Simple PID Controller Tuning Method for Processes with Inverse

Ind. Eng. Chem. Res. 2003, 42, 4461-4477

4461

Simple PID Controller Tuning Method for Processes with Inverse Response Plus Dead Time or Large Overshoot Response Plus Dead Time I-Lung Chien,* Yu-Cheng Chung, Bo-Shuo Chen, and Cheng-Yuan Chuang Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 10672, Taiwan

Whenever a controlled variable encounters two (or more) competing dynamic effects with different time constants from the same manipulated variable, the resulting composite dynamic behavior of this process can exhibit a troublesome inverse response or a large overshoot response. These unusual dynamic behaviors together with the process dead time can cause problems in PID controller tuning. These types of responses are observed in some particle control loops in the chemical industry or can be observed more frequently in multiloop systems. In this paper, a simple PID controller tuning method will be proposed to handle these two difficult types of control loops. The proposed tuning method is derived from a direct synthesis controller design method. Both PID and PI-only tuning rules will be proposed in this paper. 1. Introduction The problematic dynamics of an inverse response system or a large overshoot system are the result of two (or more) opposing dynamic effects. Let us take the following simple two opposing first-order systems (Figure 1) as an example. The combined process model for this system is

G(s) )

y(s) ) u(s)

K1 K2 ) τ1s + 1 τ2s + 1

(K1 - K2)

[

K1τ2 - K2τ1 s+1 K1 - K2

(τ1s + 1)(τ2s + 1)

]

Figure 1. Block diagram of a composite process.

(1)

When the faster “opposing” effect has the smaller magnitude (τ2 much smaller than τ1 and K2 < K1), the resulting dynamic behavior can have the initial “wrong way” response and is eventually overwhelmed by the slower effect having the larger magnitude. This kind of dynamic behavior is called “inverse response” with the following condition:

K1τ2 - K2τ1 τ2) to reduce the order of the closed-loop characteristic equation, eq 7 can be simplified to obtain

τI 0.1τIτD 2 s + s + (-τzs + 1)e-Ds ) 0 KcKp KcKp

(8)

Using the first-order Taylor approximation, e-Ds ≈ -Ds + 1, eq 8 becomes

(

) (

)

0.1τIτD τI + τzD s2 + - τz - D s + 1 ) 0 (9) KcKp KcKp

By setting the closed-loop characteristic equation to match a desired no-overshoot critically damped closedloop system such as

τcl2s2 + 2τcls + 1 ) 0

(10)

one can obtain the following relationship for Kc and τcl by equating the coefficients in s2 and s terms to be

Kc )

τI

(11)

Kp(2τcl + τz + D)

τcl ) 0.1τD + 0.5x4τzD + 0.4τDτz + 0.4τDD + 0.04τD2 (12)

2. Inverse Response Plus Deadtime Process Depending on the overdamped or underdamped nature of the studied process, the transfer function models for the inverse response plus dead time process are assumed to be as follows:

Kc )

Overdamped system G(s) )

-Ds y(s) Kp(-τzs + 1)e ) u(s) (τ1s + 1)(τ2s + 1)

(4)

Underdamped system -Ds

y(s) Kp(-τzs + 1)e ) 2 2 G(s) ) u(s) τ s + 2ζτs + 1

(5)

2.1. PID Tuning for an Overdamped System. Assuming the PID controller having the common “series” type PID1 form as in Luyben,13

(

Gc(s) ) Kc 1 +

In summary, with the process model as in eq 4, the overall PID tuning rules for “series” type PID1 (eq 6) are derived to be

)(

)

1 τDs + 1 ≡ PID1 τIs RτDs + 1

(6)

with the derivative filter parameter R having a fixed value of typically 0.1. The closed-loop characteristic equation becomes

1 + Gc(s) G(s) ) (τIs + 1)(τDs + 1) Kp(-τzs + 1)e-Ds 1+ ) 0 (7) τI (τ1s + 1)(τ2s + 1) s(0.1τDs + 1) Kc

τ1

; τI ) τ1; τD ) τ2 Kp(2τcl + τz + D)

(13)

with τcl calculated as in eq 12. For practical implementation of the PID1 form in industry, it is preferable to use the “no derivative kick” PID1 form to avoid excess manipulated variable moves due to large τD values. The “no derivative kick” PID1 form is as follows:

(

u ) Kc 1 +

)[

τDs + 1 1 ysp y τIs 0.1τDs + 1

]

(14)

The closed-loop characteristic equation will be the same as eq 7 with the above “no derivative kick” implementation. 2.2. PID Tuning for an Underdamped System. Assuming the PID controller has the other common “parallel” type PID2 form as in work by Luyben,13

(

Gc(s) ) Kc 1 +

)

τDs 1 + ≡ PID2 τIs RτDs + 1

(15)

With the typical selection of R as 0.1, the closed-loop characteristic equation is

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4463

1 + Gc(s) G(s) )

While the model transfer function is guessed wrong with underestimations of the process gain, inverse response parameter (τz), and process dead time, the model transfer function can be seen in previous eq 4. With the proposed PID tuning rules in eq 13, the characteristic equation of the closed-loop system can be obtained as

2

1+

1.1τIτDs + (τI + 0.1τD)s + 1 τI s(0.1τDs + 1) Kc Kp(-τzs + 1)e-Ds τ2s2 + 2ζτs + 1

) 0 (16)

1+

When τI and τD are selected according to the following relationship to simplify the above equation,

1.1τIτDs2 + (τI + 0.1τD)s + 1 ≡ τ2s2 + 2ζτs + 1

(17)

τI and τD can be solved to have

τD )

2.2ζτ - x4.84ζ2τ2 - 0.44τ2 0.22

(18)

τI ) 2ζτ - 0.1τD

(19)

Note that, in eq 18, the open-loop damping coefficient ζ has to be larger than 0.302 in order to obtain real value of τD. For some rare occasions when the open-loop system is too highly oscillatory to have ζ < 0.302, the real value of τD can still be achieved by selecting a smaller value of R in eq 15 and following the same derivation. With the above tuning rules for τI and τD, the characteristic equation can be simplified to be the same as eq 8. Following the same derivation after eq 8, the overall PID tuning rules are as follows:

Kc )

τI

;

Kp(2τcl + τz + D)

τI and τD using eqs 18 and 19 with τcl calculated the same way as eq 12. Notice that this PID tuning rule can only be applied to a PID controller with the form of PID2. No tuning rules can be derived if the form of PID1 is used. Again, for the same reason as that in the previous section, the “no derivative kick” PID2 form should be implemented to avoid excess manipulated variable moves. The “no derivative kick” PID 2 form is as follows:

[(

u ) Kc 1 +

) (

)]

τDs 1 sp 1 + y - 1+ y τIs τIs 0.1τDs + 1

(20)

2.3. Model Mismatch Consideration. Assuming there is model mismatch between the “true” process transfer function and the “fitted” model transfer function, because the most important process parameters to consider are the process gain, inverse response parameter (τz), and process dead time, we will examine the closed-loop stability property of the proposed controller tuning method with some model mismatches. The stability analysis of the overdamped system will be illustrated in what follow. The underdamped system will give the same result. Assume that the “true” transfer function of the process is as follows:

Gp(s) )

(Kp + ∆Kp)[-(τz + ∆τz)s + 1]e-(D+∆D)s (τ1s + 1)(τ2s + 1)

(21)

(

)(

τ1s + 1 τ2s + 1 0.1τ2s + 1 Kp(2τcl + τz + D) τ1s τ1

)

(Kp + ∆Kp)[-(τz + ∆τz)s + 1]e-(D+∆D)s (τ1s + 1)(τ2s + 1)

) 0 (22)

Using the first-order Taylor approximation for the deadtime term and after some simplification, the above equation becomes

[ [

]

0.1Kpτ2(2τcl + τz + D) + (τz + ∆τz)(D + ∆D) s2 + Kp + ∆Kp Kp(2τcl + τz + D) - (τz + ∆τz) - (D + ∆D) s + 1 ) 0 Kp + ∆Kp (23)

]

Applying the Routh stability criterion, the stability lower bound of the τcl parameter can be obtained as follows:

[

τcl > 0.5 ∆τz + ∆D +

]

∆Kp (τ + ∆τz + D + ∆D) Kp z

(24)

Notice that the τcl lower bound is more stringent when the uncertainty bounds of all three model parameters become larger. This is in perfect agreement with intuition. Although the above analysis is not exact because of the first-order Taylor approximation of the dead-time term, through many numerical simulations we find eq 24 to be quite useful in providing a conservative estimation of the stability lower bound of the τcl parameter. If the selection of the τcl parameter according to eq 12 does not meet the stability requirement of eq 24, a larger τcl parameter has to be selected. Using the same PID tuning rules as those in eq 13 with larger τcl in these circumstances, a slower closed-loop system will be obtained. The new closed-loop characteristic equation will be different from the critically damped system in eq 10, with a larger value for the average of the two second-order time constants and also with a larger closed-loop damping coefficient. For the underdamped system, the same conclusion as in eq 24 on the stability lower bound of the τcl parameter can be derived. 2.4. Simulation Results. Example 1. The same numerical example as in that work by Luyben9 will be used here to test the closed-loop performance of the proposed PID controller tuning method. The transfer function model of this example is

G(s) )

(-τzs + 1)e-Ds (s + 1)2

(25)

Two values of the dead time (D ) 0.2 and 1.6) and also two values of the τz parameter (τz ) 0.2 and 1.6) are tested, so there are four subexamples. Because in work by Luyben9 no PID settings are derived, although unfair we will compare the proposed tuning method in this paper with the PI settings in work by Luyben.9 The proposed method will also be compared to the ZN-PID

4464 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

tuning rules as suggested by Waller and Nygardas.5 Table 1 gives the tuning constants used in the simulation. The PID tuning constants for the proposed method can easily be calculated from eq 13 with the τcl parameter specified via eq 12. The closed-loop setpoint change using the above three tuning methods can be seen in Figure 2. Case A gives a result when τz ) 0.2 and D ) 0.2, case B gives a result when τz ) 1.6 and D ) 0.2, case C gives a result when τz ) 0.2 and D ) 1.6, and case D gives a result when τz ) 1.6 and D ) 1.6. Notice that the ZN-PID settings give quite oscillatory and overshoot responses for all four different cases. On the other hand, the proposed method and the method by Luyben9 give very satisfactory closed-loop responses despite the changes in the τz and D model parameters. From the figure, we can see that the proposed method, although it results in a slower rise time than that of Luyben,9 gives almost no overshoot response and faster settling time. Because the method by Luyben9 is based on an empirical relationship, the extrapolation of this correlation beyond the suggested model parameter ranges should be cautious. Also the empirical relationship is based on the assumption that τ1 ) τ2 ) 1, more general cases with different values of τ1 and τ2 may require a different empirical relationship. The advantage of the proposed method is that it is derived from the closed-loop characteristic equation; thus, the tuning method should be valid for much wider model parameter value ranges. A simulation run is also made to demonstrate the use of eq 24 under model mismatch condition. Assuming the PID tuning constants are calculated based on the model parameters used in case D, the actual process has 30% model mismatches in Kp, τz, and also D parameters. From the estimation of the stability lower bound of the τcl parameter in eq 24, the value of this parameter should be greater than 1.104. Because the actual τcl is specified according to eq 12 with the value of 1.8, which is greater than 1.104, the closed-loop system should be stable. Closed-loop simulation in Figure 3 verifies this conjecture. The proposed method gives less oscillatory response in comparison to the method by Luyben.9 Example 2. A 3 × 3 system will be used here to demonstrate the occurrence of the inverse response plus dead time system under a multiloop situation. The Tyreus case 1 in work by Luyben14 is a 3 × 3 system with the following open-loop process transfer function equation:

[] y1 y2 y3

[

-1.986e-0.71s 66.67s + 1 -0.59s ) 0.0204e (7.14s + 1)2 0.374e-7.75s 22.22s + 1

5.984e-2.24s 14.29s + 1 2.38e-0.42s (1.43s + 1)2 -9.811e-1.59s 11.36s + 1

][ ]

0.422e-8.72s (250s + 1)2 0.513e

-s

-2.368e-27.33s 33.3s + 1 u1 u2 (26) u3

We are concerned about what PID tuning constants should be used in the more difficult to control loop 3 under the multiloop PID control configuration. To describe the process transfer function under the multiloop situation, a unit step test on loop 3 is performed while the other two loops are on automatic control. The tuning constants for the other two loops are assumed

Table 1. PID Settings for Example 1 process parameters

proposed method

Luyben9

ZN-PID

τz

D

Kc

τI

τD

Kc

τI

τD

Kc

τI

0.2 1.6 0.2 1.6

0.2 0.2 1.6 1.6

0.83 0.29 0.29 0.15

1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0

3.23 0.68 1.07 0.58

1.41 2.50 3.41 4.54

0.35 0.62 0.85 1.14

1.39 0.38 0.31 0.26

1.95 1.56 1.14 1.50

to be designed based on the TLC settings in work by Luyben and Luyben15 for the diagonal elements of the transfer function matrix in eq 26. Figure 4 shows the dynamic simulation result for the open-loop test on loop 3. An inverse response plus dead time system is observed. Using a standard model identification tool, a model transfer function for PID tuning purposes can be obtained as

() y3 u3

)

other loop closed

-0.500(-159s + 1)e-1.5s (27) (22.6s + 1)(18.7s + 1)

The apparent dead time of 1.5 is selected to minimize the sums of the square errors between the true data and the model prediction. The PID settings can easily be calculated using the proposed tuning rules in eq 13. A τcl value of 25.2 is calculated according to eq 12. Notice also that, because the value of the τz parameter is extremely large and also the values of the τ1 and τ2 parameters are not the same, no PI settings according to Luyben9 can be calculated. The closed-loop setpoint change for loop 3, with the setpoints of the other two loops remaining the same, can be seen in Figure 5. For comparison purposes, the response using BLT+6 tuning constants proposed by Luyben14 is also shown in Figure 5. When we concentrate on observing the loop 3 setpoint response tuned by the proposed method in this paper, the proposed method gives an excellent closed-loop result. 2.5. PI-Only Tuning Rules. Using a similar proposed method in this section, PI-only tuning rules can also be derived for inverse response plus dead time processes. For an overdamped system with a PI-only controller, the closed-loop characteristic equation is

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

τIs + 1 Kp(-τzs + 1)e-Ds ) 0 (28) τI (τ1s + 1)(τ2s + 1) s Kc

By setting τI ) τ1 (with τ1 > τ2) to reduce the order of the closed-loop characteristic equation to become secondorder and also using the first-order Taylor approximation on the dead-time term, eq 28 can be simplified to become

(

) (

)

τ1τ2 τ1 + τzD s2 + - τz - D s + 1 ) 0 KcKp KcKp

(29)

By setting the closed-loop characteristic equation to match a desired closed-loop system (with little overshoot) such as

τcl2s2 + 1.414τcls + 1 ) 0

(30)

we can obtain the following relationship for Kc and τcl by equating the coefficients in s2 and s terms as


Figure 2. Closed-loop response for example 1.

Kc )

τ1 Kp(1.414τcl + τz + D)

τcl ) 0.707τ2 + 0.5x4(τzD + τ2τz + τ2D) + 2τ22

(31) (32)

The overall PI-only tuning rules are

Kc )

τ1 Kp(1.414τcl + τz + D)

and

τI ) τ1

(33)

with τcl calculated as in eq 32. The reason for specifying the closed-loop characteristic equation as in eq 30 is because a closed-loop damping coefficient of 0.707 in eq 30 will give a faster closed-loop response than a critically damped system with very little overshoot (about 5% overshoot). The proposed PI-only tuning rules are applied to example 1; the resulting PI-only tuning parameters and the associated τcl parameters are listed in Table 2. For comparison purposes, the parameters for the proposed PID tuning rules are also listed in this table. From the table, one can observe that the second-order closed-loop time constants (τcl) of PI-only settings are all greater than the proposed PID settings, resulting in slower closed-loop dynamic response. Also, the PI-only setting will give about 5% overshoot while the proposed PID setting will give almost no overshoot as seen previously in Figure 2. This observation agrees with the intuition because a PI-only controller has only two tuning parameters; thus, there will be some limit in pushing the closed-loop performance. On the other hand, a PID controller has three tuning parameters; thus, it can obviously perform better than the PI-only controller if

tuned correctly. If the industrial practitioners are concerned with the more excessive control valve movements from the PID controller due to noisy measurement from the controlled variable and elect to use the PI-only controller, this paper also provides guidance on the PI-only tuning rules as in eqs 32 and 33. For the underdamped system, the derivation of the PI-only tuning rules becomes more complicated. The closed-loop characteristic equation for the underdamped system is

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

τIs + 1 Kp(-τzs + 1)e-Ds ) 0 (34) τI τ2s2 + 2ζτs + 1 s Kc

No simplification of the characteristic equation can be made by the proper selection of τI. Using the first-order Taylor approximation for the dead-time term, the previous equation becomes

(

) ( (

)

τIτ2 2τIζτ + τIτzD s3 + + τzD - τIτz - τID s2 + KcKp KcKp

)

τI + τI - τz - D s + 1 ) 0 (35) KcKp

By setting the closed-loop characteristic equation to match a desired third-order closed-loop system (with little overshoot) such as

(τcls + 1)(τcl2s2 + 1.414τcls + 1) ) 0

(36)


are assumed to be as follows:

Overdamped system G(s) )

Kp(τzs + 1)e-Ds y(s) ) u(s) (τ1s + 1)(τ2s + 1)

(39)

Underdamped system G(s) )

-Ds y(s) Kp(τzs + 1)e ) 2 2 u(s) τ s + 2ζτs + 1

(40)

3.1. PID Tuning for an Overdamped System. Assuming the PID controller has the “series” type PID form with an extra filter such as

(

Gc(s) ) Kc 1 +

)(

)

1 τDs + 1 ≡ PID3 τIs τFs + 1

(41)

this form is similar to PID1 in eq 6. When RτD ) τF, these two PIDs are exactly the same. The closed-loop characteristic equation can easily be calculated as

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

(τIs + 1)(τDs + 1) Kp(τzs + 1)e-Ds ) 0 (42) τI (τ1s + 1)(τ2s + 1) s(τ s + 1) Kc F

By letting τI ) τ1, τD ) τ2, and also τF ) τz for simplification purposes, eq 42 can be simplified to obtain

τI s + e-Ds ) 0 KcKp

Figure 3. Model mismatch run for example 1. Table 2. Proposed PID vs PI Settings for Example 1 process parameters proposed PID method proposed PI method τz

D

τcl

Kc

τI

τD

τcl

Kc

τI

0.2 1.6 0.2 1.6

0.2 0.2 1.6 1.6

0.40 0.81 0.81 1.8

0.83 0.29 0.29 0.15

1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0

1.7 2.3 2.3 3.2

0.36 0.20 0.20 0.13

1.0 1.0 1.0 1.0

we can obtain the parameters Kc, τI, and τcl by equating the coefficients in s3, s2, and s terms. The tuning rules for Kc and τI are

Kc )

τI )

τI Kp(2.414τcl + τz + D - τI)

4.828ζττcl + 2ζτ(τz + D) + τzD - 2.414τcl2 2ζτ + τz + D

Using the second-order Taylor approximation, e-Ds ≈ (D2/2)s2 - Ds + 1, eq 43 becomes

(

Depending on the overdamped or underdamped nature of the studied process, the transfer function models for the large overshoot response plus dead time process

(44)

By setting the closed-loop characteristic equation to match a desired no overshoot critically damped closedloop system as in previous eq 10, one can obtain the following relationship for Kc and τcl by equating the coefficients in s2 and s terms as

Kc )

τI Kp(2τcl + D)

τcl )

(38)

3. Large Overshoot Response Plus Dead Time Process

)

τI D2 2 -D s+1)0 s + 2 KcKp

(37)

The resulting equation for τcl is a cubic equation with quite complicated coefficients. The real root of τcl cannot be expressed explicitly and has to be solved by a numerical method. Although the tuning formula is complicated, obtaining the PI tuning parameters is straightforward for a particular underdamped example.

(43)

1 D x2

(45) (46)

In summary, with the process model as eq 39, the overall PID tuning rules for “series” type PID3 (eq 41) are derived as

Kc )

τI

; τI ) τ1; τD ) τ2; τF ) τz Kp(2τcl + D)

(47)

with τcl calculated as eq 46. If the dead time is approximated by the first-order Pade´ approximation, e-Ds ≈ [(-D/2)s + 1]/[(D/2)s + 1], slightly different PID tuning rules (compared to eqs 46 and 47) will be obtained. Note that, because for the large overshoot system the value of τF is quite large already,


Figure 4. Open-loop step response for example 2 (loops 1 and 2, automatic mode; loop 3, open loop).

it is not necessary to use the “no derivative kick” PID form in real industrial implementation. 3.2. PID Tuning for an Underdamped System. Assuming that the PID controller has the “parallel” type PID with an extra filter such as

(

Gc(s) ) Kc 1 +

)(

)

1 1 + τDs ≡ PID4 (48) τIs τFs + 1

this form can also be seen in work by Luyben.13 The closed-loop characteristic equation is

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

τIτDs2 + τIs + 1 Kp(τzs + 1)e-Ds ) 0 (49) τI τ2s2 + 2ζτs + 1 s(τ s + 1) Kc F

By selecting τF ) τz and also selecting τI and τD according to the following relationship to simplify the above equation

τIτDs2 + τIs + 1 ≡ τ2s2 + 2ζτs + 1

(50)

τI and τD can be solved to have

τI ) 2ζτ

(51)

τD ) τ/2ζ

(52)

The characteristic equation can be simplified to be the same as eq 43. Following the same derivation after eq 43, the overall PID tuning rules are as follows:

Kc )

τI

τ ; τI ) 2ζτ; τD ) ; τF ) τz 2ζ Kp(2τcl + D)

(53)

with τcl selected to be the same as that in eq 46. Notice that this PID tuning rule can only apply to the PID controller with the form of PID4. For the PID form similar to the “parallel” form in PID2,

(

Gc(s) ) Kc 1 +

)

τDs 1 ≡ PID5 + τIs τFs + 1

(54)

Following a derivation similar to the previous one, we can obtain the PID tuning rules as


Figure 5. Closed-loop response for example 2 (loops 1 and 2, setpoints remain; loop 3, setpoint change).

Kc )

τI

; τI ) 2ζτ - τz; Kp(2τcl + D) τD )

Gp(s) ) (Kp + ∆Kp)[(τz + ∆τz)s + 1]e-(D+∆D)s

2

(τ1s + 1)(τ2s + 1)

τ - τz; τF ) τz (55) 2ζτ - τz

If the process has a large overshoot (τz > 2ζτ), all Kc, τI, and τD parameters will have wrong signs, which is not feasible to use. For a process having a smaller overshoot (τz < 2ζτ), using tuning rules in eq 55 with the PID form of PID5 will obtain the same result as using tuning rules in eq 53 with the PID form of PID4. 3.3. Model Mismatch Consideration. Assume that there is model mismatch between the “true” process transfer function and the “fitted” model transfer function. Because the most important process parameters to consider are the process gain, lead time constant (τz), and process dead time, we will examine the closed-loop stability property of the proposed controller tuning method with some model mismatches. Stability analysis of the overdamped system will be illustrated next. The underdamped system will give exactly the same result. Assume the “true” transfer function of the process is as follows:

(56)

While the model transfer function is guessed wrong with underestimations of the process gain, lead time constant (τz), and process dead time, the model transfer function can be seen in previous eq 39. With the proposed PID tuning rules in eq 47, the characteristic equation of the closed-loop system can be obtained as

1+

(

)(

)

τ1s + 1 τ2s + 1 τzs + 1 Kp(2τcl + D) τ1s τ1

(Kp + ∆Kp)[(τz + ∆τz)s + 1]e-(D+∆D)s (τ1s + 1)(τ2s + 1)

) 0 (57)

Using the first-order Taylor approximation for the deadtime term and after some simplification, the previous equation becomes


Figure 6. Open-loop step response for example 3 (loop 1, open loop; loop 2, automatic mode).

[ [

]

Kpτz(2τcl + D) - (τz + ∆τz)(D + ∆D) s2 + Kp + ∆Kp Kp(2τcl + D) + (τz + ∆τz) - (D + ∆D) s + 1 ) 0 (58) Kp + ∆Kp

]

Applying the Routh stability criterion, we obtain two stability regions for the τcl parameter for the coefficients in s2 and also the s terms. The overall stability lower bound of τcl is

[

(

)

∆Kp ∆τz D + ∆D + D + τcl > 0.5 ∆D + Kp τz ∆Kp ∆τz ∆τz (D + ∆D) + ∆D (59) τz Kp τ z

]

Notice that the τcl lower bound is more stringent when the uncertainty bounds of all three model parameters become larger. This is in perfect agreement with intuition. Although the above analysis is not exact because of the first-order Taylor approximation of the dead-time term, through many numerical simulations we find eq 59 to be quite useful in providing a conservative estimation of the lower bound of the τcl parameter. For the underdamped system, the same conclusion as that in eq 59 on the lower bound of the τcl parameter can be derived. The overall proposed tuning method for the inverse response plus dead time system and also the large overshoot response plus dead time system can be summarized in Table 3. 3.4. Simulation Results. Example 3. A 2 × 2 system will be used here to demonstrate the occurrence of the large overshoot response plus dead time system under a multiloop situation. The Tyreus stabilizer example in work by Tyreus16 and also used by Luyben14 is a 2 × 2 system with the following open-loop process

[

transfer function equation:

[] y1 y2

]

-0.1153(10s + 1)e-0.1s 0.2429e-2s (33s + 1)2 (4s + 1)3 ) -12.6s 0.2429e-0.17s -0.0887e (43s + 1)(22s + 1) (44s + 1)(20s + 1) u1 u2 (60)

[ ]

We are concerned about how to tune two PID controllers under the interactive two-loop PID control configuration. Assume that we are interested in tuning loop 1 first. To describe the process transfer function of loop 1 under the two-loop situation, a unit step test on loop 1 is performed while the other loop is on automatic control. The tuning constants for the other loop are assumed to be designed based on the TLC settings for the diagonal elements of the transfer function matrix in eq 60. Because the TLC-PI settings result in an oscillatory response, TLC-PID settings for loop 2 are used here. Figure 6 shows the dynamic simulation result for the open-loop test on loop 1. A large overshoot response plus dead time system is observed. Using a standard model identification tool, a model transfer function for PID tuning purposes can be obtained as

() y1 u1

loop2 closed

)

-0.0250(382s + 1)e-1.5s (62.8s + 1)(4.59s + 1)

(61)

The PID settings can easily be calculated from the proposed tuning rules in eq 47 using the PID controller with the form of PID3. A τcl value of 1.06 is calculated according to eq 46. To further fine-tune the PID settings for loop 2, an open-loop test procedure similar to the previous one can


Table 3. Proposed Tuning Method model form

Kp(-τzs + 1)e-Ds (τ1s + 1)(τ2s + 1)

PID form

(

u ) Kc 1 +

tuning rules

)[

τ Ds + 1 1 ysp y τIs 0.1τDs + 1

]

Kc )

τ1 Kp(2τcl + τz + D)

τI ) τ1 τD ) τ2

τcl selection

τcl ) 0.1τD + 0.5x4τzD + 0.4τDτz + 0.4τDD + 0.04τD2 and also satisfying

[

τcl > 0.5 ∆τz + ∆D + Kp(-τzs + 1)e-Ds 2 2

τ s + 2ζτs + 1

[(

u ) Kc 1 +

) (

)]

τ Ds 1 sp 1 y - 1+ y + τI s τIs 0.1τDs + 1

Kc )

τI ) 2ζτ - 0.1τD

τD ) Kp(τzs + 1)e-Ds (τ1s + 1)(τ2s + 1)

(

u ) Kc 1 +

)(

)

1 τDs + 1 sp (y - y) τIs τFs + 1

τI Kp(2τcl + τz + D)

Kc )

2.2ζτ - x4.84ζ2τ2 - 0.44τ2 0.22 τ1 Kp(2τcl + D)

τI ) τ1 τD ) τ2 τF ) τz

Kp(τzs + 1)e-Ds 2 2

τ s + 2ζτs + 1

(

u ) Kc 1 +

)(

)

1 1 (ysp - y) + τ Ds τIs τFs + 1

Kc )

τI ) 2ζτ τD ) τ/2ζ τF ) τz

and also satisfying

[

τcl > 0.5 ∆τz + ∆D + τcl )

[

τcl )

∆Kp (τ + ∆τz + D + ∆D) Kp z

]

1 D x2

τcl > 0.5 ∆D + τI

]

τcl ) 0.1τD + 0.5x4τzD + 0.4τDτz + 0.4τDD + 0.04τD2

and also satisfying

Kp(2τcl + D)

∆Kp (τ + ∆τz + D + ∆D) Kp z

(

)

]

(

)

]

∆Kp ∆τz ∆τz ∆Kp ∆τz D + ∆D + D + (D + ∆D) + ∆D Kp τz τz K p τz

1 D x2

and also satisfying

[

τcl > 0.5 ∆D +

∆Kp ∆τz ∆τz ∆Kp ∆τz D + ∆D + D + (D + ∆D) + ∆D Kp τz τz Kp τz


Figure 7. Open-loop step response for example 3 (loop 1, automatic mode; loop 2, open loop).

be performed for loop 2 with loop 1 on automatic control with the TLC settings. The dynamic step test result can be seen in Figure 7. A model transfer function for loop 2 can be obtained as

() y2 u2

loop1 closed

)

0.0586(268s + 1)e-12.4s 57.22s2 + 2(0.763)(57.2)s + 1

(62)

Again, the PID settings can easily be calculated. Notice that the apparent dead time for loop 2 has to be quite large in order to obtain a better fit of the overall dynamic response. After we obtain the PID settings for both loops, the next step is to see how the closed-loop performs with both loops in automatic mode. The closed-loop setpoint change for loop 1 can be seen in Figure 8 while the setpoint of loop 2 remains the same. Also, the closedloop setpoint change for loop 2 can be seen in Figure 9 while the setpoint of loop 1 remains the same. For comparison purposes, the responses using BLT tuning constants proposed by Luyben14 are also shown in Figures 8 and 9. In Figure 8, although the BLT tuning method gives a better initial response in loop 2, neither loop reached the setpoint value in 500 min. On the contrary, the proposed method gives very smooth closedloop responses, and the settling time is much smaller. The proposed method in this paper also gives a much better closed-loop result in Figure 9 while the BLT tuning method gives overshoot and oscillatory responses. In previous example 3 and also in previous example 2, an open-loop “step test” is used to obtain the model parameters for illustration purposes. Other types of identification tests, such as the relay-feedback test described by Yu,17 can also be used to obtain the model parameters.

3.5. PI Plus First-Order Filter Controller Tuning Rules. Using a method similar to the one proposed in this section, PI plus filter controller tuning rules can also be derived for large overshoot plus dead time processes. A PI plus first-order filter controller structure has to be used to control this system. For an overdamped system with PI plus filter controller, the closedloop characteristic equation is

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

(τIs + 1)

Kp(τzs + 1)e-Ds

τI (τ1s + 1)(τ2s + 1) s(τ s + 1) Kc F

) 0 (63)

By setting τI ) τ1 (with τ1 > τ2) and τF ) τz to reduce the order of the closed-loop characteristic equation to become second-order and also using the second-order Taylor approximation on the dead-time term, eq 63 can be simplified to become

(

) (

)

τ1τ2 τ1 D2 2 + s + -D s+1)0 KcKp 2 KcKp

(64)

By setting the closed-loop characteristic equation to match a desired closed-loop system (with little overshoot) as in previous eq 30, we can obtain the following relationship for Kc and τcl by equating the coefficients in s2 and s terms as

Kc )

τ1 Kp(1.414τcl + D)

τcl ) 0.707τ2 + 0.5x2(τ22 + D2) + 4τ2D

(65) (66)


Figure 8. Closed-loop response for example 3 (loop 1, setpoint change; loop 2, setpoints remain).

Figure 9. Closed-loop response for example 3 (loop 1, setpoints remain; loop 2, setpoint change).

The overall PI plus filter controller tuning rules are

Kc )

τ1

; τI ) τ1; τF ) τz

Kp(1.414τcl + τz + D)

(67)

with τcl calculated as in eq 66. The reason for specifying

the closed-loop characteristic equation as in eq 30 is the same as that in section 2.5 to speed up the closed-loop response. For the underdamped system, the derivation of the PI plus filter controller tuning rules becomes more complicated. The closed-loop characteristic equation for the underdamped system is


Figure 10. Column open-loop step responses.

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

(τIs + 1) τI s(τ s + 1) Kc F

4. Nonlinear Rigorous Column Example

Kp(τzs + 1)e-Ds τ2s2 + 2ζτs + 1

) 0 (68)

By setting τF ) τz to cancel the process numerator dynamics, no further simplification of the characteristic equation can be made by the proper selection of τI. Using the second-order Taylor approximation for the dead-time term, the previous equation becomes

(

) (

)

τID2 3 τIτ2 2τIζτ D2 + s + + - τID s2 + KcKp 2 KcKp 2

(

)

τI + τI - D s + 1 ) 0 (69) KcKp

By setting the closed-loop characteristic equation to match a desired third-order closed-loop system (with little overshoot) as in eq 36 of section 2.5, we can obtain the parameters Kc, τI, and τcl by equating the coefficients in s3, s2, and s terms. The tuning rules for Kc and τI are

Kc )

τI Kp(2.414τcl + D - τI)

(70)

4.828ζττcl + 2ζτD + 0.5D2 - 2.414τcl2 (71) τI ) 2ζτ + D The resulting equation for τcl is again a cubic equation with rather complicated coefficients. The real root of τcl cannot be expressed explicitly and has to be solved by a numerical method. Although the tuning formula is complicated, obtaining the PI tuning parameters is straightforward for a particular underdamped example.

A 2 × 2 nonlinear distillation column control system will be used here to demonstrate the usage of the proposed tuning method in a more real industrial situation with possible mismatches in the model structure. The rigorous distillation column model was developed by Weischedel and McAvoy.18 The column separates mixtures of methanol and ethanol, with 27 theoretical trays and a product split of 0.99/0.01. For more detailed information of the dynamic model, please refer to ref 18. The control configuration is the one used by Sriniwas et al.19 and Chien et al.,20,21 in which the temperatures near the top (21st tray) and bottom (7th tray) of the distillation column are the controlled variables and the reflux flow rate and vapor flow rate from the reboiler are the manipulated variables. The controlled and manipulated variables used are defined in dimensionless form as deviation variables divided by the measurement spans (cf. work by Sriniwas et al.19). In the multiloop PID implementation, 21st tray temperature is controlled by manipulating the reflux flow rate, and 7th tray temperature is controlled by manipulating the vapor flow rate. An additional 1 min of dead time is assumed in each control loop to account for all of the other time delays that are not considered in the rigorous column dynamic simulation (such as temperature sensor delays, pseudo-steady-state assumptions for the energy balances, control valve dynamics, and so on). Figure 10 shows the process open-loop responses for (1% and (5% step changes in the dimensionless reflux (u1) and steam flows (u2). Notice that the process is highly nonlinear and also very interactive. To find the PID tuning constants for these two nonlinear control loops, a procedure similar to the one used in example 3 is used here. The initial crude PID settings were calculated via the TLC tuning method of the “average” diagonal transfer function elements of this 2 × 2 control system. The “average” diagonal transfer function ele-


Figure 11. Column dynamic responses (loop 1, open-loop (1% step changes; loop 2, automatic mode).

Figure 12. Column dynamic responses (loop 1, automatic mode; loop 2, open-loop (1% step changes).

ments are obtained from a second-order plus dead time model fitting of the open-loop responses in Figure 10. The initial crude TLC-PI settings are Kc1 ) -1.20, τI1 ) 19.4, Kc2 ) 1.14, and τI2 ) 14.0. Figure 11 shows the (1% open-loop step changes in the dimensionless reflux flow rate while the other 7th tray temperature control loop is kept in automatic control mode. One interesting observation is that the open-loop response (while the other loop in automatic control) results in much more linear responses in comparison to the nonlinear re-

sponses in Figure 10, thus making good closed-loop responses from the linear PID controller a real possibility. A standard identification technique was applied to the +1% and -1% open-loop data in Figure 11, and then the average model of this overshoot system was obtained as

() y1 u1

loop2 closed

)

-0.772(36.8s + 1)e-1.4s (21.4s + 1)(4.61s + 1)

(72)


Figure 13. Column closed-loop response (loop 1, setpoint (0.05 step changes; loop 2, setpoints remain).

Figure 14. Column closed-loop response (loop 1, setpoints remain; loop 2, setpoint (0.05 step changes).

The predicted responses from the average model are also shown in Figure 11. The average model gives quite a good fit of the process data. The PID settings can easily be calculated from the proposed tuning rules in eq 47 using the PID controller with the form of PID3. When the same procedure is applied to loop 2, the (1% open-loop step changes in the dimensionless steam flow rate can be seen in Figure 12 while the other 21st tray temperature control loop is kept in an automatic control mode. The average model is also an overshoot system such as

() y2 u2

loop1 closed

)

0.688(31.6s + 1)e-1.3s (14.1s + 1)(2.43s + 1)

(73)

Again, the PID settings can easily be calculated. To test the performance of the proposed tuning method, several closed-loop tests are performed. The first test is to make setpoint step changes of the 21st tray temperature with the magnitude of (1.5 °F (e.g., y1 (5% changes) while keeping the setpoint of the 7th tray temperature the same. The second test is to make setpoint step changes of the 7th tray temperature with


Figure 15. Column closed-loop feed composition (20% load changes with loop 1 and 2 setpoints remaining.

the magnitude of (1.5 °F (e.g., y2 (5% changes) while keeping the setpoint of the 21st tray temperature the same. The final test is to make (20% changes in the feed methanol composition while keeping the two setpoints at the same value. These three closed-loop responses can be seen in Figures 13-15, respectively. The proposed tuning method gives a very satisfactory closed-loop performance for this highly nonlinear system. 5. Conclusions The PID tuning rules based on the direct synthesis controller design method for inverse response plus dead time and also the large overshoot response plus dead time systems are derived in this paper. Both overdamped and underdamped systems are treated in this paper. Depending on the process model form, a different PID form should be used in order to achieve the desired closed-loop response. The stability lower bound of the τcl parameter in the tuning rules can also be estimated with a given estimation of the uncertainty bound of the important model parameters. If a slower closed-loop performance is permissible, PI-only tuning rules are also derived for these two difficultly controlled systems. Several examples including 2 × 2 and 3 × 3 multiloop control systems and a nonlinear distillation column system are used to demonstrate the usefulness of this proposed tuning method. Acknowledgment This work is supported by China Petroleum Co. and National Science Council of R.O.C. under Grant NSC 89-CPC-7-011-001. Helpful comments and suggestions from the paper reviewers are also gratefully acknowledged.

Literature Cited (1) Stephanopoulos, G. Chemical Process Control; Prentice Hall: Englewood Cliffs, NJ, 1984. (2) Seborg, D. E.; Edgar, T. F.; Mellichanp, D. A. Process Dynamics and Control; John Wiley & Sons: New York, 1989. (3) Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill: New York, 1990. (4) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling, and Control; Oxford University Press: Oxford, U.K., 1994. (5) Waller, K. V. T.; Nygardas, C. G. On Inverse Response in Process Control. Ind. Eng. Chem. Fundam. 1975, 14 (3), 221223. (6) Iinoya, K.; Altpeter, R. J. Inverse Response in Process Control. Ind. Eng. Chem. 1962, 54 (7), 39-43. (7) Scali, C.; Rachid, A. Analytical Design of ProportionalIntegral-Derivative Controllers for Inverse Response Processes. Ind. Eng. Chem. Res. 1998, 37 (4), 1372-1379. (8) Zhang, W.; Xu, X.; Sun, Y. Quantitative Performance Design for Inverse-Response Processes. Ind. Eng. Chem. Res. 2000, 39 (6), 2056-2061. (9) Luyben, W. L. Tuning Proportional-Integral Controllers for Processes with Both Inverse Response and Deadtime. Ind. Eng. Chem. Res. 2000, 39 (4), 973-976. (10) Papastathopoulou, H. S.; Luyben, W. L. Control of a Binary Sidestream Distillation Column. Ind. Eng. Chem. Res. 1991, 30 (4), 705-713. (11) Chang, D. M.; Yu, C. C.; Chien, I. L. Identification and Control of an Overshoot Lead-Lag Plant. J. Chin. Inst. Chem. Eng. 1997, 28 (2), 79-89. (12) Marlin, T. E. Process Control; McGraw-Hill: New York, 2000. (13) Luyben, W. L. Effect of Derivative Algorithm and Tuning Selection on the PID Control of Dead-time Processes. Ind. Eng. Chem. Res. 2001, 40 (16), 3605-3611. (14) Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25 (3), 654-660. (15) Luyben, M. L.; Luyben, W. L. Essentials of process control; McGraw-Hill: New York, 1997. (16) Tyreus, B. D. Multivariable Control System Design for an Industrial Distillation Column. Ind. Eng. Chem. Process Des. Dev. 1979, 18 (1), 177-182. (17) Yu, C. C. Autotuning of PID Controllers; Springer-Verlag: London, 1999.

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4477 (18) Weischedel, K.; McAvoy, T. J. Feasibility of Decoupling in Conventionally Controlled Distillation Columns. Ind. Eng. Chem. Fundam. 1980, 19 (4), 379-384. (19) Sriniwas, G. R.; Arkun, Y.; Chien, I. L.; Ogunnaike, B. A. Nonlinear Identification and Control of a High-Purity Distillation Column: a Case Study. J. Process Control 1995, 5 (3), 149-162. (20) Chien, I. L. Simple Empirical Nonlinear Model for Temperature-Based High-Purity Distillation Columns. AIChE J. 1996, 42 (6), 2692-2697.

(21) Chien, I. L.; Tang, Y. T.; Chang, T. S. Simple Nonlinear Controller for High-Purity Distillation Columns. AIChE J. 1997, 43 (11), 3111-3116.

Received for review September 17, 2002 Revised manuscript received December 3, 2002 Accepted December 6, 2002 IE020726Z

Simple PID Controller Tuning Method for Processes with Inverse

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