process, plain P control with sufficiently large gain is sufficient to yield very good performance. Furthermore, we will use the commonly used first-order with deadtime transfer function for the process model, i.e.,

gpo(s) )

K e-sL Ts + 1

(5)

if gp(jw*) is known. Assume that

gp(jw*) ) Rp + jβp and

g/s (jw*) ) Rs + jβs (5) thus becomes

1 (1 - e-jw/-L) ) ∆R + j∆β T(jw*) + 1

(6)

where ∆R ) Rs - Rp and ∆β ) βs - βp. Solving the complex equation (eq 6), we obtain T and L as follows:

T) 2 2 2 2 2 1 -∆β ( x∆β - (∆R + ∆β )(∆R + ∆β - 2∆R) w* (∆R2 + ∆β2) (7)

L ) (1/w*)[arccos(1 + T∆βw* - ∆R)]

(8)

4.2. Primary Controller Design Based on SC Process. A primary PI controller is used which is described by

(

gc(s) ) Kc 1 +

)

1 Tis

Furthermore, denote the actual process as

N(s)e-sLp gp(s) ) Kp D(s)

Θ λ

0 0.5

0.44 1.05

1.2 1.72

1.8 2.15

1.97 2.55

2.32 2.71

3.21 3.56

3.55 4.1

where N(s) and D(s) are polynomials with N(0)/D(0) ) 1. The desired closed-loop transfer function is thus given by

g/yr(s) )

KcK(Tis + 1)(Tss + 1)N(s)e-sLp Tis(Tss + 1) + KcK(Tis + 1)D(s)

Choosing Ti ) Ts, we have

Assuming the static gain Kp of the process is known and letting K ) Ks ) Kp, we may obtain the remaining two parameters T and L of the process model gpo(s) by solving the complex equation

gso(jw*) ) gro(jw*) - gpo(jw*) + gp(jw*)

Table 1. Relation between Θ and λ

(9)

g/yr(s) )

(Tss + 1)N(s)e-sL Tss + 1 D(s) KcK

(

)

Therefore, by appropriately choosing Kc, additional phase lead or phase lag compensation may be introduced through the shaping of the zero-pole pair (Tss + 1)/[(Tss/KcK) + 1]. It is thus appropriate that the choice of Kc be related to the amount of phase lag of the actual process. A suitable characterization for the amount of phase lag of the actual process may be derived from the normalized deadtime of the process Θ (Astrom, 1991), which is defined as the ratio of equivalent deadtime to a time constant of a process. For a process with a large Θ, it has a relatively large phase lag at the same frequency and more lead compensation will be necessary. We design

Kc ) 0.4λ where λ is related to Θ. We obtain this relationship by extensive simulation on a first-order with deadtime system for a range of Θ and obtain the corresponding optimum value of λ, which will yield good transient response properties. This is reasonable since the system has dynamical representation of most practical processes. The value of λ and Θ are tabulated in Table 1, and the relationship is shown in Figure 3. From Figure 3, we derive the empirical relationship as

λ ) Θ + 0.5 5. Proposed Robust Online Relay Autotuning Method Relay autotuning of controllers has been actively researched for the past decade after Astrom’s inaugural application of the method to tune simple three-term PID controllers. Since then, the method has been extended to autotuning of deadtime compensators, as in the Smith


predictor control and finite spectrum assignment control, for time-delay systems. The method has also been successfully applied to autotuning of multiloop controllers and, more recently, autotuning of full multivariable controllers for multivariable processes. The main attraction of the pioneer method appears to be the viability of automation on a large scale for control tuning, and this is particularly useful for the process control industry where numbers of control loops on the order of several hundreds and thousands are commonly encountered. One of the main features of the relay autotuning method, which probably accounts for its success more than any other associated feature, is that it is a closed-loop method, and therefore, an onoff regulation of the process may be maintained even when the relay experiment is being conducted. However, the approach has several important practical constraints that have remained, in large proportion, unresolved to date. First, it has a sensitivity problem in the presence of disturbance signals, which may be real process perturbation signals or equivalent ones arising from varying process dynamics, nonlinearities, and uncertainties present in the process. For small and constant disturbances, given that stationary conditions are known, an iterative solution has been proposed, essentially by adjusting the relay bias until symmetrical limit cycle oscillations are obtained. However, for general disturbance signals, there has been no effective solution to date. Second, relating partly to the first problem, relay tuning may only begin after stationary conditions are reached in the input and output signals, so that the relay switching levels may be determined with respect to these conditions and the static gain of the process. In practice, under open-loop conditions, it is difficult to determine when these conditions are satisfied and therefore when the relay experiment may begin. Third, the relay autotuning method is not applicable to certain classes of processes which are not relay-stabilizable, such as the double integrator and runaway processes. For these processes, relay feedback is not able to effectively induce stable limit cycle oscillations. Finally, the basic relay method is an offline tuning method, i.e., some information on the process is first extracted with the process under relay feedback and detached from the controller. The information is subsequently used to commission the controller. Offline tuning has associated implications in the tuning-control transfer, affecting operational process regulation which may not be acceptable for certain critical applications. Indeed, in certain key process control areas (e.g., vacuum control, environment control, etc.) directly affecting downstream processes, it may be just too expensive or dangerous for the control loop to be broken for tuning purposes, and tuning under tight continuous closed-loop control (not the on-off type) is necessary. Recently, a new online robust relay automatic tuning has been developed that is effective against many of the above-mentioned constraints of the basic relay autotuning method, while still retaining the simplicity of the original method. In the new proposed configuration, the relay is applied to a controller-stabilized process in the usual manner. The initial settings of the controller can be conservative and intended primarily for the purpose of stabilizing the process. They may be based on simple prior information about the process or default settings may be used. Practical applications of controller automatic tuning methods have been used mainly to derive

Figure 4. Configuration for online tuning of the Smith controller.

more efficient updates of current or default control settings. Thus, the proposed configuration does not post additional and stringent prerequisites for its usage, but rather, it uses information on the system which is already available in many cases. In Tan et al., 1999, the method has been successfully applied to automatic tuning of PID controllers for a large class of processes, and its potential in applications over the original method is largely evident from the extensive simulation and real-time experiments presented. In this section, the new proposed robust online relay automatic tuning method is systematically described in details and applied to autotuning of the Smith predictor controller. 5.1. Configuration. The configuration of the proposed robust online relay autotuning method is given in Figure 4. Under this configuration, the set-point yref also serves as the direct current (dc) level of the relay (which accounts for the direct link to the output of a relay with a zero dc level). Essentially, in this configuration, the relay is applied to an inner control loop comprising the actual Smith control system. If the system settles in a steady-state limit cycle oscillation at frequency wu (ultimate frequency), then gyr(jwu) can be obtained from amplitude and phase response considerations in the usual manner. It then follows that one point of the frequency response of gp(jw) may be obtained at w ) wu as

gp(jwu) ) gyr(jwu)[gc(jwu)gpo(jwu) - gc(jwu)gro(jwu) - 1] gc(jwu)[gyr(jwu) - 1]

(10)

Given gp(jw) thus obtained, the new Smith control design method described in section 4 may then be invoked with w* ) wu. Note that as with the original relay method, a lower oscillating frequency wu can be obtained other than the ultimate frequency of the closedloop by additional dynamical or adaptive elements connected in series to the relay (Lee et al., 1996b). 6. Robustness Analysis for the New Smith Control Design The Smith predictor controller is a model-based control system, and as such, it will be subject to a modeling sensitivity problem, when the model used in the primary controller design deviates from the actual process. In this section, we will consider the robust stability and robust performance of the new Smith control design in the face of inevitable modeling uncertainty.


6.1. Robust Stability. The Smith predictor controller is referred to as being robustly stable if it is designed such that the closed-loop is stable for all members of a family of possible processes. Robust stability results are already available for SISO feedback control (Morari and Zafiriou, 1989). It is therefore straightforward to extend the available results to the equivalent SISO structure of the Smith control system. The following assumptions are made. Assumption 1. (i) The actual SC process gs(s) and the ideal SC process g/s (s) do not have any unstable poles. (ii) The actual process gp(s) is stable. (iii) The nominal closed-loop system g/vr(s), between the set-point and output from gs(s), is stable. Theorem 1. Assume that the family of stable SC processes ∏ with norm-bounded uncertainty is described by

∏

{

) gs:

gs(jw) - g/s (jw)

|

| |

|

}

) lm(jw) e ˜lm(w)

max||g/vdW||∞ ) max sup|g/vd(jw)W(jw)| < 1 (13) gs∈Π

gs∈Π

w

where g/vd ) 1 - g/vr is the equivalent sensitivity function and W is the performance weight. In general, W-1 provides a bound on the maximum peak of the function g/vd and imposes a minimum bandwidth constraint on the closed-loop. For the simple choice of W-1 ) MP, robust performance only requires the maximum peak of the sensitivity function to be less than MP with no bandwidth constraint. The reader may refer to Laughlin et al. (1986) for more details on the choice of the performance weight, W. As for robust stability, the result on robust performance may be directly induced from the available result for SISO feedback system. Theorem 2. Assume that the family of stable processes ∏ is described by (11). Then under assumption 1, the Smith predictor controller of Figure 4 will meet the performance specification (13) if and only if

(11)

|g/vr(jw)|l˜m(w) + |[1 - g/vr(jw)]W(jw)| < 1 ∀w (14)

where ˜lm(w) is the bound on the multiplicative uncertainty lm(jw). Then, the Smith control system of Figure 4 is robustly stable if and only if g/vr(s) ) gc(s)g/s (s)/1 + gc(s)g/s (s) satisfy the following bound

Remark 3. Equation 14 is necessary and sufficient for the norm-bounded uncertainty described in (11). In general, however, it is only sufficient when the true uncertainty region is only a portion of the uncertainty region described in (11). Equation 14 can also be rewritten as

g/s (jw)

|g/vr(jw)|l˜m(w) < 1 ∀w

(12)

Proof. The theorem follows directly from the robust stability result for SISO feedback system (Morari and Zafiriou, 1989), since the Smith control system has been posed in the same form. Remark 1. By the condition being necessary and sufficient, it implies that if condition in (12) is violated, then in the set Π defined by (11) there exists a SC process gs for which the closed-loop is unstable. Equation 12 can also be rewritten as

|g/vr(jw)| < ˜l-1 m (w) ∀w Given the uncertainty bound ˜lm(w), the robust stability of the closed-loop Smith predictor controller can thus be ensured by a proper design of the primary controller (choice of Kc and Ti) so that the magnitude-frequency response of |g/vr(jw)| lies below that of ˜l-1 m (w), and (12) is satisfied. Remark 2. Equation 12 further reiterates the point that the Smith control designed around an imperfect process model is not necessarily worse than one based on the perfect model. For good robust stability to controller design, all that is needed is that gs(jw) is close to g/s (jw) for as large w as possible. It is not necessary that gpo(jw) ) gp(jw). 6.2. Robust Performance. Robust stability is desirable in a practical environment where model uncertainty is an important issue. However, robust stability alone is not enough. Even if (12) is satisfied for the family ∏, there will exist a “worst-case” SC process in ∏ for which the closed-loop system is on the verge of instability and for which the performance is arbitrarily poor. Thus, it is necessary to ensure that some performance specifications are met for all SC processes in the family ∏. Performance specifications stated in the H∞ framework (Morari and Zafiriou, 1989) require

|g/vr(jw)| 1 - |[1 -

g/vr(jw)]W(jw)|

< ˜l-1 m (w)

Given the uncertainty bound ˜lm(w) and the performance weight W(jw), the robust performance design is thus to design a proper ideal closed-loop transfer function g/vr(s) so that the magnitude-frequency response of | g/vr(jw)|/{1 - |[1 - g/vr(jw)]W(jw)|} lies below that of lm(w) and the ˜l-1 m (w) to satisfy (14). Depending on ˜ specification W(jw), there may not be any g/vr(jw) which satisfy (14); then, W(jw) may be too tight for the uncertainty present, and it may have to be relaxed. 7. Simulation Study Several simulation examples are presented here to illustrate the effectiveness of the proposed new Smith predictor controller design for a wide variety of process dynamics commonly encountered in the process control industry. Example 1: Fifth-Order Process. Consider a highorder process described by

gp(s) )

1 e-4s 5 (s + 1)

Assuming a perfect model, the SC process is thus given by

gp(s) ) 1/(s + 1)5 The PI controller gc(s) is first designed with respect to this SC process based on the rules of Hagglund (1991). The set point changes at t ) 20, and at t ) 130, a load disturbance seeps into the process. Figure 5 shows the control performance based on a perfect model. Subsequently, the online relay experiment is conducted from


Figure 5. Closed-loop performance in example 1. Equation 1 is based on a perfect model, and (2) is the proposed method.

Figure 6. Comparison of the closed-loop performance between (2), the proposed method, and (3), the method of Wang (1993).

t ) 300 to 480, and the desired SC is specified as g/s (s) ) 1/(2.81s + 1), which yields the recommended process model as

gpo(s) )

Figure 7. Magnitude frequency of example 1, (1) ˜lm(w), and (2) | g/vr(jw)|/{1 - |[1 - g/vr(jw)]W(jw)|}.

Figure 8. Comparison of the IMC closed-loop performance from (1), reduced-order model, and (2), perfect model.

1 e-7.014s (2.32s + 1)

Note that the model would have been

gpo(s) )

1 e-8.27s (2.63s + 1)

if we had attempted to perfectly match the process and model at the oscillation frequency of wu ) 0.3306. The normalized deadtime is Θ ) 3.14, and therefore, the controller gc(s) is commissioned with Kc ) 1.4 and Ti ) 2.81. The subsequent set point change at t ) 375, and a load disturbance at t ) 420 shows clearly the enhanced performance of the proposed tuning method. We have compared our new method with that of Wang et al. (1993), where the first-order plus deadtime model was also used. The better performance of our new method is shown in Figure 6. The robustness condition (14) is satisfied with performance weight at W-1 ) 5 since the magnitude-frequency response of

|g/vr(jw)| 1 - |[1 - g/vr(jw)]W(jw)|

< ˜l-1 m (w)

lies below that of ˜l-1 m (w), as shown in Figure 7. Next, for variety, we will show that use of a common reduced order need not result in a compromise in control performance for standard IMC design. We consider the same process. In the perfect modeling case, the value of IMC controller q(s) is often chosen as

q(s) ) (s + 1)5/(s + 1)5 where is tuned for an acceptable balance between performance and robustness. In this example, ) 3. Simulation is carried out with a step disturbance at the

Figure 9. Closed-loop performance in example 2. Equation 1 is based on a perfect model and (2) is based on the proposed method.

input to the process at t ) 65. An IMC controller based on the reduced-order model obtained above is also simulated with the same value of . The closed-loop responses for both cases are shown in Figure 8, where it is evident that IMC based on a reduced-order model achieves a better performance and exhibits better noise/ disturbance suppression characteristics. Example 2: Third-Order Process. Consider a third-order system described by

gp(s) )

1 e-6s (s + 1)3

The simulation sequence and formula are similar to example 1. The recommended process model according to section 4.2 is

gpo(s) )

1 e-6.623 (2.27s + 1)

A load disturbance enters the system at t ) 140 and 430. Enhanced performance from the new tuning method is observed and robustness is verified in Figures 9 and 10, respectively. Example 3: Second-Order Underdamped Process. The first-order with deadtime model is adequate for processes with overdamped step response. It is


Figure 10. Magnitude frequency of example 2, (1) ˜lm(w), and (2) |g/vr(jw)|/{1 - |[1 - g/vr(jw)]W(jw)|}.

Figure 13. Closed-loop performance in example 4. Equation 1 is based on perfect model, and (2) is the proposed method.

Figure 11. Closed-loop performance in example 3. Equation 1 is based on a perfect model, and (2) is the proposed method.


8. Conclusion


interesting to observe the performance when the actual process is an underdamped one. Consider an underdamped second-order process:

gp(s) )

1 e-3s (s + 2s + 4) 2

The following recommended process model is obtained:

gpo(s) )

1 e-2.37s (8.52s + 4)

Good performance is observed and robustness is verified in Figures 11 and 12, respectively. Example 4: First-Order Process. This example will more directly illustrate the mismatch introduction in the model. Consider a first-order process given by

gpo(s) )

1 (4s + 1)e-7s

The recommended model is obtained as

gpo(s) )

1 e-7.8s (1.8s + 1)

Enhanced performance is again observed and robustness is verified in Figures 13 and 14, respectively.

A new Smith control design is developed in this paper that does not discriminate against model mismatch, but rather the process modeling procedure is viewed as a process precompensation phase. The Smith controller is subsequently designed with respect to the compensated process. The entire procedure is automated via a new online relay automatic tuning method applied to the Smith predictor control system under operation. The robust stability and the robust performance of the thus designed and tuned system are analyzed and conditions for robustness are provided. Simulations verify the effectiveness and applicability of the new design and automatic tuning methods developed. Literature Cited Astrom, K. J. Frequency domain properties of Otto Smith regulators. Int. J. Control 1977, 26, 307. Astrom, K. J.; Hagglund, T. Automatic tuning of simple regulators with specification on phase and amplitude margins. Automatica 1984, 20 (5), 645. Astrom, K. J. Assessment of achievable performance of simple feedback loops. Int. J. Adapt. Control Signal Proc. 1991, 5, 3. Hagglund, T.; Astrom, K. J. Industrial adaptive controllers based on frequency response techniques. Automatica 1991, 27, 599. Halevi, Y. Optimal reduced order models with delay. Proceedings of the 30th Conference on Decision and Control, Brighton, England, 1991, 602. Hang, C. C.; Chin, D. Reduced order process modelling in selftuning control. Automatica 1991, 27 (3), 529. Laughlin, D. L.; Jordan, K. G.; Morari, M. Internal model control and process uncertainty: Mapping uncertainty regions for SISO controller design. Int. J. Control 1986, 44, 1675. Lee, T. H.; Wang, Q. G.; Tan, K. K. A robust Smith-predictor controller for uncertain delay systems. AIChE J. 1996a, 42 (4), 1033. Lee, T. H.; Wang, Q. G.; Tan, K. K. Enhanced automatic tuning procedure for process control of PI/PID controllers. AIChE J. 1996b, 42 (9), 2555. Marshall, J. E.; Gorecki, H.; Walton, K.; Korytowski, A. Time-Delay Systems: Stability and Performance Criteria with Applications; Ellis Horwood: New York, 1992. Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989.

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3445 Palmor, Z. J. Stability properties of Smith dead-time compensator controllers. Int. J. Control 1980, 53, 937. Santacesaria, C.; Scattolini, R. Easy tuning of Smith predictor in presence of delay uncertainty. Automatica 1993, 29 (6), 1595. Smith, O. J. M. Closer control of loops with dead time. Chem. Eng. Progr. 1957, 53, 217. Tan, K. K.; Wang, Q. G.; Lee, T. H.; Bi, Q. New approach to analysis and design of Smith-predictor controllers. AIChE J. 1996, 42 (6), 1793. Tan, K. K.; Lee, T. H.; Jiang, X. Robust on-line relay automatic tuning method for PID control system. IEEE Trans. Control Technol. 1999, submitted for publication. Wang, Q. G.; Hang, C. C.; Chee, F. S. Autotuning of Smith predictors for processes with long dead time. 1st Chinese World

Congress on Intelligent Control and Automation, Beijing, August 25-29, 1993, 2624-2629. Yamanaka, K.; Shimemura, E. Effects of mismatched Smith controller on stability in systems with time delay. Automatica 1987, 23 (6), 787. Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1943, 65, 433.

Received for review January 27, 1999 Revised manuscript received June 4, 1999 Accepted June 7, 1999 IE990059I

New Approach for Design and Automatic Tuning of the Smith Predictor

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