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Double First Order Plus Time Delay Models to Tune PI Controllers Yongjeh Lee, Dae Ryook Yang, Jietae Lee, and Thomas F. Edgar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b01585 • Publication Date (Web): 06 Sep 2016 Downloaded from http://pubs.acs.org on September 7, 2016
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Double First Order Plus Time Delay Models to Tune PI Controllers Yongjeh Leea, Dae Ryook Yanga, Jietae Leeb*, Thomas F. Edgarc a
Department of Chemical and Biological Engineering, Korea University, Seoul 136-713, Korea Department of Chemical Engineering, Kyungpook National University, Daegu 702-701, Korea c Department of Chemical Engineering, University of Texas at Austin, Austin, TX78712, U. S. A. b
*Corresponding author. Tel: +82-53-950-5620, Fax: +82-53-950-6615, E-mail:
[email protected] Abstract: The first order plus time delay (FOPTD) model-based method is a standard approach to tune PI controllers in plants. The FOPTD model can be obtained easily from step responses. However, due to their structural limitations, FOPTD models suffer from difficulties in approximating step responses for some processes including processes with overshoot, resulting in PI controllers with unacceptable performance. To remove these drawbacks, models combining two FOPTD models that can be obtained easily from step responses are proposed to tune PI controllers. Several simulations and experimental examples are given, illustrating the improved performance of the proposed method.
Keywords: First order plus time delay model, Proportional integral control, Step response, Graphical identification, Tuning
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1. Introduction Simple proportional and integral (PI) controllers exhibit satisfactory performance for a wide range of processes.1-2 PI Controllers are often designed based on the empirical first order plus time delay (FOPTD) models. Many tuning rules are expressed in terms of the FOPTD model parameters.1-6 However, due to structural limitations, FOPTD models cannot be used for some processes. There are processes where dynamic elements are connected in parallel. When time constants of theses dynamic elements are different much from each other, processes can have dominant zeros in their transfer functions and can show overshoot, undershoot and long tail in their step responses. For example, when two first order elements with positive and negative steady state gains are connected in parallel, the processes can have positive zeros, showing undershoots in their step responses (inverse responses).1 FOPTD models cannot be used to approximate dynamics of these processes. Non-parametric models7-8 and high order (or fractional order) plus time delay models9-11 can be used for processes with overshoot, undershoot and long tail. When full high order models are available, model reduction methods6 can be used to obtain FOPTD models and to tune PI controllers. However, they need complicated computations to identify, especially when dead times exist. Here a simpler empirical method without iterative computations is proposed to tune PI controllers. Instead of finding high order models that fit step responses, we divide step responses into two parts. FOPTD models for both regions are obtained and then are combined to tune PI controllers. Each FOPTD model can be obtained easily from the step response. The proposed method can be used for processes with overshoot, undershoot and long tail in their step responses. Complementing the FOPTD model-based method, it will cover almost all the chemical processes. Several simulations and experimental examples are given, illustrating the improved performance of the proposed method.
2. Double FOPTD Models Consider a linear dynamical process;
Y (s) = G(s)U (s)
(1)
Here G(s) is the process transfer function and U(s) and Y(s) are Laplace transforms of the process input
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and output variables, respectively. To tune PI controllers, first order plus time delay (FOPTD) models approximating the process step responses are often used. However, there are processes for which FOPTD models suffer from difficulties in approximating the process step responses due to the structural limitations of FOPTD models. Figure 1 shows three typical process step responses for which FOPTD models cannot be used. For example, when G(s) has a leading negative zero, there is overshoot in the step response (Fig. 1(a)) and the first order plus time delay (FOPTD) model cannot be used to describe the step response well. High order plus time delay models with zeros should be used. However, identification of high order plus time delay model requires somewhat complicated computations or iterative computations. To design PI controllers, a simpler model may be sufficient. A model combining two FOPTD models is considered. In Fig. 1, for the response before tp,
GI ( s ) =
k1
τ 1s + 1
e −θ1 s
(2)
and, for the response after tp,
G II ( s ) =
k2
τ 2s + 1
e −θ 2 s
(3)
Three parameters for Eq. (2) can be obtained graphically as5 k1 = y p / M
τ 1 = 1 . 5( t b − t a ) θ 1 = t b −τ 1
(4)
Similarly, three parameters for Eq. (3) are k 2 = ( y inf − y p ) / M 1 .5(t d − t c ), Fig.1(a) and Fig. 1(c) Fig. 1(b)
τ2 = t d − t p ,
(5)
θ 2 = t d −τ 2 Here yp and yinf are given in Fig. 1. Measurements of ta, tb, tc, and td are times at the points of a to d in Fig. 1, respectively. M is the step size of input change. The time point tp separating GI(s) and GII(s) is located where the response changes its shape. For a process of Fig. 1(a) or Fig. 1(c), tp is the point where the response shows the maximum or the minimum. For some processes of Fig. 1(b) whose step responses have long tails, tp is the point where slopes changes sharply. Otherwise, it is not obvious how to find the time point tp. For such processes, FOPTD models can be used. When the proposed method is preferred,
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iteration may be required to find an appropriate tp and, for this, a line search for tp can be used to minimize the sum of square errors (SSE) between the process and model responses. Fig. 2 shows sum of square errors and the fitting results as the point tp varies for the process of G ( s) =
(6 s + 1)e −2 s . (8s + 1)( s + 1)(0.1s + 1)
The optimal tp is around 7.02. For this process, tp around the optimal one can be obtained without iterations by inspecting the shape of step response. When the process output is corrupted with measurement noise, yp and yinf are the averaged outputs around tp and the final time, respectively. Measurements of ta, tb, tc, and td are times at the center points where the output levels of a to d in Fig. 1 cut the process responses, respectively. Each model of Eq. (2) or Eq. (3) can describe well each part of the response in Fig. 1. Here, to cover the whole response, two FOPTD models are combined (we call this a double first order plus time delay (dFOPTD) model);
GdFOPTD(s) =
k1
τ1s +1
e−θ1s +
k2
τ 2 s +1
e−θ2s + k1e
−(t p −θ1) /τ1
(1−
1
)e
−t ps
τ1s +1
(6)
The last term in Eq. (6) is to cancel the effects of GI(s) after tp.
3. Pade Approximation and PI Controller Tuning The model of Eq. (6) is simplified further. The empirical model of Eq. (6) is rewritten as
GdFOPTD(s) =
Q(s) (τ 1s + 1)(τ 2 s + 1)
Q(s) = k1 (τ 2 s + 1)e
−θ1s
+ k 2 (τ 1s + 1)e
(7) −θ2 s
+ k1e
−(t p −θ1 ) / τ1
τ 1s(τ 2 s + 1)e
−t p s
Here, to tune the PI controller
k 1 = kC + I C(s) = kC 1 + s τIs
(8)
Q(s) is reduced to the form of
Q(s) ≈ k(τ 3s +1)e−θs
(9)
The approximate model becomes
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G2 P ( s ) =
k (τ 3 s + 1)e −θs ≈ GdFOPTD (s) (τ 1s + 1)(τ 2 s + 1)
(10)
For this, the Pade approximation is applied. The Taylor series expansion of Q(s) is given as
Q(s) = q0 + q1s + q2 s 2 + L = k1 + k2 + (k1 (τ 2 − θ1 ) + k2 (τ 1 − θ 2 ) + k1τ 1e 2
−(t p −θ1 ) /τ1
2
(k1 (θ1 / 2 − θ1τ 2 ) + k2 (θ 2 / 2 − θ 2τ 1 ) + k1τ 1e
)s +
−(t p −θ1 ) / τ1
(11)
(τ 2 − t p ))s + L 2
Here
q0 = k1 + k 2 q1 = k1 (τ 2 − θ1 ) + k 2 (τ 1 − θ 2 ) + k1τ 1e
−(t p −θ1 ) / τ1
(12)
q2 = k1 (θ1 / 2 − θ1τ 2 ) + k 2 (θ 2 / 2 − θ 2τ 1 ) + k1τ 1e 2
2
−(t p −θ1 ) / τ1
(τ 2 − t p )
Since
Q(s) ≈ k (τ 3 s + 1)e −θs = k (1 + τ 3s)(1 − θs + (θs) 2 / 2 + L) = k + k (τ 3 − θ )s + k (θ 2 / 2 −τ 3θ )s 2 + L
(13)
parameters equating both series of Eqs. (11) and (13) up to the s2 terms are
k = q0
τ3 = ±
q12 − 2q0 q2
(14)
q0
θ = τ 3 − q1 / q0 Here the sign of τ3 used is positive for processes of Fig. 1(a) and Fig. 1(b) and negative for processes of Fig. 1(c). Because the Taylor series of Eqs. (11) and (13) are accurate around s=0, the approximate model of Eq. (10) with Eq. (14) is effective for the low frequency region and hence it is useful for slow responses. Tuning rules for processes of Eq. (10) are available.2,6,12-13 For processes with positive τ3 (Fig. 1(a) or Fig. 1(b)), tuning rules in Lee et al.13 are used here. Applying the tuning rules in Lee et al.13 to the model of Eq. (10), the PI controller is
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kC =
1 τ 1τ 2 2k τ 3θ
kI =
1 2k
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w 2 (1 + τ 12 w 2 )(1 + τ 2 2 w 2 ) (1 + τ 3 2 w 2 )(1 + θ 2 w 2 )
(15)
2
w2 =
P + P 2 + 4τ 1τ 2τ 3θ
P = τ 1τ 2 + θ (τ 1 + τ 2 − τ 3 ) − τ 3 (τ 1 + τ 2 ) For processes with negative τ3 (Fig. 1(c)), the SIMC tuning rules6 are used. It uses the model reduction of, for negative τ3,
G2 P ( s ) =
k (τ 3 s + 1)e −θs ke −(θ −τ 3 +τ1 / 2) s ≈ (τ 1s + 1)(τ 2 s + 1) (τ 2 + τ 1 / 2)s + 1
(16)
The PI controller parameters6 are
kC =
τˆ kλˆ
(
(17)
)
τ I = min τˆ, 4λˆ , τˆ = τ 2 + τ 1 / 2, λˆ = 2(θ − τ 3 + τ 1 / 2)
4. Frequency Response Approximation and PI Controller Tuning The above model reduction is based on the Taylor series expansion of Q(s) that is accurate in the low frequency region and will be effective to design slow control systems. When the estimated time delay θ1 is much smaller than the time constant τ1, fast control systems for processes of Fig. 1(a) and Fig. 1(b) can be designed without worrying about stability. For a fast control, the frequency response matching of
Q( jωb ) ≈ k ( jτ 3ωb + 1)e − jθωb
(18)
is tried for processes of Fig. 1(a) and Fig. 1(b). Here the angular frequency ωb is chosen to be the bandwidth of closed-loop system approximately. From Eq. (18) with k=Q(0),
G2F (s) =
k (τ 3 s + 1)e −θs (τ 1s + 1)(τ 2 s + 1)
k = k1 + k 2
τ3 = θ=
1
ωb 1
ωb
(19)
| Q( jωb ) |2 / k 2 − 1
(tan
−1
(τ 3ωb ) − ∠Q( jωb )
) 6
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The approximate model of Eq. (19) is useful for a small time delay such that
min(τ 2 ,τ 3 ) > 5θ
(20)
For such cases, we can design the closed-loop system whose bandwidth is
ωb
