A Saturation-Based Tuning Method for Fuzzy PID ... - IEEE Xplore

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Xiao-Gang Duan, Member, IEEE, Hua Deng, Member, IEEE, and Han-Xiong Li, ... X.-G. Duan is with the State Key Laboratory of High-Performance Complex.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 11, NOVEMBER 2013

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A Saturation-Based Tuning Method for Fuzzy PID Controller Xiao-Gang Duan, Member, IEEE, Hua Deng, Member, IEEE, and Han-Xiong Li, Fellow, IEEE

Abstract—In this paper, a saturation-based tuning method for fuzzy proportional–integral–derivative (PID) controller is proposed. The key feature is that this tuning method adopts an inherent saturation property resulting from finite rules in practical application. Based on the saturation, fuzzy PID controller can be expressed as a sliding-mode controller that has two nonlinear terms: One plays as an equivalent control, and the other acts as a switching control. A nominal tuning is first presented to design a stable equivalent control using gain margin and phase margin. Then, a robust tuning is presented to design the switching control by using maximum sensitivity function or compensation sensitivity function. The maximum bound of uncertainty is given by the robust analysis. Finally, this proposed tuning method is used to control a clamp rotation of a forging manipulator. The simulations and real-time experiments demonstrate the effectiveness of the proposed tuning method. Index Terms—Fuzzy proportional–integral–derivative (PID) controller, saturation, tuning.

I. I NTRODUCTION

G

ENERALLY speaking, conventional proportional– integral–derivative (PID) controllers may not perform well for the complex process, such as the high-order and time delay systems, nonlinear systems, complex and vague systems without precise mathematical models, and systems with uncertainties. It has been found that fuzzy-logic-based PID controllers have better capabilities of handling the aforesaid systems [1]–[6]. There are too many variations of fuzzy PID controllers, 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 much information on the derivative action, and the three-input fuzzy PID controller may cause exponential growth of rules. The two-input fuzzy PID controller, as we used

Manuscript received April 16, 2012; revised July 11, 2012; accepted September 4, 2012. Date of publication October 4, 2012; date of current version June 6, 2013. This work was supported in part by the National Basic Research Program 973 of China under Grants 2011CB013302 and 2011CB013104, by Research Grant Council of Hong Kong under General Research Fund Grant CityU117310, by National Science Foundation of China under Grant 51175519, and by the Postdoctoral Foundation of Central South University. X.-G. Duan is with the State Key Laboratory of High-Performance Complex Manufacturing and the School of Mechanical and Electrical Engineering, Central South University, Changsha 410078, China (e-mail: [email protected]). H. Deng is with the School of Mechanical and Electrical Engineering, Central South University, Changsha 410078, China (e-mail: [email protected]). H.-X. Li is with the Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2012.2222858

in this paper, has a proper structure and the most practical use and, thus, is the most popular type of fuzzy PID used in various research works and applications [7]–[9]. Despite the fact that industry shows greater and greater interest in the applications of fuzzy PID controller, it is still a highly controversial topic from the point of view of mainstream control engineering community. One of the key reasons is that the fundamental theory for tuning methods of fuzzy PID controller is still lacking [10]. The difficulty in using fuzzy logic controller (FLC) is to properly design membership function (MF) and rule base. The most popular approach used before is to select them based on the expert’s experience. A linguistic variable can mean different things to different people [11], resulting in different interpretation for MFs and rules, and thus, different control performance may be produced [12]. To meet the challenge, a scaling gain and a phase-plane principle are introduced to design MF and rule base [13], [14], respectively. The scaling gain and the phaseplane principle may reduce the design complexity of fuzzy controllers. By adjusting the scaling gain, a fuzzy set universe of discourse is changed, which causes that the MF and the knowledge base are also changed correspondingly. It implies that the challenge of fuzzy controllers can be transformed to design their scaling gains, as we used in this paper. Based on the phase-plane principle, analytical models of fuzzy PID controllers can be obtained through mathematical derivation. Traditional control theory can be used to guide the design of scaling gain. Generally, the fuzzy PID controller can be tuned qualitatively by two-level tuning [14]–[17], i.e., the scaling gains and the knowledge base are tuned simultaneously. However, it is difficult to tune the knowledge base parameters for industrial applications. Moreover, the knowledge base parameters can be adjusted by tuning their scaling gains. Thus, it is preferred to leave the design and tuning exercises to scaling gains. The tuning methods of scaling gains are mainly classified as two kinds, a nonanalytical tuning method [18]–[24] and a PID-based tuning method [25]–[28]. By comparison with the nonanalytical tuning method, the PID-based tuning method is easy to use for industrial processes because it can borrow many matured PID tuning rules. However, not all PID tuning methods can be used for fuzzy PID controllers. Moreover, the PIDbased tuning methods neglect the inherent property of fuzzy PID controller, which may produce a poor control performance. Recently, a saturation of the FLC is revealed due to the finite fuzzy rules used [29]. Compared with the conventional PID controller, the fuzzy PID controller has an inherent capability to suppress impulse signal and integral windup without additional

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Fig. 1. SISO feedback control system.

Fig. 2. Structure of the fuzzy PID controller, with Kd = αKe and K1 = βK0 .

filter or structure modification. Parameters of the fuzzy PID controller can be designed based on the saturation. In this paper, a saturation-based tuning method is proposed for fuzzy PID controllers. Based on the inherent saturation, a fuzzy PID controller can be expressed as one kind of slidingmode controller, which has two nonlinear terms: One plays as an equivalent control, and the other acts as a switching control. Thus, a tuning method is proposed to have a nominal tuning for achieving the equivalent control and a robust tuning for achieving the switching control based on the saturation. In the nominal tuning, the equivalent control can be designed by using gain margin and phase margin. In the robust tuning, the robust parameter can be obtained by using the sensitivity function or compensation sensitivity function. The maximum bound of uncertainty is discussed in the robust analysis. Finally, the effectiveness of the tuning methodology is demonstrated by numerical simulations and real-time experiments on a forging manipulator. II. P ROBLEM F ORMULATION Consider a single-input–single-output (SISO) feedback control system, as shown in Fig. 1, where P is a plant, C is a fuzzy PID controller, u is the control action, r and d are the reference input and the disturbance, respectively, and y is the output of the plant. Many industrial processes can be modeled by a first-order plus delay time (FOPDT). The FOPDT model is one of the most common and adequate ones used, particularly in the process control industries [30]. It is used in this paper for simplicity. The transfer function of the FOPDT model is given by P(s) =

K e−Ls Ts + 1

(1)

where P denotes a nominal model of the plant and K, T , and L are the steady-state gain, the time constant, and the time delay, respectively. A. PID Controller The ideal PID controller is described by the following time model [30], [31]:    1 (2) edt + Td e˙ UPID = KP e + Ti where UPID is the control signal acting on the error signal e, KP is the proportional gain, and Ti and Td are integral time constant and derivative time constant, respectively.

B. Fuzzy PID Controller The detail design of a fuzzy PID controller was discussed in [28] and [32]. Here, only the structure of the fuzzy PID controller is shown in Fig. 2. Only one 2-D rule base is used, which is normally chosen as a linear rule base [32]. The model of the fuzzy PID controller is given by [28], [32] U F z = uL + uN with

 u L = Kf u N = K0

e+

1 α+β

 e+

(3)

αβ e˙ α+β



 B β(1 − γ)sat(kA − σ) A   + (1 − γ)sat(kA − σ)dt

(4)

(5)

˙ uL is a PID-type controller, where σ = E + R = Ke e + Kd e, Kf = Ke K0 (α + β)B/A is a proportional gain, (α + β) and αβ/(α + β) are integral time constant and derivative time constant, respectively; uN is a nonlinear item; E = iA + e∗ , R = jA + r∗ , and k = i + j + 1; A and B represent half of the spread of input and out member function, respectively; e∗ and r∗ are relative input data in the inference cell IC(i, j); γ is a nonlinear parameter whose value depends on input data in the inference cell IC(i, j); and k is an index number depending on the inference cell under use. The larger the integer, the farther away the state (e, e) ˙ is from the origin in the phase plane (see details in [32]). The basic principle of the PID-based tuning methods is to set uL = UPID [25]–[28], which yields  Kf = KP α, β = 0.5Ti (1 ± 1 − 4Td /Ti ). (6) Ti ≥ 4Td is required to guarantee that (6) has a real solution. However, not all PID tuning methods can meet this condition. Moreover, this PID-based tuning method may not have desired performance because it neglects the nonlinear item uN . III. T UNING OF F UZZY PID C ONTROLLER Rules of the fuzzy PID controller are always finite in realworld application. Finite rules imply a saturated output u , as shown in Fig. 2. The output model of the rule base becomes (for details, see the Appendix or [29]) sgn(σ), |σ| > 1 u  = sat(σ) = (7) g(σ), |σ| ≤ 1

DUAN et al.: SATURATION-BASED TUNING METHOD FOR FUZZY PID CONTROLLER

with g(σ) = σ + (1 − γ)(kA − σ)

(8)

where Ke = 1/| max(e)|, Kd = αKe , and K1 = βK0 . Thus, the mathematical model of the fuzzy PID controller can be easily derived from Fig. 2 as    = K0 sat(σ) dt + K1 sat(σ) (9)  dt + K1 u U F z = K0 u where u  is given in (7). Equation (9) can be rewritten as UF z = ue + K1 sat(σ)

(10)

with  u e = K0

sat(σ) dt

(11)

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on the basic definitions of gain margin and phase margin [30], [33], the following equations are obtained:

φm = arg Gf (jωg )P (jωg ) + π (16) (17) Gf (jωg )P(jωg ) = 1 1 Am = Gf (jωp )P (jωp )

arg Gf (jωp )P (jωp ) = −π

A. Nominal Tuning for Equivalent Control By substituting (7) into (11), one can obtain e + Δu ue = u

(12)

where  u e = αK0 Ke  Δu = K0

1 e+ α



 e dt

sat(σ) dt − αK0 Ke

(13) 

1 e+ α



 e dt . (14)

Equation (13) is a proportional integral (PI)-type controller, where αK0 Ke is the proportional gain and α is the integral time constant. The transfer function of (13) is expressed by   Ue (s) 1 = αK0 Ke 1 + Gf (s) = (15) E(s) αs e and e, where Ue (s) and E(s) are Laplace transformations of u respectively. The controller C and the plant P , as shown in Fig. 1, are replaced by Gf and the nominal model P, respectively. Based

(19)

where Am ∈ (0, ∞) and φm ∈ (0, π/2) denote gain margin and phase margin, respectively, the frequency ωp ∈ (0, ∞) at which the Nyquist curve has a phase of −π is known in classical control terminology as phase crossover frequency, and the frequency ωg ∈ (0, ∞) at which the Nyquist curve has an amplitude of one is known as the gain crossover frequency. Substituting (1) and (15) into (16)–(19) yields π + arctan(ωg α) − arctan(ωg T ) − ωg L 2

1 + ωg2 T 2 ωg K0 = Ke K 1 + ωg2 α2

1 + ωp2 T 2 ωp K0 = Am Ke K 1 + ωp2 α2 φm =

where UF z can be considered as a sliding-mode controller, ue is an equivalent control to compensate the known dynamics, and K1 sat(σ) is a switching control to compensate unmodeled dynamics. Based on (10), a tuning strategy is proposed to design the equivalent control and the switching control, respectively. A nominal tuning is first presented to design a stable equivalent control using gain margin and phase margin. Then, a robust tuning is presented to design the switching control, which can guarantee certain robustness by using sensitivity function or compensation sensitivity function.

(18)

π + arctan(ωp α) − arctan(ωp T ) − ωp L = 0. 2

(20) (21)

(22) (23)

In order to solve (20) and (23) analytically, we use the following function to approximate the “arctan” function [33] π x, 0 1. 2 4x For ωp in (22), we consider the following four cases. Case 1) (ωp α ≤ 1 and ωp T > 1): There is   π π πωp α π + − − − ωp L = 0. 2 4 2 4ωp T Solving this equation, one can obtain  π . ωp = 4T L − πT α

(25)

(26)

Case 2) (ωp α ≤ 1 and ωp T < 1): There is π πωp α πωp T + − − ωp L = 0. 2 4 4

(27)

Solving this equation, one can obtain ωp =

2π . 4L + π(T − α)

Case 3) (ωp α > 1 and ωp T > 1): There is   π π π π π + − − − − ωp L = 0. 2 2 4ωp α 2 4ωp T

(28)

(29)

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It yields ωp =



π 1± 4L

 1−

4L(T − α) πT α

B. Robust Tuning for Switching Control

 .

(30)

π π π πωp T + − − − ωp L = 0. 2 2 4ωp α 4

(31)

Case 4) (ωp α > 1 and ωp T < 1): There is

It yields

   2π πT + 4L ωp = 1± 1− . πT + 4L 4πα

(32)

Equations (25)–(32) give six possible approximate solutions for ωp of which one is valid. The valid solution can be obtained by substituting each value of ωp into the residue function π εp = + arctan(ωp α) − arctan(ωp T ) − ωp L (33) 2 and choosing the value that gives the smallest |εp |. Similarly, the value of ωg can be obtained from (20). We consider the following cases. Case 1) (ωg α ≤ 1 and ωg T > 1):

  π(πα − 4L) 2φm ωg = 1± 1− . (34) π(α − L) 4T φ2m Case 2) (ωg α ≤ 1 and ωg T ≤ 1): ωg =

2π − 4φm . 4L + π(T − α)

Case 3) (ωg α > 1 and ωg T > 1):

  π − 2φm 4πL(T − α) ωg = 1± 1− . 4L T α(π − 2φm )2 Case 4) (ωg α > 1 and ωg T ≤ 1):

  π(πT + 4L) 2(π − φm ) ωg = 1± 1− . πT + 4L 4α(π − φm )2

(35)

(36)

Similarly, by substituting (7) into (9) and neglecting the error item, one obtains    (39) UF z ≈ K0 Ke (α + β)e + edt + αβ e˙ which can be further expressed by the following transfer function:   F z (s) U 1 ≈ K0 Ke (α + β) + + αβs (40) GF z (s) = E(s) s F z (s) and E(s) are Laplace transformations of UF z where U and e, respectively. Using GF z (s) and replacing controller C in Fig. 1, a sensitivity function or a compensation sensitivity function is used to determine the β value. The sensitivity function and the compensation sensitivity function are given by 1 (41) MS = max |S(jω)| = max 0≤ω