Lag Compensator Design for Unstable Delay ... - ACS Publications

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Ind. Eng. Chem. Res. 2011, 50, 1330–1337

Lead/Lag Compensator Design for Unstable Delay Processes Based on New Gain and Phase Margin Specifications Zhuo-Yun Nie,†,‡ Qing-Guo Wang,*,‡ Min Wu,† Yong He,† and Qin Qin‡ School of Information Science and Engineering, Central South UniVersity, Changsha 410083, China, Department of Electrical and Computer Engineering, National UniVersity of Singapore, 119260, Singapore

This paper considers a lead/lag compensator design problem for a class of unstable delay processes based on a new set of gain and phase margin specifications. Due to the nature of the unstable system, both upper and lower gain margins are required to measure the true stability robustness with regard to gain change. In addition to a phase margin, such a combined margin problem leads to a set of nonlinear and coupled equations which have no analytical solution. Thus, an effective graphical method is developed such that the solution is determined from the intersections of the curves constructed by a transformed set of nonlinear equations. The tuning procedure is presented and examples are given for its illustration and comparison. 1. Introduction Unstable processes are encountered in industry, and examples include batch chemical reactors and the combination of a feed/ effluent heat exchanger.1 They are known to be very difficult to control, because of the nature of the open loop instability. This control problem has attracted a lot of research attention recently, typically by using the proportional, integral, and derivative (PID) controller or its special cases.2-8 A common control scheme for such processes consists of inner and outer loops,9,10 which is shown in Figure 1. The PD controller in the inner loop plays the role of stabilizing a given unstable process. Then, the PI controller of the outer loop is tuned for performance. As all know, the PD controller, kp + kds, is not a proper transfer function and thus not physically realizable. In order to obtain a proper and thus physically realizable controller, one has to multiply with a first-order filter 1/(Nkds + 1) where 0 < N e 0.01, to the derivative term of the PD controller, so that the practical D action becomes kds/(Nkds + 1). Then, the resulting implementable PD controller is given by [(Nkp + 1)kds + kp]/(Nkds + 1), which is actually in the form of a lead/lag compensator. Further, the parameter N in the filter is chosen on the basis of experience with the present art. Therefore, these two steps may bring in significant design errors which may make the implemented control system’s stability and performance deviate substantially from the designed system. In view of the above observations, it is more reasonable and accurate to use the lead/lag compensator in the inner loop and design it for proper stability margins without approximations, leaving the outer loop to cater to performance as usual. Note that for the control of an unstable process, the first and foremost problem is to stabilize it. No performance can be achieved without closedloop stability. Therefore, this paper addresses the stabilization of an unstable delay process by the lead/lag compensator. Stability robustness is a key issue in the process control and is often measured by gain and phase margins. The controller design based on gain and phase margin specifications is common and is widely used in practice.11-17 Note that a lead/lag compensator has some parameters in the denominator and differs from a PID controller in which all the parameters appear linearly. * Author to whom correspondence should be addressed. Tel.: (+65)6516 2282. Fax: (+65)6779 1103. E-mail: [email protected]. † Central South University. ‡ National University of Singapore.

Thus, the effective tuning techniques for PID control11-17 are not applicable to lead/lag compensators. There have been reported works on gain and phase margin design methods for the tuning of lead/lag compensators, but they were developed only for the stable processes.18-21 To the best of our knowledge, no method is available in the literature to design a lead/lag compensator, which can make the control system achieve the desired gain and phase margins for unstable processes. But this paper is not just to make some simple extension of the existing tuning methods of lead/lag compensators using gain and phase margins for unstable processes. Some fundamental change is needed to formulate a meaningful gain and phase margin design problem for unstable processes. This is because a single gain margin specification which has been used in the literature exclusively is inadequate to measure stability robustness for an unstable process, see Figure 2b, where the left intersection point of the open-loop Nyquist curve with the real axis also contributes to the stability of the closed loop. Without knowing its location, no stability or margin can be determined. Hence, we here propose for the first time that the gain margin specifications j for an unstable process should be two-sided: an upper limit A

Figure 1. The inner-outer loop feedback control.

Figure 2. (a) Normal stable open loop. (b) Some unstable open loop.

10.1021/ie1007322  2011 American Chemical Society Published on Web 11/09/2010

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sees from Figure 2 that in order to form such an anticlockwise encirclement, the three frequencies should satisfy ωp1 < ωg < ωp2. 3. The Proposed Solution

Figure 3. Unity output feedback system.

for gain increase, which is the normal-sense gain margin, and j for gain decrease, which is normally called gain a lower limit A reduction. This new formulation leads to a stabilizing gain j ), meaning that if the open-loop transfer function interval (A _ ,A is multiplied by a gain within this interval, the closed loop is guaranteed to remain stable. This is different from a normal stable process case, as shown in Figure 2a. In this paper, we formulate a controller design problem for unstable delay processes based on specifications of a gain margin, a gain reduction, and a single phase margin and present an effective graphical tuning method for lead/lag compensators which is to be placed in the inner loop of the control system for stabilization purposes. The rest of the paper is organized as follows. In section 2, the problem formulation is given. In section 3, the tuning method for a class of unstable processes is developed. In section 4, examples are given for illustration and comparison. Finally, section 5 draws conclusions.

A graphical method is useful to solve some nonlinear problems for which no analytical solution can be derived. Wang et al.20 developed a graphical method for lead compensator tuning to achieve gain and phase margins exactly, regardless of the plant order, time delay, or damping nature. However, that method assumes stable processes and is thus not suitable for our present case of unstable processes nor with one more condition, gain reduction. In this section, we develop a new graphical solution for the problem stated in section 2. The typical open-loop Nyquist curve of G(s)C(s) is given in Figure 4. There are two cases for its initial part from zero frequency. One is for m ) 0 with ωp1 ) 0, as shown in Figure 4a. In this case, the Nyquist curve starts from the real axis and moves immediately downward in the anticlockwise direction to encircle the critical point. Since the phase of G(jω)C(jω) is always larger than -π at frequency ω ∈(0,ωg], we have [∠G(j0+)C(j0+) + π][∠G(jωg)C(jωg) + π] > 0

2. Problem Formulation and Preliminaries Take for consideration only the inner loop in Figure 1, which gives the unity output feedback configuration depicted in Figure 3. Denote the process and compensator transfer functions by G(s) and C(s), respectively. Let the unstable process be G(s) )

g(s) exp(-Ls) sm(s - 1)

C(s) ) Kc

1 + T1s 1 + T 2s

[∠G(j0+)C(j0+) + π][∠G(jωg)C(jωg) + π] < 0 We first consider the case of ωp1 ) 0. Submit ωp1 ) 0 into eq 3; then we can get Kc )

(2)

where T2 ) RT1 > 0 and 0 < R < 1 for the case of the lead compensator and 1 < R for the case of the lag compensator. j ) and phase Given the specifications of the gain interval (A _ ,A margin φm (0 < φm < π/2), we may choose a lead/lag compensator to meet three resulting margin conditions. This imposes the following complex equations: Kc

1 + jωp1T1 1 G(jωp1) ) 1 + jωp1T2 A _

(3)

Kc

1 + jωp2T1 1 G(jωp2) ) j 1 + jωp2T2 A

(4)

1 + jωgT1 Kc G(jωg) ) -exp(jφm) 1 + jωgT2

The other case is ωp1 * 0, as shown in Figure 4b, where the Nyquist curve first moves upward in the clockwise direction for m ) 0, or m ) 1 in eq 1. One can see that the phase of G(jω)C(jω) is always less than -π at frequency ω ∈(0,ωp1) but larger than -π at frequency ω ∈(ωp1,ωg], which leads to

(1)

where g(s) is a minimum phase and stable rational transfer function, m ) 0, or m ) 1, and L > 0. The controller is the lead/lag compensator:

1 -A _ G(0)

(7)

Complex eqs 4 and 5 are equivalent to the following four real equations:

[

Re -

[

Im -

]

[

]

1 1 - T2ωp2 Im )1 j j KcAG(jωp2) KcAG(jωp2)

]

[

]

(8)

1 1 + T2ωp2 Re ) ωp2T1 j j KcAG(jωp2) KcAG(jωp2)

[

Re -

(5)

where ωp1 and ωp2 are two phase crossover frequencies and ωg is the gain crossover frequency. Three equations involve six unknowns, that is, three compensator’s parameters and three crossover frequencies. Since an unstable pole is present in the process (eq 1), an anticlockwise encirclement of the critical point by the open-loop Nyquist curve is necessary for closed-loop stability, according to the Nyquist stability theorem. One readily

(6)

[

Im -

]

[

]

exp(jφm) exp(jφm) - T2ωg Im )1 KcG(jωg) KcG(jωg)

]

[

]

(9)

(10)

exp(jφm) exp(jφm) + T2ωg Re ) ωgT1 KcG(jωg) KcG(jωg)

(11)

Observe that eqs 8-11 are linear equations with regard to T1 and T2. We solve two equations each to express T1 and T2 as functions of ωp2 and ωg, respectively, as follows

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{ {

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

Figure 4. (a) G(s)C(s) with ωp1 ) 0. (b) G(s)C(s) with ωp1 * 0.

( [

]

[

1 1 1 Im2 + Re2 j G(jωp2) G(jωp2) KcA T1 ) 1 ωp2Im G(jωp2) 1 1 Re -1 j G(jωp2) KcA T2 ) ωp2 1 Im j G(jωp2) KcA

[

]

[

( [

[

]

]

[

- Re -

] [ [

1 G(jωp2)

-

Re -

[

{

]

[

]

1 1 - T2ωp1 Im )1 KcA _ G(jωp1) KcA _ G(jωp1)

]

[

(16)

]

1 1 + T2ωp1 Re ) ωp1T1 Kc A _ G(jωp1) KcA _ G(jωp1) (17)

and solve for T1 and T2 in terms of ωp1 as

(12)

]) [ ]

exp(jφm) - Re G(jωg)

]

]

(13)

One set of two equations, say 12, gives a relationship between T1 and T2 via intermediate variables, ωp2 and ωg, and can be plotted as a curve in a plane with T1 as the horizontal axis and T2 as the vertical axis. We do so for two sets in eqs 12 and 13 and plot the resulting two curves in the same plane. The intersection point will meet both sets, which is the solution of our design problem. The values of designed T1 and T2 can be read off directly from the intersection point. The corresponding intermediate variables ωp2 and ωg are obtained as well. Furthermore, the computational efforts can be substantially reduced by limiting possible frequency ranges for ωp2 and ωg via which eqs 12 and 13 are plotted. Note that the controller (eq 2) with non-negative parameters can only give a phase change in the range from -π/2 to +π/2. It follows from eqs 4 and 5 that the solution will occur only in the frequency ranges of ωp2 and ωg, bounded implicitly by 3π π < G(jωp2) < 2 2

(14)

3π π + φm < G(jωg) < - + φm 2 2

(15)

-

[

]

Im -

]

exp(jφm) exp(jφm) 1 Im2 + Re2 Kc G(jωg) G(jωg) T1 ) exp(jφm) ωgIm G(jωg) exp(jφm) 1 Re -1 Kc G(jωg) T2 ) ωg exp(jφm) Im Kc G(jωg)

[

]) [

respectively. Thus, we plot eqs 12 and 13 with respect to ωp2 and ωg in the ranges given in eqs 14 and 15. By a similar treatment to that of eqs 4 and 5 as done above, we transform the complex eq 3 into the two real equations:

( [

]

[ [

]) [ ]

1 1 1 Im2 + Re2 KcA _ G(jωp1) G(jωp1) T1 ) 1 ωp1 Im G(jωp1) 1 1 -1 Re KcA _ G(jωp1) T2 ) ωp1 1 Im KcA _ G(jωp1)

[

]

[

]

- Re -

1 G(jωp1)

]

(18)

And the possible frequency range of ωp1 for the solution is limited by -

3π π < G(jωp1) < 2 2

(19)

However, unlike the case of ωp1 ) 0, this new case leaves Kc unknown yet. Thus, one needs to find a suitable Kc in order to plot eqs 12, 13, and 18 in the same plane to enable us to find the solution if any. To this end, we choose a stabilizing gain as a good initial value and then refine it through some simple iteration. To set the initial value for Kc, one sees in the Introduction that the PD controller is implemented by a lead/lag compensator. Theoretically, we can design a PD controller kp(1 + kds) to stabilize the process (eq 1), say using the method of Nie et al.22 Then, Kc ) kp is taken as the initial value, which will be much better than a blindly selected value, say the default value of zero in most search algorithms. In some sense, we start with a theoretical controller and try to find a practical controller by some search algorithm. With the above chosen value of Kc, one can plot the graphs of eqs 12, 13, and 18 in the same diagram. Three curves usually do not have a common intersection but yield three intersections each for a pair of curves. Thus, we need to adjust Kc in such a way that the three intersections move close to each other and eventually merge to one point, which is the desired solution. It is easily seen that the functions defined by eqs 12, 13, and 18

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j , and have the same structure but with different parameters A _, A exp(jφm), implying that the three curves will move in the same direction when Kc changes. In addition, since the parameters j < |exp(jφm)| < 1/A _ , the curve for eq 18 will move satisfy 1/A faster than other two with the same amount of change of Kc. In the T1/T2 plane, the curve for eq 18, which achieves gain reduction A _ exactly, will divide the plane into two parts. One part is larger than A _ , denoted by A _ +, and the other part is smaller _ have the than A _ , denoted by A _ . Note that changes of Kc and A same effect on eq 18. Thus, the curve for eq 18 will move to the A _ - side when Kc decreases. If the intersection of eqs 12 and 13 lies on the A _ - side, one should reduce Kc; otherwise, one should increase it. In this way, the curve for eq 18 will move closer to the intersection of eqs 12 and 13, until the three intersections merge to one. Recall that there are two possible cases for our design problem, ωp1 ) 0 and ωp1 * 0. Obviously, an integral process falls at ωp1 * 0 since the Nyquist curve of its open loop with the compensator does not start from the real axis. To determine which case a nonintegral process is in, we first assume the case ωp1 ) 0 and find its solution. Then, we check if eq 6 holds true. If the answer is YES, then it is indeed the case of ωp1 ) 0, and the design ends. Otherwise, we have the case ωp1 * 0 and find the solution accordingly. It should be pointed out that the procedure developed above is to find the solution of eqs 3-5. If the solution exists with ωp1 < ωg < ωp2, T1 > 0, T2 > 0, and the magnitude of the process frequency response decreases monotonically after ωp2, then the compensator obtained stabilizes the process, indeed, and those prespecified stability margins are actually achieved. Otherwise, one may need to adjust specifications and try it again. Recall that for a stable process, the typical gain and phase margins used to tune a PID controller are between 2 and 5, and between 30° and 60°,14 respectively. However, these margins would be difficult or impossible to achieve for an unstable process because the feedback is first to stabilize the process and then to have stability margins. It is worth mentioning that in certain situations, such as with a very large delay in the process, no PID or lead/ lag compensator exists for stabilization, and thus there will be no solution for design problems with gain and phase margin specifications. Hence, the process (eq 1) discussed in this paper must be stabilizable in order for any control design to have a solution. One can use, for instance, the methods given by Lee et al.5,23-25 to check if the process can be stabilized. If yes, then one can proceed to our design. Otherwise, there is no solution. For the stabilizable case, we find that the phase margin between 15° and 30° is appropriate for an unstable process, that is, half for the stable case. Further, it follows from Nie et al.’s method22 that one can calculate the stabilizing gain range for a given unstable process and choose suitable gain margins for the given controller. With the chosen gain and phase margins, our design method is invoked to find the controller solution. 4. Examples Example 1: Consider the process26,27 given by G(s) )

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

The gain and phase margins are set to (0.5, 2) and 37°, respectively. Assume the case of ωp1 ) 0 first. Kc ) 2 is computed by eq 7. The graphs of eqs 12 and 13 are plotted with respect to ωp2 and ωg in Figure 5, exhibiting one intersection P(0.7462, 0.143). Its corresponding frequencies are

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Figure 5. Plot of functions 12 and 13.

Figure 6. Nyquist plot of G(s)C(s).

ωp2 ) 2.2436 and ωg ) 0.982. Check eq 16, and see that it holds. Thus, the solution of eqs 3-5 is T1 ) 0.7462, T2 ) 0.143, and Kc ) 2, which gives the resultant compensator as C(s) ) 2

0.7462s + 1 0.143s + 1

Obviously, the solution satisfies T1 > 0, T2 > 0, and ωp1 < ωg < ωp2, and the process frequency response has a monotonic decreasing magnitude over the frequency. The closed loop should meet the Nyquist criterion and is stable. This is the case, indeed, as shown in Figure 6, where all of the prespecified margins are achieved exactly by the proposed method. Under the control structure in Figure 1, we use C(s) as the inner loop controller and design the PI controller for the outer loop one using Fung et al.’s method,28 which is also based on the performance specifications in terms of gain and phase margins. Set the gain and phase margins to be 3 and 60°, respectively. The outer controller is computed as Co(s) ) 0.155 +

0.314 s

The output time responses for the unit step set-point and disturbance with a magnitude of 0.1 are shown in Figure 7. It is observed that the proposed method gives a much better response in terms of a smaller overshoot, shorter settling time, and better disturbance rejection compared with Park et al.’s26

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in the case of ωp1 * 0. With Nie et al.’s22 method, we find Kc ) 0.5 and set it as the initial value. The curves of eqs 12, 13, and 18 are plotted with dash and dot lines in Figure 11 and are marked in green, red, and blue, respectively. To find the right _ to 0.32, and plot the direction of search for Kc, try to reduce A curve of eq 18 again with the blue thin line. One can find that the intersection point P of eqs 12 and 13 lies in the A _ + region, and we need to increase Kc to let the curve for eq 18 move close to the point P. Set the step size ∆Kc ) 0.1 and increase Kc to 0.5 + ∆Kc. Plot the curves of 12, 13, and 18 again, which are marked by a dash line in Figure 11. The three intersection points move closer than before. Then, continue this procedure with a few iterations. The results are shown in Table 2. In Figure 11, the three intersections almost merge as one point P(4.772, 0.0046) when Kc ) 0.7. This yields the resultant controller as Figure 7. Step responses for example 1 (nominal).

C(s) ) 0.7

4.772s + 1 0.0046s + 1

Check that the solution meets T1 > 0, T2 > 0, and ωp1 < ωg < ωp2. The Nyquist curve of G(s)C(s) is given in Figure 12. The achieved gain and phase margins are (0.3296, 2.15) and 32°, respectively. Once again, we design the outer-loop PI controller using Fung et al.’s28 method and with gain and phase margins set 1.8 and 55°, respectively. It produces Co(s) ) 0.3026 +

Figure 8. Step response for example 1 (perturbed).

method and Jacob and Chidambaram’s27 method. Suppose that the model is not accurate and the plant static gain changes to 0.8. The results are given in Figure 8. The three methods are all robust against this gain change with the proposed method having the best stability. Example 2: Consider an unstable process,11 G(s) ) 1/(s 1) e-0.2s. We compare the achieved gain and phase margins with Ho and Xu’s method, which is based on PI controller tuning for unstable processes. The results are depicted in Table 1, from which one sees that the proposed method can exactly achieve the specifications, whereas approximations are involved with Ho and Xu’s15 method. For example, a PI control, C ) 3.2(1 + 1/(1.16s)), results with Ho and Xu’s method in specifications of the gain margin of 2 and the phase margin of 20°, whereas a lag compensator C(s) ) 3.333(0.1005s + 1)/(0.25025s + 1) is obtained with the proposed method for the gain margins of (0.3, 2) and a phase margin of 20°. The Nyquist curves of 2C(s)G(s) and exp(-[(20°π)/180°]i)C(s)G(s) are plotted in Figures 9 and 10, respectively, where our case passes the critical point but his case does not. In addition, the gain reduction is not considered at all in Ho and Xu’s method. Without this measure, stability robustness is not guaranteed for unstable open loops, and instability in the closed loop may occur with small process perturbations. Example 3: Consider the integral and unstable process:29 G(s) )

1 e-0.2s s(s - 1)

In this example, the gain and phase margins are set to (0.33, 2.15) and 32°, respectively. This is an integral process and falls

1.3236 s

The output time responses for the step set point and disturbance are compared with those of Lee et al.29 in Figure 13, indicating that the proposed method has a faster response and better disturbance rejection than the latter. To test for robustness, suppose that the open-loop gain changes to 0.5. The results are given in Figure 14, which shows that the closedloop system from our design remains stable with a large margin, whereas it becomes unstable from Lee et al.’s method. Example 4: Consider a high-order unstable time delay process: G(s) )

(0.5s + 1) e-0.2s (s + 1)4(5s - 1)

The existing control design methods for unstable processes in the literature usually assume a low-order one and need to use model reduction for their applications. Note that model reduction causes errors in the frequency response, and such an error should be avoided if possible, since the stabilizing gain range for an unstable process is very limited. The proposed method has no restriction on process order and it utilizes frequency response directly without a need for any approximation. This example demonstrates such an advantage. Suppose the gain and phase margins to be (0.7, 2) and 20°, respectively. Assume the case of ωp1 ) 0 first. We obtain Kc ) 1.4286 from eq 7. Plot the graphs of eqs 12 and 13, as in Figure 15, which yields the intersection point P(2.278, 0.678). The two additional crossover frequencies are ωg ) 0.209 and ωp2 ) 0502, which satisfy ωp1 < ωg < ωp2. By checking eq 16, we find it is indeed the case of ωp1 ) 0. Then, we need not have any further iteration on Kc, and the above intersection produces the solution controller as C(s) ) 1.4286

2.278s + 1 0.678s + 1

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Table 1. Achieved Gain and Phase Margins Specified j) (A _ ,A

φm

T1

Our method T2

Kc

j ′) (A _ ′,A

φm′

j ′) (A _ ′,A

Ho and Xu’s method φm′

(0.3, 2) (0.3, 2) (0.3, 3) (0.3, 3) (0.5, 4) (0.5, 4) (0.6, 5)

20° 30° 30° 35° 30° 35° 30°

0.1005 0.0524 0.5941 1.3315 0.1678 0.2593 0.2118

0.25025 0.0919 0.8896 1.8925 0.2873 0.3225 0.3207

3.3333 3.3333 3.3333 3.3333 2 2 1.6667

(0.3, 2) (0.3, 2) (0.3, 3) (0.3, 3) (0.5, 4) (0.5, 4) (0.6, 5)

20° 30° 30° 35° 30° 35° 30°

(/, 2.01) (/, 2.02) (/, 3.03) (/, 3.03) (/, 4.04) (/, 4.04) (/, 5.05)

20.97° 31.25° 31.01° 36.38° 30.28° 36.05° 29.07°

The Nyquist curve of G(s)C(s) in Figure 16 shows that both gain and phase margins are exactly achieved. As in the previous examples, the outer-loop PI controller is designed with Fund et al.’s28 method under gain and phase margins of 2 and 50°, respectively, and found to be Co(s) ) 0.335 +

0.0478 s

loop gain changes to 0.8, the results are given in Figure 18, showing that the process is still well stabilized. 5. Conclusion This paper has for the first time formulated a controller design problem with a set of three frequency response specifications, namely, gain reduction plus normal gain and phase margins. This set of new specifications is necessary to ensure true stability

The output responses for the step set point and disturbance with a magnitude of 0.1 are shown in Figure 17. When the open-

Figure 11. Plot of functions 12, 13, and 18.

Figure 9. Nquist plot of 2C(s)G(s).

Figure 12. Nyquist plot of G(s)C(s). Table 2. Search for Kc

Figure 10. Nquist plot of exp(-[(20°π)/180°]i)C(s)G(s).

Kc

T1

T2

ωp1

ωg

ωp2

A _′

0.5 0.6 0.7

6.11 5.212 4.572

0.01549 0.0098 0.0046

0.504 0.54 0.612

2.888 2.969 3.049

6.461 6.638 6.808

0.3419 0.3356 0.3296

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Figure 13. Step response for example 3 (nominal).

Figure 16. Nyquist plot of G(s)C(s).

Figure 14. Step response for example 3 (perturbed).

Figure 17. Step response for example 4 (nominal).

Figure 15. Plot of functions 12 and 13.

Figure 18. Step response for example 4 (perturbed).

robustness for conditionally stable feedback systems, which is the case if the process is unstable, the case of consideration in this paper. We have further shown that a lead/lag compensator is preferred over a PD controller for stabilization purposes, as the former is directly implementable whereas the latter is not. We have also pointed out that any approximation for either the

process for simpler control design or the controller for physical realization is highly undesirable and should be avoided since the stabilizing controller gain range is limited for unstable delay processes. The resulting design problem leads to three nonlinear and coupled complex equations which have no analytical solution in general. Instead, an effective graphic method has been presented to find the solution from some intersection points

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of suitably constructed curves based on the frequency response of the process. The proposed method is intensively demonstrated with examples. Acknowledgment This work was supported in part by the National Science Foundation of China under grants 60974045 and 60425310, in part by the Hunan Provincial Natural Science Foundation of China under grant 08JJ1010, in part by the Program for New Century Excellent Talents in University under grant NCET-060679, in part by the Doctor Subject Foundation of China under grant 200805330004, and in part by a scholarship under the State Scholarship Fund, China. Literature Cited (1) William, L. L. External versus internal open-loop unstable processes. Ind. Eng. Chem. Res. 1998, 37, 2713–2720. (2) Xiang, C.; Wang, Q. G.; Lu, X.; Nguyen, L. A.; Lee, T. H. Stabilization of second-order unstable delay processes by simple controllers. J. Proc. Control 2007, 17, 675–682. (3) Shafiei, Z.; Shenton, A. T. Tuning of PID-type controllers for stable and unstable systems with time-delay. Automatica 1994, 30, 1609–1615. (4) Lu, X.; Yang, Y. S.; Wang, Q. G.; Zheng, W. X. A double twodegree-of-freedom control scheme for improved control of unstable delay processes. J. Proc. Control 2005, 15, 605–614. (5) Lee, S. C.; Wang, Q. G.; Xiang, C. Stabilization of all-pole unstable delay processes by simple controllers. J. Proc. Control 2010, 20, 235–239. (6) Uma, S.; Chidambaram, M.; Rao, A. S. Enhanced control of unstable cascade processes with time delay using a modified simith predictor. Ind. Eng. Chem. Res. 2009, 48, 3098–3111. (7) Kwak, H. J.; Sung, S. W.; Lee, I. B. On-line process identification and autotuning for integrating processes. Ind. Eng. Chem. Res. 1997, 36, 5329–5338. (8) Rao, A. S.; Chidambaram, M. Enhanced two-degrees-of-freedom control strategy for second-order unstable processes with time delay. Ind. Eng. Chem. Res. 2006, 45, 3604–3614. (9) Kaya, I. A PI-PD controller design for control of unstable and integrating processes. ISA Trans. 2003, 42, 111–121. (10) Padhy, P. K.; Majhi, S. Relay based PI-PD design for stable and unstable FOPDT processes. Comput. Chem. Eng. 2006, 30, 790–796. (11) Ho, W. K.; Xu, W. PID tuning for unstable processes based on gain and phase-margin specifications. IEEE Proc. Control Theory Appl. 1998, 145, 392–396. (12) Astrom, K. J.; Hagglund, T. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica 1984, 20, 645–651.

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ReceiVed for reView March 26, 2010 ReVised manuscript receiVed July 23, 2010 Accepted October 18, 2010 IE1007322