Proportional-integral based fuzzy sliding mode control of the milling head

Control Engineering Practice 53 (2016) 1–13

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Proportional-integral based fuzzy sliding mode control of the milling head Pengbing Zhao a,n, Yaoyao Shi b, Jin Huang a a

The Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi’an, Shaanxi 710071, China The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology Ministry of Education, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 24 November 2015 Received in revised form 19 April 2016 Accepted 19 April 2016

A-axis (that is, the milling head) is an essential assembly in the five-axis CNC machine tools, positioning precision of which directly affects the machining accuracy and surface qualities of the processed parts. Considering the influence of nonlinear friction and uncertain cutting force on the control precision of the A-axis, a novel fuzzy sliding mode control (FSMC) based on the proportional-integral (PI) control is designed according to the parameters adaptation. Main idea of the control scheme is employing the fuzzy systems to approximate the unknown nonlinear functions and adopting the PI control to eliminate the input chattering. Simulation analyses and experimental results illustrate that the designed control strategy is robust to the uncertain load and the parameters perturbation. & 2016 Elsevier Ltd. All rights reserved.

Keywords: A-axis Positioning control Parameters adaptation Sliding mode control Proportional-integral control

1. Introduction The five-axis CNC machine tools are widely applied in manufacturing the parts of aeronautics and astronautics, the turbine wheels and some special molds which typically have the complicated geometries (Harik, Gong, & Bernard, 2013; Zhou, Chen, & Yang, 2015). As the essential component in five-axis CNC machine tools, the A-axis (that is, the milling head) has the characteristics of structural complexity, components diversity, transmission compactness and weak stiffness (Pengbing & Yaoyao, 2014). Research of the A-axis focuses on how to improve the tracking and positioning precision, increase the drive torque and enhance the system stiffness (Bi et al., 2015). Recently, all aspects of the system performance have been improved, but there are almost no cases that can take all the indicators into consideration, therefore, it is important to improve the positioning precision while enhance the driving torque of the existing A-axis (Zhao & Shi, 2014). The A-axis researched in this paper is mainly based on the worm gear transmission. Compared with the rolling engagement of the ordinary gear transmission, engagement of the worm gear is pure sliding. Thus, sliding friction influences significantly on the positioning precision of the A-axis, generally, friction is considered to be a kind of interference for the control system. In addition, the uncertain cutting load during the machining process impacts the n

Corresponding author. E-mail address: [email protected] (P. Zhao).

http://dx.doi.org/10.1016/j.conengprac.2016.04.012 0967-0661/& 2016 Elsevier Ltd. All rights reserved.

tracking precision, thus, surface qualities of the machined parts will be deteriorated. To maintain the perfect dynamic performance of the A-axis during the milling process, a controller that is robust to the nonlinear friction and external disturbance needs to be constructed. The sliding mode control (SMC) has a complete self-adaptability to the uncertainties and external disturbances (Utkin, Guldner, Shi, & Mode, 2009), however, the input chattering of the SMC will undermine the control precision, increase the energy consumption, stimulate the unmodeled dynamics, deteriorate the system function and even damage the controller (Boiko, 2013; ElSousy, 2013; Shtessel, Edwards, & Fridman, 2014). Therefore, the most important problem in designing SMC is how to eliminate the chattering, and the commonly used methods are the quasi-SMC and the approaching laws (Chakrabarty & Bandyopadhyay, 2015; Huang, Liao, Chen, & Yan, 2012; Wang, Jia, & Dong, 2013; Zhao, Qiao, & Wu, 2013). However, these methods can only reduce the input chattering to a certain extent, to further eliminate the chattering, many investigators introduce the artificial intelligence into the SMC. A fuzzy sliding mode control (FSMC) algorithm based on the boundary layer is presented in Saghafinia, Ping, and Uddin (2014), in which a fuzzy system is used to eliminate the chattering in spite of the system uncertainties. A SMC based on the adaptive neural network is designed in Zou and Lei (2015) to treat the model uncertainties and external disturbances of a stable inertial platform. A backstepping SMC is proposed in Dong and Tang (2014) to inhibit the vibration of a flexible ball screw drives induced by the time-varying parametric uncertainties and external

2

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

approximated by the fuzzy systems. By replacing the switching part of the traditional sliding mode control (TSMC) with the PI controller, input chattering of the TSMC is eliminated. Stability and convergence of the control system can be guaranteed by the Lyapunov theory and Barbalat Lemma. Simulation analyses and experimental verifications illustrate that the designed control algorithm is robust to the uncertain external load and the system parameters perturbation. Compared with the conventional PID control, the designed FSMC can undermine the topping phenomenon and the deadzone in the displacement and speed tracking respectively, and compared with the TSMC, the FSMC based on the PI control not only can improve the positioning and tracking precision, but also can reduce the input chattering. The rest of the paper is organized as follows: mechanical structure, control system and mathematical model of the A-axis are constructed in Section 2. In Section 3, basic principles of the FSMC based on PI control is elaborated. Simulation and experimental verification are carried out in Section 4. In Section 5, we draw the conclusions.

disturbances. A novel adaptive backstepping SMC with fuzzy monitoring strategy is proposed in Song and Sun (2014) for the tracking control of the nonlinear mechanical system. An adaptive SMC with the recurrent radial basis function network (RRBFN) is proposed in El-Sousy (2013) for the indirect field orientation control of an induction motor and the RRBFN is employed as an uncertainty observer. In Hwang, Chiang, and Yeh (2014), trajectory tracking of the under-actuated nonlinear dynamic system with uncertainty is tackled by an adaptive fuzzy hierarchical SMC. An intelligent second-order SMC using a wavelet fuzzy neural network with an asymmetric membership function estimator is proposed in Lin, Hung, and Ruan (2014) to control a six-phase permanent magnet synchronous motor for an electric power steering system. An adaptive controller using a fuzzy compensator for MEMS triaxial gyroscope is presented in Fei and Zhou (2012), which can well compensate the model uncertainties and the external disturbances. An adaptive controller of MEMS gyroscope using global fast terminal SMC and fuzzy neural network is presented in Fei and Yan (2014). A robust adaptive SMC using the RBF neural network for a class of time varying system in the presence of model uncertainties and external disturbance is proposed in Fei and Ding (2012). As for the fuzzy system and the fuzzy control, an observerbased adaptive controller using a simplified type-II fuzzy neural network and a three dimensional type-II membership function is investigated in Mohammadzadeh and Hashemzadeh (2015). A novel fuzzy system based strategy for modeling both rate-independent and rate-dependent hysteresis in the piezoelectric actuator is proposed in Li, Yan, and Ge (2013). An adaptive fuzzy backstepping control and H1 performance analysis for a class of nonlinear systems with sampled and delayed measurements are investigated in Wang, Zhang, Qiu, and Gao (2015). A novel composite control based on fuzzy system and disturbance observer is proposed in Xu, Shi, and Yang (2015) for the uncertain nonlinear systems with actuator saturation and external disturbances. In Tong, Sui, and Li (2015), a tracking error constrained fuzzy output-feedback dynamic surface control scheme is proposed for a class of uncertain MIMO nonlinear systems. H1 controllers of interval type-II T–S fuzzy systems via dynamic output feedback control are designed in Zhao, Xiao, Sheng, and Wang (2015). In Wang, Wang, and Chai (2009), an adaptive fuzzy system is employed to approximate the nonlinear friction and estimation of the friction is applied in PD control to enhance the control performance. Considering the nonlinear dynamics and the uncertain cutting load during the machining process of the A-axis, a novel FSMC based on PI control is proposed. Parameters uncertainty of the Aaxis and external disturbance during the milling process is

2. Mechanical structure, control system and mathematical model of the A-axis The A-axis with high-power, high-torque and high stiffness used for processing the titanium alloy, superalloy and other difficult to machine materials is illustrated in Fig. 1. Backlash of the drive mechanism can be adjusted by the dual lead worm gear, and with a high-resolution encoder mounted on the right side of the shaft, the A-axis can be controlled in a closed-loop system. The hydraulic locking mechanism installed on the left side of the shaft can make the motorized spindle locked at any position within the angle range [  120°, 30°] (Pengbing & Yaoyao, 2014; Zhao & Shi, 2014). Principle of the A-axis control system is shown in Fig. 2, and parameters of the AC servo motor, the motorized spindle and the angular encoder are illustrated in Tables 1–3 respectively. As part of the control system, an industrial computer (ADVANTECH IPC-610-H) is used as the host computer which is utilized to edit and execute the control programs, achieve human– computer interaction and control various motions of the A-axis via the motion controller. PMAC2A PC-104 is adopted as the slave computer which is used to drive the AC motor to complete the desired motions and monitor the change of each state variable. Edit the control programs with the C þ þ language, which includes the traditional PID control, the TSMC and the FSMC, conversion programs of the A/D and D/A, the sampling program and

1 2

5 8 6

3

7

4

8

9 10

Fig. 1. Mechanical structure of the A-axis. 1- Support; 2-Left shield; 3-Spindle box; 4-Motorized spindle; 5-Servomotor; 6-Right shield; 7-Lockingmechanism; 8-Pulleys; 9-Wormgear; 10-Encoder.

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

3

ACServomotor Powersupply Drivingsystem

Motorized spindle

Servodriver

Worm

Gear Measurement system

30° 120°

Angularencoder

IPC

Control system

Fig. 2. Control system of the A-axis.

Table 1 Parameters of the AC servo motor (1FT7105-1AC71-1NG1). Rated speed Rated power

Static torque

Rated torque

Rated current

Inertia

2000 rpm

50 Nm

38 Nm

15 A

1.78  10  2 kg m2

7.96 kW

Fig. 4. Experimental equipment of the A-axis. Table 2 Parameters of the motorized spindle (RT-TM300-29/5000-G-B-E). Power

Rated torque

Rated speed

Maximum speed

Rated current

29.3 kW

200 Nm

1400 r/min

6500 r/min

168 A

Table 3 Parameters of the angular encoder (HEIDENHAIN RCN228). Number of lines

System precision

Measuring step

Allowed speed

16,384

7 2.5″

0.0001°

r 3000 r/min

automatically downloaded from the PMAC and the user-defined algorithm will become the firmware program of PMAC. The data acquisition, conversion, and output will be implemented by the data acquisition card (ADVANTECH PCL818L), the isolated digital card (PCL730) and the analog output card (PCL726). In the experiments, the metal block with mass m is placed at the end of the motorized spindle to simulate the cutting load, that is τ^l ¼mgd cos θg, where, m ¼20 kg, d¼ 0.5 m, θg∈[  π/6, 2π/3] and the gravitational acceleration g ¼9.80665 m/s2, as shown in Fig. 3, the designed experimental equipment of the A-axis control system is illustrated in Fig. 4. The mathematical model of the A-axis is represented as follows (Zhao & Shi, 2014):

⎧ (J i wg + CJ ) θ¨g = τm ib + Cτ l w g ⎪ ⎪ ⎨ Jw = Jw0 + Jb2 + (Jm + Je + Jb1 ) ib2 ⎪ ⎪ ⎩ Jg = Jg 0 + Js + Jc + Jf

(1)

where Jw0, Jm, Je, Jb1 and Jb2 are the inertias of the worm, the motor, the expansion sleeve, the small pulley and the big pulley respectively. Jg0, Js, Jc and Jf are the inertias of the gear, the motorized spindle, the support shaft and the fixed portion of the braking mechanism respectively. Jw and Jg are the equivalent inertias of the worm and the gear respectively. τm is the motor torque and τl is the load torque. ib is the transmission ratio of the timing belt. iwg is the transmission ratio between the worm and the gear. Parameter C in (1) can be determined based on the following conditions.

Fig. 3. Schematic of the experimental program.

measurement program of the angle encoder. PEWIN32 PRO is used to compile the edited program, and the compiled machine code is downloaded into RAM of the PMAC user program, then, use the command SAVE to copy the program to the flash memory. When the system is power on or reset, the control programs will be

(1) (2) (3) (4)

If If If If

(dθg/dt)4 0, (dθg/dt)4 0, (dθg/dt)o 0, (dθg/dt)o 0,

τmib r τl(Jwiwg)/Jg, then C ¼C1. τmib 4 τl(Jwiwg)/Jg, then C ¼C2. τmib r τl(Jwiwg)/Jg, then C ¼C2. τmib 4 τl(Jwiwg)/Jg, then C ¼C1.

rw (cos α sin λ +μs cos λ) rw (cos α sin λ −μs cos λ) , C2 = , α is the where C1 = rg (cos α cos λ −μs sin λ) rg (cos α cos λ +μs sin λ) pressure angle, λ is the lead angle, θg is the rotation angle of the gear, μs is the coefficient of the static friction, rw and rg are the pitch circle radius of the worm and the gear respectively, and

4

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

Table 4 Parameters of the A-axis. Parameter (kg m2)

Value

Parameter

Value

Jw0 Jb2 Jm Je Jb1 Jg0 Js Jc

97.29  10  4 194.34  10  4 178  10  4 8.79  10  4 11.46  10  4 790.80  10  4 6.60 5253.68  10  4

Jf (kg m2) ib iwg α (deg) λ (deg) rw (m) rg (m) μs

1890  10  4 2 90 20 5 0.04 0.18 0.15 Fig. 5. Structure of the FSMC based on PI control.

system parameters of the A-axis are illustrated in Table 4. To design the SMC, the control input u = τb − Jg−1 τ^l (Jw i wg ) is introduced, τ^l ¼mgd cos θg is the estimation of load torque. When

1

(dθg/dt)40, the following dynamics equations can be derived according to (1):

where

τ˜l = τl − τ^l ,

g1 = (Jw i wg + C1Jg )−1,

(2) g2 = (Jw i wg + C2 Jg )−1,

d1 = d2 = Jg−1, e1 = C1τ˜l (Jw i wg + C1Jg )−1, e2 = C2 τ˜l (Jw i wg + C2 Jg )−1. When (dθg/dt)o0, the similar dynamics equations can be derived.

3. Design of the FSMC based on PI control

0.6 0.5 0.4 0.3 0.2

-0.6

-0.4

-0.2 x i (i=1,2);

(3)

where, f(x,t) and g(x,t) are the unknown but bounded nonlinear functions, x∈Rn is the measurable variables, u∈R is the control input, y∈R is the system output, d(t) is the bounded disturbance that cannot be measured, that is |d(t)| rD o1. The system is controllable and the control gain g(x,t)≠0, without losing generality, suppose g(x,t) 40. Control objective of the nonlinear system is to design an appropriate control law, by which the state vector can track a desired T state trajectory x d = ⎡⎣xd , xd , … , xd(n − 1) ⎤⎦ , and for M40, ||xd|| oM. Define e¼x xd is the tracking error, then, the error vector is

0

0.2

0.4

0.6

0.8

y l (l=1,2,…,m)

Fig. 6. Membership functions of the fuzzy input and output variables.

⎡ n− 1 ⎤ u = g −1 (x, t ) ⎢ − ∑ ci e (i) − f (x, t ) + xd(n) − k sgn (s ) ⎥ ⎢⎣ i = 1 ⎥⎦

(7)

where, the switching gain k4 0 is a constant. Define the Lyapunov function as V¼ s2/2, then

V1̇ = ss ̇ = s (c1e ̇ + c2 e¨ + ⋯ + cn − 1e (n − 1) + x (n) − xd(n) ) ⎛ n− 1 ⎞ = s ⎜⎜ ∑ ci e (i) + f (x, t ) + g (x, t ) u (t ) + d (t ) − xd(n) ⎟⎟ ≤ − η s ⎝ i=1 ⎠

(8)

(4)

Define the sliding surface in error state space is as follows:

s = c1e + c2 e + ⋯ + cn − 1e (n − 2) + e (n − 1) = cTe

0.7

0 -0.8

Consider the following n-order SISO nonlinear system

T e = x − x d = ⎡⎣ e , e , …, e (n − 1) ⎤⎦

0.8

0.1

3.1. Traditional sliding mode control

⎧ x (n) = f (x, t ) + g (x, t ) u (t ) + d (t ) ⎨ ⎩y=x

Degree of input membership function

⎧ θ¨ = g u + d τ^ + e , u < J−1 τ˜ (J i ) 1 l 1 l w wg ⎪ g 1 g ⎨ 1 − ¨ ^ ⎪ ⎩ θg = g2 u + d2 τ l + e2 , u > Jg τ˜l (Jw i wg )

0.9

(5)

Generally, the control law (7) can be written as u ¼ueq  usw, where

where c ¼[c1, c2,…,cn  1, 1] is the coefficient vector of Hurwitz polynomial h(λ) ¼c1 þ⋯ þcn  1λ(n  2) þ λ(n  1). For the initial condition e(0)¼0, the tracking control problem (x ¼xd) is considered to make the error vector slide on s ¼0. The sufficient condition that can guarantee the error trajectory to reach the sliding surface is to design an appropriate u to satisfy the condition (6).

⎡ n− 1 ⎤ ueq = g −1 (x, t ) ⎢ − ∑ ci e (i) − f (x, t ) + xd(n) ⎥ ⎢⎣ i = 1 ⎥⎦

1d 2 s ≤ − ηs, η > 0 2 dt

where ueq is the equivalent control and usw is the switching control. In the actual system, f(x,t) and g(x,t) are usually unknown, therefore, it is difficult to apply the control law (7) to the nonlinear system. Furthermore, the switching control usw can cause the input chattering. To solve these problems, a novel FSMC is constructed by introducing the fuzzy systems and the PI controller.

T

(6)

The sliding motion includes the reaching phase (s≠0) and the sliding phase (s ¼0), and the SMC can guarantee the system convergence, that is, when t-1, e(t)-0. For system (3), the control law (7) can satisfy the condition (6) if f(x,t) and g(x,t) are known in advance.

usw = g −1 (x, t ) k sgn (s )

(9)

(10)

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

3.2. Basic fuzzy system

μ A l (xi ) is the membership function of xi. i

Suppose the fuzzy system is composed by the following fuzzy rules with IF-THEN form (Roopaei, Zolghadri, & Meshksar, 2009)

R (l): IF x1 is A1l and⋯ and xn is A nl , THEN y is Bl

(11)

where Ail and Bl are the fuzzy sets within the universe Ui ⊂R and V ⊂R, x¼ (x1, x2,…, xn)T∈U and y∈V are the input and output variables respectively. Suppose m is the total number of fuzzy rules, that is, l ¼1, 2,…,m. Actually, there are numerous fuzzy inference engines and fuzzy logic operators, and they can constitute a large number of fuzzy systems. As far as the control application is concerned, the most commonly used fuzzy system is employed here. Suppose that Bl in (11) is normal and its center is y¯ l , and the fuzzy system with fuzzy rules base (11), product inference engine, singleton fuzzifier and center average defuzzifier can be expressed as follows: (Cerman & Hušek, 2012; Wang, 1997)

(

m ∑l = 1 y¯ l ∏in= 1 μ A l (xi ) m ∑l = 1

(

i

n ∏i = 1

μ A l (xi ) i

)

)

(12)

where x∈U⊂R and y(x)∈V⊂R are the system input and output, and

Angle displacement (rad)

y (x ) = θTξ (x )

(13)

where θ ¼(y ,…, y ) is the parameter vector, and ξ(x)¼(ξ (x),… ,ξm(x))T is a regressive vector with the regressor defined as 1

m T

1

n

ξ i (x) =

∏i = 1 μ A l (xi ) i

m

(

∑l = 1 ∏in= 1 μ A l (xi ) i

)

. (14)

3.3. FSMC based on PI control For the TSMC, only when f(x,t) and g(x,t) are known, it can be available, however, f(x,t) and g(x,t) are generally unknown in practical control problems. Thus, the fuzzy rules (11) and the fuzzy ^ systems (13) and (14) are adopted, fuzzy system output f (x θf ) and g^ (x θ ) are used to approximate the f(x,t) and g(x,t), and the PI g

n

control is used to reduce the input chattering and improve the steady state performance. The continuous PI controller can be expressed as

30

1 Desired Response

Output Response

0.5 20 0 -0.5 -1

10 0

1

2

3

4

5 Time(s)

6

7

8

9

10

x 10

Tracking error (rad)

To simplify the expression of (12), introduce the fuzzy basis function and (12) can be written as

1

u (Nm)

y (x) =

5

0

-10

0 -20 -1 0

1

2

3

4

5

6

7

8

9

Time(s)

(a) Tracking response and control error

10

-30

0

1

2

3

4

5

6

7

Time(s)

(b) Control input

Fig. 7. TSMC with the sign function (simulation). (a) Tracking response and control error. (b) Control input.

Fig. 8. TSMC with the saturation function (simulation). (a) Tracking response and control error. (b) Control input.

8

9

10

6

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

Fig. 9. FSMC with the PI control (simulation). (a) Tracking response and control error. (b) Control input. (c) Adaptive process of θp1. (d) Adaptive process of θp2.

Fig. 10. Traditional PID control (experiment). (a) Step response and control error. (b) Control input.

up = θ p1 z1 + θ p2 z2

(15)

where up is the controller output, z1 and z2 are the input variables and z1 = s, z2̇ = s , θp1 and θp2 are the control gains that need to be designed, and (15) can be rewritten as

p^ (z θp ) = θpT ϕ (z )

(16)

where θp ¼ [θp1, θp2]T∈R2 is an adjustable parameter vector and φT (z) ¼[z1, z2]∈R2 is a regression vector. ^ Define θ and θ are the free parameters in f (x, t ) and g^ (x, t ), f

g

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

7

Fig. 11. TSMC with the sign function (experiment). (a) Step response and control error. (b) Control input.

^ ^ then, f (x, t ) = f (x θf ) and g^ (x, t ) = g^ (x θg ). Replacing f(x,t) and g(x, ^ t) in (7) by the fuzzy systems f (x θf ) and g^ (x θg ), and replacing the switching part ksgn(s) by the PI control p^ (z θ ). Then, the following p

control law can be obtained:

⎤ ⎡ n− 1 −1 ^ u = g^ (x θg ) ⎢ − ∑ ci e (i) − f (x θf ) + xd(n) − p^ (z θp ) ⎥ ⎥⎦ ⎢⎣ i = 1

(17)

^ f (x|θf ) = θfT ξ (x),

(18)

g^ (x|θg ) = θgT ξ (x),

p^ (z θp ) = θpT ϕ (z )

To avoid the chattering, when |s| o Δ, use the PI control to replace the switching control; when |s| Z Δ, the control input keeps at the saturation value (Cupertino, Naso, Mininno, & Turchiano, 2009). Thus, when |s| Z Δ, suppose p^ (z θp ) = D + k , where Δ is the boundary layer thickness. Design the adaptive law as

θḟ = r1sξ (x), θġ = r2 sξ (x) u (t ),

θṗ = r3 sφ (z )

(19)

where r1 4 0, r2 40 and r3 40 are the adjusting rates. Stability of the designed control system can be guaranteed by the assumption of parameters boundaries in (26). To ensure the parameters boundaries, the following improvements can be made for the adaptive laws (19) according to the projection algorithm (Wang, 1997). Define Mf, Mg and Mp are the pre-specified boundaries of the estimated parameters. For θf, the following adaptive laws can be adopted:

⎧ (|θf | < Mf ) or (|θf | = Mf and sθfT ξ (x) ≤ 0) ⎪ r1sξ (x ) θf = ⎨ T ⎪P ⎩ f { r1sξ (x) } (|θf | = Mf and sθf ξ (x) > 0)

(20)

where Mf is the pre-specified boundary of the estimated parameter θf. For θg, when some element θgi ¼ ε, the following adaptive laws can be adopted:

⎧ r2 sξ (x) u (t ) (sξ (x) u (t ) < 0) i θgi̇ = ⎨ (sξi (x) u (t ) ≥ 0) ⎩0

For

θp, the following adaptive laws can be adopted:

⎧ (|θp | < Mp ) or (|θp | = Mp and sθpT φ (z ) ≥ 0) ⎪ r 3 sφ (z ) θp = ⎨ T ⎪P ⎩ p { r3 sφ (z ) } (|θp | = Mp and sθp φ (z ) < 0)

(23)

where Mp is the pre-specified boundary of the estimated parameter θp. The projection operators Pi{*} (i¼ f, g, p) are defined as follows:

⎧ P r sξ (x) = r sξ (x) − r sθ θ T ξ (x)|θ |−2 1 1 f f f ⎪ f 1 ⎪ ⎨ Pg r2 sξ (x) u (t ) = r2 sξ (x) u (t ) − r2 sθg θgT ξ (x) u |θg |−2 ⎪ ⎪ Pp r3 sϕ (z ) = r3 sϕ (z ) − r3 sθp θpT ϕ (z )|θp |−2 ⎩

{ { {

}

}

}

(24)

Structure of the designed adaptive FSMC based on PI control is illustrated as Fig. 5. In conclusion, design steps of the FSMC based on PI control can be summarized as follows: Step1 Select the appropriate initial values for PI control; Step2 Determine the coefficients c1, c2,…,cn  1 for the sliding surface s¼cTe; Step3 Select the adjusting rates r1, r2 and r3 for the adaptive laws (19); Step4 Define the fuzzy sets for the linguistic variables and design the membership functions for the fuzzy sets; ^ Step5 Establish the fuzzy control rules for f (x|θ ) and g^ (x|θ ); f

Step6

Establish

the

fuzzy system

^ f (x|θf ) = θfT ξ (x )

g

and

g^ (x θg ) = θgT ξ (x ); Step7 Establish the control law (17) according to the adaptive laws (20)–(24); and Step8 Apply the designed the control algorithm to the controlled object and adjust the parameter vectors θf, θg and θp according to the adaptive laws (20)–(24). 3.4. Analyses of the stability and convergence

⎪

⎪

(21)

where ξi(x) is the i-th element in ξ(x). Otherwise, the following adaptive laws can be adopted: ⎧ r sξ (x ) u (t ) (|θg | < Mg ) or (|θg | = Mg and sθgT ξ (x ) u (t ) ≤ 0) ⎪ 2 θg = ⎨ T ⎪ ⎩ Pg { r2 sξ (x ) u (t ) } (|θg | = Mg and sθg ξ (x ) u (t ) > 0)

(22)

where Mg is the pre-specified boundary of the estimated parameter θg.

The FSMC based on PI control has been designed with the Lyapunov synthetic method, however, performance of the adaptive control system has not been analyzed in detail. There are two fundamental issues in performance analysis: stability and convergence. Stability means that the involved variables are bounded and convergence means that the tracking errors converge to zero. The following theorem will prove that the system parameters are bounded and the tracking errors converge to zero. Theorem. For the nonlinear system (3), if the control law (17) is

8

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

Fig. 12. Traditional PID control (experiment). (a) Displacement response and tracking error. (b) Speed response and tracking error. (c) Control input.

^ used, the functions f , g^ , p^ are approximated by (18) and the parameters vector θf, θg and θp are adjusted by (19), signals of the closedloop control system will be bounded and the tracking errors will converge to zero asymptotically.

The existence of Mf, Mg and Mp is guaranteed by Theorem 25.1 of Wang (1997), and Mf, Mg and Mp can be determined from (20), (22) and (23). n− 1

s=

Proof. The optimal parameter vectors are defined as follows:

n− 1

∑ ci e(i) + x (n) − xd(n) = ∑ ci e(i) + f (x, t ) + g (x, t ) u (t ) i=1

i=1 n− 1

+ d (t ) − xd(n) =

i=1

⎡ ⎤ ^ θ *f = arg min ⎢ sup f (x θ f ) − f (x , t ) ⎥ θf ∈ Ωf ⎢ ⎣ x ∈ Rn ⎦⎥

n− 1

+ (g (x, t ) − g^ (x|θg )) u (t ) −

⎡ ⎤ θg* = arg min ⎢ sup g^ (x θ g ) − g (x , t ) ⎥ θg ∈ Ωg ⎢ ⎣ x ∈ Rn ⎦⎥ ⎡ ⎤ θ p* = arg min ⎢ sup p^ z θ p − usw ⎥ ⎥⎦ θp ∈Ωp ⎢ ⎣ z ∈ R2

(

)

Ωg Ωp

f

g

p

n

θf ≤ Mf

n0

2

}

< ε ≤ θg ≤ Mg

θ p ≤ Mp

}

^ + d (t ) = f (x, t ) − f (x|θf ) + (g (x, t ) − g^ (x|θg )) u (t ) − p^ (z|θp ) ^ ^ + d (t ) + w = f (x|θ*f ) − f (x|θf ) + (g^ (x|θg* ) − g^ (x|θg )) u (t )

(25)

optimal values. Then, define the sets of θf, θg and θp are as follows:

{θ ∈ R = {θ ∈ R = {θ ∈ R

∑ ci e(i) − p^ (z|θp ) i=1

where Ωf, Ωg and Ωp are the sets of θf, θg and θp respectively.Since ^ f (x|θ*f ) and g^ (x|θg* ) are the best approximations of f(x,t) and g(x,t) respectively, define the minimum approximation error ^ w = f (x, t ) − f (x|θ*f ) + (g (x, t ) − g^ (x|θg* )) u. In this case, w-0 as θf * * - θf , θg- θg and θp- θp*, that is, when θf, θg and θp attain their Ωf =

∑ ci e(i) + f (x, t ) − f^ (x|θf )

− p^ (z|θp ) + p^ (z|θp* ) − p^ (z|θp* ) + d (t ) + w = ϕfT ξ (x) + ϕgT ξ (x) u (t ) + ϕpT φ (z ) − p^ (z|θp* ) + d (t ) + w

where ϕf = θ*f − θf , ϕg = θg* − θg , ϕp = θp* − θp .Define the following Lyapunov function:

V=

} (26)

(27)

⎞ 1⎛ 2 1 1 1 ⎜ s + φfT φf + φgT φg + φpT φp ⎟ ⎠ 2⎝ r1 r2 r3

Differentiate V along the error state trajectory, then

(28)

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

9

Fig. 13. TSMC with the sign function (experiment). (a) Displacement response and tracking error. (b) Speed response and tracking error. (c) Control input. (d) Switching function.

V = ss +

V̇ ≤ sw − k s ≤ 0

1 T 1 1 φ φ + φgT φg + φpT φp r1 f f r2 r3

From the universal approximation theorem, the fuzzy systems can enable the minimum approximation error w to be very small, therefore, V̇ r 0. From (30), s, θf, θg and θp are bounded. Since s¼ cT e, if e(0) is bounded, then, e(t) is bounded. If the reference trajectory xd is bounded, then, x is bounded.Suppose |s| r ηs, since kZ η 4 0, then, the following can be derived from (30):

= s (φfT ξ (x) + φgT ξ (x) u (t ) + φpT ϕ (z ) − p^ (z|θp* ) + d (t ) + w ) 1 T 1 1 φ φ + φgT φg + φpT φp r1 f f r2 r3 1 1 = sφfT ξ (x) + φfT φf + sφgT ξ (x) u (t ) + φgT φg + sφpT ϕ (z ) r1 r2 1 + φpT φp − sp^ (z|θp* ) + s (d (t ) + w ) r3 1 1 = φfT (r1sξ (x) + φf ) + φgT (r2 sξ (x) u (t ) + φg ) r1 r2 1 + φpT (sϕ (z ) + φp ) − sp^ (z|θp* ) + s (d (t ) + w ) r3 1 1 ≤ φfT (r1sξ (x) + φf ) + φgT (r2 sξ (x) u (t ) + φg ) r1 r2 1 + φpT (sϕ (z ) + φp ) − s (D + k ) sgn (s ) + sd (t ) + sw r3 1 1 < φfT (r1sξ (x) + φf ) + φgT (r2 sξ (x) u (t ) + φg ) r1 r2 1 + φpT (sϕ (z ) + φp ) − k s + sw r3

(30)

+

V̇ ≤ s w − s η ≤ ηs w − s η

(31)

Integrating two sides of (31) and transposing, then

∫0

t

s (τ ) dτ ≤

η 1 [V (0) + V (t )] + s η η

∫0

t

w dτ

(32)

If w∈L1, then, from (32), s∈L1. According to (30), s is bounded, thus, s∈L1. Since variables on right side of (27) are bounded, then, s ̇ ∈ L∞. According to Barbalat's Lemma (Slotine, 1991), when t-1, s(t)-0 and e(t)-0. Therefore, the designed FSMC based on PI control is stable and convergent. ◽

4. Simulation analyses and experimental verifications

(29)

where ϕḟ = − θḟ , ϕġ = − θġ , ϕṗ = − θṗ , substitute the adaptive law (19) into (29), then

Set the desired command xd(t) ¼0.5 sin(πt), then, x¨ d r0.5π2, the sliding surface s¼ 15eþ e ̇, the initial conditions x(0) ¼[0.5,0] and θp(0) ¼[4,10]T, the sampling time is 0.01 s. There is no

10

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

Fig. 14. TSMC with the saturation function (experiment). (a) Displacement response and tracking error. (b) Speed response and tracking error. (c) Control input. (d) Switching function.

standard and objective method in the design of fuzzy sets and membership functions, and most approaches are based on the experiences and experiments. For simplicity, the same fuzzy sets are used to characterize the behaviors of the input and output variables. Theoretically, increase the elements of the fuzzy sets and the subdivisions of the discourse domain can improve the control precision, however, the calculation complexity will increase. Conversely, too few fuzzy sets cannot cover the whole domain. Thus, seven elements are selected for each of the fuzzy set. Define the fuzzy sets of the system inputs and outputs are {NB, NM, NS, ZO, PS, PM, PB}. To achieve a balance between the system stability and the control precision, the Gaussian membership functions are employed. Mathematical expression of the membership functions are as follows, which can be illustrated as Fig. 6.

μNB ( xi ) = exp [ − ((xi + π /4)/(π /24))2⎤⎦, μNM ( xi ) = exp ⎡⎣ − ((xi + π /6)/(π /24))2], μNS ( xi ) = exp [ − ((xi + π /12)/(π /24))2⎤⎦, μZO ( xi ) = exp ⎡⎣ − (xi /(π /24))2⎤⎦, μPS ( xi ) = exp ⎡⎣ − ((xi − π /12)/(π /24))2], μPM ( xi ) = exp [ − ((xi − π /6)/(π /24))2⎤⎦, μPB ( xi ) = exp ⎡⎣ − ((xi − π /4)/(π /24))2]. Initial values of the elements in θfT and θgT are 0.1, and the

adjusting rates r1 ¼5, r2 ¼1, r3 ¼10. Switching gain k¼ 5, thickness of the boundary layer Δ ¼0.05 and Mf ¼ 20, Mg ¼ 20, Mp ¼80. Simulation results of the TSMC with sign function, the TSMC with saturation function and the FSMC with PI control are illustrated in Fig. 7, Fig. 8 and Fig. 9 respectively. From Fig. 7(a), Fig. 8 (a) and Fig. 9(a), the system tracking errors are 8.9303  10  5 rad, 1.1596  10  4 rad and 6.7569  10  5 rad respectively. Compared with another two, the FSMC with PI control has improved the tracking precision by 24.34% and 41.73%. From Fig. 7(b), Fig. 8 (b) and Fig. 9(b), the FSMC with PI control has the minimum chattering. Adaptive adjustment processes of the control gains θp1 and θp2 are illustrated in Fig. 9(c) and (d). Therefore, the designed control scheme not only can achieve high control precision, but also can effectively suppress the input chattering. To verify the superiority of the designed controller, control experiments of the FSMC are compared with that of the PID control and the TSMC. For the step response of the PID control, set the desired command xd(t)¼ 0.5 rad, after repeated testing, select the control parameters kp ¼ 60, ki ¼1 and kd ¼3. For sine response, set xd(t)¼0.5 sin(πt) rad, kp ¼20, ki ¼0.01, kd ¼ 5 and x(0) ¼[0, 0]. For the TSMC and FSMC, set xd(t) ¼0.5 sin(πt) rad, x¨ d r0.5π2 ¼ M, the sliding surface s ¼15eþ e ̇, x(0)¼[0.5,0], the switching gain k¼ 5, the boundary thickness Δ ¼0.05, the initial elements in θfT and

θgT are 0.1, the adjusting rates r1 ¼ 5, r2 ¼1 and r3 ¼10, the initial control gains θp(0)¼ [4,10]T, the boundary parameters Mf ¼20, Mg ¼20, Mp ¼80, and the sampling time is 0.01 s.

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

11

Fig. 15. FSMC with the PI control (experiment). (a) Displacement response and tracking error. (b) Speed response and tracking error. (c) Control input. (d) Switching function. (e) Adaptive process of the θp1. (f) Adaptive process of the θp2.

Because of the elastic deformation, nonlinear friction and other uncertainties in the transmission mechanism of the A-axis, deadzone effect of the control system impacts significantly on the output response. Due to no response in the deadzone, limit loop or system instability will be triggered. From Fig. 10(a), under the action of deadzone, static error of the PID control is up to 2.533  10  4 rad. Because of the self-adaptability to the uncertainties and the external disturbances, the TSMC with sign function can effectively undermine the the static error, as shown

in Fig. 11(a), the maximum error is 1.472  10  4 rad which reduced 41.89% compared with the PID control. However, input chattering of the TSMC (Fig. 11(b)) is much more significant than that of the PID control (Fig. 10(b)). There exists the flat top and the deadzone in the displacement and the speed responses respectively, as shown in Fig. 12(a) and (b), and the tracking errors are 2.462  10  4 rad and 1.638  10  3 rad/s, Fig. 12(c) is the control input. Compared with the PID control, displacement tracking error of

12

P. Zhao et al. / Control Engineering Practice 53 (2016) 1–13

Funds for the Central Universities of China (No. JB150409).

Table 5 Performance comparison of the different control algorithms. Traditional PID TSMC (sign)

Displacement error (rad) Speed error (rad/s) Chattering extent

TSMC (saturation)

FSMC (PI control)

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.conengprac.2016.04.012.

2.462  10  4

1.705  10  4 2.329  10  4

1.228  10  4

1.638  10  3

1.138  10  3

1.411  10  3

8.631  10  4

Smaller

Larger

Moderate

Smaller

the TSMC with sign function is 1.705  10  4 rad which has been reduced 30.75% (Fig. 13(a)), simultaneously, the speed tracking error is 1.138  10  3 rad/s, which has decreased 30.53% (Fig. 13(b)). However, chattering of the TSMC with sign function is more severe, as illustrated in Fig. 13(c), and variation of the switching function (Fig. 13(d)) also reflects the chattering to some extent. From Fig. 7 and Fig. 13, the experimental results verify the simulations well. To reduce the chattering of TSMC, saturation function is used to replace the sign function. From Fig. 14(a) and (b), tracking errors of the displacement and speed are 2.329  10  4 rad and 1.411  10  3 rad/s, which have been increased 36.60% and 23.99% respectively compared with that of the TSMC with sign function. However, compared Fig. 14 (c) and Fig. 13(c), chattering of the TSMC with saturation function has been reduced significantly. Therefore, saturation function can reduce the input chattering while it will lose some control precision. From Fig. 15(a) and (b), tracking errors of the displacement and speed are 1.228  10  4 rad and 8.631  10  4 rad/s, which have been reduced 27.98% and 24.16% compared with the TSMC with sign function. From Fig. 15(c), chattering of the FSMC with PI control has significantly been decreased. Therefore, the FSMC with PI control not only can improve the control precision, but also can suppress the chattering. Fig. 15(e) and (f) are the adaptive processes of θp1 and θp2 respectively. Comparison of the control precision and the input chattering of the PID, TSMC with sign function, TSMC with saturation function and FSMC with PI control are illustrated in Table 5.

5. Conclusions The A-axis with high-power, high-torque and high-stiffness used for processing the titanium alloy, superalloy and other difficult to machine materials is presented in this paper. Considering the influence of the nonlinear friction and uncertain cutting force on the control precision of the A-axis, a robust FSMC with PI control is proposed based on the parameters adaptation. The fuzzy systems are used to approximate the unknown nonlinear functions and the PI control is employed to undermine the chattering. The global stability and asymptotic convergence of the designed control algorithm has been guaranteed by the Lyapunov theory. Simulation analyses and experimental verifications illustrate that the designed FSMC with PI control is robust to the uncertain load and parameters perturbation, and it not only can improve the control precision, but also can reduce the input chattering.

Acknowledgment This work has been supported by the National Natural Science Foundation of China (No. 51505356), the National Science and Technology Major Project of the Ministry of Science and Technology of China (No. 2013ZX04001081) and the Fundamental Research

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