IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
753
Robust Backstepping Control of Nonlinear Systems
Using Neural Networks
Chiman Kwan, Senior Member, IEEE and F. L. Lewis, Fellow, IEEE
Abstract—A controller is proposed for the robust backstepping
control of a class of general nonlinear systems using neural networks (NNs). A new tuning scheme is proposed which can guarantee the boundedness of tracking error and weight updates. Compared with adaptive backstepping control schemes, we do not require the unknown parameters to be linear parametrizable. No regression matrices are needed, so no preliminary dynamical analysis
is needed. One salient feature of our NN approach is that there is
no need for the off-line learning phase. Three nonlinear systems, including a one-link robot, an induction motor, and a rigid-link flexible-joint robot, were used to demonstrate the effectiveness of the
proposed scheme.
Index Terms—Adaptive, backstepping control, neural networks,
nonlinear systems, robust.
I. INTRODUCTION
I
N RECENT adaptive and robust control literature, there has
been a tremendous amount of activity on a special control
scheme known as “backstepping” [20], [23], [24]. When used
under some mild assumptions, many existing robust and adaptive control techniques can be extended to wide classes of applications [23]. Recent papers in [4], [12], and [29] have applied
such techniques to various kinds of robotic control schemes with
the inclusion of motor dynamics. A major problem with backstepping approaches is that certain functions must be “linear in
the unknown parameters,” and some very tedious analysis is
needed to determine “regression matrices.” For instance, even
for two robots within the same class (same number of links,
revolute joints) but with a different number of unknown parameters, minor changes in link lengths and masses, etc., one has to
restart the whole tedious process of determining the regression
matrix again. For a robot with six links, the job becomes even
more difficult. Although symbolic computation may offer some
help, one still has to manually manipulate and combine a lot of
terms in the dynamical equations. In the case of backstepping
adaptive control, the problem of determining and computing the
regression matrices becomes even more acute. The complexity
of the regression matrices and the number of unknown parameters increase with each step of the backstepping process. If one
looks at a recent paper [13], which talks about the application of
backstepping technique to a simple DC motor control, one will
notice that the regression matrix almost covers one full page
in the Transactions on Control System Technology. In addition,
the so-called linearity-in-the-parameter assumption may not be
true in many practical situations. For example, friction in a robot
is a complicated nonlinear process that is hard to model as a
linear-in-the-parameter process.
Parallel to fast development in adaptive and robust control
techniques, neural networks (NNs) have been applied to system
identification [7], [18] or identification-based control [5], [34],
[35]. Uncertainty on how to initialize the NN weights leads to
the necessity for “preliminary off-line tuning” [5], [10]. Recently, many NN controllers have been proposed for various
control applications that can provide closed-loop stability [6],
[17], [26]–[28], [31], [36], [38]–[40], [44].
In this paper, a unified and general approach to backstepping
control of nonlinear systems using NN is presented. We will
use neural nets in each stage of the backstepping procedure
to estimate certain nonlinear functions. This means that linearity-in-the-parameter assumption is not needed, and no
regression matrices need be found. Thus, a major problem
with backstepping is cured. Recent papers in [26]–[28] and
[31] have initially applied this new idea to robots and motors.
The objective of this paper is to further generalize our work
to more general nonlinear systems with the goal of retaining
the advantage of systematic design in backstepping control,
while eliminating its tedious and lengthy procedure of finding
the regression matrices. Compared with other NN approaches,
the NN weights here are tuned on-line, with no learning phase
required. Most importantly, we can guarantee the boundedness
of tracking error and weight updates.
The paper is organized, as follows. In Section II, we will
give a description of a class of nonlinear systems, system
stability, and an example of standard backstepping design.
Then, in Section III, we will introduce our NN backstepping
controller. Closed-loop stability of NN will also be stated and
proven in Section III. Several practical applications, including a
one-link robot tracking, speed control of induction motors, and
rigid-link flexible-joint robot trajectory control, will be given
in Section IV. Finally, conclusions will be given in Section V.
II. PRELIMINARIES
A. System Description
Manuscript received April 1, 1999; revised October 13, 2000. This work was
supported by NSF Grant IRI-9216545. This paper was recommended by Associate Editor S. Lakshmivarahan.
C. Kwan is with Intelligent Automation Inc., Rockville, MD 20850 USA
(e-mail: ckwan@i-a-i.com).
F. L. Lewis is with the Automation and Robotics Research Institute, The University of Texas at Arlington, TX 76118 USA.
Publisher Item Identifier S 1083-4427(00)11082-3.
Robust control of nonlinear systems with uncertainties is of
prime importance in many industrial applications. The model of
many practical nonlinear systems can be expressed in a special
state-space form
1083–4427/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
(2.1)
design can be easily used to tackle this problem. The procedure
consists of two steps.
Step 1—Treat
as a Fictitious Control Input to (2.3a): We
can simply choose
to be of the form
(2.4)
denote the states
is the vector of control inputs,
are nonlinear functions that
contain both parametric and nonparametric uncertainties, and
’s are known and invertible. Many systems can be expressed
in the above form. For example, rigid robots and motors [4],
[12]–[15], [26]–[31], power converters [37], and jet engines
[24].
The requirement that ’s be invertible and known may be
stringent. However, as will be illustrated in Section IV, one can
eliminate this requirement by exploiting physical properties of
a given nonlinear system.
Equation (2.1) is known as strict-feedback form [24]. The
depend
reason for this name is that the nonlinearities
, that is, on state variables that are “fed
only on
back.”
Other more general nonlinear systems such as “pure-feedback” and “block-strict-feedback” forms [24] will be dealt with
in future papers.
to follow certain desired
The control objective is to make
trajectory
. In the case of robotic control, denotes the joint
angles of the robot.
where
of the system,
Substituting (2.4) into (2.3a) yields
(2.5)
Define the following Lyapunov function candidate
(2.6)
Differentiating (2.4) and using the dynamics of (2.3) gives
Hence, from Lyapunov stability theory,
is globally asymptotically stable. If the nonlinear function
in (2.4) contains
unknown parameters, we can use a neural network to approximate this function.
Step 2—Realize the Fictitious Control Signal : Since
is only a fictitious control signal, we need to find
,
a way to realize this. Let us denote this desired signal by
. Define
i.e.,
(2.7)
as the error between
and
. Differentiating
B. Stability of Systems [31]
gives
(2.8)
Consider the following nonlinear system
Choosing the following controller
(2.2)
. We say the solution is uniformly ultiwith state
mately bounded (UUB) if there exists a compact set
such that for all
, there exists an
and a
such that
for all
.
number
(2.9)
will make the error to go to zero exponentially since, after
substituting (2.9) into (2.8), the resulting equation becomes
(2.10)
C. An Example of Backstepping Design
To illustrate the backstepping design procedure, let us consider a very simple nonlinear system in “strict-feedback” form.
The system is a second-order nonlinear system described by
(2.3a)
(2.3b)
where
,
scalar state variables;
nonlinear functions with
for all states;
control input.
The control objective is to make both
to go to zero despite
the presence of nonlinearities. One important observation of this
system is that the linearized system is uncontrollable since the
linearized equation for (2.3a) is of the form
which is clearly uncontrollable. Hence, linear control techniques cannot be used for (2.3). On the contrary, backstepping
will go to zero. Therefore, (2.5) will be valid
Hence,
will go to zero. Here again, if (2.9) contains significant
and
nonlinearities due to the functions and , we can also use
a second neural network to approximate them. Details of extending the above ideas to more general nonlinear systems will
be described in Section III.
Although the above backstepping procedure becomes more
complicated when there exist parametric uncertainties in the
systems, the basic idea remains the same. The complications are
due to the following problems with the existing robust and adaptive procedures. First, “regression matrices” in each step of the
backstepping design must be determined. For example, it is well
known that, in the first step of designing adaptive controllers
for robots, one has to determine the regression matrix which is
a very tedious and time consuming task. This procedure gets
even more involved as the number of backstepping increases.
Second, one basic assumption in the current robust and adaptive backstepping methods is that the unknown system parameters must satisfy the so-called linearity-in-the-parameter as-
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
sumption. This may not be true in many practical situations. For
example, friction in robots is a complicated nonlinear process
which is hard to model as a linear-in-the-parameter process. Another example will be that certain nonlinear functions may not
where is an unbe linear parametrizable, i.e., if
known system uncertainty.
The goal of this paper is to present a unified approach of backstepping control to a class of nonlinear systems using neural networks. While retaining the merit of systematic design in backstepping control, we strive to alleviate the disadvantage of the
tedious and lengthy process of determining and computing the
regression matrices. The proposed controller is also re-usable
in a sense that it is applicable to a class of, for instance, 6-link
revolute joint robots.
III. ROBUST BACKSTEPPING CONTROLLER DESIGN USING NNs
A. NN Basics
Let denote the real numbers,
the real -vectors,
the real
matrices. Let be a compact simply connected
. With map
, define
the functional
set of
space such that is continuous. We denote by
any suitable
vector norm. When it is required to be specific we denote the
. Define
as the collection of NN weights.
p-norm by
Then the net output is
(3.1)
A general nonlinear function
be approximated by an NN as
,
Fig. 1.
755
Two-layer NN.
to
. Each
is designed with the aim
controllers for
in the previous design
to reduce error between
stage. Second, we design an actual controller for to force the
error between
and
as small as possible. In each step of
the design process, NNs are used to approximate the nonlinear
functions in the error dynamics. The overall control structure
in shown in Fig. 2. Third, we perform an overall closed-loop
stability and performance analysis of the on-line weight-tuning
algorithm.
, and
Step 1—Design Fictitious Controllers for
: First of all, we design the fictitious controller for
. Recalling that
can
(3.2)
a NN functional reconstruction error vector. The
with
structure of a two-layer NN is shown in Fig. 1.
must satisfy
For suitable NN approximation properties,
some conditions.
Definition 1 [38]: Let be a compact simply connected set
, and
be integrable and bounded. Then
of
is said to provide a basis for
if
1) A constant function on can be expressed as (3.2) for
.
finite
for
2) The functional range of NN (3.2) is dense in
.
countable
is not difficult to find. The
It is emphasized that a basis
radial basis functions, for instance, provide a universal basis for
all smooth nonlinear functions [40]. Barron [1] has shown that
the approximation error in (3.2) can never be made smaller than
where is order of the input space. Despite
order
, the tracking error will be
the lower bound achievable for
proven to be bounded by a term as shown in (3.23), and the
bound can be made small by increasing the gain of certain coefficients in the controller.
B. Controller Structure
to
Referring to (2.1), our control objective is to make
. The idea of backstepping is
follow a desired trajectory
up to
as fictitious control siglike this. First, we treat
nals. In this stage, we use NN approach to design the fictitious
(3.3)
Choosing the following fictitious controller
(3.4)
a design parameter,
the estimate of
and
with
substituting (3.4) into subsystem (3.3) yields the error dynamics
(3.5)
. The form of will be discussed in the next
with
section. The usual adaptive backstepping approach is to assume
[24] are linear parametrizthat the unknown parameters in
able (LP) so that standard adaptive control can be used in this
stage. As we will see in a moment, we will use a two-layer NN
to approximate . The advantage is that no linearity-in-the-unknown-system-parameters assumption is needed and no regression matrix need be found.
The next step of backstepping design is to make the error beas small as possible. Differentiating defined
tween and
in (3.5) gives
(3.6)
A fictitious controller for
of the form
(3.7)
in (3.5).
can be chosen. Note that there is a coupling term
in (3.7) is to compensate the
The purpose of the term
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
Fig. 2. Backstepping NN control of nonlinear systems in the “strict-feedback” form.
effect of the coupling due to
controller (3.7) into (3.6) gives
. Substituting the fictitious
with
,
a design parameter and
the
estimate of .
In a similar fashion, we can design a fictitious controller for
to make the error
as small as
possible, i.e.,
(3.8)
The dynamics of
is then governed by
(3.9)
,
a design parameter and
.
backstepping
design
assumes
that
are LP in unknown system parameters.
.
Regression matrices are needed for all ’s,
This is a very lengthy and tedious process which needs to be
repeated for each new system. As we will see in Section III-C,
our design procedure uses NNs to approximate the complicated
. As a result, no renonlinear functions ’s,
gression matrices are needed and the controller is re-usable for
different systems within the same class of nonlinear systems.
Step 2—Design of Actual Control : After the fictitious conis designed, we need to find a way to realize them.
troller
Differentiating
defined in (3.9) yields
with
the estimate of
Conventional
(3.10)
Choosing the controller of the form
(3.11)
with
a design parameter and
the estimate of
.
Similar to Step 1, the usual backstepping design procedure is
to be LP in unknown system parameters. Howto assume
ever, in our controller design here, we will use a two-layer NN
which means no LP or regression matrix
to approximate
is
requirement is needed. Also note that a term
added in (3.9) which is necessary to compensate the coupling
effects introduced in (3.9).
Step 3—Closed-Loop Stability and Performance Analysis of
NN Weight Tuning Algorithm: We will perform a detailed treatment of stability and performance analysis of a weight-tuning
algorithm in Section III-C. Using Lyapunov stability theory we
will carry out the stability analysis. We can show that all signals
including tracking error, NN weights are all UUB. The overall
control structure is shown in Fig. 2. It is important to note the
simplicity of NN control when compared to adaptive backstepping control. In adaptive backstepping control, it is assumed that
(3.5), (3.6), (3.9), (3.10) are linear in terms of
known regression matrices. These regression matrices are very
tedious to find and must be computed for each specific system.
In fact, for some systems the LP assumption may not hold. For
example, friction in robot is a complicated nonlinear process
that is hard to model as a linear-in-the parameter process. Another simple example is that the nonlinear function may be in
which is clearly not LP. On the other hand, if
the form of
in (3.13) are appropriately chosen, then
bases
NN equations (3.13) are valid. No regression matrices need be
computed. It has been shown that the sigmoid can form a basis
set [1], [11], [19]. In [40], it was shown that the radial basis
functions can form a basis. In [8], it was shown that a basis set is
particularly easy to choose for CMAC (Cerebellar Model Arithmetic Computer) neural network.
C. Bounding Assumptions, Error Dynamics, and Weight
Tuning Algorithms
Assume that the nonlinear functions ’s,
in (3.5), (3.6), (3.9), and (3.12) can be represented by
2-layer neural nets for some constant “ideal” weights
, i.e.,
gives the following dynamics for error
(3.12)
(3.13)
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
757
where ’s provide suitable basis functions for the NNs. The
are bounded by known constants
net reconstruction errors
.
Define the NN functional estimate of
in (3.13) by
Theorem 1: Suppose Assumptions 1 and 2 are satisfied. Take
the control input (3.11) with NN weight tuning be provided by
(3.14)
,
with constant matrices
and scalar positive constant
. Then the errors
, and NN weight estimates are UUB. The errors
, can be kept as small as possible by
in (3.16).
increasing gains
Proof: Let the NN approximation property [i.e., (3.13)]
’s for all in the compact set
holds with given accuracy
with a positive constant. Let
.
Now consider the following Lyapunov function candidate
the current NN weight estimates provided by the tuning
with
algorithms. Then the error dynamics (3.5), (3.6), (3.9), (3.12)
become
(3.18)
(3.19)
(3.15)
Differentiating (3.19) and using (3.16) gives
Define
diag
diag
(3.20)
Applying the following inequality (also known as Schwartz inequality in [16]) to (3.20)
diag
(3.21)
we have
(3.22)
The error dynamics (3.15) can be expressed in terms of the
above quantities as
(3.16)
denotes the couplings between the error
Note that the term
is skew-symmetric. The
dynamics in (3.16). The matrix
closed-loop stability analysis and the weight tuning algorithms
will be discussed in the next section.
Two standard assumptions, which are quite common in the
neural networks literature [17], [26]–[28], [31], [44] are stated
next.
Assumption 1: The ideal weights are bounded by known positive values so that
which is negative as long as the term in square bracket is posis the minimum eigenvalue of . Completing
itive. Here
the square for the term inside the square bracket in (3.22) yields
which is positive as long as
(3.23)
or
(3.24)
is negative outside a compact set. The form of the
Thus,
right-hand side of (3.24) shows that the control gain , which
, can be selected large enough so that
are contained in
or equivalently
(3.17)
diag
and
is known. The
where
denotes the Frobenius norm, i.e., given a matrix
symbol
, the Frobenius norm is given by
Assumption 2: The desired trajectory
th order are bounded.
up to the
and its derivatives
According to a standard Lyapunov theorem extension (Narendra
and Annaswamy 1987), this demonstrates the UUB of both
and
.
Q.E.D.
Remarks:
a) A comparison with -modification [33] shows that the
increases the bounds on
NN reconstruction error
and
in a very interesting way. Note, however, that small tracking error bounds may be achieved
by selecting large control gain . On the other hand,
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the NN weight error is fundamentally bounded by
,
the known bound on the ideal weights . The parameter
offers a design tradeoff between the relative eventual
and
; a smaller
yields a
magnitudes of
and a larger
, and vice versa.
smaller
b) Similar to -modification in [33], no persistency of excitation (PE) is needed to establish the bounds on NN
weight errors with the weight tuning algorithm (3.18).
c) The contrast between the NN approximation property
(3.2) and the adaptive control linear-in-the-parameter
(LIP) assumption should be understood.
1) Both are linear in the tunable parameters, but the
former is linear in the tunable NN weights, while the
latter is linear in the unknown system parameters.
in
,
2) The former holds for all functions
.
while the latter holds only for a specific function
suffices
3) In the NN property, the same basis set
, while in the LIP assumpfor all
tion the regression matrix
depends on
and
. That is, for
must be recomputed for different
for each different
instance, one must recompute
type of robot arm. Therefore, the two-layer NN controller is significantly more powerful than adaptive
controllers; it provides a universal controller for robot
arms within the same class. An example of “class” is
the class of 2-link revolute robot arms with flexible
joints.
d) It should be emphasized that our NN controller design
procedure was motivated by a technique known as backstepping control [24]. In backstepping design procedures,
preliminary dynamical analysis to determine regression
matrices is crucial. The procedure becomes very tedious
if we are dealing with a robot with multiple degrees
of freedom. In [29], a backstepping design for RLFJ
was given using sliding mode and adaptive control. The
derivation of regression matrices, even for the one-link
RLFJ simulation model, was very time-consuming and
tedious. An immediate advantage of NN design is the no
regression analysis is needed and the controller structure
is reusable for different robots with different masses and
lengths within the same class.
e) Note that the problem of neural net weight initialization
’s are taken as zeroes the
does not arise, since if
stabilizes the
linear proportional control term
system on an interim basis. Similar to other NN methods,
’s will
our control scheme does not guarantee that
’s. All we can say is that we can
converge to the true
’s.
guarantee the boundedness of
’s are known and
f) In Section II, it was assumed that
invertible. Although this may sound restrictive, it should
be emphasized that, in many practical applications,
the above mentioned restriction can be alleviated by
exploiting the physical properties of the system. Two
practical applications will be described in Section IV:
one for induction motor control and the other one for
rigid-link flexible-joint robot control. Although the
details of controller derivation and simulation results of
these applications have been included in [27] and [28],
we present some partial results here to illustrate that
stability is still achievable even ’s are unknown.
To deal with more general systems with unknown ’s,
we propose to use an approach described in [31, p. 287].
This research is still underway.
IV. APPLICATIONS
In this section, we present three applications. The first one is
a one-link robot system with the inclusion of motor dynamics.
As pointed out by Tarn et al. [43], the effects of motor dynamics will affect the performance of overall robot tracking. In
this case, the ’s are assumed to be known. The backstepping
NN theory described in Section III can be directly applied. The
second application is the robust control of an induction motor.
Here, the ’s are actually unknown. However, by exploiting
the physical properties of the motor dynamics, we circumvented
the problem of unknown ’s. In the third application, we applied the theory of backstepping NN control to rigid-link flexible-joint system. Properties of robot dynamics were used to alproblems.
leviate the unknown
These applications, especially the last two, clearly demonstrate that the proposed backstepping control using NN has great
potential in many diverse applications.
A. One-Link Robot Tracking
Consider a one-link manipulator with the inclusion of motor
dynamics. The robot model is given by
(4.1)
Equation (4.1) can be expressed in the form (2.1) by noting that
The parameter values with appropriate units are given by
,
,
,
,
,
. The
. The design procedure in
desired trajectory is
Section III was modified slightly. First, we defined a filtered
with
and
.
tracking error
Second, we design a fictitious NN controller for , namely ,
which drives to zero. Third, we design a second NN controller
for u to drive the error between and
to zero. The con,
,
, and
troller parameters are
. The number of neurons used in each of the two 2-layer
NNs is 10. We used sigmoids for . The initial conditions for
are 0.1, 6.28, and 0, respectively. The robust tuning
algorithms (3.18) are used for the simulations. Simulation results are shown in Fig. 3. The performance is very good. The
tracking errors are reduced significantly when NNs are used.
B. NN Control of Induction Motors
The nomenclature of induction motors can be found in [32]
and details of controller derivation, proof, and simulations can
be found in [27].
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
759
Fig. 3. Performance of robust NN backstepping controller for 1-link robot tracking.
(4.4b)
with
,
,
,
. Since
is unknown, , are also unand
known. However, and are known. Blaschke [2] also developed a feedback controller to control (4.4). Other nonlinear approaches include [3], [21], [22], and [25]. Marino et al. [32] used
adaptive input–output decoupling technique to tackle the control
problem. Our work here uses the same field-oriented model and
follows the same assumptions as those in [32]. In the next section, we will make use of a special structure of the above model
to perform our NN controller design.
2) Controller Structure, Error Dynamics, and Weight Update
and
, then (4.4) is in
Rules: If we define
strict-feedback form (2.1). We first treat , as the ideal fictitious control signals for a subsystem consisting of (4.4a)–(4.4b).
We design an NN controller for , . Finally, we use a second
2-layer NN to realize these fictitious signals. It should be noted
that special physical properties of the motor dynamics were exploited so that the problem of unknown ’s can be eliminated.
Our control objective is to regulate the rotor speed and the magas the desired reference
netic flux magnitude. Denote and
levels of and , respectively.
Step 1—Selection of Desired and to Control Subsystem
(4.4a) and (4.4b): First, we rewrite (4.4a)–(4.4b) as
(4.4c)
(4.5a)
1) Model of Induction Motor: This model is known as fieldoriented model, which was introduced by Blaschke [2]. It into a
volves a transformation from the stator fixed frame
frame
, which rotates along the flux vector
. The
transformations between currents and flux magnitudes in different frames are given by
(4.2)
(4.3)
where
The field-oriented model of induction motor in
given by [2]
frame is
(4.4a)
(4.5b)
(4.4d)
(4.4e)
where
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Fig. 4.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
NN backstepping controller for induction motor.
Since in (4.5b) is unknown, this will cause some difficulties
using the results of backstepping NN described in Section II. To
alleviate this difficulty, we apply the following trick. Dividing
yields
both sides of (4.5b) by
where
(4.6)
Now the coefficient of
rewrite (4.5) as
is unity and known. Then we can
(4.7)
where
It should be noted that
is exactly known. To make as small
as possible, the following control is chosen:
(4.12)
It should be noted that
is exactly known and invertible. By
treating as a fictitious input, we design a controller for the ideal
as
(4.8)
a design parameter,
, and
with
estimate of . Substituting (4.8) into (4.7) gives
the
(4.9)
Note that is the estimate of the unknown function . Similar
to Step 1, we will use another two-layer NN to approximate
. Also note that a term
is added in (4.12) which is
in (4.9) so that we will be
necessary to cancel the effect of
able to prove the closed-loop stability.
Step 3—Closed-Loop Stability Analysis and On-Line WeightTuning Algorithm: We now perform a detailed treatment of stability and performance analysis of a weight-tuning algorithm.
The overall control scheme is shown in Fig. 4.
and
yields the following error dyUsing (3.13) for
namics for
:
(4.13)
(4.14)
. The form of
is given by (3.13), which is the
where
output of a two-layer NN.
Step 2—Realization of the Desired Reference Signals in
(3.8): In order to achieve the desirable result in Step 1, i.e., the
ideal fictitious control signal in (4.8), we need to find the error
dynamics of which is defined as
and
be
Theorem 2: Let the desired trajectories
bounded. Take the control input (4.12) with NN weight tuning
be provided by
(4.10)
(4.15a)
(4.15b)
Differentiating (4.10) and using the dynamics in (4.4) yields
(4.11)
,
with constant matrices
positive constant . Then the errors
, and scalar
are UUB. NN
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
Fig. 5.
761
Performance of NN backstepping controller for induction motor.
weight estimates are bounded. The errors
can be kept
as small as desired by increasing gains
diag
in
(4.13) and (4.14).
Proof: Consider the following Lyapunov function candidate:
with defined in (4.13) and is a
identity matrix. Then
the proof follows the same procedure described in Section III.
Details of the proof can also be found in [27].
Q.E.D.
is related to the actual control ,
Finally,
through the following relation:
(4.16)
Simulation Results: Using the data in [32], we simulate the
robust backstepping NN controller. The model we used was
the original motor model before the state transformation from
stator frame to rotator frame was applied. In other words, the
field-oriented model was only used for controller design. The
results are shown in Fig. 5. We used four and ten neurons in the
, respectively. The inputs
two NNs which approximate
to NN1 consist of , , , and . The inputs to NN2 consists , , , , ,
. The reference trais zero
jectories are the same as those in [32]. Reference
from 0 to 0.3 s., 220 r/s from 0.3 to 5 s., and 350 r/s from 5 s.
is 1.3 Wb from 0 to 5 s. and 0.8 Wb
onwards. Reference
after 5 s. The discontinuities are smoothed by linear interpolations. A load disturbance of 40 Nm is added at
s. We
set
diag
,
diag
,
,
,
. The applied voltage
has the same magnitude as that of [32] and is well within inverter limits.
are not due to
It should be noted that the plots of ,
switching in sliding mode control as it appears to be. There is
no switching term in the NN controller. Similar waveforms have
also been observed in [32]. This phenomenon is just a characteristic of induction motor dynamics.
C. NN Backstepping Control of N-DOF Rigid-Link
Flexible-Joint Robots
1) RLFJ Robot Model and Its Properties: The model for an
-link RLFJ robot is given by [42]
(4.17a)
(4.17b)
denoting the link position, velocity, and acwith
celeration vectors, respectively,
the inertia mathe centripetal-Coriolis matrix,
trix,
the gravity vector,
representing the friction
the additive bounded disturbance,
terms,
the motor shaft angle, velocity, acceleration, respectively,
the difference between motor and joint
the positive definite constant diagonal maangles,
a positrix which characterizes the joint flexibility,
tive definite constant diagonal matrix denoting the motor inertia,
representing the natural damping term, the control
reprevector used to represent the motor torque, and
senting an additive bounded torque disturbance.
762
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
If we define
,
,
,
, then (4.17)
is indeed in the form of (2.1). However, due to the presence of
uncertainties in the robot dynamics, we cannot directly apply the
backstepping NN control theory to RLFJ expressed in “strictfeedback” form of (2.1). We need to exploit the following robot
properties in order to effectively control the robot system.
The rigid dynamics (4.17a) has the following properties [9],
[41]:
Property 1—Boundedness of the Inertia Matrix: The inertia
is symmetric and positive definite, and satisfies the
matrix
following inequalities:
(4.18)
and
are known positive constants, and
where
notes the standard Euclidean norm.
Property 2—Skew Symmetry: The inertia and
tripetal-Coriolis matrices have the following property:
decen(4.19)
is the time derivative of the inertia matrix.
where
is bounded by
The joint elasticity matrix
for arbitrary vector
(4.21)
where and are positive scalar bounding constants.
Property 1 is very important in generating a positive definite
function to prove stability of the closed-loop system. Property
2 will help in simplifying the controller. Many robust methods
have incorporated Properties 1 and 2 in their controller designs
[9], [41]. It should be emphasized here that, unlike standard robust and adaptive control schemes, we do not require linearity
in the unknown robot parameter assumption.
2) Control Objective and Central Ideas of Our Controller
Design: The control objective is to develop a link positiontracking controller for the RLFJ robot dynamics given by (4.17)
based on inexact knowledge of manipulator dynamics. To accomplish this purpose, we first define the link position tracking
as
error
(4.22)
denotes the desired link position trajectory. It
where
is assumed that and its derivatives up to the fourth order are
bounded. In addition, we also define a filtered tracking error as
(4.23)
is a diagonal, positive definite control gain
where
matrix. Using (4.23) and (4.17a), we can derive the equation
(4.24)
where the complicated nonlinear function
Our controller design can be considered as consisting of three
steps. The first step is to treat , the difference between motor
shaft angle and joint angle , as a fictitious control signal to
the error dynamics (4.24). We call this fictitious signal . Then
(4.24) can be rewritten as
(4.26)
(4.20)
and
are positive scalar bounding constants. The
where
motor inertia matrix is also bounded by
for arbitrary vector
Fig. 6. Two-link robot with joint flexibility.
is defined as
(4.25)
is an error signal which we will try to make
where
as small as possible in the second step. The control objective of
the first step is to design an NN controller for to make (and
hence tracking error ) as small as possible. The structure of the
controller will be described below. The objective of the second
step is to design a second NN controller for another fictitious
such that the error signal
is as small as possible.
signal
To achieve this, we need to derive the dynamic equation for .
and using (4.17b) yields
Differentiating
(4.27)
and
is a very complicated nonlinear
where
, and . Now we need to derive a controller
function of
as small as possible. The error dynamics for
for u to make
is obtained by differentiating
and multiplying the final
expression by
(4.28)
is another very complicated nonlinear function
, and . It should be noted that link acceleration
is not needed in our controller. The reason is that, whenever shows up in (4.27) and (4.28), it will be replaced by
. Finally, in the
third step, we will perform an overall stability analysis using
Lyapunov stability theory.
Now (4.26)–(4.28) are in modified “strict-feedback” form
(2.1). The problem of unknown ’s is eliminated.
3) Control Design Procedure:
Step 1—Design of NN Controller for : To design an NN
controller for the fictitious signal , we select the following
structure:
where
(4.29)
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
763
Fig. 7. NN backstepping controller for RLFJ.
where
,
is a positive definite matrix,
is a robustifying term to be defined shortly, and is defined in (4.20).
will depend on certain update algorithms to be deNote that
scribed in the next section. Substituting (4.29) into (4.26) gives
Differentiating
yields
(4.37)
where
. Control
is chosen to be
(4.38)
(4.30)
The form of
et al. 1995)
is chosen to be (Dawson et al. 1992 and Dawson
is a control gain. Substituting
(4.31)
(4.39)
(4.32)
Step 3—Overall Stability Analysis: The stability analysis
will be proved by using Lyapunov stability theory. We can
show that all signals including tracking error, NN weights are
all UUB. The overall control structure is shown in Fig. 7.
and its derivative
Theorem 3: Let the desired trajectory
up to the fourth order be bounded. Let
,
the control input be given by (4.29), (4.31), (4.34) and weight
tuning provided by
where
(sufficiently small number)
and
where
(4.38) into (4.37) gives
(4.33)
in (4.32) stands for the upper bound of
.
and the Actual Control : For the
Step 2—Design of
design of fictitious signal , we choose the following structure:
(4.34)
where
(4.40)
. The weights
will be generated from some
and
update algorithms to be described below. Inserting (4.34) into
(4.27) gives
,
with any constant symmetric matrices
,
, and scalar positive constant . Then
the errors
,
,
, and NN weight estimates are. The
,
,
can be kept as small as desired by inerrors
in (4.29), (4.31), (4.34).
creasing gains , ,
Proof: Consider the Lyapunov function candidate
(4.36)
(3.23)
(4.35)
764
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
Fig. 8. Performance of NN backstepping controller to RLFJ.
is defined as
diag
and , are
where
defined in (4.17), is the identity matrix. Then the proof follows
a similar procedure as was described in Theorem 1. Details can
also be found in [28]. Hence it is omitted.
Q.E.D.
Simulation Results: Consider a simple 2-link manipulator
shown in Fig. 6. The model for this robot system can be described in the form of (4.17) with [30]
eters are
diag
diag
,
diag
. The inputs to the NNs are given by
,
where
,
, is chosen as diag
.
Here we use sigmoid functions in the NNs. The gains of the
,
. The
NNs are chosen as
diag
,
diag
,
gains are
diag
. Fig. 8 shows the results using the NN controller. Both tracking errors go to small values. The desired and
actual trajectory almost overlap with each other. In addition, we
do not even need the explicit expressions for those three highly
complicated nonlinear functions , and in (4.26)–(4.28),
respectively. This is a significant advantage since our controller
can be applied to any type of RLFJ robots of different masses
and lengths within the same class.
V. CONCLUSIONS
The parameter values are
kg,
kg,
m,
m,
m/s . The flexible-joint param-
We have presented a general NN controller for the robust
backstepping control of a class of nonlinear systems. The
method does not require the system dynamics to be exactly
known. Compared with adaptive backstepping control, linearity
KWAN AND LEWIS: ROBUST BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS
in unknown parameters is not needed. A major problem with
backstepping is corrected in that no tedious computation of
“regression matrices” is needed. Compared with other NN
approaches, we do not require an off-line “training phase.” All
errors and weight are guaranteed to be bounded. The tracking
error can be reduced to arbitrarily small values by choosing
certain gains large enough.
Several practical systems, including an induction motor and
a RLFJ robot, were used to demonstrate the effectiveness of the
proposed controller.
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000
Chiman Kwan (S’85–M’93–SM’98) was born on
February 19, 1966, in Jilian, China. He received the
B.S. degree in electronics (with honors) from the
Chinese University of Hong Kong in 1988 and the
M.S. and Ph.D. degrees in electrical engineering
from the University of Texas at Arlington in 1989
and 1993, respectively.
From April 1991 to February 1994, he was with
the Beam Instrumentation Department of the Superconducting Super Collider Laboratory (SSC), Dallas,
TX, where he was heavily involved in the modeling,
simulation, and design of modern digital controllers and signal processing algorithms for the beam control and synchronization system. He later joined the Automation and Robotics Research Institute, Fort Worth, TX, where he applied intelligent control methods such as neural networks and fuzzy logic to the control
of power systems, robots, and motors. Since July 1995, he has been the Director
of Robotics Research at Intelligent Automation, Inc., Rockville, MD, where he
has been Principal Investigator/Program Manager for more than 20 different
projects such as modeling and control of advanced machine tools, digital control
of high-precision electron microscope, enhancement of microscope images, and
adaptive antenna arrays for beam forming, automatic target recognition of FLIR
and SAR images, fast flow control in communication networks, vibration management of gun pointing system, health monitoring of flight critical systems,
high-speed piezoelectric actuator control, fault tolerant missile control, active
speech enhancement, fault detection isolation of various electromechanical systems, and underwater vehicle control. His primary research areas include fault
detection and isolation, robust and adaptive control methods, signal and image
processing, communications, neural networks, and fuzzy logic applications.
Dr. Kwan is listed in the New Millennium edition of Who’s Who in Science
and Engineering and is a member of Tau Beta Pi. He received an invention award
for his work at SSC.
F. L. Lewis (S’78–M’81–SM’86–F’94) was born
in Würzburg, Germany, and subsequently studied in
Chile and at the Gordonstoun School in Scotland.
He received the B.S. degree in physics/electrical
engineering from Rice University, Houston, TX, and
the M.S. degree in electrical engineering from Rice
University, both in 1971. After spending six years
in the U.S. Navy, serving as Navigator aboard the
frigate USS Trippe (FF-1075) and Executive Office
and Acting Commanding Officer aboard the USS
Salinan (ATF-161), he received the M.S. degree in
aeronautical engineering from the University of West Florida, Pensacola, in
1977 and the Ph.D. degree from the Georgia Institute of Technology (Georgia
Tech), Atlanta, in 1981.
From 1981 to 1990, he was a Professor at Georgia Tech, where he is currently
an Adjunct Professor. He is also a Professor of electrical engineering at The University of Texas, Arlington (UTA), where he was awarded the Moncrief-O’Donnell Endowed Chair in 1990 at the Automation and Robotics Research Institute.
He has studied the geometric, analytic, and structural properties of dynamical
systems and feedback control automation. His current interests include robotics,
intelligent control, neural and fuzzy systems, nonlinear systems, and manufacturing process control. He is the author/coauthor of two U.S. patents, 124 journal
papers, 20 chapters and encyclopedia articles, 210 refereed conference papers,
and seven books. He was selected to the Editorial Boards of the International
Journal of Control, Neural Computing and Applications, and the International
Journal of Intelligent Control Systems. He is currently an Editor for the flagship
journal Automatica.
Dr. Lewis is a Registered Professional Engineer in the State of Texas. He is the
recipient of an NSF Research Initiation Grant and has been continuously funded
by NSF since 1982. Since 1991, he has received $1.8 million in funding from
NSF and upwards of $1 million in SBIR/industry/state funding. He has received
a Fulbright Research Award, the American Society of Engineering Education F.
E. Terman Award, three Sigma Xi Research Awards, the UTA Halliburton Engineering Research Award, the UTA University-Wide Distinguished Research
Award, the ARRI Patent Award, various Best Paper Awards, the IEEE Control
Systems Society Best Chapter Award (as Founding Chairman), and the National
Sigma Xi Award for Outstanding Chapter (as President). He was selected as Engineer of the Year in 1994 by the Fort Worth IEEE Section. He was appointed
to the NAE Committee on Space Station in 1995 and to the IEEE Control Systems Society Board of Governors in 1996. In 1998, he was selected as an IEEE
Control Systems Society Distinguished Lecturer. He is a Founding Member of
the Board of Governors of the Mediterranean Control Association.