PI Design Method Based on Sensitivity - Industrial & Engineering

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Ind. Eng. Chem. Res. 2006, 45, 3174-3181

PI Design Method Based on Sensitivity Ming-Da Ma* and Xin-Jian Zhu Fuel Cell Research Institute, Shanghai Jiaotong UniVersity, Shanghai, People’s Republic of China 200030

A new robust proportional-integral (PI) autotuning method based on the maximum sensitivity is proposed in this paper which can be applied to unknown, stable, and minimum-phase plants. The presented algorithm chooses the maximum sensitivity as the design parameter. The PI controller parameters are obtained by making the Nyquist curve of the loop transfer function tangentially touch the sensitivity circle at a specified frequency. The process frequency response at the frequency where the maximum sensitivity is achieved is identified by relay feedback test, and the derivatives of amplitude and phase of the plant with respect to frequency can be estimated by process frequency response by introducing Bode’s integrals. No intermediate model is needed. Therefore, the presented method does not suffer from the modeling error. Simulation examples for different kinds of processes illustrate the simplicity and the effectiveness of the proposed method. 1. Introduction Proportional-integral-derivative (PID) control is a control strategy that has been successfully used over many years. Simplicity, robustness, a wide range of applicability, and nearoptimal performance are some of the reasons that have made PID control so popular. In process control, more than 90% of the control loops are PID type; most loops are actually PI control loops.2 The PID autotuning method, proposed by Åstro¨m and Ha¨gglund,1 has received much attention from process-control practitioners. The basic idea is to approximately estimate the critical gain and the critical frequency from the relay feedback tests, and then controllers are tuned accordingly. 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. This is highly appreciated for nonlinear processes. Some modifications and extensions of relay autotuning technique have been proposed to yield better performance.3,4 Recent progress in relay feedback is summarized by Yu11 and Wang et al.12 The tuning of PI controllers based on gain- and phase-margin specifications has been studied.5-7 They developed simple analytical formulas to tune PI and PID controllers for commonly used first-order and second-order plus dead time plant models to meet gain- and phase-margin specifications. However, the gain- and phase-margin specifications can sometimes give poor results when the shape of the frequency response curve is unusual. This is because each of these criteria measures the closeness of the loop transfer function to the (-1, 0) point at only one particular spot. The maximum sensitivity does not have this problem, since it directly measures the closeness of the loop transfer function to the (-1, 0) point at all frequencies. It places a bound on the sensitivity at all frequencies, not just at the two frequencies associated with the gain and phase margins. There are some controller design methods which are based on the sensitivity. Yaniv and Nagurka13 proposed a PID controller design method satisfying gain margin and sensitivity constraints on a set of plants. Åstro¨m and co-workers14,15 proposed PI(D) controller design methods by a numerical method based on optimization of load-disturbance rejection with constraints on sensitivity and complementary sensitivity simul* To whom correspondence should be addressed. E-mail: [email protected].

Figure 1. Closed-loop control system.

taneously. However, in all theses cases, transfer function models are required and the sensitivity is a constraint for an optimization problem. In this paper, a robust PI autotuner is proposed. The presented algorithm chooses the maximum sensitivity and the frequency at which the maximum sensitivity is obtained as design parameters. The controller parameters are obtained by making the Nyquist curve of the loop transfer function tangentially touch the sensitivity circle at the specified frequency. The tuning procedure is automated by relay feedback test. The presented method does not suffer from the modeling error because no intermediate model is used. Simulation examples are given to illustrate the simplicity and the effectiveness of the proposed method. This paper is organized as follows. The controller design method is presented in Section 2. In Section 3, a method to identify the process frequency response at a specified frequency by relay feedback tests is introduced. Autotuning procedures are presented in Section 4. Simulation examples follow in Section 5, and conclusions are drawn in Section 6. 2. PI Controller Design Based on the Maximum Sensitivity Consider the closed-loop system shown in Figure 1, in which Gc(s) and G(s) represent the controller and plant transfer functions. The signals r, e, d, u, and y are the reference, error, disturbance, control, and plant output, respectively. For this control system, the sensitivity function S(s) is defined as

S(s) )

1 1 ) 1 + G(s)Gc(s) 1 + L(s)

(1)

where L(s) ) G(s)Gc(s) is the open-loop transfer function. The maximum sensitivity is then given by

Ms(s) ) max |S(jω)|

10.1021/ie0505817 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/25/2006

ω

(2)

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3175

Ms is a useful measure of both performance and robust stability of the closed-loop system. In terms of performance, Ms is a measure of the worst-case disturbance amplification. At the frequency where Ms is obtained, the plant output disturbances are amplified the most. It has also been pointed out that Ms is good design parameter for set-point response behavior, and it is recommended that Ms should be in range 1.2-2.0.14 Approximately, Ms values 1.2-2.0 correspond to damping constant values 0.7-0.3. Therefore, increasing Ms will speed up the output response and vice versa. In terms of robust stability, the inverse value of Ms is the shortest distance between the Nyquist curve of the loop transfer function and the critical point (-1, 0). By assigning Ms the classical measures of robust stability, the gain and phase margins are bounded to the values

GM g 20 log10

( ) Ms Ms - 1

(3)

( ) 1 2Ms

(4)

PM g 2 arcsin

Ms values 1.2-2.0 correspond to the bounds of 6.0-2.0 of gain margin and 49.2-29.0 degrees of phase margin. Increasing Ms will speed up the output response and improve the loaddisturbance rejection, but the time response tends to be oscillatory. Therefore, the choice of Ms is based on the balance of the performance and robust stability. The PI controller transfer function is given as

Gc(s) ) k +

ki s

( ) (

f′(ω) ) 2R′k + 2rr′ k2 +

Equations 10 and 11 can be solved with respect to k, ki > 0 when frequency ω is fixed. An applicable frequency range is up to the critical frequency, where the process has a phase lag of π. Remark: Initial conditions are very important for solving nonlinear equations. Equations 10 and 11 can be solved iteratively with the Newton-Raphson method, which converges very fast if suitable initial conditions are given. There may be several solutions which can be found by starting the iteration from different initial conditions. As an initial guess, (k, ki) obtained by the Ziegler-Nichols method can lead to convergence in most cases. In the following section, we will show that the derivative of R, β, and r with respect to frequency can be calculated from the process frequency response, that is, the amplitude and phase at that frequency. The following notations are defined as follows16

sa(ω0) )

(6)

Let R(ω) and β(ω) be the real and imaginary parts of the process transfer function. Hence,

G(jω) ) R(ω) + jβ(ω) ) r(ω) ejφ(ω)

(7)

where

sp(ω0) ) ω0

d∠G(jω) |ω0 dω

(13)

d ln |G(jω)| 2 |ω0 ≈ ∠G(jω0) dω π

d∠G(jω) 2 |ω0 ≈ ∠G(jω0) + [ln |Kg| - ln |G dω π (jω0)|] (15)

G(s) ) Gm(s) e-τs

β(ω) ) r(ω) sin φ(ω) To achieve the specific maximum sensitivity at a certain frequency, the following two nonlinear equations with respect to the proportional gain k and integral gain ki are obtained

1 )0 Ms2

(8)

∂ f(ω) ) f′(ω) ) 0 ∂ω

(9)

After a straightforward calculation (for simplicity, ω as an argument is omitted)

(14)

where Kg is the static gain of the plant and |G(jω0)| and ∠G(jω0) are the amplitude and phase of the plant at the frequency ω0, respectively. The estimation of sp and sa can be easily extended to plants with integrators and/or time delay. Consider a system with time delay

R(ω) ) r(ω) cos φ(ω)

f(ω) ) |1 + L(jω)|2 -

(12)

The approximation of sp and sa for a stable and minimum-phase plant can be calculated as follows by introducing the Bode’s integrals

sa(ω0) ) ω0

ki Gc(jω) ) k - j ω

d ln|G(jω)| | ω0 dυ

sp(ω0) ) ω0

(5)

Hence,

)

2 ki β r ki + 2 β′ )0 ω ω ω2 ω2 (11)

ki2

(16)

where Gm(s) is a stable, minimum-phase system. Then sa(ω0) and sp(ω0) for a system including a pure time delay are computed as follows

sa(ω0) ) ω0

d ln |Gm(jω)| 2 d ln |G(jω)| |ω0 ) ω0 ≈ (∠G dω dω π (jω0) + τω0) (17)

sp(ω0) ) ω0

d(∠Gm(jω) - τω) 2 |ω0 ≈ ∠G(jω0) + [ln |Kg| dω π ln |G(jω0)|] (18)

Consider a system with integrators

r2 1 β f(ω) ) 1 + 2Rk + 2 ki + r2k2 + 2ki2 - 2 ) 0 ω ω M s

(10)

G(s) )

Gm(s) sm

(19)

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Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006

sa(ω0) ) ω0

d(ln |Gm(jω)| - m ln ω) 2 |ω0 ≈ ∠G(jω0) + dω π mπ 2 - m ) ∠G(jω0) (20) 2 π

(

)

d∠G(jω) |ω0 ) ω0 sp(ω0) ) ω0 dω

(

d ∠Gm(jω) -

mπ 2

dω

)|

ω0

)

G(s) )

d∠Gm(jω) ω0 |ω0 (21) dω Therefore, it is interesting to find out that time delay has no effect on the calculation of sp for a system with time delay and that integrators have no effect on the calculation of sa for a system with integrators. However, the time delay should be known for the calculation of sa for a system with time delay, and sp should be estimated according to the systems without any integrator for a system with integrators. It should be noted that τ represents the pure time delay of the system, which is usually related to the mass-transport delay and is often negligible or can be easily measured. This value should not be confounded with the time delay that is used to approximate a high-order system as a first-order plus time delay system. It is easy to determine whether a plant is an integrating process from a step test or prior knowledge of the process dynamics. The precision of the estimates of sa and sp is studied.16 The estimates are proven to be accurate for the typical models of industrial plants, and the absolute normalized error of the estimates is in an acceptable range. For oscillatory systems, the estimation may be poor at certain frequencies, but in general, the results remain satisfactory. However, for nonminimum-phase systems, the estimates are no longer valid. From the estimates of sa and sp, the derivatives of amplitude and phase of the process with respect to frequency ω0 can be calculated as

r′ )

d ln |G(jω)| ∂|G(jω)| | ) |G(jω0)| |ω0 ) {|G(jω0)|sa ∂ω ω0 dω (ω0)}/{ω0} (22) φ′ )

d∠G(jω) |ω0 ) {sp(ω0)}/{ω0} dω

(23)

Therefore, the derivatives of R and β can be calculated as

R′ ) (r cos φ)′ ) r′ cos φ - r sin φφ′

(24)

β′ ) (r sin φ)′ ) r′ sin φ + r cos φφ′

(25)

3. Relay Feedback Test Denote the frequency at which the maximum sensitivity is achieved as ωd. As shown in the previous section, the controller parameters can be calculated from process frequency responses |G(jωd)| and ∠G(jωd). The relay feedback test is a simple procedure to identify process frequency response. If an ideal relay is used, the critical gain and critical frequency can be calculated as

4h 2π K c ) , ωc ) πa Tc

obtained by a single relay test.21 The process frequency response at a specified frequency can also be obtained. In a relay test, the process output y(t) and input u(t) are neither periodic nor absolutely integrable. Therefore, they are decomposed into the periodic stationary cycle parts ys(t) or us(t) and the transient parts ∆y(t) or ∆u(t). For a process G(s) ) Y(s)/U(s), we have

where h is the relay output amplitude, a is the amplitude of the limit cycles, and Tc is the period of the limit cycles. With the fast Fourier transform (FFT) identification method, multiple points on the process frequency response can be

(27)

∆U(s) + Us(s)

where ∆Y(s) and ∆U(s) are the Laplace transforms of the transient parts ∆y(t) and ∆u(t), respectively. Ys(s) and Us(s) are the Laplace transforms of the periodic parts ys and us, respectively. At s ) jωd, eq 27 can be calculated as

∫0T ys(t) e-jω t dt

1 1 - e-jωdTc G(jωd) ) 1 ∆U(jωd) + 1 - e-jωdTc ∆Y(jωd) +

c

d

∫0T us(t) e-jω t dt c

d

(28)

Ys(jωd) and Us(jωd) can be calculated using a digital integral. ∆Y(jωd) and ∆U(jωd) can be computed with the standard FFT technique. The static gain Kp or Kg is easy to measure. It can be determined from a separate step test or a biased relay test.10 Here, the static gain is assumed to be known for convenience. Noise is always present in output measurements and is inevitable in process identification. Hysteresis in the relay is a simple way to reduce the influence of measurement noise. Before a relay test is performed, the noise band in the process output measurements can be estimated using simple statistics. The hysteresis width should be greater than the noise band to avoid wrong switchings in the relay output and is usually chosen to be twice the noise band. 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 to be 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 the averaging method.12 4. Autotuning Procedures The selection of ωd at which the maximum sensitivity is obtained has an obvious effect on the performance that can be achieved. Figure 2 presents the PI controller parameters in the (k, ki) domain for different values of Ms when ω varies. All the (k, ki) value pairs from a boundary yield the same Ms but a different frequency at which it is obtained. Therefore, it is possible to select the (k, ki) pair with some time-domain performance criteria (e.g., IAE/ISE/ITAE for set-point or load disturbance). For a PID control system, the integral error is inversely proportional to the integral gain2

IE ) (26)

∆Y(s) + Ys(s)

∫0∞ e(t) dt ) k1i

(29)

This corresponds to the peaks of the curve in the Figure 3. A very efficient algorithm was provided by Åstro¨m et al.14 to find the frequency where the integral gain is maximized. However, precise transfer function models of the plants are required. To extend the proposed method to the unknown systems and

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3177 Table 1. Parameters in Equation 31 for Different Ms Values a b c

Ms ) 1.4

Ms ) 1.6

Ms ) 2.0

1.51 1.56 1.68

-0.89 -0.96 -1.10

1.09 1.13 1.18

Figure 2. PI controller’s maximum sensitivity boundaries in the (k, ki) domain for G(s) ) (s + 1)-1 e-s.

Figure 4. The frequency ωd plotted versus the critical frequency ωc for integrating processes with Ms ) 1.4.

The process dynamics is characterized by the critical gain Kc, the critical frequency ωc, and the gain ratio κ ) 1/KpKc.17 Figure 3 shows the relation between ωd and the process dynamics for Ms ) 1.4. In Figure 3, the y-axis is scaled by ωd/ωc. The figure also indicates that it is possible to express ωd/ωc as a function of κ. The relationship between ωd/ωc and κ can be well-approximated by the functions having the form

ωd/ωc ) f(κ) ) ln(aκ0.5 + bκ + c)

Figure 3. The normalized frequency ωd/ωc plotted versus the gain ratio κ for stable processes with Ms ) 1.4.

facilitate autotuning, we study the benchmark systems,18 which are typical process models that are encountered in process control, to find out the relationship between the system dynamics and ωd at which the integral gain is maximized with guaranteed Ms. Processes with the following transfer functions are used

e-s , 1 + sT T ) 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 1, 2, 3, 5, 10, 20, 100

G1(s) )

(31)

The corresponding graph is shown in a solid line in the figure. Many other functions can also be used. Table 1 gives the coefficients a, b, and c of functions of the form shown in eq 31 for stable systems with typical Ms values. The transfer functions (eq 30) are all stable. Integrating processes are obtained by adding an integrator to all the transfer functions given in eq 30. Processes with integration are treated separately because the static gain does not exist for this kind of processes. Figure 4 shows the relation between ωd and the critical frequency ωc with Ms ) 1.4 for the benchmarks. It can be approximated by a straight line, e.g.,

ωd ) λωc

(32)

-s

G2(s) )

e , (1 + sT)2 T ) 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1, 1.5, 2, 3, 5, 8, 10, 20, 100 G3(s) )

G4(s) )

1 , (s + 1)n

n ) 2, 3, 4, 5, 6, 7, 8

(30)

1 , (1 + s)(1 + Rs)(1 + R2s)(1 + R3s) R ) 0.1, 0.2, 0.5, 0.7

G5(s) )

1 , (s + 1)(s + 2ξs + 1) 2

ξ ) 0.5, 0.7, 0.9

For Ms ) 1.4, 1.6, and 2.0, λ is equal to 0.338, 0.389, and 0.457, respectively. Therefore, the autotuning procedures can be summarized as follows: (1) Choose the design parameter Ms for a given process. (2) Perform the relay feedback test. Calculate Kc, ωc, and κ from the measured limit cycle data. (3) Choose the frequency ωd at which the maximum sensitivity is obtained by eq 31 or 32. Calculate the process frequency response at the frequency ωd from eq 28. (4) Calculate r′, φ′, R′, and β′ from eqs 22-25. (5) Solve k, ki from nonlinear eqs 10 and 11 using, for example, the Gauss-Newton method.

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Figure 5. Ms boundary in the (k, ki) domain with Ms ) 1.4 for example 1.


performance, some simulation comparisons are done among the Ziegler-Nichols (ZN) method, a PI controller based on gain and phase margins (GPM), Luyben’s method which determines a first-order plus dead time (FOPDT) model by running a relay feedback test and applying IMC tuning rules,20 and the method proposed in this paper. As indicated above, the reasonable range for the maximum sensitivity is Ms ) 1.2-TO 2.0; lower values give better robustness but poorer rejection of load disturbance. We have chosen Ms ) 1.4 in the simulation examples if there is no additional indication, since it provides a good compromise between performance and robustness. With this value, the closed-loop response typically does not have oscillations. The gain and phase margins used in the GPM method are 3 and π/3, respectively. Example 1: Consider a First-Order Plus Dead Time Plant.

Figure 6. Sensitivity function magnitudes for example 1.

G(s) )

e-s s+1

It is a time-delay-dominated process. Perform the proposed autotuning procedures. The frequency at which the maximum sensitivity is obtained is selected as 1.08 from eq 31. The optimal value is 1.03 if an accurate model is known. Perform the autotuning procedures presented in the previous section, and then the following PI controller settings are obtained

Gc(s) ) 0.368 + 0.372/s

Figure 7. Step response of the closed-loop system (solid line ) proposed, dashed line ) ZN method, dotted line ) GPM method, and dashed-dotted line ) Luyben’s method).

(6) Apply the controller and evaluate the performance achieved. If the performance is not satisfactory, choose a new frequency and go back to step 3. 5. Simulation Examples A few examples will be given in this section to illustrate the proposed design method. To verify the proposed controller’s

The Ms boundary in the (k, ki) domain for Ms ) 1.4 and the controller parameters obtained by the proposed method are plotted in Figure 5. The controller parameters are very close to the optimal values. It should be noted that we obtain this result without knowing the transfer-function model of the plant. The sensitivity function magnitudes and set-point and disturbance responses are plotted in Figure 6 and Figure 7, respectively. The gain and phase margins for the control system with the obtained controller are 4.2 and 69°. The gain and phase margins for the control system with the ZN method are 2.0 and 80°. This is too aggressive. Luyben’s method gives good set-point and disturbance responses because the model identified by the relay test is accurate. The proposed method, together with the GPM method, also works well for this kind of process. Example 2: Consider a High-Order System with Multiple Equal Poles.

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3179

G(s) )

1 (s + 1)4

After the autotuning procedures are performed, the frequency ωd calculated by eq 31 is 0.484 and the optimal value is 0.499. The controller obtained by the proposed method is

Gc(s) ) 0.477 + 0.194

1 s

The Ms boundary in the (k, ki) domain for Ms ) 1.4 and the controller parameters obtained by the proposed method are plotted in Figure 8. The sensitivity function magnitudes and the set-point and disturbance responses are plotted in Figure 9 and Figure 10, respectively. Example 3: Consider an Integrating System.

G(s) )

e-s s(s + 1)3


After the autotuning procedures are performed, the maximum sensitivity is chosen as Ms ) 2.0 to improve the disturbance rejection. The frequency ωd calculated by eq 32 is 0.187, and the optimal value is 0.188. The controller obtained by the proposed method is

Gc(s) ) 0.138 + 0.00876

1 s

The Ms boundary in the (k, ki) domain for Ms ) 2.0 and the controller parameters obtained by the proposed method are plotted in Figure 11. The sensitivity function magnitudes and the set-point and disturbance responses are plotted in Figure 12 and Figure 13, respectively. Generally speaking, the ZN method has a fast step response and it gives closed-loop systems with poor damping and poor robustness. However, it is not surprising that the ZN method gives a good disturbance response, because it is designed according to the optimization of disturbance rejection. Example 4: Consider an Oscillatory System.

G(s) )


e-0.3s (s2 + 2s + 3)(s + 3)

After the autotuning procedures are performed, the frequency ωd calculated by eq 31 is 1.144 and the optimal value is 1.207. The controller obtained by the proposed method is

Gc(s) ) 2.19 + 3.18

1 s

The Ms boundary in the (k, ki) domain for Ms ) 1.4 and the controller parameters obtained by the proposed method are plotted in Figure 14. The approximation error for sa and sp at the frequency ωd leads to the difference between the controller parameters obtained by the proposed method and the optimal values. The sensitivity function magnitudes and the set-point and disturbance responses are plotted in Figure 15 and Figure 16, respectively. We can see that Luyben’s method gives poor performance, because this plant cannot be well-approximated by the FOPDT model. From the simulation results above, we can see that the ZN method is usually too aggressive. The performances of the GPM method and the IMC method are closely related with the models that were used. When the accuracy of the identified model is bad, the performance of the control system is poor. However,


the proposed method does not suffer from the modeling error, and it works well for different types of processes. The controller parameters are very close to the optimal values, though the process models are unknown. The proposed method can make the system have the specified maximum sensitivity. Therefore, the robustness and performance of the control system can be ensured.

3180



Figure 13. Step response of the closed-loop system (solid line ) proposed and dashed line ) ZN method).



phase of the plant with respect to frequency can be estimated by introducing Bode’s integrals from process frequency response, which can be identified by the relay feedback test. The presented method does not suffer from the modeling error. Simulation examples for different kinds of processes are given to verify the simplicity and effectiveness of the presented method. Literature Cited

Figure 14. Ms boundary in the (k,ki) domain with Ms ) 1.4 for example 4.

6. Conclusions A new robust PI autotuning method based on the maximum sensitivity is proposed. The method can be widely applied to unknown, stable, and minimum-phase plants. The controller parameters are obtained by making the Nyquist curve of the loop transfer function tangentially touch the sensitivity circle at the specified frequency. The derivatives of amplitude and

(1) Åstro¨m, K. J.; Ha¨gglund, T. Automatic tuning of simple controllers with specification on phase and amplitude margins. Automatica 1984, 20, 645. (2) Åstro¨m, K. J.; Ha¨gglund, T. PID Controllers: Theory, Design, and Tuning, 2nd ed.; Instrument Society of America: Research Triangle Park, NC, 1995. (3) Tan, K. K.; Lee, T. H.; Wang, Q. G. An enhanced automatic tuning procedure for PI/PID controllers for process control. AIChE J. 1996, 42, 2555. (4) Friman, M.; Waller, K. V. A two-channel relay for autotuning. Ind. Eng. Chem. Res. 1997, 36, 2662. (5) Ho, W. K.; Hang, C. C.; Cao, L. S. Tuning of PID controllers based on gain and phase margin specifications. Automatica 1995, 31, 497. (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) Wang, Y. G.; Shao, H. H. PID autotuner based on gain and phase margin specifications. Ind. Eng. Chem. Res. 1999, 38, 3007. (8) Wei, L.; Eskinat, E.; Luyben, W. L. An improved autotune identification method. Ind. Eng. Chem. Res. 1991, 30, 1530.

Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 3181 (9) 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. (10) Shen, S.; Wu, J. S.; Yu, C. C. Use of biased-relay feedback for system identification. AIChE J. 1996, 42, 1174. (11) Yu, C. C. Autotuning of PID controllers; Springer-Verlag: London, 1999. (12) Wang, Q. G.; Lee, T. H.; Lin, C. Relay Feedback: Analysis, Identification and Control; Springer-Verlag: London, 2003. (13) Yaniv, O.; Nagurka, M. Design of PID controllers satisfying gain margin and sensitivity constraints on a set of plants. Automatica 2004, 40, 111. (14) Åstro¨m, K. J.; Panagopoulos, H.; Ha¨gglund, T. Design of PI controllers based on nonconvex optimization. Automatica 1998, 34, 585. (15) Panagopoulos, H.; Åstro¨m, K. J.; Ha¨gglund, T. Design of PID controllers based on constrained optimization. IEE Proc., Part D: Control Theory Appl. 2002, 149, 216. (16) Karimi, A.; Garcia, D.; Longchamp, R. PID controller tuning using Bode’s integrals. IEEE Trans. Control Syst. Technol. 2003, 11, 812.

(17) Hang, C. C.; Åstro¨m, K. J.; Ho, W. K. Refinements of the ZieglerNichols tuning formula. IEE Proc., Part D: Control Theory Appl. 1991, 138, 111. (18) Åstro¨m, K. J.; Ha¨gglund, T. Benchmark systems for PID control. In IFAC Workshop on Digital Control. Past, Present, and Future of PID Control, Terrassa, Spain, 2000; p 181. (19) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759. (20) Luyben, W. L. Getting more information from relay-feedback tests. Ind. Eng. Chem. Res. 2001, 40, 4391. (21) Wang, Q. G.; Hang, C. C.; Bi, Q. A Technique for Frequency Response Identification from Relay Feedback. IEEE Trans. Control Syst. Technol. 1999, 7, 122.

ReceiVed for reView May 17, 2005 ReVised manuscript receiVed January 15, 2006 Accepted February 24, 2006 IE0505817

PI Design Method Based on Sensitivity - Industrial & Engineering

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