Adaptive Fuzzy Torque Control of Passive Torque Servo Systems

Chinese Journal of Aeronautics 25 (2012) 906-916

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Chinese Journal of Aeronautics journal homepage: www.elsevier.com/locate/cja

Adaptive Fuzzy Torque Control of Passive Torque Servo Systems Based on Small Gain Theorem and Input-to-state Stability WANG Xingjiana,b,*, WANG Shaopinga,b, ZHAO Pana,b a

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China b

Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, China Received 21 July 2011; revised 4 September 2011; accepted 9 October 2011

Abstract Passive torque servo system (PTSS) simulates aerodynamic load and exerts the load on actuation system, but PTSS endures position coupling disturbance from active motion of actuation system, and this inherent disturbance is called extra torque. The most important issue for PTSS controller design is how to eliminate the influence of extra torque. Using backstepping technique, adaptive fuzzy torque control (AFTC) algorithm is proposed for PTSS in this paper, which reflects the essential characteristics of PTSS and guarantees transient tracking performance as well as final tracking accuracy. Takagi-Sugeno (T-S) fuzzy logic system is utilized to compensate parametric uncertainties and unstructured uncertainties. The output velocity of actuator identified model is introduced into AFTC aiming to eliminate extra torque. The closed-loop stability is studied using small gain theorem and the control system is proved to be semi-globally uniformly ultimately bounded. The proposed AFTC algorithm is applied to an electric load simulator (ELS), and the comparative experimental results indicate that AFTC controller is effective for PTSS. Keywords: flight simulation; adaptive control; fuzzy control; passive torque servo system; electric load simulator; extra torque; small gain theorem; input-to-state stability

1. Introduction1 Passive torque servo system (PTSS) is often required in modern avionic area, which is called load simulator or loading system. It is an important equipment in hardware-in-loop (HIL) simulation for aircraft control system [1], which can exert aerodynamic load on aircraft actuation system to verify the performance of flight control system on ground [2]. Usually, in order to simulate aerodynamic load, PTSS connects actuation system directly with rigid shaft coupling. It is obvious that PTSS inevitably endures strong position coupling disturbance from active *Corresponding author. Tel.: +86-10-82338917. E-mail address: [email protected] Foundation items: National High-tech Research and Development Program of China (2009AA04Z412); “111” Project; BUAA Fund of Graduate Education and Development 1000-9361/$ - see front matter © 2012 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(11)60461-5

rotary motion of actuation system [3]. This disturbance is called extra torque, and it is an inherent disturbance which may even be bigger than the desired torque output sometimes. Much literature focuses on designing appropriate mechanical structure or compensation controller to eliminate extra torque. The basic idea of mechanical structure optimization [4] is to decouple torque servo and position tracking by introducing a synchronizing motor between system base and loading motor. The stator of synchronizing motor is connected to system base while the rotator is connected to loading motor [5]. In this system, the stator of loading motor is used to realize position tracking while the rotator is used to realize torque tracking. But this structure makes system more complex and increases cost. For compensation controller, constant structure control and velocity synchronization are widely used in application of load simulator [1]. Constant structure control was designed by Liu [6] whose idea was feedforward actuator velocity to eliminate extra torque. This method

No.6

WANG Xingjian et al. / Chinese Journal of Aeronautics 25(2012) 906-916

depends on velocity measurement, so phase delay influence is inevitable. Velocity synchronizing compensation was presented by Jiao, et al. in Ref. [1], which foreknows the actuator motion by collecting its servo-valve signal and compensated extra torque in advance. Feedforward and compensation were introduced [7] to eliminate extra torque in electro-hydraulic load simulator. Plummer [8] presented a valve crosscompensation design approach to improve force tracing accuracy, which was also based on velocity synchronizing control essentially. Besides, there are many robustness improvement methods considering actuator movement as disturbance. Nam and Hong [2] proposed a robust force control system using quantitative feedback theory designed for loading system to enhance the robustness of force control system. Chantranuwathana and Peng [9] presented a modular adaptive robust control technique to improve the force control performance of vehicle active suspensions. A hybrid control scheme of dynamic loading system for ground testing of high-speed aerospace actuator was introduced in Ref. [10], which included velocity compensation, torque input feedforward and PID closed-loop control. Truong and Ahn [11] proposed a self-tuning grey predictor combined with a fuzzy PID controller to improve the robustness of force control of hydraulic load simulator. Additionally, although many nonlinearities such as unknown system parameters, unstructured uncertainties and external disturbance, are neglected in system modeling, they really exist in practical system. Therefore, it is difficult to achieve high precision torque tracking and good robustness without effective control method. Adaptive backstepping design by Lyapunov function was proposed for parametric strict-feedback systems in Ref. [12]. Besides Lyapunov method, input-to-state stability (ISS) approach is also a useful tool to design controller and analyze system stability, which was firstly proposed by Sontag [13]. Furthermore, small gain theorem was proved [14] and the synthesis of adaptive fuzzy tracking controller was developed [15-16], where Takagi-Sugeno (T-S) fuzzy system is utilized to approximate nonlinear unstructured uncertainties. In this paper, adaptive fuzzy torque control (AFTC) of PTSS is proposed to guarantee transient tracking performance as well as final tracking accuracy. T-S fuzzy logic system is used to compensate nonlinear unknown system function caused by parametric uncertainties and unstructured uncertainties. Only one parameter needs to be tuned online at each step and the proposed controller structure is simple and easy to be implemented in practical PTSS. Application on electric load simulator (ELS) shows that the method presented in this paper is effective. 2. Mathematical Preliminaries In this section, the concepts of ISS and small gain

· 907 ·

theorem [13-14, 17] will be introduced, which have been widely used in nonlinear control problems for stability analysis [16]. The class K, K ∞ and KL functions [18] will be firstly reviewed. Then the structure of fuzzy logic system will be briefly described. 2.1. ISS and small gain theorem Definition 1 A function α : (0, ∞) → (0, ∞) is said to belong to class K if it is continuous, strictly increasing and α (0) = 0 . It is said to belong to class K ∞ if additionally α is unbounded. Definition 2 A function β : (0, ∞) × (0, ∞) → (0, ∞) is of class KL if β (i, t ) is said to belong to class K for each fixed t ≥ 0 , and β (r , i) is strictly decreasing for each fixed r > 0 and lim β (r , t ) = 0 . t →∞

Definition 3 For a nonlinear system x = f ( x , u) , it is said to be input-to-state practically stable (ISpS) if there exist a class KL function β and a class K function γ , such that for any initial condition x (0) , each bounded control input u(t) defined for all t > 0 and a constant d > 0 , the associated solution x (t ) ∈ [0, ∞) satisfies x (t ) ≤ β ( x (0) , t ) + γ ( ut

where

ut

∞

∞

)+d

(1)

= max | u(τ ) | and u( τ ) is a truncated 0 ≤τ ≤t

function defined as ⎧u(τ ) 0 ≤ τ ≤ t ut (τ ) = ⎨ τ < 0 or τ > t ⎩0

(2)

When d = 0 in Eq. (1), the ISpS property becomes the ISS property [13, 19]. Definition 4 A function V is said to be an ISpS-Lyapunov function for the system x = f ( x, u) if 1) there exist class K ∞ functions α1 , α 2 , such that α1 ( x )≤V ( x )≤α 2 ( x ) , ∀x ∈ R n (3) 2) there exist class K functions α 3 , α 4 and a constant d > 0 , such that ∂V ( x ) f ( x , u) ≤ −α 3 ( x ) + α 4 ( u ) + d ∂x

(4)

When d=0 in Eq. (4), V is said to be an ISSLyapunov function [19]. Then nonlinear L∞ gain γ in Eq. (1) can be chosen as

γ ( s ) = α1−1 ( s ) α 2 ( s ) α 3−1 ( s ) α 4 ( s ),

∀s > 0

(5)

Proposition 1 The nonlinear system x = f ( x, u) is ISpS if and only if there exists an ISpS-Lyapunov

· 908 ·


function [16]. Theorem 1 (Small Gain Theorem) Consider a system in composite feedback form of two ISpS systems [14, 16]: ⎧ x = f (t , x, ω ) ⎩ e = H (t , x , ω )

Σ eω : ⎨

⎧ y = g (t , y, e )

(7)

where x and e are state vector and output vector of system Σ eω while y and ω are state vector and output vector of system Σ ω e . In particular, if there exist class KL functions βω , β e , class K functions γ e , γ ω , and two constants d1 ≥ 0 , d 2 ≥ 0 , for all t ≥ 0 , the solutions X ( x; ω , t ) and Y ( y; e , t ) satisfy H ( X ( x; ω , t )) ≤ βω ( x (0) , t ) + γ e ( ω

) + d1 ∞

xn ]T ,

x = [ x1 x2

x = [1 x T ]T

⎡ a10 a11 ⎢a a21 20 Af = ⎢ ⎢ ⎢ ⎣⎢aq0 aq1

(6)

Σ ωe : ⎨ ⎩ω = K (t , y, e )

No.6

a1n ⎤ a2n ⎥⎥ ⎥ ⎥ aqn ⎦⎥

ξ f ( x ) and ξ fi ( x ) are fuzzy basis vector and fuzzy basis function respectively, and can be expressed as

ξ f ( x ) = ⎡⎣ξ f1 ( x ) ξf 2 ( x ) n

ξ fi ( x ) =

∏ μM j =1

q

⎛

i j

ξfq ( x ) ⎤⎦

(x j )

n

∑ ⎜⎜ ∏ μM i =1 ⎝ j =1

i j

⎞ (x j ) ⎟ ⎟ ⎠

where μ M i ( x j ) is the membership function of fuzzy set j

(8)

M ij

and usually is given as

K (Y ( y; e , t )) ≤ β e ( y (0) , t ) + γ ω ( e ∞ ) + d 2

i j

(9) If

(i = 1, 2,

γ e (γ ω ( s)) < s (or γ ω (γ e ( s)) < s), ∀s > 0

(10)

Then the solution of the composite systems (6) and (7) is ISpS.

and xn is M ni

THEN yi = ai 0 + ai1 x1 + ai 2 x2 +

where M ij ( j = 1, 2,

+ ain xn

, n) is fuzzy subsets, ai 0 , ai1 ,

, ain denote unknown constant parameters in fuzzy

rule conclusion step, yi is fuzzy output, and q the total number of fuzzy rules. By using singleton fuzzifier production inference engine and center-average defuzzifier, the output of T-S fuzzy system can be given as q

y = ∑ yiξ fi ( x ) = ξ f ( x ) Af x i =1

where

of μ M i ( x j ) [19]. j

For any continuous function f f ( x )

defined in a compact set x ∈ U x and ∀ε > 0 , there

In the past years, fuzzy logic system has been widely researched. In this paper, it will be used as a practical function approximator [16] in adaptive fuzzy torque controller design. T-S fuzzy system [20] can be described as the following form which consists of q rules. Rule Ri : IF x1 is M 1i and x2 is M 2i and

(12)

where bij ∈ R and hij > 0 are adjustable parameters

Lemma 1

2.2. Fuzzy logic system

⎡ ( x j − bij )2 ⎤ ⎥ i 2 ⎢⎣ 2(h j ) ⎥⎦ , q ; j = 1, 2, , n)

μM ( x j ) = exp ⎢−

(11)

exists an ideal fuzzy logic such that [21] sup y − f f ( x ) ≤ ε

x∈U x

(13)

3. Dynamic Models and Basic Assumptions 3.1. Dynamic models of PTSS PTSS is used to simulate aerodynamic load of aircraft rudder for ground testing. In HIL simulation process, the actuation system is controlled by onboard flight control computer while the relevant aerodynamic loading torque is calculated by simulation computer according to height, speed and posture of aircraft. PTSS is designed to exert this loading torque to actuation system. Generally, PTSS connects to actuation system directly in order to simulate aerodynamic load as shown in Fig. 1, in which the right side is the actuation system (i.e., position servo system) and the left side is loading system (i.e., torque servo system).

No.6


· 909 ·

θ m and θ a are rotary displacements of torque motor and actuation system, respectively. Then differentiating Eq. (16) leads to the following function: Tl = K s ( ωm − ωa ) + d n

(17)

where ωa represents actuator velocity and it is the source of coupling disturbance of PTSS.

3.2. Model design Fig. 1

Passive torque servo system structure.

It is obvious that PTSS and actuation system are two mutually coupling systems. For actuation system, this position servo system is required to follow desired angle trajectory in the presence of external disturbances, so loading torque can be treated as external disturbance acting on the position closed-loop system. On the other hand, torque servo system needs to exert desired loading torque on actuation system when actuator operates, and active motion of actuator can be considered as an external disturbance to the torque closed-loop system. A good PTSS design is that PTSS could catch up with actuator motion and put corresponding aerodynamic load to actuation system without extra torque. For a kind of electric-torque-motor-driven PTSS, the dynamics of motor amplifier can be ignored when PTSS operates in normal condition. Thus, the relationship of driving torque Tm versus input control voltage u to motor amplifier can be represented by the following equation [22]: Tm = K mu

(14)

where K m is proportional coefficient between input control voltage u and driving torque Tm . Considering friction torque and external disturbances, torque balance equation of loading system can be given as

Tm + Td = J mωm + Bmωm + Tl + T f

(15)

where J m is total inertia of motor rotator and loading shaft, Bm combined coefficient of the damping and viscous friction on the load, ωm velocity of torque motor, Td the lumped effect of external disturbances, Tl loading torque, T f the combined effect of stiction, Coulomb friction and Stribeck effect, and it will be formulated later. In addition, torque sensor is a key component which connects actuation system and load simulator. Since its inertia is very small, it can be considered as an elastic model: Tl = Ks (θ m − θ a ) + d n

(16)

where Ks is total stiffness coefficient of torque sensor and loading shaft, dn represents measurement noise,

Define loading torque, velocity of PTSS as state variables, i.e. x = [ x1 x2 ] T [ Tl ωm ] T , and noting Eq. (15) and Eq. (17), then the entire PTSS system can be expressed in state space form as ⎧ ⎪ x1 = K s x2 − K sωa + d n ⎪ (18) ⎨ K B 1 1 1 m m ⎪x = u− x2 − x1 − Tf + Td ⎪⎩ 2 J m Jm Jm Jm Jm

From previous analysis, it is clear that active motion of actuator is the main disturbance to PTSS, so it is necessary to predict actuator motion. For this purpose, this paper utilizes the output velocity ωˆ a of an identified actuation system model as the estimated value of actuator velocity. For the convenience of system analysis and controller design, the mathematical model of PTSS can be written in a general model of typical strict-feedback nonlinear system as ⎧ x1 = g1 x2 − Φ1 + f1 + Δ1 (19) ⎨ ⎩ x2 = g 2 u − Φ2 + f 2 + Δ2 where g1 and g 2 are unknown control gains with lower bounds g1 min and g 2 min respectively,

Φ1 = g1 Kˆ sωˆ a g1 min

and Φ2 = g 2 x1 ( g 2 min Jˆm )

two

parts which can be compensated, Δ1 = d n and Δ2 = Td / J m time-variant disturbance uncertainties, f1 and f 2 unknown smooth functions which represent the lumped effects of compensation errors, parametric uncertainties and unstructured uncertainties, and they are given as g Kˆ f1 = 1 s ωˆ a − K sωa (20) g1 min f2 =

g2

g 2 min Jˆ m

x1 −

B 1 1 x1 − m x2 − Tf Jm Jm Jm

(21)

where Kˆ s and Jˆ m are off-line identified parameters for K s and J m , respectively.

3.3. Basic assumptions The following basic assumptions are introduced. Assumption 1 The uncertain control gains are

· 910 ·


No.6

e1 = g1 ( e2 + μ1 ) + f1 + Δ1 − xd

bounded, such that 0 < g1 min < g1 < g1 max

(22)

0 < g 2 min < g 2 < g 2 max

(23)

where gi min and gi max (i=1, 2) are lower and up-

Since f1 is an unknown continuous function, T-S fuzzy logic system can be used to approximate the uncertain term f1 according to Lemma 1. Therefore, f1 can be described as

per bounds of gi . Assumption 2 In Eq. (19), the disturbance uncertainties Δi ( i = 1, 2 ), have the property:

Δ i < piφi ( xi )

(24)

where pi are unknown positive constants, φi ( xi ) are known positive smooth functions and xi = [ x1 x2 xi ] T .

f1 = ξ1 Α 1[1 x1 ]T + δ1 ( x1 ) =

ξ1Α 01 + ξ1Α11e1 + ξ1 Α11 xd + δ1 ( x1 ) 1 ⎡ a10 ⎢ 2 ⎢a where Α 1 = ⎢ 10 ⎢ ⎢aq ⎣ 10

ξ1 = ⎡⎣ξ11 ξ12

4. Controller Design and Stability Analysis Considering a PTSS in the form of Eq. (19), the desired loading torque is given as xd ( t ) = Tr ( t ) , which is assumed to be known and bounded and its first and second derivatives xd , xd are also continuous and bounded. The control objective is to design a control input u such that output y = x1 tracks xd ( t ) as closely as possible in spite of coupling disturbance from actuator.

e1 = x1 − xd

(25)

e2 = x2 − μ

(26)

Step 1 Design a virtual control law μ for the first subsystem of Eq. (19) as

μ = μ1 + μ2

(27)

(28)

Considering control law Eq. (27) and the first equation of Eq. (19), the time derivative of e1 can be given as e1 = g1 x2 − Φ1 + f1 + Δ1 − xd

Considering x2 = e2 + μ (29) can be rewritten as

(29)

and μ = μ1 + μ2 , Eq.

1 1 ⎤ 1 ⎡ a10 ⎤ ⎡ a11 ⎤ a11 ⎢ ⎥ ⎥ ⎢ ⎥ 2 2 2 a11 ⎥ ⎢ a10 ⎥ ⎢ a11 ⎥ 0 1 ⎥ , Α1 = ⎢ ⎥ , Α1 = ⎢ ⎥ , ⎢ ⎥ ⎥ ⎢ ⎥ q ⎥ q ⎥ ⎢ ⎢aq ⎥ a11 ⎦ ⎣ 11 ⎦ ⎣ a10 ⎦ ξ1q ⎤⎦ . δ1 ( x1 ) represents fuzzy ap-

here ε1 is unknown but bounded positive constant. Let c1 = n Α11 , Α*1 = c1−1 Α 11 . It is clear that

Α *1 ≤ 1/ n . Then let ω1 = e1 Α*1 , Eq. (31) can be rewritten as f1 = ξ1 Α 01 + c1ξ1ω1 + ξ 1 Α 11 xd + δ 1 ( x1 )

(32)

Substituting Eq. (32) into Eq. (30) leads to e1 = g1 (e2 + μ1 ) + c1ξ1ω1 + v1

(33)

v1 = ξ1Α 01 + ξ1Α11 xd + δ1 ( x1 ) + Δ 1 − xd

(34)

v1 ≤ θ1q1

(35)

where

and

{

}

where θ1 = max Α 01 , Α11 , ε1 , p1 , 1

is a positive

constant and q1 = (1 + xd ) ξ1 + 1 + φ1 ( x1 ) + xd . Design the adaptive fuzzy control law μ1 as

where μ1 is adaptive fuzzy control law which will be designed in the following text, and μ2 is velocity synchronization compensation law which is given as Kˆ μ2 = s ωˆ a g1 min

(31)

proximating error, and for any x1 ∈ U x1 , δ1 ≤ ε1 ;

4.1. Adaptive fuzzy torque controller design In this subsection, AFTC laws will be designed by using backstepping technique based on small gain theorem and ISpS. The following error variables are defined:

(30)

where ψ1 =

ˆ e μ1 = − k1e1 − λψ 1 1 1

(36)

λˆ1 = Γ 1 ⎡⎣ψ 1e12 − σ 1 (λˆ1 − λ10 ) ⎤⎦

(37)

ξ1 4γ

2

2 1

+

q12 4ρ12

,

{

1 1 λ1 = max g1−min c12 , g1−min θ12

}

,

λˆ1 is an online estimate of unknown parameter λ1 with estimate error λ1 = λ1 − λˆ1 . k1 > 0 , γ 1 > 0 , ρ1 > 0 , Γ 1 > 0 , σ 1 > 0 and λ10 > 0 are positive design parameters. Design a Lyapunov function as V1 =

1 2 1 e1 + g1 min Γ 1−1λ12 2 2

(38)

No.6


Considering λ1 = −λˆ1 , the time derivative of V1 can be given as V = e e − g Γ −1λ λˆ = 1

1 1

1 min

1

1 1

g1e1e2 + g1e1μ1 + c1ξ1ω1e1 + v1e1 − g1 min Γ 1−1λ1λˆ1

(39) Noting that v1 ≤ θ1q1 and applying Young’s inequality, the following inequality can be obtained: 2

2 × 2γ 12

ξ1

+

2γ 12 ω1

+

2

2

4γ 12

2

c12 e12 +

q12

(θ1q1 ) 2 e12

(40)

2

4 ρ12

θ12 e12 + γ 12 ω1 + ρ12

w2 =

e2 = g 2 u1 + f 2 + Δ2 − w2

ξ 2 Α 02 [1 ∂μ / ∂x1 xd w2 ]T + ξ 2 Α12 e2 + ξ 2 Α12 [0 μ ]T + δ 2 ( χ 2 )

lowing inequality holds: c1ξ1ω1e1 + v1e1 ≤ g1 min ˆ1 1e12

λψ

+

(49)

f 2 = ξ 2 Α 2 [1 χ 2T ]T + δ 2 ( χ 2 ) =

}

λψ

∂μ x1 ∂x1

to Lemma 1, and f 2 can be describe as

1 1 λ1 = max g1−min c12 , g1−min θ12 , from Eq. (40), the fol-

+γ

2 1

2

where

ω1 + ρ

λψ

g1 min 1 1e12

+γ

2 1=

2 1

2

ω1 + ρ (41) 2 1

Substituting Eqs. (36)-(37) and Eq. (41) into Eq. (39) leads to V1 ≤ g1e1e2 − g1 min k1e12 + γ 12 ω1

2

+ ρ12 +

g1 minσ 1λ1 (λˆ1 − λ10 )≤ g1e1e2 − g1 min k1e12 − 1 2 g1 minσ 1λ12 + γ 12 ω1 + d1 2 where d1 = g1 minσ 1 (λ1 − λ10 ) 2 / 2 + ρ12 .

(42)

Step 2 Design an actual control law u for the second subsystem of Eq. (19) as u = u1 + u2 (43) where u1 is adaptive fuzzy control law and u2 model compensation law given as 1 (44) u2 = x1 g 2 min Jˆm Noting control law Eq. (43) and the second equation of Eq. (19), the time derivative of e2 can be given as e2 = g 2 u − Φ2 + f 2 + Δ2 − μ

(45)

The time derivative of μ can be given as

χ2 = [e1 x2 ∂μ / ∂x1 xd w2 ]T ,

⎡a120 a121 ⎢ 2 a2 ⎢a Α 2 = ⎢ 20 21 ⎢ ⎢ a q aq ⎣ 20 21 ⎡ a121 ⎢ 2 ⎢a 1 Α 2 = ⎢ 21 ⎢ ⎢a q ⎣ 21

(50)

e2 = [ e1 e2 ] T ,

⎡a120 a123 a124 a125 ⎤ a125 ⎤ ⎥ ⎢ 2 ⎥ 2 2 2 2 a25 a25 a24 ⎥ ⎢a20 a23 ⎥ 0 ⎥ , ⎥ , Α2 = ⎢ ⎢ ⎥ ⎥ q ⎥ ⎢aq aq aq aq ⎥ a25 ⎣ 20 23 24 25 ⎦ ⎦

a122 ⎤ ⎥ 2 a22 ⎥ ⎥ , ξ 2 = ⎡⎣ξ 21 ξ 22 ⎥ q ⎥ a22 ⎦

ξ 2q ⎤⎦ , δ 2 ( χ 2 )

represents fuzzy approximating error, and for any χ 2 ∈ U χ 2 , δ 2 ( χ 2 ) ≤ε 2 ; here ε 2 is unknown but bounded positive constant. Let c2 = n Α12 , Α*2 = c2−1 Α 12 . It is clear that

Α *2 ≤ 1/ n . Then let ω 2 = e2 Α*2 , Eq. (50) can be rewritten as f 2 = ξ 2 Α 02 [1 ∂μ / ∂x1 xd w2 ]T + c2ξ 2ω 2 + ξ 2 Α12 [0 μ ]T + δ 2 ( χ 2 )

(51)

Substituting Eq. (51) into Eq. (48) leads to e2 = g 2 u1 + c2ξ 2 ( x2 )ω 2 + v2

(52)

where v2 = ξ 2 Α 02 [1 ∂μ / ∂x1 xd ]T + ξ 2 Α12 [0 μ ]T +

∂μ ∂μ ˆ ∂μ μ= λ1 + x1 + xd + μ 2 = ∂x1 ∂xd ∂λˆ1 ∂μ x1 + w2 ∂x1

(48)

Then T-S fuzzy logic system can be used to approximate the uncertain nonlinear term f 2 according

2

g1 min 1 1e12

(47)

is a computable intermediate variable. Substitute Eq. (43) and Eq. (46) into Eq. (45), then e2 can be rewritten as

Considering that ψ 1 = ξ1 / (4γ 12 ) + q12 /(4 ρ12 ) , and

{

∂ μ ˆ ∂μ xd + μ2 λ1 + ∂xd ∂λˆ1

f2 = f2 +

2ρ 2 + 1 = 2

2 × 2 ρ12

where

where

c1ξ1ω1e1 + v1e1 ≤ c1ξ1ω1e1 + v1e1 ≤ c12 ξ1 e12

· 911 ·

(46)

δ 2 ( χ 2 ) + Δ2 − w2

(53)

v2 ≤ θ 2 q2

(54)

and

· 912 ·


(

)

where θ 2 = max Α 02 , Α12 , p1 , ε1 , 1 constant

is a positive

q2 = ( [1 ∂μ / ∂x1 xd ]T + μ ) ξ1 +

and

φ2 ( x2 ) + 1 + w2 . Design the adaptive fuzzy control law u1 as

where ψ 2 =

u1 = − k2 e2 − λˆ2ψ 2 e2

(55)

λˆ2 = Γ 2 ⎡⎣ψ 2 e22 − σ 2 (λˆ2 − λ20 ) ⎤⎦

(56)

ξ2 4γ

2 2 2

+

q22 4ρ

2 2

choosing the gain of subsystem Σ eω as 0 < γ < 1 , and k1 > 1/ g1 min and k2 > 1/ g 2 min satisfy Assumptions 1-2, then the closed-loop control system is semiglobally uniformly ultimately bounded in the sense that all signals in this system are bounded. If control parameters are chosen suitably, the tracking error can be smaller than a prescribed error bound, which means the tracking error asymptotically converges to zero. Proof Rewrite the closed-loop system into two composited subsystems, viz. Σ eω and Σ ω e : ⎧ e1 ⎪e ⎪ 2 ⎪λ ⎨ 1 ⎪ ⎪λ2 ⎪ ⎩E

, λ2 = max (g2−1min c22 , g2−1minθ22 ) ,

λˆ2 is an online estimate of unknown parameter λ2 with estimate error λ2 = λ2 − λˆ2 . k2 > 0 , γ 2 > 0 ,

ρ 2 > 0 , Γ 2 > 0 , σ 2 > 0 and λ20 > 0 are positive design parameters. Design a Lyapunov function V2 as (57)

V2 = V1 + e2 e2 − g 2 min Γ 2−1λ2 λˆ2 =

Σ eω

[e1

and

g 2 min Γ 2−1λ2 λˆ2

(58)

c2ξ 2ω 2 e2 + v2 e2 ≤ 4γ 22

c22 e22 +

q22

θ 2 e2 + γ 22 ω 2 2 2 2

2

4 ρ2

2

+ ρ 22 (59)

Substituting Eqs. (55)-(56) and Eq. (59) into Eq. (58) leads to

1 g 2 minσ 2 λ22 + γ 12 ω1 2

2

2

is output,

E = ⎡⎣ eT λ T ⎤⎦

+ d2

T

, e=

T

e2 ]

T

and λ = ⎡⎣λ1 λ2 ⎤⎦ . Design a Lyapunov function as V = V2

(62) and γ =(γ12 +

γ 22 )1/ 2 , and noting Eq. (60), the time derivative of V can be obtained: V ≤− E

2

+γ 2 ω

2

+ d2

(63)

According to Definitions 3-4, the subsystem Σ eω satisfies ISpS; there exist class K ∞ functions α1 ( s ) ,

α 3 ( s ) = s 2 , α 4 ( s ) = γ 2 s 2 , then the gain of subsystem Σ eω can be obtained:

γ z ( s ) = α1−1 ( s ) α 2 ( s ) α 3−1 ( s ) α 4 ( s ) , ∀s > 0 (64) Consider subsystem Σ ω e :

1 g1 minσ 1λ12 − 2

+ γ 22 ω 2

is given as input of subsystem

α 2 ( s) , such that α1 ( E ) ≤ V ( E ) ≤ α 2 ( E ) , and

+ ρ 22 ≤

g 2 min λˆ2ψ 2 e22 + g 2 min λ2ψ 2 e22 + γ 22 ω 2

V2 ≤− g1 min k1e12 − g 2 min k2 e22 −

= E(E ) = E

E

Noting v2 ≤ θ 2 q2 and applying the same technique as Eqs. (40)-(41), the following inequality can be obtained:

(61)

= −Γ 2 ⎡⎣ψ 2 e22 − σ 2 (λˆ2 − λ20 ) ⎤⎦

Choosing k1 > 1 / g1 min , k2 > 1/ g2 min

V1 + g 2 u1e2 + c2ξ 2ω 2 e2 + v2 e2 −

ξ2

= −Γ 1 ⎡⎣ψ 1e12 − σ 1 (λˆ1 − λ10 ) ⎦⎤

T

Considering λ2 = −λˆ2 , the time derivative of V2 can be written as

2

= g1 (e2 + μ1 ) + c1ξ1 ( x1 )ω1 + v1 = g 2 u1 + c2ξ 2 ( x2 )ω 2 + v2

where ω = [ω1 ω 2 ]

1 1 V2 = V1 + e22 + g 2 min Γ 2−1λ22 2 2

No.6

⎧⎪ ω1 = Α *1e1 ⇒ ⎨ * * T ⎪⎩ω 2 = Α2 e2 = Α2 [e1 e2 ] ω = Α e = ΑΕ = ΑΕ

(60)

where d 2 = d1 + g 2 minσ 2 (λ2 − λ20 ) 2 / 2 + ρ 22 .

where Ε is the input of subsystem, and the output is

Theorem 2 Consider the closed-loop system consisting of system (18), controller (43), adaption laws (37) and (56). For bounded initial conditions, if

T

ω = [ω1 ω2 ] . Ε = Ε = ⎡⎣eT λ T ⎤⎦ , e = e2 = [e1 e2 ]T , T

4.2. Closed-loop system stability analysis

(65)

⎡ Α*1 0 ⎤ T *2 ⎤ ⎢ ⎥ , Α , = λ = ⎡⎣λ1 λ2 ⎤⎦ , Α *2= ⎡⎣Α *1 Α 2 2 ⎦ *1 ⎢⎣ Α 2 Α *2 ⎥ 2 ⎦

Α = [ Α 0] . Considering

Α *i ≤ 1/ n (i = 1, 2) , the

No.6


following inequality can be obtained: Α *1 0 2 2 1/ 2 = ⎛⎜ Α1* + Α2* ⎞⎟ ≤1 Α = Α = ⎠ Α *1 Α *2 ⎝ 2

2

(66)

Let γ ' = Α , then we have:

ω ≤ Α Ε = γ' Ε

(67)

Therefore γ ' ≤ 1 , and the gain of subsystem Σ ω e satisfies γ ω = γ ' ≤ 1 . According to small gain theorem, if γ e (γ ω ( s )) < s , then the closed-loop system (61) is ISpS. Then,

γ e (γ ω ( s)) < s

⇒

γ γ' < 1

⎣⎡ e (t ) λ (t ) ⎦⎤ T

T

(

)

T

≤ β ⎣⎡e (0) λ (0) ⎦⎤ , t + δ 0 T

T

(69) Therefore,

e (t ) ∈ L∞ ,

x (t ) ∈ L∞ . There exist

σ 0 > 0 and T > 0 , such that

x( t ) < σ 0 , ∀t ≥ T ,

i.e. the closed-loop system is uniformly ultimately bounded. Furthermore, according to Eq. (63), if d 2 = 0 , the subsystem Σ eω is ISS and the closed-loop system is also ISS. There exists a class KL function such that ⎡⎣ e T (t ) λ T (t ) ⎤⎦

T

(

with Heidenhain PC counter card IK220 is used to measure the rotary displacement of torque motor and the velocity signal is obtained by the difference of rotary displacement. A 16 bit AD/DA card PCI-1716 by Advantech is used to sample torque signal and to send out control voltage. Original designed real-time control program based on RTX real-time operating system and Labwindows/CVI is applied to the system and its sampling frequency is 2 kHz.

(68)

Because 0 < γ ' ≤ 1 , if choosing 0 < γ < 1 , the closed-loop system satisfies ISpS conditions. Therefore, there exist a class KL function and a positive condition δ 0 such that T

· 913 ·

T

≤ β ⎡⎣e T (0) λ T (0) ⎤⎦ , t

)

(70) That means lim e1 (t ) = 0 , i.e., the asymptotic outt →∞

Fig. 2

Experimental setup of ELS.

Parameter identification is performed to get the nominal values of system parameters and the results are shown in Table 1. Table 1

Identified values of ELS parameters

Parameter

Value

Km/(N·m·V−1)

37.7

Ks/(N·m·rad−1)

1 964.7

Jm/(kg·m2)

0.045

Bm/(N·m·(rad·s−1) −1)

2.16

As shown in Fig. 3, the membership functions (NB, NM, NS, ZERO, PS, PM and PB) of fuzzy set is chosen as Eq. (12), in which bij = −3, − 2, , 3 and hij = 0.424 7 .

put tracking is achieved. 5. Application Example

The proposed AFTC controller is implemented on an ELS which is a typical application of PTSS driven by electric torque motor. The goal is to further illustrate that AFTC algorithm can effectively cope with extra torque and achieve desired torque tracking in a practical PTSS. Then comparative experimental results are carried out as follows. 5.1. Experiment setup

As shown in Fig. 2, a direct drive rotary torque motor D143M by Danaher is used as drive component in this ELS system and it is driven by a Danaher digital servo amplifier S620. A torque sensor AKC-17 with measurement range of 300 N·m is used to measure the loading torque. A Heidenhain rotary encoder ECN113

Fig. 3

Membership functions of fuzzy set.

According to ELS parameters in Table 1 and control performance of practical experiment, AFTC controller parameters are chosen as: k1 = 1.1 , k2 = 0.5 , Γ 1 = 0.2 , Γ 2 = 0.6 , σ 1 = 0.15 , σ 2 = 0.25 , γ 1 = 0.5 , γ 2 = 0.5 , ρ1 = 10 , ρ 2 = 8.5 , λ10 = 0.4 , and λ20 = 0.5 . In the following text, three typical experiments for ELS are carried out. 5.2. Static loading experiment

As a basic performance test item for PTSS, static

· 914 ·


loading experiment is carried out firstly. In this condition, actuation system is absent and output shaft of torque motor is fixed directly, thus estimated actuator velocity can be set to zero, i.e., ωˆ a = 0 and μ2 = 0 . This experiment is to investigate torque tracking performance without velocity disturbance. Three algorithms, PID with kP=0.7, kI=0.1, kD=0.000 1, fuzzy control and AFTC, will be compared in this experiment. Here, fuzzy control uses the same membership functions as AFTC. The desired loading torque is given by a 1.0 Hz sinusoidal signal with 80 N·m amplitude. Then tracking errors with three controllers are shown in Fig. 4.

Fig. 5

No.6

Tracking errors of high-frequency in static loading experiment.

As shown in Fig. 5, fuzzy cannot get a better performance than PID at high-frequency but AFTC still have a good performance at high-frequency. 5.3. Eliminating extra torque experiment

The ability to eliminate extra torque is a key issue for ELS, so it is important to measure exact extra torque. An aviation actuator is used in this experiment and its shaft is connected to ELS by a coupling. Assume ELS desired loading torque xd = 0 and the actuator operates with the desired sinusoidal angle command, then the measured torque output of torque motor is so-called extra torque. Since extra torque is caused by actuation system, the estimated actuator velocity ωˆ a will be used in AFTC controller in this experiment. To achieve this goal, off-line model identification by frequency sweep technique for actuation system is carried out and the identification results are shown in Fig. 6.

Fig. 4

Tracking errors of low-frequency in static loading experiment.

It is obvious that the tracking error of AFTC is much smaller than that of PID and smaller than fuzzy due to well-designed and excellent performance of AFTC. In order to verify AFTC performance under high-frequency desired loading torque, experiment is also conducted for high-frequency desired signal which is given by a 6.0 Hz sinusoidal signal with 40 N·m amplitude, then the tracking errors with the above three controllers are shown in Fig. 5.

Fig. 6

Frequency sweep model and identified models of actuation system.

As shown in Fig. 6, the 5th order model fits the frequency sweep model well and it is given as 3.4×109 s2 + 2.0×1013 s + 2.8×1016 s + 4.8×10 s +1.2×108 s3 + 6.7×1011s2 + 5.8×1014 s + 2.4×1016 But it is impossible to apply this model in practical control algorithm due to its high order. Therefore, a 2th order model is found instead of the 5th order model: 5

4 4

Ga ( s ) =

7 328 s + 108 s + 7 372 2

(71)

No.6


Although the 2nd order model cannot fit frequency sweep model well in high frequency band, it can fit very well in low frequency band, especially under 80 rad/s (about 13 Hz). So it is reasonable to use the 2nd order model for compensation because both ELS and actuation system operate at 10 Hz. Since the desired angle signal θ d is easy to obtain and without noises, it is input into identified model Ga ( s ) , then the estimated actuator velocity ωˆ a can be obtained by differentiating the angle output of Ga ( s ) . Set the desired loading torque of ELS as xd = 0 and the desired angle trajectory of actuation system as θ d = 0.1sin( 8.0 πt ) rad . First, ELS operates without any control algorithm, i.e. open-loop, then two control algorithms, fuzzy and AFTC, which are mentioned in Section 5.2, are used to eliminate extra torque, respectively. The extra torque comparison with three control strategies is shown in Fig. 7.

5.4. Gradient loading experiment

Gradient loading is the most common use of PTSS, in which loading torque command is proportional to actuator angle command. For example, the desired command of actuator is θ d = 0.2 sin( 4.0 πt ) rad and torque gradient coefficient is K TG = 100 N ⋅ m/rad , thus the desired loading torque can be calculated by xd = KTGθ d as shown in Fig. 8. Fuzzy and AFTC are also compared in this experiment, actual loading torque outputs are also shown in Fig. 8, while torque tracking errors are shown in Fig. 9.

Fig. 8

Fig. 7

· 915 ·

System commands and torque outputs in gradient loading experiment.

Tracking errors in experiment of eliminating extra torque.

In Fig. 7, it is obvious that AFTC controller has excellent performance in eliminating extra torque. To verify controller performance, detailed values of extra torque with these three strategies are given in Table 2. Table 2

Detailed values of extra torque N·m

Controller

eM

eF

Open-loop

22.256

12.461

Fuzzy

8. 615

6. 846

AFTC

3. 076

2. 231

In Table 2, eM = max{| e1 ( t )|} is the maximum absolute value of tracking error, and eF = max {| e1 (t ) |} is the maximum absolute value of tf − 2tp ≤t ≤tf

tracking error during the last two periods of experiment, t f represents total running time and t p is periodic time. According to the final tracking error eF, fuzzy control can only eliminate about 45% of extra torque while AFTC control can eliminate about 82%. Clearly, AFTC controller has a good performance in eliminating the extra torque in practical PTSS.

Fig. 9

Tracking errors in gradient loading experiment.

The experimental results indicate that AFTC with estimated actuator velocity ωˆ a is very effective in gradient loading testing for actuator. The above experimental results indicate that well-designed AFTC controller can achieve perfect tracking performance for ELS under coupling disturbance from actuator. 6. Conclusions

The adaptive fuzzy torque control is proposed in this paper and the closed-loop system is proved to be semi-globally uniformly ultimately bounded by small gain theorem. The AFTC is applied to ELS and experimental results show its good control performance. Parametric uncertainties and unstructured uncertainties in PTSS are effectively compensated by T-S fuzzy logic system. The AFTC is convenient to be implemented in practical PTSS because only one parameter needs to be turned online at each step. The output velocity of actuator identified model is utilized in AFTC algorithm to eliminate extra torque. This technique is easy to be realized, the compensation signal is smooth without noise, and the experiment of eliminating extra

· 916 ·


torque also illustrates the effectiveness of this technique. References [1] [2] [3]

[4] [5] [6] [7]

[8]

[9]

[10] [11] [12]

[13] [14]

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[15] Yang Y S. Direct robust adaptive fuzzy control (DRAFC) for uncertain nonlinear systems using small gain theorem. Fuzzy Sets and Systems 2005; 151(1): 79-97. [16] Yang Y S, Zhou C J. Adaptive fuzzy H∞ stabilization for strict-feedback canonical nonlinear systems via backstepping and small-gain approach. IEEE Transactions on Fuzzy Systems 2005; 13(1): 104-114. [17] Sontag E D. Further facts about input to state stabilization. IEEE Transactions on Automatic Control 1990; 35(4): 473-476. [18] Khalil H K. Nonlinear system. 3rd ed. New York: Prentice Hall, 2002. [19] Zhang T P, Wen H, Zhu Q. Adaptive fuzzy control of nonlinear systems in pure feedback form based on input-to-state stability. IEEE Transactions on Fuzzy Systems 2010; 18(1): 80-93. [20] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics 1985; 15(1): 116-132. [21] Wang L X. A course in fuzzy systems and control. New York: Prentice Hall, 1997. [22] Wang X J, Wang S P, Wang X D. Electrical load simulator based on velocity-loop compensation and improved fuzzy-PID. Proceeding of IEEE International Symposium on Industrial Electronics, 2009; 238-243.

Biographies: WANG Xingjian received his B.E. degree in mechatronic engineering from Beihang University in 2006. He is currently a Ph.D. student in the School of Automation Science and Electrical Engineering at Beihang University. His research interests are nonlinear control, fault diagnosis, prognosis and health management. E-mail: [email protected] WANG Shaoping received her Ph.D., M.E. and B.E. degrees in mechatronis engineering from Beihang University in 1994, 1991 and 1988. She was promoted to the rank of professor in 2000. Her research interests are reliability, fault diagnosis, prognosis and health management. E-mail: [email protected] ZHAO Pan received his B.E. degree in mechatronic engineering from Beihang University in 2009. He is currently a postgraduate student in the School of Automation Science and Electrical Engineering at Beihang University. His research interests are system identification and robot control. E-mail: [email protected]