Ind. Eng. Chem. Res. 2008, 47, 8317–8323
8317
Effective Tuning Method for Fuzzy PID with Internal Model Control Xiao-Gang Duan,*,† Han-Xiong Li,†,‡ and Hua Deng† School of Mechanical and Electrical Engineering, Central South UniVersity, Changsha 410083, China, and Department of Manufacturing Engineering and Engineering Management, City UniVersity of Hong Kong, Hong Kong
An internal model control (IMC) based tuning method is proposed to autotune the fuzzy proportional integral derivative (PID) controller in this paper. An analytical model of the fuzzy PID controller is first derived, which consists of a linear PID controller and a nonlinear compensation item. The nonlinear compensation item can be considered as a process disturbance, and then parameters of the fuzzy PID controller can be analytically determined on the basis of the IMC structure. The stability of the fuzzy PID control system is analyzed using the Lyapunov stability theory. The simulation results demonstrate the effectiveness of the proposed tuning method. 1. Introduction Generally speaking, conventional proportional integral derivative (PID) controllers may not perform well for the complex process, such as the high-order and time delay systems. Under this complex environment, it is well-known that the fuzzy controller can have a better performance due to its inherent robustness. Thus, over the past three decades, fuzzy controllers, especially, fuzzy PID controllers have been widely used for industrial processes due to their heuristic natures associated with simplicity and effectiveness for both linear and nonlinear systems.1-4 There are too many variations of fuzzy PID controllers,5 such as, one-input, two-input, and three-input PID type fuzzy controllers. In general, there is no standard benchmark. The one-input may miss more information on the derivative action, and the three-input fuzzy PID controllers may cause exponential growth of rules. The two-input fuzzy PID, as we used in the paper, has a proper structure and the most practical use, and thus is the most popular type of fuzzy PID used in various research and application. Despite the fact that industry shows greater and greater interest in the applications of fuzzy PID, it is still a highly controversial topic from the point of view of the mainstream control engineering community. One reason is that the fundamental theory for the analytical tuning methods of fuzzy PID is still missing. Therefore, fuzzy PID controllers had to be tuned qualitatively by two-level tuning.6,7 At a lower level, the tuning is performed by adjusting the scaling gains to obtain overall linear control performance. At a higher level, the tuning is performed by changing the knowledge base parameters to enhance the control performance. However, it is difficult to tune the knowledge base parameters. Moreover, it is hard to improve the transient response by changing the member function.8 As the knowledge base conveys a general control policy, it is preferred to keep the member function unchanged and to leave the design and tuning exercises to scaling gains. However, the tuning mechanism of scaling factors and the stability analysis are still difficult tasks due to the complexity of the nonlinear control surface that is generated by fuzzy PID controllers. If the nonlinearity can be suitably utilized, fuzzy PID controllers may pose the potential to achieve better system performance than conventional PID controllers. Some nonanalytical tuning methods were introduced.9-12 Al* To whom correspondence should be addressed. E-mail: dxg509@ yahoo.com.cn. † Central South University. ‡ City University of Hong Kong.
though the nonlinearity was considered on the basis of gain margin and phase margin specifications, the fuzzy PID controller may produce higher gains than conventional PID controllers due to the nonlinear factor.13,14 A high gain could deteriorate the stability of the control system.15 The conventional PID controller is easy to implement, and lots of tuning rules are available to cover a wide range of process specifications. Among tuning methods of the conventional PID controller, the internal model control (IMC) based tuning is one of the popular methods in commercial PID software packages16 because only one tuning parameter is required and better setpoint response can be produced.17 An analytical tuning method based on IMC to tune fuzzy PID controllers is proposed in this paper. The fuzzy PID controller is first decomposed as a linear PID controller plus a nonlinear compensation item. When the nonlinear compensation item is approximated as a process disturbance, the fuzzy PID scaling parameters can then be analytically designed using the IMC scheme. The stability analysis of the fuzzy PID controllers is given on the basis of the Lyapunov stability theory. Finally, the effectiveness of the tuning methodology is demonstrated by simulations. 2. Problem Formulation A. Conventional PID Controller. The conventional PID controller is often described by the following equation:20,21
∫ ∫
UPID )KPe + KI e dt + KDe˙ 1 e dt + Tde˙ )KP e + Ti
(
)
(1)
where e is the tracking error, KP is the proportional gain, KI is the integral gain, KD is the derivative gain, and Ti and Td are the integral time constant and the derivative time constant, respectively. The relationships between these control parameters are KI ) KP/Ti and KD ) KPTd. The transfer function of the PID controller (1) can be expressed as follows: Gc(s) ) Kc
(tis + 1)(tds + 1) s
where Kc )
KP ; ti + td
ti, td )
10.1021/ie800485j CCC: $40.75 2008 American Chemical Society Published on Web 10/01/2008
[ (
Ti 4Td 1( 12 Ti
(2)
)] 2
8318 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008
controllers, resulting in the changing of the control performance. As the control actions are fuzzily coupled, the contribution of each Ke, Kd, K0, and K1 to different control actions is still not very clear, which makes the practical design and tuning process rather difficult. 3. Tuning Fuzzy PID Based on the IMC To tune the fuzzy PID controller based on the IMC method, an analytical model of the fuzzy PID controller is obtained first by simple derivation. Then, the parameters of the fuzzy PID controller can be determined on the basis of the IMC principle. Suppose that an industrial process can be modeled by a firstorder plus delay time (FOPDT) structure that has the transfer function as follows:
Figure 1. IMC configuration.
Figure 2. Fuzzy-PID controller structure with Kd ) RKe and K1 ) βK0.
On the root-locus plane, the PID controller has two zeros ti and td, and one pole at the origin. The condition to have real zeros is that Ti g 4Td. B. Principle of IMC. The basic IMC principle22,23 is shown in Figure 1a, where P is the plant, P˜ is a nominal model of the plant, C is a controller; r and d are the set point and the disturbance, and y and k y are the outputs of the plant and its nominal model, respectively. The IMC structure is equivalent to the classical single-loop feedback controller shown in Figure 1b. If the single-loop controller CIMC is given by CIMC(s) )
C(s) 1 - C(s) P˜(s)
(3)
P˜(s) )
K -Ls e Ts + 1
(7)
where K, T, and L are the steady-state gain, the time constant, and the time delay, respectively. The estimation of these parameters using the step response method, frequency response, and closed-loop relay feedback, etc., is well-described.20,21 The FOPDT model is one of the most common and adequate ones used, especially in the process control industries.24 A. Analytical Model of Fuzzy PID Controller. By substituting k ) i + j + 1, where i ) (E - e*)/A and j ) (R - r*)/A, into the eq 6, one obtains 1 B UPID ) (S + δ) K0 + K1 A p
(
)
(8)
where 1/p ) ∫ dt, and
with
δ ) (1 - γ)(A - e* - r*) ) (1 - γ)σ C(s) )
1 P˜-(s)
f(s)
(4)
where P˜(s) ) P˜-(s)P˜+(s), P˜-(s) is the minimum phase part of the plant model, P˜+(s) contains any time delays and right-halfplane zeros, and f(s) is a low-pass filter with a steady-state gain of one, which typically has the form 1 f(s) ) (1 + tcs)n
(
(10) σ ) A - e* - r* Using S ) E + R ) ke(e + Re˙), and letting β ) K1/K0, eq 8 becomes
{
B 1 UPID )KeK0(R + β) A e + R + β
B e˙ + K (βδ + ∫ δ) ∫ e + RRβ +β } A 0
)uPID + uN
(5)
(11)
The tuning parameter tc is the desired closed-loop time constant, and n is a positive integer to be determined. C. Model of Fuzzy PID Controller. The fuzzy PID controller,18,19 as shown in Figure 2, is described as follows: 1 UPID ) u K0 + K1 p
(9)
with
)
(6)
with
with uPID ) KeK0(R + β)
e˙ ∫ e + RRβ +β }
(12)
∫
B uN ) K0 (βδ + δ dt) (13) A The complex domain of (11) is shown as follows by Laplace transform:
B u ) kB(1 - γ) + γS A where 1/p ) ∫dt, S ) Kee + Kd˙ e ) E + R, Ke ) 1, E ) iA + e*, R ) jA + r*, k ) i + j + 1, γ is a nonlinear timevarying parameter (2/3 e γ e 1), A and B are half of the spread of each input and out member function, respectively, e* and r* are relative input data in the inference cell IC(i,j), and k is an index number depending on the inference cell under use. The large the integer, the farther away the state (e, e˙) is from the origin in the phase plane. The fuzzy PID control actually has two levels of gains.6 The scaling gains (Ke, Kd, K0, and K1) are at the lower level. The tuning of these scaling gains will affect the gains of fuzzy PID
{
B 1 e+ A R+β
UPID(s) ) uPID(s) + uN(s)
(14)
where
{
}
Rβ B 1 uPID(s) )KeK0(R + β) A 1 + (R + β)s + R + β s E(s) (15) B (Rs + 1)(βs + 1) E(s) )KeK0 A s uN(s) ) K0
B 1 β + δ(s) A s
(
)
(16)
with δ(s) being a nonlinear term without an explicit analytical expression. Equation 15 can be rewritten as a transfer function form:
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8319
B (Rs + 1)(βs + 1) u (s) ) KeK0 (17) E(s) A s Obviously, the fuzzy PID control can be considered as a conventional PID with a nonlinear compensation. The conventional PID control term is uPID(s) and the nonlinear compensation is uN(s). B. Tuning of Fuzzy PID Controller Based on IMC. If we consider the nonlinear compensation uN as a process disturbance and set Gf(s) ) CIMC(s), which is shown in Figure 3, the IMCbased tuning for fuzzy PID controllers can be simplified as follows. By the first-order Pade´ approximation, the delay time is approximated as follows: Gf(s) )
PID
-Ls
e
L 1- s 2 ≈ L 1+ s 2
(18)
Therefore, the P˜(s) can be factorized as P˜(s) ) P˜+(s)P˜-(s), where P˜+(s) ) 1 - Ls/2 and K
P˜-(s) )
(19) L (Ts + 1) 1 + s 2 Substituting P˜_(s) into (3) and setting n ) 1 in filter (5), one has
(
)
L (Ts + 1) 1 + s 2 CIMC(s) ) K(1 + tcs)
(
)
(1 + L2 s)(Ts + 1) L K(t + )s 2
Figure 4. Filter property of fuzzy PID, p ) d/dt.
is obtained from S through a sequence of first-order lowpass filter25 (Figure 4). The bandwidth of the fuzzy PID at the kth level can be controlled by adjusting R. A small value of R gives wide bandwidth and fast response; otherwise, it gives a low bandwidth and sluggish response. To improve the rise time, the value of R should be small. Therefore, the two parameters R and β can be determined as R ) min(T, L ⁄ 2)β ) max(T, L ⁄ 2) (26) Remark: The fuzzy-PID control (11) is actually a conventional PID control uPID plus a pseudo-sliding mode control δ. Because the sliding mode control is a robust control,26 the fuzzy PID control is more robust than a conventional PID control.
(20)
4. Stability Analysis
(21)
Consider a class of industrial processes modeled by eq 7, which are controlled by a fuzzy PID controller whose parameters are given in (22) and (26). If the delay time is approximated by the first-order Pade´ approximation (18), the process model can be described by the following two-order model:
Comparing (20) with (17), one obtains B (Rs + 1)(βs + 1) ) KeK0 A s
Figure 3. Equivalent format of FPID control in the closed loop.
c
In terms of (21), parameters K0, R, and β can be given as follows: K0 )
A B
1
(
KKe tc +
L 2
)
R ) T, β ) L ⁄ 2 or R ) L ⁄ 2, β ) T
(22)
(23)
As R and β are coupling, they are not easy to determine. However, the following procedure can be used to determine the two parameters. As E ) iA + e* and R ) jA + r*, eq 10 can be rewritten as σ ) kA - E - R ) kA - S (24) The nonlinear item in eq 9 can be expressed as follows σ δ ) A(1 - γ) sat. A
()
(25)
where sat.(σ ⁄ A) )
{
sgn(σ), |σ| g A σ ⁄ A, |σ| < A
According to (25), the nonlinear δ is actually a sliding mode control at the kth level (see details in ref 18). The up and low bound of the S at the kth level are (k + 1)A and (k 1)A, respectively. Therefore, the bound on S can be translated into bound on the tracking performance at the kth level. By defining S ) Kee + Kde ) Ke(1 + Rp)e, the tracking error e
y¨ ) a1y˙ + a0y + bu + w
(27)
where w denotes the ummodeled dynamics and first-order Pade´ approximation error of the delay time. It is assumed that |w| e W0 with W0 being a finite positive number. Defining e ) r - y, where r is the reference input and y is the output, and using (11), one has
{
∫ ∫ )
B 1 e+ e dt + A R+β B Rβ e˙ + bK0 βδ + δ - w - fd (28) R+β A where fd ) br + ar˙ - r¨ It is assumed that |fd| e F0 with F0 being a finite positive number. Letting z ) [∫edt e e˙]T, (28) becomes
e¨ ) a1e˙ + a0e - bKeK0(R + β)
}
(
z˙ ) A˜z + B˜[uN - w - fd] where
[
0 1 0 0 1 ˜A ) 0 B B B -bKeK0 a - KeK0(R + β) -bKeK0Rβ A 0 A A B˜ ) [0 0 1 ]T
∫
(29)
]
B uN ) bK0 (βδ + δ dt) A The parameters K0, Ke, R, β, B, and A in the fuzzy PID controller can be chosen so that the nominal system is stable.
8320 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008
This implies that A˜ is a stable matrix and the following Lyapunov equation holds: A˜TP + PA˜ ) -I
(30)
where I is the unit matrix. Theorem 1. If parameters K0, Ke, R, β, B, and A in the fuzzy ˜ PID controller are chosen as in eqs 22 and 26 so that matrix A is stable, the error dynamics (28) driven by controller 11 is globally asymptotically stable. Proof: The proof is divided into two steps. The control error is first proven to be bounded. And, then, it is proven that the control system is globally asymptotically stable. Step 1: Define the Lyapunov function
1 1 1 F0 ) W02 + F02 + u02 η η η The solution of (37) is given as follows: V(t) e
(
) (
)
F0λmax(P) F0λmax(P) F + V(0) t exp F F λmax(P)
(38) For state z, one has |z| e
(
) (
)
F0λmax(P) F0λmax(P) F + V(0) t exp Fλmin(P) Fλmin(P) λmax(P) (39)
Equation 39 implies that V ) zTPz The derivative of V is given by
(31) lim 98 |z(t)| e
T T T N T T V˙ )z (A˜ P + PA˜)z + 2z PB˜u - 2z PB˜w - 2z PB˜fd (32) 2 T ˜ N T ˜ T ˜ )-|z| + 2z PBu - 2z PBw - 2z PBfd
Since |w| e W0 and |fd| e F0, one has 1 1 -2zTPB˜w e ηzTPB˜B˜TPz + w2 e ηzTPB˜B˜TPz + W02 η η 1 1 -2zTPB˜fd e ηzTPB˜B˜TPz + fd2 e ηzTPB˜B˜TPz + F02 (33) η η where η is an arbitrary positive constant. For the nonlinear compensation term δ in uN, an envelope function defined in Appendix I can be used to replace it. The envelope function is shown as follows
tf∞
i.e., control error is bounded. Step 2: A Lyapunov function at the kth layer can be chosen as V(σ) )
∫
{
)
PID
1
1
0
d
0
d
e
0
(43)
∫ sin[ω (e
*
0
}
+ r*)] dt | e u0
where u0 is a finite positive number. Now the term 2z PB˜uN in (32) can be expressed approximately as T
1 1 2zTPB˜uN e ηzTPB˜B˜TPz + (uN)2 e ηzTPB˜B˜TPz + u02 η η
where
Assume that the maximum and minimum eigenvalues of matrix PB˜B˜TP is λmax and λmin, respectively. Then, (32) becomes 1 2 1 2 1 2 2 2 V˙ e-|z| + 3ηλmax|z| + η W0 + η F0 + η u0 (36) 1 1 1 )-(1 - 3ηλmax)|z|2 + W02 + F02 + u02 η η η Let F ) 1 - 3ηλmax and choose η to meet 0 < η < (3ηλmax)-1. From eq 31, one obtains λmin(P)|z|2 e V e λmax(P)|z|2, where λmax(P) and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively. Thus, (36) becomes 1 F 1 1 F V + W02 + F02 + u02 ) V + F0 λmax(P) η η η λmax(P) (37)
∫
B δ dt A According to step 1, the control error is bounded, which means that Ue can compensate for most of the undesirable effects. Thus, one has Ue ) uPID + K0
|
(35)
where
(42)
with σ˙ )Kd
(34)
V˙ e -
(41)
1 V˙(σ) ) σσ˙ Kd
d
(
1 2 σ 2Kd
The derivative of V(σ) is given as
Thus, one has B |u | ) | bK0 βδ + δ dt | A e|0.1bK0B β sin[ω0(e* + r*)] +
(40)
( R1 e˙ + a e˙ + a e - w - f - bU ) 1 σ )K [( e˙ + a e˙ + a e - w - f - bU ) - bK Bβ sat.( )] R A
δ ) 0.1A sin[ω0(e* + r*)]
N
F0λmax(P) Fλmin(P)
|
1 e˙ + a1e˙ + a0e - f(y, y˙) - fd - bUe e ∆F R
(44)
where ∆F is a small positive number. By using the ideal function sgn(σ/A) instead of the approximation sat.(σ/A) in (25), (43) can be expressed as follows: σ˙ ) Kd[∆F - bK0Bβ sgn(σ)] Then 1 V˙(σ) ) σσ˙ e ∆Fσ - bK0Bβ|σ| Kd If the control at every layer can be determined so that bK0Bβ ) ∆F + η0, with η0 > 0
(45)
then the closed-loop system stability is guaranteed by V˙(σ) e -η0|σ|
(46)
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8321 Table 1. Conventional PID Parameters and Performance Indices for Example 1
Table 3. Conventional PID Parameters and Performance Indices for the Process e-0.6s/(s + 1)
Kp
Ti
Td
N
tc
IAE
ITAE
Kp
Ti
Td
N
tc
IAE
ITAE
1.91
1.05
0.91
10
0.5T
0.824
2.910
1.625
1.3
0.231
10
0.5T
1.783
5.717
Table 2. Fuzzy PID Parameters and Performance Indices for Example 1 with tc ) 0.5T
Table 4. Fuzzy PID Parameters and Performance Indices for the Process e-0.6s/(s + 1) with tc ) 0.5T
K0
R
β
IAE
ITAE
K0
R
β
IAE
ITAE
1.82
0.05 1
1 0.05
0.863 1.810
2.955 4.446
1.25
0.3 1
1 0.3
1.491 2.194
4.695 6.20
From (46), the control system is globally asymptotically stable. 5. Control Simulations In this section, the control performance of fuzzy PID tuned by the proposed method is compared with that of conventional PID control. Quantitative criteria for measuring the performance are chosen as IAE and ITAE. Smaller numbers imply better performance. IAE )
∫ |e| dt,
ITAE )
∫ t|e| dt
In all control simulations, parameters of conventional PID control are determined by IMC-based method20,22 and the parameters of fuzzy PID control are determined by the proposed tuning method. Example 1. Consider an industrial process that is approximately described by a first-order rational transfer function model with a delay time as follows: 1 -0.1s e s+1 The linear part is the dominant process. The small delay time implies weak nonlinear features. Parameters of the conventional PID control with control performance indices are shown in Table 1. The corresponding fuzzy PID control parameters and control performance are shown in Table 2. As shown in Figure 5, little difference is observed between the conventional PID control and fuzzy PID control due to the small delay time. However, when the delay time is increased to L ) 0.6, there will be large model error caused by approximating the delay time with a first-order Pade´ approximation in (18). Parameters of the conventional PID control with control performance indices P(s) )
are shown in Table 3. The corresponding fuzzy PID control parameters and control performance are shown in Table 4. As shown in Figure 6, fuzzy PID control achieves better control performance than conventional PID control. Morever, the gain of the fuzzy PID controller is lower than that of conventional PID controller. Example 2. Assume that an industrial process is described by 1 (47) (s + a)8 where a ) 1. Suppose that there is no modeling error in the process (47). On the basis of step response and Nyquist curves of the industrial process (47), the approximation model can be obtained as follows:21 P(s) )
1 e-4.3s 4.3s + 1 Parameters of the conventional PID control with control performance indices are shown in Table 5. The corresponding fuzzy PID control parameters and control performance are shown in Table 6. As shown in Figure 7, little difference is P˜(s) )
Figure 6. Performance of fuzzy PID and PID for delay L ) 0.6, Fuzzy PID (solid line) and conventional PID (dotted line). Table 5. Conventional PID Parameters and Control Performance Indices for the Process 1/(s + 1) 8 Kp
Ti
Td
N
tc
IAE
ITAE
0.465
6.45
1.433
10
0.25T
20.85
1009.8
Table 6. Fuzzy PID Parameters and Performance Indices for the Process 1/(s + 1)8 with tc ) 0.25T
Figure 5. Control performance of fuzzy PID and PID for example 1, fuzzy PID (solid line), and conventional PID (dotted line).
K0
R
β
IAE
ITAE
0.0721
2.15 4.3
4.3 2.15
22.52 24.62
1009.4 1035.2
8322 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008
Figure 9. Numerical simulation of nonlinear compensation term δ.
Figure 7. Control performance of fuzzy PID and PID for a ) 1. Fuzzy PID (solid line) and conventional PID (dotted line).
scaling gains R and β are coupled, a procedure is used to decouple them on the basis of the sliding mode control. The stability analysis shows that the control system is globally asymptotically stable. Fuzzy PID controllers tuned by the proposed method are more robust than the conventional PID controller. The simulation results show that fuzzy PID controllers tuned by the proposed method achieve better control performance in both the transient and steady states and are more robust than conventional PID controllers. Acknowledgment We would like to thank the anonymous reviewers for their useful comments. The work is partially supported by a grant from NSF China (50775224), a grant from RGC Hong Kong SAR (CityU: 116406), and a grant from the National Basic Research Program of China (2006CB705400). Appendix 1
Figure 8. Control performance of fuzzy PID and PID for process a ) 0.95. Fuzzy PID (solid line) and conventional PID (dotted line). Figure A1. Numerical simulation of nonlinear compensation term δ.
The nonlinear control term δ can be simulated by numerical simulation, which is shown in Figure 9. From figure A1, the maximum and minimum values of δ occur when e* equals r*. As dδ/de* ) 0 and dδ/dr* ) 0, the maximum and minimum values of δ can be obtained as follows min(δ) )
Table 7. Parameters of fuzzy PID and PID controller with a ) 0.95 PID fuzzy PID
IAE/ITAE 20.49/1159.6 IAE/ITAE: 18.14/878.2
observed between the conventional PID control and fuzzy PID control because the model is accurate. However, suppose that there is modeling error and the practical value of the parameter a is 0.95. The controller parameters of fuzzy PID and conventional PID are unchanged and shown in Tables 5 and 6. As shown in Figure 8 and Table 7, fuzzy PID control achieves better control performance than conventional PID control. Morever, the gain of the fuzzy PID controller is lower than that of the conventional PID controller, which is shown in Figure 8. 6. Conclusion An effective tuning method for fuzzy PID controllers based on IMC is presented in this paper. An analytical model is first developed for the tuning of fuzzy PID controllers. The analytical model includes a linear PID control and a nonlinear compensation item. On the basis of the IMC method, the parameters of fuzzy PID controller can be analytically determined by regarding the compensation item as a process disturbance. Although the
3√13 - 11 A ≈ -0.09167A 2
(A1)
with e* ) r* ) max(δ) )
5 - √13 A 2
11 - 3√13 A ≈ 0.09167A 2
(A2)
with
√13 - 3 A 2 Therefore, an envelope function can be used to envelop the nonlinear control term. The envelope function is defined as follows. e* ) r* )
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ReceiVed for reView March 27, 2008 ReVised manuscript receiVed July 17, 2008 Accepted July 28, 2008 IE800485J
