Performance Assessment and Controller Design Based on Modified

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Ind. Eng. Chem. Res. 2005, 44, 3538-3546

Performance Assessment and Controller Design Based on Modified Relay Feedback Ming-Da Ma* and Xin-Jian Zhu Fuel Cell Research Institute, Shanghai Jiaotong University, Shanghai, People’s Republic of China 200030

A performance assessment procedure and a controller design method based on modified relay feedback are proposed in this paper. The well-known robustness measurements gain and phase margins are used to assess the performance of the control system. The proposed scheme can online estimate the gain and phase margins by two successive relay tests to indicate the appropriateness of the controller parameters and to determine whether a retuning is necessary. A proportional-integral-derivative controller is tuned based on a least-squares fit of the actual to the desired closed-loop dynamic characteristic. Performance monitoring and controller design can be done simultaneously, which ensures a good performance of the control system. The illustrative simulation verifies the effectiveness of the presented algorithm. 1. Introduction The proportional-integral-derivative (PID) controller is widely used in process industries because of its simple structure and robustness to the modeling error. In process control, more than 90% of the control loops are PID type; most loops are actually proportionalintegral (PI) control loops.2 In the past decades, there have been numerous papers dealing with the tuning of PID controllers for different processes. However, according to the survey of Ender,23 the process control performance is, indeed, “not as good as you think”. Performance assessment and regular controller retuning are necessary. A natural question arises: how can the performance of a controller be assessed? In process control, minimum variance has been used as a criterion for assessing the closed-loop performance for decades.24,25 This criterion is a valuable measurement of the system performance, but it pays little attention to the traditional performance such as set-point tracking and disturbance rejection. Besides, another important factor of system performance, robustness, is not addressed directly. The performance measurement in a time domain is generally based on the errors in the controlled variable (e.g., integral absolute error or integral square error). However, time domain signals do not give a clear indication of the stability of the closed-loop system. Gain and phase margins have always served as important measurements of robustness. It is also known from classical control theories that the phase margin is related to the damping of the system and can, therefore, also serve as a performance measurement. The recommended ranges of gain and phase margins are between 2 and 5° and between 30 and 60°, respectively.3 Their solutions are normally obtained numerically or graphically by trial-and-error use of Bode plots. Ho et al.6,7 made a comparison of the gain-phase margins of some well-known PI and PID tuning methods. The gain and phase margins given by many methods vary as the process dead time to the time constant ratio increases from 0.1 to 1. Hence, their performances vary greatly for different processes. Over the years, many controller * To whom correspondence should be addressed. Tel.: +8621-62932154. Fax: +86-21-62932154. E-mail: [email protected].

tuning methods are available for achieving the specified gain and phase margins.5,6 Most of them use simple models such as the first-order plus dead-time (FOPDT) model and the second-order plus dead-time (SOPDT) model. Recently, much research effort has been focused on the automatic tuning of PID controllers, which was first proposed by A° stro¨m and Ha¨gglund.1 The standard relay feedback system is shown in Figure 1. The relay feedback test is carried out under closed-loop control so that, with an appropriate choice of the relay parameters, the process can be kept close to the operating point. Thus, the tuning procedure almost does not disturb the normal working of the system. It is quite efficient and easily understood. Therefore, it is widely used in process control. Many modifications and extensions of relay autotuning technique have been proposed. Enhanced autotuning techniques for the PID controller were developed20-22 to yield better performance. Relay feedback methods have been used for process identification too. One of the earliest applications of the autotuning procedure to chemical processes was by Luyben,11 who proposed a method (known as the ATV method) to obtain transfer function models for highly nonlinear distillation columns. The improved ATV method of Wei et al.12 removes the need for the steady-state gain to be known a priori, by using one or more additional relay tests. Scali et al.13 extend the improved ATV method to completely unknown processes. It is known as the ATV+ method. Shen et al.18 and Wang et al.19 derived loworder models using biased relay and analytical methods. Autotuners for advanced controllers such as the Smith predictor controller and a finite spectrum assignment controller were developed.9,10 These controllers are useful for processes with complex dynamics, e.g., long dead-time, oscillatory, and unstable dynamics. There are also attempts to extend the autotuning technique to multivariable systems. Methods proposed by Loh et al.14 and Shen and Yu15 are a combination of sequential loop closing and relay tuning. The basic idea is that the multivariable design problem is treated as a sequence of single-input single-output design problems. Plamor et al.16 and Wang et al.17 proposed the tuning of multivariable systems by decentralized relay feedback

10.1021/ie048831r CCC: $30.25 © 2005 American Chemical Society Published on Web 04/12/2005

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3539

Figure 1. Standard relay feedback system.

Figure 2. Modified relay feedback system.

(DRF). DRF is a complete closed-loop test compared with sequential relay feedback. Closed-loop testing is more preferable. However, the response of the multivariable system under DRF is very complex and was not completely clear until now. Recent progresses in relay feedback are summarized by Yu.26 In this paper, a performance assessment procedure based on modified relay feedback is proposed. Gain and phase margins are estimated by two relay tests. A linear assumption about the amplitude of the limit cycle is made, and the accuracy of the assumption is examined. A retuning of the controller is performed if the performance is poor. The PID controller is tuned based on a least-squares fit of the actual to the desired closed-loop dynamic characteristic. It is assumed that a stable (and conservative) controller is established prior to the retuning. Several examples are given to illustrate the effectiveness of the proposed performance assessment procedure and controller design method. This paper is organized as follows. Section 2 introduces the modified relay feedback structure. An online performance assessment procedure is proposed in section 3. Gain and phase margins are estimated by two relay tests. Problems such as the accuracy of the proposed algorithm and limitations are also discussed. Section 4 presents a PID controller design method based on the least-squares method. Simulation examples follow in section 5 to demonstrate the proposed performance assessment procedure and PID controller design method. Conclusions are drawn in section 6. 2. Modified Relay Feedback Structure The proposed modified relay feedback structure is shown in Figure 2. The signals r, ui, u, and y are the reference, relay output, control, and plant output, respectively. In the modified relay feedback structure, the controller is always connected in line with the process and we do not have to add an offset signal at the process input to achieve the desired set point. It can ensure limit-cycle oscillation even for a low-order process. Compared with standard relay feedback, the most important characteristic of the proposed scheme is that

it can assess the performance of the closed-loop system by online estimatation of gain and phase margins to determine whether a retuning of the controller is necessary when the performance of the control system deteriorates because of changes in the operating conditions or for other reasons, as demonstrated in section 3. 3. Performance Assessment Generally speaking, the controller is designed according to a linear model of the process around the operating point. However, chemical processes are actually nonlinear in nature, and they are often operated at different operating conditions. It is difficult for a fixed controller to work well at all operating points. Therefore, regular performance assessment and controller retuning are necessary. As mentioned above, gain and phase margins are good measurements for the performance and robustness. It is very tedious to calculate gain and phase margins in traditional ways. It is highly desirable to find a procedure for online monitoring of gain and phase margins. Here we provide an algorithm that can online estimate gain and phase margins by relay tests to assess the performance of the control system. 3.1. Gain Margin Estimation. Consider the modified relay feedback system shown in Figure 2. Let the loop transfer function be GL(s) ) Gc(s) Gp(s), where Gc(s) and Gp(s) are the controller and process transfer functions. The system starts to oscillate if the system has a phase lag of at least 180°. The closed-loop characteristic equation is

1 + N(a) GL(jω) ) 0

(1)

where N(a) is the describing function of the nonlinear element and a is the amplitude of the limit cycles. For an ideal relay, the describing function is N(a) ) 4d/πa, where d is the relay output amplitude. The oscillating point is the intersection of the process Nyquist curve and the negative real axis in the complex plane, which is traditionally called the critical point, as shown in

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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005

Figure 5. Estimated phase margin with the hysteresis relay.

Figure 3. Gain margin estimation.

However, the method presented above can only estimate the phase margin roughly. Here we propose a more accurate method that can estimate the phase margin by two relay tests. For the modified relay feedback system shown in Figure 2, the condition for the limit cycle is now given as

4d -j sin-1(/a) GL(jω0) ) -1 e πa

(5)

Hence Figure 4. Input-output characteristic of the hysteresis relay.

Figure 3. The critical gain and critical frequency can be calculated as1

Kc )

1 4d ) , ωc ) 2π/Tp |GL(jω)| πa

(2)

where Tp is the period of the limit cycle. Kc is equal in size to the gain margin Am. 3.2. Phase Margin Estimation. Using a relay with hysteresis, we can estimate the phase margin approximately with one relay test. The input-output characteristic of the hysteresis relay is shown in Figure 4. The negative inverse of the describing function of a relay with hysteresis is

-1 πxa - π )-j 4d 4d N(a) 2

2

(3)

where represents hysteresis of the relay. Assume the desired phase margin is φm. From eq 3 and Figure 5, we can calculate

)

4d sin φm π

|GL(jω0)| )

πa , ∠GL(jω0) ) -π + sin-1(/a) (6) 4d

According to the definition, the phase margin is determined by the intersection point of the process Nyquist curve and the unit cycle. Therefore, if we can determine the settings of the relay that make the amplitude of the oscillation point equal to 1, then we can calculate the phase margin from the relay settings. Chiang and Yu8 used relay tests to identify the maximum closed-loop log modulus. They assumed that the amplitude of the limit cycle is a function of as well as GL(s)

a ) m + a0 where m is the slope, which is characterized by GL(s), and a0 is the amplitude of the oscillation when an ideal relay is applied. Exact expressions for the periods and amplitudes of the limit cycles under relay feedback are derived for processes that may be modeled by FOPDT dynamics Gp(s) ) Ke-Ls/(Ts + 1).19 The expression for a is given by

a ) dK(1 - e-L/T) + e-L/T

(7)

(4)

Perform a relay test with the above-calculated hysteresis value. If the process phase margin is bigger than desired, Gp(jω) and the describing function will intersect at point C, and if the process phase margin is smaller than desired, the intersection point would be D, as shown in Figure 5. The positions of C, D, and B can be determined by a comparison of the real part of the intersection point χ ) -πxa2 - 2/4d and the real part of point B -cos φm.

From eq 7, we can see that the amplitude of the limit cycles is determined by relay parameters d and and the process itself, K(1 - e-L/T), e-L/T for the FOPDT process. Therefore, we assume that

a ) R + βd

(8)

for the general process, where the parameters R and β are characterized by the process. This assumption turns out to be accurate for most processes. The simulation below proves eq 8 to be reasonable and accurate.


The amplitude of the limit cycles a is a function of both d and according to eq 8 for a specific process. Hence, we can change d to adjust a. On the other hand, we can also change to adjust a. Here we fix d and adjust a by changing . The parameters R and β can be measured by relay tests. After R and β are determined, we can calculate d from the corresponding ad, which makes the amplitude of the oscillating point equal to 1. Then the phase margin of the control system can be calculated. Therefore, the procedures to estimate the phase margin can be summarized as follows. (1) Perform an ideal relay test ( ) 1 ) 0) and record a1. Calculate β from a1 and d.

β ) a1/d

(9)

Table 1. Phase Margin Estimation for Different Processes phase margin process

real

estimated A

estimated a

-0.5s

e s(s + 2)

62.43

62.92

53.23

0.3e-0.9s s(s + 1)2

44.53

42.51

37.30

0.5e-0.5s s(s + 1)3

3.94

4.96

3.79

0.7e-0.1s s(100s + 1)

6.36

7.27

7.48

19.68

17.60

17.56

5e-0.1s s +s+5 2

(2) Perform a second relay feedback experiment with hysteresis ) 2 and record the amplitude a2. Calculate R from a2, d, β, and 2.

R)

a2 - βd 2

(10)

(3) Calculate d from the corresponding ad, which makes the amplitude of the oscillating point equal to 1

d )

ad - βd R

(11)

where

ad ) 4d/π

(12)

(4) Calculate phase margin φˆ m from the data obtained above.

φˆ m ) sin-1(d/ad)

(13)

3.3. Improving the Accuracy of the Algorithm. The error brought by the algorithm is mainly from two sides. One is from the linear assumption (8). The other is from the use of the describing function. Although the linear assumption is made based on the FOPDT model, it turns out to be accurate for different processes from extensive study. The error brought by linear assumption (8) is quite acceptable. The describing function is an approximation method, and the error is large when it is applied to high-order or large-delay processes. It is assumed that the input signal is a sine wave in the describing function analysis. In fact, the system input is a square-wave signal, which consists of the main harmonic component at oscillating frequency ω0 and other higher harmonic components:

ui(t) )

4d

∞

1

sin[(2n + 1)ω0t] ) ∑ π n)02n + 1

(

4d π

sin ω0t +

1 3

)

sin 3ω0t + ... (14)

Therefore, process output y(t) is not a pure sine wave. It contains responses on all harmonic components of the input signal. Usually, we assume that the process has low-pass characteristics and the effect of higher harmonics is negligible. In some cases, this will make the accuracy of the describing function poor.

To detect the main harmonic (n ) 0), we have to filter out higher harmonic components, which is, in fact, Fourier transformation of y(t) at the frequency ω0

Are )

2 Tp

∫0T y(t) sin(ω0t) dt, p

Aim )

2 Tp

∫0T y(t) cos(ω0t) dt p

A ) xAre2 + Aim2

(15) (16)

where Are and Aim are real and imaginary components of the amplitude, respectively, and A is the amplitude of the first harmonic. When a is replaced by A in the above equations, the accuracy of the proposed algorithm is greatly improved. This can be shown from the simulations below. Phase margin estimation for different processes with A and a is given by Table 1. 3.4. Selection of Relay Settings. The relay output amplitude d has an effect on the amplitude of oscillation. Therefore, the choice of d should make the oscillations obvious and, in the meantime, keep the oscillation in the range of permission. The relay hysteresis 2 is determined by the process. The closer 2 is to d, the more accurate the result we get. However, d is kept unknown before the second relay test. This is a contradiction. The recommended range of 2/d is 0.1-0.4. It is shown from a large number of simulations that the error of the phase margin between real and estimated values is usually less than (5° with 2 unknown. In most cases, the error is less than (3°. This precision is acceptable for an approximate estimation method. Noise is always present in output measurements and is inevitable in a practical monitoring procedure. Hysteresis in the relay is a simple way to reduce the influence of measurement noise. However, zero hysteresis in the first relay test is used. Filtering is another possible antinoise measure. Note that noise is usually of high frequency while most processes are of low-pass nature. A low-pass filter may be employed to preprocess noisy output. The filter bandwidth is usually set as 3-5 times higher than the process critical frequency. Another antinoise measure is to utilize multiple periods of limit cycles instead of a single period so as to filter out noise by an averaging method. 3.5. Limitations. The proposed scheme can successfully estimate gain and phase margins of the control system in most cases. However, several limitations of

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the proposed method should be noticed. Because all of the calculations depend on the data measured from relay tests, we must first ensure the existence of successful relay tests. Because a negative value of the relay hysteresis is not physically realizable, the online search of the phase margin is limited to the third quadrant, which is the most common case for the control system. Certainly, we can extend the application range by inserting an artificial element, e.g., a derivative or a time delay, in the feedback loop. Although the proposed method is quite accurate in the estimation of gain and phase margins, it is an approximated method in nature because of the use of the describing function. It has inherent shortcomings of the describing function. The main harmonic of the measured signal is used here to improve the estimation. Compared with a standard relay autotuning technique, the proposed procedure requires two relay experiments instead of one. We can get two points on the process Nyquist curve from two relay tests. Therefore, the proposed method can extract more useful information of the control system and assess the performance more efficiently. It is also very helpful for the controller design because it is obvious that the closed-loop response will be improved by considering more than just one frequency. Of course, this is at the cost of longer tuning time. 4. PID Tuning Method After the completion of two relay tests, the gain and phase margins are estimated online to assess the performance of the control system. If the performance is poor, retuning of the controller is needed. For the modified relay feedback system shown in Figure 2, the oscillation is periodic when the system settles down. Then Fourier analysis can be applied to the periodic signals y(t) and u(t) to yield estimates of the process frequency response at the oscillating frequency i

pi

i

(17)

where Tpi is the period of the stationary oscillations and ωi ) 2π/Tpi is the corresponding frequency. We can identify low-order transfer function models from the process frequency response, and then a modelbased controller design method can be used. Here we design the controller directly from the process frequency response. No intermediate model is needed. The PID controller transfer function is given as

1 Gc(s) ) Kp + KI + Kds s

(18)

1 Gc(jω) ) Kp + j Kdω - KI ω

)

(19)

Let the desired closed-loop transfer function be GCL(s). Then, the corresponding desired open-loop GOL(jω) is given by

GOL(jω) )

GCL(jω) 1 - GCL(jω)

GOL(jω) ) ψθ

(21)

x(ω) + jy(ω) ) ψθ

(22)

or

where ψ ) [Gp(jω), j(-1/ω)Gp(jω), jωGp(jω)] and θ ) [Kp, KI, Kd]T is the controller parameter vector. x(ω) and y(ω) are real and imaginary parts of GOL(jω), respectively. Splitting ψ into real and imaginary parts, ψr and ψi, respectively, and then equating the real and imaginary parts on both sides of eq 22

[ ] [ ]

ψ (ω) x(ω) ) r θ ψi(ω) y(ω)

(20)

(23)

For N different measured frequency response data, we obtain N sets of the above equation

(24)

To estimate the controller parameter vector in the least-squares sense, we minimize the following cost function:

JLS ) T ) (Y - φθˆ )T(Y - φˆ )

(25)

Because JLS is a nonnegative definite and quadratic, it has a unique global minimum precisely when the matrix φ has full column rank. Column rank deficiency occurs if 2N < 3. Here N is equal to 2 and the uniqueness of the solution can be guaranteed. The least-squares estimate of parameter vector θˆ is given by

θˆ ) (φTφ)-1φY

Hence

(

Let GOL(jω) ) Gc(jω) Gp(jω) and organize it as

Y2N×1 ) φ2N×3θ3×1

∫0T y(t) e-jω t dt Gp(jωi) ) T ∫0 u(t) e-jω t dt pi

Figure 6. Examination of eq 8 (-A ) R + βd; O, measured amplitude by relay).

(26)

5. Simulation Examples Example 1. Consider a FOPDT Gp(s) ) e-s/(s + 1). The controller is Gc(s) ) 0.768 + 1.239(1/s). The calculated gain and phase margins are Am ) 1.38 and φm ) 17.95°. The relay settings are d ) 1, 1 ) 0, and 2 ) 0.1. After two relay tests, d ) 0.338, and the estimated gain and phase margins are A ˆ m ) 1.35 and φˆ m ) 15.38°, respectively. The phase margin estimated with a is 7.29°. The examination of eq 8 is shown in Figure 6. In Figure 6, the y axis is A and the x axis is scaled by /d.


Figure 7. Step response before retuning.

Figure 8. Step response after retuning.

Because the phase margin of the original control system is only 18°, the performance is poor. Retuning of the controller is needed. We choose the desired closedloop transfer function GCL(s) as GCL(s) ) 1/(τcl2s2 + 2ζτcls + 1)e-Ls. Generally speaking, the time delay L is selected to equal the “apparent” delay of the process. Damping factor ζ is selected to achieve a good damp property. The time constant τcl determines the speed of the closed-loop response. These parameters can be specified by the designer according to the expected characteristics. This desired closed-loop characteristic is also used in examples 2 and 3. The time delay L ) 1, damping factor ζ ) 0.707, and time constant τcl ) 0.5 are chosen. Using the PID controller design method described above yields the following PID controller: Gc(s) ) 0.738 + 0.584(1/s) + 0.045s. The step responses of the control system before and after retuning are shown in Figures 7 and 8. To verify the proposed controller’s performance, some simulation comparisons are done among the A° stro¨m-Ha¨gglund (A-H) method, the Ziegler-Nichols (Z-N) method, the Smith predictor, and the method proposed in this paper. For the sake of convenience, the PID controller in the Smith predictor is tuned by the Z-N method. The model used in the Smith predictor is the FOPDT model, which can be identified through two relay tests. Example 2. Consider a SOPDT Gp(s) ) e-s/(10s + 1)(2s + 1). The controller is Gc(s) ) 2 + 0.667(1/s). The


calculated gain and phase margins are Am ) 2.3 and φm ) 16°. The relay settings are d ) 1, 1 ) 0, and 2 ) 0.2. After two relay tests, d ) 0.345, and the estimated gain and phase margins are A ˆ m ) 2.48 and φˆ m ) 15.70°. The examination of eq 8 is shown in Figure 9. Retune the controller. The parameters of the desired closed-loop transfer function are chosen as L ) 1.2, ζ ) 0.707, and τcl ) 3. Using the PID controller design

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method described above yields the following PID controller: Gc(s) ) 1.95 + 0.177(1/s) + 0.099s. The step responses of the control system before and after retuning are shown in Figures 10 and 11. Example 3. Consider a high-order process with multiple equal poles Gp(s) ) 1/(s + 1)8. The controller is Gc(s) ) 0.769 + 0.229(1/s). The calculated gain and phase margins are Am ) 1.33 and φm ) 22.56°. The relay settings are d ) 1, 1 ) 0, and 2 ) 0.2. After two relay tests, d ) 0.439, and the estimated gain and phase margins are A ˆ m ) 1.329 and φˆ m ) 20.18°. The phase margin estimated with a is 13.48°. The examination of eq 8 is shown in Figure 12. Retune the controller. The parameters of the desired closed-loop transfer function are chosen as L ) 2, ζ ) 0.707, and τcl ) 4. Using the controller design method described above yields the following PID controller: Gc(s) ) 0.516 + 0.132(1/s) + 0.855s. The step responses of the control system before and after retuning are shown in Figures 13 and 14. From Figures 8, 11, and 14, we can see that the Z-N method has a fast step response; however, the overshoot is large, and the response is very oscillatory. The A-H method is only suitable for a small-time-delay system and not for a large-time-delay system. The Smith predictor has a good performance, while it is sensitive to modeling error, and its tuning is not trivial and requires the determination of a relatively large number


of parameters; the proposed controller has better performance because it gives a small overshoot and welldamped step response. Example 4. Consider a process with large dead time Gp(s) ) e-10s/(s + 1)(s + 2)(s + 3). The controller is Gc(s) ) 0.5245 + 0.3098(1/s) + 0.2331s. The calculated gain and phase margins are Am ) 2.99 and φm ) 60°. The relay settings are d ) 1, 1 ) 0, and 2 ) 0.2. After two






relay tests, d ) 1.05. The estimated gain and phase margins are A ˆ m ) 3.07 and φˆ m ) 56.12°. The control system is in good performance. The linear assumption (8) is quite accurate, as shown in Figure 15. Example 5. Consider a process with right-half-plane (RHP) zero Gp(s) ) (1 - Rs)/(s + 1)3 and R ) 1. The

controller is Gc(s) ) 1 + 0.5(1/s). The calculated gain and phase margins are Am ) 1.28 and φm ) 19.7°. The estimated gain and phase margins using the proposed method are A ˆ m ) 1.19 and φˆ m ) 12.7°. The examination of eq 8 is shown in Figure 16. When R ) 0.1, the calculated gain and phase margins are Am ) 3.48 and

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φm ) 51.9° and the estimated gain and phase margins are A ˆ m ) 3.29 and φˆ m ) 41.8°. For the RHP process, the error between real and estimated phase margins is relatively large and the accuracy of the estimated value decreases with a decrease in R. The same phenomenon exists when the proposed method is applied to the oscillatory process. The accuracy of the estimated phase margin decreases with a decrease in the damping ratio. 6. Conclusions A performance assessment procedure and PID controller design method based on modified relay feedback are proposed in this paper. The modified relay feedback structure can ensure sustained oscillations for the loworder process. The performance of the closed-loop system is assessed by an online estimation of the gain and phase margins, which are calculated by two relay tests. The effectiveness and accuracy of the proposed method are investigated. When the performance deteriorates, the PID controller is retuned based on a least-squares fit of the actual to the desired closed-loop dynamic characteristic. No intermediate model is needed. Performance monitoring and controller design can be done simultaneously, which can keep the control system in good performance. Examples are given to verify the effectiveness of the presented algorithm. Literature Cited (1) A° stro¨m, K. J.; Ha¨gglund, T. Automatic tuning of simple controllers with specification on phase and amplitude margins. Automatica 1984, 20, 645. (2) A° stro¨m, K. J.; Ha¨gglund, T. PID Controllers: Theory, Design, and Tuning, 2nd ed.; Instrument Society of America: Research Triangle Park, NC, 1995. (3) Ha¨gglund, T.; A° stro¨m, K. J. Industrial adaptive controllers based on frequency response techniques. Automatica 1991, 27, 599. (4) Ho, W. K.; Hang, C. C.; Cao, L. S. Tuning of PID controllers based on gain and phase margin specifications. Automatica 1995, 31, 497. (5) Wang, Y. G.; Shao, H. H. PID autotuner based on gain and phase margin specifications. Ind. Eng. Chem. Res. 1999, 38, 3007. (6) Ho, W. K.; Hang, C. C.; Zhou, J. H. Performance and gain and phase margins of well-known PI tuning formulas. IEEE Trans. Control Syst. Technol. 1995, 3, 245. (7) Ho, W. K.; Gan, O. P.; Tay, E. B.; Ang, E. L. Performance and gain and phase margins of well-known PID tuning formulas. IEEE Trans. Control Syst. Technol. 1996, 4, 473. (8) Chiang, R. C.; Yu, C. C. Monitoring procedure for intelligent control: On-line identification of maximum closed loop log modulus. Ind. Eng. Chem. Res. 1993, 32, 90.

(9) Plamor, Z. J.; Blau, M. An auto-tuner for Smith dead time compensator. Int. J. Control 1994, 60, 117. (10) Tan, K. K.; Wang, Q. G.; Lee, T. H. Finite spectrum assignment control of unstable time delay processes with relay tuning. Ind. Eng. Chem. Res. 1998, 37, 1353. (11) Luyben, W. L. Derivation of transfer functions for highly nonlinear distillation columns. Ind. Eng. Chem. Res. 1987, 26, 2490. (12) Wei, L.; Eskinat, E.; Luyben, W. L. An improved autotune identification method. Ind. Eng. Chem. Res. 1991, 30, 1530. (13) Scali, C.; Marchetti, G.; Semino, D. Relay and additional delay for identification and autotuning of completely unknown processes. Ind. Eng. Chem. Res. 1999, 38, 1987. (14) Loh, A. P.; Hang, C. C.; Quek, C. K.; Vsanani, V. U. Autotuning of multi-loop proportional-integral controllers using relay feedback. Ind. Eng. Chem. Res. 1993, 32, 1102. (15) Shen, S.; Yu, C. C. Use of relay-feedback test for automatic tuning of multivariable systems. AIChE J. 1994, 40, 627. (16) Plamor, Z. J.; Halevi, Y.; Krasney, N. Automatic tuning of decentralized PID controllers for TITO processes. Automatica 1995, 31, 1001. (17) Wang, Q. G.; Zou, B.; Lee, T. H.; Bi, Q. Auto-tuning of multivariable PID controllers from decentralized relay feedback. Automatica 1997, 33, 319. (18) Shen, S.; Wu, J. S.; Yu, C. C. Use of biased-relay feedback for system identification. AIChE J. 1996, 42, 1174. (19) Wang, Q. G.; Hang, C. C.; Zou, B. Low-order modeling from relay feedback. Ind. Eng. Chem. Res. 1997, 36, 375. (20) Leva, A. PID auto-tuning algorithm based on relay feedback. IEE Proc. D 1993, 140, 328. (21) Schei, T. S. A method for closed loop automatic tuning of PID controllers. Automatica 1992, 28, 587. (22) Friman, M.; Waller, K. V. A two-channel relay for autotuning. Ind. Eng. Chem. Res. 1997, 36, 2662. (23) Ender, D. B. Process control performance: Not as good as you think. Control Eng. 1993, 40, 180. (24) Harris, T. Assessment of closed loop performance. Can. J. Chem. Eng. 1989, 67, 856. (25) Huang, B.; Shah, S. L. Performance assessment of control loops: theory and applications; Springer-Verlag: London, 1999. (26) Yu, C. C. Autotuning of PID controllers; Springer-Verlag: London, 1999. (27) Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1991. (28) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759. (29) A° stro¨m, K. J.; Ha¨gglund, T. Benchmark systems for PID control. IFAC Workshop Digital Control, Past, Present, Future PID Control, Terrassa, Spain 2000, 181.

Received for review December 2, 2004 Revised manuscript received March 20, 2005 Accepted March 23, 2005 IE048831R

Performance Assessment and Controller Design Based on Modified

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