Robust Adaptive Integral Backstepping Control and its Implementation on Motion Control

System

Shubhobrata Rudra

Assistant Prof. of Electrical Engineering

Calcutta Institute of Engineering and Management

Kasturi Ghosh

Student of Electrical Engineering

Calcutta Institute of Engineering and Management

Monalisa Das

Student of Electrical Engineering

Calcutta Institute of Engineering and Management

Abstract—In this paper a design methodology for a novel robust adaptive integral backstepping controller for the motion control system, has been presented in a systematic manner. Backstepping is a realistic nonlinear control design algorithm based on Lyapunov design approach, as a consequence it automatically ensures the convergence of the regulated variable to zero. Adaptation schemes are designed to estimate the inertia variation and load disturbance in the motion control systems. Integral action is being used to enhance the control action of the controller in steady state against the disturbances. We explore the concept of continuous switching function in parameter update law to ensure the robustness of the adaptive design. The effectiveness of the proposed algorithm has been demonstrated in simulation studies. The controller design has been evaluated not only for the tracking performance but also for the parameters convergence rate of the system. It is quite interesting to note that during the simulation it does not require any prior information about the parameters of the mathematical model of the motion control system.

Key Words: Robust Adaptive Backstepping, Integral action, continuous switching function, Lyapunov Function, Motion Control Model, Global stability.

I. INTRODUCTION

Generally in industrial motion control systems where three control loops typically position, velocity, and torque are nested to form a multiple-loop system. The major concern of the designs is the position and velocity loops. It is apparent that the performance of the position and velocity control loops would definitely be limited by the physical ability of motor drives and the load, because they are directly dealing with the system load and mechanical parts [1]. The bandwidth of the position and velocity control loops restricted the performance of the overall motion control system. Hence to achieve improve bandwidth of the system design of position and velocity control

loops are very crucial in overall design perspective [1].

In the last decade, significant developments have been reported in the field of nonlinear control. Backstepping is one of the most accepted and efficient control algorithms among those nonlinear control design algorithms [2, 3]. The main idea of the backstepping method is that the overall dynamic system is partitioned into two series cascaded subsystems. Therefore, the states of the first subsystem are the control variables for the second. In backstepping approach, first of all, the desired control input for the second subsystem is computed and then the control input for the first subsystem is computed so as to realize the desired state, which is the desired control input for the second subsystem. Inspired by the efficiency of backstepping control algorithm, we have proposed a new control schemes that explore the nonlinear backstepping design and are aimed at dealing with the parameter variations, load disturbance and friction presented in the motion control systems. The design of an adaptive integral backstepping controller for motion control system have been reported on [1,4,7,8,10]. However, when the knowledge of the parameters of the system is incomplete or approximate system data are available, approximation errors creep into the feedback loop which makes the system difficult to achieve its desired control performance. Although integral action enhance the control effort of the controller in steady state [1,4,7,8,9,10] but it is insufficient to mitigate the problem of adaptive systems originated due to high rate of adaptation, parameter drift etc. This type of problems can be addressed with the help of robust adaptive controller [5,6].

In this paper, authors have addressed this very challenging problem of designing an adaptive controller for a motion control loop without any prior knowledge of the parameters of the motion control model. In this paper, the authors have proposed a

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robust adaptive nonlinear control scheme to achieve the desired tracking performance of the motion control system. The robust adaptive control law has been derived in two steps. First of all an adaptive integral backstepping controller has been designed for the motion control model, and at next step we are robustifying those parameter update law of uncertain parameter using continuous switching function. The analysis of the system stability and asymptotic position tracking performance is conducted through the Lyapunov function. The efficacy of the controller is extensively verified through the performance study of the proposed controller in simulation environment.

II. PROBLEM FORMULATION

We use simplified second-order differential equations for the motion control model to explain the adaptive backstepping control design [1]:

ωθ =dt

d

Lq TTdt

dJ −=ω

(1)

where � is the motor angular position, � is the motor angular velocity, J is the total effective inertia, Tq be the acting torque generated by the motor drive, and TL is the load torque. In our design, Tq is our control input to the motion control system. In order to confine the inertia variation and the load torque disturbance, we assume that J and TL are unknown positive constants. We will develop adaptation schemes to estimate the effects of these unknown parameters and compensate them in our backstepping control system.

III. DESIGN OF ROBUST ADAPTIVE INTEGRAL BACKSTEPPING CONTROL LAW

In this section, the proposed robust adaptive integral backstepping control law has been derived in two steps. At first, the adaptive integral backstepping control law has been derived by considering the Lyapunov stability criterion. Then, the robust adaptive integral backstepping control law has been formulated by incorporating necessary robustification measures in the previously designed adaptive backstepping control law. This decoupled two-stage design procedure is simple and straight forward but offers robust stability and tracking performance.

A. Adaptive Integral Backstepping Control Law

The primary objective of the motion control system is to track a continuous bounded reference signal �ref, while keeping all the internal states of the system bounded. Here, an adaptive integral backstepping control law will be derived that will

generate a control input to the system so that the tracking error becomes equal to zero asymptotically. The mathematical model of the motion control system has already been discussed in previous section which is given by

Lq TTJ −=θ�� (2)

Now let us assume two state of the system,

θ=1z and θ�=2z the equation (2) can be written in

state space form as

(3)

For sake of simplicity and ease of computation we introduce a normalized variable h where h is

given byJ

Th L= , the above equation becomes

( ) qThzJ =+2� (4)

Now, let us define an error variable e1 as

θθ −= rerfe1 (5)

where �ref is the reference signal, it is a piecewise continuous and bounded function of time. Now, the objective is to design a virtual control law zref which makes e1�0. Let us consider the following control Lyapunov function

211 2

1eV =

(6)

The first order time derivative of V1 becomes

( ) ( )211111 zeeeeV refref −=−== θθθ ����� (7)

Now, the following error variable has been defined as:

22 zze ref −= (8)

where zref is a virtual control law for equation (7). In this step the objective of the design is to find out a suitable virtual control law for equation (7) which would make the above mentioned first order system stabilizable. To introduce an integral action in steady state a suitable choice of virtual control law is given below in equation (9)

1111 χλθ ++= refref ecz �

(9)

where c1 and �1 are positive design constants, and

( )�=t

dtte0

11χ which guarantees the asymptotic

�

�

21 zz =� �

Lq TTzJ −=2� �

stability of the system. Then the time derivative of e2

becomes

(((( ))))(((( )))) hJ

Teeecc

hJ

Teczz

dt

de

qref

qrefref

++++−−−−++++++++−−−−++++−−−−====

++++−−−−++++++++====−−−−====

11112111

111122

λθχλ

χλθ

��

������

(10)

Now, from equation (10) the expression for Tq

can be found as

( ) ( )( )hceccecJT refq

ˆˆ ++−+++−= θχλλ ��11122111

211 (11)

In the above equation c2 is a positive design constant. Although the expression of the torque is determined by equation (11), the control design is still incomplete, because the parameter adaptation law is not yet determined. Now, the following parameter errors have been defined:

hhh

JJJ

ˆ

ˆ

−=

−= (12)

Substituting the expression of torque Tq from equation (11) in the expression (10), the following error dynamics has been obtained:

heec

hceccecJ

J

dt

deref

+−−

++−+++−=

122

1112211121

2 1

}ˆ)(){( θχλλ ��

(13)

In order to find out the parameter update laws for the parameter J and h the following augmented Lyapunov function has been formulated for the closed loop system:

2

2

2

1

22

21

21

12 2

1

2

1

2

1

2

1

2hJ

JeeV

γγχλ ++++= (14)

Here, the augmented Lyapunov function includes all error variables as well as the parameter errors. The time derivative of V2 is given by:

)ˆ

( }ˆ

)ˆ)(

)(({

ˆˆ

ref dt

hdeh

dt

Jdhecc

ceceJ

Jecec

dt

hd

h

h

dt

Jd

J

JeeeeV

22

1221

11111212

222

211

2122111112

11

1

γγθ

χλλ

γγχχλ

−+−+++

+−+−+−−=

���

����

�−+�

��

����

�−+++=

��

����

(15)

From the above equation the following parameter

updates law can be constructed for the parameters J

and �

)ˆ)()((ˆ

ref hceccecedt

Jd ++−+++−= θχλλγ ��11122111

2121 1

(16)

22edt

hd γ=ˆ

(17)

The above parameter update laws make the derivative of the augmented Lyapunov function V2

negative definite as give below:

222

2112 ececV −−−−−−−−====�

(18)

Now, it is evident from equation (18) that the error dynamics of the system is asymptotically stable. Therefore, the derived adaptive backstepping control law stabilizes system.

B. Robust Adaptive Backstepping Control Law

During the design of adaptive integral backstepping control law in the last section, it has been assumed that the model of the plant is free from the un-modeled dynamics, parameters drift, and noise. However, in actual systems these assumptions may not be valid. When the knowledge of the parameters of the motion control system is incomplete or approximate system data is available, approximation errors creep into the feedback loop [5, 6]. Introduction of Integral action may enhance the steady state control effort of the controller in steady state [1] but that is not sufficient enough to serve the problems of instability originated from the parameter drift or instability causes due to high rate of adaptation. Therefore, while using a high rate of parameter adaptation gain, a modification of the above control law is necessary so that the stability of the overall system is not affected. In order to mitigate the above problem a continuous switching function [5] has been introduced in the parameter adaption law. Now, the modified adaptation law can be written as:

( ) ( )( )J

hcecceceJ

Js

ref

ˆ

ˆˆ

σγ

θχλλγ

1

111221112121 1

−

++−+++−= ���

(19)

heh hsˆˆ σγγ 222 −=�

(20)

Where �gs & �hs are called the continuous switching function [5] and are represented as:

�

�

≥

≤≤���

�

�

���

�

� −

<

=

0

00

00

0

2

2

0

J

JJJ

JJ

J

JJs

J if

ˆJ if

ˆ

J if

J0

0

σ

σσ

(21)

�

�

≥

≤≤���

�

�

���

�

� −

<

=

0

00

0

0

0

2

2

0

h

hhh

hh

h

hhs

h if

ˆh if

ˆ

h if

h0

0

σ

σσ

(22)

In equation (21) & (22), J0, �0, �Jo & �ho are the design constants. Now, the magnitude of the constants in equation (21) and (22) may be chosen in actual systems by trial and error.

C. Stability Analysis

The fact that 0<<<<V� from (18) implies that

)0()( VtV ≤ , and therefore, that e1 and e2 are

bounded. Now, the following new function has been defined:

( ) 222

211 ecectN += (23)

Now, integrating (18) gives

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ττ

ττ

dNhJeV

dVhJeVtV

t

t

�

�

−=

+=

0

0

000

000

,,

,, �

(24)

Thus

( ) ( ) ( ) ( )( ) ( )tVhJeVdNt

−=� 0000

,,ττ (25)

Considering 0<<<<V� and ( ) 0>tV the

following results can be easily derived:

( ) ∞<�→∞

t

tdN

0

ττlim (26)

To use Barbalat’s lemma, let us check the

uniform continuity of ( )tV� . The derivative of ( )tV�is

( ) [ ]2221112 eeceectV ���� +=

(27)

This shows that ( )tV�� is bounded, because

e1 & e2 are bounded. Therefore ( )tV� is uniformly

continuous. Through Barbalat’s lemma, it can be shown that e1 and e2 converge to zero as t��. Furthermore, the presence of continuous switching

function makes V� more negative whenever the parameter adaptation of J and h becomes higher than a specified value of J0 & h0 respectively. Hence it is

quite obvious that the parameter estimates hJ ˆ&ˆ

will always converge to their real parameter valuesasymptotically.

IV. SIMULATION RESULTS

In this section the performance of the robust adaptive integral backstepping control law has been verified in simulation studies. Our simulation focuses on the following aspects

i) Investigating the effectiveness of Robust Adaptive integral Backstepping controller with respect to the conventional adaptive integral backstepping controller.

ii) Demonstrating the convergence of the parameter estimates to their real values and ability of the controller to handle the inertia variation and load disturbance.

In order to achieve the convergence of the

parameter estimation errors J and h to zero the given position reference signal should satisfy sufficient richness condition [1]. We select the following reference signal for our simulation

���

���=

50sin)

2sin(10

tpi

tpirefθ

The reference input and system response are shown in fig. 1. Tracking error has been plotted on fig. 2. Fig. 3.a.demonstate the variation of estimated inertia using robust adaptive integral backstepping control, while fig. 3.b demonstrates the variation of estimated inertia using ordinary adaptive integral backstepping control. Similarly fig. 4.a demonstrates the estimated variation of load torque using robust adaptive integral backstepping control, while fig. 4.b demonstrates the estimated load torque variation using ordinary adaptive integral backstepping control.

Fig.1 Reference Trajectory and Response of the System with time

Fig.2 Tracking Error of the System

Fig. 3.a. Estimation of Inertia variation of the system with time using Robust Adaptive Integral Backstepping Control Scheme

Fig. 3.b. Estimation of Inertia variation of the system with time using ordinary Adaptive Integral Backstepping Control Scheme

Fig. 4.a. Estimation of Load torque variation of the system with time using Robust Adaptive Integral Backstepping Control Scheme

Fig. 4.b. Estimation of Load torque disturbance variation of the system with time using ordinary Adaptive Integral

Backstepping Control Scheme

From fig.1 we can see that the system response closely follows the given reference signal �ref while the maximum tracking error is less than 0.1rad. Even with considerable variations in inertia and load torque

the controller is able to predict there variation and compensate them by increasing the control torque effort. From fig. 3 and fig. 4 it is clearly evident that robust adaptive integral backstepping controller is able to estimate the parameter variation in a smooth way. Parameter estimation error is more with ordinary adaptive integral backstepping scheme. This robust adaptive controller offers a smart estimation of the parameters variation. And the parameter estimation errors are small and the sudden change in inertia and load torque is not able to affect the parameter estimation of the controller.

V. CONCLUSION

In this paper, a robust adaptive integral backstepping control law has been proposed to solve the motion control problem of a servo system. The control algorithm exhibits a stable control performance in the presence of unknown parameters. Authors have used normalized adaptive backstepping control law which reduces the number of uncertain terms in control law. Therefore, the control algorithm is simple and considerably reduces the computational complexity. The control algorithm also uses continuous switching function to prevent the controller becoming unstable due to high rate of change of parameters during adaptation. The simulation results clearly reveal that the performance of the controller is very good in presence of large inertia variation. The controller also exhibits a stable response and error free estimation with a large amount of load torque disturbance. The robust adaptive integral backstepping control method has the capability of quickly achieving the control objectives and exhibits excellent ability of tracking the reference signal while the parameter estimation is also quite smooth and error free.

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