IMC-Based Control System Design for Unstable Processes

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PROCESS DESIGN AND CONTROL IMC-Based Control System Design for Unstable Processes Xue-Ping Yang, Qing-Guo Wang,* C. C. Hang, and Chong Lin Department of Electrical & Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

This paper studies an internal model control (IMC) based single-loop controller design method to find the feedback controller in either proportional-integral-derivative or high-order form for unstable processes with time delay. The performance limitation of implementing the IMC controller in an equivalent feedback structure for an open-loop unstable process has been addressed. The proposed method is generally applicable to a wide range of unstable processes and can be made automatic for online tuning. Introduction It is well-known that control system design for an open-loop unstable process is more difficult than that for a stable one because of the unstable nature of the dynamics. The tuning of controllers for a time-delay unstable process has been an active area of research in the literature1-5 Reference 6 surveyed the previous works which are related to the control of unstable processes. The tuning rules for proportional-integral (PI) and proportional-integral-derivative (PID) controllers in conventional feedback structure7-10 show an excessive overshoot and large settling time. To overcome this drawback, authors in ref 11 have proposed an enhanced PID-P control strategy, while those in ref 12 have proposed a PI-PD one. Both of them include inner feedback loops and can yield acceptable overshoot and small settling times. However, their works are only applicable to first-order plus dead time (FOPDT) and second-order plus dead time (SOPDT) processes. During the past decade, internal model control (IMC)13 has demonstrated its effectiveness in the process industry. Many efforts have been made to exploit the IMC principle to tune conventional PID controllers. Satisfactory results have been shown in SISO applications,14,15 whereas the development for unstable systems is not straightforward. Authors in ref 16 developed a simple PI and PID controller tuning for an unstable process based on the IMC principle, but they are only applicable to specific types of processes such as FOPDT and SOPDT processes. In their work, a first-order Pade´ approximation is applied to derive the tuning rules. This inevitably imposes some limitations on the applicability of the methods and the performance of the PID controller designed. Moreover, the performance limitation of implementing the IMC controller in an equivalent feedback structure for an open-loop unstable process has not been addressed. For a process whose Nyquist curve exhibits a strange shape, especially around the crossover frequency, a PID controller could be difficult to shape * To whom all correspondence should be addressed. Email: [email protected]. Tel: (+65) 874 2282. Fax: (+65) 779 1103.

satisfactorily. For an open-loop stable process, one can simply detune the PID controller by trading off some performance to a sufficient extent to generate a stable closed loop. However, for an open-loop unstable process, there is a limit to how much we can detune the controller; some minimal controller gain is needed just to stabilize the systems.13 We are thus motivated to develop a new method for IMC-based single-loop controller design and consider, in addition to PID, a highorder controller as well for complex processes where PID controllers are not adequate. In this paper, an IMC-based single-loop controller design method is given to find the feedback controller in either PID or high-order form. The proposed method is generally applicable to a wide range of unstable processes and can be made automatic for online tuning. Compared with the existing methods presented for the unstable process, the proposed design shows a much better control performance. This paper is organized as follows. The controller design is detailed in section 2. The respective PID and high-order controller cases are detailed in section 3. Finally, some concluding remarks are drawn in section 4. 2. Controller Design Methodology The schematic of the IMC system is depicted in Figure 1, where G(s) is the given unstable process to be controlled, G ˆ (s) a model of the process, and C(s) the IMC primary controller. It is well-known that the IMC structure is internally unstable if the process G(s) is unstable13 and has to be abandoned for implementation. However, authors in ref 13 suggested that one can still design the controller using the IMC method and then implement the controller in an equivalent feedback structure as follows. In the nominal case (the uncertain case will be discussed below) of G(s) ) G ˆ (s), the equivalent single-loop feedback system in Figure 2 is derived from the IMC system in Figure 1 with

K ) C(1 - CG ˆ )-1

10.1021/ie010812j CCC: $22.00 © 2002 American Chemical Society Published on Web 07/19/2002

(1)

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4289

Figure 1. IMC control system.

Figure 2. Single-loop control system.

Figure 3. Typical relationship between τjmin and L h.

The system is internally stable if

investigate the root locations of

C and (1 - G ˆ C)G ˆ are both analytic in the RHP

δ(s,τ,L) ) (τs + 1)m+1 - (Rs + 1)e-Ls

(2)

(8)

Generally, the feedback controller K in (1) should include an integrator to eliminate the steady-state error and keep the rest stable. Thus, we require

Normalize L, τ, a, and s as L h ) L/T, τj ) τ/T, R j ) R/T, and sj ) sT. Equation 8 becomes

K (after all possible pole-zero cancellations) has no pole in the closed RHP except s ) 0 (3)

j sj + 1)e-Lh δ(sj, τj, L h ) ) (τjsj + 1)m+1 - (R

Let us consider a class of unstable processes with a single RHP pole only:

G ˆ (s) )

1 G ˆ (s) e-Ls 1 - Ts -

C ) (1 - Ts)G ˆ - -1f

(5)

Q(sj,L h ) ) A(sj) + B(sj) e-Lh

Rs + 1 , R ) T(τ/T + 1)m+1eL/T - T m+1 (τs + 1)

K(s,τ) )

(1 - Ts)(Rs + 1) G ˆ -[(τs + 1)

m+1

-Ls

- (Rs + 1)e

]

{ {

h ) ) Re cos(ω0L (6)

where m is an integer sufficiently large to guarantee that the IMC controller C is proper. Thus, τ > 0 is the only tuning parameter to be selected by the user to meet (3), to achieve the appropriate tradeoff between performance and robustness, and to keep the manipulated variable within bounds. We now address them separately. For G ˆ in (4) and C in (5), K in (1) becomes

(7)

It can be verified that s ) 1/T is both a zero and a pole of K(s). It should be canceled to form the final K(s) for actual implementation. s ) 0 is another pole of K(s), which is desired to eliminate the steady-state error. Equation 3 requires that no roots of the denominator of K(s,τ) lie in the closed RHP, except s ) 1/T and s ) 0. Because G ˆ - is of minimum phase, we only need to

sh

(10)

where A(sj) and B(sj) are polynomials in sj. Authors in ref 17 proposed a method to study the movement of the roots of (10) with respect to a certain parameter, and those method can be employed to minimum τjmin at which some roots of (10) are lying on the imaginary axis. This τjmin should meet

where f is a user-specified low-pass filter and is chosen as

f(s,τ) )

(9)

Equation 9 is a quasi-polynomial and can be written into a standard form

(4)

where G ˆ -(s) is rational, stable, and of minimum phase. The optimal H2 IMC controller for step inputs is

sh

} }

(jω0τjmin + 1)m+1 jω0R j +1

h ) ) Im sin(ω0L

(jω0τjmin + 1)m+1 jω0R j +1

(11)

(12)

where

j 2 + 1) ) ω0 ) {min(ω0)|(ω02τjmin2 + 1)m+1 - (ω02R 0, ω02 ∈ R+, ω0 > 0} (13) Thus, for a given L, τ should be chosen to satisfy

τj > τjmin

(14)

For a given m, τjmin depends on L h . When m ) 1-3, the typical relationship between τjmin and L h is shown in Figure 3. It is interesting to note that when τjmin tends to infinity, L h tends to be a constant, L h max, which indicates that the process is stabilizable only if L h eL h max. L h max is determined by m only and can be obtained as

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{

Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002

{max(L h )|eLh L h m < (3π/2)m}, m ) 4l + 1 L h m h < πm}, m ) 4l + 2 {max(L h )|e L L h max ) h m < (π/2)m}, m ) 4l + 3 {max(L h )|eLh L h m < (2π)m}, m ) 4l + 4 {max(L h )|eLh L

}

(15)

where l ) 0, 1, 2, .... For system performance, with controller K(s) in (7), the closed-loop function is

η(s) )

G(s) K(s) Rs + 1 -Ls e )G ˆ +(s) f(s) ) 1 + G(s) K(s) (τs + 1)m+1 (16)

The maximum of the magnitude of η is related directly to L h and τj as

||η||∞ )

x{ [

1 R j2 -m-1 +1 m τj 1 τj 2 1 - (m + 1) +1 m R j

[( )

Figure 4. ||η∞|| and τj.

] ( )] }

m+1

(17)

A typical relation between |η|∞ and τj is shown in Figure 4. The large amplitude of |η|∞ usually produces a peak overshoot in the step response in the time domain perspective.18 To eliminate the overshoot, a prefilter F ) 1/(Rs + 1) is added with R given in (6), as shown in Figure 2. For system robustness, let the actual process be G(s) )G ˆ (s) [1 + ∆G(s)] and |∆G| e δG(ω). In implementation, the presence of dead time in the denominator of K(s) increases the complexity of the controller. Moreover, because of the fact that the denominator of K(s) is not in a polynomial form, it is not possible to cancel s ) 1/T in K(s) explicitly. Thus, model reduction will be applied to obtain the best approximation K ˆ to K, where K ˆ (s) ) K(s) [1 + ∆K(s)] and |∆K| e δK(ω). In our context, δk e ERR < , where  is the threshold for the model reduction error and is specified prior to the design and usually taken as 5%. With the standard assumption that K ˆ has the same number of unstable poles as K, it follows from the stability robustness theorem19 and some algebra that the uncertain feedback system remains stable for all ∆ ) diag{∆K, ∆G} if

x{ [

1 R j 2 -m-1 +1 m τj 1 τj 2 1 - (m + 1) +1 m R j

[( )

] ( )] }

m+1

<

|

xδK2(ω) + δG2(ω) + 4δK2(ω) δG2(ω) - 2δK(ω) δG(ω) δK2(ω) + δG2(ω)

ω)ωp

} δp (21)

which gives a range of τ to achieve robust stability. To consider the performance limitation imposed by input constraint, we use a frequency-by-frequency analysis.20 Assume that at each frequency |U(jω)| e U h and the reference signal satisfies |R(jω)| e R h . The manipulated variable is

U(s) ) F(s) )

K(s,τ) R(s) 1 + K(s,τ) G(s)

1 - Ts G ˆ --1(s) R(s) (τs+1)m+1

One requires

1 - Tjω G ˆ --1(jω) R(jω) e U h (τjω + 1)m+1

(22)

δK2(ω) |η(jω)|2 + δG2(ω) |η(jω)|2 + 4δK(ω) δG(ω) |η(jω)| < 1, ∀ ω (18)

Considering the worst case references (|R(jω)| ) R h ), we require

It can be noted from (16) that |η(jω)| has a peak value at a low frequency ωp and decays quickly for higher frequencies. Then (18) is likely to hold if

1 1 U h | |e| G ˆ -(jω)| m+1 1 Tjω R h (τjω + 1)

δK2(ωp)||η||∞2 + δG2(ωp)||η||∞2 + 4δK(ωp) δG(ωp)||η||∞ < 1 (19)

To derive an inequality on τ imposed by the input constraint, let ω ) ωob, where ωob is the open-loop bandwidth, and notice that |[1/(1 - Tjωob)]G ˆ -(jωob)| ) 1/x2, we require

An upper bound of ||η||∞ can be obtained from (19):

h 1 U 1 |e | m+1 R h (τjωob + 1) x2

||η||∞ <

ω)ωp

(20)

Combining (17) and (20) yields

(24)

i.e.,

x2δK2(ω) + δG2(ω) + 4δK2(ω) δG2(ω) - 4δK(ω) δG(ω)| 2[δK2(ω) + δG2(ω)]

(23)

τg

x( ) 2R h2 U h2

1/(m+1)

ωob

-1 (25)

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In short, for a single-loop controller design, the tuning parameter τ in the filter (6) should be, in general, chosen to meet (14), (21), and (25) simultaneously, and this will determine a suitable range, τ ∈(τmin, τmax). 3. Controller Implementation For practical issues in the implementation, our idea is to apply a suitable model reduction to obtain the best approximation K ˆ to K, so that the resulting single-loop performance can be better guaranteed and well predicted from the IMC counterpart. If the user specifies the type of K ˆ (say, PID), then the model reduction algorithm will generate its parameters. If the approximation accuracy is satisfactory, the design is completed; otherwise, the algorithm will adjust the IMC controller performance via τ within a specified interval to achieve the best single-loop approximation. On the other hand, if the user has no preference on controller structure, our algorithm starts with a PID type and gradually increases the controller complexity such that the simplest approximation K ˆ is attained with the guaranteed accuracy to K. This allows a unified treatment of all cases and facilitates autotuning application. In the subsequent two subsections, PID and general controllers will be considered, respectively. 3.1. PID Controller. Because of its simple structure, the PID controller is the most widely used controller in the process industry, even though many advanced control algorithms have been introduced. Consider a PID controller in the form

KPID ) kp +

ki + kds s

(26)

where kp is the proportional gain, ki the integral gain (units of time), and kd the derivative gain (units of time). Our task is to find the three PID parameters to match K ˆ ) KPID to K ) C/(1 - CG ˆ ) as well as possible. This objective can be realized by minimizing the loss function

Figure 5. Step response for example 1: (s) proposed; (- -) Majhi and Atherton; (-‚-) Park et al.

where k represents the kth iteration and ζ is an adjustable factor reflecting the approximation accuracy of the present iteration and is set at 1/4, 1/2, and 1 when 5% < E e 20%, 20% < E e 100%, and 100% < E, respectively. The iteration continues until the accuracy bound is fulfilled or τk+1 > τmax. If (28) cannot be fulfilled when τk+1 > τmax, a more complex controller than PID is necessary. We now present some simulation examples to demonstrate our PID tuning algorithm, and the performance is compared with the results of refs 11, 12, and 21. The ideal PID controller in (26) used for our algorithm development is not physically realizable and thus is replaced by

KPID ) kp +

m

∑

min J } min |KPID(jωi) - K(jωi)|2, kp, ki, kd > 0 KPIDi)1 KPID (27) whose solution is obtained by the standard nonnegative least squares to give the optimal PID parameters as [kp*, ki*, kd*]T ) θ*. Our studies suggest that the frequency range [ω1, ωM] in the optimal fitting (27) be chosen as (0.1ωb, ωb) with the step of (1/100-1/10)ωb, where ωb is the desired closed-loop bandwidth. Once a PID controller is found, the following criterion should be used to validate the solution:

K ˆ (jω) - K(jω) |e E ) max | K(jω) ω∈[0,ωb]

(30)

In implementation, N is suggested to be chosen within refs 5 and 20. In simulation examples of this subsection, we will consider the nominal case and (21) is not used. Normally, (25) gives a smaller lower bound on τ, and thus only (14) is utilized to derive τmin and τ0 ) τmin is set in examples 1-3. Example 1. Consider an unstable process:21

G)

4e-2s 1 - 4s

From (14), one obtains τ0 ) 1.7. This results in

(28) KPID ) 0.6407 +

where  is the user-specified fitting error threshold.  is specified according to the desired degree of performance, or accuracy of the SL approximation to the IMC one. Usually  may be set as 5%. If (28) holds true, the design is completed. On the other hand, if the given threshold cannot be met, one can detune the PID controller by relaxing the IMC specification, i.e., increasing τ. Our detuning rule is

τk+1 ) τk + ζkL

ki k ds + s kd s+1 N

(29)

0.0626 0.5633s + s 0.5633 s+1 N

(31)

This controller has the approximation error E ) 1.80%, which can fulfill the accuracy threshold. The PI-PD controllers of ref 12 are kp[1 + (1/Tis)] ) 0.131[1 + (1/ 2s)] and Kf(Tds + 1) ) 0.5(s + 1). The PID-P controllers of ref 11 are Kp[1 + (1/Tis) + Tds] ) 0.068[1 + (1/1.885s) + 4.296s] and Kf ) 0.350. The closed-loop responses are shown in Figure 5. It can be seen that the proposed method shows a much better performance than the other designs. Because refs 16 and 21 used only con-

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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002

Figure 6. Step response for example 2: (s) proposed; (- -) Majhi and Atherton; (-‚-) Park et al.

Figure 7. Step response for example 3: (‚‚‚) proposed PID (τ ) 1); (s) proposed high order (τ ) 1).

ventional PID controllers, they show very poor performances in terms of an overshoot and a settling time. Example 2. Consider an unstable process:11

The closed-loop responses are shown in Figure 7, where a setpoint filter of 1/(5.78s + 1) is added to the method of ref 21. It can be seen from the simulation study that the proposed method always yields a PID controller with a much better performance than the other methods, regardless of what τ is chosen. Our experience indicates, that for FOPDT and SOPDT processes and a slow closed-loop response requirement, the proposed method can always achieve E < 5%, and thus the closed-loop performance can be well predicted from the corresponding IMC design. It is, however, noticed for high-order processes with fast responses that none of the above methods is able to generate PID systems with good performance. This implies that a controller in the PID form is insufficient to obtain the desired performance. In this case, a higher order of the controller has to be considered for a better fitting and performance, which will be discussed in the next section. 3.2. High-Order Controller. In this subsection, model reduction is employed to find the lowest order of the controller which can achieve a specified approximation accuracy. A number of methods for rational approximation are surveyed in ref 23. The recursive least squares (RLS) algorithm is suitable for our application and is briefly described as follows. The problem at hand is to find an nth-order rational function approximation

G)

e-0.5s (1 - 2s)(0.5s + 1)

The proposed design gives τ0 ) 0.6 and

KPID ) 3.1308 +

1.5651s 0.7452 + s 1.5651 s+1 N

(32)

with E ) 0.47%. The PI-PD controllers of ref 12 are 0.937[1 + (1/1.339s)] and 2.328(0.53s + 1). The PID-P controllers of ref 11 are 0.561[1 + (1/1.165s) + 1.478s] and Kf ) 1.687. The closed-loop responses are shown in Figure 6. Example 3. Consider an unstable process:22

G)

e-0.5s (1 - 5s)(2s + 1)(0.5s + 1)

It follows that τ0 ) 1 and

KPID ) 4.3794 +

0.5106 7.2571s + s 7.2571 s+1 N

(33)

K ˆ )

with E ) 26.07%, which cannot fulfill the accuracy threshold, and the closed-loop response is very poor. Then τ is adjusted to τ1 ) τ0 + L ) 1.5 according to the proposed tuning rule (29). The new τ results in

KPID ) 2.9886 +

4.6668s 0.2335 + s 4.6668 s+1 N

(34)

which has E ) 2.06% and meets the specified approximation accuracy E e 5%. The corresponding controller in ref 21 is

(

KPID ) 6.1859 1 +

1.4724s 0.1395 + s 1.4724 s+1 N

)

(35)

bnsn + bn-1sn-1 + ... + b1s + b0 sn + an-1sn-1 + ... + a1s

(36)

with an integrator such that M

J(k) }

|W(jωi) [K ˆ (jωi) - K(jωi)]|2 ∑ i)1

(37)

is minimized, where k denotes the index for the kth recursion in the iterative weighted linear least-squares method, and let W h (jω) ) W(jωi)/(jωi)n + an-1(k-1)(jωi)n-1 + ... + a1(k-1)(jωi), which operates as a weighting function in the standard least-squares problem and depends on the parameters generated in the last recursion. In this paper, the W h is chosen as 1 and standard LS is applied in each iteration. On convergence, the resultant parameters will form one solution to (37).

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Figure 8. Robust performance with a gain uncertainty of 50% for example 3: (s) proposed high order (τ ) 1); (- -) Huang and Chen.

Like the LS algorithm, the frequency range in RLS is also chosen as (0.1ωb, ωb) with the step of (1/100-1/ 10)ωb. In this range, RLS yields satisfactory fitting results in the frequency domain. The above algorithm deals with the problem of approximating a given, probably nonrational, transfer function by a rational function. Error bounds for such an approximation have been investigated.24,25 Authors in ref 25 proposed the approach based on weighted leastsquares estimation and provided hard frequency-domain transfer function error bounds. However, it is not easy to calculate such a bound, and the convergence of estimation has not been addressed. In our work, we use a maximum likelihood index E to evaluate the approximation accuracy and assume that the accuracy threshold can be achieved when the controller order is high enough. When a τ is chosen, we first find the PID controller with the standard least-squares method and evaluate the corresponding approximation error E in (28) as described in the preceding section. If E cannot achieve the specified approximation accuracy  (usually 5%), we recommend the high-order controller in (36) and start from a controller order of 2 until the smallest integer n such that E e . Tuning Procedure. Step 1. Find the smallest τmin from (14) and (25) and let τ0 ) τmin. Find the largest τmax from (21) if the plant uncertainty δG is given. Step 2. Find the PID controller from (27) and evaluate the corresponding approximation error E in (28). If E achieves the specified approximation accuracy  (usually 5%), end the design. Step 3. Otherwise, we have two ways to solve this problem: if the PID controller is desired, update τ by (29) and go to step 2 when τ < τmax; otherwise, go to step 4. Step 4. Adopt the high-order controller in (36) and start from a controller order of 2 until the smallest integer n is reached with E e . Example 3 (continued). Reconsider

G)

e-0.5s (1 - 5s)(2s + 1)(0.5s + 1)

for which τ0 ) 1 and a PID has been obtained there with

Figure 9. Step response for example 4 with τ ) 2.7: (s) proposed high order; (‚‚‚) proposed PID.

E ) 17.46%. For a high-order controller, our procedure ends with

K ˆ )

29.8188s3 + 75.8869s2 + 39.4556s + 4.3412 s3 + 3.2042s2 + 8.5984s (38)

with the fitting error E less than  ) 5%. The closedloop step responses are shown in Figure 7. We can see that the new controller K ˆ restores the IMC performance, while the previous PID controllers in both (33) and (35) are not capable of that under such a tight performance specification. If a 50% perturbation in gain is introduced, i.e., δG ) 0.5, (21) results in δp ) 1.8804 with δk ) 0.03. The proposed tuning procedure gives τmin ) 1 and τmax ) 2.62, so that the high-order controller in (38) with τ0 ) 1 can achieve robust stability. Figure 8 shows the resulting performance, which is still stable and more robust than that of the method of ref 21. Example 4. Consider an unstable process:

G)

e-1.2s (1 - s)(0.5s + 1)

It follows that τ0 ) 2.7 and

KPID ) 1.0134 +

1.0155s 0.0063 + s 1.0155 s+1 N

(39)

with E ) 18.50%, which cannot fulfill the accuracy threshold, and the closed-loop response is very poor. One can detune the PID controller by increasing τ. However, this results in a sluggish response. For a high-order controller, our procedure ends with K ˆ ) 5.2221s4 + 41.6265s3 + 128.3411s2 + 131.1146s + 0.7798 s4 + 14.0061s3 + 9.6026s2 + 123.0987s (40)

with the fitting error E less than  ) 5%. The closedloop step responses are shown in Figure 9. We can see that the high-order controller makes a significant improvement over the PID controller. To the authors’

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knowledge, the PID controller design methods in the literature do not apply to this example.11,12,21 4. Conclusion An IMC-based single-loop controller design method is given to find the feedback controller in either PID or high-order form. The performance limitation of implementing the IMC controller in an equivalent feedback structure for an open-loop unstable process has been addressed. The proposed method is generally applicable to a wide range of processes and can be made automatic for online tuning. Extensive simulations have been performed to show that the proposed method gives a consistent and satisfactory performance over a large class of processes. Literature Cited (1) De Paor, A. M.; O’Malley, M. Controllers of Ziegler-Nichols type for unstable process with time-delay. Int. J. Control 1989, 49 (4), 1273-1284. (2) Leonard, F. Optimum PIDs controllers, an alternative for unstable delayed systems. Proceedings of the IEEE International Conference on Control Applications, Glasgow, U.K., 1994; Vol. 2, pp 1207-1210. (3) Shafiei, Z.; Shenton, A. T. Tuning of PID-type controllers for stable and unstable systems with time-delay. Automatica 1994, 30 (10), 1609-1615. (4) Venkatashankar, V.; Chidambaram, M. Design of P and PI controllers for unstable first-order plus time-delay systems. Int. J. Control 1994, 60 (1), 137-144. (5) Poulin, E.; Pomerleau, A. PID tuning for integrating and unstable processes. IEE Proc. Control Theory Appl. 1996, 143 (5), 429-435. (6) Chidambaram, M. Control of unstable systems: a review. J. Energy, Heat Mass Transfer 1997, 19, 49-56. (7) Ho, W. K.; Xu, W. PID tuning for unstable processes based on gain and phase-margin specifications. IEE Proc. Control Theory Appl. 1998, 145 (5), 392-396. (8) Tan, W.; Yuan, Y.; Niu, Y. Tuning of PID controller for unstable process. Proceedings of the IEEE International Conference on Control Applications, Kohala Coast, HI, 1999; Vol. 1, pp 121124. (9) Datta, A.; Ho, M.-T.; Bhattacharyya, S. P. Structure and Synthesis of PID Controllers; Springer: New York, 1999. (10) Visioli, A. Optimal tuning of PID controllers for integral

and unstable processes. IEE Proc. Control Theory Appl. 2001, 148 (2), 180-184. (11) Park, J. H.; Sung, S. W.; LEE, I. B. An enhanced PID control strategy for unstable processes. Automatica 1998, 34 (6), 751-756. (12) Majhi, S.; Atherton, D. P. Online tuning of controllers for an unstable FOPDT process. IEE Proc. Control Theory Appl. 2000, 147 (4), 421-427. (13) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: London, 1989. (14) Chien, I. L. IMC-PID controller designsan extension. IFAC Proceedings Series; IFAC: 1988; Vol. 6, pp 147-152. (15) Wang, Q.-G.; Hang, C. C.; Yang, X.-P. Single-loop controller design via IMC principles. Automatica 2001, 37 (12), 2041-2048. (16) Rotstein, G. E.; Lewin, D. R. Simple PI and PID tuning for open-loop unstable systems. Ind. Eng. Chem. Res. 1991, 30 (8), 1864-1869. (17) Walton, K.; Marshall, J. E. Direct method for TDS stability analysis. Proc. IEE 1987, 134 (part D), 101-107. (18) Kuo, B. C. Automatic Control Systems; Prentice Hall: Englewood Cliffs, NJ, 1995. (19) Doyle, J. C.; Wall, J. E.; Stein, G. Performance and robustness analysis for structured uncertainty. Proceedings of the IEEE Conference on Decision and Control, Orlando, FL, 1982; Vol. 2, pp 629-636. (20) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control: Analysis and Design; Wiley: New York, 1996. (21) Huang, H.-P.; Chen, C.-C. Control-system synthesis for open-loop unstable process with time delay. IEE Proc. Control Theory Appl. 1997, 144 (4), 334-346. (22) Huang, C. T.; Lin, Y. S. Tuning PID controller for openloop unstable process with time delay. Chem. Eng. Commun. 1995, 33, 11-30. (23) Pintelon, R.; Guillaume, P.; Rolain, Y.; Schoukens, J.; Van Hamme, H. Parametric identification of transfer functions in the frequency domain-a survey. IEEE Trans. Autom. Control 1994, 39 (11), 2245-2260. (24) Yan, W.-Y.; Lam, J. An approximate approach to H2 optimal model reduction. IEEE Trans. Autom. Control 1999, 44 (7), 1341-1358. (25) Wahlberg, B.; Ljung, L. Hard frequency-domain model error bounds from least-squares like identification techniques. IEEE Trans. Autom. Control 1992, 37 (7), 900-912.

Received for review October 1, 2001 Revised manuscript received May 22, 2002 Accepted May 22, 2002 IE010812J