AS A POPULAR approach for compensating external

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 137 A Novel Robust Nonlinear Motion Controller With Disturbance Observer Zi-Jiang Yang, Hiroshi Tsubakihara, Shunshoku Kanae, Member, IEEE, Kiyoshi Wada, Member, IEEE, and Chun-Yi Su, Senior Member, IEEE Abstract In this brief, a novel robust nonlinear motion controller with disturbance observer (DOB) for positioning control of a nonlinear single-input single-output (SISO) mechanical system is proposed. The controller is designed in a backstepping manner. First, a proportional integral (PI) controller is designed to stabilize the position error. Consequently, a novel robust nonlinear velocity controller with DOB is designed to stabilize the velocity error. With the help of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proven, which leads to a major contribution of construction of a theoretically guaranteed robust nonlinear controller with DOB. The performance of the proposed controller is verified through application to a magnetic levitation system. Comparative studies with an adaptive robust nonlinear controller are also carried out. It is shown that the proposed novel controller while being simple is superior over the adaptive robust nonlinear controller for the experimental setup under study. Index Terms Backstepping, disturbance observer (DOB), input-to-state stability (ISS), motion control, robust nonlinear control. I. INTRODUCTION AS A POPULAR approach for compensating external disturbances and model mismatch, a disturbance observer (DOB) is often included into a motion controller. The DOB-based motion controllers have been widely accepted in the industrial side, due to their simplicity and transparency of design, and excellent disturbance compensation ability. The key point of a DOB is to pass the external disturbances and model mismatch, lumped as an error term of the motion equation, through a low-pass filter and then to compensate the external disturbance and model mismatch by the output of the low-pass filter. The filter s output is viewed as the estimate of the lumped disturbance. This results in an inner-loop around the controlled plant such that the inner-loop approximates a simple nominal plant model at low-frequencies. Hence, a simple controller can be designed for the approximated nominal model. So far, many papers have been published on the DOB-based motion controllers [1] [5], [7], [9], [10]. A major problem of the DOB-based controller is that the inner-loop approximates a simple nominal plant model only at low-frequencies, therefore, the disturbances and model mismatch at high-frequencies Manuscript received April 22, 2005; revised May 8, 2006. Manuscript received in final form February 2, 2007. Recommended by Associate Editor B. de Jager. Z. J. Yang, H. Tsubakihara, S. Kanae, and K. Wada are with the Department of Electrical and Electronic Systems Engineering, Graduate School of Information Science and Electrical Engineering, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan (e-mail: yoh@ees.kyushu-u.ac.jp). C.-Y. Su is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC H3G 1M8, Canada. Color versions of one or more of the figures in this brief are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2007.903091 may degenerate the control performance and even destroy the closed-loop stability. It was pointed out in [12] theoretically and experimentally that the DOB-based controllers may not be robust to large model mismatch. Many efforts have been made to handle this problem. A robust two-degree-of-freedom (2DOF) control system, including a DOB, was proposed in [10]; a DOB-based controller was designed using the linear control theory in [7]; and in [4], a sliding mode controller was introduced to suppress the disturbance estimation error under the assumption that the linear nominal model is stabilized by an inner 2DOF controller. On the other hand, some improvements of the disturbance estimation performance of the DOBs have also been reported. In [5], high-order DOBs were proposed for various types of disturbances, and in [2] and [3], the variable structure system theory was applied to the DOBs so that the disturbance estimation error can be made sufficiently small by some design parameters. Despite the works mentioned before, it should be commented here that the DOB-based motion controllers are usually designed according to the linear control theory, even if the actual controlled plant is strongly nonlinear. Unfortunately, the rigorous stability of these controllers for nonlinear systems has not been well studied in the literature except [1], the equivalence between a passivity-based robot controller and a DOB-based robot controller was investigated. It was proven that the disturbed system of the velocity-loop is input/output stable with respect to the pair of lumped disturbance (as the input) and velocity-loop error (as the output). However, no active efforts were put to suppress the disturbance estimation error. Also, it is still unclear if the lumped disturbance itself is bounded in the case some internal signals associated with large model mismatch are included in the lumped disturbance. In this brief, a theoretically guaranteed robust nonlinear motion controller with a DOB for positioning control of a nonlinear single-input single-output (SISO) mechanical system is proposed. The controller is designed in a backstepping manner. First, a proportional integral (PI) controller is initially designed to stabilize the position error. Then, a novel robust nonlinear velocity controller with the DOB is consequently designed to stabilize the velocity error. With the use of nonlinear damping terms, the input-to-state stability (ISS) property of the overall nonlinear control system is proven, which leads to a major contribution of construction of a theoretically guaranteed robust nonlinear controller with the DOB. By exploiting the controller structure, the complicated-looking controller can be explained as hierarchical modifications of the conventional PI motion controller with minor-loop, by adding the feedforward module, nonlinear damping module and DOB module to it. Therefore, it is believed that the proposed controller may gain wide acceptance of the engineers owing to its simplicity of structure and transparency of design. The performance of the 1063-6536/$25.00 2007 IEEE

138 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 proposed controller is verified through application to a magnetic levitation system. Comparative studies with an adaptive robust nonlinear controller are also carried out. It is shown that the proposed novel controller, while being simple, is superior over the adaptive robust nonlinear controller for the experimental setup under study. II. STATEMENT OF THE PROBLEM Consider the following SISO nonlinear mechanical system: (1a) (1b),, and are the position and velocity, respectively, is the control input; and are modelable nonlinear functions with known nominal functions; and is the completely unknown term composed of unmodelled nonlinearities and disturbances. Denoting the nominal nonlinearities based on a priori knowledge as and, we can model and as (2) III. CONTROLLER DESIGN Since the system (1) is of relative degree 2, the proposed controller is designed in a backstepping procedure, composed of two steps. First, a PI controller is designed to stabilize the position error. Second, a novel robust nonlinear velocity controller with a DOB is designed to stabilize the velocity error. The nonlinear damping terms are employed to yield a theoretically guaranteed robust nonlinear controller with the DOB. In this section, the design procedure is presented as follows. Step 1) Define the position error signal and velocity error signal, respectively, as is the virtual input to stabilize. Then, from (1a), we have subsystem The virtual input technique (6) (7) is designed based on the common PI control (8) and denote the modelling errors. We impose the following standing assumptions. Assumption 1: and are bounded away from zero with the same known sign, for, is the desired domain of operation. Assumption 2: There exist finite positive, but not necessarily known constants and known continuous functions and such that the following inequalities hold for : (3),. Notice that Step 2) From (1b), (2), and (6), we have subsystem The next step is to stabilize the velocity error. Denoting the error terms lumped disturbance,wehave (9) (10) as the and are appropriate known functions (bounding functions) which will be used for construction of nonlinear damping terms [6], [12]. The inequalities in (4) mean that the nominal function and the error function do not grow in a higher order than itself. Assumption 3: The reference trajectory is appropriately chosen as a sufficiently smooth function such that and are known and (5) To generate a smooth reference trajectory, we can pass the command trajectory through a low-pass filter:. The task is to design a theoretically guaranteed high performance controller to effectively counteract the modelling errors and disturbances so that the position tracks the reference trajectory accurately. (4) (11a) (11b) Therefore, the lumped disturbance can be obtained as (11a). However, since calculation of by direct differentiation is usually contaminated with high frequency noise, we have to pass (11a) through a low-pass filter to obtain the estimate of as (12) This is the so-called DOB studied extensively in the literature [1] [5], [7], [9], [10]. Although a high-order filter may improve the estimation performance of the DOB [5], in this brief, we adopt a simple first-order filter is ob- The benefit of compensating the control input by vious. Replacing in (10) by (13)

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 139 and assuming model:, we have the following simple nominal (14) is a nominal linear input. This implies that an inner-loop around the controlled plant is formed such that the inner-loop approximates a simple nominal plant model at low frequencies. A simple controller can, therefore, be designed for the approximated nominal model. The simplest design is, for example, to let. However, it should be pointed out that we can only expect at low frequencies. If the disturbance and model mismatch are fast changing, the estimation error cannot be neglected and can even destroy the stability of the closed-loop in the case of large model mismatch [12]. Although many efforts have been done to improve the robustness of the DOB-based motion controllers [2] [5], [7], [10], the DOB-based motion controllers are usually designed according to the linear control theory, even if the actual controlled plant may be strongly nonlinear. Unfortunately, the rigorous stability of these controllers for nonlinear systems has not been well studied in the literature. To stabilize the subsystem, we design the following controller: If, it is easy to show that is asymptotically stabilized to zero. However, we can only expect that at low frequencies. At high frequencies, we cannot neglect the effects of. Therefore, it is necessary to investigate if the internal signals of the closed-loop is bounded. IV. STABILITY ANALYSIS We now present the stability analysis taking into account the effects of. A. Step 1 Applying the virtual input to the subsystem,wehave In the transfer function form, the subsystem as (18) can be expressed Equation (19) can be rewritten into the state-space form (19) (20) (15) (21) The ISS property of the subsystem can be described in the following lemma [8]. Lemma 1: If the virtual input is applied to subsystem and if is made uniformly bounded at the next step, then is ISS, i.e., for, (16) and ; is a feedback controller with model compensation; is a compensating term by the disturbance observer s output;,, and are nonlinear damping terms [6] to counteract,, and, respectively; is a nonlinear damping term to ensure boundedness of when is used. are designed as time-varying control gains so that they grow at least as the same order as the corresponding uncertain terms to be counteracted. Applying the designed control input to the subsystem, we have B. Step 2 Next, we show that the boundedness and transient performance of can be achieved by the nonlinear damping terms. From (17), we have (22) (17) (23)

140 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 Finally, we have (24) According to the inequalities in Assumption 2, it is obvious that each term in the numerator of is counteracted by a nonlinear damping term in the denominator which grows at least as the same order as the corresponding term in the numerator, so that is uniformly bounded. Basic concepts of the nonlinear damping terms may refer to the [6, ch. 3]. Moreover, from (22), we have (25) and, hence (31) (32) The theoretical results are summarized in the following lemma. Lemma 2: Let Assumptions 1 3 hold. If the control input is applied to the subsystem, then the subsystem is ISS and the error signal is uniformly bounded as and, hence (26) Therefore, the uniform boundedness of can be ensured by the nonlinear damping terms. Thus, Lemma 1 holds, which implies is bounded. Since the reference trajectory,, and are uniformly bounded (Assumption 3), we can, therefore, conclude that all the internal signals of the two subsystems are uniformly bounded. In the previous analysis, the main attention is to show the boundedness of the internal signals of the closed-loop. No analysis can yet be done for the attenuation effects of. Without such an analysis, we cannot clearly see how the DOB s output can bring improvement. We now attempt to make such an effort. Keeping that all the internal signals are bounded in mind, we rewrite (22) according to the last line of (17) Remark 1: Owing to the nonlinear damping terms, the ISS property described by (26) or (32) still holds when the DOB is not used [see in (23) or in (28)]. C. ISS Property of the Overall Error System Lemmas 1 and 2 imply that the overall error system is a cascade of two ISS subsystems. Define the error signal vector (33) Then along the same lines of the [6, Proof of Lemma C.4], we have the following results: (34) (27) Notice that (28) (29) (30) (35) Furthermore, according to (19), we can see that due to the PI controller for the first subsystem, the steady offset component (zero frequency component) of converges to zero. The results are summarized into the following theorem. Theorem 1: Let Assumptions 1 3 hold. All the internal signals are uniformly bounded and the following results hold. 1) The overall error system is ISS such that The attenuation effect of is reflected by. Recall that all the internal signals are uniformly bounded, it is trivial to verify that each term in the numerator of is counteracted by a nonlinear damping term in the denominator. Furthermore, it should be commented here that has very transparent physical meaning. At low frequencies, we can expect. Any nonzero at high frequencies is counteracted by so that is quite robust against. 2) The steady offset of converges to zero. Remark 2: The second conclusion of Theorem 1 does not mean itself converges to zero since can only be made small around the origin. However, due to the integral action of PI control, we can make the zero frequency component (steady offset) of converges to zero.

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 141 Fig. 1. Evolutionary development of controller structure. (a) Conventional nonlinear motion controller with minor-loop. (b) Nominal nonlinear motion controller by backstepping design. (c) Nominal nonlinear motion controller with DOB by backstepping design. (d) Robust nonlinear motion controller by backstepping design. (e) Robust nonlinear motion controller with DOB by backstepping design. Remark 3: According to the results of Lemma 1, Lemma 2 and Theorem 1, the initial error signals and influence the transient performance significantly. To achieve satisfactory transient performance, it is recommended to initialize the reference trajectory appropriately [6]. This can be simply achieved by appropriately initializing the output of the reference

142 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 Fig. 2. Diagram of the magnetic levitation system. filter as,,so that the initial error signals become. Also, notice that can be made arbitrarily small by increasing the nonlinear damping term gains,,, in the denominator of sufficiently. V. COMMENTS ON THE CONTROLLER STRUCTURE Although the robust nonlinear motion controller with the DOB is designed in a backstepping manner, it should be pointed out that the complicated-looking controller designed in Section III can be, however, explained as hierarchical modifications of the conventional PI controller with minor-loop. Therefore, it is believed that the proposed controller may gain wide acceptance of the engineers of the industrial side. The conventional PI controller with minor-loop and nominal model compensation can be illustrated in Fig. 1(a). This structure is easy to understand and is widely used in the industrial side. To improve trajectory tracking performance, we can add feedforward terms to each control loop, as shown in Fig. 1(b). If neither modelling error nor disturbance exists, the system is strictly linearized and the output will track perfectly. This diagram exactly matches the backstepping approach studied extensively by the community of adaptive control theory and gives an explanation of the close relation between the backstepping approach with the conventional PI controller with minor-loop. With the consideration of the modelling errors or disturbances, a straightforward and popular method to compensate them is to introduce a DOB to the control system as shown in Fig. 1(c). However, the robustness of this control system is not theoretically guaranteed. To suppress the modelling errors or disturbances, we can adopt the nonlinear damping terms by setting in controller (15). Then the control system can be illustrated by Fig. 1(d). In this case, the ISS property is achieved by the nonlinear damping terms. To actively compensate the modelling errors or disturbances, we can adopt the DOB together with the nonlinear damping terms by setting in controller (15). Then the control system becomes Fig. 1(e). This is the proposed novel robust nonlinear controller with the DOB. Therefore, the complicated-looking controller can be explained as hierarchical modifications of the conventional PI motion controller with minor-loop, by adding the feedforward module, nonlinear damping module, and DOB module to it. VI. APPLICATION TO A MAGNETIC LEVITATION SYSTEM To demonstrate the efficient performance of the proposed robust nonlinear controller with the DOB, extensive simulations and experiments on a 1DOF magnetic levitation system have been performed. In this section, we will show how the simulation and experimental results reflect the theoretical analysis. A. Experimental System Description Extensive experiments have been performed on the setup shown in Fig. 2, whose dynamics is governed by (36) (37) is the air gap (vertical position) of the steel ball; is the velocity of the steel ball; is the coil current; is the gravity acceleration; is the mass of the steel ball; and are the positive constants determined by the characteristics of the coil, magnetic core, and steel ball. The unknown disturbance is a sinusoidal signal artificially added to the control input to test the robustness of the proposed approach (38)

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 143 TABLE I PARAMETERS OF THE MAGNETIC LEVITATION SYSTEM Denoting the nominal physical parameters as,,, and, we have the nominal nonlinear functions and modelling errors, respectively, as follows (39) (40) The physical parameters shown in Table I were identified through closed-loop operational data and are thought to be reliable. The physically allowable operating region of the steel ball shown in Fig. 1 is limited to 0 m 0.013 m. The vertical position of the steel ball is measured by a laser distance sensor (KEYENCE LB-60). The levitated steel ball is controlled by a digital control system that consists of a PC with an 1.0-GHz Intel Pentium III Processor which is loaded with Windows 2000 OS, 12 bits analog-to-digital (A/D) and digital-to-analog (D/A) converters, and a current feedback power amplifier. The control algorithm is coded in Borland C++ language and discretized with a sampling interval of 0.5 ms. The velocity is measured by pseudo-differentiation of the measured position as. B. Design of the Controllers The controllers given in Fig. 1(b) (e) for illustrating the evolutionary development of controller structures will be applied to show the distinguished control performance of the proposed robust nonlinear controller with the DOB. The following nominal system parameters with large errors are used to verify the robust performance of the controllers: 0.30 kg 0.0020 m 0.0015 Hm (41) The bounding functions satisfying Assumption 2 used for nonlinear damping terms are given as (42) The designed controller parameters are selected as follows. Controller 1 [see Fig. 1(b)]: A nominal nonlinear motion controller (43) Controller 2 [see Fig. 1(c)]: A nonlinear motion controller with the DOB (44) Fig. 3. Simulation results of the nominal nonlinear motion controller (controller 1). Controller 3 [see Fig. 1(d)]: A robust nonlinear motion controller without the DOB (45) Controller 4 [see Fig. 1(e)]: A robust nonlinear motion controller with the DOB (46) The steel ball is initially at rest with 13 mm and 0 [mm/s], i.e., the steel ball is held on the steel plate shown in Fig. 2 before the controller s start. C. Simulation and Experimental Results Since the robustness of the first two controllers is not theoretically guaranteed, the results of the first two controllers were only verified by numerical simulations, to avoid destroying the experimental setup. To mimic a noisy position sensor, a uniformly distributed stochastic noise between -0.1 mm and 0.1 mm was added to the measured position. The results of the first two controllers are shown in Figs. 3 and 4, respectively. In each figure, from the top to the bottom are, respectively, the position, velocity, error signals of subsystems and, i.e., and, control input. In the figures of and, the dotted lines indicate their reference values, as the solid lines indicate their exact values. It can be found from Fig. 3 that in the presence of considerable large errors of the physical parameters and significant disturbance,

144 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 Fig. 4. Simulation results of the nominal nonlinear motion controller with DOB (controller 2). Fig. 6. Experimental results of the robust nonlinear motion controller with DOB (controller 4). Fig. 5. Experimental results of the robust nonlinear motion controller without DOB (controller 3). Fig. 7. Experimental results of the adaptive robust nonlinear controller. the control results by the nominal nonlinear controller are not acceptable at all. The steel ball hits the steel plate under it (at the position of 13 mm). Also, it can be seen from Fig. 4 that the nominal nonlinear controller with a DOB does not bring any further improvement for such a strongly nonlinear, open-loop unstable system with large modelling errors and disturbance. The ball even sometimes sits on the steel plate under it. Fig. 5 shows the experimental results of controller 3. It can be seen that by virtue of the nonlinear damping terms, the effects of the large modelling errors and disturbance are much suppressed and the control performance is improved much. Still, it is difficult to further reduce the control error signals only by the nonlinear damping terms. However, by applying our proposed novel controller 4, it can be seen from Fig. 6 that an excellent control performance is achieved despite of the large modelling errors and disturbance. The results match the theoretical analysis quite well. D. Comparative Studies With an Adaptive Robust Nonlinear Controller For comparative studies, we also performed the adaptive robust nonlinear controller studied in [11], which is slightly modified and redesigned in Appendix I. The design parameters are also given in Appendix I. The results are shown in Fig. 7. A comparison of Figs. 6 and 7 show that the proposed new controller is superior over the adaptive robust nonlinear controller. This is mainly due to the fact that the adaptive robust nonlinear controller while being able to cope with uncertain parameters cannot compensate unknown external disturbance actively. E. Robustness Against a Large Time-Constant of the DOB So far, it has been emphasized in the literature [1] [5], [7], [9], [10], [12] that for a DOB-based motion controller, the timeconstant of the DOB should be chosen carefully. A very small

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 145 Fig. 8. Experimental results of the robust nonlinear motion controller with DOB by a large time constant ( =0:05). VII. CONCLUSION In this brief, a theoretically guaranteed robust nonlinear motion controller with the DOB for positioning control of a nonlinear SISO mechanical system is proposed. Rigorous stability analysis was performed as well. By investigations of the controller structures, it is commented that the complicated-looking controller can be explained as hierarchical modifications of the conventional PI motion controller with minor-loop, by adding the a feedforward module, a nonlinear damping module, and a DOB module to it. Therefore, it is believed that the proposed controller may gain wide acceptance of the engineers owing to its simplicity of structure and transparency of design. The theoretical results were verified through simulation and experimental studies on a magnetic levitation system. Comparative studies with an adaptive robust nonlinear controller were carried out as well. It is concluded that at least for the magnetic levitation system, the proposed novel controller while being simple is superior over the adaptive robust nonlinear controller. APPENDIX DESIGN OF THE ADAPTIVE ROBUST NONLINEAR CONTROLLER FOR THE MAGNETIC LEVITATION SYSTEM The adaptive robust nonlinear controller studied in [11] is slightly modified and redesigned here. The nonlinear damping terms are introduced to counteract the modelling errors and disturbance, and the adaptive laws are introduced to reduce ultimate error signals. Only the design process is presented here. Theoretical analysis is similar to that in [11]. Modelling of the Uncertain Nonlinear Function: The modeling error is approximated by a radial basis function (RBF) network which is linear-in-the-parameters, as the following: (47) Fig. 9. Experimental results of the robust nonlinear motion controller with DOB by a very large time constant ( =0:1). (48) time-constant may amplify the measurement noise and hence may cause oscillations of the internal signals, as a large time-constant may lead to significant degeneration of the disturbance compensation performance and hence may make the control performance unacceptable. We should emphasize here that owing to the established ISS property of the proposed control system, the control performance is not sensitive to the time-constant of the DOB. To show this extreme, we performed some experiments by intentionally using large values of the time-constant. Two extreme cases are studied and shown in Figs. 8 and 9. It is verified that although the time-constant is unusually large in both examples, the control performance in the extreme cases does not degenerate too much. The control performances even in the extreme cases, as shown in Figs. 8 and 9, are at least comparable to the performance by the adaptive robust nonlinear controller shown in Fig. 7. This again shows the superiority of the new proposed controller. (49) is a Gaussian basis function, in which denotes the center of the th basis function, and determines its width. In this study, the basis functions are equidistantly located in, i.e., the physically allowable operating region of (see Fig. 2). When the distance between two adjacent centers is determined, can be determined as [11] (50) The network can approximate within a sufficient accuracy if is sufficiently large, i.e., there exists a desired such that (51)

146 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 1, JANUARY 2008 Fig. 10. Adaptive robust nonlinear controller by backstepping design. is a sufficiently small positive real number, and (52) The lower and upper bounds of the weight parameter vector and the gravity acceleration are assumed to be known a priori, i.e., (53) damping terms to counteract,, and, respectively. To further reduce the effects of the parameter errors, we adopt the following adaptive laws with projection which ensures the adaptive parameters stay in a prescribed range: for, for, otherwise (58) The upper bound should be chosen such that (54) is the adaptively updated parameter at time instant. Backstepping Design of the Adaptive Robust Nonlinear Controller: Step 1),, subsystem and are the same as those in Section III, and are, therefore, not repeated here. Step 2) The second subsystem is obtained as To stabilize the subsystem we design the control input as (55) for, for, otherwise (59),. In the case of, the controller is reduced to the robust nonlinear controller [see Fig. 1(d)]. The block diagram of the adaptive robust nonlinear controller is shown in Fig. 10. A. Design Parameters Five Gaussian basis functions were used and the centers of the basis functions were set as (56) The upper and lower unknown parameters were given as (60) (61) (57) and ; is a feedback controller with model compensation;,, and are nonlinear. It can be verified that (54) is satisfied. The initial values of adaptive parameters were given as (62)

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