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Abstract—A controller is proposed for the robust backstepping control of a class of general nonlinear systems using neural net- works (NNs). A new tuning ...
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 30, NO. 6, NOVEMBER 2000

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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: [email protected]). 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

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(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 to be of the form can simply choose (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 . In the case of robotic control, denotes the joint trajectory 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 asympin (2.4) contains totically stable. If the nonlinear function 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. to go to zero despite The control objective is to make both 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 the real -vectors, Let denote the real numbers, the real matrices. Let be a compact simply connected . With map , define the functional set of any suitable space such that is continuous. We denote by 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.

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Two-layer NN.

controllers for to . Each is designed with the aim in the previous design to reduce error between stage. Second, we design an actual controller for to force the and as small as possible. In each step of error between 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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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 defined in (3.9) yields Differentiating 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)

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where ’s provide suitable basis functions for the NNs. The are bounded by known constants net reconstruction errors . in (3.13) by Define the NN functional estimate of

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 . Then the errors and scalar positive constant , 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 depends on and tion the regression matrix . 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 to zero. The confor u to drive the error between and , , , 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].

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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 , which rotates along the flux vector . The frame 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.

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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

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Fig. 5.

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Performance of NN backstepping controller for induction motor.

weight estimates are bounded. The errors can be kept diag in as small as desired by increasing gains (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 the inertia maceleration vectors, respectively, the 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.

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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 , , , and NN weight estimates are. The the errors , , 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)

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Fig. 8. Performance of NN backstepping controller to RLFJ.

where is defined as diag and , are 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.

REFERENCES [1] A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. PAGE NOS?–, 1993. [2] F. Blaschke, “The principle of field orientation applied to the new transvector closed-loop control system for rotating field machines,” Siemens Rev., vol. 39, pp. 217–220, 1972. [3] M. Bodson, J. Chiasson, and R. Novotnak, “High performance induction motor control via input–output linearization,” IEEE Control Syst. Mag., vol. 14, no. 4, pp. 25–33, 1994. [4] T. C. Burg, D. M. Dawson, J. Hu, and P. Vedagarbha, “Velocity tracking control for a separately excited DC motor without velocity measurements,” in Proc. Amer. Contr. Conf., 1994, pp. 1051–1056. [5] F. C. Chen and H. K. Khalil, “Adaptive control of nonlinear systems using neural networks,” Int. J. Control, vol. 55, no. 6, pp. 1299–1317, 1992. [6] F.-C. Chen and C.-C. Liu, “Adaptively controlling nonlinear continuous-time systems using multilayer neural networks,” IEEE Trans. Automat. Contr., vol. 39, pp. 1306–1310, 1994. [7] S. R. Chu and R. Shoureshi, “Neural-based adaptive nonlinear system identification,” in Intelligent Control Systems, ASME Winter Annu. Meet., vol. DSC-45, 1992. [8] S. Commuri and F. L. Lewis, “CMAC neural networks for control of nonlinear dynamical systems: Structure, stability and passivity,” in Proc. IEEE Int. Symp. Intelligent Control, 1995, pp. 123–129. [9] J. J. Craig, Adaptive Control of Mechanical Manipulators. New York: Wiley, 1986. [10] X. Cui and K. G. Shin, “Direct control and coordination using neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 23, no. 3, pp. PAGE NOS?–, 1993. [11] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals, and Systems, vol. 2, no. 4, pp. 303–314, 1989. [12] D. Dawson, Z. Qu, and J. Hu, “Robust tracking control of an induction motor,” in Proc. Amer. Contr. Conf., 1993, pp. 648–652. [13] D. Dawson, J. J. Carroll, and M. Schneider, “Integrator backstepping control of a brush DC motor turning a robotic load,” IEEE Trans. Contr. Syst. Technol., vol. 2, no. 3, pp. PAGE NOS?–, 1994. [14] D. Dawson, Z. Qu, and M. M. Bridges, “Hybrid adaptive control for tracking of rigid-link flexible-joint robots,” Proc. Inst. Elect. Eng., vol. 140, pt. D, no. 3, pp. 155–159, 1992. [15] D. Dawson, M. M. Bridges, and Z. Qu, Nonlinear Control of Robotic Systems for Environmental and Waste Restoration. Englewood Cliffs, NJ: Prentice-Hall, 1995. [16] F. R. Gantmacher, The Theory of Matrices. New York: Chelsea, 1959. [17] S. S. Ge and C. C. Hang, “Direct adaptive neural network control of robots,” Int. J. Syst. Sci., vol. 27, no. 6, pp. 533–542, 1996. [18] B. Horn, D. Hush, and C. Abdallah, “The state space recurrent neural network for robot identification,” in Advanced Control Issues for Robot Manipulators, ASME Winter Annu. Meet., vol. DSC-39, 1992. [19] K. Hornik, M. Stinchombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw., vol. 2, pp. 359–366, 1989.

765

[20] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, 1991. [21] G. S. Kim, I. J. Ha, and M. S. Ko, “Control of induction motors via feedback linearization with input–output decoupling,” Int. J. Control, vol. 51, no. 4, pp. 863–883, 1990. [22] G. S. Kim, I. J. Ha, and M. S. Ko, “Control of induction motors for both high dynamic performance and power efficiency,” IEEE Trans. Ind. Applicat., vol. 39, no. 4, pp. 323–333, 1992. [23] P. V. Kokotovic, “Bode lecture: The joy of feedback,” IEEE Contr. Syst. Mag., no. 3, pp. 7–17, June 1992. [24] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley Interscience, 1995. [25] Z. Krzeminski, “Nonlinear control of the induction motor,” in 10th IFAC World Congress, 1987, pp. 349–354. [26] C. M. Kwan, F. L. Lewis, and D. Dawson, “Robust neural-network control of rigid-link electrically driven robots,” IEEE Trans. Neural Networks, vol. 9, no. 4, pp. PAGE NOS?–, 1998. [27] C. M. Kwan and F. L. Lewis, “Robust backstepping control of induction motors using neural networks,” IEEE Trans. Neural Networks (AUTHOR: UPDATE?), September 2000. [28] C. M. Kwan, F. L. Lewis, and Y. H. Kim, “Robust neural network control of rigid link flexible-joint robots,” Asian J. Control, vol. 1, no. 3, pp. 188–197, 1999. [29] C. M. Kwan and K. S. Yeung, “Robust adaptive control of revolute flexible-joint manipulators using sliding technique,” Syst. Control Lett., vol. 20, pp. 279–288, 1989. [30] F. L. Lewis, C. Abdallah, and D. Dawson, Control of Robot Manipulators. New York: MacMillan, 1993. [31] F. L. Lewis, S. Jaganathan, and A. Yesildirek, Neural Network Control of Robot Manipulators and Nonlinear Systems. New York: Taylor and Francis, 1998. [32] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearization control of induction motor,” IEEE Trans. Automat. Contr., vol. 38, pp. 208–221, 1993. [33] K. S. Narendra and A. M. Annaswamy, “A new adaptive law for robust adaptation without persisitent excitation,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 134–145, 1987. [34] K. S. Narendra, “Adaptive control using neural networks,” in Neural Networks for Control. Cambridge, MA: MIT Press, 1991, pp. 115–142. [35] T. Ozaki, T. Suzuki, T. Furuhashi, S. Okuma, and Y. Ushikawa, “Trajectory control of robotic manipulators,” IEEE Trans. Ind. Electron., vol. 38, pp. 195–202, 1991. [36] M. M. Polycarpou and P. A. Ioannou, “Identification and control using neural network models: design and stability analysis,” Dept. Elect. Eng. Sys., Univ. of Southern California, Tech. Rep. 91-09-01, September 1991. [37] A. Sabanovic, N. Sabanovic, and K. Ohnishi, “Sliding modes in power converters and motion control systems,” Int. J. Control, vol. 57, pp. 1237–1259, 1993. [38] N. Sadegh, “Nonlinear identification and control via neural networks,” in Control Systems with Inexact Dynamic Models, ASME Winter Annu. Meet., vol. DSC-vol. 33, 1991. [39] G. A. Rovithakis and M. A. Christodoulou, “Adaptive control of unknown plants using dynamical neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 24, pp. 400–412, 1994. [40] R. M. Sanner and J. J. Slotine, “Stable adaptive control and recursive identification using radial Gaussian networks,” in Proc. IEEE Conf. Decision and Control, 1991. [41] J. J. Slotine and W. Li, “Adaptive manipulator control: A case study,” IEEE Trans. Automat. Contr., vol. 33, no. 11, pp. 995–1003, 1988. [42] M. W. Spong, “Adaptive control of flexible joint manipulators,” Syst. Control Lett., vol. 13, pp. 15–21, 1989. [43] T. J. Tarn, A. K. Bejczy, X. Yun, and Z. Li, “Effects of motor dynamics on nonlinear feedback robot arm control,” IEEE Trans. Robot. Automat., vol. 7, no. ISSUE NO?, pp. PAGE NOS?–, 1991. [44] T. Zhang, S. Ge, and C. C. Hang, “Direct adaptive control of nonaffine nonlinear systems using multilayer neural networks,” in Amer. Contr. Conf., 1998, pp. PAGE NOS?–.

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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.