ADAPTIVE NEURAL POSITION TRACKING CONTROL FOR …Chiaverini and Fusco describe speed and rotor ux norm tracking H1 controllers with unknown load torque disturbances for current-fed

International Journal of InnovativeComputing, Information and Control ICIC International c⃝2011 ISSN 1349-4198Volume 7, Number 7(B), July 2011 pp. 4503–4516

ADAPTIVE NEURAL POSITION TRACKING CONTROLFOR INDUCTION MOTORS VIA BACKSTEPPING

Jinpeng Yu, Yumei Ma, Bing Chen, Haisheng Yu and Songfeng Pan

Institute of Complexity ScienceQingdao University

No. 308, Ningxia Road, Qingdao 266071, P. R. [email protected]; [email protected]; [email protected]

yu.hs; pansongfeng @163.com

Received March 2010; revised July 2010

Abstract. The position tracking control of induction motors with parameter uncer-tainties and load torque disturbance is addressed. Neural networks are employed to ap-proximate the nonlinearities and an adaptive backstepping technique is used to constructcontrollers. The proposed adaptive neural controllers guarantee that the tracking errorconverges into a small neighborhood of the origin. Compared with the conventional back-stepping, the designed neural controllers’ structure is very simple. Simulation resultsillustrate the effectiveness of the proposed control scheme.Keywords: Nonlinear system, Neural networks, Adaptive control, Induction motor,Uncertainty, Backstepping

1. Introduction. During the past few decades, the control of the induction motor con-stitutes a theoretically challenging problem since the dynamical system is usually multi-variable, coupled and highly nonlinear. One of the most significant developments in thisarea is the field-oriented control (FOC) proposed by Blaschke [1] during the early 1970s,which is based on decoupling of the torque and flux producing components of the statorcurrent [2, 3]. Unfortunately, this control approach suffers from sensitivity to the motorparameter variations and load disturbances.

In order to cope with the above drawback, much research has been done and someadvanced control techniques are applied to the speed or position control of IMs and refer-ences therein. Wai et al. [4] develop a sliding-mode controller for field-oriented inductionmotor servo drive which can overcome the common drawback of FOC. Theoretically, thesliding motion is smooth if the switching frequency of a system is infinite. However, inpractice, the switching frequency is finite, and thus chattering comes out along the slidingsurface [4-7]. Marino et al. [8] develop an adaptive input-output linearizing control of IM.A new approach to dynamic feedback linearization control for an induction motor wasaddressed by Chiasson in [9]. However, the employed method of feedback linearizationrequires the exact mathematical model, so the controller requires the desired dynamics toreplace the system at the d − q axis stator currents [10]. Chiaverini and Fusco describespeed and rotor flux norm trackingH∞ controllers with unknown load torque disturbancesfor current-fed induction motors [11]. The key of the H∞ control is to synthesize a feed-back law that renders the closed-loop system to satisfy a prescribed H∞-norm constraintwhich represents desired stability or tracking requirements. However, in order to ensurerobustness under large uncertainty perturbations, the H∞ design usually brings a solutionwith high control gain which employs this approach not feasible in practical application.

4503

4504 J. YU, Y. MA, B. CHEN, H. YU AND S. PAN

Backstepping is a newly developed technique to control the nonlinear systems withparameter uncertainty, particularly those systems in which the uncertainties do not satisfymatching conditions [12-15]. Though the conventional backstepping is successfully appliedto the control of IM drivers recently, a major problem with backstepping approaches isthat certain functions must be “linear in the unknown system parameters”, and somevery tedious analysis is needed to determine a “regression matrix” [16]. Dawson et al. [17]used conventional backstepping technique to control a dc motor, but one will notice thatthe regression matrix almost covers one full page. Another drawback in the conventionalbackstepping control method is called the problem of “explosion of terms” caused by therepeatedly differing virtual control signals [18]. For example, the adaptive backsteppingcontroller developed in [19] to control the induction motors is a representative “explosionof terms” problem. The proposed controller contains too many nonlinear terms and it isdifficult to be implemented in practice.In recent years, neural network (NN) method has attracted considerable attention be-

cause of its inherent capability for modelling and controlling highly uncertain, nonlinearand complex systems [20]. Kwan and Lewis presented a new robust backstepping speedcontroller for IMs using NNs [16], where a two-layer NN was used to design the fictitiouscontroller, and a second NN was utilized to robustly realize the fictitious NN signals. Waiand Chang [21] address a newly designed decoupling system and a backstepping waveletneural network (WNN) control system for achieving high-precision position-tracking per-formance of an indirect field-oriented IM drives. The radial basis function (RBF) neuralnetwork (NN) [22, 23] is considered as a two-layer network, which contains the hiddenand the output layers. Therefore, neural networks can be used to handle uncertain infor-mation, and furthermore, be applied to control the systems which are ill-defined or toocomplex to have a mathematical model. It has been found one of the popular and con-ventional tools in functional approximations provides an effective way to design controlsystem that is one of important applications in the area of control engineering.In this paper, an adaptive neural control approach is proposed for position tracking

control of IMs via backstepping. During the controller design process, the RBF neuralnetwork is employed to approximate the unknown nonlinearities; adaptive technique andbackstepping are used to construct NN controllers. This means that the uncertain pa-rameters can be taken into account, and that no regression matrices need to be foundand that the problem of “explosion of terms” is overcome. Thus, the major problemswith the traditional backstepping method are cured. To verify the advantage of the pro-posed control method, a comparison between the adaptive neural control design and theclassical backstepping design is given. Moreover, the proposed controllers guarantee thatthe tracking error converges into a small neighborhood of the origin and all the closed-loop signals are bounded. Simulation results demonstrate the effectiveness and robustnessagainst the parameter uncertainties and load disturbances.

2. Mathematical Model of the IM Drive System and Preliminaries. Under theassumptions of equal mutual inductance and a linear magnetic circuit and through thefield-oriented transformation, a fifth-order induction motor, which includes both the elec-trical and mechanical dynamics, can be described in the well known (d − q) frame asfollows [2, 8].

dθ

dt= ω,

dω

dt=npLm

LrJψdiq −

TLJ,

diqdt

= −L2mRr + L2

rRs

σLsL2r

iq −npLm

σLsLr

ωψd − npωid −LmRr

Lr

iqidψd

+1

σLs

uq,

NEURAL CONTROL FOR INDUCTION MOTORS 4505

dψd

dt= −Rr

Lr

ψd +LmRr

Lr

id,

diddt

= −L2mRr + L2

rRs

σLsL2r

id +LmRr

σLsL2r

ψd + npωiq +LmRr

Lr

i2qψd

+1

σLs

ud,

where σ = 1− L2m

LsLr. θ, ω, Lm, np, J , TL and ψd denote the rotor position, rotor angular

velocity, mutual inductance, pole pairs, inertia, load torque and rotor flux linkage. id andiq stand for the d− q axis currents. ud and uq are the d− q axis voltages. Rs and Ls meanthe resistance, inductance of the stator. Rr and Lr denote the resistance, inductance ofthe rotor. For simplicity, the following notations are introduced:

x1 = θ, x2 = ω, x3 = iq, x4 = ψd, x5 = id, a1 =npLm

Lr

, b1 = −L2mRr + L2

rRs

σLsL2r

,

b2 = − npLm

σLsLr

, b3 = np, b4 =LmRr

Lr

, b5 =1

σLs

, c1 = −Rr

Lr

, d2 =LmRr

σLsL2r

.

By using these notations, the dynamic model of IMs can be described by the followingdifferential equations:

x1 = x2,

x2 =a1Jx3x4 −

TLJ,

x3 = b1x3 + b2x2x4 − b3x2x5 − b4x3x5x4

+ b5uq,

x4 = c1x4 + b4x5,

x5 = b1x5 + d2x4 + b3x2x3 + b4x23x4

+ b5ud. (1)

The control objective is to design an adaptive NN controller such that the state variablexi (i = 1, 4) follows the given reference signal xid and all the closed-loop signals arebounded. In control engineering, the RBF neural network is usually used as a tool formodeling unknown nonlinear functions because of its good approximation capability. Inthis paper, the RBF neural network will be used to approximate the unknown continuousfunction φ(z) : Rq → R as φ(z) = ϕ∗TP (z) where z ∈ Ωz ⊂ Rq is the input vector with qbeing the neural network input dimension, ϕ∗ = [Φ∗

1, . . . ,Φn]T ∈ Rn, is the weight vector,

n > 1 is the neural network node number, and P (z) = [p1(z), . . . , pn(z)]T ∈ Rn is the

basis function vector with pi(z) chosen as the commonly used Gaussian function in the

following form: pi(z) = exp[−(z−νi)

T (z−νi)

k2i

], i = 1, 2, . . . , n where νi = [νi1, . . . , xνiq]

T is

the center of the receptive field and ki is the width of the Gaussian function. It has beenproven in [24, 25] that, for given scalar ε > 0, by choosing sufficiently large l, the RBFneural network can approximate any continuous function over a compact set Ωz ∈ Rq toarbitrary accuracy as φ(z) = ϕTP (z)+δ(z) ∀z ∈ Ωz ⊂ Rq where δ(z) is the approximationerror, satisfying |δ(z)| ≤ ε and ϕ is an unknown ideal constant weight vector, which is anartificial quantity required for analytical purpose. Typically, ϕ is chosen as the value of

ϕ∗ that minimizes |δ(z)| for all z ∈ Ωz, i.e., ϕ := arg minϕ∗∈ Rn

supz∈Ωz

∣∣φ(z)− ϕ∗TP (z)∣∣.

3. Adaptive NN Controller Design. This section is devoted to developing a noveladaptive NN control design procedure. The system (1) leads a simplified system structurewith two approximately decoupled subsystems, namely, the subsystem with state variables(x1, x2, x3) and control signal uq, and the subsystem with (x4, x5) as state variables and


ud as control input. The backstepping design procedure contains 5 steps. At each designstep, a virtual control function αi (i = 1, 2, 3) will be constructed by using an appropriateLyapunov function. At the final step, the real controller is constructed to control thesystem.Step 1: For the reference signal x1d, define the tracking error variable as z1 = x1−x1d.

From the first differential equation of (1), the error dynamic system is given by z1 =x2− x1d. Consider the Lyapunov function candidate as V1 =

12z21 , then the time derivative

of V1 is computed by

V1 = z1z1 = z1(x2 − x1d). (2)

Construct the virtual control law α1 as:

α1(x1, x1d, x1d) = −k1z1 + x1d (3)

with k1 > 0 being a design parameter. By using (3), (2) can be rewritten as V1 =−k1z21 + z1z2 with z2 = x2 − α1.Step 2: Differentiating z2 gives z2 = x2 − α1 = a1

Jx3x4 − TL

J− α1. Now, choose the

Lyapunov function candidate as V2 = V1 +J2z22 . Obviously, the time derivative of V2 is

given by

V2 = V1 +J

2z2z2 = −k1z21 + z2(z1 + a1x3x4 − TL − Jα1). (4)

Remark 3.1. Due to the parameter TL being bounded in practice system, we assume theTL is unknown but its upper bound is d > 0, which maybe unknown, namely, 0 ≤ TL ≤ d.

Obviously, z2TL ≤ 12ε22z22+

12ε22d

2, where ε2 is an arbitrary small positive constant. Then,

the time derivative of V2 satisfies the following inequality:

V2 ≤ −k1z21 + z2

(z1 +

1

2ε22z2 + a1x3x4 − Jα1

)+

1

2ε22d

2. (5)

Remark 3.2. In the realistic model of IMs, the system inertia J may be unknown, so itcannot be used to construct the control signal unless we specify it by a corresponding adap-tation law. Thus, we introduce the adaptive scalar parameter, i.e., J , be the estimationof J , in this letter.

Then, the virtual control α2 is constructed as:

α2(Z2) =1

a1x4

(−k2z2 −

1

2ε22z2 − z1 + J α1

)=

1

a1x4

(−k2z2 − z1 + J α1

), (6)

where k2 = k2+1

2ε22> 0 is a positive design parameter and Z2 =

[x1, x2, x1d, x1d, x1d, J

]T.

Adding and subtracting α2 in the bracket in (5) shows that V2 = −k1z21−k2z22+a1z2z3x4+z2(J − J)α1 +

12ε22d

2 with z3 = x3 − α2.Step 3: Differentiating z3 results in the following differential equation.

z3 = x3 − α2 = b1x3 + b2x2x4 − b3x2x5 − b4x3x5x4

+ b5uq − α2.


Choose the Lyapunov function candidate as V3 = V2 +12z23 . Furthermore, differentiating

V3 yields

V3 = V2 + z3z3 = V2 + z3

(b1x3 + b2x2x4 − b3x2x5 − b4

x3x5x4

+ b5uq − α2

)= −k1z21 − k2z

22 +

1

2ε22d

2 + z2

(J − J

)α1 + z3(f3 + b5uq), (7)

where

α1 = −k1(x2 − x1d) + x1d,

α2 =2∑

i=1

∂α2

∂xixi +

2∑i=0

∂α2

∂x(i)1d

x(i+1)1d +

∂α2

∂J

˙J +

∂α2

∂x4x4

=∂α2

∂x1x2 +

∂α2

∂x2

(a1Jx3x4 −

TLJ

)+

2∑i=0

∂α2

∂x(i)1d

x(i+1)1d +

∂α2

∂J

˙J +

∂α2

∂x4(c1x4 + b4x5),

f3(Z) = a1z2x4 + b1x3 + b2x2x4 − b3x2x5 − b4x3x5x4

− α2,

Z =[x1, x2, x3, x4, x5, x1d, x1d, x1d, J

]T. (8)

Notice that f3 contains the derivative of α2, therewithal, the unknown parameter Jappears in the expression of f3. This will make the classical adaptive backstepping designvery complex and troublesome, and the designed control law uq would have a complexstructure. To avoid this trouble and simplify the control signal structure, we will employthe RBF neural network to approximate the nonlinear function f3. As shown later, thedesign procedure of uq becomes simple and uq is of a simple structure. Thus, accordingto the RBF neural network approximation property, for given ε3 > 0, there exists RBFneural network ϕT

3 P (Z) such that

f3(Z) = ϕT3 P3(Z) + δ3(Z), (9)

where δ3(Z) is the approximation error and satisfies |δ3| ≤ ε3. Consequently, a straight-forward calculation produces the following inequality:

z3f3(Z) = z3(ϕT3 P3(Z) + δ3(Z)

)≤ 1

2l23z23 ∥ϕ3∥2 P T

3 (Z)P3(Z) +1

2l23 +

1

2z23 +

1

2ε23. (10)

Thus, it follows immediately from substituting (10) into (7) that

V3 ≤ −k1z21 −k2z22 +z2(J − J

)α1+

1

2ε22d

2+1

2l23z23 ∥ϕ3∥2 P T

3 P3+1

2l23+

1

2z23 +

1

2ε23+ b5z3uq.

At this present stage, the control law uq is designed as:

uq =1

b5

(−k3z3 −

1

2z3 −

1

2l23z3θP

T3 P3

), (11)

where θ is the estimation of the unknown constant θ which will be specified later. Fur-thermore, using the equality (11), it can be verified easily that

V3 ≤ −3∑

i=1

kiz2i + z2

(J − J

)α1 +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε22d

2 +1

2ε23.


Step 4: For the reference signal x4d, define the tracking error variable as z4 = x4−x4d.Then, one has z4 = x4 − x4d.Choose the Lyapunov function candidate as V4 = V3+

12z24 . From the fourth differential

equation of (1), the derivative of V4 is given by

V4 ≤ −3∑

i=1

kiz2i +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε22d

2 +1

2ε23

+z2(J − J)α1 + z4 (c1x4 + b4x5 − x4d) . (12)

Now, construct the virtual control law α3 as:

α3(x4, x4d, x4d) =1

b4(−k4z4 − c1x4 + x4d) (13)

with k4 > 0 being a design parameter. Define z5 = x5 − α3. By using (13), the inequality(12) can be expressed as:

V4 ≤ −4∑

i=1

kiz2i +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε23 +

1

2ε22d

2 + z2

(J − J

)α1 + z4z5.

Step 5: At this step, we will construct the control law ud. To this end, choose theLyapunov function candidate as V5 = V4 +

12z25 . Then, the derivative of V5 is given by

V5 ≤ −4∑

i=1

kiz2i +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε23

+1

2ε22d

2 + z2

(J − J

)α1 + z5(f5 + b5ud), (14)

where f5(Z) = z4 + b1x5 + d2x4 + b3x2x3 + b4x23

x4− α3. Similarly, according to the RBF

neural network approximation property, the RBF neural network ϕT5 P5(Z) is utilized to

approximate the nonlinear function f5 such that for given ε5 > 0,

z5f5(Z) ≤1

2l25z25 ∥ϕ5∥2 P T

5 P5 +1

2l25 +

1

2z25 +

1

2ε25. (15)

Substituting (15) into (14) gives

V5 ≤ −5∑

i=1

kiz2i +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε23 +

1

2l25z25

(∥ϕ5∥2 − θ

)P T5 P5

+1

2l25 +

1

2ε25 +

1

2ε22d

2 + z2

(J − J

)α1 + z5b5ud.

Now, design ud as:

ud =−1

b5

(k5z5 +

1

2z5 +

1

2l25z5θP

T5 P5

)(16)

and define θ = max∥ϕ3∥2 , ∥ϕ5∥2

. Furthermore, using the equality (16), the derivative

of V5 can be verified easily that

V5 ≤ −5∑

i=1

kiz2i +

1

2l23z23

(∥ϕ3∥2 − θ

)P T3 P3 +

1

2l23 +

1

2ε23 +

1

2ε22d

2

+z2

(J − J

)α1 +

1

2l25z25

(∥ϕ5∥2 − θ

)P T5 P5 +

1

2l25 +

1

2ε25. (17)


Introduce variables J and θ as J = J − J , θ = θ − θ, and choose the Lyapunov functioncandidate as:

V = V5 +1

2r1J2 +

1

2r2θ2, (18)

where ri, i = 1, 2 are positive constants. By differentiating V and taking (17) and (18)into account, one has

V ≤ −5∑

i=1

kiz2i +

1

2l23 +

1

2ε23 +

1

2l25 +

1

2ε25 +

1

2ε22d

2 +1

r1J(r1z2α1 +

˙J)

+1

r2θ

[− r22l25

z25PT5 P5 −

r22l23

z23PT3 P3 +

˙θ

]. (19)

According to (19), the corresponding adaptive laws are chosen as follows:

˙J = −r1z2α1 −m1J ,

˙θ =

r22l23

z23PT3 P3 +

r22l25

z25PT5 P5 −m2θ, (20)

where mi, for i = 1, 2, l3 and l5 are positive constant.

Remark 3.3. Form the above adaptive NN controllers, it is clearly seen that the proposedcontrollers have simpler structure. This means that the proposed adaptive NN controllersare easy to be implemented in practical engineering. For this, a comparison between theproposed controller and the classical one will be given in Appendix B.

Theorem 3.1. Consider the system (1) satisfying Assumptions 1 and 2 and the givenreference signal xd. Then under the action of the adaptive NN controllers (11) and (16),the tracking error of the closed-loop controlled system will converge into a sufficiently smallneighborhood of the origin and all the closed-loop signals will be bounded. The detailedproof is given in Appendix A.

4. Simulation. To give the further comparison, the proposed adaptive NN controllers(11) and (16) and the classical backstepping controllers (27) and (31) will be used tocontrol the following real system under the initial condition of x1 = x2 = x3 = x4 = x5 =0.1, respectively. The system parameters are listed as follows: J = 0.0586Kgm2, Rs =0.1Ω, Rr = 0.15Ω, Ls = Lr = 0.0699H, Lm = 0.068H, np = 1. The reference signals are

taken as x1d = 0.5 sin(t) + sin(0.5t) and x4d = 1 with TL being TL =

1.5, 0 ≤ t ≤ 20,3, t ≥ 20.

The proposed adaptive NN controllers are used to control this induction motor. Thecontrol parameters are chosen as follows:

k1 = 150, k2 = 100, k3 = 200, k4 = k5 = 100, r1 = r2 = 0.05,

m1 = m2 = 0.05, l3 = l5 = 0.5.

And RBF neural networks are chosen in the following way. Neural network ϕT3 P3(Z)

contains eleven nodes with centers spaced evenly in the interval [–5, 5] and widths beingequal to 2. Neural network ϕT

5 P5(Z) is chosen as same as ϕT3 P3(Z). The simulation

for adaptive fuzzy control is run under the assumption that the system parameters andthe nonlinear functions are unknown. In addition, to illustrate the effectiveness of theproposed adaptive neural control, the classical controllers (27) and (31) are also used tocontrol this system, the chosen controller parameters are:

k1 = 50, k2 = 50, k3 = 40, k4 = 50, k5 = 20.

All the simulation results for the cases of adaptive NN control and classical backsteppingcontrol are shown by Figures 1-10. Figures 1-4 display the system output responses and


the reference signals for both control approaches, Figures 5-8 show the control inputsignals. From Figures 1-4, it is seen clearly that under the actions of controllers (11) and(16) and the controllers (27) and (31), the system outputs follow the desired referencesignals well. Figures 9 and 10 display the curve of id and iq. From the above simulationresults, it is seen clearly that under the actions of the proposed controllers, the systemoutputs follow the desired reference signals well even if system dynamics containing theunknown nonlinear functions.

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

Time(sec)

Pos

ition

(rad)

x1x1d

Figure 1. x1 and x1d for adaptive NN control

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

Time(sec)

Pos

ition

(rad)

x1x1d

Figure 2. x1 and x1d for classical backstepping

0 10 20 30 40 500.982

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

Time(sec)

Rot

or fl

ux lin

kage

(wb)

x4x4d

Figure 3. x4 and x4d for adaptive NN control


0 10 20 30 40 500.982

0.984

0.986

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

Time(sec)

Rot

or fl

ux li

nkag

e(w

b)

x4x4d

Figure 4. x4 and x4d for classical backstepping

0 10 20 30 40 50−500

−400

−300

−200

−100

0

100

200

300

400

500

Time(sec)

uq(

v)

uq

Figure 5. uq for adaptive NN control

0 10 20 30 40 50−500

−400

−300

−200

−100

0

100

200

300

400

500

Time(sec)

uq(

v)

uq

Figure 6. uq for classical backstepping

Remark 4.1. From the Figures 1-4, it is clearly seen that under the action of controller(11) and (16) the tracking performance of the closed-loop system is perfect and very similarto the one under the classical backstepping control. Then, it should be pointed out thatthe controller is constructed under the condition that the nonlinear functions in systemdynamics are unknown. And the proposed controller has much simpler structure thanthe classical backstepping controller. This implies that the proposed controller is easilyimplemented in practice.

Remark 4.2. The controllers (11) and (16) are constructed under the assumption thatthe nonlinear functions in system dynamics are unknown. Therefore, the developed controlstrategy can be used to control the nonlinear uncertain systems. Since the controllers (27)


0 10 20 30 40 50−200

−150

−100

−50

0

50

100

150

200

Time(sec)

ud(

v)

ud

Figure 7. ud for adaptive NN control

0 10 20 30 40 50−200

−150

−100

−50

0

50

100

150

200

Time(sec)

ud(

v)

ud

Figure 8. ud for classical backstepping

0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

2

Time(sec)

Id(A

), Iq

(A)

idiq

Figure 9. id and iq for adaptive NN control

and (31) require the precise information on the nonlinear functions, theoretically, whenthe functions are unknown the classical backstepping will be invalid to control the unknownsystems.

5. Conclusion. Based on the backstepping technique, an adaptive NN control schemeis proposed to control induction motors. The proposed controllers overcome the majorproblems of the traditional backstepping and guarantee that the tracking error convergesinto a small neighborhood of the origin and all the closed-loop signals are bounded.Simulation results are provided to demonstrate the effectiveness and robustness againstthe parameter uncertainties and load disturbances.


0 10 20 30 40 50−1

−0.5

0

0.5

1

1.5

2

Time(sec)

Id(A

), Iq

(A)

idiq

Figure 10. id and iq for classical backstepping

Acknowledgment. This work is partially supported by the Natural Science Foundationof China (61074008, 60774027, 60774047), the Natural Science Foundation of ShandongProvince (ZR2009GM034), the cultivating plan of excellent degree thesis (Qingdao Uni-versity) and Shandong Province Domestic Visitor Foundation.

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Appendix A.To address the stability analysis of the resulting closed-loop system, substitute (20)

into (19) to obtain that

V ≤ −5∑

i=1

kiz2i +

1

2l23 +

1

2ε23 +

1

2l25 +

1

2ε25 +

1

2ε22d

2 − m1

r1J J − m2

r2θθ. (21)

For the term −J J , one has −J J ≤ −J(J + J) ≤ −12J2 + 1

2J2. Similarly, −θθ ≤

−12θ2 + 1

2θ2 holds. Consequently, by using these inequalities (21) can be rewritten in the

following form.

V ≤ −5∑

i=1

kiz2i −

m1

2r1J2 − m2

2r2θ2 +

1

2l23 +

1

2ε23 +

1

2l25 +

1

2ε25 +

1

2ε22d

2 +m1

2r1J2 +

m2

2r2θ2

≤ −a0V + b0, (22)

where a0 = min2k1,

2k2J, 2k3,2k4,2k5,m1,m2

and b0 = 1

2l23 +

12ε23 +

12l25 +

12ε25 +

12ε22d

2 +m1

2r1J2 + m2

2r2θ2. Furthermore, (22) implies that

V (t) ≤(V (t0)−

b0a0

)e−a0(t−t0) +

b0a0

≤ V (t0) +b0a0, ∀t ≥ t0. (23)

As a result, all zi(i = 1, 2, 3, 4, 5), J and θ belong to the compact set

Ω =

(zi, J , θ)|V ≤ V (t0) +

b0a0,∀t ≥ t0

.

Namely, all the signals in the closed-loop system are bounded. Especially, from (23), wehave

limt→∞

z21 ≤ 2b0a0.

From the definitions of a0 and b0, it is clear that to get a small tracking error by takingri sufficiently large and li and εi small enough after giving the parameters ki and mi.


Appendix B.This section devotes to design controllers by classical backstepping approach.Step 1: For the reference signal x1d, define the tracking error variable as z1 = x1−x1d.

From the first differential equation of (1), the error dynamic system is given by z1 =x2 − x1d. Choose Lyapunov function candidate as V1 = 1

2z21 , then the time derivative of

V1 is computed by

V1 = z1z1 = z1(x2 − x1d). (24)

Construct the virtual control law α1 as α1 = −k1z1 + x1d with k1 > 0 being a designparameter and z2 = x2 −α1. Then, Equation (24) can be rewritten as V1 = −k1z21 + z1z2.

Step 2: Differentiating z2 gives z2 = x2 − α1 = a1Jx3x4 − TL

J− α1. Now, choose the

Lyapunov function candidate as V2 = V1 +J2z22 . Obviously, the time derivative of V2 is

given by V2 = V1 +J2z2z2 = −k1z21 + z2(z1 + a1x3x4 − TL − Jα1). The virtual control α2

is constructed as:

α2 =1

a1x4(−k2z2 − z1 + TL + Jα1), (25)

where k2 > 0 is a positive design parameter and α1 = −k1(x2 − x1d) + x1d. Then, thetime derivative of V2 becomes V2 = −k1z21 − k2z

22 + a1z2z3x4, with z3 = x3 − α2.

Step 3: Differentiating z3 results in the following differential equation.

z3 = x3 − α2 = b1x3 + b2x2x4 − b3x2x5 − b4x3x5x4

+ b5uq − α2.

Choose the Lyapunov function candidate as V3 = V2 +12z23 . Furthermore, differentiating

V3 yields

V3 = V2 + z3z3 = −k1z21 − k2z22

+ z3

(a1z2x4 + b1x3 + b2x2x4 − b3x2x5 − b4

x3x5x4

+ b5uq − α2

),

(26)

where

α2 =2∑

i=1

∂α2

∂xixi +

2∑i=0

∂α2

∂x(i)1d

x(i+1)1d +

∂α2

∂x4x4

=∂α2

∂x1x2 +

∂α2

∂x2

(a1Jx3x4 −

TLJ

)+

2∑i=0

∂α2

∂x(i)1d

x(i+1)1d +

∂α2

∂x4(c1x4 + b4x5).

And the control law uq is designed as:

uq = − 1

b5

(k3z3 + a1z2x4 + b1x3 + b2x2x4 − b3x2x5 − b4

x3x5x4

)+

1

b5

(∂α2

∂x1x2 +

∂α2

∂x2

(a1Jx3x4 −

TLJ

)+

2∑i=0

∂α2

∂x(i)1d

x(i+1)1d +

∂α2

∂x4(c1x4 + b4x5)

).(27)

Furthermore, using the equality (27), it can be verified easily that V3 ≤ −∑3

i=1 kiz2i .

Step 4: For the reference signal x4d, define the tracking error variable as z4 = x4−x4d.Then one has z4 = x4 − x4d. Choose the Lyapunov function candidate as V4 = V3 +

12z24 .

Then, the derivative of V4 is given by

V4 = V3 + z4z4 ≤ −3∑

i=1

kiz2i + z4 (c1x4 + b4x5 − x4d) . (28)


Now, construct the virtual control law α3 as:

α3 =1

b4(−k4z4 − c1x4 + x4d) (29)

with k4 > 0 being a design parameter. Define z5 = x5 − α3. By using (29), (28) can beexpressed as V4 ≤ −

∑4i=1 kiz

2i + z4z5.

Step 5: At this step, we will construct the control law ud. To this end, choose theLyapunov function candidate as V5 = V4 +

12z25 . Then the derivative of V5 is given by

V5 ≤ −4∑

i=1

kiz2i + z5

(z4 + b1x5 + d2x4 + b3x2x3 + b4

x23x4

− α3 + b5ud

), (30)

Now, design ud as:

ud =−1

b5

(k5z5 + z4 + b1x5 + d2x4 + b3x2x3 + b4

x23x4

− α3

)=

−1

b5

(k5z5 + z4 + b1x5 + d2x4 + b3x2x3 + b4

x23x4

)+

1

b4b5[(−k4 − c1) x4 + k4x4d + x4d] (31)

with k5 > 0.

Remark 5.1. By comparing the adaptive fuzzy controllers Equations (11) and (16) withthe conventional backstepping controllers Equations (27) and (31), it can be seen clearlythat the expression of backstepping controller (27) and (31) would be much more compli-cated than that of the new controller (11) and (16). The number of terms in the expressionof (27) and (31) is much greater. This drawback was called “explosion of terms” [18].

ADAPTIVE NEURAL POSITION TRACKING CONTROL FOR …Chiaverini and Fusco describe speed and rotor ux norm tracking H1 controllers with unknown load torque disturbances for current-fed

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