PI/PID Controller Design Based on Direct Synthesis andDisturbance Rejection
Dan Chen and Dale E. Seborg*
Department of Chemical Engineering, University of California, Santa Barbara, California 93106
A design method for PID controllers based on the direct synthesis approach and specification ofthe desired closed-loop transfer function for disturbances is proposed. Analytical expressionsfor PID controllers are derived for several common types of process models, including first-order and second-order plus time delay models and an integrator plus time delay model. Althoughthe controllers are designed for disturbance rejection, the set-point responses are usuallysatisfactory and can be tuned independently via a set-point weighting factor. Nine simulationexamples demonstrate that the proposed design method results in very good control for a widevariety of processes including those with integrating and/or nonminimum phase characteristics.The simulations show that the proposed design method provides better disturbance rejectionthan the standard direct synthesis and internal model control methods when the controllersare tuned to have the same degree of robustness.
1. Introduction
The ubiquitous PID controller has continued to be themost widely used process control technique for manydecades. Although advanced control techniques such asmodel predictive control can provide significant im-provements, a PID controller that is properly designedand tuned has proved to be satisfactory for the vastmajority of industrial control loops.1,2 The enormousliterature on PID controllers includes a wide variety ofdesign and tuning methods based on different perfor-mance criteria.3-6 Two early and well-known designmethods were reported by Ziegler and Nichols (ZN)7 andCohen and Coon.8 Both methods were developed toprovide a closed-loop response with a quarter decayratio. Other well-known formulas for PI controllerdesign include design relations based on integral errorcriteria9-11 and gain and phase margin formulas.12
The design methods for PID controllers are typicallybased on a time-domain or frequency-domain perfor-mance criterion. However, the relationships between thedynamic behavior of the closed-loop system and theseperformance indices are not straightforward. In thedirect synthesis (DS) approach,13-15 however, the con-troller design is based on a desired closed-loop transferfunction. Then, the controller is calculated analyticallyso that the closed-loop set-point response matches thedesired response. The obvious advantage of the directsynthesis approach is that performance requirementsare incorporated directly through specification of theclosed-loop transfer function. One way to specify theclosed-loop transfer function is to choose the closed-looppoles. This pole placement method4,13 can be interpretedas a special type of direct synthesis.
In general, controllers designed using the DS methoddo not necessarily have a PID control structure. How-ever, a PI or PID controller can be derived for simpleprocess models such as first- or second-order plus timedelay models by choosing appropriate closed-loop trans-fer functions.15,16 For example, the λ-tuning method was
originally proposed by Dahlin17 and is widely used inthe process industries. It is based on a first-order plustime delay model that has a relatively large time delay.The resulting controller is a PI controller with time-delay compensation.4 Also, the well-known internalmodel control (IMC) design method1,18-20 is closelyrelated to the DS method and produces identical PIDcontrollers for a wide range of problems. For higher-order systems, a model reduction technique and IMCcan be used to synthesize PID controllers.21 Alterna-tively, a high-order controller can be designed and thenreduced to PID form by a series expansion.22
DS design methods are usually based on specificationof the desired closed-loop transfer function for set-pointchanges. Consequently, the resulting DS controllerstend to perform well for set-point changes, but thedisturbance response might not be satisfactory. Forexample, the IMC-PID controller provides good set-point tracking but very sluggish disturbance responsesfor processes with a small time-delay/time-constantratio.1 However, for many process control applications,disturbance rejection is much more important than set-point tracking. Therefore, controller design that em-phasizes disturbance rejection, rather than set-pointtracking, is an important design problem that hasreceived renewed interest recently.
Middleton and Graebe23 have investigated the rela-tionship between input disturbance responses androbustness. They concluded that the decision to cancel,rather than shift, slow stable open-loop poles involvesa design tradeoff between input disturbance rejectionand robustness. Lee et al.22 extended the IMC designapproach for two degree of freedom controllers toimprove disturbance performance. Their controller is acombination of two controllers, a standard IMC control-ler for set-point changes and a second IMC type ofcontroller designed to shape the disturbance response.Their control system also includes a set-point filter thatis specified as the inverse of the IMC controller fordisturbances. This design provides a set-point responsethat is identical to that for the standard IMC controller.This novel control scheme can provide improved dy-
namic performance over standard IMC controllers, butthe design procedure is more complicated and mightresult in unstable controllers.
It is somewhat surprising that the development ofdirect synthesis design methods for disturbance rejec-tion has received relatively little attention. Early designmethods for sampled-data systems were based onspecifying the z transform of the desired closed-loopresponse to a particular disturbance.14 However, thisapproach is sensitive to the assumed disturbance anddoes not necessarily produce a PID controller. A morepromising approach was proposed recently by Szita andSanathanan.24-26 They specify the desired disturbancerejection characteristics in terms of a closed-loop trans-fer function for disturbances. The resulting controllerusually is not a PI or PID controller and might be ofhigh order. The authors propose approximating thehigh-order controller by a low-order controller usingerror minimization in the frequency domain.
In this paper, analytical expressions for PI and PIDcontrollers are derived for common process modelsthrough the direct synthesis method and disturbancerejection. The proposed design method has a singledesign parameter, the desired closed-loop time constant,τc. The performance-robustness tradeoff involved inspecifying τc is analyzed. A simple set-point weightingfactor is used to improve controller performance for set-point changes without affecting the response to distur-bances. Simulation results for nine examples demon-strate that the proposed design method provides robustPID controllers that perform well for both disturbanceand set-point changes.
2. Direct Synthesis Method Based on Set-PointResponses
In the direct synthesis approach, an analytical ex-pression for the feedback controller is derived from aprocess model and a desired closed-loop response. Inmost of the DS literature, the desired closed-loopresponse is expressed as a closed-loop transfer functionfor set-point changes. Consequently, this popular ver-sion of the direct synthesis method will be brieflyintroduced in the next section.
2.1. Direct Synthesis for Set-Point Tracking(DS). Consider a feedback control system with thestandard block diagram in Figure 1a. Assume that Gp(s)is a model of the process, measuring element, transmit-ter, and control valve.
The closed-loop transfer function for set-point changesis derived as
Rearranging gives an expression for the feedback con-troller
Let the desired closed-loop transfer function for set-pointchanges be specified as (y/r)d, and assume that a processmodel Gp(s) is available. Replacing the unknown (y/r)and Gp(s) by (y/r)d and Gp(s), respectively, gives a designequation for Gc(s)
Because the characteristics of (y/r)d have a direct impacton the resulting controller, (y/r)d should be chosen sothat the closed-loop performance is satisfactory and theresulting controller is physically realizable.
The DS controller in eq 3 results in the followingclosed-loop transfer functions
For the ideal case where the process model is perfect(i.e., Gp ) Gp), the closed-loop transfer functions become
respectively.2.2. Comparison with Internal Model Control
(IMC). A well-known control system design strategy,internal model control (IMC) was developed by Morariand co-workers20 and is closely related to the directsynthesis approach. Like the DS method, the IMCmethod is based on an assumed process model andrelates the controller settings to the model parametersin a straightforward manner. The IMC approach hasthe advantages that it makes the consideration of modeluncertainty and the making of tradeoffs between controlsystem performance and robustness easier.
The IMC approach has the simplified block diagramshown in Figure 1b, where Gp(s) is the process modeland Gc
/(s) is the IMC controller. The IMC controllerdesign involves two steps:
Step 1. The process model Gp(s) is factored as
Figure 1. Feedback control strategies. (a) Classical feedbackcontrol. (b) Internal model control.
yr
)Gp(s) Gc(s)
1 + Gp(s) Gc(s)(1)
Gc(s) )(yr)
Gp(s)[1 - (yr)](2)
Gc(s) )(yr)d
Gp(s)[1 - (yr)d](3)
(yr)DS)
Gp(yr)d
Gp + (yr)d(Gp - Gp)
(4)
(yd)DS)
GpGd[1 - (yr)d]Gp + (yr)d
(Gp - Gp)(5)
(yr)DS) (yr)d
(6)
(yd)DS) Gd[1 - (yr)d] (7)
Gp(s) ) Gp+(s) Gp-(s) (8)
4808 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002
where Gp+(s) contains any time delays and right-half-plane zeros. It is specified so that its steady-state gainis 1.
Step 2. The IMC controller is specified as
where f is a low-pass filter with a steady-state gain of1. The IMC filter f typically has the form
where τc is the desired closed-loop time constant.Parameter r is a positive integer that is selected so thateither Gc
/ is a proper transfer function or the order ofits numerator exceeds the order of the denominator by1, if ideal derivative action is allowed.
The IMC structure, Figure 1b, can be converted intothe conventional feedback control structure, Figure 1a.27
Comparing the resulting controllers and the closed-loopresponses of the IMC and direct synthesis (DS) ap-proaches, it is obvious that these two approachesproduce equivalent controllers and identical closed-loopperformances in certain situations. For example, if thedesired closed-loop response for set-point change isspecified as (y/r)d ) Gp+ f, then the DS controller isequivalent to the IMC controller, and identical closed-loop performance results, even when modeling errorsare present.
2.3. Direct Synthesis for PI/PID Controllers. Ingeneral, both the direct synthesis and IMC methods donot necessarily result in PI/PID controllers. However,by choosing the appropriate desired closed-loop responseand using either a Pade approximation or a power-seriesapproximation for the time delay, PI/PID controllers canbe derived for process models that are commonly usedin industrial applications.
Choose the desired closed-loop transfer function as
where θ is the time delay of the system and τc is thedesign parameter. Then, the DS design eq 3 and atruncated power-series expansion for the time delayterm in the denominator, e-θs ≈ 1 - θs, gives
For systems that can be described by first-order andsecond-order plus time delay models, a PI or PIDcontroller can be obtained from eq 12. For a first-orderplus time delay model
eq 12 reduces to
Equation 14 can be expressed as an ideal PI controller
with the following controller settings
For a second-order plus time delay model
substituting into eq 12 gives an ideal PID controller
with the following settings
Identical PI/PID settings have been obtained using theIMC approach.1,18
3. Direct Synthesis Design for DisturbanceRejection
The PI/PID settings obtained from the DS and IMCapproaches are based on specifying the closed-looptransfer function for set-point changes. For processeswith small time-delay/time-constant ratios, these PI/PIDcontrollers provide very sluggish disturbance responses.1Therefore, it is worthwhile to develop a modified directsynthesis approach based on disturbance rejection. Thenew design method will be denoted by “DS-d”.
Consider a control system with the standard blockdiagram shown in Figure 1a. The closed-loop transferfunction for disturbances is given by
Rearranging gives an expression for the feedback con-troller
Let the desired closed-loop transfer function for distur-bances be specified as (y/d)d, and assume that a processmodel Gp(s) and a disturbance model Gd(s) are available.Replacing the unknown (y/d), Gp(s), and Gd(s) by (y/d)d,Gp(s), and Gd(s), respectively, gives a design equationfor Gc(s)
Gc/(s) ) 1
Gp-(s)f (9)
f ) 1(τcs + 1)r
(10)
(yr)d) e-θs
τcs + 1(11)
Gc ) 1Gp
e-θs
(τc + θ)s(12)
Gp(s) ) Ke-θs
τs + 1(13)
Gc ) τs + 1K(τc + θ)s
(14)
GPI(s) ) Kc(1 + 1τIs) (15)
Kc ) 1K
ττc + θ
(16)
τI ) τ (17)
Gp(s) ) Ke-θs
(τ1s + 1)(τ2s + 1)(18)
GPID(s) ) Kc(1 + 1τIs
+ τDs) (19)
Kc ) 1K
τ1 + τ2
θ + τc(20)
τI ) τ1 + τ2 (21)
τD )τ1τ2
τ1 + τ2(22)
yd
)Gd(s)
1 + Gp(s) Gc(s)(23)
Gc(s) )Gd(s)
(yd)Gp(s)- 1
Gp(s)(24)
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4809
For the DS-d controller in eq 25, the closed-loop transferfunctions are
For the ideal case where the model is perfect (i.e., Gp )Gp and Gd ) Gd), the closed-loop transfer functionsbecome
respectively.The DS-d design method does not necessarily produce
a PI or PID controller. The structure and order of thecontroller depend on the specification of the desiredclosed-loop response and the process model. In thissection, the new DS-d design method is used to derivePI/PID controllers for simple process models that arewidely used.
3.1. DS-d PI Settings. The DS-d methods will nowbe used to design PI controllers for a first-order plustime delay model and then an integrator plus time delaymodel.
3.1.1. First-Order Plus Time Delay Model. As-sume that the process is described by a first-order plustime-delay model
and that Gd(s) ) Gp(s). (This latter assumption will beremoved in section 3.3). Thus, if the PI controller in eq15 is used, the closed-loop transfer function in eq 23 canbe expressed as
Approximating the time delay term in the denominatorby a first-order power-series expansion, e-θs ≈ 1 - θs,and rearranging gives
Therefore, for PI controller design, it is reasonable tospecify the desired closed-loop transfer function as
with
The DS-d design equation, eq 25, produces a standardPI controller if the time delay in the denominator isapproximated by, e-θs ≈ 1 - θs. The resulting PIcontroller parameters are
and Kd is given by
From eqs 35 and 36, it is apparent that, for largevalues of τc, anomalous results can occur because Kc andK can have opposite signs and τI can become negative.Both of these undesirable situations can be avoided,however, if the design parameter τc satisfies
This constraint on τc is not restrictive at all becausedirect synthesis controllers are typically designed sothat τc < 2τ.
For this PI controller and Gp ) Gp, the closed-looptransfer function for set-point changes is
3.1.2. Integrator Plus Time Delay Model. Pro-cesses with integrating characteristics are quite com-mon in the process industries. Assume that the processis described by
and that Gd(s) ) Gp(s). If a PI controller is used, theclosed-loop transfer function for disturbances in eq 23becomes
Approximating the time delay term in the denominatorby e-θs ≈ 1 - θs gives
Gc(s) )Gd(s)
(yd)dGp(s)
- 1Gp(s)
(25)
(yr)DS-d)
Gp[Gd - (yd)d]GpGd + (yd)d
(Gp - Gp)(26)
(yd)DS-d)
GpGd(yd)d
GpGd + (yd)d(Gp - Gp)
(27)
(yr)DS-d) 1 -
(yd)d
Gd(s)(28)
(yd)DS-d) (yd)d
(29)
Gp(s) ) Ke-θs
τs + 1(30)
yd
)
Ke-θs
τs + 1
1 + Ke-θs
τs + 1Kc(1 + 1
τIs)(31)
(yd) ≈
τI
Kcse-θs
( τK
τI
Kc- τIθ)s2 + ( τI
KKc+ τI - θ)s + 1
(32)
(yd)d)
Kdse-θs
(τcs + 1)2(33)
Kd )τI
Kc(34)
Kc ) 1K
τθ + 2ττc - τc2
(τc + θ)2(35)
τI )τθ + 2ττc - τc
2
τ + θ(36)
Kd ) K(τc + θ)2
τ+θ(37)
0 < τc < τ + xτ2 + τθ (38)
(yr)DS-d ≈ τIs + 1
(τcs + 1)2e-θs (39)
Gp(s) ) Ke-θs
s(40)
yd
)
Ke-θs
s
1 + Ke-θs
sKc(1 + 1
τIs)(41)
4810 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002
Therefore, if the desired closed-loop transfer functionfor disturbances is specified as
with Kd ) τI/Kc, then the controller obtained using theDS-d method can be rearranged to give a standard PIcontroller with
and Kd is given by
For this controller and Gp ) Gp, the closed-loopresponse for set-point changes is
3.2. DS-d PID Settings. In this section, the proposedDS-d method is used to design PID controllers for somecommonly used process models, including first-order andsecond-order plus time delay models and an integratorplus time delay model.
3.2.1. First-Order Plus Time Delay Model. As-sume that the process is described by a first-order plustime delay model
and that Gd(s) ) Gp(s). Thus, if the PID controller in eq19 is used, the closed-loop transfer function in eq 23 canbe expressed as
Approximating the time delay term in the denominatorby a first-order Pade approximation
and rearranging gives
Thus, for PID controller design, it is reasonable tospecify the desired closed-loop transfer function as
with Kd ) τI/Kc. Then, the controller obtained from theDS-d method, eq 25, can be rearranged to give astandard PID controller. The resulting PID controllerparameters are
and Kd is given by
It is apparent from the above equations that KKc, τI,and τD can be negative for large values of τc. However,simulation experience has demonstrated that this po-tential problem does not occur if the closed-loop timeconstant τc is chosen in a reasonable manner.
For this PID controller and Gp ) Gp, the closed-looptransfer function for set-point changes is obtained as
3.2.2. Integrator Plus Time Delay Model. Nowassume that the process is described by
and that Gd(s) ) Gp(s). If a PID controller is used, theclosed-loop transfer function in eq 23 becomes
(yd) ≈ [ τI
Kcs(1 + θ
2s) e-θs]/[( θττI
2KKc- θ
2τIτD)s3 +
((τ + θ/2)τI
KKc+ τIτD - θ
2τI)s2 + ( τI
KKc+ τI - θ
2)s + 1](51)
(yd)d)
Kds(1 + θ2s)e-θs
(τcs + 1)3(52)
Kc ) 1K
(2τθ + θ2
2 )(3τc + θ2) - 2τc
3 - 3τc2θ
2τc3 + 3τc
2θ + θ2
2 (3τc + θ2)
(53)
τI )(2τθ + θ2
2 )(3τc + θ2) - 2τc
3 - 3τc2θ
(2τ + θ)θ(54)
τD )3τc
2τθ + τθ2
2 (3τc + θ2) - 2(τ + θ)τc
3
(2τθ + θ2
2 )(3τc + θ2) - 2τc
3 - 3τc2θ
(55)
Kd ) K2τc
3 + 3τc2θ + θ2
2 (3τc + θ2)
(2τ + θ)θ(56)
(yr)DS-d ≈(τIτDs2 + τIs + 1)(1 + θ
2s)
(τcs + 1)3e-θs (57)
Gp(s) ) Ke-θs
s(58)
yd
)
Ke-θs
s
1 + Ke-θs
sKc(1 + 1
τIs+ τDs)
(59)
(yd) ≈
τI
Kcse-θs
( τI
KKc- θτI)s2 + (τI - θ)s + 1
(42)
(yd)d)
Kdse-θs
(τcs + 1)2(43)
Kc ) 1K
2τc + θ
(τc + θ)2(44)
τI ) 2τc + θ (45)
Kd ) K(τc + θ)2 (46)
(yr)DS-d ≈ τIs + 1
(τcs + 1)2e-θs (47)
Gp(s) ) Ke-θs
τs + 1(48)
yd
)
Ke-θs
τs + 1
1 + Ke-θs
τs + 1Kc(1 + 1
τIs+ τDs)
(49)
e-θs ≈1 - θ
2s
1 + θ2s
(50)
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4811
Approximating the time delay term in the denominatorby eq 50 gives
Therefore, if the desired closed-loop transfer functionfor disturbances is specified as
with Kd ) τI/Kc, then the controller obtained using theDS-d method can be rearranged to give a standard PIDcontroller. The resulting PID controller parameters are
and Kd is given by
For this controller and Gp ) Gp, the closed-loop responsefor set-point change is obtained as
3.2.3. Second-Order Plus Time Delay Model.Assume that the process is described by a second-orderplus time delay model
and that Gd(s) ) Gp(s). Thus, if a PID controller is used,the closed-loop transfer function in eq 23 can beexpressed as
Approximating the time delay term in the denominatorby a first-order power-series expansion, e-θs ≈ 1 - θs,and rearranging gives
Therefore, if the desired closed-loop transfer functionfor disturbance is specified as
with Kd ) τI/Kc, then the controller obtained from theDS-d method, eq 25, can be rearranged to give astandard PID controller. The resulting PID controllerparameters are
and Kd is given by
For this PID controller and Gp ) Gp, the closed-looptransfer function for set-point changes is
3.2.4. First-Order with an Integrator Plus TimeDelay Model. Assume that the process is described by
and that Gd(s) ) Gp(s). If a PID controller is used, thenthe closed-loop transfer function in eq 23 becomes
Approximating the time delay term in the denominatorby e-θs ≈ 1 - θs gives
Therefore, if the desired closed-loop transfer function
(yd) ≈ [ τI
Kcs(1 + θ
2s) e-θs]/[( θτI
2KKc- θ
2τIτD)s3 +
( τI
KKc+ τIτD - θ
2τI)s2 + (τI - θ
2)s + 1] (60)
(yd)d)
Kds(1 + θ2s)e-θs
(τcs + 1)3(61)
Kc ) 1K
θ(3τc + θ2)
(τc + θ2)3
(62)
τI ) 3τc + θ2
(63)
τD )
32
τc2θ + 3
4τcθ
2 + θ3
8- τc
3
θ(3τc + θ2)
(64)
Kd ) K(τc + θ
2)3
θ(65)
(yr)DS-d ≈(τIτDs2 + τIs + 1)(1 + θ
2s)
(τcs + 1)3e-θs (66)
Gp(s) ) Ke-θs
(τ1s + 1)(τ2s + 1)(67)
yd
)
Ke-θs
(τ1s + 1)(τ2s + 1)
1 + Ke-θs
(τ1s + 1)(τ2s + 1)Kc(1 + 1
τIs+ τDs)
(68)
(yd) ≈ [ τI
Kcse-θs]/[(τ1τ2τI
KKc- θτIτD)s3 +
((τ1 + τ2)τI
KKc+ τIτD - θτI)s2 + ( τI
KKc+ τI - θ)s + 1]
(69)
(yd)d)
Kdse-θs
(τcs + 1)3(70)
Kc ) 1K
[(τ1 + τ2)θ + τ1τ2](3τc + θ) - τc3 - 3τc
2θ
(τc + θ)3(71)
τI )[(τ1 + τ2)θ + τ1τ2](3τc + θ) - τc
3 - 3τc2θ
τ1τ2 + (τ1 + τ2 + θ)θ(72)
τD )3τc
2τ1τ2 + τ1τ2θ(3τc + θ) - (τ1 + τ2 + θ)τc3
[(τ1 + τ2)θ + τ1τ2](3τc + θ) - τc3 - 3τc
2θ(73)
Kd ) K(τc + θ)3
τ1τ2 + (τ1 + τ2 + θ)θ(74)
(yr)DS-d ≈ (τIτDs2 + τIs + 1)
(τcs + 1)3e-θs (75)
Gp(s) ) Ke-θs
s(τs + 1)(76)
yd
)
Ke-θs
s(τs + 1)
1 + Ke-θs
s(τs + 1)Kc(1 + 1
τIs+ τDs)
(77)
(yd) ≈ [ τI
Kcse-θs]/[( ττI
KKc- θτIτD)s3 +
( τI
KKc+ τIτD - θτI)s2 + (τI - θ)s + 1] (78)
4812 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002
for disturbance is specified as
with Kd ) τI/Kc, then the controller obtained from theDS-d method can be rearranged to give a standard PIDcontroller. The resulting PID controller parameters are
and Kd is give by
For this PID controller and Gp ) Gp, the closed-looptransfer function for set-point changes is
3.3. Discussion. In the previous sections, the DS-dmethod has been used to design PI/PID controllers forwidely used process models. The resulting PI/PIDcontroller settings are shown in Table 1.
Remark 1. The only design parameter, τc, is directlyrelated to the closed-loop time constant. As τc decreases,the closed-loop response becomes faster.
Remark 2. Larger values of τc give larger values ofKd because Kd ) τI/Kc.
For a unit step disturbance at the process input,A° strom and Hagglund4 derived the following relationfor the integral error (IE) associated with PI/PID control
Thus
This expression means that smaller τc values providesmaller IE values for step disturbances.
Remark 3. Although smaller values of τc providebetter performance for disturbance and set-point changes,the control system robustness is worse. Therefore, thesystem robustness should be considered when τc is beingchosen.
Remark 4. The PI/PID tuning rules in Table 1 werederived based on the assumption that Gd ) Gp. For themore general case where Gd * Gp, the desired closed-
Table 1. PI/PID Controller Settings for the DS-d Design Method
where (y/d)d is given in Table 1. Thus, the PI/PIDsettings and the closed-loop responses for set-pointchanges are the same as for the special case where Gd) Gp.
Remark 5. The proposed PI tuning rules for integrat-ing processes are equivalent to the IMC PI settings ofChien and Fruehauf.1
3.4. Set-Point and Derivative Weighting. Equa-tions 15 and 19 are conventional PI and PID controllers.A more flexible control structure that includes set-pointweighting and derivative weighting is given by A° stromand Hagglund4
where the set-point weighting coefficient b is boundedby 0 e b e 1 and the derivative weighting coefficient cis also bounded by 0 e c e 1. The overshoot for set-point changes decreases with increasing b.
The controllers obtained for different values of b andc respond to disturbances and measurement noise in thesame way as conventional PI/PID controllers, i.e., dif-ferent values of b and c do not change the closed-loopresponse for disturbances. Therefore, the same PI/PIDtuning rules developed here using the DS-d method arealso applicable for the modified PI/PID controller in eq88. However, the set-point response does depend on thevalues of b and c. If set-point weighting and derivativeweighting are used, the closed-loop transfer function forset-point changes is given by
where Gc(s) is the conventional PID controller given byeq 19.
4. Simulation Results
Several simulation examples are used to demonstratethe proposed PI/PID tuning rules for the DS-d method.In practice, the derivative weighting factor c is usuallyset to zero to avoid a large derivative kick. Thus, in thispaper, c is chosen to be zero for all of the simulationexamples. Furthermore, the PID controller is imple-mented in the widely used “parallel form”
The derivative filter parameter R is specified as R ) 0.1.Other implementations of PID control, such as the seriesform, are also widely used. The controller settings forone form can easily be converted to other forms.3
For each example, the DS-d, DS, ZN, and/or someother methods were used to design PI/PID controllers.As mentioned in section 2.3, the DS and IMC methodscan provide identical PI/PID settings if the same closed-
loop transfer function is specified and the same ap-proximation is used for time delay term. For someprocess models, however, the IMC tuning rules are verywell-known. Thus, the IMC method was used insteadof the DS method for a few examples.
The following robustness and performance metricswere used as evaluation criteria for the comparison ofthe PI/PID controllers:
Robustness Metric. The peak value of the sensitiv-ity function, MS,28 has been widely used as a measureof system robustness. Recommended values of MS aretypically in the range of 1.2-2.0.29
To provide fair comparisons, the model-based control-lers (DS-d, DS, and IMC) were tuned by adjusting τc sothat the MS values were very close. This tuning facili-tated a comparison of controller performance for dis-turbance and set-point changes for controllers that hadthe same degree of robustness.
Performance Metrics. Two metrics were used toevaluate controller performance. The integrated abso-lute error (IAE) is defined as15
To evaluate the required control effort, the total varia-tion (TV) of the manipulated input u was calculated
The total variation is a good measure of the “smooth-ness” of a signal and should be as small as possible.30
4.1. Example 1. Consider the following model witha step disturbance acting at the plant input31
The DS-d, IMC, and ZN methods were used to designthe PID controllers shown in Table 2. For the DS-dmethod, a value of τc ) 1.2 was chosen so that MS )1.94. To obtain a fair comparison, τc ) 0.85 was selectedfor the IMC method so that MS ) 1.94.
The simulation results in Figure 2 and the IAE andTV values in Table 2 indicate that the disturbanceresponse for the DS-d controller is much better andfaster than the IMC response, whereas the movementsof these two controller outputs are similar. The distur-bance response of the ZN controller has a smaller peakvalue but is more oscillatory than the DS-d response.The set-point response of the DS-d controller is moresluggish and has a larger overshoot than the IMCcontroller. However, the overshoot for the DS-d control-ler can be eliminated, without affecting the disturbanceresponse, by setting b ) 0.5. By comparing the responsesand the MS, IAE, and TV values of all three controllers,
(yd)d
/
) (yd)d
Gd
Gp(87)
u(t) ) Kc([br(t) - y(t)] + 1τI∫0
t[r(τ) - y(τ)] dτ +
τDd[cr(t) - y(t)]
dt ) (88)
yr
)cτIτDs2 + bτIs + 1
τIτDs2 + τIs+1
Gp(s) Gc(s)
1 + Gp(s) Gc(s)( Gsp(s) (89)
Gc(s) ) Kc(1 + 1τIs
+τDs
RτDs + 1) (90)
Table 2. PID Controller Settings for Example 1 (θ/τ )0.01)
it can be concluded that the DS-d controller providesthe best performance without using excessive controleffort.
4.2. Example 2. Consider a process is described by
The PI controller characteristics for the DS-d, DS, andZN controllers are shown in Table 3.
Unit step changes were introduced in the set point(at t ) 0) and in the disturbance (at t ) 4 min). Thesimulation results in Figure 3 indicate that the distur-bance response of the DS-d PI controller is faster thanthat of the DS PI controller and less oscillatory thanthat of the ZN PI controller. The set-point response ofthe DS-d PI controller exhibits overshoot, but it can bereduced by setting b ) 0.5. These conclusions can beconfirmed by the IAE and TV values in Table 3.
Figure 4 shows the simulation results for the practicalsituation where there are inequality constraints on themanipulated variable u: -2 e u e 2. A comparison ofFigures 3 and 4 indicates that the initial set-pointresponses of the DS-d (b ) 1), DS, and ZN PI controllerswere slower because of controller saturation. Also, theoscillations for the DS and ZN PI controllers weredamped. The disturbance responses were not affectedbecause inequality constraints on u were not active.
Examples 1 and 2 demonstrate that the DS-d methodprovides better performance than the DS and IMCmethods for processes with small θ/τ ratios. The follow-
Figure 2. Simulation results of PID controllers for example 1(θ/τ ) 0.01). (a) Response to a unit step set-point change. (b)Response to a unit step disturbance.
Table 3. PI Controller Settings for Example 2 (θ/τ ) 0.25)
Figure 3. Simulation results without u constraints for example2 (θ/τ ) 0.25). (a) Controlled variable y. (b) Manipulated variableu.
Figure 4. Simulation results with u constraints for example 2(θ/τ ) 0.25). (a) Controlled variable y. (b) Manipulated variable u.
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4815
ing example is used to illustrate how the proposed DS-dmethod works for processes with larger θ/τ ratios.
4.3. Example 3. Different Values of the θ/τ Ratio.Consider the process model
In this example, two values of θ/τ ratio are considered:θ/τ ) 1 and θ/τ ) 5.
For θ/τ ) 1, PI and PID controllers were designedusing the DS-d, DS/IMC, and ZN methods. The control-ler characteristics are shown in Tables 4 and 5. Thesimulation results in Figures 5 and 6 indicate that thePI and PID responses for the DS-d and DS/IMC methodsare very similar. The DS-d and DS/IMC controllersprovide better performance than the ZN controllers.
For the process with θ/τ ) 5, the DS-d, DS/IMC, andZN methods were used to design PI and PID controllers.Luyben2 has used this large-time-delay example to
compare the IMC, Ciancone-Marlin (CM),34 and ZNmethods. Consequently, the CM PI and PID settingswere also considered for comparison. The CM tuningrules were derived using an optimization procedure thatincorporates considerations of performance, robustness,and saturation of the manipulated variable. The CMtuning rules are valid only for first-order plus time delayprocesses and are presented graphically.34
The controller characteristics are given in Tables 6and 7. The responses to unit step changes in the setpoint and disturbance are shown in Figures 7 and 8.The DS-d and DS PI controllers give the fastest PIresponses with small overshoots that are very similar.The responses of the ZN PI controller are very sluggish.The CM PI controller provides the best performanceamong all four PI controllers. For the PID controllers,the responses of the DS-d, IMC, and CM controllers arerelatively close. The ZN PID controller gives very erratic
Figure 5. Simulation results of PI controllers for example 3 (θ/τ) 1).
Figure 6. Simulation results of PID controllers for example 3(θ/τ ) 1).
Table 4. PI Controller Settings for Example 3 (θ/τ ) 1)
responses that are not shown in the plot but can befound in Luyben.2 Its MS value in Table 7 is extremelylarge.
The results for these two θ/τ values illustrate that theDS-d method provides better disturbance rejection thanthe DS method for processes with small values of θ/τ.As θ/τ becomes larger, the controller settings and theclosed-loop performance for the DS-d method becomecloser to those for the conventional DS method.
4.4. Example 4. Effect of τc. The DS-d method hasa single tuning parameter, τc, that is directly related tothe speed of the closed-loop response. In this example,the effect of τc is analyzed. Consider the general model
For three different values of θ/τ (0.25, 0.5, 1), DS-d PIcontroller settings were calculated for different τc values.
Figure 9 shows the DS-d PI controller settings fordifferent values of τc. As τc increases, Kc decreases, andthe integral time, τI, increases if τc e 1 but decreases ifτc g 1. The symmetry of τI around τc ) τ is confirmedby eq 36.
Similarly, for three different θ/τ values (0.25, 0.5,0.75), the DS-d PID controller settings were calculatedfor different τc values and are shown in Figure 10. Asτc increases, Kc decreases, while τI and τD first increase
and then decrease. If τc is very large, τI and τD becomenegative. Thus, for this example, the upper bounds onτc for positive Kc, τI, and τD values obtained from Figure10 are
For θ/τ ) 0.25, the DS-d PID controllers were de-signed for four values of τc (0.25, 0.3, 0.35, 0.4), and the
Figure 9. DS-d PI controller settings for different τc values(example 4).
Gp(s) ) Gd(s) ) e-θs
s + 1(96)
Figure 10. DS-d PID controller settings for different τc values(example 4).
τc e 0.4 for θ/τ ) 0.25 (97)
τc e 0.7 for θ/τ ) 0.5 (98)
τc e 0.935 for θ/τ ) 0.75 (99)
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4817
corresponding simulation results are shown in Figure11. The simulation results demonstrate that the DS-dcontrollers for smaller τc values provide disturbanceresponses with smaller peaks and smaller IE values.
4.5. Example 5. Different Disturbance TimeConstants. In this example, a process described by thesame transfer function as in example 2
but disturbance transfer functions with different valuesof τd is considered
PID controllers were designed using the DS-d, IMC, andZN methods and assuming that τd ) τ. The controllercharacteristics are shown in Table 8. The DS-d control-ler was modified for other values of τd by using eq 87.After substituting the transfer functions for the processand disturbance, the closed-loop transfer function forthe DS-d PID controller in eq 87 becomes
Thus, when τd/τ increases, the disturbance response ofthe DS-d PID controller has a smaller peak value andis more sluggish.
Because τd does not affect the set-point response, onlythe disturbance responses were simulated. The simula-tion results for four τd values (0.25, 0.5, 1, 2) shown inFigure 12 confirm that DS-d controllers provide distur-bance responses with smaller peak values but longersettling times as τd/τ increases. For small τd/τ values,the disturbance performance of the IMC controller isbetter than that of the DS-d controller.
4.6. Example 6. Consider a second-order plus timedelay system described by Seborg et al.15
The DS-d, DS, and ZN methods were used to design PIDcontrollers, and the resulting controller settings areshown in Table 9.
The simulation results for a unit step change in theset point at t ) 0 min and a unit step disturbance at t) 50 min are presented in Figure 13. The disturbanceresponse for the DS-d PID controller is fast and has asmall peak value, but the set-point response has anovershoot. By using a set-point weighting factor of b )0.5, the overshoot is eliminated. The set-point anddisturbance responses for the DS PID controller arequite sluggish, whereas the ZN PID controller providesvery oscillatory responses. Again, the IAE and TVvalues in Table 9 confirm that the DS-d controllerprovides superior performance without using excessivecontrol efforts.
4.7. Example 7. Distillation Column Model. Adistillation column that separates a small amount of alow-boiling material from the final product was consid-ered by Chien and Fruehauf.1 The bottom level of thedistillation column is controlled by adjusting the steamflow rate. The process model for the level control systemis an integrator with a time delay
The DS controller obtained from eq 12 has the form
Because the DS method results in a proportional-onlycontroller for the integrating process, only the DS-dtuning methods was used to design PI controllers. Notethat the DS-d and IMC methods provide identical PItuning rules for this type of process model. The resultingcontroller settings from the DS-d method are shown inTable 10. The IMC PI controller settings with τc ) 8used by Chien and Fruehauf1 are included in the table.Furthermore, for integrator plus time delay processes,Tyreus and Luyben35 have developed a design methodthat yields the best PI settings attainable for a specifieddegree of closed-loop damping. Their tuning rule can beexpressed in terms of the ultimate gain Ku and ultimatefrequency Pu as Kc ) Ku/3.22 and τI ) 2.2Pu. Thus, it isa modified version of ZN tuning. The TL PI controllersettings35 are included in Table 10 for comparison.
The simulation results for a unit step change in theset point (at t ) 0) and a 0.5 step disturbance (at t )150 min) are shown in Figure 14. The disturbanceperformance of the DS-d controller is good, but the set-point response has a large overshoot that can beeliminated by setting b ) 0.5. The IMC PI controllerdesigned by Chien and Fruehauf1 provides a fasterdisturbance response than the DS-d method, but the set-point response is too aggressive, as confirmed by thelarge MS and TV values in Table 10. The use of set-point weighting can reduce the large set-point overshootfor the IMC controller, but it would not affect theoscillatory nature of the response. The large MS value
Figure 11. DS-d PID controllers for different τc values and b )1 (example 4, θ/τ ) 0.25).
Table 8. PID Controller Settings for Example 5 (θ/τ )0.25)
indicates that the τc value used by Chien and Fruehauf1
is so small that the resulting system has poor robust-ness. The responses of the TL PI controller are verysluggish because of the large τI value.
4.8. Example 8. Level Control Problem. Considera level control problem given by Seborg et al.15 Theliquid level in a reboiler of a steam-heated distillationcolumn is to be controlled by adjusting the control valve
Figure 12. Disturbance responses for PID control and different τd values (example 5, θ/τ ) 0.25). (a) τd ) 0.25. (b) τd ) 0.5. (c) τd ) 1.(d) τd ) 2.
on the steam line. The process transfer function is givenby
The PID settings obtained from the DS-d, IMC, and ZNmethods are shown in Table 11.
The simulation results for a unit step change in theset point at t ) 0 min and a unit step disturbance at t) 50 min are presented in Figure 15. The disturbanceresponse for the DS-d PID controller is good, but theset-point response has an overshoot. With a set-pointweighting factor set at b ) 0.5, the DS-d controllerprovides a fast set-point response without overshoot.The performance of the IMC PID controller is close tothat of the DS-d PID controller, which is confirmed bythe similar PID settings. The ZN controller producesvery oscillatory responses.
4.9. Example 9. Fourth-Order Process. Considera fourth-order process described by29,30
For high-order processes, the DS-d, DS, and IMC designmethods do not yield PI/PID controllers directly. Thus,the model order must be reduced, or the resultingcontroller must be approximated by a PI/PID controller.Skogestad30 has proposed a simple method of ap-proximating high-order models with low-order models.He also derived modified IMC rules, which were named“simple control” or “Skogestad IMC” (SIMC). For thefirst-order model in eq 30, the PI controller obtainedfrom SIMC has the following parameters
For the second-order model in eq 67 with τ1 > τ2, theSIMC method provides the following rules for PIDcontrollers with the series structure
Using Skogestad’s approximation method, the fourth-order process in eq 107 can be approximated as a first-order plus time delay model
or as a second-order plus time delay model
Both models provide accurate approximations, but thesecond-order model is more accurate.
PI controllers were designed using the approximatefirst-order model and the DS-d, DS, and SIMC methods.Values of τc ) 0.4 and 0.148 were selected for the DS-dand DS methods, respectively, so that their MS valueswere very close to the 1.59 value for the SIMC controllerreported by Skogestad.30 For τc ) 0.148, the same PIcontroller settings were calculated using the DS andSIMC methods. Also, the TL method,35 which wasdiscussed in example 7, was used to design a PIcontroller for the process. The PI settings are shown inTable 12.
PID controllers were designed using the approximatesecond-order model and the DS-d, IMC, and SIMCmethods. The τc values for these methods were selectedto give MS values close to the 1.58 value of Table 13.Because the tuning rules for the SIMC method are forthe series PID structure, these PID settings wereconverted to the parallel structure, which was then usedfor the simulation. Also, the ZN PID settings werecalculated for the fourth-order model. The PID controllercharacteristics are shown in Table 13.
The simulation responses for a unit step change inthe set point at t ) 0 min and a step disturbance (d )3) at t ) 5 min are given in Figures 16 and 17 for PIand PID controllers, respectively. The values of MS, IAE,and TV for all of these controllers are presented inTables 12 and 13. Figure 16 indicates that the distur-bance response of the DS-d PI controller is better andfaster than the responses of the DS and SIMC control-lers. The IAE values for the TL PI controller are smallerthan those of the other PI controllers, but its responsesare oscillatory and its MS and TV values are large. A
comparison of the PID controllers in Figure 17 indicatesthat the DS-d PID controller provides better and fasterdisturbance responses than the other PID controllers.The large overshoot for the set-point response of theDS-d PID controller can be eliminated by setting b )0.5. The superior performance of the DS-d PID controlleris also confirmed by the IAE and TV values in Table13. Furthermore, a significant improvement was ob-tained using PID control instead of PI control, becausethis process is a dominant second-order process.
An alternative approach for high-order systems is todesign the DS-d controller using eq 25 and then reducethe resulting high-order controller using a series expan-sion22 or a frequency domain approximation.24-26 Be-cause this approach is more complicated than Skoges-tad’s model reduction approach, it was not applied here.
5. Conclusions
A new direct synthesis method for controller designbased on disturbance rejection (DS-d), rather than set-point tracking, has been developed. By specifying thedesired closed-loop transfer function properly, PI/PIDcontrollers can be synthesized for widely used processmodels such as first-order and second-order plus timedelay models and integrator plus time delay models. Forhigher-order models, PID controllers can be derived byapproximating the high-order model with a low-ordermodel or by approximating the high-order controllerusing either a series expansion or a frequency domainapproximation.
In the proposed DS-d design method, the closed-looptime constant τc is the only design parameter, and ithas a straightforward relation to the disturbance rejec-
tion characteristics. Thus, the proposed design proce-dure is simple and easy to implement. Although the PI/PID controllers are designed for disturbance rejection,the set-point responses are usually satisfactory and canbe independently tuned via a standard set-point weight-ing factor or a set-point filter constant. The set-pointtuning does not affect the disturbance response.
Nine simulation examples have been used to comparealternative design methods and to illustrate the effectof τc. The simulation results demonstrate that the DS-dmethod provides better disturbance performance thanstandard DS and IMC methods and that satisfactoryresponses to set-point changes can be obtained by simplytuning the set-point weighting factor b. The DS-dmethod furnishes a convenient and flexible designmethod that provides good performance in terms ofdisturbance rejection and set-point tracking.
Acknowledgment
The authors acknowledge the UCSB Process ControlConsortium for financial support and Professor SigurdSkogestad for his helpful comments and suggestions.
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Received for review September 7, 2001Revised manuscript received January 22, 2002
