Robust Proportional-Integral-Derivative Controller Design for Stable

Article pubs.acs.org/IECR

Robust Proportional-Integral-Derivative Controller Design for Stable/ Integrating Processes with Inverse Response and Time Delay Jyh-Cheng Jeng* and Sheng-Wen Lin Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei 106, Taiwan ABSTRACT: Processes that exhibit dynamic behaviors of inverse response and time delay increase the difficulty of control system design. This paper presents the design concepts that lead to robust tuning of a proportional-integral-derivative (PID) controller for stable/integrating processes with inverse response and time delay. The proposed control system design is based on a Smith-type compensator for nonminimum phase dynamics, which aims to remove these elements from the feedback loop. In this control scheme, it is necessary to factorize the process model into minimum phase and nonminimum phase parts. By investigating and comparing different factorization methods, this study shows that a system based on the direct factorization method can achieve better trade-off between control performance and system robustness. Furthermore, this study approximates the equivalent feedback controller for the proposed control configuration as a classical PID controller based on Maclaurin-series approach. Analytical tuning rules for the PID controller are developed, and the analysis of robust stability is provided. Adjusting a tuning parameter can achieve a superior trade-off between nominal performance and robust stability of the closed-loop system. Simulation results confirm the superiority of the proposed control design method.

1. INTRODUCTION A process exhibits an inverse response when its initial response to an input is in the opposite direction of the final steady-state value. A process generally yields an inverse response due to conflict between two opposite dynamic effects from the same manipulated variable. This phenomenon frequently appears in chemical process industries,1,2 such as the level of drum boiler in a distillation column and the exit temperature of a tubular exothermic reactor. Integrating processes that exhibit inverse response are also frequently encountered in the level control of a boiler steam drum.3 The transfer function of process with inverse response generally has an odd number of zeros in the open right-half-plane (RHP).4 The RHP zeros and time delay normally contribute phase lag to the whole system. As a result, these dynamic characteristics are called nonminimum phase (NMP) elements. The NMP dynamics of the process generally impose fundamental limitations on achievable control performance, and the condition of perfect control cannot be obtained by any stable and causal controller.5 According to root locus analysis, the closed-loop poles move toward the position of open-loop zeros as the controller gain approaches infinity. One important limitation due to the presence of RHP zero is the high gain instability. Another control limitation is associated with the bandwidth.6 These factors usually preclude tight process control. Because of the difficulties in control of processes with inverse response, many researchers have attempted to develop appropriate control design methods. There are two categories of control structures. The first category uses a proportionalintegral (PI) or proportional-integral-derivative (PID) controller with various tuning methods. Waller and Nygardas7 demonstrated that a PID controller with Ziegler−Nichols (ZN)8 tuning achieves acceptable control of inverse response systems. Luyben9 proposed a new tuning method in which the PI tuning parameters are functions of RHP zero and time delay. © 2012 American Chemical Society

However, the PI setting is empirical and causes large oscillation and overshoot. Chien et al.10 derived a PID tuning method from a direct synthesis controller design method, which achieves smooth output response. The good results achieved by PID controllers are due to a positive feature of the derivative action in “anticipating” the wrong direction of the system’s response.1 Unfortunately, the controller parameters in the aforementioned methods are not conveniently adjustable to maintain the stability margin. On the contrary, an internal model control (IMC)11 design makes it possible to conveniently adjust the system’s robustness using a single tuning parameter. Based on the IMC framework, Scali and Rachid5 proposed an analytical design of a PID controller for inverse response processes without time delay. Chen et al.12 proposed a PID tuning method for inverse response processes by approximating the RHP zero with time delay using the Pade approximation. However, relatively few studies discuss the control design for integrating processes with inverse response and time delay. For this type of system, Luyben13 presented an empirical PI/PID controller tuning method that adopts the frequencydomain technique. Gu et al.3 developed a PI/PID controller design method based on H∞ optimization and IMC theory. Pai et al.14 recently presented PID tuning rules based on the method of direct synthesis for disturbance rejection (DS-d),15 but the resulting set-point response is not satisfactory. The second category utilizes model-based regulators, called inverse response compensators. Most of these inverse response compensators are based on the Smith predictor,16 known as the time delay compensator, and attempt to remove the RHP zero from of the feedback loop so that the controller just faces the Received: Revised: Accepted: Published: 2652

July 7, 2011 December 30, 2011 January 4, 2012 January 4, 2012 dx.doi.org/10.1021/ie201449m | Ind. Eng.Chem. Res. 2012, 51, 2652−2665

Industrial & Engineering Chemistry Research

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minimum phase dynamics. This in turn makes the controller design much easier. Iinoya and Altpeter17 proposed a compensation scheme to cope with inverse response of a process, but their controller design is empirical. Zhang et al.18 modified Iinoya and Altpeter’s scheme to avoid unnecessary complexity and used H∞ control theory for controller design. Alcantara et al.19 recently presented a Smith-type predictor scheme that also uses the H∞ design. However, all of these inverse response compensators only consider inverse response systems without time delay. In addition, any implementation of the Smith-type compensation schemes (e.g., Alcantara et al.19) must factorize the process model into minimum phase and nonminimum phase parts, as this is also necessary in IMC design. Although the method of all-pass factorization (i.e., factorizing the nonminimum phase part as all-pass form) appears frequently in the literature,5,12,19 the effects of different factorization methods on system performance and robustness have never been systematically studied. There are plenty of methods in the literature for the control system design of processes with inverse response or time delay, but few of them consider both factors in a system, especially for the design of inverse response compensator. This study proposes a Smith-type compensator scheme for processes with both inverse response and time delay. The proposed scheme considers two types of process with the following transfer functions: (i) Stable process with inverse response with time delay

G (s) =

K p( − αs + 1)

(τ1s + 1)(τ2s + 1) α > 0; τ1, τ2 , θ > 0

Figure 1. Structure of proposed Smith-type compensator.

controller, respectively. The signals r, d, y, and u denote the setpoint, disturbance, process output, and controller output, respectively. In this structure, the controller C(s) sees the actual process G(s) in parallel with a model G̃ (s). Let the process model, G̃ (s), be factorized into a minimum phase part, G̃ M(s), and a nonminimum phase part, G̃ N(s), such that

G̃ (s) = G̃ M(s)G̃ N(s)

The output of the minimum phase block, ξ, is fed back to generate the controller input. Residual signal y − ỹ is fed back in the main feedback loop to take into account the effects of the disturbances and modeling errors. The closed-loop transfer functions that relate the input signals (i.e., set-point and disturbance) and the process output can be derived as

y(s) =

e−θs ,

τ, θ > 0

CG r (s) 1 + CG̃ M + C(G − G̃ ) G(1 + CG̃ M − CG̃ ) + d (s) 1 + CG̃ M + C(G − G̃ )

(4)

If the model is perfect, then eq 4 can be simplified as (1)

y(s) =

(ii) Integrating process with inverse response with time delay

K p( − αs + 1) −θs G (s) = e , s(τs + 1)

(3)

⎛ CG̃ MG̃ N CG̃ MG̃ N ⎞ r (s ) + G ⎜ 1 − ⎟ d (s) 1 + CG̃ M 1 + CG̃ M ⎠ ⎝

(5)

In the case of nominal condition (i.e., G̃ (s) = G(s)) and no disturbance (d = 0), Figure 2 shows the net result of the

α > 0; (2)

where Kp is the steady-state gain, τ1, τ2, and τ are the time constants, and θ is the time delay. This study systematically investigates the effects of different factorization methods on the system performance and robustness using the proposed scheme. For practicability, this study also approximates the equivalent classical feedback controller of the proposed control scheme as a PID controller using the Maclaurin-series approach20 and derives the analytical PID tuning rules. Compared with the existing PID design methods, the proposed method achieves a superior trade-off between control performance and system robustness. The rest of this paper is organized as follows. Section 2 presents the configuration and design method of the proposed compensator. Sections 3 and 4 compare the control systems resulting from different model factorization methods for stable and integrating processes, respectively. Section 5 presents the tuning rules of the PID controller in the unity feedback scheme, and analyzes system robustness. Section 6 provides examples showing the effectiveness of the proposed approach. Finally, section 7 makes concluding remarks.

Figure 2. Ideal net result of proposed compensator in the nominal condition.

proposed compensator, according to eq 5. Note that, in view of the ideal scheme, the objective of the proposed control system is to get the nonminimum phase dynamics out of the feedback loop. As a result, the controller C(s) only faces the minimum phase part G̃ M(s), and some of the control limitations imposed by RHP zero and time delay, such as high gain instability, disappear. Figure 2 shows that the controller C(s) can be designed by considering only the minimum phase portion of the process, which makes the controller design easier. Consequently, this study uses the direct synthesis (DS) method to design the controller C(s). Denote the closed-loop transfer function of the dashed box in Figure 2 as H(s), where

2. CONFIGURATION AND DESIGN METHOD OF SMITH-TYPE COMPENSATOR Figure 1 shows the structure of proposed Smith-type compensator, where G(s) and C(s) represent the process and

H (s) =

C(s)G̃ M(s) 1 + C(s)G̃ M(s)

(6)

By specifying a desired form of H(s) in eq 6, the controller C(s) can be synthesized as 2653

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C(s) =

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resulting in the following commercial PID controller:

H (s) ̃ GM(s)[1 − H(s)]

(7)

C(s) =

The complementary sensitivity function of the control system is given as

y(s) T (s) = = H(s)G̃ N(s) r (s)

(8)

(τ1s + 1)(τ2s + 1) G̃ N(s) = ( −αs + 1) e−θs

e−θs λ1εs 2 + λ1s + 1 (15) The all-pass factorization method uses the same H(s) as given in eq 11. Therefore, the complementary sensitivity function in the case of all-pass factorization (by choosing λ = λ2) is −αs + 1 Tallpass(s) = e−θs (λ 2s + 1)(αs + 1) (16) The corresponding controller computed from eq 7 becomes

C(s) =

K p(αs + 1) (τ1s + 1)(τ2s + 1)

( −αs + 1) −θs G̃ N(s) = e (αs + 1)

TH∞(s) =

;

;

C(s) =

−αs + 1 (λ3s + 1)2

e−θs (18)

(τ1s + 1)(τ2s + 1) K ps(λ32s + 2λ3 − α)

(19)

3.2. Comparisons of System Performance and Robustness. This section compares the performance and robustness of control systems resulting from these two factorization methods. Performance is evaluated by the integral of the squared error (ISE), defined as

(10)

1 (11) λs + 1 The parameter λ can be chosen to make a trade-off between control performance and system robustness. By increasing λ, the robustness margins increase at the expense of a slower response. Equation 8 shows that the complementary sensitivity function T(s) in the case of direct factorization (by choosing λ = λ1) is −αs + 1 −θs Tdirect(s) = e λ1s + 1 (12) H (s) =

ISE = || e ||22 =

∫0

∞

e 2 (t ) d t

(20)

where the error is given by e = r − y. The robustness is measured by the maximum of the sensitivity function (MS), defined as

MS = max|S(j ω)| = max|1 − T (j ω)| ω

ω

(21)

where S denotes the sensitivity function with S + T = 1. The value MS can be interpreted as the inverse of the shortest distance between loop transfer function and the critical point in the Nyquist plot. Thus, as MS decreases, the dynamic response of closed-loop system becomes more robust. Note that MS < 2 is a traditional robustness indicator.6 Considering the limit cases, λ1 = α in eq 12 and λ2 = 0 in eq 16, both systems become identical and would be H2 optimal (i.e., with minimum ISE) to a step input.11 In practical applications, however, it is necessary to choose λ1 > α and λ2 > 0 due to the robust stability considerations. This implies that a system resulting from direct factorization (eq 12) would be

In this case, the controller synthesized from eq 7 becomes

(τ1s + 1)(τ2s + 1) K p λ1s

(17)

The complementary sensitivity function above is obtained by extending the work of Alcantara et al.,19 who used the H∞ design method with all-pass factorization, to inverse response processes with time delay. The corresponding controller C(s) is then given as follows:

(9)

For the method of direct factorization, the desired form of H(s) is specified as

C(s) =

(τ1s + 1)(τ2s + 1) K p λ 2s(αs + 1)

which has the form of a commercial PID controller. In addition to the two systems described above, this study includes a third system for comparison:

The second method is the all-pass factorization (AF), where G̃ (s) is factorized as

G̃ M(s) =

−αs + 1

Tdirect(s) =

3. COMPARISON OF CONTROLLER DESIGN METHODS FOR STABLE PROCESSES This section considers the process of eq 1 and compares the performance and robustness of the systems resulting from different model factorization methods. 3.1. Controller Design Using Different Factorization Methods. There are two ways to factorize the process model given in eq 1. The first one is the direct factorization (DF) method, where G̃ (s) is factorized as G̃ M(s) =

(14)

The term ε in eq 14 is the filter time constant with small value. Through extensive simulations, the value of ε is suggested as ε = 0.1(α + θ) in this work. Consequently, the resulting complementary sensitivity function is a little different from that in eq 12 and is rolled off at high frequency as given by

Theoretically, only the nonminimum phase portion of the process, which cannot be eliminated by any stable and causal regulator, imposes the control performance limitations. In addition, eq 8 indicates that the control performance that can be achieved by the proposed control structure depends on how the process is factorized (i.e., depends on G̃ N(s)). Therefore, one of the objectives of this study is to determine which type of model factorization achieves the best system performance and robustness.

Kp

(τ1s + 1)(τ2s + 1) K p λ1s(εs + 1)

(13)

which has the form of an ideal PID controller. For physical realization, a first-order filter can be augmented to C(s), 2654



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method. Specifically, the direct factorization system has the lowest MS value (most robust) when all systems are tuned with the same ISE performance. On the other hand, it also has the smallest ISE when all systems are tuned with the same degree of robustness (the same MS). This advantage becomes more apparent when the parameter α is large (i.e., cases B and D). This makes sense because the effect of different factorization methods becomes more significant as α increases. To illustrate the above performance/robust analysis is independent from the choice of the performance index, another performance index, integral of the time-weighted absolute error (ITAE), is adopted for repeating the analysis. The ITAE is defined as:

more robust than that resulting from all-pass factorization (eq 16). This is because achieving similar control performance (in terms of ISE) for these two systems requires that λ1 > λ2. Note that a larger value of λ indicates increased robustness. The following comparative studies justify this inference. The ISE to a step set-point change of these systems can be derived analytically using the following equation

ISE = =

∫0 ∫0

∞

[e(t )]2 dt 2 1 ⎤ −1 1 − L T ( s ) ⎢⎣ ⎥ dt s ⎦

∞⎡

{

}

(22)

where L−1 denote the operator of inverse Laplace transform. The results are given, for T(s) shown in eqs 12, 16, and 18, respectively, as follows.

ISEdirect =

(λ1 + α)2 +θ 2λ1

(23)

ISEallpass =

λ2 + 2α + θ 2

(24)

ISE H∞ =

5λ32 + 4αλ3 + α 2 +θ 4λ3

ITAE =

∫0

∞

t |e(t )| dt

(27)

Figure 4 shows the relationship between ITAE and MS for the considered three control systems, from which similar results can be concluded. This study also simulates closed-loop responses for these three control designs using the scheme in Figure 1. Figure 5 shows the

(25)

The value of MS can also be computed numerically from eq 21 using T(s) given in eqs 12, 16, and 18. For the purpose of comparison, this study simulated the following process with different parameters as in the work by Luyben:9

G (s) =

−αs + 1 2

(s + 1)

e−θs (26)

Four cases were tested: case (A) α = 0.2 and θ = 0.2; case (B) α = 1.6 and θ = 0.2; case (C) α = 0.2 and θ = 1.6; case (D) α = 1.6 and θ = 1.6. Figure 3 depicts the relationship between

Figure 4. Relationship between ITAE and MS for systems using different design methods (stable process).

responses in the nominal condition, where a unit step set-point change is added at t = 0 and a step disturbance with magnitude −1 is added to the process input later. To ensure a fair comparison, the tuning parameters (i.e., λ1, λ2, and λ3) were selected such that the ISE value for each system is similar to that for the classical PID control system tuned by the method given in Chien et al.10 Robustness is more important than the nominal performance in practice. Next, consider the case of model-mismatch. Suppose that the parameters of real process deviate by +20% in Kp, α, θ, and −20% in τ1, τ2 from the model (i.e., the worst case). Figure 6 shows the responses of the perturbed system. The system resulting from direct factorization method still maintains good performance (the responses are less oscillatory and with smaller overshoot), and is therefore more robust than the others. Table 1 summarizes the tuning parameters and indices of performance and robustness. Though the all-pass factorization method is usually used in the literature,5,12,19 the comparative studies above indicate that the direct factorization method is a better choice for the proposed compensator scheme for stable processes with inverse response and time delay.

Figure 3. Relationship between ISE and MS for systems using different design methods (stable process).

ISE and MS for three control systems by tuning the parameter λ in each case. These plots clearly show that the control system resulting from the direct factorization method achieves a better trade-off between performance and robustness than the other two systems, which resulted from the all-pass factorization 2655



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Table 1. Tuning Parameters and Indices of Performance and Robustness for Process of eq 26 ISE (set-point + disturbance) case

method

λ

MS

nominal

perturbed

A

DS-DFa DS-AFb H∞-AFc DS-DF DS-AF H∞-AF DS-DF DS-AF H∞-AF DS-DF DS-AF H∞-AF

0.933 0.647 0.405 2.212 0.351 0.783 2.278 2.076 1.004 4.425 2.111 1.847

1.318 1.446 1.490 1.736 1.900 2.127 1.423 1.471 1.596 1.485 1.690 1.694

1.248 1.248 1.248 6.822 6.822 6.822 5.163 5.163 5.163 11.65 11.65 11.65

1.196 1.233 1.256 18.19 24.73 89.86 5.886 5.888 6.140 15.42 17.52 17.55

B

C

D

a

Direct synthesis design with direct factorization. bDirect synthesis design with all-pass factorization. cH∞ design with all-pass factorization.

Figure 5. Closed-loop responses of nominal system (stable process).

In both cases, the desired form of H(s) is specified as

γs + 1

H (s) =

(λs + 1)2

(30)

Equation 8 shows that the complementary sensitivity function T(s) in the case of direct factorization (by choosing λ = λ1) is

(γs + 1)( −αs + 1) −θs e (λ1s + 1)2

Tdirect(s) =

(31)

Consequently, the controller computed from eq 7 becomes

C(s) =

(γs + 1)(τs + 1) K p(λ12s + 2λ1 − γ)

(32)

Again, a first-order filter is augmented to C(s) for physical realization as follows:

C(s) =

Figure 6. Closed-loop responses of perturbed system (stable process).

4. COMPARISON OF CONTROLLER DESIGN METHODS FOR INTEGRATING PROCESSES This section considers the process in eq 2, and compares the performance and robustness of the systems resulting from different model factorization methods. 4.1. Controller Design Using Different Factorization Methods. For integrating processes with inverse response and time delay of eq 2, direct factorization of the process model gives

G̃ M(s) =

Kp s(τs + 1)

(33)

Tdirect(s) =

(γs + 1)( −αs + 1) λ12εs3

+

(λ12

2

+ (2λ1 − γ)ε)s + 2λ1s + 1

e−θs (34)

For the all-pass factorization method, the complementary sensitivity function (by choosing λ = λ2) is

G̃ N(s) = ( −αs + 1) e−θs

Tallpass(s) =

(28)

while all-pass factorization of the process model gives

G̃ M(s) =

K p(λ12s + 2λ1 − γ)(εs + 1)

where ε = 0.1(α + θ) is suggested in this paper. A commercial PID controller can be obtained for C(s) by choosing γ = 2λ1. The resulting complementary sensitivity function is a little different from that in eq 31 and is given by

;

K p(αs + 1)

(γs + 1)(τs + 1)

(35)

The corresponding controller computed from eq 7 becomes

;

s(τs + 1) ( −αs + 1) −θs G̃ N(s) = e (αs + 1)

(γs + 1)( −αs + 1) −θs e (λ 2s + 1)2 (αs + 1)

C(s) =

(τs + 1)(γs + 1) K p(αs + 1)(λ 2 2s + 2λ 2 − γ)

(36)

Likewise, a commercial PID controller can be obtained by choosing γ = 2λ2.

(29) 2656



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4.2. Comparisons of System Performance and Robustness. By deriving eq 22, the ISE values to a step setpoint change of control systems given in eq 31 and eq 35, respectively, are obtained as follows.

ISEdirect =

5α 2 + 4αλ1 + λ12 +θ 4λ1

(37)

ISEallpass =

8α + λ 2 +θ 4λ 2

(38)

The value of MS can also be computed numerically from eq 21 using T(s) given in eq 31 and eq 35. For the purpose of comparison, this study simulates the following process with different parameters, as in the work of Gu et al.3

G (s) =

−αs + 1 −θs e s(s + 1)

(39)

Figure 8. Relationship between ITAE and MS for systems using different design methods (integrating process).

Four cases were tested: case (A) α = 0.2 and θ = 0.2; case (B) α = 1.6 and θ = 0.2; case (C) α = 0.2 and θ = 1.6; case (D) α = 1.6 and θ = 1.6. Figure 7 depicts the relationship between ISE

Figure 9. Closed-loop responses of nominal system (integrating process).

Figure 7. Relationship between ISE and MS for systems using different design methods (integrating process).

and −20% in τ from the model (i.e., the worst case). Figure 10 shows the responses of the perturbed system, indicating that the robustness of the system from direct factorization is better because its responses are less oscillatory. Table 2 summarizes the tuning parameters and indices of performance and robustness. For integrating processes with inverse response and time delay, the comparative studies above also indicate that the direct factorization method is a better choice than the all-pass factorization method in the proposed compensator scheme. Remark 1. For stable processes, the structure of the proposed Smith-type compensator can be used for the control system implementation. In the case of integrating processes, the structure of the Smith-type compensator has to be abandoned for the control system implementation because of the problem of internal instability.21 The control system must be implemented via the classical feedback structure shown in Figure 11, where the classical feedback controller GC(s) has to be approximated with a PID controller to avoid internal instability, as the following section shows. On the other hand

and MS for two control systems of eq 31 and eq 35 by tuning the parameter λ in each case. Similar results can be found with that for stable processes. That is, the control system resulting from direct factorization method achieves a better trade-off between performance and robustness than the system resulting from all-pass factorization method. In addition, this advantage is more manifest when the parameter α is large (i.e., cases B and D). Figure 8 shows the relationship between ITAE and MS for the considered two control systems. The results evidence that the method is independent from the choice of the performance index. This study also simulates the closed-loop responses for these two control designs using the scheme in Figure 1. To ensure a fair comparison, the tuning parameters (i.e., λ1 and λ2) are selected such that the ISE value for each system is similar. Figure 9 shows the responses in the nominal condition, where a unit step change is added at t = 0. Next, consider the case of model-mismatch by assuming that the parameters of real process deviate by +20% in Kp, α, θ, 2657



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Thus the desired sensitivity function possesses two zeros at s = 0 so that step load disturbances injected into the integrating process input can be counteracted. For the Tdirect(s) given in eq 31 or 34, the constraint eq 41 yields

γ = 2λ1 + α + θ

(42)

On the other hand, the constraint leads to γ = 2λ2 + 2α + θ for the Tallpass(s) given in eq 35.

5. CLASSICAL PID CONTROLLER DESIGN Despite continued advancements in control technology, classical feedback control systems that employ PID controllers are widely used in process industries because of their simple structure and ease of implementation. The proposed control system presented in the previous section should be able to, at least for integrating processes, be implemented in the classical PID control scheme in Figure 11. Using simple blocks algebra, the equivalent feedback controller for the configuration shown in Figure 1 is given by

Figure 10. Closed-loop responses of perturbed system (integrating process).

GC (s) =

Table 2. Tuning Parameters and Indices of Performance and Robustness for Process of eq 39

A B C D

method a

DS-DF DS-AFb DS-DF DS-AF DS-DF DS-AF DS-DF DS-AF

λ

MS

nominal

perturbed

2.22 1.60 4.90 1.60 2.24 2.00 6.35 2.80

1.290 1.445 1.680 1.968 1.747 1.819 1.682 1.980

0.996 0.997 3.798 3.798 2.498 2.498 5.499 5.499

1.083 1.120 4.836 5.378 3.499 3.413 7.230 7.930

a

(43)

Only the system with direct factorization design method is considered at this stage, as this type of system achieves a better trade-off between performance and robustness. 5.1. Development of PID Tuning Rule for Stable Processes. By substituting eqs 9 and 14 into eq 43, the equivalent feedback controller can be written as

ISE case

C(s) ̃ (s) − G̃ (s)] 1 + C(s)[GM

GC (s) =

(τ1s + 1)(τ2s + 1) K p[λs(εs + 1) + 1 − ( −αs + 1) e−θs]

The above controller is not in the form of PID controller. Therefore, this study uses the Maclaurin series expansion formula, which has been shown as a good approximation,20 to obtain a PID controller which approximates GC(s) given in eq 44. First, define

b

Direct synthesis design with direct factorization. Direct synthesis design with all-pass factorization.

f (s) = sGC(s)

GC (s) =

even for integrating processes, the Smith-type compensator can be used for the parametrization and design of the feedback controller GC(s). Remark 2. For integrating processes, the Smith-type compensator with the choice of γ = 2λ in H(s), leading to a PID controller for C(s), will result in an offset when a step load disturbance is injected into the process input. According to eq 5, the closed-loop transfer function for disturbance in the nominal condition can be expressed as

s→0

d (1 − T (s)) = 0 ds

⎤ f ″(0) 2 1⎡ s + ···⎥ ⎢f (0) + f ′(0)s + ⎦ s⎣ 2

(46)

The standard PID controller can be approximated using only the first three terms in eq 46 and truncating all other high-order terms. That is,

⎛ ⎞ 1 GC(s) ≈ K C⎜1 + + τDs⎟ τIs ⎝ ⎠

(47)

with

(40)

K C = f ′(0);

Because G(s) has an integral term 1/s, the following asymptotic tracking constraint has to be satisfied to eliminate the offset:

lim

(45)

Then, expanding f(s) in a Maclaurin series in s gives

Figure 11. Classical feedback control scheme.

y(s) = G(s)(1 − T (s)) d (s)

(44)

τI =

f ′(0) ; f (0)

τD =

f ″(0) 2f ′(0)

(48)

Using eq 48, the PID tuning parameters can be obtained as analytical functions of the process model parameters and the adjustable parameter λ. After some tedious derivations, the PID tuning rule can be written as

(41) 2658



KC =

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τI K p(λ + α + θ)

τI = (τ1 + τ2) +

2αθ + θ 2 − 2λε 2(λ + α + θ)

2(τ12 + τ1τ2 + τ2 2) + τD = τI −

(τ1 + τ2)(2αθ + θ2 − 2λε) (λ + α + θ)

3αθ2 + θ3 3(λ + α + θ)

2τI

(49)

The same PID tuning procedures can also be applied to the following underdamped processes with inverse response and time delay.

To implement the PID controller physically, a lag element 1/(0.1τDs + 1) is usually introduced to the derivative term. The conditions for the selection of adjustable parameter λ so that eq 49 leads to positive PID parameters have been derived. The results are given in the Appendix. In general for the models of typical chemical processes, there exists a wide range of λ in which positive PID parameters can be obtained. Figure 12 compares the closed-loop responses for the processes given in eq 26 using the Smith-type compensator and the approximated classical PID controller. It can be seen that the performance is unchanged using the approximated PID controller.

KC =

+

G (s) =

K p( −αs 2 2

+ 1)

τ s + 2τζs + 1

τ, θ > 0;

e−θs ,

α > 0;

0 0 is

parameters and the resulting performance indices. Figure 21 depicts the closed-loop responses, where a unit step setpoint input is added at t = 0 and a step disturbance with amplitude −0.5 is added to the process input at t = 30. Luyben’s method provides the smallest overshoot in the setpoint response but results in very sluggish set-point and

2(τ1 + τ2 − ε)λ > − 2(α + θ)(τ1 + τ2) − 2αθ − θ 2 2664

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Article

The condition for τD > 0 is

If τ1 + τ2 − ε ≥ 0, then τI > 0 is satisfied for any λ > 0. Otherwise, if τ1 + τ2 − ε < 0, the condition for selecting λ to obtain τI > 0 is given as

λ
0 3(λ + α + θ) ⎥⎦

(A3)

which can be further derived as (A4) a λ2 + b λ + c > 0 where a, b, and c are functions of model parameters given as

a = 2[τ1τ2 − ε(τ1 + τ2) + ε2] 1 b = [4τ1τ2 − 2ε(τ1 + τ2)](α + θ) + (τ1 + τ2 − 2ε)(2αθ + θ 2) − (3αθ 2 + θ3) 3 2 1 c = [2τ1τ2(α + θ) + (τ1 + τ2)(2αθ + θ 2)](α + θ) + α 2θ 2 + αθ3 + θ4 3 6

(9) Luyben, W. L. Tuning Proportional-Integral Controllers for Processes with Both Inverse Response and Deadtime. Ind. Eng. Chem. Res. 2000, 39, 973−976. (10) Chien, I. L.; Cheng, Y. C.; Chen, B. S.; Chuang, C. Y. Simple PID Controller Tuning Method for Processes with Inverse Response Plus Dead Time or Large Overshoot Response Plus Dead Time,. Ind. Eng. Chem. Res. 2003, 42, 4461−4477. (11) Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: New York, 1989. (12) Chen, P.; Zhang, W.; Zhu, L. Design and Tuning Method of PID Controller for A Class of Inverse Response Processes. Proceedings of the 2006 American Control Conference, Minneapolis, Minnesota, USA, 274−279, June 2006. (13) Luyben, W. L. Identification and Tuning of Integrating Processes with Deadtime and Inverse Response. Ind. Eng. Chem. Res. 2003, 42, 3030−3035. (14) Pai, N. S.; Chang, S. C.; Huang, C. T. Tuning PI/PID Controllers for Integrating Processes with Deadtime and Inverse Response by Simple Calculations. J. Process Control 2010, 20, 726− 733. (15) Chen, D.; Seborg, D. E. PI/PID Controller Design Based on Direct Synthesis and Disturbance Rejection. Ind. Eng. Chem. Res. 2002, 41, 4807−4822. (16) Smith, O. J. M. Closed Control of Loops with Deadtime. Chem. Eng. Prog. 1957, 53, 217−219. (17) Iinoya, K.; Altpeter, R. J. Inverse Response in Process Control. Ind. Eng. Chem. 1962, 54, 39−43. (18) Zhang, W.; Xu, X.; Sun, Y. Quantitative Performance Design for Inverse-Response Processes. Ind. Eng. Chem. Res. 2000, 39, 2056− 2061. (19) Alcantara, S.; Pedret, C.; Vilanova, R.; Zhang, W. D. Analytical H∞ Design for A Smith-Type Inverse-Response Compensator. 2009 American Control Conference, St. Louis, MO, USA, 1604−1609, June 2009. (20) Lee, Y.; Park, M.; Brosilow, C. PID Controller Tuning for Desired Closed-Loop Responses for SI/SO Systems. AIChE J. 1998, 44, 106−115. (21) Mirkin, L.; Palmor, Z. J. Control Issues in Systems with Loop Delays. Handbook of Networked and Embedded Control Systems; HristuVarsakelis, D., Levine, W. S., Eds.; Birkhäuser: Boston, MA, 2005; pp 627−648. (22) Bequette, B. W. Process Control: Modeling, Design and Simulation; Prentice-Hall: Upper Saddle River, NJ, 2003.

Notice that c is always positive. According to eq A4, there are several cases for the selection of adjustable parameter λ: (i) If a > 0 and b2 − 4ac < 0, then τD > 0 is always satisfied irrespective of the selection of λ. (ii) If a > 0 and b2 − 4ac ≥ 0, the condition for selecting λ to obtain τD > 0 is λ < λ or λ > λ̅, where λ and λ̅ are the two roots of aλ2 + bλ + c = 0 with λ ≤ λ̅. (iii) If a < 0, which implies b2 − 4ac > 0, the condition for selecting λ to obtain τD > 0 is λ < λ < λ̅. (iv) If a = 0 and b ≥ 0, then τD > 0 is satisfied for any λ > 0. (v) If a = 0 and b < 0, the condition for selecting λ to obtain τD > 0 is λ < −(c/b). In general for the models of typical chemical processes, there exists a wide range of λ so that the tuning rule of eq 49 leads to positive PID parameters. For eq 51, the conditions for obtaining positive PID parameters can be derived by replacing τ1τ2 and τ1 + τ2 in the above results with τ2 and 2τζ, respectively.

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REFERENCES

(1) Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; Prentice-Hall: Englewood Cliffs, NJ, 1984. (2) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling and Control; Oxford: New York, 1994. (3) Gu, D.; Ou, L.; Wang, P.; Zhang, W. Relay Feedback Autotuning Method for Integrating Processes with Inverse Response and Time Delay. Ind. Eng. Chem. Res. 2006, 45, 3119−3132. (4) Rosenbrock, H. H. State-space and Multivariable Theory; Nelson: London, U.K., 1970. (5) Scali, C.; Rachid, A. Analytical Design of Proportional-IntegralDerivative Controllers for Inverse Response Processes. Ind. Eng. Chem. Res. 1998, 37, 1372−1379. (6) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Design: Analysis and Design; Wiley: New York, 1996. (7) Waller, K. T. V.; Nygardas, C. G. On Inverse Response in Process Control. Ind. Eng. Chem. Fundam. 1975, 14, 221−223. (8) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759−768. 2665


Robust Proportional-Integral-Derivative Controller Design for Stable

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