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Analytical Proportional-Integral (PI) Controller Tuning Using Closed-Loop Setpoint Response Wuhua Hu and Gaoxi Xiao* School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore ABSTRACT: Recently, Shamsuzzoha and Skogestad proposed a proportional-integral (PI) tuning rule for a wide range of unidentified processes. [M. Shamsuzzoha and S. Skogestad, J. Process Control 2010, 20, 1220.] The rule relies on a closed-loop setpoint response (CSR) of a process and was developed from extensive numerical experiments. This work analytically derives a similar PI tuning rule using the CSR method. Simulations indicate that the two rules perform similarly if the tuning parameter is selected properly for the analytical rule. Meanwhile, a guideline is proposed for choosing the proportional (P) controller gain for the CSR experiment to result in a proper overshoot for obtaining good PI settings. Numerical examples are used to demonstrate the usefulness of the theoretical results.
1. INTRODUCTION The proportional-integral (PI) controller has been widely applied in industry for its simplicity, robustness, and wide ranges of applicability in the regulatory layer.1-5 PI controller tuning has been extensively studied in the last decades, generating a large number of PI tuning rules.3-6 However, conventional PI controller tuning requires trials and may experience instability during tuning experiment or process modeling, and the resulting closedloop performance is satisfactory only for particular classes of processes.1 To overcome these problems, Shamsuzzoha and Skogestad recently proposed using closed-loop setpoint response (CSR) to set the PI parameters, which requires a single closed-loop experiment and gives fast and robust performance for a broad range of processes typical for process control.1 An earlier CSR method was considered by Yuwana and Seborg in 1982,7 but their method leads to a more-complicated solution. In the CSR method, one carries out a closed-loop experiment with a single proportional (P) controller and then utilizes the response information to derive the PI settings. Shamsuzzoha and Skogestad considered a special case of the simple internal model control (SIMC) tuning rule with its single tuning parameter, the closedloop time constant τc, set as τc = θ (where θ is the effective time delay of a process).8,9 They derived a PI tuning rule by relating the closedloop response quantities with the SIMC settings, including the peak time, overshoot, and steady-state offset. The resulting PI tuning rule was tested on a broad range of processes and demonstrated to give comparable performance as the SIMC tuning rule. While Shamsuzzoha and Skogestad developed the PI tuning rule from series of numerical experiments, analytical derivation of a similar rule using CSR method is of interest in this work. The derivation is based on an integral plus time delay (ITD) process and then extended to a first-order plus time delay (FOTD) process. The main idea is as follows. With the CSR method, a single P controller is applied to the process and a step test of setpoint change is performed. From the closed-loop response, the steady-state offset, peak time, and overshoot or rise time are recorded. These quantities, together with the applied proportional gain and setpoint change, are used to estimate the process r 2011 American Chemical Society
parameters and consequently express the SIMC tuning rule in a new manner. The resulting PI tuning rule has a single tuning parameter (R) that controls the tradeoff between performance and robustness. This rule is tested on a broad range of processes that are typical for process control applications. The results indicate that the analytical rule gives comparable performance to Shamsuzzoha and Skogestad’s PI tuning rule1 if the detuning parameter R is chosen properly. In a sense, the analysis and derived rule provide some form of insight and support to the PI tuning rule proposed by Shamsuzzoha and Skogestad. Throughout the paper, the notation “:=” means “is defined as” and is used to introduce new symbols when necessary.
2. DERIVATION OF THE PI TUNING RULE The control system is described in Figure 1, where u is the manipulated control input, d the disturbance, y the controlled output, ys the setpoint (reference) for the controlled output, c(s) the PI controller transfer function, and g(s) the process transfer function. The PI controller takes the form of 1 cðsÞ ¼ kc 1 þ ð1Þ τI s where kc is the proportional (P) gain and τI is the integral (I) time constant. CSR Experiment. When a single P controller (c(s) = kc0) is applied to the process, a setpoint change is made. From the CSR experiment (see Figure 2), we record the following values:1 Δys: Setpoint change Δyp: Peak output change Δy¥: Steady-state output change after setpoint step test tr: Time from setpoint change to reach steady-state output for the first time Received: July 10, 2010 Accepted: January 5, 2011 Revised: December 31, 2010 Published: January 18, 2011 2461
dx.doi.org/10.1021/ie101475n | Ind. Eng. Chem. Res. 2011, 50, 2461–2466
Industrial & Engineering Chemistry Research
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Hence, the characteristic polynomial of ~(s) g is 1 1 1 2 sþ f ðsÞ := s þ 0:5Kθ2 0:5θ 0:5θ2
ð6Þ
The above f(s) term is in the standard second-order form, s2 þ 2ζωns þ ωn2, with pffiffiffi 1 1 2 -1 ð7Þ , ζ ¼ pffiffiffi ωn ¼ θ 2 Kθ
Figure 1. Block diagram of feedback control system.
In eq 7, ζ has a physical meaning of being the damping ratio of the closed-loop system.10 Equation 7 solves K as 1 K ¼ pffiffiffi ð8Þ ð 2ζ þ 1Þθ Therefore, the unit step setpoint response is 1 1 sþ 2 1 0:5θ 0:5θ Y ðsÞ ¼ ~g ðsÞys ðsÞ 1 1 1 s s2 þ sþ 22 0:5Kθ 0:5θ 0:5θ s2 -
¼
tp: Time from setpoint change to reach peak output kc0: Controller gain used in experiment. From this data, the following parameters are calculated: Δy¥ b ¼ Δys
ð2Þ
As recommended in the work of Shamsuzzoha and Skogestad,1 to derive good PI settings, the experiment should make Mp be larger than 10% (at best, it should be ∼30%). In the case that it takes a long time for the response to settle down, one may simply record the output, Δyu, when the response reaches its first minimum and compute Δy¥ as Δy¥ = 0.45(Δyp þ Δyu).1 The analytical derivation of the PI tuning rule proceeds as follows. Consider an ITD process, ke - θs gðsÞ ¼ s
where k is the process gain and θ is the time delay. With c(s) = kc0 applied to the process, the closed-loop transfer function is obtained as ~g ðsÞ :=
Assume that the initial states of the system and their derivatives are zero. By inverse Laplace transform, from eq 9, the time-domain response is derived as yðtÞ 1 þ ce - σt sin ωdt
gðsÞcðsÞ e ¼ 1 þ gðsÞcðsÞ ðs=K Þ þ e - θs
ð4Þ
where K := kkc0. The time delay component in eq 4 is usually approximated by a Pade approximation or Maclaurin expansion. Although a Pade approximation is normally more accurate, a Maclaurin expansion is adopted for the purpose of deriving a simple analytical PI tuning rule. Use a Maclaurin expansion and approximate the numerator and denominator of ~(s) g by the second-order polynomials, respectively, yielding 1 1 sþ s2 2 0:5θ 0:5θ ð5Þ ~g ðsÞ 1 1 1 s2 þ s þ 0:5Kθ2 0:5θ 0:5θ2
ð11Þ
From eq 11, the time-domain performance indices, such as the rise time (tr), the peak time (tp), and the overshoot (Mp), can all be estimated. Let the rise time be defined as the time needed for y(t) to reach a steady-state value of 1 for the first time. This means yðtr Þ ¼ 1 ¼ 1 þ ce - σtr sin ωdtr
ð12Þ
Equation 12 solves tr as
ð3Þ
- θs
ð9Þ
where σ := ζωn and ωd := ωn(1 - ζ2)1/2. The parameters ζ and ωn are those defined in eq 7, and a, b, and c are given by pffiffiffi 2ζ þ 2 1 ffiffiffiffiffiffiffiffiffiffiffiffi p a ¼ 1, b ¼ 0, c ¼ ¼ ð10Þ 0:5Kωd θ2 1 - ζ2
Figure 2. Setpoint response with proportional (P) control.1
Δyp - Δy¥ Mp ¼ , Δy¥
a bðs þ σÞ þ cωd þ 2 s s þ 2ζωn s þ ωn 2
tr ¼
π πθ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωd 2ð1 - ζ2 Þ
ð13Þ
With dy(t)/dt|t=tp = 0, the peak time tp is solved as 1 θ ðπ þ arccos ζÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tp ¼ ðπ þ arccos ζÞ ð14Þ ωd 2 1 - ζ2 Consequently, the overshoot Mp, which is defined in eq 2, is computed as Mp ¼ ðyðtp Þ - 1Þ 100% ¼ ce - σtp sin ωd tp 100% " # pffiffiffi ζ ¼ ð2ζ þ 2Þ exp - pffiffiffiffiffiffiffiffiffiffiffiffi2 ðπ þ arccos ζÞ 100% ð15Þ 1-ζ
Note that the dimensionless scalars tr/θ, tp/θ, and Mp are all functions of ζ. The relationship between Mp and ζ is shown in 2462
dx.doi.org/10.1021/ie101475n |Ind. Eng. Chem. Res. 2011, 50, 2461–2466
Industrial & Engineering Chemistry Research
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modeling of the process dynamics: only the peak time and rise time or overshoot as recorded from a single CSR experiment. This eases the PI controller tuning in practice. Next, consider an FOTD process: gðsÞ ¼
ke - θs τs þ 1
ð21Þ
where k is the process gain, θ the time delay, and τ the process time constant. During the transient of a setpoint response, since it involves mainly high-frequency response, the transfer function can be approximated as gðsÞ Figure 3. Mp-ζ curve, showing the relationship between the overshoot (Mp) and the damping ratio (ζ).
Figure 3. Hence, ζ can be read from the Mp-ζ curve after Mp has been measured from the CSR experiment. Or it can be solved from eqs 13 and 14 as πt p ð16Þ ζ ¼ cos tr Although cos[(πtp)/tr] < 0 comes from π e (πtp)/tr e (3π)/2, as deduced from eqs 13 and 14, the absolute is taken to ensure a positive ζ value, since tp and tr are measured values and the analysis here is approximate. Consequently, θ can be estimated from eq 14 as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tp 2ð1 - ζ2 Þ ð17Þ θ ¼ π þ arccos ζ (The parameter θ may also be estimated from eq 13, in terms of tr. The estimation in eq 17 is recommended because it avoids the measurement of tr if the damping ratio is read from the Mp-ζ curve.) Therefore, the process gain is solved from eq 8 as 1 ð18Þ k ¼ pffiffiffi ð 2ζ þ 1Þθkc0 Note that the SIMC tuning rule9 for an ITD process is equivalently expressed by R 4θ , τI ¼ ð19Þ kc ¼ kθ R where R is a tuning parameter that corresponds to θ/(τc þ θ) in the original SIMC rule. Applying the estimated processes parameters θ and k in eqs 17 and 18, respectively, the SIMC tuning rule for an ITD process becomes 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 pffiffiffi 4 6 2ð1 - ζ Þ 7 τI ¼ 4 kc ¼ Rð 2ζ þ 1Þkc0 , 5tp ð20Þ R π þ arccos ζ where
πt p ζ ¼ cos tr
Alternatively, ζ can be read from the Mp-ζ curve shown in Figure 3. Equation 20 is the new PI tuning rule, which requires no
k0 e - θs , s
k0: =
k τ
ð22Þ
As the transient dynamics is of main interest, where quantities such as the rise time, peak time, and overshoot are measured, approximate analysis can be made similarly to that for an ITD process. Therefore, the time delay θ is estimated in eq 17 and the gain k0 is estimated in eq 8 with K := k0 kc0. At the steady state, the process gain k satisfies kkc0 ¼
b 1-b
ð23Þ
where b is given in eq 2. Equation 23, together with eq 8, solves τ ¼
kkc0 b pffiffiffi ð 2ζ þ 1Þθ ¼ 1-b K
ð24Þ
The SIMC tuning rule9 for an FOTD process is equivalently expressed as R 4θ , τI ¼ min τ, kc ¼ ð25Þ kθ R where R is a tuning parameter. Applying the process parameter estimated in eqs 8, 17, and 24, the SIMC tuning rule for an FOTD process is rewritten as 8 ! 9
