Ind. Eng. Chem. Res. 2002, 41, 4807-4822 4807

PI/PID Controller Design Based on Direct Synthesis and Disturbance 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 of the desired closed-loop transfer function for disturbances is proposed. Analytical expressions for 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. Although the controllers are designed for disturbance rejection, the set-point responses are usually satisfactory and can be tuned independently via a set-point weighting factor. Nine simulation examples demonstrate that the proposed design method results in very good control for a wide variety of processes including those with integrating and/or nonminimum phase characteristics. The simulations show that the proposed design method provides better disturbance rejection than the standard direct synthesis and internal model control methods when the controllers are tuned to have the same degree of robustness.

1. Introduction originally proposed by Dahlin17 and is widely used in the process industries. It is based on a first-order plus The ubiquitous PID controller has continued to be the time delay model that has a relatively large time delay. most widely used process control technique for many The resulting controller is a PI controller with time- decades. Although advanced control techniques such as delay compensation.4 Also, the well-known internal model predictive control can provide significant im- model control (IMC) design method1,18-20 is closely provements, a PID controller that is properly designed related to the DS method and produces identical PID and tuned has proved to be satisfactory for the vast 1,2 controllers for a wide range of problems. For higher- majority of industrial control loops. The enormous order systems, a model reduction technique and IMC literature on PID controllers includes a wide variety of can be used to synthesize PID controllers.21 Alterna- design and tuning methods based on different perfor- tively, a high-order controller can be designed and then mance criteria.3-6 Two early and well-known design reduced to PID form by a series expansion.22 methods were reported by Ziegler and Nichols (ZN)7 and Cohen and Coon.8 Both methods were developed to DS design methods are usually based on specification provide a closed-loop response with a quarter decay of the desired closed-loop transfer function for set-point ratio. Other well-known formulas for PI controller changes. Consequently, the resulting DS controllers design include design relations based on integral error tend to perform well for set-point changes, but the criteria9-11 and gain and phase margin formulas.12 disturbance response might not be satisfactory. For - The design methods for PID controllers are typically example, the IMC PID controller provides good set- based on a time-domain or frequency-domain perfor- point tracking but very sluggish disturbance responses mance criterion. However, the relationships between the for processes with a small time-delay/time-constant dynamic behavior of the closed-loop system and these ratio.1 However, for many process control applications, performance indices are not straightforward. In the disturbance rejection is much more important than set- direct synthesis (DS) approach,13-15 however, the con- point tracking. Therefore, controller design that em- troller design is based on a desired closed-loop transfer phasizes disturbance rejection, rather than set-point function. Then, the controller is calculated analytically tracking, is an important design problem that has so that the closed-loop set-point response matches the received renewed interest recently. desired response. The obvious advantage of the direct Middleton and Graebe23 have investigated the rela- synthesis approach is that performance requirements tionship between input disturbance responses and are incorporated directly through specification of the robustness. They concluded that the decision to cancel, closed-loop transfer function. One way to specify the rather than shift, slow stable open-loop poles involves closed-loop transfer function is to choose the closed-loop a design tradeoff between input disturbance rejection poles. This pole placement method4,13 can be interpreted and robustness. Lee et al.22 extended the IMC design as a special type of direct synthesis. approach for two degree of freedom controllers to In general, controllers designed using the DS method improve disturbance performance. Their controller is a do not necessarily have a PID control structure. How- combination of two controllers, a standard IMC control- ever, a PI or PID controller can be derived for simple ler for set-point changes and a second IMC type of process models such as first- or second-order plus time controller designed to shape the disturbance response. delay models by choosing appropriate closed-loop trans- Their control system also includes a set-point filter that fer functions.15,16 For example, the λ-tuning method was is specified as the inverse of the IMC controller for disturbances. This design provides a set-point response * Corresponding author. Tel.: (805) 893-3352. Fax: (805) that is identical to that for the standard IMC controller. 893-4731. E-mail: [email protected]. This novel control scheme can provide improved dy-

10.1021/ie010756m CCC: $22.00 © 2002 American Chemical Society Published on Web 08/16/2002 4808 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

G (s) G (s) y ) p c + (1) r 1 Gp(s) Gc(s)

Rearranging gives an expression for the feedback con- troller y (r) G (s) ) (2) c y G (s) 1 - p [ (r)] Let the desired closed-loop transfer function for set-point changes be specified as (y/r)d, and assume that a process model G˜ p(s) is available. Replacing the unknown (y/r) Figure 1. Feedback control strategies. (a) Classical feedback ˜ control. (b) Internal model control. and Gp(s)by(y/r)d and Gp(s), respectively, gives a design equation for Gc(s) namic performance over standard IMC controllers, but y the design procedure is more complicated and might ) (r)d result in unstable controllers. Gc(s) (3) It is somewhat surprising that the development of ˜ - y Gp(s) 1 direct synthesis design methods for disturbance rejec- [ (r)d] tion has received relatively little attention. Early design methods for sampled-data systems were based on Because the characteristics of (y/r)d have a direct impact specifying the z transform of the desired closed-loop on the resulting controller, (y/r)d should be chosen so response to a particular disturbance.14 However, this that the closed-loop performance is satisfactory and the approach is sensitive to the assumed disturbance and resulting controller is physically realizable. does not necessarily produce a PID controller. A more The DS controller in eq 3 results in the following promising approach was proposed recently by Szita and closed-loop transfer functions 24-26 Sanathanan. They specify the desired disturbance y rejection characteristics in terms of a closed-loop trans- Gp y DS (r)d fer function for disturbances. The resulting controller ) (4) usually is not a PI or PID controller and might be of (r) ˜ + y - ˜ Gp (Gp Gp) high order. The authors propose approximating the (r)d high-order controller by a low-order controller using error minimization in the frequency domain. ˜ - y GpGd 1 y DS [ (r)d] In this paper, analytical expressions for PI and PID ) (5) controllers are derived for common process models (d) ˜ + y - ˜ Gp (Gp Gp) through the direct synthesis method and disturbance (r)d rejection. The proposed design method has a single design parameter, the desired closed-loop time constant, For the ideal case where the process model is perfect ) ˜ τc. The performance-robustness tradeoff involved in (i.e., Gp Gp), the closed-loop transfer functions become specifying τc is analyzed. A simple set-point weighting y DS y factor is used to improve controller performance for set- ) (6) point changes without affecting the response to distur- (r) (r)d bances. Simulation results for nine examples demon- DS strate that the proposed design method provides robust y ) - y Gd 1 (7) PID controllers that perform well for both disturbance (d) [ (r)d] and set-point changes. respectively. 2.2. Comparison with Internal Model Control 2. Direct Synthesis Method Based on Set-Point (IMC). A well-known control system design strategy, Responses internal model control (IMC) was developed by Morari In the direct synthesis approach, an analytical ex- and co-workers20 and is closely related to the direct pression for the feedback controller is derived from a synthesis approach. Like the DS method, the IMC process model and a desired closed-loop response. In method is based on an assumed process model and most of the DS literature, the desired closed-loop relates the controller settings to the model parameters response is expressed as a closed-loop transfer function in a straightforward manner. The IMC approach has for set-point changes. Consequently, this popular ver- the advantages that it makes the consideration of model sion of the direct synthesis method will be briefly uncertainty and the making of tradeoffs between control introduced in the next section. system performance and robustness easier. 2.1. Direct Synthesis for Set-Point Tracking The IMC approach has the simplified block diagram shown in Figure 1b, where G˜ p(s) is the process model (DS). Consider a feedback control system with the / standard block diagram in Figure 1a. Assume that Gp(s) and Gc(s) is the IMC controller. The IMC controller is a model of the process, measuring element, transmit- design involves two steps: ter, and control valve. Step 1. The process model G˜ p(s) is factored as The closed-loop transfer function for set-point changes ˜ ) ˜ ˜ is derived as Gp(s) Gp+(s) Gp-(s) (8) Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4809 ˜ where Gp+(s) contains any time delays and right-half- ) + 1 plane zeros. It is specified so that its steady-state gain GPI(s) Kc(1 ) (15) τIs is 1. Step 2. The IMC controller is specified as with the following controller settings

/ 1 G (s) ) f (9) 1 τ c ˜ K ) (16) Gp-(s) c + K τc θ where f is a low-pass filter with a steady-state gain of τ ) τ (17) 1. The IMC filter f typically has the form I 1 For a second-order plus time delay model f ) (10) + r - (τcs 1) Ke θs G˜ (s) ) (18) p + + (τ1s 1)(τ2s 1) where τc is the desired closed-loop time constant. Parameter r is a positive integer that is selected so that / substituting into eq 12 gives an ideal PID controller either Gc is a proper transfer function or the order of its numerator exceeds the order of the denominator by 1, if ideal derivative action is allowed. ) + 1 + GPID(s) Kc(1 τDs) (19) The IMC structure, Figure 1b, can be converted into τIs the conventional feedback control structure, Figure 1a.27 Comparing the resulting controllers and the closed-loop with the following settings responses of the IMC and direct synthesis (DS) ap- + proaches, it is obvious that these two approaches 1 τ1 τ2 K ) (20) produce equivalent controllers and identical closed-loop c K θ + τ performances in certain situations. For example, if the c desired closed-loop response for set-point change is ) + τI τ1 τ2 (21) specified as (y/r)d ) G˜ p+ f, then the DS controller is equivalent to the IMC controller, and identical closed- τ τ loop performance results, even when modeling errors ) 1 2 τD + (22) are present. τ1 τ2 2.3. Direct Synthesis for PI/PID Controllers. In general, both the direct synthesis and IMC methods do Identical PI/PID settings have been obtained using the not necessarily result in PI/PID controllers. However, IMC approach.1,18 by choosing the appropriate desired closed-loop response and using either a Pade´ approximation or a power-series 3. Direct Synthesis Design for Disturbance approximation for the time delay, PI/PID controllers can Rejection be derived for process models that are commonly used in industrial applications. The PI/PID settings obtained from the DS and IMC Choose the desired closed-loop transfer function as approaches are based on specifying the closed-loop transfer function for set-point changes. For processes - y e θs with small time-delay/time-constant ratios, these PI/PID ) (11) 1 (r)d τ s + 1 controllers provide very sluggish disturbance responses. c Therefore, it is worthwhile to develop a modified direct synthesis approach based on disturbance rejection. The where θ is the time delay of the system and τc is the design parameter. Then, the DS design eq 3 and a new design method will be denoted by “DS-d”. truncated power-series expansion for the time delay Consider a control system with the standard block term in the denominator, e-θs ≈ 1 - θs, gives diagram shown in Figure 1a. The closed-loop transfer function for disturbances is given by -θs ) 1 e Gc (12) y Gd(s) G˜ (τ + θ)s ) p c + (23) d 1 Gp(s) Gc(s) For systems that can be described by first-order and second-order plus time delay models, a PI or PID Rearranging gives an expression for the feedback con- controller can be obtained from eq 12. For a first-order troller plus time delay model Gd(s) 1 -θs G (s) ) - (24) Ke c y G (s) G˜ (s) ) (13) G (s) p p τs + 1 (d) p eq 12 reduces to Let the desired closed-loop transfer function for distur- bances be specified as (y/d)d, and assume that a process τs + 1 ˜ ˜ G ) (14) model Gp(s) and a disturbance model Gd(s) are available. c + K(τc θ)s Replacing the unknown (y/d), Gp(s), and Gd(s)by(y/d)d, G˜ p(s), and G˜ d(s), respectively, gives a design equation Equation 14 can be expressed as an ideal PI controller for Gc(s) 4810 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

˜ Therefore, for PI controller design, it is reasonable to Gd(s) 1 G (s) ) - (25) c ˜ specify the desired closed-loop transfer function as y ˜ Gp(s) Gp(s) (d)d -θs y Kdse ) (33) For the DS-d controller in eq 25, the closed-loop transfer (d)d (τ s + 1)2 functions are c y with G G˜ - DS-d p d y ) [ (d)d] τ (26) ) I (r) ˜ + y ˜ - Kd (34) GpGd (Gp Gp) Kc (d)d The DS-d design equation, eq 25, produces a standard ˜ y - GpGd PI controller if the time delay in the denominator is y DS d (d)d ) (27) approximated by, e-θs ≈ 1 - θs. The resulting PI (d) ˜ + y ˜ - controller parameters are GpGd (Gp Gp) (d)d + - 2 ˜ ) 1 τθ 2ττc τc For the ideal case where the model is perfect (i.e., Gp K ) (35) c 2 Gp and G˜ d ) Gd), the closed-loop transfer functions K + (τc θ) become + - 2 y τθ 2ττc τc τ ) (36) DS-d I + y ) - (d)d τ θ ( ) 1 ˜ (28) r Gd(s) and Kd is given by y DS-d y ) (29) (τ + θ)2 (d) (d)d K ) K c (37) d τ+θ respectively. The DS-d design method does not necessarily produce From eqs 35 and 36, it is apparent that, for large a PI or PID controller. The structure and order of the values of τc, anomalous results can occur because Kc and controller depend on the specification of the desired K can have opposite signs and τI can become negative. closed-loop response and the process model. In this Both of these undesirable situations can be avoided, section, the new DS-d design method is used to derive however, if the design parameter τc satisfies PI/PID controllers for simple process models that are widely used. 0 < τ < τ + xτ2 + τθ (38) 3.1. DS-d PI Settings. The DS-d methods will now c be used to design PI controllers for a first-order plus This constraint on τc is not restrictive at all because time delay model and then an integrator plus time delay direct synthesis controllers are typically designed so model. that τ < 2τ. 3.1.1. First-Order Plus Time Delay Model. As- c For this PI controller and G˜ p ) Gp, the closed-loop sume that the process is described by a first-order plus transfer function for set-point changes is time-delay model - + - y DS d τIs 1 - Ke θs ≈ e θs (39) G (s) ) (30) (r) + 2 p τs + 1 (τcs 1) and that Gd(s) ) Gp(s). (This latter assumption will be 3.1.2. Integrator Plus Time Delay Model. Pro- removed in section 3.3). Thus, if the PI controller in eq cesses with integrating characteristics are quite com- 15 is used, the closed-loop transfer function in eq 23 can mon in the process industries. Assume that the process be expressed as is described by

-θs -θs Ke ) Ke + Gp(s) (40) y ) τs 1 s - (31) d Ke θs 1 + + and that G (s) ) G (s). If a PI controller is used, the 1 + Kc(1 ) d p τs 1 τIs closed-loop transfer function for disturbances in eq 23 becomes Approximating the time delay term in the denominator -θs ≈ - - by a first-order power-series expansion, e 1 θs, Ke θs and rearranging gives y s ) (41) τ d -θs I -θs + Ke + 1 se 1 Kc(1 ) y Kc s τIs ≈ (32) (d) τ τ τ I - 2 + I + - + Approximating the time delay term in the denominator ( τIθ)s ( τI θ)s 1 - K Kc KKc by e θs ≈ 1 - θs gives Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4811

τI -θs y τI θ - θττI θ se ( ) ≈ s 1 + s e θs / - τ τ s3 + [ ] [( I D) y Kc d Kc ( 2 ) 2KKc 2 ≈ (42) (d) τI 2 + - θτ s + (τ - θ)s + 1 (τ θ/2)τI θ τI θ (KK I) I ( + τ τ - τ )s2 + + τ - s + 1 I D I ( I ) c ] KKc 2 KKc 2 (51) Therefore, if the desired closed-loop transfer function for disturbances is specified as Thus, for PID controller design, it is reasonable to specify the desired closed-loop transfer function as -θs y Kdse ) (43) θ -θs 2 K s 1 + s e (d)d (τ s + 1) y d ( 2 ) c ) (52) (d)d + 3 (τcs 1) with Kd ) τI/Kc, then the controller obtained using the DS-d method can be rearranged to give a standard PI with Kd ) τI/Kc. Then, the controller obtained from the controller with DS-d method, eq 25, can be rearranged to give a standard PID controller. The resulting PID controller + 1 2τc θ parameters are K ) (44) c 2 K (τ + θ) 2 c θ θ (2τθ + )(3τ + ) - 2τ 3 - 3τ 2θ ) + 1 2 c 2 c c τI 2τc θ (45) K ) (53) c K 2 3 + 2 + θ + θ 2τc 3τc θ 3τc and Kd is given by 2 ( 2)

2 ) + 2 θ θ 3 2 Kd K(τc θ) (46) (2τθ + )(3τ + ) - 2τ - 3τ θ 2 c 2 c c τ ) (54) I (2τ + θ)θ For this controller and G˜ p ) Gp, the closed-loop response for set-point changes is 2 2 + τθ + θ - + 3 3τc τθ 3τc 2(τ θ)τc DS-d τ s + 1 2 ( 2) y I -θs τ ) (55) ≈ e (47) D 2 2 θ θ (r) (τ s + 1) (2τθ + )(3τ + ) - 2τ 3 - 3τ 2θ c 2 c 2 c c

3.2. DS-d PID Settings. In this section, the proposed and Kd is given by DS-d method is used to design PID controllers for some commonly used process models, including first-order and θ2 θ 2τ 3 + 3τ 2θ + 3τ + second-order plus time delay models and an integrator c c 2 ( c 2) K ) K (56) plus time delay model. d + 3.2.1. First-Order Plus Time Delay Model. As- (2τ θ)θ sume that the process is described by a first-order plus It is apparent from the above equations that KKc, τI, time delay model and τD can be negative for large values of τc. However, simulation experience has demonstrated that this po- - Ke θs tential problem does not occur if the closed-loop time G (s) ) (48) p τs + 1 constant τc is chosen in a reasonable manner. For this PID controller and G˜ p ) Gp, the closed-loop and that Gd(s) ) Gp(s). Thus, if the PID controller in eq transfer function for set-point changes is obtained as 19 is used, the closed-loop transfer function in eq 23 can 2 + + + θ be expressed as - (τIτDs τIs 1) 1 s y DS d ( 2 ) - ≈ e θs (57) -θs (r) + 3 Ke (τcs 1) + y ) τs 1 - (49) 3.2.2. Integrator Plus Time Delay Model. Now d Ke θs 1 1 + K 1 + + τ s assume that the process is described by τs + 1 c( τ s D ) I - Ke θs G (s) ) (58) Approximating the time delay term in the denominator p s by a first-order Pade´ approximation and that Gd(s) ) Gp(s). If a PID controller is used, the θ closed-loop transfer function in eq 23 becomes 1 - s - 2 -θs e θs ≈ (50) Ke + θ y s 1 s ) (59) 2 d -θs + Ke + 1 + 1 Kc(1 τDs) and rearranging gives s τIs 4812 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

Approximating the time delay term in the denominator y τI -θs τ1τ2τI 3 by eq 50 gives ( ) ≈ se / - θτ τ s + [ ] [( I D) d Kc KKc τ θτ y I θ -θs I θ 3 + ≈ s 1 + s e / - τ τ s + (τ1 τ2)τI τI (d) [K ( 2 ) ] [(2KK 2 I D) + τ τ - θτ s2 + + τ - θ s + 1 c c ( KK I D I) (KK I ) ] τ c c I + - θ 2 + - θ + (69) ( τIτD τI)s τI s 1] (60) KKc 2 ( 2) Therefore, if the desired closed-loop transfer function Therefore, if the desired closed-loop transfer function for disturbance is specified as for disturbances is specified as - K se θs y ) d θ -θs (70) K s 1 + s e (d)d ( s + 1)3 y d ( 2 ) τc ) (61) (d)d + 3 (τcs 1) with Kd ) τI/Kc, then the controller obtained from the DS-d method, eq 25, can be rearranged to give a with Kd ) τI/Kc, then the controller obtained using the standard PID controller. The resulting PID controller DS-d method can be rearranged to give a standard PID parameters are controller. The resulting PID controller parameters are [(τ + τ )θ + τ τ ](3τ + θ) - τ 3 - 3τ 2θ ) 1 1 2 1 2 c c c + θ Kc (71) θ 3τc K + 3 1 ( 2) (τc θ) K ) (62) c K θ 3 + 3 2 τc + + + - - ( 2) [(τ1 τ2)θ τ1τ2](3τc θ) τc 3τc θ ) (72) τI + + + τ1τ2 (τ1 τ2 θ)θ ) + θ τI 3τc (63) 2 2 3 3τ τ τ + τ τ θ(3τ + θ) - (τ + τ + θ)τ ) c 1 2 1 2 c 1 2 c (73) 3 τD 3 2 3 2 3 2 θ 3 [( + ) + ](3 + ) - - 3 τ θ + τ θ + - τ τ1 τ2 θ τ1τ2 τc θ τc τc θ 2 c 4 c 8 c τ ) (64) D θ and Kd is given by θ 3τ + ( c 2) (τ + θ)3 K ) K c (74) and Kd is given by d + + + τ1τ2 (τ1 τ2 θ)θ θ 3 τ + For this PID controller and G˜ ) G , the closed-loop ( c 2) p p K ) K (65) transfer function for set-point changes is d θ 2 DS-d (τ τ s + τ s + 1) ˜ ) y I D I -θs For this controller and Gp Gp, the closed-loop response ≈ e (75) (r) + 3 for set-point change is obtained as (τcs 1)

2 + + + θ 3.2.4. First-Order with an Integrator Plus Time - (τIτDs τIs 1) 1 s y DS d ( 2 ) - ≈ e θs (66) Delay Model. Assume that the process is described by (r) 3 (τ s + 1) - c Ke θs G (s) ) (76) 3.2.3. Second-Order Plus Time Delay Model. p s(τs + 1) Assume that the process is described by a second-order ) plus time delay model and that Gd(s) Gp(s). If a PID controller is used, then the closed-loop transfer function in eq 23 becomes -θs Ke - G (s) ) (67) Ke θs p (τ s + 1)(τ s + 1) 1 2 y s(τs + 1) ) (77) -θs and that Gd(s) ) Gp(s). Thus, if a PID controller is used, d Ke 1 1 + K 1 + + τ s the closed-loop transfer function in eq 23 can be s(τs + 1) c( τ s D ) expressed as I

-θs Approximating the time delay term in the denominator Ke by e-θs ≈ 1 - θs gives (τ s + 1)(τ s + 1) y ) 1 2 - (68) y τI - ττI d Ke θs 1 ≈ se θs / - θτ τ s3 + 1 + K 1 + + τ s (d) [K ] [(KK I D) (τ s + 1)(τ s + 1) c( τ s D ) c c 1 2 I τ I + - 2 + - + ( τIτD θτI)s (τI θ)s 1] (78) Approximating the time delay term in the denominator KKc by a first-order power-series expansion, e-θs ≈ 1 - θs, and rearranging gives Therefore, if the desired closed-loop transfer function Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4813

Table 1. PI/PID Controller Settings for the DS-d Design Method

a,b case model KKc τI τD

A -θs 2 + - - 2 2 + - - 2 - Ke τ τθ (τc τ) τ τθ (τc τ) + τs 1 + 2 τ + θ (τc θ)

- B Ke θs θ2 θ θ2 θ τθ2 θ (2τθ + )(3τ + ) - 2τ 3 - 3τ 2θ (2τθ + )(3τ + ) - 2τ 3 - 3τ 2θ 3τ 2τθ + 3τ + - 2(τ + θ)τ 3 τs + 1 2 c 2 c c 2 c 2 c c c 2 ( c 2) c + / 3 + 2 2(τc θ 2) (2τ θ)θ θ θ (2τθ + )(3τ + ) - 2τ 3 - 3τ 2θ 2 c 2 c c -θs + + - C Ke 2τc θ 2τc θ s + 2 (τc θ)

- s 3 D Ke θ + θ + θ + θ - 3 θ 3τc 3τc τc 2τc s ( 2) 2 ( 2) θ 3 θ τ + θ 3τ + ( c 2) ( c 2)

-θs 2 3 2 E (3τ + θ)(τ + θ) 3τc + θ + - + Ke c 3τc τ 3τcτθ τc τθ s(τs + 1) + 3 + + (τc θ) (3τc θ)(τ θ)

2 3 2 F K(τ s + 1) (3τ - τ )(τ - τ ) 3τc - τa - - + a c a a 3τc τ 3τcττa τc ττa s(τs + 1) - 3 - - (τc τa) (3τc τa)(τ τa)

G -θs + + + - 3 - 2 + + + - 3 - 2 2 + + - + + 3 Ke [(τ1 τ2)θ τ1τ2](3τc θ) τc 3τc θ [(τ1 τ2)θ τ1τ2](3τc θ) τc 3τc θ 3τc τ1τ2 τ1τ2θ(3τc θ) (τ1 τ2 θ)τc + + (τ1s 1)(τ2s 1) + 3 τ τ + (τ + τ + θ)θ + + + - 3 - 2 (τc θ) 1 2 1 2 [(τ1 τ2)θ τ1τ2](3τc θ) τc 3τc θ

H -θs + 2 + - 3 - 2 + 2 + - 3 - 2 2 2 + 2 + - + 3 Ke (2úτθ τ )(3τc θ) τc 3τc θ (2úτθ τ )(3τc θ) τc 3τc θ 3τc τ τ θ(3τc θ) (2úτ θ)τc τ2s + 2úτs + 1 + 3 2 + + + 2 + - 3 - 2 (τc θ) τ (2úτ θ)θ (2úτθ τ )(3τc θ) τc 3τc θ

I K(τ s + 1) 2 + - + - - 3 2 + - + - - 3 - - 3 + 2 - - a 3τc τa [τ1τ2 (τ1 τ2)τa](3τc τa) τc 3τc τa [τ1τ2 (τ1 τ2)τa](3τc τa) τc (τa τ1 τ2)τc 3τc τ1τ2 τ1τ2τa(3τc τa) + + 3 2 3 (τ1s 1)(τ2s 1) - τ τ - (τ + τ - τ )τ + - + - - (τc τa) 1 2 1 2 a a 3τc τa [τ1τ2 (τ1 τ2)τa](3τc τa) τc

θ -θs - s + - s θ Kds 1 s e θ y Kdse y ( 2 ) y Kdse a Cases A and C, ) ; cases B and D, ) ; cases E, G, and H, ) ; cases F and I, (d)d + 2 (d)d + 3 (d)d + 3 (τcs 1) (τcs 1) (τcs 1) K s(τ s + 1) y d a b ) . Kd ) τI/Kc. (d)d + 3 (τcs 1) for disturbance is specified as 3.3. Discussion. In the previous sections, the DS-d method has been used to design PI/PID controllers for -θs y Kdse widely used process models. The resulting PI/PID ) (79) controller settings are shown in Table 1. (d)d (τ s + 1)3 c Remark 1. The only design parameter, τc, is directly related to the closed-loop time constant. As τ decreases, ) c with Kd τI/Kc, then the controller obtained from the the closed-loop response becomes faster. DS-d method can be rearranged to give a standard PID Remark 2. Larger values of τc give larger values of controller. The resulting PID controller parameters are Kd because Kd ) τI/Kc. + + For a unit step disturbance at the process input, 1 (3τc θ)(τ θ) ° 4 K ) (80) Astro¨m and Ha¨gglund derived the following relation c K + 3 for the integral error (IE) associated with PI/PID control (τc θ)

) + ∞ τI τI 3τc θ (81) IE ) ∫ [r(t) - y(t)] dt ) (85) 0 Kc 3τ 2τ + 3τ τθ + τθ2 - τ 3 ) c c c (82) Thus τD + + (3τc θ)(τ θ) ) IE Kd (86) and Kd is give by This expression means that smaller τ values provide 3 c (τ + θ) smaller IE values for step disturbances. K ) K c (83) d τ + θ Remark 3. Although smaller values of τc provide better performance for disturbance and set-point changes, For this PID controller and G˜ p ) Gp, the closed-loop the control system robustness is worse. Therefore, the transfer function for set-point changes is system robustness should be considered when τc is being chosen. 2 DS-d + + Remark 4. The PI/PID tuning rules in Table 1 were y (τIτDs τIs 1) - s ≈ e θ (84) derived based on the assumption that G ) G . For the (r) + 3 d p (τcs 1) more general case where Gd * Gp, the desired closed- 4814 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 loop response for disturbances is specified as Table 2. PID Controller Settings for Example 1 (θ/τ ) 0.01) / y y Gd set point disturbance ) (87) (d)d (d)d Gp tuning method Kc τI τD MS IAE TV IAE TV

DS-d (τc ) 1.2, b ) 1) 0.829 4.05 0.354 1.94 3.06 1.46 4.89 1.89 ) ) where (y/d)d is given in Table 1. Thus, the PI/PID DS-d (τc 1.2, b 0.5) 0.829 4.05 0.354 1.94 2.19 0.82 4.89 1.89 ) settings and the closed-loop responses for set-point IMC (τc 0.85) 0.744 100.5 0.498 1.94 1.88 1.10 84.4 1.59 ZN 0.948 1.99 0.498 2.30 3.52 2.82 3.22 3.06 changes are the same as for the special case where Gd ) Gp. Remark 5. The proposed PI tuning rules for integrat- loop transfer function is specified and the same ap- ing processes are equivalent to the IMC PI settings of proximation is used for time delay term. For some Chien and Fruehauf.1 process models, however, the IMC tuning rules are very 3.4. Set-Point and Derivative Weighting. Equa- well-known. Thus, the IMC method was used instead tions 15 and 19 are conventional PI and PID controllers. of the DS method for a few examples. A more flexible control structure that includes set-point The following robustness and performance metrics weighting and derivative weighting is given by A° stro¨m were used as evaluation criteria for the comparison of and Ha¨gglund4 the PI/PID controllers: Robustness Metric. The peak value of the sensitiv- 28 1 t ity function, MS, has been widely used as a measure u(t) ) K [br(t) - y(t)] + ∫ [r(τ) - y(τ)] dτ + of system robustness. Recommended values of M are c( τ 0 S I typically in the range of 1.2-2.0.29 d[cr(t) - y(t)] To provide fair comparisons, the model-based control- τD ) (88) dt lers (DS-d, DS, and IMC) were tuned by adjusting τc so that the MS values were very close. This tuning facili- where the set-point weighting coefficient b is bounded tated a comparison of controller performance for dis- by 0 e b e 1 and the derivative weighting coefficient c turbance and set-point changes for controllers that had is also bounded by 0 e c e 1. The overshoot for set- the same degree of robustness. point changes decreases with increasing b. Performance Metrics. Two metrics were used to The controllers obtained for different values of b and evaluate controller performance. The integrated abso- c respond to disturbances and measurement noise in the lute error (IAE) is defined as15 same way as conventional PI/PID controllers, i.e., dif- ferent values of b and c do not change the closed-loop ∞ IAE ( ∫ | r(t) - y(t)| dt (91) response for disturbances. Therefore, the same PI/PID 0 tuning rules developed here using the DS-d method are also applicable for the modified PI/PID controller in eq To evaluate the required control effort, the total varia- 88. However, the set-point response does depend on the tion (TV) of the manipulated input u was calculated values of b and c. If set-point weighting and derivative weighting are used, the closed-loop transfer function for ∞ set-point changes is given by TV ( ∑ |u(k+1) - u(k)| (92) k)1 cτ τ s2 + bτ s + 1 G (s) G (s) y ) I D I p c ( Gsp(s) (89) r τ τ s2 + τ s+1 1 + G (s) G (s) The total variation is a good measure of the “smooth- I D I p c ness” of a signal and should be as small as possible.30 4.1. Example 1. Consider the following model with where Gc(s) is the conventional PID controller given by 31 eq 19. a step disturbance acting at the plant input

100 - 4. Simulation Results G (s) ) G (s) ) e s (93) p d 100s + 1 Several simulation examples are used to demonstrate the proposed PI/PID tuning rules for the DS-d method. The DS-d, IMC, and ZN methods were used to design In practice, the derivative weighting factor c is usually the PID controllers shown in Table 2. For the DS-d set to zero to avoid a large derivative kick. Thus, in this method, a value of τc ) 1.2 was chosen so that MS ) paper, c is chosen to be zero for all of the simulation 1.94. To obtain a fair comparison, τc ) 0.85 was selected examples. Furthermore, the PID controller is imple- for the IMC method so that MS ) 1.94. mented in the widely used “parallel form” The simulation results in Figure 2 and the IAE and TV values in Table 2 indicate that the disturbance τ s ) + 1 + D response for the DS-d controller is much better and Gc(s) Kc(1 R + ) (90) τIs τDs 1 faster than the IMC response, whereas the movements of these two controller outputs are similar. The distur- The derivative filter parameter R is specified as R)0.1. bance response of the ZN controller has a smaller peak Other implementations of PID control, such as the series value but is more oscillatory than the DS-d response. form, are also widely used. The controller settings for The set-point response of the DS-d controller is more one form can easily be converted to other forms.3 sluggish and has a larger overshoot than the IMC For each example, the DS-d, DS, ZN, and/or some controller. However, the overshoot for the DS-d control- other methods were used to design PI/PID controllers. ler can be eliminated, without affecting the disturbance As mentioned in section 2.3, the DS and IMC methods response, by setting b ) 0.5. By comparing the responses can provide identical PI/PID settings if the same closed- and the MS, IAE, and TV values of all three controllers, Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4815

Figure 2. Simulation results of PID controllers for example 1 (θ/τ ) 0.01). (a) Response to a unit step set-point change. (b) Figure 3. Simulation results without u constraints for example ) Response to a unit step disturbance. 2(θ/τ 0.25). (a) Controlled variable y. (b) Manipulated variable u. Table 3. PI Controller Settings for Example 2 (θ/τ ) 0.25) set point disturbance

tuning method Kc τI MS IAE TV IAE TV

DS-d (τc ) 0.35, b ) 1) 2.30 0.662 1.88 0.635 3.64 0.288 1.54 DS-d (τc ) 0.35, b ) 0.5) 2.30 0.662 1.88 0.630 2.10 0.288 1.54 DS (τc ) 0.13) 2.63 1 1.90 0.532 3.80 0.37 1.40 ZN 3.12 0.763 2.37 0.632 6.63 0.244 2.02 it can be concluded that the DS-d controller provides the best performance without using excessive control effort. 4.2. Example 2. Consider a process is described by

- e 0.25s G (s) ) G (s) ) (94) p d s + 1 The PI controller characteristics for the DS-d, DS, and ZN 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). The simulation results in Figure 3 indicate that the distur- bance response of the DS-d PI controller is faster than that of the DS PI controller and less oscillatory than that of the ZN PI controller. The set-point response of the DS-d PI controller exhibits overshoot, but it can be reduced by setting b ) 0.5. These conclusions can be confirmed by the IAE and TV values in Table 3. Figure 4 shows the simulation results for the practical situation where there are inequality constraints on the manipulated variable u: -2 e u e 2. A comparison of Figures 3 and 4 indicates that the initial set-point Figure 4. Simulation results with u constraints for example 2 responses of the DS-d (b ) 1), DS, and ZN PI controllers (θ/τ ) 0.25). (a) Controlled variable y. (b) Manipulated variable u. were slower because of controller saturation. Also, the oscillations for the DS and ZN PI controllers were Examples 1 and 2 demonstrate that the DS-d method damped. The disturbance responses were not affected provides better performance than the DS and IMC because inequality constraints on u were not active. methods for processes with small θ/τ ratios. The follow- 4816 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

Figure 5. Simulation results of PI controllers for example 3 (θ/τ Figure 7. Simulation results of PI controllers for example 3 (θ/τ ) 1). ) 5).

Figure 6. Simulation results of PID controllers for example 3 (θ/τ ) 1).

Table 4. PI Controller Settings for Example 3 (θ/τ ) 1) Figure 8. Simulation results of PID controllers for example 3 (θ/τ ) 5). set point disturbance ) tuning method Kc τI MS IAE TV IAE TV Table 6. PI Controller Settings for Example 3 (θ/τ 5)

DS-d (τc ) 0.8, b ) 1) 0.60 0.98 1.80 2.13 1.09 1.80 1.30 set point disturbance DS-d (τc ) 0.8, b ) 0.5) 0.60 0.98 1.80 2.34 1.09 1.80 1.30 tuning method Kc τI MS IAE TV IAE TV DS (τc ) 0.62) 0.62 1.00 1.81 2.11 1.09 1.80 1.30 ZN 1.02 2.58 2.05 2.52 1.62 2.50 1.45 DS-d (τc ) 1.9) 0.11 0.87 1.86 10.9 1.27 10.8 1.37 DS (τc ) 2.8) 0.13 1.00 1.86 10.6 1.24 10.5 1.36 Table 5. PID Controller Settings for Example 3 (θ/τ ) 1) CM 0.35 3.3 1.68 9.43 0.68 9.43 1.01 ZN 0.515 9.8 1.89 18.8 0.96 18.5 1.34 set point disturbance ) tuning method Kc τI τD MS IAE TV IAE TV Table 7. PID Controller Settings for Example 3 (θ/τ 5)

DS-d (τc ) 0.75, b ) 1) 1.11 1.45 0.317 1.92 1.68 2.18 1.30 1.46 set point disturbance DS-d (τc ) 0.75, b ) 0.7) 1.11 1.45 0.317 1.92 1.74 1.70 1.30 1.46 tuning method Kc τI τD MS IAE TV IAE TV IMC (τc ) 0.85) 1.11 1.50 0.333 1.94 1.65 2.21 1.35 1.51 ZN 1.36 1.55 0.387 2.59 1.81 4.46 1.16 2.79 DS-d (τc ) 2.5) 0.4 2.86 0.313 1.86 7.69 0.939 7.39 1.18 IMC (τc ) 4.5) 0.5 3.5 0.714 1.87 7.38 1.28 6.99 1.48 ing example is used to illustrate how the proposed DS-d CM 0.4 3.0 1.2 1.92 7.60 1.20 7.52 1.57 method works for processes with larger θ/τ ratios. ZN 0.66 5.9 1.48 42.2 9.48 5.94 9.26 9.64 4.3. Example 3. Different Values of the / Ratio. θ τ - 34 Consider the process model compare the IMC, Ciancone Marlin (CM), and ZN methods. Consequently, the CM PI and PID settings - were also considered for comparison. The CM tuning e θs G (s) ) G (s) ) (95) rules were derived using an optimization procedure that p d s + 1 incorporates considerations of performance, robustness, and saturation of the manipulated variable. The CM In this example, two values of θ/τ ratio are considered: tuning rules are valid only for first-order plus time delay θ/τ ) 1 and θ/τ ) 5. processes and are presented graphically.34 For θ/τ ) 1, PI and PID controllers were designed The controller characteristics are given in Tables 6 using the DS-d, DS/IMC, and ZN methods. The control- and 7. The responses to unit step changes in the set ler characteristics are shown in Tables 4 and 5. The point and disturbance are shown in Figures 7 and 8. simulation results in Figures 5 and 6 indicate that the The DS-d and DS PI controllers give the fastest PI PI and PID responses for the DS-d and DS/IMC methods responses with small overshoots that are very similar. are very similar. The DS-d and DS/IMC controllers The responses of the ZN PI controller are very sluggish. provide better performance than the ZN controllers. The CM PI controller provides the best performance For the process with θ/τ ) 5, the DS-d, DS/IMC, and among all four PI controllers. For the PID controllers, ZN methods were used to design PI and PID controllers. the responses of the DS-d, IMC, and CM controllers are Luyben2 has used this large-time-delay example to relatively close. The ZN PID controller gives very erratic Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4817

Figure 9. DS-d PI controller settings for different τc values (example 4). responses that are not shown in the plot but can be 2 found in Luyben. Its MS value in Table 7 is extremely large. The results for these two θ/τ values illustrate that the DS-d method provides better disturbance rejection than the DS method for processes with small values of θ/τ. As θ/τ becomes larger, the controller settings and the closed-loop performance for the DS-d method become closer to those for the conventional DS method. 4.4. Example 4. Effect of τc. The DS-d method has a single tuning parameter, τc, that is directly related to the speed of the closed-loop response. In this example, the effect of τ is analyzed. Consider the general model c Figure 10. DS-d PID controller settings for different τc values (example 4). - e θs G (s) ) G (s) ) (96) p d s + 1 and then decrease. If τc is very large, τI and τD become negative. Thus, for this example, the upper bounds on For three different values of θ/τ (0.25, 0.5, 1), DS-d PI τc for positive Kc, τI, and τD values obtained from Figure controller settings were calculated for different τc values. 10 are Figure 9 shows the DS-d PI controller settings for different values of τc.Asτc increases, Kc decreases, and e ) τc 0.4 for θ/τ 0.25 (97) the integral time, τI, increases if τc e 1 but decreases if g ) e ) τc 1. The symmetry of τI around τc τ is confirmed τc 0.7 for θ/τ 0.5 (98) by eq 36. e ) Similarly, for three different θ/τ values (0.25, 0.5, τc 0.935 for θ/τ 0.75 (99) 0.75), the DS-d PID controller settings were calculated for different τc values and are shown in Figure 10. As For θ/τ ) 0.25, the DS-d PID controllers were de- τc increases, Kc decreases, while τI and τD first increase signed for four values of τc (0.25, 0.3, 0.35, 0.4), and the 4818 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

4.6. Example 6. Consider a second-order plus time delay system described by Seborg et al.15

- 2e s G (s) ) G (s) ) (103) p d (10s + 1)(5s + 1)

The DS-d, DS, and ZN methods were used to design PID controllers, and the resulting controller settings are shown in Table 9. The simulation results for a unit step change in the set point at t ) 0 min and a unit step disturbance at t ) 50 min are presented in Figure 13. The disturbance response for the DS-d PID controller is fast and has a small peak value, but the set-point response has an ) Figure 11. DS-d PID controllers for different τc values and b overshoot. By using a set-point weighting factor of b ) ) 1 (example 4, θ/τ 0.25). 0.5, the overshoot is eliminated. The set-point and disturbance responses for the DS PID controller are Table 8. PID Controller Settings for Example 5 (θ/τ ) quite sluggish, whereas the ZN PID controller provides 0.25) very oscillatory responses. Again, the IAE and TV

tuning method Kc τI τD MS values in Table 9 confirm that the DS-d controller provides superior performance without using excessive DS-d (τ ) 0.26, b ) 1) 3.46 0.702 0.0887 1.89 c control efforts. DS-d (τc ) 0.26, b ) 0.5) 3.46 0.702 0.0887 1.89 IMC (τc ) 0.22) 3.26 1.13 0.111 1.90 4.7. Example 7. Distillation Column Model. A ZN 4.16 0.458 0.115 2.37 distillation column that separates a small amount of a low-boiling material from the final product was consid- corresponding simulation results are shown in Figure ered by Chien and Fruehauf.1 The bottom level of the 11. The simulation results demonstrate that the DS-d distillation column is controlled by adjusting the steam controllers for smaller τc values provide disturbance flow rate. The process model for the level control system responses with smaller peaks and smaller IE values. is an integrator with a time delay

4.5. Example 5. Different Disturbance Time - 0.2e 7.4s Constants. In this example, a process described by the G (s) ) G (s) ) (104) same transfer function as in example 2 p d s

- The DS controller obtained from eq 12 has the form e 0.25s G (s) ) (100) p s + 1 1 G (s) ) (105) c K(τ + θ) but disturbance transfer functions with different values c of is considered τd Because the DS method results in a proportional-only controller for the integrating process, only the DS-d -0.25s ) e tuning methods was used to design PI controllers. Note Gd(s) + (101) τds 1 that the DS-d and IMC methods provide identical PI tuning rules for this type of process model. The resulting controller settings from the DS-d method are shown in PID controllers were designed using the DS-d, IMC, and Table 10. The IMC PI controller settings with τ ) 8 ZN methods and assuming that τ ) τ. The controller c d used by Chien and Fruehauf1 are included in the table. characteristics are shown in Table 8. The DS-d control- Furthermore, for integrator plus time delay processes, ler was modified for other values of τ by using eq 87. d Tyreus and Luyben35 have developed a design method After substituting the transfer functions for the process that yields the best PI settings attainable for a specified and disturbance, the closed-loop transfer function for degree of closed-loop damping. Their tuning rule can be the DS-d PID controller in eq 87 becomes expressed in terms of the ultimate gain Ku and ultimate frequency Pu as Kc ) Ku/3.22 and τI ) 2.2Pu. Thus, it is -0.25s y 1.125s(s + 1)e a modified version of ZN tuning. The TL PI controller * ≈ (102) 35 (d) (τ s + 1)(τ s + 1)3 settings are included in Table 10 for comparison. d c The simulation results for a unit step change in the set point (at t ) 0) and a 0.5 step disturbance (at t ) Thus, when τd/τ increases, the disturbance response of 150 min) are shown in Figure 14. The disturbance the DS-d PID controller has a smaller peak value and performance of the DS-d controller is good, but the set- is more sluggish. point response has a large overshoot that can be Because τd does not affect the set-point response, only eliminated by setting b ) 0.5. The IMC PI controller the disturbance responses were simulated. The simula- designed by Chien and Fruehauf1 provides a faster tion results for four τd values (0.25, 0.5, 1, 2) shown in disturbance response than the DS-d method, but the set- Figure 12 confirm that DS-d controllers provide distur- point response is too aggressive, as confirmed by the bance responses with smaller peak values but longer large MS and TV values in Table 10. The use of set- settling times as τd/τ increases. For small τd/τ values, point weighting can reduce the large set-point overshoot the disturbance performance of the IMC controller is for the IMC controller, but it would not affect the better than that of the DS-d controller. oscillatory nature of the response. The large MS value Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4819

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.

Figure 13. Simulation results for example 6. Figure 14. Simulation results for example 7.

Table 9. PID Controller Settings for Example 6 Table 10. PI Controller Settings for Example 7 set point disturbance set point disturbance

tuning method Kc τI τD MS IAE TV IAE TV tuning method Kc τI MS IAE TV IAE TV ) ) DS-d (τc 2.4, b 1) 6.3 7.60 2.10 1.87 5.59 13.3 1.19 2.10 DS-d (τc ) 15, b ) 1) 0.373 37.4 1.94 27.1 0.675 50.1 0.932 ) ) DS-d (τc 2.4, b 0.5) 6.3 7.60 2.10 1.87 4.58 6.78 1.19 2.10 DS-d (τc ) 15, b ) 0.5) 0.373 37.4 1.94 19.6 0.354 50.1 0.932 ) DS (τc 0.5) 5 15 3.33 1.92 6.25 11.7 3.03 2.34 IMC (τc ) 8) 0.49 23 3.06 30.9 1.63 30.9 1.58 ZN 4.72 5.83 1.46 2.27 8.41 12.9 1.74 2.77 TL 0.33 64.7 1.67 28.4 0.455 93.2 0.742

1 indicates that the τc value used by Chien and Fruehauf 4.8. Example 8. Level Control Problem. Consider is so small that the resulting system has poor robust- a level control problem given by Seborg et al.15 The ness. The responses of the TL PI controller are very liquid level in a reboiler of a steam-heated distillation sluggish because of the large τI value. column is to be controlled by adjusting the control valve 4820 Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

Table 11. PID Controller Settings for Example 8 set point disturbance

tuning method Kc τI τD MS IAE TV IAE TV

DS-d (τc ) 1.6, b ) 1) -1.25 5.3 1.45 1.94 3.42 3.39 4.33 2.86 DS-d (τc ) 1.6, b ) 0.5) -1.25 5.3 1.45 1.94 2.82 1.62 4.33 2.86 IMC (τc ) 1.25) -1.22 6.0 1.50 1.96 3.48 3.29 4.99 2.83 ZN -0.752 3.84 0.961 2.76 7.17 2.86 9.31 3.91 on the steam line. The process transfer function is given by

-1.6(-0.5s + 1) G (s) ) G (s) ) (106) p d s(3s + 1)

The PID settings obtained from the DS-d, IMC, and ZN Figure 15. Simulation results for example 8. methods are shown in Table 11. The simulation results for a unit step change in the Table 12. PI Controller Settings for Example 9 set point at t ) 0 min and a unit step disturbance at t set point disturbance ) 50 min are presented in Figure 15. The disturbance tuning method K τ M IAE TV IAE TV response for the DS-d PID controller is good, but the c I S DS-d (τc ) 0.4, b ) 1) 2.94 0.707 1.61 0.553 3.66 0.721 4.56 set-point response has an overshoot. With a set-point ) ) weighting factor set at b ) 0.5, the DS-d controller DS-d (τc 0.4, b 0.5) 2.94 0.707 1.61 0.594 2.08 0.721 4.56 DS (τc ) 0.148) 3.72 1.1 1.59 0.450 4.45 0.874 4.22 provides a fast set-point response without overshoot. SIMC (τc ) 0.148) 3.72 1.1 1.59 0.450 4.45 0.874 4.22 The performance of the IMC PID controller is close to TL 9.46 1.24 2.72 0.498 25.9 0.387 8.67 that of the DS-d PID controller, which is confirmed by the similar PID settings. The ZN controller produces Table 13. PID Controller Settings for Example 9 very oscillatory responses. set point disturbance

4.9. Example 9. Fourth-Order Process. Consider tuning method Kc τI τD MS IAE TV IAE TV a fourth-order process described by29,30 DS-d (τc ) 0.135, b ) 1) 22.4 0.415 0.106 1.58 0.276 38.0 0.056 5.56 DS-d (τc ) 0.135, b ) 0.5) 22.4 0.415 0.106 1.58 0.231 18.3 0.056 5.56 ) ) ) Gp(s) Gd(s) IMC (τc 0.025) 22.6 1.2 0.167 1.58 0.311 48.3 0.160 6.57 ) 1 SIMC (τc 0.028) 21.8 1.22 0.18 1.58 0.333 48.7 0.168 6.85 (107) ZN 18.1 0.281 0.07 2.38 0.423 52.0 0.070 9.03 (s + 1)(0.2s + 1)(0.04s + 1)(0.008s + 1) Both models provide accurate approximations, but the For high-order processes, the DS-d, DS, and IMC design second-order model is more accurate. methods do not yield PI/PID controllers directly. Thus, PI controllers were designed using the approximate the model order must be reduced, or the resulting first-order model and the DS-d, DS, and SIMC methods. controller must be approximated by a PI/PID controller. Values of τ ) 0.4 and 0.148 were selected for the DS-d Skogestad30 has proposed a simple method of ap- c and DS methods, respectively, so that their M values proximating high-order models with low-order models. S were very close to the 1.59 value for the SIMC controller He also derived modified IMC rules, which were named reported by Skogestad.30 For τ ) 0.148, the same PI “simple control” or “Skogestad IMC” (SIMC). For the c controller settings were calculated using the DS and first-order model in eq 30, the PI controller obtained SIMC methods. Also, the TL method,35 which was from SIMC has the following parameters discussed in example 7, was used to design a PI τ controller for the process. The PI settings are shown in K ) , τ ) min{τ,4(τ + θ)} (108) Table 12. c K(τ + θ) I c c PID controllers were designed using the approximate second-order model and the DS-d, IMC, and SIMC For the second-order model in eq 67 with τ > τ , the 1 2 methods. The τ values for these methods were selected SIMC method provides the following rules for PID c to give M values close to the 1.58 value of Table 13. controllers with the series structure S Because the tuning rules for the SIMC method are for τ the series PID structure, these PID settings were K ) 1 , ) min{ ,4( + )}, ) converted to the parallel structure, which was then used c + τI τ1 τc θ τD τ2 K(τc θ) for the simulation. Also, the ZN PID settings were (109) calculated for the fourth-order model. The PID controller characteristics are shown in Table 13. Using Skogestad’s approximation method, the fourth- The simulation responses for a unit step change in order process in eq 107 can be approximated as a first- the set point at t ) 0 min and a step disturbance (d ) order plus time delay model 3) at t ) 5 min are given in Figures 16 and 17 for PI - and PID controllers, respectively. The values of MS, IAE, e 0.148s G (s) ) (110) and TV for all of these controllers are presented in 1 1.1s + 1 Tables 12 and 13. Figure 16 indicates that the distur- bance response of the DS-d PI controller is better and or as a second-order plus time delay model faster than the responses of the DS and SIMC control- -0.028s lers. The IAE values for the TL PI controller are smaller ) e than those of the other PI controllers, but its responses G2(s) (111) (s + 1)(0.22s + 1) are oscillatory and its MS and TV values are large. A Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4821

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 can be independently tuned via a standard set-point weight- ing factor or a set-point filter constant. The set-point tuning does not affect the disturbance response. Nine simulation examples have been used to compare alternative design methods and to illustrate the effect of τc. The simulation results demonstrate that the DS-d method provides better disturbance performance than standard DS and IMC methods and that satisfactory responses to set-point changes can be obtained by simply tuning the set-point weighting factor b. The DS-d Figure 16. Simulation results of PI controllers for example 9. method furnishes a convenient and flexible design method that provides good performance in terms of disturbance rejection and set-point tracking.

Acknowledgment The authors acknowledge the UCSB Process Control Consortium for financial support and Professor Sigurd Skogestad for his helpful comments and suggestions.

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