Deadzone Compensation in Motion Control Systems Using Neural Networks

Transcription

1 602 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 Deadzone Compensation in Motion Control Systems Using Neural Networks Rastko R. Selmić and Frank L. Lewis Abstract A compensation scheme is presented for general nonlinear actuator deadzones of unknown width. The compensator uses two neural networks (NN s), one to estimate the unknown deadzone and another to provide adaptive compensation in the feedforward path. The compensator NN has a special augmented form containing extra neurons whose activation functions provide a jump function basis set for approximating piecewise continuous functions. Rigorous proofs of closed-loop stability for the deadzone compensator are provided and yield tuning algorithms for the weights of the two NN s. The technique provides a general procedure for using NN s to determine the preinverse of an unknown right-invertible function. I. INTRODUCTION AGENERAL class of industrial motion control systems has the structure of a dynamical system, usually of the Lagrangian form, preceded by some nonlinearities in the actuator, either deadzone, backlash, saturation, etc. [7]. This includes -positioning tables [17], robot manipulators [13], overhead crane mechanisms, and more. The problems are particularly exacerbated when the required accuracy is high, as in micropositioning devices. Due to the nonanalytic nature of the actuator nonlinearities and the fact that their exact nonlinear functions are unknown, such systems present a challenge for the control design engineer. Proportional-derivative (PD) controllers have been observed to result in limit cycles if the actuators have deadzones. Rigorous results for motion tracking of such systems are notably sparse, though ad hoc techniques relying on simulations for verification of effectiveness are prolific. Stability proofs and design of deadzone compensator for industrial positioning systems using a fuzzy logic controller is given in [15]. Fuzzy logic deadzone compensator is given in [11] and [15]. The experimental evolutionary programming is used to obtain the fuzzy rules in [11]. Recently, in seminal work several rigorously derived adaptive schemes have been given [33]. Backlash compensation is considered in [30] and [31] and hysteresis in [33]. Compensation for nonsymmetric deadzones is considered in [29] and [32] for linear systems and in [23] for nonlinear systems in Brunovsky form with known nonlinear functions. All of the known approaches in deadzone compensation assume that the Manuscript received December 19, 1997; revised October 5, 1998, February 11, 1999, and March 14, Recommended by Associate Editor, J. Farrell. This work was supported in part by the NSF under Grant ECS , Texas ATP under Grant , and ARO under Grant DAAD The authors are with the Automation and Robotics Research Institute, The University of Texas at Arlington, Fort Worth, TX USA ( rselmic@arrirs04.uta.edu). Publisher Item Identifier S (00) deadzone function can be parameterized using a few parameters such as deadzone width, slope, etc. Standard adaptive technique requires that assumption. Neural networks (NN s) have been used extensively in feedback control systems. Most applications are ad hoc with no demonstrations of stability. The stability proofs that do exist rely almost invariably on the universal approximation property for NN s [2], [14], [21], [24], [25]. However, in most real industrial control systems there are nonsmooth functions (piecewise continuous) for which approximation results in the literature are sparse. Examples include deadzone, friction, backlash, and so on. Though there do exist some results for piecewise continuous functions, it is found that attempts to approximate jump functions using smooth activation functions require many NN nodes and many training iterations and still do not yield very good results. Here is assumed a general model of the deadzone. It is not required to be symmetric, and the function outside the dead-band may not be a linear function. Moreover, the proposed method can be applied for compensation of any invertible, bounded, unknown, nonlinear function. The generality of the method and its applicability to a broad range of nonlinear functions make this approach a useful tool for compensation of deadzone, hysteresis, etc. The deadzone compensator consists of two NN s, one used as an estimator of the nonlinear deadzone function and the other used for the compensation itself. The NN used for deadzone compensation is a modified multilayer perceptron [27], capable of approximating the piecewise continuous functions of the sort that appear in deadzone, backlash, friction, and other motion control actuator nonlinearities. It is found that to approximate such functions suitably, it is necessary to augment the standard NN that uses smooth activation functions with extra nodes containing a certain jump function approximation basis set of nonsmooth activation functions. II. BACKGROUND Let be a compact simply connected set of. With map, define as the space such that is continuous. The space of functions whose th derivative is continuous is denoted by, and the space of smooth functions is. By is denoted any suitable vector norm. When it is required to be specific we denote the -norm by. The supremum norm of, over, is defined as (1) /00$ IEEE

2 SELMIĆ AND LEWIS: DEADZONE COMPENSATION 603 Given, the Frobenius norm is defined by with the trace. The associated inner product is. The Frobenius norm is compatible with the 2-norm so that. When is a function of time we use the standard norms. It is said that is bounded if its norm is bounded. Matrix is bounded if its induced matrix -norm is bounded. Consider the nonlinear system with state. The equilibrium point is said to be uniformly ultimately bounded (UUB) if there exists a compact set, so that for all there exists a and a number such that for all. That is, after a transition period, the state remains within the ball of radius around. A. Dynamics of Mechanical Motion Tracking Systems The dynamics of mechanical systems with no vibratory modes can be written [13] as where is a vector describing position and orientation, is the inertia matrix, is the coriolis/centripetal matrix, are the friction terms, is the gravity vector, and represents disturbances. The dynamics (4) satisfy some important physical properties as a consequence of the fact that they are a Lagrangian system. These properties are important in control system design and are as follows. Property 1: (2) (3) (4) is a positive definite symmetric matrix bounded by, where, are known positive constants. Property 2: The norm of the matrix is bounded by with known. Property 3: The matrix is skew-symmetric. This is equivalent to the fact that the internal forces do no work. Property 4: The unknown disturbance satisfies, with a known positive constant. To design a motion controller that causes the mechanical system to track a prescribed trajectory, define the tracking error by and the filtered tracking error by where is a design parameter matrix. Common usage is to select diagonal with large positive entries. Then, (6) is a stable system so that is bounded as long as the (5) (6) controller guarantees that the filtered error fact is bounded. In with the minimum singular value of. Differentiating (6) and invoking (4) it is seen that the robot dynamics are expressed in terms of the filtered error as where the nonlinear robot function is Vector contains all the time signals needed to compute and may be defined for instance as. It is noted that the function contains all the potentially unknown functions, except for, appearing in (8); these latter terms cancel out in the stability proof. III. NEURAL NETWORKS FOR DEADZONE COMPENSATION In this section we review some background in NN and present a new result on NN approximation of piecewise continuous functions. It is found that to approximate such functions suitably, it is necessary to augment the standard NN that uses smooth activation functions with extra nodes containing a certain jump function approximation basis set of nonsmooth activation functions. A. Background on Neural Networks The two-layer NN consists of two layers of tunable weights. The hidden layer has neurons, and the output layer has neurons. The multilayer NN is a nonlinear mapping from input space into output space. If is the th input to the network, and the weight between the th input and th hidden node, then the output of the output layer is given by (7) (8) (9) (10) where hidden layer activation function; first-layer weights; bias weights for the first layer; output layer activation function; second-layer weights; bias weights for the second layer. There are many different ways to choose the activation functions, including sigmoid, hyperbolic tangent, etc. We choose to be the sigmoid function. The output layer activation function is usually chosen to be a linear function. Define the weight matrices for the first and second layers as, ; and ;. Define

3 604 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 vectors,, and one can write (10) in matrix form as (11) where the vector of the thresholds is defined as. In order to include the thresholds in the matrix, the vector activation function is defined as, where. Tuning of the weights then includes tuning of the thresholds too. B. NN Approximation of Continuous Functions Many well-known results say that any sufficiently smooth function can be approximated arbitrarily closely on a compact set using a two-layer NN with appropriate weights [5], [8]. For instance, Cybenko s result [5] for continuous function approximation says that given any function, with compact subset of, and any, there exists a sum of the form for some,,, such that (12) (13) for all. Function could be any continuous sigmoidal function [5]. This result shows that any continuous function can be approximated arbitrarily well using a linear combination of sigmoidal functions. This is known as the NN universal approximation property. Combining (11) (13) one has (14) where the is the NN approximation error. The reconstruction error is bounded on a compact set by. Moreover, for any one can find an NN such that for all. The first layer weights are selected randomly and will not be tuned. The second layer weights are tunable. The approximation holds [10] for such an NN, with approximation error convergence to zero of order, where is the number of hidden layer nodes (basis functions), and is independent of. The approximating weights are ideal target weights, and it is assumed that they are bounded such that. C. NN Approximation of Jump Functions Results for approximation of piecewise continuous functions or functions with jumps are given in [27]. It is found that to approximate such functions suitably, it is necessary to augment the set of functions used for approximation. In addition to continuous sigmoidal functions, one requires a set of discontinuous basis functions. We propose the sigmoidal jump approximation functions in the next result. Theorem 1 (Approximation Using Sigmoid Jump Basis Functions): Let there be given any bounded function which is continuous and analytic on a compact set, except at where function has a finite jump and is continuous from the right. Then, given any, there exists a sum of the form such that (15) (16) for every in, where is a function in, and the sigmoid jump approximation basis functions are defined as for for (17) Proof: See [27]. Theorem 1 says that any bounded function with a finite jump can be approximated arbitrarily well by a sum of a function with continuous first derivatives and a linear combination of jump basis functions. Using the known result of Cybenko for continuous function approximation, one can formulate the following result for approximation of functions with a finite jump. Theorem 2 (General Approximation Result): Let there be given bounded function which is continuous and analytic on a compact set, except at where function has a finite jump and is continuous from the right. Given any, there exists a sum of the form such that (18) (19) for every in, where is a sigmoid function, and the sigmoidal jump approximation functions are defined as (17). Proof: See [27]. Similarly as in the Theorem 1 it can be shown that instead of the sigmoidal jump approximation functions, one can use either for for or jump basis functions based on the hyperbolic tangent where. for for (20) (21)

4 SELMIĆ AND LEWIS: DEADZONE COMPENSATION 605 Fig. 1. Nonsymmetric deadzone nonlinearity. Fig. 2. Deadzone inverse. Note that the above result requires that the point of discontinuity ( ) is known. This is the case in many industrial motion control system nonlinearities: friction ( ), deadzone inverse ( ), backlash inverse (discontinuous at ). Therefore, the augmented NN is a useful tool for compensation of such nonlinearities and will be used here for deadzone compensation. IV. COMPENSATION OF DEADZONE NONLINEARITY In this section an NN precompensator for a general model of the deadzone is given. It is not required to be symmetric, and the function outside the dead-band may not be a linear function. The proposed method can be applied for compensation of any preinvertible, bounded, unknown, nonlinear function. The generality of the method and applicability to a broad range of nonlinear functions make this approach a potentially useful tool for compensation of backlash, hysteresis, and other nonlinearities. A. Deadzone Nonlinearity Fig. 1 shows a nonsymmetric deadzone nonlinearity where and are scalars. In general, and are vectors. It is assumed that the deadzone has a nonlinear form, which is more general than in [29], [31], and [32]. A mathematical model for the deadzone characteristic of Fig. 1 is given by (22) Functions, are smooth, nonlinear functions, so this describes a very general class of. All of,,, and are unknown, so that compensation is difficult. Assumption 1: The functions and are smooth and invertible continuous functions. Most application schemes cover only the case of symmetric deadzones, where, or the case where functions and are linear. These assumptions are required because of the limitations of standard adaptive control techniques. In our approach, the general deadzone model is analyzed, and Assumption 1 requires only that the deadzone nonlinearities be invertible. B. NN Deadzone Precompensator To offset the deleterious effects of deadzone, one may place a precompensator as illustrated in Fig. 3. There, the desired function of the precompensator is to cause the composite throughput from to to be unity. In order to accomplish this, it is necessary to generate the inverse of the deadzone nonlinearity [23], [32]. By assumption, the function (22) is invertible; therefore, there exist, such that (23) The function is shown in Fig. 2. The mathematical model for the function shown in Fig. 2 is given by The deadzone inverse form as where the modified deadzone inverse (24) can be expressed in equivalent is given by. (25) (26) Equation (25) is a direct feedforward term plus a correction term. Function is discontinuous at zero, as is. Based on the NN approximation property, one can approximate the deadzone function by (27) Using the modified NN with activation functions shown in Section III-C, one can design an NN for the approximation of the modified inverse function given in (26) by (28) In these equations, are the NN reconstruction error and,, are ideal target weights. The reconstruction error is bounded by,, where is equal to

5 606 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 Fig. 3. Deadzone compensation scheme. and, respectively. In this paper we consider restricted to a compact set, and in that case these bounds are constant, i.e.,,. The case where these bounds are not constant and not restricted to the compact set is treated in [15], and in that case an additional saturation term has to be added to the robustifying signal. The first-layer weights,,, in both (27) and (28) are fixed, and if they are properly chosen, the approximation property of the NN is still valid [10]. Note we use two NN. The structure of the NN deadzone precompensator and deadzone estimator are shown in Fig. 3. The first NN is used as deadzone estimator, while the second is used as a deadzone compensator. Note that only output of the NN II is directly affecting the input, while the NN I is higher level performance evaluator, and is used for tuning the NN II. Define, as estimates of the ideal NN weights, which are given by the NN tuning algorithms. Define the weight estimation errors as (29) and the approximations of the nonlinear deadzone and modified deadzone inverse functions as (30) (31) The next result shows the effectiveness of the proposed NN structure, by providing an expression for the composite throughput error of the compensator plus deadzone. It shows that, as the estimates, approach the actual neural network parameters,, the NN precompensator effectively provides a preinverse for the deadzone nonlinearity. It is shown in Section V how to tune (30) and (31) so that tracking error is small, and and are close to,, i.e., the weight estimation errors defined by (29) are bounded. Theorem 3 (Throughput Error Using NN Deadzone Compensation): Given the NN deadzone compensator (31), (32), and the NN observer (30), the throughput of the compensator plus the deadzone is given by where the modeling mismatch term with defined in the proof. Proof: From (27) and (32), one has is given by (33) (34) (35) Note that expressions (30) and (31) represent, respectively, an NN approximation of the deadzone function (22) and of the modified deadzone inverse (26). Signal is used for the deadzone compensation, and represents the estimated value of signal. Note that (32) Assumption 2 (Bounded Ideal NN Weights): The ideal weights, are bounded such that,, with, and known bounds. From (23) follows whence, by (28) (36) (37)

6 SELMIĆ AND LEWIS: DEADZONE COMPENSATION 607 Using the Taylor series expansion one has where,,, are computable constants. Proof: From (34) one has (43) (38) where is the remainder of the first Taylor polynomial. Regrouping the terms one has From definition (40) follows: (44) (45) (39) where is given by (46) (40) Combining (40) and (32) gives (47) (48) (41) which combined with (35) gives (33). The form of (33) is crucial in Section V in deriving the NN tuning laws that guarantee closed-loop stability. The first term has known factors multiplying, the second term has known factors multiplying, and a suitable bound can be found for. The form of (33) is similar to the form in [29], but instead of parameterizing the direct and the inverse functions with the same parameters, we use different function approximators for the direct and inverse functions together with crucial fact (23), which is actually a connection between them. The expression (23) couples the information inherent in NN I and NN II. Intuitively, compensating the unknown effect (NN II) depends on what one observes (NN I), and vice versa; observation of the unknown effect (NN I) depends on how one modifies the system (NN II). This will later, in Section V, be very clearly seen in the tuning laws derived for, and, where the differential equations for tuning NN I and NN II are mutually coupled. In general, this proposed NN compensation scheme can be used for inverting any continuous invertible function. Therefore, it is a powerful result for compensation of general actuator nonlinearities in motion control systems. The next result gives us the upper bound of the norm of. It is an important result used in the stability proof. Lemma 1: The norm of the modeling mismatching term in (33) is bounded on a compact set by (42) where,, and are computable constants. Combining (40) and (47) one gets (42). V. NN CONTROLLER WITH DEADZONE COMPENSATION Deadzone compensation is considered in [23] for nonlinear systems in Brunovsky form with known nonlinear functions and for linear systems in [29] and [32]. In this section we show how to provide NN deadzone compensation for deadzones in mechanical systems, including robotic systems, with inexactly known nonlinearities. The deadzone precompensator given by [15], [17] used fuzzy logic. In this section it is shown how to tune or learn the weights of the NN in (30), (31) on-line so that the tracking error is guaranteed small and all internal states are bounded. It is assumed, of course, that the actuator output is not measurable. A. Tracking Controller with NN Deadzone Compensation If in (8) is unknown, it can be estimated using adaptive control techniques [13], [28] or the neural network controller in [14]. A robust compensation scheme for unknown terms in is provided by selecting the tracking controller (49) with an estimate for the nonlinear terms and a robustifying term to be selected for the disturbance rejection. The estimate is fixed in this paper and will not be adapted,

7 608 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 Fig. 4. Tracking controller with NN deadzone compensation. as is common in robust control techniques [4], [13]. The feedback gain matrix is often selected diagonal. Deadzone compensation is provided using (50) The multiloop control structure implied by this scheme is shown in Fig. 4, where,,. The controller has a PD tracking loop with gains, where the deadzone effect is ameliorated by the NN feedforward compensator. The estimate is computed by an inner nonlinear control loop. In order to design an NN system such that the tracking error is bounded and all internal states are stable, one must examine the error dynamics. Substituting (49) and (33) into (8) yields the closed-loop error dynamics (51) where the nonlinear function estimation error is given by. Assumption 3 (Bounded Estimation Error): The nonlinear function is assumed to be unknown, but a fixed estimate is assumed known such that the functional estimation error satisfies (52) for some known bounding function. This is not unreasonable [4], [13] as in practical systems the bound can be computed knowing the upper bound on payload masses, frictional effects, and so on. Assumption 4 (Bounded Reference Trajectory): The desired trajectory is bounded so that (53) with a known scalar bound. The next theorem provides algorithms for tuning the NN weights for the deadzone precompensator with guaranteed closed-loop stability. Theorem 4 (Tuning of NN Adaptive Deadzone Compensator): Given the system in (8) and Assumptions 1 3, select the tracking control law (49), plus the deadzone compensator (50). Choose the robustifying signal as (54) where the and are bounds on functional estimation error and disturbance, respectively. Let the estimated NN weights be provided by the NN tuning algorithm (55) (56) with any constant matrices,, and small scalar design parameters. Then the filtered tracking error and NN weight estimates, are UUB, with bounds given by (76) (78). Moreover, the tracking error may be kept as small as desired by increasing the gains.

8 SELMIĆ AND LEWIS: DEADZONE COMPENSATION 609 Proof: Select the Lyapunov function candidate (57) Differentiating yields (58) and using (51) yields (65) (59) Applying Property 3 and the tuning rules one has (66) (60) (61) Using the inequality (62) (67) and Lemma 1, the expression (61) can be modified as (68) (63) where the function is defined as (69) Let the constant be defined by (70) Defining (64) (71)

9 610 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 one has for every. Then, (68) is equivalent to (72) Fig. 5. Two-link robot arm. Therefore, is guaranteed negative as long as or or The last three inequalities are equivalent to (73) (74) (75) (76) for the second-order modeling mismatching term in (42). The right-hand side of (76) can be taken as a practical bound on the tracking error in the sense that will never stray far above it. Note that the stability radius may be decreased any amount by increasing the PD gain. It is noted that PD control without deadzone compensation requires much higher gain in order to achieve the similar performance that is, eliminating the NN feedforward compensator will result in degraded performance. Moreover, it is difficult to guarantee the stability of such highly nonlinear system using only PD. Using the NN deadzone compensation, stability of the system is proven, and the tracking error can be kept arbitrarily small by increasing the gain. The NN weight errors are fundamentally bounded in terms of,. The tuning parameters, offer a design tradeoff between the relative eventual magnitudes of and,. The weights,, are set to random values. It is shown in [13] that for such NN, termed random variable functional link (RVFL) NN s, the approximation property holds. The weights are initialized to random values, are initialized at zero. Then the PD loop in Fig. 4 holds the system stable until the NN begins to learn. (77) (78) The mutual dependence between NN I and NN II results in coupled tuning law equations. This mathematical result followed from the fact that the information stored in NN I and NN II are dependent on each other. In fact, the proposed NN compensator with two NN s can be viewed as an NN compensator of second order. The first terms of (55) and (56), are modified versions of the standard backpropagation algorithm. The terms correspond to the -modification [18], to guarantee bounded parameter estimates. Note that the term corresponding to in (56) is a second order e-modification, which is an efficient way to compensate VI. SIMULATION OF NN DEADZONE COMPENSATOR To illustrate the performance of the NN deadzone compensator, the two link robot arm (Fig. 5) is used. The model of the system shown in Fig. 5 is given in [13]. The system parameters are chosen as,,,. The manipulator is acting in horizontal plane. In order to examine the effects of the deadzone, the gravity is not included in the system model. The NN I has hidden-layer nodes with sigmoidal activation functions. The first-layer weights are selected randomly [10]. They are uniformly randomly distributed between and. These weights represent the stiffness of the sigmoidal activation function. The threshold weights for the first layer are uniformly randomly distributed between and. The threshold weights represent the bias in activation functions positions. Therefore, they should cover the range of the deadzone. Since the deadzone width is not known, it is recommended that this range is large enough so it covers the deadzone width. The tuning law requires that the second-layer weights and cannot both be initialized at zero, because it is clear from (55) and (56) that in that case NN weights would stay at zero forever. Therefore, the second-layer weights for the NN I are

10 SELMIĆ AND LEWIS: DEADZONE COMPENSATION 611 Fig. 6. Position error for 1. Joint: without deadzone compensation (dash) and with NN deadzone compensator (full). uniformly randomly initialized between 50 and 50. Note that this weight initialization will not affect system stability since the weights are initialized at zero, and therefore there is initially no input to the system except for the PD loop. The NN II is augmented for approximation of discontinuous functions and has hidden-layer nodes with sigmoidal activation functions and four additional nodes with jump function basis. The first-layer weights are uniformly randomly distributed between and as in NN I, and the thresholds weights -uniformly randomly distributed between and. The second-layer weights are initialized at zero. We simulated the two link robot arm with deadzones in both links. The size of the NN s should be selected to satisfy the system performance, but not too large. The right way to do it is to select smaller number of the hidden-layer neurons, and then increase the number in successive simulations until we get satisfactory performance of the system behavior. To focus on the deadzone compensation, we selected the disturbance as and so the robust term. We simulated several cases. A. Nonsymmetric Deadzone with Different Slopes, Step Input The step signal is applied at the input. The NN weight tuning parameters are chosen as,,,. The controller parameters are chosen as,. The deadzone is assumed to have linear functions with the same slopes outside the deadband, i.e.,,,,. The position errors to unit step inputs for the first and second joints are shown in Figs. 6 and 7 using only PD without deadzone compensation [using (50) with ] and using PD plus NN compensator. Control signal is shown in Figs. 8 and 9. B. Nonsymmetric Deadzone with Different Slopes, Sinusoidal Input The same compensator is simulated when the sinusoidal input is applied. The structure of the compensator and controller is the same as in the previous case. The tracking errors for the first and second joints are shown in Figs. 10 and 11 using only Fig. 7. Position error for 2. Joint: without deadzone compensation (dash), and with NN deadzone compensator (full). Fig. 8. Control signal u(t) for the first joint: without deadzone compensation (dash), and with NN deadzone compensator (full). Fig. 9. Control signal u(t) for the second joint: without deadzone compensation (dash), and with NN deadzone compensator (full). PD without NN deadzone compensation and using PD plus NN compensator. Control signal is shown in Figs. 12 and 13. One can see that after the transient period of 1.5 s, NN s adapt their weights in order to decrease the filter tracking error. It is interesting to note that the control has a superimposed high-

11 612 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 4, APRIL 2000 Fig. 10. Tracking error for 1. Joint: without deadzone compensation (dash), and with NN deadzone compensator (full). Fig. 13. Control signal u(t) for the second joint: without deadzone compensation (dash), and with NN deadzone compensator (full). From this simulation it is clear that the proposed NN deadzone compensator is an efficient way to compensate for deadzone nonlinearities of all kind, without any restrictive assumptions on the deadzone model itself. Fig. 11. Tracking error for 2. Joint: without deadzone compensation (dash), and with NN deadzone compensator (full). Fig. 12. Control signal u(t) for the first joint: without deadzone compensation (dash), and with NN deadzone compensator (full). frequency component that is very similar to that injected using dithering techniques. In fact, the signal from NN is injected at the same position in the control loop that dithering signals are injected. Therefore, one could consider adaptive NN deadzone compensator as an adaptive dithering technique. VII. CONCLUSION A new technique for inverting an unknown right-invertible function is presented. It is applied to actuator deadzone compensation and does not require any restrictive assumptions on the deadzone nonlinearity (e.g., linearity outside the deadband). A compensator scheme consisting of an NN estimator and an NN compensator is developed. The NN controller does not require preliminary off line training. Rigorous stability proofs are given using Lyapunov theory. Simulation results show that the proposed compensation scheme is efficient for both symmetrical and unsymmetrical deadzone nonlinearities. REFERENCES [1] R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory, vol. 39, pp , May [2] F.-C. Chen and H. K. Khalil, Adaptive control of nonlinear systems using neural networks, Int. J. Contr., vol. 55, no. 6, pp , [3] S. Commuri and F. L. Lewis, CMAC neural networks for control of nonlinear dynamical systems: Structure, stability and passivity, in Proc. IEEE Int. Symp. Intell. Contr., Monterey, CA, Aug. 1995, pp [4] M. J. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Contr., vol. 26, pp , May [5] G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Contr. Signals, Syst., vol. 2, no. 4, pp , [6] C. A. Desoer and S. M. Shahruz, Stability of dithered nonlinear systems with backlash or hysteresis, Int. J. Contr., vol. 43, no. 4, pp , [7] B. Friedl, Advanced Control System Design. Englewood Cliffs, NJ: Prentice-Hall, [8] K. Funahashi, On the approximate realization of continuous mappings by neural networks, Neural Networks, vol. 2, pp , [9] J. W. Gilbart and G. C. Winston, Adaptive compensation for an optical tracking telescope, Automatica, vol. 10, pp , [10] B. Igelnik and Y. H. Pao, Stochastic choice of basis functions in adaptive function approximation and the functional-link net, IEEE Trans. Neural Networks, vol. 6, pp , Nov [11] J.-Y. Jeon, J.-H. Kim, and K. Koh, Experimental evolutionary programming-based high-precision control, IEEE Contr. Syst. Mag., vol. 17, pp , Apr

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Automat. Contr., vol. 40, pp , Feb [31], Continuous-time adaptive control of systems with unknown backlash, IEEE Trans. Automat. Contr., vol. 40, pp , June [32], Discrete-time adaptive control of systems with unknown deadzones, Int. J. Contr., vol. 61, no. 1, pp. 1 17, [33], Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York: Wiley, [34] L.-X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Englewood Cliffs, NJ: Prentice-Hall, Rastko R. Selmić was born in Belgrade, Serbia, Yugoslavia, in He received the B.S. degree in electrical engineering from the University of Belgrade in 1994 and the M.S. degree in electrical engineering from the University of Texas, Arlington in Since 1995, he has been a Ph.D. student and Graduate Research Assistant at the Automation and Robotics Research Institute of the University of Texas, Arlington. In 1996 and 1998, he worked for Signalogic, Inc., in Dallas, TX, developing DSP drivers for Motorola processors and implementing new DSP functions. His research interests include nonlinear control, adaptive control, neural networks, friction, and deadzone, and backlash compensation using intelligent control tools. Mr. Selmić received the first prize in the IEEE Fort Worth Section Graduate Paper Contest for 1999, the ARRI Best Student Paper Award for 1997, and a scholarship from Signalogic in He is a member of Sigma Xi. Frank L. Lewis was born in Würzburg, Germany, subsequently studying in Chile and Gordonstoun School in Scotland. He obtained the B.S. degree in physics/electrical engineering and the M.S. degree in electrical engineering at Rice University, In 1977, he received the M.S. degree in Aeronautical Engineering from the University of West Florida. In 1981, he obtained the Ph.D. degree at the Georgia Institute of Technology, Atlanta. He spent six years in the U.S. Navy, serving as Navigator aboard the frigate USS Trippe (FF-1075), and Executive Officer and Acting Commanding Officer aboard the USS Salinan (ATF-161). He was employed as a Professor from 1981 to 1990 and is currently an Adjunct Professor at the Georgia Institute of Tech. He is a Professor of Electrical Engineering at the University of Texas, Arlington. His current research interests include robotics, intelligent control, neural and fuzzy systems, nonlinear systems, and manufacturing process control. He is the author/co-author of two U.S. patents, 124 journal papers, 20 chapters and encyclopedia articles, 210 refereed conference papers, and seven books: Optimal Control, Optimal Estimation, Applied Optimal Control and Estimation, Aircraft Control and Simulation, Control of Robot Manipulators, Neural Network Control, High-Level Feedback Control with Neural Networks, and the IEEE reprint volume Robot Control. Dr. Lewis is a registered Professional Engineering in the State of Texas and was selected to the Editorial Boards of International Journal of Control, Neural Computing and Applications, and Int. J. Intelligent Control Systems. He is the recipient of an NSF Research Initiation Grant. He 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 ARRI Patent Award, various Best Paper Awards, the IEEE Control Systems Society Best Chapter Award (as Founding Chairman), the National Sigma Xi Award for Outstanding Chapter (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 Ft. Worth IEEE Section. He was appointed to the NAE Committee on Space Section in 1995 and to the IEEE Control Systems Society Board of Governors in 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 and was awarded the Moncrief-O Donnell Endowed Chair in 1990 at the Automation and Robotics Research Institute.