Abstract
This paper presents a PID tuning method for unstable processes using an adaptive-network-based-fuzzy-inference system (ANFIS) for given gain and phase margin (GPM) speci)cations. PID tuning methods are widely used to control stable processes. However, PID controller for unstable processes is less common. In this paper, the PID controller parameters can be determined by the ANFIS. Because the de)nitions ofgain and phase margin equations are complex, an analytical tuning method for achieving speci)ed the gain and phase margins is not yet available. In this paper, the ANFIS is adopted to identify the relationship between the gain-phase margin speci)cations and the PID controller parameters. Then, it is used to automatically tune the PID controller parameters for di;erent gain and phase margin speci)cations so that neither numerical methods nor graphical methods need be used. A simple method is also developed to estimate the stabilizing region of PID controller parameters and valid region for gain-phase margin. Even for unreasonable speci)cations, out of the valid region, the ANFIS can still )nd suitable PID controller to guarantee the stability ofthe closed-loop system. Simulation results show that the ANFIS can achieve the speci)ed values e=ciently. c 2002 Elsevier Science B.V. All
rights reserved.
Keywords: ANFIS; Unstable process control; PID controller tuning; Gain and phase margin
1. Introduction
Several methods for determining PID controller parameters have been developed over the past 50 years. Some employ information about open-loop step response, for example, the Coon–Cohen reaction curve method [7]; other methods use knowledge of the Nyquist curve, for example, the Ziegler–Nichols frequency-response method. However, these tuning methods use only a small amount ofinformation about the dynamic behavior ofthe system, and often do not provide good tuning. It is known that gain margin and phase margin have served as important measures ofrobustness. From classical control theories, phase margin is related to the
∗Corresponding author. Tel.: +886-3-5712121=ext: 54345; fax: +886-3-5715998.
E-mail addresses: [email protected] (C.-H. Lee), [email protected] (C.-C. Teng). 0165-0114/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.
damping of the system, and therefore also serves as a performance measurement. Their solutions are nor-mally obtained numerically or graphically by trial-and-error use ofBode plots. Controllers designed to satisfy gain margin and phase margin (GP=GM) criteria are not new approaches [2,8–11,19]. In 1984, Astrom and Hagglund )rst proposed a tuning method for PID controllers based on phase and amplitude speci)cations [1]. Then, Ho et al. presented a tuning method for stable and unstable processes [9–11]. They adopt linear equations to approximate the arctan function as to simplify the gain-phase margin formulas. Due to the ap-proximation of arctan function, this method may result in unstable controllers or unstable systems for some speci)cations. In this paper, a fuzzy neural network approach is presented to solve this problem. The ap-proach determines the PID controller parameters that guarantee the stability ofcontroller and the closed-loop system.
With the development offuzzy logic controllers and the more recent hybrid controllers which use both fuzzy logic and neural network methodology, the possibility exists that one or both of these methods could perform as a feedback controller [5,6,15,17,18]. The fuzzy logic toolbox [16] implements one of the hybrid schemes known as the adaptive-network-based-fuzzy-inference system (ANFIS). The ANFIS has proven to be an excellent function approximation tool and can be as good or better than a plain feedforward neural network for some situations. Although various kinds of fuzzy logic controllers (FLCs) [20,21] are widely used nowadays and have certain advantages over conventional PID controllers, relatively few theoretical analysis that explain why they can achieve better performance are available. In literature [6], we have presented a tuning method that uses the ANFIS based on gain and phase margin speci)cations, to tune the PI controller parameters processes e=ciently. This approach enjoys the advantage offunctionally mapping the ANFIS, and gives better performance than GPM [9]. The purpose of this paper is to extend this approach to unstable processes and solve the unsuitable results of[11]. The stabilizing region ofcontroller parameters and the valid region ofspeci)cations (Am; m) for PID controller are also estimated.
The arrangement ofthis paper is as follows. In Section 2, we brieKy introduce gain and phase margins, and the used fuzzy neural network (ANFIS). Section 3 proposes the structure of PID controller using the ANFIS and the tuning method. Section 4 describes a procedure for estimating the stabilizing region of controller parameters and valid region ofGPM speci)cations for PID controller. Section 5 gives the simulation results and discusses the advantages ofthe proposed approach as compared with other methods. Finally, conclusions are summarized in Section 6.
2. Preliminaries
2.1. Gain margin and phase margin
Consider the n-order unstable process with time-delay Gp(s) = K(1 + wn1s)
n1(1 + w
n2s)n2· · · (1 + wnqs)nq
(1 + wd1s)d1(1 + wd2s)d2· · · (1 + wdps)dp e
−Ls; (1)
where at least one of wdiis negative and n =pi=1di. The open-loop step response ofthe process is unbounded,
since it has a pole in the right-halfplane. Figs. 1(a) and 1(b) show the Bode and Nyquist diagrams ofan unstable )rst-order plus delay process with PI control. Note that unstable plant have more than one GM=PM. As the de)nitions ofthe GM=PM [14], Am1 and Am2 are called gain margin (or upward gain margin) and gain
reduction margin (or downward margin). In addition, applying the Nyquist criterion for stability, the Nyquist diagram should encircle the point (−1; 0) [in the G(jw) plane] exactly once in the anti-clockwise direction. Based on the stability criterion, the gain margin Am1 is chosen in this literature. Here, the PID tuning for
The PID controller given by
Gc(s) = KP+KsI + KDs; (2)
must be used to satisfy the Nyquist criterion. Let the speci)ed gain and phase margins be denoted by Am and
m, respectively. The formulas for gain and phase margins are as follows:
arg[GC(jwp)GP(jwP)] = − (3)
Am=|G 1
C(jwP)GP(jwP)|; (4)
|GC(jwg)GP(jwg)| = 1; (5)
m= arg[GC(jwg)GP(jwg)] + ; (6)
where the gain margin is de)ned by Eqs. (3) and (4), and the phase margin by Eqs. (5) and (6). Here wp
and wg denote the phase crossover frequency and gain crossover frequency, respectively. The loop transfer
function is obtained from
Gc(s)Gp(s) = K(KI+ KPs + KDs
2)(1 + wn1s)n1(1 + wn2s)n2· · · (1 + wnqs)nq
s(1 + wd1s)d1(1 + wd2s)d2· · · (1 + wdps)dp e
−Ls: Substituting the above equation into Eqs. (3)–(6), we have
1
2 + tan−1(wpwc1) + tan−1(wpwc2) + n1tan−1(wpwn1) + · · · + nqtan−1(wpwnq) − wpL
−d1tan−1(wpwd1) − d2tan−1(wpwd2) − · · · − dptan−1(wpwdp) = 0; (7)
AmK = wp (1 + w2 pw2n1)n1 (1 + w2 pw2n2)n2· · · (1 + w2 pw2nq)nq 1 + w2 pwc12 1 + w2 pw2c2 (1 + w2 pwd12 )d1· · · (1 + w2 pwdp2 )dp ; (8) K = wg (1 + w2 gw2d1)d1 (1 + w2 gwd22 )d2· · · (1 + w2 gwdp2 )dp (1 + w2 gw2c1) 1 + w2 gwc22 (1 + w2 gw2n1)n1· · · (1 + w2 gwnq2 )nq ; (9)
m=12 + tan−1(wgwc1) + tan−1(wgwc2) + n1tan−1(wgwn1) + · · · + nqtan−1(wgwnq) − wgL
− d1tan−1(wgwd1) − d2tan−1(wgwd2) − · · · − dptan−1(wgwdp); (10)
where wc1 and wc2 are the roots of(KI+ KPs + KDs2). For a given process (K; wn1; : : : ; wnq; wd1; : : : ; wdp; L)
and speci)cations (Am; m), Eqs. (7)–(10) can be solved for the PID controller parameters (KP; KI; KD) and
crossover frequencies (wg; wp) numerically but not analytically because ofthe presence ofthe tan−1 function.
For stable processes, controllers such as the IMC [4] and GPM [9,10] that are based on gain and phase margins cannot e=ciently meet speci)cations within a 10% error margin owing to the approximation ofthe tan−1 function. In addition, a similar controller based on GPM for an unstable process improves performance but still can only meet the speci)cations within 5% error [11]. Using this approximated method [11], unstable results (unstable controller or unstable closed-loop system) occurred due to the approximation oftan−1 (for details, see Remark 3). Therefore, another approach using the ANFIS for general processes is considered here. This approach yields high accurate tuning formulas for controllers including P; PI; PD and PID controllers ofstable and unstable processes with time delay.
Fig. 2. The architecture ofthe ANFIS.
Remark 1. It is well known that the model G(s) = K
1 + sT e−sL (11)
is the most common process model used in paper on PID controller tuning [2]. As the statement of[2,13] the following processes were chosen that are representative for the dynamics of typical industrial processes:
G1(s) = e −s (1 + sT)2; T = 0:1; : : : ; 10; G2(s) = (1 + s)1 n; n = 3; 4; 8; G3(s) = (1 + s)(1 + s)(1 + 1 2s)(1 + 3s); = 0:2; 0:5; 0:7; G4(s) = (1 + s)1 − s3; = 0:1; 0:2; 0:5; 1:2: (12)
The test batch (12) does not include the transfer function (11) because this model is not representative for typical industrial processes [2]. Therefore, we present here our approach in transfer function (1) that includes the test batch (12) and model (11) as model (11) is the most common process model used in the paper on PID controller tuning.
2.2. Fuzzy neural network (ANFIS)
The used ANFIS [15–17] architecture is shown in Fig. 2. The inputs are given by (x; y) and have Ri (i =
1; : : : ; n2) implications, then the value of f is implied as follows.
Layer 1: Here we denote the output node i in this layer as Ol; i. Every node is an adaptive node with a node
output de)ned by
O1; i = Ai(x) f or i = 1; : : : ; n
where x is the input and Ai is a fuzzy set associated with this node. In other words, outputs of
this layer are the membership values ofthe premise part. Here the membership function can be characterized by the generalized bell-shaped function:
Ai(x) =
1
1 + [(xici=ai)2]bi;
where {ai; bi; ci} is in the parameter set. Parameters in this layer are referred to as premise
parameters.
Layer 2: Every node in this layer is a )xed node labeled , which multiplies the incoming signals and outputs the product,
O2; k = wk= Ai(x) × Aj(y); i; j = 1; : : : ; n; k = 1; : : : ; n2:
Each node output represents the )ring strength ofa rule.
Layer 3: Every node in this layer is a )xed node labeled N. The ith node calculates the ratio ofthe ith rule’s )ring strength to the sum ofall rule’s )ring strengths:
O3; i = Owi=w wi
1+ · · · + wn2; i = 1; : : : ; n
2:
For convenience, the outputs from this layer are called normalized :ring strengths. Layer 4: Every node in layer 4 is an adaptive node with a node function
O4; i = Owifi= Owi(p1ix + pi2y + pi0); i = 1; : : : ; n2;
where Owi is the output oflayer 3 and {pi0; pi1; pi2} is in the parameter set. Parameters in this layer
are called as consequent parameters.
Layer 5: The single node in this layer is a )xed node labeled that computes the overall outputs as the summation ofall incoming signals, i.e.,
f = O5;1= i Owifi= iwifi iwi ; i = 1; : : : ; n 2:
3. PID controller using the ANFIS
To obtain parameters (KP; KI; KD) for a PID controller more exactly, without using the approximation of
arctan functions, we use the ANFIS [15–17] based on gain and phase margins to model these equations analytically.
Considering the nonlinear coupled Eqs. (7)–(10), we )nd that there are )ve parameters (wp; wg; KP; KI; KD)
in those four equations. If we are given gain margin and phase margin speci)cations (Am; m), it may not be
possible to solve for the )ve parameters analytically because the equations are nonlinear. Now, let us consider another approach. First, it is possible to give randomly controller parameters (KP; KI; KD) as the input of
these equations. Using Eq. (7), we can solve for wp then substitute it into Eq. (8) to get Am. And using
Eq. (9), we can calculate wg then substitute it into Eq. (10) to obtain m. Hence we obtain the parameters
(wp; wg; Am; m) that correspond to the controller parameters (KP; KI; KD), respectively. Fig. 3 summarizes
the approach. In preparation for training the ANFIS, we assign randomly points (KP; KI; KD), obtain the
corresponding (Am; m) points, and set them as the training data. That is, the input data are (Am; m) and the
output are (KP; KI; KD). Note that the training data satisfy the stability condition, i.e., Am¿0 and m¿0. Thus,
we can get our training data for the ANFIS. This approach avoids the possibility of not )nding a solution to nonlinear Eqs. (7)–(10), and reduces the overall task. Furthermore, this approach is useful for all processes
(stable, unstable, higher-order, under-damped response, etc.). In Section 5, simulation results demonstrate the e;ectiveness ofthis approach.
Fig. 3 illustrates the block diagram ofthe function mapping ofEqs. (7)–(10) using the ANFIS. Suppose we are given (Am; m) and have Ri (i = 1; : : : ; n2) implications, then the value of y ∈ {KP; KI; KD} is implied
as follows.
3.1. Tuning of the ANFIS
We note that when the values ofthe premise parameters are )xed, the overall output f = {KP; KI; KD} can
be expressed as a linear combination ofconsequent parameters. In symbols, the output f in Fig. 2 can be written as f =w w1 1+ · · · + wn2f1+ · · · + wn2 w1+ · · · + wn2 fn2 = Ow1f1+ · · · + Own2fn2 = ( Ow1Am)p11+ ( Ow1m)p12+ ( Ow1)p10+ · · · + ( Own2Am)pn12+ ( Own2m)p2n2+ ( Own2)pn02;
which is linear in the consequent parameters {p1
0; p11; p12; : : : ; pn 2 0; pn 2 1; pn 2
2 }. Note that ifa fuzzy neural network
output or its transformation is linear in some of the network’s parameters, then we can identify these linear parameters using the well-known linear least-squares method [12]. Therefore, we use an o;-line learning (the recursive least-square algorithm) to update the parameters of ANFIS. After the parameters are updated for each data presentation, we have an on-line learning scheme. This learning strategy [3] is vital to on-line parameter identi)cation by systems with changing characteristics. In this learning scheme, we use back-propagation learning [21] to update the premise parameters {ai; bi; ci}. Details for tuning the ANFIS can be found in
[6,15–17].
4. Stabilizing region and valid region for PID controller
Some of the equations that appeared in the derivation are useful for assessing what is achievable by PID control. Firstly there are some restrictions on the choice ofthe gain and phase margins. One usual requirement is that the controller parameters KP¿0; KI¿0 and KD¿0. Therefore, the suitable choice (speci)cation) of
gain and phase margins for the unstable process must be determined. In literature [10], Ho and Xu used the linear equation to approximate the tan−1 function that reduces the complex of Eqs. (7)–(10). Therefore, they found the relationship between Am; m, time-delay, and unstable-pole. An analysis method was developed
to )nd the suitable choice ofgain-phase margins. However, there are unsuitable speci)cations by using the result of[11], see Example 1. A simple method is proposed to estimate the valid region ofAm; m for PID
controller and the stabilizing region.
4.1. Procedure for estimating the stabilizing region and valid region
Step 1: Estimate the stabilizing range $ roughly for controller parameters (KP¿0; KI¿0 and KD¿0).
Step 2: Choose randomly (or uniformly) the testing data (Ki
P; KIi; KDi; i = 1; : : : ; n) from the region $ and
calculate the corresponding gain and phase margins using Eqs. (7)–(10).
Step 3: Find the data set $P that every testing data in $ results a stable closed-loop system (Am¿0 and
m¿0).
Step 4: Estimate the stabilizing region $∗
Pofparameter (KP; KI; KD) from these points, i.e., )nd the boundary
of $P (here we can omit the isolated point that are far from the grouped points).
Step 5: Choose randomly on the closure of $∗
P and calculate the corresponding Am and m using
Eqs. (7)–(10). Then, the valid region ofgain and phase margin speci)cation for the PID controller can be obtained.
Remark 2. It is known that choosing proper training data is important for neural network system. In this case, gain and phase margins are input data while the PID parameters are the output data. In the preceding discussion, we explained that we get our training data by giving the PID parameters randomly to derive the desired output (gain margin and phase margin). The stability ofthe closed-loop system depends on the PID parameters we choose. Thus, the above method for )nding the stabilizing region that guarantees the validity ofthe training data.
Example 1. Unstable plant Gp(s) = e−0:2s=s − 1 with PI controller [11].
First, we roughly give the stabilizing range of KP and KI as [0; 10] and [0 10]. Then the testing data are
chosen randomly. For each pair (KP; KI), the corresponding gain and phase margin can be obtained. Then,
)nd the points set that satisfy Am¿0 and m¿0. We omit the isolated points that far from the grouped points
and estimate the stabilizing region $∗
P ofparameter (KP; KI) from these points. Fig. 4 shows the estimated
stabilizing region ofPI controller for the unstable plant Gp(s) = e−0:2s=s − 1. Finally, we would estimate the
valid region ofgain and phase margins using the information provided by the stabilizing region. Points chosen randomly on the closure ofthe stabilizing region (KP; KI) are used to calculate the corresponding gain and
phase margins using Eqs. (7)–(10). Then, the valid region ofgain and phase margin speci)cation for the PI controller can be obtained. Fig. 5 shows the estimated valid range for the unstable plant Gp(s) = e−0:2s=s − 1
with PI controller. Note that, the form of the controller Ho et al. [9–11] used was Gc(s) = Kc(1 + (1=sTI)).
By comparing these two forms of controllers, we have KC= KP and TI= KP=KI.
Remark 3. Denote $1 (dash–dotted line) and $2 (solid-line) as the estimated valid regions for the result in
[11] and our approach. From Fig. 5 and Table 1, it is clear that P1; P2; P3 =∈ ($1∪ $2) and P4; : : : ; P10∈ $1
but P4; : : : ; P10 =∈ $2. By testing these data, we obtain that these 10 speci)cations by using the approximated
method [11] give unavailable results (at least one of KC; TI; Am; m is negative, see the shadow items in
Table 1) because ofthe approximation oftan−1 function. Since P
1; P2; P3 =∈ $1, we got TI¡0. On the other
hand, P4; : : : ; P10∈ $1 we have parameters KC; TI¿0 and wrong phase margin (m¡0 or ∞). Note that,
even for these unreasonable speci)cations, the ANFIS provides suitable controller parameters that guarantee the closed-loop system stability. In this paper, the system plant is directly used to design the controller. Therefore, we avoid the above results using the ANFIS. In the following section, we will show the comparison ofsimulations between the results of[11] and our approach.
Fig. 4. Estimated stabilizing region of(KP; KI).
Fig. 5. Valid region of(Am; m) for PI controller: solid line, our result; dash–dotted line: result of [10] (P1–P10are outside the estimated valid region).
Table 1
Comparison result ofunreasonable speci)cations
Speci)cations Results of[11] ANFIS results
Am m KC TI Am m KC TI P1 (3; 40) 3.0420 41.8603 2.3998 −17:3642 3.0536 39.9748 2.3818 96.6836 P2 (4; 40) 4.0559 42.0461 1.8035 −13:0428 3.9295 38.3010 1.8476 37.5771 P3 (5; 35) 5.0700 35.9405 1.4363 −29:5354 4.8060 33.1555 1.5076 21.7432 P4 (7; 15) 7.0952 −2:6003 1.0090 8.5340 6.9303 9.7452 1.0478 40.3184 P5 (7:5; 15) 7.6038 −13:3516 0.9477 16.4203 6.9303 9.7452 1.0478 40.3184 P6 (8; 15) 8.1117 ∞ 0.8934 101.3638 6.9303 9.7452 1.0478 40.3184 P7 (7:5; 10) 7.6012 −12:6772 0.9398 7.3770 6.9303 9.7452 1.0478 40.3184 P8 (8; 5) 8.1072 −20:2941 0.8796 6.6594 6.9303 9.7452 1.0478 40.3184 P9 (8:5; 10) 8.6184 ∞ 0.8390 32.6130 6.9303 9.7452 1.0478 40.3184 P10 (9; 5) 9.1249 ∞ 0.7909 20.7626 6.9303 9.7452 1.0478 40.3184 Table 2
Di;erent PI controllers for Gp(s) = 100e−0:01s=(s2+ 10s − 5)
Speci)cations Result Error
Am ◦ m KP KI A∗m m∗ wg wp Error of Am (%) Error of ◦m (%) 2 30 0.5894 0.003808 1.9356 29.8200 8.3540 5.0432 3.220 0.600 3 45 0.3741 0.0024667 3.0500 44.5976 8.3539 3.3631 1.667 0.894 4 50 0.2784 0.0176454 4.0734 49.8999 8.3170 2.5372 1.835 0.200 5. Simulation results
In this section, we give a speci)c performance comparison with GPM [11] because both were designed based on gain and phase margin speci)cations.
Example 2. PI controller for a second-order process. The process is given as follows:
Gp(s) = 100e
−0:05s s2+ 10s − 5:
Since the process is not a )rst-order type, the GPM cannot be applied. Various gain and phase margins are speci)ed for this model in Table 2. The ANFIS yields less than 3.5% and 0.9% for desired gain and phase margin speci)cations.
Example 3. PI controller for a )rst-order with time-delay process. The process is given as
Gp(s) = e
−0:2s
s − 1; L=' = 0:2¡1:
The results for di;erent speci)cations in this example are illustrated in Table 3, which shows that even when the plant is )rst-order with time-delay, the proposed ANFIS approach has better performance than GPM.
Table 4
Results for Gp(s) = e−0:2s=(s − 1) with PID controllers under di;erent speci)cations
Speci)cations Results Errors
Am m KP KI KD A∗
m ∗m Error of Am (%) Error of ◦m (%)
2 20 5.1641 0.1081 0.2915 2.0278 20.3724 1.390 1.862
3 30 5.5592 0.0135 0.3102 2.9998 30.3809 0.007 1.270
4 40 2.6842 0.1798 0.0684 3.9787 40.5597 0.5325 1.400
ANFIS yields less than a 2% error but GMP’s is greater than 5%. Also, Table 1 shows the comparison with the result of[11] for unreasonable speci)cations.
Remark 4. It is clear that the results ofANFIS all satisfy the stability conditions KC; TI¿0 and Am; m¿0.
Even ifsomeone gives unreasonable speci)cations (outside the valid region $2) for PI controller, we have a
stable system using the ANFIS. This guarantees the stabilization ofthe proposed PI controller. Example 4. PID controller for a )rst-order with time-delay process.
The process is given as Gp(s) = e
−0:2s
s − 1; L=' = 0:2¡1:
In this example, we use the PID controller to compensate the )rst-order unstable process. ANFIS also yields less than a 2% error in this case. Table 4 shows the simulation results for di;erent speci)cations. However, the ANFIS gives acceptable errors for the speci)ed gain and phase margins.
6. Conclusion
This paper has investigated the PID tuning method using fuzzy neural system (ANFIS) based on gain and phase margin speci)cations. The proposed method has been generalized to determine the PID controller parameters for general processes that include the test batch and common used model of the typical industrial processes. There are two advantages to use the ANFIS for formulating gain and phase margin problems. First, the trained ANFIS automatically tunes the PID controller parameters for di;erent gain and phase margin
speci)cations so that neither numerical methods nor graphical methods need be used. Second, the ANFIS can also )nd the relationship between PID controllers (KP; KI; KD) and speci)cations (Am; m) in the
weight-ing parameters in the networks. Therefore, the proposed method is simple and systematic in reducweight-ing the complexity ofthe problem presented in this paper. A simple method was also developed to estimate the stabilizing region ofcontroller parameters and valid region for gain-phase margin speci)cation. The ANFIS can still )nd suitable PID controller parameters that guarantee the stabilization even for unreasonable speci)-cations. That is, the ANFIS can provide controller parameters for guaranteeing the stability of the closed-loop system. Simulation results have shown that the ANFIS can achieve the speci)ed values e=ciently.
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