Abstract
In this study, we study the performance guaranteed tracking problem for the uncertain MEMS gyroscope dynamics. A fixed-time nonlinear function and the transformed error are introduced to establish the preset time control property for the closed-loop system. Then, a funnel sliding mode controller is presented to stabilize the system. To compensate for the uncertainty and external disturbance, a novel lumped disturbance observer is proposed. Different from most existing compensation control results for the MEMS gyroscope with antidisturbance that can only achieve uniformly ultimately bounded, the proposed control scheme is able to ensure that the system errors converge to the preassigned compact sets within a given time rather than infinite time and the system outputs track the desired command asymptotically. Simulation verification also confirms the effectiveness of the proposed approach.
1. Introduction
Micromechanical system (MEMS) gyroscope plays a significant role in angular rate measurement application such as UAV attitude adjustment, robot servo control, and smart wearable system owing to their outstanding advantages, including high power-to-weight ratios and small size [1–3]. It should be pointed out that due to the influence of manufacturing errors and the external disturbance factors in practical applications, the frequency mismatch of the MEMS is likely to occur and further reduce the sensitivity of the gyroscope [4–6]. Therefore, it is particularly important to compensate the adverse influence caused by parameter uncertainty and suppress external disturbance. Generally speaking, in order to improve the measurement accuracy of MEMS, it can be realized with the help of the drive control system. Over the past decades, the studies on driving control methods have attracted much attention, and extensive studies have been devoted to solve the complex nonlinear problems with plentiful advanced control methods appeared.
Among the numerous control methods, the sliding mode control framework has received great attention due to the remarkable robustness to the external disturbances and parameter uncertainty. In [7], an adaptive fuzzy sliding mode control strategy with bound estimation was proposed to control the position of a microelectromechanical systems gyroscope in the presence of model uncertainties and external disturbances. In [8], a neural sliding mode controller was developed to adjust the sliding gain using a radial basis function neural network for the tracking control of the MEMS gyroscope. To enhance the adaptive ability of the closed-loop system, a new interval type-2 fuzzy sliding mode control [9] was designed for the MEMS gyroscope subjected to modeling uncertainties without explicit upper bound information of the uncertain nonlinearities. In addition, the robust sliding mode control with the backstepping technique in [10–12] can achieve asymptotic tracking performance even in the presence of uncertain nonlinearities with a continuous control input. But it is worth noting that the abovementioned sliding mode control approach against uncertainty or external disturbances is achieved by introducing a discontinuity term with large gain into the controller, which may result in high-frequency chattering behavior [13, 14]. Although some functions (e.g., saturation function or with being the sliding mode variable and being a small positive constant) can replace the signum function, the performance of the control system is also degraded.
On the one hand, the abovementioned approaches achieve only asymptotic stability, which implies that the system errors cannot converge to the equilibrium in finite time. In some special control requirements, it is usually necessary to consider preset performance and tracking error convergence time in the controller design in order to improve the robustness of the closed-loop system. For achieving the control performance, some important finite-time control strategies are presented in [15–18]. In [19], a nonsingular terminal sliding mode (NTSM) controller was proposed by introducing a nonsingular terminal sliding mode controller, which ensures the control system could reach the sliding surface and converge to equilibrium point in a finite period of time. Zhang et al. [20, 21] proposed an integral terminal sliding mode control scheme with the neural network for the MEMS gyroscope in the presence of system nonlinearity. By applying the finite-time stability theory and online-estimation technique, the adaptive finite-time control issue of the MEMS gyroscope with uncertain dynamics and the lumped disturbances was addressed [22]. In the abovementioned terminal sliding mode control strategies, the estimate bound of setting time is regularly dependent on the initial states; unfortunately, the information of initial condition is difficult to obtain in practical. Besides, the aforementioned finite-time results can be concluded from the inequality in the form of , in which is a Lyapunov function; , , and are the design parameters; is a uncertain parameter, and is unknown but bounded function. Obviously, if the parameters and are unknown, an accurate error convergence region cannot be obtained.
Motivated by the aforementioned observation, an adaptive prescribed performance sliding mode control strategy with the disturbance compensation system is proposed for the MEMS gyroscope in this article. The main contributions of the proposed method are summarized as follows:(1)By combining the funnel control technique with preset performance functions, a funnel sliding mode controller is developed skillfully, which ensures the predesigned transient behavior(2)A nonlinear lumped disturbance observer is constructed to estimate the lumped disturbance. Based on the proposed disturbance observer, the designed controller can ensure that the system errors reach the prescribed control accuracy in a specified time.(3)The proposed control scheme is an adaptive robust control method, which does not need to know the precise information of the system model, such as the boundary value of external disturbance and the precise system parameters.
2. System Model and Problem Formulation
2.1. System Model
Consider the dynamics of the MEMS gyroscope described by the Cartesian coordinate system [23, 24]:where is the displacement of the gyroscope’s reference point; is the control input vector; is the lumped disturbance; and are the velocity and acceleration of the gyroscope’s reference point; , , and denote the system parameter matrixes expressed as follows:where denote the angular velocity about the z-axis; , , , and are the bounded damping ratios; and , , , and are the bounded spring parameters. In practical engineering, the damping ratios and the spring constants are difficult to obtain accurately, and therefore, and with these bounded parameters are treated as the uncertain system matrixes in this study. The MEMS gyroscope system structure is shown in Figure 1.
To complete the control system design, the following assumptions are need.
Assumption 1. The desired command signals and are second-order differentiable and bounded.
Assumption 2. The each element of the lumped disturbance vector and are bounded, and therefore, there exists two unknown constants and , such that and .
Assumption 3. The system initial states ( and ) are bounded, and there exist positive constant satisfying
2.2. Preliminaries
For analyzing the stability of the closed-loop system, the following lemmas are given.
Lemma 1 (see [25]). Consider the variable defined aswhere is a positive design parameter and is an error variable. If then and
Lemma 2 (see [26]). Consider the system for positive definite function if there exist scalars , , and , which make
Then, the trajectory of system is practical finite-time stable, i.e.,and the finite time can be estimated as
2.3. Funnel Control
From [27, 28], funnel control strategy can not only ensure the stability of the controlled system but also make the tracking error evolve within a prespecified performance funnel. Define the bounded domain of the performance funnel aswhere is the system error, and is an arbitrarily bounded and continuous function satisfying and . According to [29], if can be selected reasonably, the funnel control can force the system error to be constrained inside the funnel formed by the boundaries related with . It is worth mentioning that the funnel variable in traditional funnel control is defined as . Unfortunately, such funnel variable is not differentiable at the point ; it means that such definition way of funnel variable does not meet the design requirements of some nonlinear control methods. To solve this problem, the following funnel variable will be utilized in the controller design.where is a system error vector and is a performance function ( will be given later). In the light of (9), the time derivative of can be calculated asin which . Based on (10), we can obtain
It is worth noting that nonlinear function in (9) is called performance function [29], and is chosen as follows:where the related design parameters satisfy , , and . It is obvious that the function has the following properties:(1) and (2), , and (3), , and are bounded
The control goal is to design a new fixed-time sliding mode controller to steer the system output to track the reference trajectory, while the system errors can converge to a small neighborhood of zero in a fixed time. Moreover, the control performance indexes, including the control accuracy and steady-state time, can be preset.
3. Main Results
3.1. The Prescribed Performance Controller Design
Before designing the fixed-time controller, the following new variables are defined as and , and let ; then, (1) can be rewritten as
Let , and the error transformation is defined as
Combining (14) and (15), a sliding mode surface vector is introduced bywhere is a design parameter. Furthermore, the derivative of isin which
From (17), the corresponding form of can be expressed as
For subsequent analysis, a compact set is defined aswhere is a positive constant.
Remark 1. To ensure initial conditions satisfying , the design parameters in (12) can be configured by
Remark 2. From Assumptions 1 and 2 and the boundedness of , , and , it is inferred that are bounded in compact set . Hence, there exist positive constants such that .
In (18), the nonlinear functions contain uncertain parameters and external disturbance terms. In this study, a lumped disturbance observer (21) is constructed to estimate , i.e.,where and can be obtained bywith positive constant .
Next, the control laws , are designed aswhere and are the design parameters.
Let , and the expression of and for two cases is as follows. Case 1. . Differentiating with respect to time, it yields Meanwhile, by substituting (23) into (18), one has Case 2. . Similar to the mathematical operation for case , we obtain that
3.2. Stability Analysis
Now, the main results of this study are summarized as the following theorem.
Theorem 1. Consider the MEMS (1) under Assumptions 1–3; then, for any initial error in , if the control strategy (23) is applied, the control parameters , and are appropriately selected as
Then, the following control objectives are achieved.(1)All signals in the closed-loop control system are ultimately uniformly bounded(2)The errors can converge to a predefined residual set in a predefined time
Proof. The following two cases need to be further considered in order to obtain the theoretical results.
Case 1. .
Let and . Based on (24) and (25), we obtainFrom (28), it is obvious thatwhere , , , and
Choose the following Lyapunov function candidate:It follows from taking (29) into account, differentiating respect to time, one hasNote that the design parameters (, , , and ) are selected by (27), the positive definiteness of the matrices , , , and is guaranteed; moreover, with the help of Young’s inequality and Remark 2, the following relationships can be established, i.e.,Using (32), we haveDue tothen for it follows thatCombining with (35) and (33), it can be rewritten asFrom Lemma 2, it is seen that and are finite-time stable. It means that the sliding mode variables , are bounded. According to Lemma 1 and the definition of , it implies that is bounded, and therefore, there exists positive constants , such thatand we further havenamely,
Case 2. .
Let and . One getsFrom (40), we can derive thatwhere and .
Consider the Lyapunov function as follows:It follows from (41) that the time derivative of can be expressed asBased on the condition (27), it follows that , , , and , and we can rewrite (43) aswhich implies that , , , and are bounded; by virtue of Lemma 1 and the definition of , the result (38) can be obtained; it results inCombining with (37) and (45), it follows from the definition of thatThe above formula implies that the system error converges to a prescribed compact set within fixed time . All these complete the proof.
Remark 3. In view of (46), the size of system errors depends on the parameters and . These two parameters ensure that the control system can reach the preset accuracy within a fixed time . Nevertheless, cannot be infinitesimal, which should be greater than the sampling time of the control system.
4. Numerical Simulation
In this section, numerical simulation is performed to validate the superiority of the proposed method. The system parameters of the MEMS can be decomposed into nominal and uncertain parts, i.e.,
The nominal parts are given as follows:
To verify the robustness of the proposed control system, the following two cases including uncertain parameters and the external disturbances are considered in simulation.
Case 1.
Case 2. The design parameters and the initial condition are chosen as , , , , , , , , , , , and . The desired command signals are given as .
The tracking performance for states and with the proposed controller is shown in Figures 2 and 3; the actual motion trajectories of the two states can well track the desired command signals. As the results are shown in Figures 4 and 5, we can see that the proposed method can guarantee that the tracking error resides in the region bounded within a fixed time ( and for ). As predicted by the theoretical analysis, the proposed method provides fast response and robustness for the system despite the effects of external disturbances and uncertain parameters. The response figures of sliding mode variables and control inputs are shown in Figures 6 and 7, respectively. Obviously, these signals are bounded stable. To further examine the performance of the proposed control method in compensating the effect caused by external disturbance and parameter uncertainty in a wide amplitude range, case 2 is considered. Without loss of generality, the other control parameters are the same as those in case 1. From Figures 8–13, when uncertainty and disturbance increase, consistent with the theoretical result, the remarkable fixed-time tracking is still achieved by the proposed controller. As the results show (Figures 10 and 11), we see that the proposed method guarantees the tracking error converge into bounded region in fixed time ( and for ).
5. Conclusion
This article has studied the prescribed performance tracking problem for uncertain MEMS gyroscope systems with unknown parameters and the external disturbance. By combining the funnel control technique with barrier Lyapunov functions, a robust sliding mode controller with adaptation compensation strategy is established, which achieves the fixed-time prescribed performance for the closed-loop system. Finally, a MEMS gyroscope simulation is performed to verify the effectiveness of the proposed control method. As our future research, the proposed approach will extend for the multiagent system.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the comprehensive reform project of undergraduate education and teaching of Dalian University of Technology (China) (JGLX2021119).