Article pubs.acs.org/IECR
Tuning Optimal Proportional−Integral−Derivative Controllers for Desired Closed-Loop Response Using the Method of Moments S. Sundaramoorthy† and M. Ramasamy*,‡ †
Department of Chemical Engineering, Pondicherry Engineering College, Pondicherry, 605014, India Chemical Engineering Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Perak Malaysia
‡
S Supporting Information *
ABSTRACT: In the direct synthesis method of controller design, the optimality of the controller parameters are strongly affected by the approximations involved in developing the process models and the choice of desired closed-loop response characteristics. This work (i) captures the process dynamic characteristics through the method of moments; (ii) proposes a tuning parameter λ in the direct synthesis method, which is optimized through the minimization of integrated absolute error (IAE) subject to the constraint on peak sensitivity; and (iii) provides correlations for optimal λ values as a function of process parameters. First-order plus delay time (FOPDT), second-order plus delay time (SOPDT), and second-order with inverse response (SOIR) models are used to characterize the process dynamics and as desired closed-loop transfer functions. The efficacy of the proposed method is illustrated with the help of several types of processes. The proposed controller tuning method performs equally well as compared to other methods for plants approximated with FOPDT models and has been shown to be very effective for SOPDT and SOIR models. where τc is the desired closed-loop time constant which is the design or the tuning parameter. Sometimes, the symbol λ is used instead of τc and the direct synthesis method is referred to as the lambda-tuning method.12 λ-tuning methods are wellaccepted in designing and tuning such model-based controllers. The parameter λ is to be chosen optimally to obtain optimal PID controller parameters as a trade-off between (i) fast response and good disturbance rejection and (ii) stability, robustness, and small input variations. Skogestad13 has recommended an optimal value of λ equal to the effective time delay of the process, whereas other authors, such as Chen and Seborg,9 Rivera et al.,10 and Wang et al.,14 have provided only limits on the values of λ. A few studies have attempted to provide correlations for the optimal λ values in terms of process parameters and desired closed-loop response behavior. Abbas15 correlated the closedloop gain to the process time delay/time constant ratio and the desired overshoot of the closed-loop response for P, PI, and PID controllers for first-order plus delay time (FOPDT) processes with the desired closed-loop response as in eq 1. However, no explicit correlations for optimal λ in terms of process parameters were provided and no higher-order systems were considered. Shamsuzzoha and Lee16 used the internal model control (IMC) technique to design enhanced PID controllers for integrating plus first-order unstable processes with time delay and provided a guideline for selecting λ over a range of timedelay/time-constant ratios in a graphical form. Direct synthesis controllers have been reported for integrating systems with time delay using higher-order desired closed-loop transfer functions with numerator dynamics.17 In their study, the closed-loop time
1. INTRODUCTION Tuning optimal proportional−integral−derivative (PID) controllers has been the subject of intense research in recent years because of the fact that PID controllers still remain the workhorse in all major process plants, and even advanced process control strategies such as model predictive controllers employ PID controllers to implement their control actions in the plant. Optimal tuning of PID controllers, therefore, is essential for fully realizing the benefits of advanced process control strategies. There are two major approaches in the literature adopted to arrive at optimal PID parameters. One of the approaches involves formulation and solution of nonlinear optimization problems with appropriate objective function subject to certain constraints. The optimization-based methods express the optimal PID controller parameters as a function of process parameters obtained through the minimization of certain performance criteria such as integrated absolute error (IAE), integral of timeweighted absolute error (ITAE), integral square error (ISE), etc. subject to stability and robustness constraints.1−8 The other approach involves design of model-based controllers such as direct synthesis and internal model controllers.9,10 For certain classes of process models, the resulting model-based controllers can be realized through PID controllers. In the design of direct synthesis controllers, the controller performance largely depends on the selection of appropriate desired closed-loop responses. An evaluation of PID controller design techniques using internal model control, direct synthesis, and optimization-based methods has been reported with recommendations for selecting optimal tuning rules.11 Generally, a first-order with or without time delay is specified as the desired closed-loop response as ⎛Y ⎞ e ⎜ ⎟ = ⎝ R ⎠d τcs + 1
Received: Revised: Accepted: Published:
−θs
(1) © 2014 American Chemical Society
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Table 1. Moments Equations for Estimation of Model Parameters equations for GP′ , GP″, and GP‴
plant model GP(s) FOPDT model:
G P(s) =
G P′ = − (θP + τP) G P″ = (θP + τP)2 + τP2
KPe−θ Ps τPs + 1
SOPDT model:
K e−θ Ps G P(s) = 2 2 P τP s + 2τPζPs + 1
G P′ = − z z = (θP + 2τPζP) G P″ = z 2 + 2τP2(2ζP2 − 1) G P‴ = − (z 3 + 6τP2(2ζP2 − 1)z + 4τP3ζP(4ζP2 − 3))
SOIR model: K (1 − θPs) G P(s) = 2 2 P τP s + 2τPζPs + 1
G P′ = − z z = (θP + 2τPζP) G P″ = 4τPζPz − 2τP2 G P‴ = 6τP2(1 − 4ζP2)z + 12τP3ζP
Figure 1. Block diagram of a typical feedback control system.
constant λ has been recommended to be in the range of 0.8θ to 3.0θ, where θ is the process time delay. Pai et al.18 have reported tuning of PI/PID controllers for integrating processes with dead time and inverse response using the DS-d approach of Chen and Seborg.9 In their approach, the optimum λ values that minimize the IAE were correlated with the dimensionless process parameters without any constraint on the robustness measures. Generally, the optimality of PID controller parameters is affected by (i) the approximations in the development of the reduced-order models to represent the processes and (ii) the choice of the desired closed-loop response and the tuning parameter(s). In a two-step procedure suggested by Skogestad,13 the first step involves obtaining a reasonably accurate reduced-order dynamic model using model reduction techniques. It may be noted that not all processes can be approximated by FOPDT models, especially the processes with inverse response characteristics and numerator dynamics. It is also well-known that controllers designed using higher-order models such as second-order plus delay time (SOPDT) lead to more optimal controllers. Our earlier work19 has shown that a variety of processes can be captured reasonably accurately using the moments of impulse response. The same work has also provided expressions for the controller parameters in terms of moments of impulse response derived using the Maclaurin series of sGc(s) where Gc(s) is the controller transfer function. In the direct synthesis approach of designing feedback controllers, the choice of desired closed-loop response is also critical in achieving optimal PID controllers. Generally, a firstorder plus time delay response has been utilized in most cases as the desired closed-loop response.11−13,20 As much as the use of SOPDT process models leads to better controllers, the use of a SOPDT desired closed-loop response is also expected to improve the control system performance. Similarly, desired closed-loop responses expressed as second-order plus inverse response (SOIR) models will also lead to the design of better controllers for SOIR processes. However, there is no literature on designing direct synthesis controllers with SOPDT/SOIR desired closed-loop responses.
In this study, the process dynamic characteristics are captured through the method of moments and appropriately represented by FOPDT, SOPDT, or SOIR models. Desired closed-loop responses are chosen similar to that of the respective process model for designing direct synthesis controllers. Correlations for optimal tuning parameter λ have been obtained through extensive simulation of control systems over a wide range of process parameters and optimized through the minimization of IAE subject to the constraint on peak sensitivity to ensure robustness of the controllers. A brief introduction to method of moments and characterization of process dynamics are provided in Section 2. Design of PID controllers for FOPDT, SOPDT, and SOIR processes using the moments of the impulse response is explained in Section 3. Correlation of optimal λ with the process parameters is explained in Section 4. Design and evaluation of the PID controllers using the proposed technique is illustrated with the help of a number of examples in Section 5.
2. METHOD OF MOMENTS Let Y(t) be the unit impulse response of a stable plant whose transfer function is GP(s). The Laplace transform of Y(t) is Y (s ) =
∫0
∞
Y (t )e−st dt = G P(s)
(2)
Writing e−st term as a Maclaurin series in s, we have ∞
Y (s) =
∫0
s3 3!
∫0
−
Y (t ) dt − s ∞
∫0
∞
tY (t ) dt +
t 3Y (t ) dt + ...
s2 2!
∫0
∞
t 2Y (t ) dt
(3)
Expanding GP(s) as a Maclaurin series in s G P(s) = G P(0) + sG P′ (0) +
s2 s3 G P″(0) + G P‴(0) + ... 2! 3! (4)
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Table 2. Moments Equations for the Desired Closed-Loop Transfer Function Gd(s) equations for Gd′ (0), Gd″(0) and Gd‴(0)
desired response Gd(s) FOPDT model:
Gd′ (0) = − (θ + τ ) Gd″(0) = (θ + τ )2 + τ 2 Gd‴(0) = − (θ 3 + 3θ 2τ + 6θτ 2 + 6τ 3)
e−θs τs + 1 SOPDT model:
Gd(s) =
e−θs Gd(s) = 2 2 τ s + 2τζs + 1
Gd′ (0) = − z z = (θ + 2τζ ) Gd″(0) = z 2 + 2τ 2(2ζ 2 − 1) Gd‴(0) = − (z 3 + 6τ 2(2ζ 2 − 1)z + 4τ 3ζ(4ζ 2 − 3))
SOIR model: (1 − θs) Gd(s) = 2 2 τ s + 2τζs + 1
Gd′ (0) = − z z = (θ + 2τζ ) Gd″(0) = 4τζz − 2τ 2 Gd‴(0) = 6τ 2(1 − 4ζ 2)z + 12τ 3ζ
Table 3. PID Controller Tuning Formulae Obtained Using the Method of Moments desired response Gd(s)
PID controller equations
FOPDT model:
KC =
e−θs Gd(s) = τs + 1
τI KP
z
z = (θ + τ )
2
τI =
θ − G P′ − θ 2z
τD = (τI + G P′ + θ) +
τI z = (θ + 2τζ ) KPz 2 (θ − 2τ 2) τI = − G P′ − θ 2z (G P′ )2 + (G P′ + θ)2 − G P″ θ3 τD = (τI + G P′ + θ) + − 2τI 6τIz
SOPDT model:
KC =
−θs
Gd(s) =
(G P′ )2 + (G P′ + θ)2 − G P″ θ3 − 2τI 6τIz
e τ 2s 2 + 2τζs + 1
τI z = (θ + 2τζ ) KPz 2 τ τI = − − G P′ − θ z (G P′ )2 + (G P′ + θ)2 − G P″ θ2 τD = (τI + G P′ + θ) + − 2τI 2τI
SOIR model: (1 − θs) Gd(s) = 2 2 τ s + 2τζs + 1
KC =
Table 4. Selection of Desired Closed-Loop Transfer Function Gd(s) and Its Parameters desired Gd(s)
plant model
tuning parameters
G P(s) =
KPe τPs + 1
Gd(s) =
e τs + 1
θ = θP
τ = λτP
G P(s) =
KPe−θ Ps τP s + 2τPζPs + 1
Gd(s) =
e−θs τ s + 2τζs + 1
θ = θP ζ=1
τ = λτP
G P(s) =
KP(1 − θPs) τP2s 2 + 2τPζPs + 1
Gd(s) =
(1 − θs) τ 2s 2 + 2τζs + 1
θ = θP ζ=1
τ = λτP
−θs
−θ Ps
2 2
2 2
where the integral term is the nth moment of Y(t) . Given the unit impulse response Y(t) of the plant GP(s), the moments of Y(t) can be obtained by numerical integration. The zeroth moment of Y(t) is the plant gain KP, i.e.
where G P(0) = G P(s)|s = 0 ,
G P′ (0) =
⎞ ⎛ d2 G P″(0) = ⎜ 2 G P(s)⎟ ⎠ ⎝ ds
⎛d ⎞ ⎜ G P ( s )⎟ ⎝ ds ⎠
, s=0
G P(0) = KP =
, ... (5)
s=0
= ( −1)n s=0
G P′ =
Y (t ) d t
(7)
G P′ (0) , G P(0)
G P″ =
G P″(0) , G P(0)
G P‴ =
G P‴(0) , ... G P(0)
(8)
In the “method of moments” adopted in this study, the values of GP′ , GP″, and GP‴ are used for characterization of process dynamics and for the design of PID controller. If the
0
∫∞ t nY (t ) dt
∞
and define
Comparing the coefficients of powers of s in eqs 3 and 4, we have ⎛ dn ⎞ G Pn(0) = ⎜ n Gp(s)⎟ ⎠ ⎝ ds
∫0
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Table 5. Correlations for the Calculation of Optimal Tuning Parameter λ plant model FOPDT model:
controller
correlation for lambda λ
PI
λ = a + bθ ̃ + cθ ̃ θp θ̃ = τp
PID
a = 0.0547 b = 0.8037 c = − 0.0522 0.1 ≤ θ ̃ ≤ 5.0
0.999
λ = a + bθ ̃ + cθ ̃ θp θ̃ = τp
2
a = 0.0240 b = 0.2881 c = − 0.0075 0.1 ≤ θ ̃ ≤ 5.0
0.999
PID
2 λ = a + bθ ̃ + cθ ̃ θp θ̃ = τp a = a1ζP + a0 b = b1ζP + b0 c = c1ζP + c0
a0 = 0.0140 a1 = 0.0483 b0 = 0.7816 b1 = − 0.4948 c0 = − 0.1280 c1 = 0.1044 0.1 ≤ θ ̃ ≤ 5.0 0.2 ≤ ζP ≤ 1.0
0.992
PID
2 λ = a + bθ ̃ + cθ ̃ θp θ̃ = τp a = a1ζP + a0 b = b1ζP + b0 c = c1ζP + c0
a0 = 0.0129 a1 = − 0.0113 b0 = 0.4734 b1 = − 0.0601 c0 = 0.0123 c1 = 0.0146 for 0.7 ≤ ζP ≤ 1.0 0.1 ≤ θ ̃ ≤ 5.0 for 0.2 ≤ ζP < 0.7 0.1 ≤ θ ̃ ≤ 2.0
0.998
−θ Ps
Ke G P(s) = 2 2 P τP s + 2τPζPs + 1
SOIR model: K (1 − θPs) G P(s) = 2 2 P τP s + 2τPζPs + 1
R2
2
K e−θ Ps G P(s) = P τPs + 1
SOPDT model:
parameter values
Figure 2. Plot showing the scatter of estimated λ values (correlation) around the actual λ values for SOPDT plant with PID controller.
unit impulse response Y(t) of the plant is given, then G′P, G″P , and G‴ P can be calculated by evaluating the first, second, and third moments of Y(t). However, an ideal impulse signal cannot be given to the open-loop process as it is not possible in practice to produce a perfect impulse signal to serve as test input. Sometimes, a brief pulse of larger magnitude is used as an approximation of an impulse. However, in many systems, a very short strong pulse may drive the system into a nonlinear regime.
Therefore, in practice, the system is driven with either a step input or a pseudorandom sequence, and the impulse response is computed from the input and output signals. Alternatively, if the plant transfer function GP(s) is known then the values of G′P, G″P , and G‴ P can be calculated through the analytical expressions for the first, second, and third derivatives of GP(s) at s = 0. For more details of derivation, please refer to Ramasamy and Sundaramoorthy.19 17406
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Figure 3. Comparison of the performance (a) and robustness (b) of the proposed controller tuning method with other selected controller tuning methods for a FOPDT process (KP = 1, θP = 8) as the controllability index θ̃ increases.
Similarly, systems that exhibit nonmonotonic response with overshoot are represented by second-order plus delay time model:
For the purpose of controller design, stable processes are classified as systems that exhibit “monotonic response” and “nonmonotonic response” to step inputs. Among the systems that exhibit nonmonotonic response to step inputs are systems with “overshoot” and “inverse response” characteristics. Systems that exhibit monotonic response to step inputs are approximated by the first-order plus delay time model: G P (s ) =
KPe−θPs τPs + 1
G P (s ) =
KPe−θPs τP 2s 2 + 2τPζPs + 1
(10)
where τP is the natural period of oscillation and ζP the damping coefficient. Systems with inverse response characteristics are approximated with the second-order with inverse response model:
(9)
where KP is the plant gain, θP the delay time, and τP the time constant.
G P (s ) = 17407
KP(1 − θPs) 2 2
τP s + 2τPζPs + 1
(11)
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Figure 4. Performance (a) and robustness (b) of the proposed controller tuning method for a SOPDT process (KP = 1, θP = 5 and ζP = 0.9,0.7,0.5,0.2) as the controllability index θ̃ increases.
In the block diagram, R(s) refers to the set point, Y(s) the controlled output, d(s) the disturbance, GP(s) the process transfer function, and GC(s) the PID controller whose transfer function is given by
Analytical expressions for the moments are derived for FOPDT, SOPDT, and SOIR models and listed in Table 1. With the calculated values of G′P, G″P , and G‴ P from the impulse response Y(t) of the plant, the analytical equations listed in Table 1 are solved to obtain the estimated values of corresponding process model parameters, i.e., KP, τP, and θP for the FOPDT model and KP, τP, ζP, and θP for the SOPDT and SOIR models.
⎞ ⎛ 1 GC(s) = Kc⎜1 + + τDs⎟ τIs ⎠ ⎝
(12)
where Kc, τI and τD are the proportional gain, the integral time constant and the derivative time constant, respectively. The PID controllers are designed using a direct synthesis method in which the controller parameters are calculated by equating the actual closed-loop transfer function GA(s) derived
3. DESIGN OF PID CONTROLLERS A block diagram of a SISO feedback control system is shown in Figure 1. 17408
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Figure 5. Performance (a) and robustness (b) of the proposed controller tuning method for a SOIR process (KP = 1, θP = 5 and ζP = 0.9,0.7,0.5,0.2) as the controllability index θ̃ increases.
of the process transfer function GP(s) with the steady-state gain as 1, i.e.
for the servo control of the process to the desired closed-loop transfer function specified by Gd(s), i.e.
GA (s) = Gd(s)
⎧ e−θs ⎪ for FOPDT processes ⎪ τs + 1 ⎪ ⎪ e−θs Gd(s) = ⎨ 2 2 for SOPDT processes ⎪ τ s + 2τζs + 1 ⎪ (1 − θs) ⎪ ⎪ τ 2s 2 + 2τζs + 1 for SOIR processes ⎩
(13)
The actual closed-loop transfer function GA(s) for the servo control problem of the plant GP(s) in Figure 1 is given by GA (s) =
GC(s)G P(s) 1 + GC(s)G P(s)
(14)
The desired closed-loop transfer function Gd(s) represents the closed-loop servo response that the PID controller is designed to achieve. In this work, we assume that the desired closed-loop transfer function Gd(s) takes the same form as that
(15)
Desired closed-loop response and robustness can be achieved by choosing appropriate values for the parameters τ, θ, and ζ. 17409
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Table 6. Identification of Model Parameters by the Method of Moments moments process model GP(s)
system
model parameters
GP″
GP‴
model
KP
θP
τP
1.14 × 104
−1.492 × 106
FOPDT
1
62.58
37.42
−
0.1735
−23.0
933.0
−5.606 × 104
FOPDT
1
2.9
20.1
−
0.0014
GP′ −100
ζP
ISE
E1
e (10s + 1)(20s + 1)(30s + 1)
E2
e −s (20s + 1)(2s + 1)
E3
1 (s + 1)4
−4.0
20.0
−120.0
FOPDT
1
2
2
−
0.0181
E4
1 (s + 1)(0.2s + 1)(0.04s + 1)(0.0008s + 1)
−1.24
2.58
−7.80
FOPDT
1
0.220
1.02
−
0.0004
E5
e−5s (100s + 6s + 1)(5s + 1)
−16.0
117.0
5494.0
SOPDT
1
9.53
9.51
0.34
0.1148
E6
e−5s (100s 2 + 10s + 1)(5s + 1)
−20.0
325.0
250.0
SOPDT
1
9.28
9.74
0.55
0.0250
E7
e−5s (100s + 14s + 1)(5s + 1)
−24.0
597.0
−1.267 × 104
SOPDT
1
8.46
10.5
0.74
0.0056
e−5s (100s + 18s + 1)(5s + 1)
−28.0
933.0
−3.558 × 104
SOPDT
1
8.27
10.96
0.90
0.0024
−5.0
24.0
−132.0
FOPDT SOIR
1 1
15.79 2.08
12.2 1.62
− 0.90
0.1417 0.0223
−40s
2
2
E8
2
E9
(1 − 2s) (s + 1)3
E10
(1 − 5s) (100s 2 + 10s + 1)
−15.0
100.0
6000.0
SOIR
1
5.0
10.0
0.5
3.0 × 10−9
E11
(1 − 10s) (400s + 12s + 1)(5s + 1)
−27.0
98.0
5.654 × 104
SOIR
1
11.2
19.4
0.41
1.1335
(1 − s)e−s (6s + 1)(2s + 1)2
−12.0
187.0
−3.742 × 103
SOIR
1
2.13
5.0
0.99
0.0111
FOPDT
1
5.44
6.56
−
0.017
2
E12
⎛ τ ⎞2 ⎛ K K ⎞ GA‴(0) = 3⎜ I ⎟ ⎜ C P G P″ + 2K CKPG P′ + 2K CKPτD⎟ ⎠ ⎝ K CKP ⎠ ⎝ τI
Expanding GA(s) and Gd(s) using Maclaurin series, we get GA (s) = GA (0) + sGA′ (0) +
s2 s3 GA″ (0) + GA‴(0) + ... 2! 3!
2 ⎛ τI ⎞3⎛ K CKP ⎞ G P′ ⎟ − 6⎜ ⎟ ⎜1 + K CKP + τI ⎠ ⎝ K CKP ⎠ ⎝
(16)
s2 s3 Gd(s) = Gd(0) + sGd′ (0) + Gd″(0) + Gd‴(0) + ... 2! 3!
Similarly, analytical expressions for Gd′ (0), Gd″(0), and Gd‴(0) were derived for FOPDT, SOPDT, and SOIR models and listed in Table 2. Equation 13 implies that the coefficients of similar powers of s in eqs 16 and 17 are equal, i.e.
(17)
where GA (0) = Gd(0) = 1 ⎛ dn ⎞ GAn (0) = ⎜ n GA (s)⎟ ⎝ ds ⎠
Gdn(0)
⎛ dn ⎞ = ⎜ n Gd(s)⎟ ⎝ ds ⎠
(18)
s=0
s=0
(19)
GA′ (0) = Gd′ (0)
(24a)
GA″ (0) = Gd″(0)
(24b)
GA‴(0) = Gd‴(0)
(24c)
and so on. Expressions for the controller parameters KC, τI, and τD can be derived by solving the set of equations 24a−24c for FOPDT, SOPDT, and SOIR models as listed in Table 3.
(20)
The analytical expressions for the first, second, and third derivatives of GA(s) at s = 0 are derived as τ GA′ (0) = − I K CKP (21) ⎛ τI ⎞2 ⎛ ⎞ K K GA″ (0) = 2⎜ ⎟ ⎜1 + K CKP + C P G P′ ⎟ τI ⎠ ⎝ K CKP ⎠ ⎝
(23)
4. OPTIMAL λ TUNING In the direct synthesis design of controller discussed in Section 3, the performance and the robustness of the PID controller designed (using the formulas listed in Table 3) for a given process depend upon the values assigned to the parameters θ, τ, and ζ of the desired closed-loop transfer function, Gd(s) (eq 15).
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Figure 6. Comparison of unit step responses of the plant and the model for SOPDT plant E5.
Table 7. Summary of Selected PID Controller Tuning Rules for FOPDT Plant controller parameters tuning rules Ziegler−Nichols
22
Smith and Corripio ITAE20
τI
KC
controller
0.9 ⎛ τP ⎞ ⎜ ⎟ KP ⎝ θP ⎠
3.33θP
−
−
PID
1.2 ⎛ τP ⎞ ⎜ ⎟ KP ⎝ θP ⎠
2.00θP
0.5θP
−
PI
0.916 0.586 ⎛ τP ⎞ ⎜ ⎟ KP ⎝ θP ⎠
−
−
−
Astrom−Hagglund21
PID
Rivera et.al.10
PI
τP
(
PID
( )) θP τP
1.03 − 0.165
τP
0.855 0.965 ⎛ τP ⎞ ⎜ ⎟ KP ⎝ θP ⎠
(0.796 − 0.147( ))
⎛ θ ⎞0.929 0.308⎜ P ⎟ ⎝ τP ⎠
α1θP + α2τP KPθP
⎛ α3θP + α4τP ⎞ ⎜ ⎟θP ⎝ θP + α5τP ⎠
α6θPτP θP + α7τP
−
τP +
θP τP
θP 2
( )
τP +
θP 2
−
≥0.2τP ≥1.7θP
τP +
θP 2
τPθP 2τP + θP
≥0.2τP ≥0.8θP
KPλ θP 2
( ) K (λ + ) τP + P
Lee et al.23
λ
PI
PID
Skogestad13
τD
θP 2
PI
τI KP(λ + θP)
min[τ1, 4(λ + θP)]
−
θP
PID
τI KP(λ + θP)
min[τ1, 4(λ + θP)]
τ2
θP
PI
τI KP(λ + θP)
τP +
θP 2(λ + θP)
−
−
PID
τI KP(λ + θP)
τP +
θP 2(λ + θP)
⎛ θP2 θ ⎞ ⎜3 − P ⎟ τI ⎠ 6(λ + θP) ⎝
−
performance metric (IAE, ISE or ITAE value of the closed-loop servo response) attains an optimum value. In our work, we choose a fixed value for time delay, θ, of the desired closed-loop
The PID controller is said to be optimally designed if the parameters θ, τ, and ζ of the desired closed-loop transfer function, Gd(s), are calculated in such a way that the closed-loop 17411
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Table 8. Performance and Robustness of the Proposed PID Controller Tuning Method for Various FOPDT Processes and Comparison with Other Selected Controller Tuning Methods controller parameters process model GP(s)
system E1
e (10s + 1)(20s + 1)(30s + 1)
E2
e−s (20s + 1)(2s + 1)
E3
1 (s + 1)4
E4
1 (s + 1)(0.2s + 1)(0.04s + 1)(0.0008s + 1)
−40s
performance indices
tuning method
KC
τI
τD
IAEservo
IAEregul.
Ms
proposed Ziegler−Nichols Smith ITAE Astrom−Hagglund Rivera et al. Skogestad Lee et al. proposed Ziegler−Nichols Smith ITAE Astrom−Hagglund Rivera et al. Skogestad Lee et al. proposed Ziegler−Nichols Smith ITAE Astrom−Hagglund Rivera et al. Skogestad Lee et al. proposed Ziegler−Nichols Smith ITAE Astrom−Hagglund Rivera et al. Skogestad Lee et al.
0.764 0.717 0.622 0.516 0.845 0.091 0.487 5.0 8.32 5.1 3.95 3.94 10.0 2.99 1.06 1.20 0.965 0.734 1.15 0.333 0.71 3.57 5.56 3.58 2.71 3.60 4.01 2.54
61.7 125.2 68 49.3 68.7 10.0 54.8 21.1 5.8 25.9 8.94 21.5 8.0 20.7 2.77 4.0 3.08 2.09 3.0 1.0 2.56 1.1 0.44 1.33 0.56 1.13 0.166 1.1
16.1 31.3 0.497 18.1 17.0 35.0 10.8 0.952 1.45 0.051 1.12 1.35 2.0 0.579 0.582 1.0 0.308 0.651 0.667 1.5 0.411 0.074 0.11 0.074 0.084 0.10 0.22 0.053
86.34 174.6 113.1 106.9 84.4 190.9 112.5 4.58 5.09 6.13 7.33 5.46 3.58 6.95 3.02 3.35 3.42 3.77 2.81 5.73 3.72 0.356 0.398 0.384 0.521 0.337 0.702 0.442
80.82 174.6 109.3 103.8 81.3 181.0 112.5 4.18 0.86 4.93 2.49 5.40 0.83 6.85 2.61 3.32 3.18 3.33 2.60 5.06 3.59 0.307 0.084 0.366 0.229 0.313 0.116 0.431
1.74 1.82 1.87 1.48 1.82 1.91 1.51 1.30 1.45 1.58 1.31 1.19 1.54 1.25 1.48 1.36 1.59 1.44 1.45 1.67 1.43 1.22 1.29 1.20 1.29 1.15 1.45 1.23
transfer function equal to the time delay of the process model θP. We also choose a value of 1 for the desired damping coefficient ζ (for SOPDT and SOIR models) to ensure a closed-loop response that is free from underdamped oscillations. Once the fixed values for the two parameters θ and ζ are assigned, the value of the third parameter τ is varied with respect to the process parameter τP such that an optimal closed-loop performance is achieved. That is, write τ as a factor λ multiplied by τp (τ = λτp), where the factor λ is the tuning parameter. Table 4 summarizes the selection of appropriate desired closed-loop transfer functions and the corresponding tuning parameters for different types of processes. With an increase in the value of λ, the value of the closedloop performance metric increases as the closed-loop step response becomes more and more sluggish and takes a longer time to settle. Even at very small values of λ, the value of the performance metric would be very high as the closed-loop response tends to become lesser and lesser damped with larger overshoots and oscillations. Thus, there exists an optimal value of λ at which the performance metric takes a minimum value. Thus, the optimal design of the PID controller gets reduced to a single-parameter optimization problem in which calculations are carried out to determine the optimal value of the tuning parameter λ for which the value of the performance metric is minimum. Although using an optimal value for λ in the formulas derived for calculation of PID controller parameters (Table 3) would result in the best closed-loop performance, the design cannot ensure robustness. A controller that is driven to achieve the
best index of performance has a tendency to push the system closer to the limits of stability and therefore is poorly robust. Thus, a good controller design is a trade-off between performance and robustness. The value of peak sensitivity, Ms, defined by eq 25 is usually taken as a measure of robustness Ms ≜ max ω|S(jω)|
(25)
where S = 1/(1 + GcGp). An upper limit of 2.7 to 3 on the value of peak sensitivity Ms is recommended for better robustness of the controller. A value of peak sensitivity lower than 1.1 or 1.2 is expected to result in a highly robust control with very sluggish closed-loop response. To ensure robustness without compromising the closed-loop performance, the optimal design of the PID controller is solved as a constrained optimization problem in which the optimal value of λ is calculated such that an objective function which is a closed-loop performance metric (such as IAE/ISE/ITAE) is minimized subject to fulfilling an inequality constraint that defines an upper bound on peak sensitivity Ms. Optimal values of λ were calculated through extensive simulation of FOPDT, SOPDT, and SOIR processes by varying the controllability index θ̃ (a dimensionless ratio of θP to τp, θ̃ = θP/τp, and damping coefficient ζP over a wide range of values. The values of θ̃ (for FOPDT, SOPDT, and SOIR processes) were varied in the range of 0.1 ≤ θ̃ ≤ 5.0, and the values of ζP (for SOPDT and SOIR processes) were varied in the range of 0.2 ≤ ζP ≤ 1.0. The optimization carried out by taking all three performance indices, namely, IAE, ISE and ITAE, as objective 17412
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Figure 7. Closed-loop servo and regulatory responses and input changes of the FOPDT plant E1.
of robustness for FOPDT and SOPDT processes. However, an upper bound of 2 on the peak gain value resulted in highly sluggish response for SOIR processes particularly for values of controllability index θ̃ smaller than 1. Hence, for SOIR processes, a higher value of 2.5 was set as the upper bound on peak gain Ms for the calculation of optimal values of λ.
functions showed that the results were better with IAE as the performance index (objective function) in most of the cases. IAE =
∫0
∞
|e(t )| dt
(26)
The simulation results showed that setting an upper bound of 2 on the value of peak gain Ms fulfilled the condition 17413
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In this work, we have proposed empirical equations (Table 5) for calculating the optimal values of λ as functions of dimensionless process parameters, namely, controllability index, θ̃, for FOPDT processes, controllability index, θ̃, and damping coeff icient, ζP, for SOPDT and SOIR processes. These equations were derived by correlating the optimal values of λ with the process parameters θ̃ and ζP. The values of λ used in the derivation of empirical calculations are the optimal λ values calculated through extensive simulations of the FOPDT, SOPDT, and SOIR processes in which θ̃ and ζP are varied over a wide range of values (0.1 ≤ θ̃ ≤ 5.0, 0.2 ≤ ζP ≤ 1.0). It is observed that the empirical equations (Table 5) proposed in this study fit the data well with the correlation coefficient R2 taking a value of 0.999 for FOPDT process, 0.992 for SOPDT process, and 0.998 for SOIR process. Figure 2 and Figures S1−S3 in Supporting Information show the scatter of data points around a straight line with slope equal to 1 on the plot of optimal λ value (actual) calculated through simulation versus optimal λ value (correlation) calculated using empirical correlations. These plots confirm the validity of the empirical equations as the points on these plots are located very close to the straight line. The optimal values of PID controller parameters can be calculated for any open-loop stable process by following the steps given below: (i) Calculate the values of G′P, G″P , and G‴ P for the given process. These values are calculated from the impulse response of the plant using the method of moments (Section 2). (ii) On the basis of the open-loop step or impulse response characteristics of the plant, categorize it as either “monotonic” or “nonmonotonic” and represent it by a suitable model (FOPDT/SOPDT/SOIR).
(iii) Estimate the process model parameters using the values of G′P, G″P , and G‴ P calculated in step i and the equations listed in Table 1. (iv) Calculate the optimal value of the tuning parameter λ using the appropriate empirical correlation listed in Table 5. Calculate the values of the parameters θ, τ, and ζ of the desired closed-loop transfer function Gd(s) (Table 4). (v) Calculate the values of the controller parameters using appropriate formulas listed in Table 3. Performance and robustness of the PID controller design proposed in this work are evaluated for typical processes represented by FOPDT, SOPDT, and SOIR models, and the results are presented in Figures 3−5. IAE and Ms are the metrics for the measure of performance and robustness of the controllers, respectively. The performance and the robustness of the proposed PID controller were evaluated for the FOPDT plant model by fixing the process parameters KP = 1 and θP = 8 and varying the controllability index θ̃ in the range 0.1 ≤ θ̃ ≤ 5.0. The results in Figure 3 show that the proposed PID controller design method compared to four other standard design methods reported in the literature yield very low IAE values with the values of Ms lying well below 2. Similarly, for plants represented by SOPDT and SOIR models, the performance and robustness of the proposed PID controller were evaluated by fixing the plant parameters KP = 1, θP = 5, and ζP = 0.9, 0.7, 0.5, 0.2 and varying the controllability index θ̃ in the range 0.1 ≤ θ̃ ≤ 5.0. It is observed from Figures 4 and 5 that the Ms values are well below 2 for SOPDT processes and exceed the value of 2 for the SOIR processes in certain cases.
5. CASE STUDIES A number of examples with varying dynamic characteristics were chosen to illustrate the efficacy of the proposed approach
Table 9. Performance and Robustness of the Proposed PID Controller Tuning Method for Various SOPDT Processes controller parameters
performance indices
system
process model GP(s)
KC
τI
τD
IAEservo
IAEregul.
Ms
E5
−5s
e (100s 2 + 6s + 1)(5s + 1)
proposed
0.367
7.36
12.2
24.03
40.10
1.78
E6
e−5s (100s + 10s + 1)(5s + 1)
proposed
0.655
12.0
8.6
20.58
23.71
1.67
e−5s (100s + 14s + 1)(5s + 1)
proposed
1.07
17.0
7.5
18.21
16.46
1.72
e−5s (100s + 18s + 1)(5s + 1)
proposed (SOPDT) proposed (FOPDT)
1.46 0.893
21.4 18.3
6.96 4.34
16.88 23.25
14.76 21.05
1.73 1.56
tuning method
2
E7
2
E8
2
Table 10. Performance and Robustness of the Proposed PID Controller Tuning Method for Various SOIR Processes controller parameters system
process model, GP(s)
performance indices
KC
τI
τD
IAEservo
IAEregul.
Ms
0.681 0.2
2.70 1.0
0.745 1.5
4.56 7.86
5.37 8.88
2.39 1.86
tuning method
E9
(1 − 2s) (s + 1)3
proposed Skogestad
E10
(1 − 5s) (100s 2 + 10s + 1)
proposed
0.975
9.43
10.0
10.23
15.04
1.96
E11
(1 − 10s) (400s 2 + 12s + 1)(5s + 1)
proposed
0.514
12.78
28.7
36.14
66.78
1.84
E12
(1 − s)e−s (6s + 1)(2s + 1)2
proposed (SOIR) proposed (FOPDT) Skogestad
2.44 1.21 1.0
9.66 8.63 6.0
7.92 7.72 9.15
4.00 7.16 7.69
2.48 1.55 1.36
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systems E9−E12 with SOIR models. Typical comparisons between the unit step responses of the original system and approximated models are shown in Figure 6 for system E5 and Figures S4, S5, and S6 in the Supporting Information for systems E1, E9, and E12, respectively. The low ISE values
in determining the optimal PID controller parameters. Table 6 summarizes the 12 examples chosen for this study; their moments GP′ , GP″, and GP‴; and their corresponding process model parameters. Systems E1−E4 are approximated with FOPDT models, systems E5−E8 with SOPDT models, and
Figure 8. Closed-loop servo and regulatory responses and input changes of the SOPDT plants E5, E6, and E7.
Figure 9. Closed-loop servo and regulatory responses and input changes of the SOPDT plant E8 with PID controllers designed using SOPDT and FOPDT models. 17415
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Skogestad,13 Smith and Corripio,20 Astrom and Hagglund,21 Ziegler and Nichols,22 and Lee et al.,23 were selected. A list of such selected controller tuning rules for FOPDT processes are given in Table 7. The optimal controller parameters are estimated for each one of the systems by the proposed method and by the design
indicate that the moments of the impulse response have captured the process dynamic characteristics very accurately for the different types of processes considered in this study. To compare the performance of the optimal PID controllers designed through the proposed approach, a number of other well-accepted controller tuning rules, such as, Rivera et al.,10
Figure 10. Closed-loop servo and regulatory responses and input changes of the SOIR plant E9.
Figure 11. Closed-loop servo and regulatory responses and input changes of the SOIR plant E12. 17416
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methods chosen for comparison. The closed-loop servo and regulatory performances of the controllers are evaluated, and the IAE values for the unit step changes in the set-point and
load variables are reported as IAEservo and IAEregul., respectively. Table 8 provides the controller parameters calculated for systems E1−E4 characterized by FOPDT models, the methods of controller design used, the servo and regulatory performance measures IAEservo and IAEregul., and the peak gain, Ms. It is very clearly seen that the proposed method performs equally well as compared to the other methods for FOPDT processes. It may be noted that in some cases, the proposed method provides better performance either in IAE or in Ms alone. Figure 7 and Figures S7, S8, and S9 in the Supporting Information show the comparison between the closed-loop responses for unit step changes in the set point and load variable for different controller design methods for systems E1, E2, E3, and E4, respectively. Corresponding changes in the system input (controller output) are also shown in these figures. From these figures, it is clearly
Table 11. Effect of Controllability Index θ̃ on the Performance and Robustness of the Proposed PID Controller Tuning Method for SOPDT and SOIR Plants SOPDT plant θ̃
IAE
0.2 0.5 1.0 3.0 5.0
20.7 19.7 19.4 14.2 10.9
SOIR plant MS
θ̃
IAE
MS
1.42 1.47 1.74 1.79 1.95
0.5 0.8 1.0 1.5 2.0
18.9 19.8 20.0 19.4 18.3
1.98 1.97 1.96 1.94 2.00
Figure 12. Closed-loop step responses of a SOPDT plant (KP = 1, θP = 8, and ζP = 0.2) for different values of controllability index θ̃.
Figure 13. Closed-loop step responses of a SOIR plant (KP = 1, θP = 8, and ζP = 0.2) for different values of controllability index θ̃. 17417
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seen that the controllers designed by the proposed method produce responses with a low rise time, minimum overshoot, and shorter settling time with no oscillations. It is also noted that the changes in the input variables are not significantly different from the closely performing controller in each case. Similar closed-loop studies are performed on systems E5−E8 approximated by SOPDT models and systems E9−E12 approximated by SOIR models. The performance and robustness measures for each one of the systems represented by SOPDT and SOIR models are listed in Tables 9 and 10, respectively. The closed-loop responses of systems E5−E7 for unit step changes in set points and load variables and their corresponding input changes are shown in Figure 8. The closed-loop servo and regulatory responses of system E8 with PID controllers designed with FOPDT and SOPDT model approximations are shown in Figure 9. It is observed that the controller designed based on the SOPDT model performs much better than that designed based on FOPDT model approximation. Figures 10 and 11 and Figures S10 and S11 in Supporting Information show the closed-loop responses of unit step change in set points and load variables and corresponding input changes for systems E9, E12, E10, and E11, respectively. Figures 10 and 11 also show the comparison between the closed-loop responses with controllers designed by the proposed method and Skogestad method for cases E9 and E12. Table 11 summarizes the performance measures of SOPDT and SOIR models for various values of the controllability index θ̃. Figures 12 and 13 show the corresponding closed-loop responses for unit step change in set point for SOPDT (KP = 1, θP = 8, and ζP = 0.2) and SOIR (KP = 1, θP = 8, and ζP = 0.2) plants, respectively. It is observed that the peak sensitivity for all cases is less than 2.0.
AUTHOR INFORMATION
Corresponding Author
*Tel.: +605 3687585. Fax: +605 365 6176. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Ciancone, R.; Marlin, T. Tune controllers to meet your performance goals. Control 1992, 50−57. (2) Madhuranthakam, C. R.; Elkamel, A.; Budman, H. Optimal tuning of PID controllers for FOPTD, SOPTD and SOPTD with lead processes. Chem. Eng. Process. 2008, 47, 251−264. (3) Davendra, D.; Zelinka, I.; Senkerik, R. Chaos driven evolutionary algorithms for the task of PID control. Comput. Math. Appl. 2010, 60, 1088−1104. (4) Herreros, A.; Baeyens, E.; Peran, J. R. Design of PID-type controllers using multi-objective genetic algorithms. ISA Trans. 2002, 41, 457−472. (5) Hwang, C.; Hsiao, C.-Y. Solution of a non-convex optimization arising in PI/PID control design. Automatica 2002, 38, 1895−1904. (6) Syrcos, G.; Kookos, I. K. PID controller tuning using mathematical programming. Chem. Eng. Process. 2005, 44, 41−49. (7) Bagis, A. Tabu search algorithm based PID controller tuning for desired system specifications. J. Franklin Inst. 2011, 348, 2795−2812. (8) Padula, F.; Visioli, A. Tuning rules for optimal PID and fractional order PID controllers. J. Process Control 2011, 21, 69−81. (9) Chen, D.; Seborg, D. E. PI/PID controller design based on direct synthesis and disturbance rejection. Ind. Eng. Chem. Res. 2002, 41, 4807−4822. (10) Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control, 4. PID controller design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252−265. (11) Lin, M. G.; Lakshminarayanan, S.; Rangaiah, G. P. A comparative study of recent/popular PID tuning rules for stable first-order plus dead time, single-input single-output processes. Ind. Eng. Chem. Res. 2008, 47, 344−368. (12) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, 2nd ed.; Wiley: New York, 2004. (13) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control. 2003, 13 (4), 291−309. (14) Wang, Q.-G.; Hang, C. C.; Yang, X.-P. Single loop controller design via IMC principles. Automatica 2001, 37, 2041−2048. (15) Abbas, A. A new set of controller tuning relations. ISA Trans. 1997, 36 (3), 183−187. (16) Shamsuzzoha, M.; Lee, M. Analytical design of enhanced PID filter controller for integrating and first order unstable processes with time delay. Chem. Eng. Sci. 2008, 63, 2717−2731. (17) Seshagiri Rao, A.; Rao, V. S. R.; Chidambaram, M. Direct synthesis-based controller design for integrating processes with time delay. J. Franklin Inst. 2009, 346, 38−56. (18) 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. (19) Ramasamy, M.; Sundaramoorthy, S. PID controller tuning for desired closed-loop responses for SISO systems using impulse response. Comput. Chem. Eng. 2008, 32, 1773−1788. (20) Smith, C. L.; Corripio, A. B. Principles and Practice of Automatic Process Control; McGraw Hill: New York, 1985. (21) Astrom, K. J.; Hagglund, T. Revisiting the Ziegler-Nichols step response method for PID control. J. Process Control. 2004, 14 (6), 635−650. (22) Ziegler, J. G.; Nichols, N. B. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759−768. (23) Lee, Y.; Park, S.; Lee, M.; Brosilow, C. PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 1998, 44 (1), 106−115.
6. CONCLUSIONS In an effort to determine optimal PID controller parameters, the following steps have been proposed and carried out: (i) approximation of process dynamics through the moments of impulse response to develop FOPDT, SOPDT and SOIR process models; (ii) developed controller design equations using the direct synthesis approach for each of the above process models with the desired closed-loop response specified by transfer functions that are the same as that of the process model; and (iii) developed correlations for the optimal tuning parameter λ in terms of the process model parameters. The use of SOPDT and SOIR models to describe the process and as the desired closedloop transfer functions in the direct synthesis approach are the contributions of this work. From the case studies, it is very clearly demonstrated that the PID controllers designed using the optimal λ values calculated from the correlations developed provide control performances that are better than or comparable to those of the other controller design methods reported in the literature.
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Article
ASSOCIATED CONTENT
S Supporting Information *
Additional graphical results that show the scatter of estimated λ values (correlation) around the actual λ values for different systems in Figures S1−S3; comparison of unit step responses of the plant and the model obtained by the method of moments for systems E1, E9, and E11 in Figures S4, S5, and S6, respectively; and closed-loop servo and regulatory responses and input changes for plants E2, E3, E4, E10, and E11 in Figures S7, S8, S9, S10, and S11, respectively. This material is available free of charge via the Internet at http://pubs.acs.org. 17418
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