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Then, we will utilize an adaptive backstepping scheme to deal with the case of the unknown parameters. Figure 2 describes the block diagram of the proposed ABC approach of a microgyroscope. The control objective for a z -axis microgyroscope is to track a reference oscillation trajectory q d as closely as possible and make all the signals in the closed-loop system be uniformly bounded.

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For the microgyroscope in 5 , the backstepping control design can be synthesized in two steps. Step 1: Treat X 2 as a virtual control force and design a control law for it to make X 1 follow the reference trajectory. Assume the first and second derivatives of the reference trajectory q d are all bounded.

Now that X 2 is treated as a control input, we naturally design the following simple virtual control law for X 2 to make e 1 converge to zero exponentially:. Due to the positive property of c 1 , tracking error e 1 will approach zero exponentially. Roughly speaking, X 1 rapidly approximates to q d. Step 2: However, X 2 is not the actual control input, but a state variable. We cannot operate X 2 directly.

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So, let us move on to the second line of 5 , which reveals the dynamics of X 2. In 10 , the actual control u appears. Select a Lyapunov function V for the whole system as:. We finally derive and design the real controller u. Some terms in 13 are definitely negative, and we shall keep them. Some terms are positive or indefinite, and we will use the control force to cancel them. Thus, we design the control effort as:.

In the following, we will develop the procedure to deal with unknown system dynamics, lumped parametric uncertainties, and disturbances. The modified controller in 13 is. Regarding the characteristics and performance of the proposed ABC strategy, we state the following theorem.

In the presence of lumped disturbances d f , the adaptive controller 15 with the adaptive estimator 16 applied to the microgyroscope model 3 guarantees that all the closed-loop signals are bounded and that state tracking errors converge to zero asymptotically. Note that 20 and 14 are identical. Thus, e 1 and e 2 converge to zero asymptotically.

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The adaptive laws that guarantee the tracking error converges to zero do not mean the parameter estimates are consistent only if the PE condition can be satisfied. Since the reference trajectories contain two distinct nonzero frequencies, the PE condition is satisfied, and the microgyroscope has sufficient persistence of excitation to permit the accurate identification of major fabrication imperfections and all the unknown system parameters.

The proposed ABC scheme was evaluated on a lumped z -axis microgyroscope sensor [ 1 , 2 ]. The physical parameters are described as:.

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  • Non-dimensionalizing the physical parameters, we obtained the following nondimensional parameter matrices defined in 3 :. The desired trajectory should be the resonance of vibration modes. For the moment, there is no disturbance. It must be noted that all of the system parameters, including the gyroscope, controller, and disturbance parameters are nondimensional herein, meaning that all of the parameters on vertical axes in the following figures are unitless. The simulation time was nondimensional, as were the simulation positions.

    Though they were nondimensional, the same class of parameters could be compared with each other, due to the unified reference physical quantity.

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    For comparison, Figure 4 depicts the tracking error using the proposed ABC approach, and Figure 5 shows the adaptation procedure of the parameter estimates. Figure 6 plots the control forces for the microgyroscope. Obviously different from the result depicted in Figure 3 , tracking errors approached zero quickly when using the proposed ABC scheme. Since the reference trajectories contained two different nonzero frequencies, the PE condition was satisfied. In Figure 5 , the parameter estimates converged to their true values, including the angular velocity. Standard adaptive controllers are not always robust in the presence of model uncertainties and external disturbances.

    For example, a step signal with an amplitude of was added at 20 s as an external disturbance. Figure 7 shows the tracking errors using the adaptive controller without the robust term. Comparing Figure 7 with Figure 8 , the robust term effectively suppressed the disturbances and the tracking error maintained a very small value. A well-known adaptive microgyroscope controller without the backstepping technique was presented in [ 2 ] by Park.

    The performance of our proposed ABC strategy was compared with the adaptive controller in [ 2 ]. Figure 9 , Figure 10 and Figure 11 show the dynamic response using the adaptive controller in [ 2 ] with the same nominal gyroscope parameters under the same model uncertainties and disturbances. Tracking errors using the adaptive controller in [ 2 ].

    Adaptation of parameter estimates using the adaptive controller in [ 2 ]. Control efforts for a microgyroscope using the adaptive controller in [ 2 ]. The tracking errors with the adaptive controller displayed quite a large overshot at the beginning, as did the control efforts. The settling time of tracking errors was also worse than our proposed adaptive backstepping controller. The advantage of our proposed controller over the adaptive controller in the performance of parameter estimation is clear. Put simply, the proposed adaptive backstepping controller could improve the dynamic and static performance of the microgyroscope.

    An adaptive control with backstepping technique for a z -axis microgyroscope was investigated and analyzed. The dynamics model of the microgyroscope was developed and transformed to aid in the backstepping control design. A backstepping approach and adaptive strategy were utilized to deal with the model uncertainties, disturbances, and unknown parameters of the microgyroscope.

    A controller was designed to recursively and progressively step back out of the subsystem, guaranteeing stability at each step until reaching the final external control step. Consistent parameter estimates, asymptotic stability, and tracking performance under the lumped disturbances were proved based on a Lyapunov analysis. Numerical simulation examples demonstrated the validity of the proposed ABC scheme, showing the improved performance and consistent parameter estimation.

    In our study, we only emphasized the proposed adaptive backstepping control algorithm on the microgyroscope model. In the next step, the proposed adaptive backstepping controller should be implemented in a practical experimental system to verify its effectiveness.


    The authors thank the anonymous reviewers for their useful comments that improved the quality of the paper. Conceptualization, J. National Center for Biotechnology Information , U. Journal List Micromachines Basel v. Micromachines Basel. Their combined citations are counted only for the first article. Merged citations. This "Cited by" count includes citations to the following articles in Scholar.

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    New articles related to this author's research. It uses fuzzy control for managing latitude movement of flying object. Besides, in order to decide about amounts of PID gains, it uses a simple phase control theory.

    In [7], fuzzy sliding mode has been used for missile autopilot. The phenomenon of chattering is studied as a main problem of these systems to overcome which the border of layer technique is used. In [8], analysis of air to air missile of autopilot has been shown by backstopping and a transferring method has been proposed to improve system operation.

    In [9], designing autopilot has been done for two spin projectile based on PI and linear feedback. In this study, Pitch and Yaw acceleration effectiveness are examined by a nonlinear simulation. In [10], nonlinear MRAC has been presented that is of use for a model of missile in managing Pitch channel which is directed by aerodynamic forces.

    Missile nonlinear movement is modelled by uncertainties.

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    Both certain and uncertain parameters are estimated, and based on this estimation controller parameters are updated in each step. In [12], the control technique of adaptive predictor has been provided for the inflection of transportation vehicle. The current offer has been operated for channel model of missile pitch.

    Dynamic system has been formulated based on aerodynamic stability derivations and construction of body vibration in the certainty form of state space. In [14], dynamic simulation of a missile surface to air has been done by one axis gimbal seeker, and autopilot model has been considered a first order dynamic.

    At last, simulated system operation has been studied during flying path. In [15], predictor control has been used for managing one axis gimbal seeker, and to evaluate its operation, both dynamical model of missile of surface to air with 6DOF and autopilot with first order dynamic are simulated.