#### Abstract

This study investigates the prescribed performance control problem for microelectro-mechanical system (MEMS) gyroscope subject to system parameters’ uncertainty. A finite-time observer is firstly designed to estimate the unmeasurable velocity state of MEMS gyroscope. Subsequently, a coordinate transformation with the performance function is introduced into an error system which will be kept bounded to ensure expected dynamic and steady-state responses. Based on the proposed finite time-velocity reconstruction system, the adaptive backstepping design procedure is further designed to deal with the lumped uncertainty term. Furthermore, when considering actuator saturation, an improved control strategy is developed with a nonlinear input updating law, and meanwhile, it is proved that the system error converges to a preset compact set around zero in a preassigned time. Simulation results show the effectiveness and reliability of the proposed methods.

#### 1. Introduction

MEMS gyroscopes have been widely used in various fields due to their small size, light weight, low power consumption, and easy integration [1–3]. However, MEMS is susceptible to external noise and temperature interference and further leads to zero drift and random error, which not only reduces the measurement accuracy of MEMS gyroscope system but also limits its application in high precision control [4, 5].

To assure that MEMS systems have excellent robustness and significant measurement accuracy, some nonlinear control methods have been devised and applied. Si et al. [6] and Shao et al. [7, 8] constructed the neural performance control method by utilizing the minimum-learning-parameter strategy, where the effect of quantization error induced by logarithmic quantizer was compensated by incorporating a robust quantized control. In [9], a robust sliding mode controller using a neural-network compensator was proposed for MEMS vibratory gyroscopes.

It should be noted that the above proposed method only guarantees the asymptotic convergence of the system error to the neighborhood of the equilibrium point. Compared with the asymptotic control strategies [10, 11], the finite-time control [12–14] can provides the faster dynamic response and higher control precision for the closed-loop system. To tackle the uncertain dynamics and to improve the dynamic response of MEMS, some finite-time control strategies have been developed in [15–18], and these results can significantly improve the robustness for MEMS. It is worth mentioning that most of the above results are obtained by using the full-state feedback way. In practice, not all states are easily measured directly, and even when they are, multisensor hardware deployments will increase the energy consumption of the system and the hardware design complexity. To solve this problem, a popular approach is to design state observers to obtain estimates of the true states. Then, these estimated states are used instead of the real states to develop the desired control law. Recently, some important results based on the combination with the respective advantages finite-time control strategies and states observer have been proposed. By applying the nonsingular terminal sliding mode theory and the extended state observer technique [19–21], the finite-time control issue of the nonlinear system with uncertain dynamics and the lumped disturbances were addressed. Moreover, by incorporating neural network or fuzzy approximator into the control law design process, many approximation-based finite-time control methods with the state observer were presented [22–24]. In [22], an adaptive neural output-feedback control scheme was presented for a class of nonlinear systems with actuator failures. In [23], the NN-based finite-time control algorithm was proposed for nonlinear quantized systems with unmeasurable states. Considering unmeasured state of multi-input and multi-output nonlinear nonstrict-feedback systems [24], a finite-time adaptive fuzzy control was explored.

Although the above nonlinear finite-time controllers with the state observer have been developed, there are still some issues that need further consideration. On the one hand, the stabilization time and tracking accuracy involved in the above control strategies are usually unknown because they are determined by the unknown positive constants appearing in the derivative inequalities of Lyapunov functions, with the result that these methods are difficult to perform in some tasks requiring specified control accuracy in a specific time. On the other hand, the existing finite-time output-feedback control schemes are involved in which the designed observers have few finite-time convergence properties. Therefore, to ensure that the whole Lyapunov function satisfies the corresponding finite-time stability condition, the inequality amplification and minification is usually used. This also makes the corresponding stability analysis more conservative.

The main contributions of this study can be summarized as(i)To solve the effects of the saturation nonlinearity, an input updating law is constructed for elimination of excessive control quantity, and the proposed control law based on backstepping does not incorporate the derivatives of virtual control law instead of using the adaption compensation method.(ii)A finite-time state observer is introduced into the closed-loop system, by which the state information of the controlled object can be effectively obtained in a finite time.(iii)Compared to traditional finite-time control methods, the proposed method ensures that the performance of the controlled plant can be directly specified according to the task requirements, and meanwhile, no any accurate model information of MEMS is known in advance.

#### 2. Problem Formulation

##### 2.1. System Model and Preliminaries

The vibratory model of MEMS gyroscope is described by [11]where represents the weight of internal moving mass, denotes the reference frequency, is the reference length, , , , and are system parameters, is angular velocity, and are the displacements of drive axis and sensitive axis, and and are control input. In addition, considering that the damping coefficients and the stiffness coefficients are difficult to obtain accurately in practical engineering, these coefficients are treated as bounded unknown parameters. The MEMS gyroscope system structure is shown in Figure 1.

Let , , , and with

Then, the controlled plant (1) can be written as

Throughout this study, the following definition and assumptions are needed.

*Definition 1 (see [25]). *The nonlinear function *(2)* is a speed function:where is the design parameter and can be described aswith initial condition and preset-time parameter , and moreover, function has some attractive features:(1) is positive and strictly increasing for , and thus, is strictly decreasing in some time interval(2) and , for (3) is *n*-order smooth function, and its *n*-order derivative is bounded for Obviously, some important properties can be obtained from :(i) (P1), and is strictly increasing with time and positive for ; moreover, and , for (P2) is continuously differentiable bounded function

*Assumption 1. *The desired command signals and are bounded, and their derivatives and are also bounded.

*Assumption 2. *For the system initial states, there exists a positive constant satisfyingThis work aims to develop a fixed-time control scheme without velocity state measurement to ensure that the system error can converge to a prescribed region within a prespecified time and that all signals in the closed-loop system are ultimately uniformly bounded [26–28].

#### 3. Main Results

This section focuses on proposing an adaptive fixed-time prescribed performance control strategy with a velocity observer to achieve our aforementioned aim. To this end, the whole design process consists of two parts. We first design an observer system for estimating the state . Then, a new fixed-time prescribed performance controller is developed.

##### 3.1. Velocity Observation System Design

Firstly, we define a velocity observation error vector aswhere is updated by the following nonlinear system:where , , and , , and are positive design parameters. Notice that if .

Lemma 1. *For the controlled plant (1), if the parameter is chosen such that , then the variable provided by (6) can estimate the system state ; in particular, can converge to zero in a finite time.*

*Proof. *With the help of (1), (5), and (6), we obtainConsider the following Lyapunov function candidate:Taking the tine derivative of (9) along (8), it follows thatwhere . Furthermore, we haveIntegrating (11) from to yieldsFrom (8) and (11), we have , for . Hence, we can further obtain due to , for . In view of , it means for . This shows that is able to reconstruct the system state , and the observer error can converge to zero after a finite time. To this end, the proof is completed.

##### 3.2. Velocity-Free Fixed-Time Prescribed Performance Control Strategy

To accomplish aforementioned control aim, the system errors and as well as the auxiliary transformed error are introduced as

In (12) and (13), is the desired command vector and is a virtual control law which will be given later. Next, a compact set is defined aswhere is a positive design parameter satisfying . The design process of control law is as follows: Step 1: from (1), (5), (12), and (13), the derivative of is

Design the virtual control law as follows:where is a positive design parameter. Substituting the virtual control law (17) into (16) yields

*Remark 1. *To ensure that the initial value satisfies , need to be met. Since , if hold under the assumption 2, then . Step 2: taking the tine derivative of (13), it follows thatwhere . It is worth noting that the virtual law contains some relevant measured signals which are likely to be mixed with interfering noise, so the differential operations of may bring harmful and limited spike pulse. In this work, the derivative term of and the function are combined as a hybrid nonlinear term, and we will utilize the adaptive compensation method to estimate the upper bound of instead of calculating directly, thus avoiding calculation for .

Now, the control law is designed aswith the parameter updating lawwhere and are positive design parameters.

Before the main result of stability is given, the following compact set is defined below, i.e.,where is a positive constant.

Note that is bounded in , and therefore, there exists a positive constant such thatIn the follow-up content, we will use to estimate , and .

From the proposed control law (20) along updating law (21), one important result is given in the following theorem.

Theorem 1. *Consider the dynamics system (1) with assumption 1 and assumption 2, if a velocity observation system is constructed in (6) with parameter condition , and the control law (20) with parameter updating law as specified in equations (21) is used; then, for any initial states in , there exist design parameters such that all signals in the closed-loop control system are ultimately uniformly bounded and that the error can converge to a predefined residual set in a predefined time.*

*Proof. *Consider the following Lyapunov function:where . Taking the derivative of with respect to time and using (9), (18), and (19), we haveSubstituting (21) and (22) into (24), one hasIt is noted that the following inequalities hold:With the help of (26), then (25) can be written aswhere and . It follows from (27) that and , and moreover, holds. From the definition of and , it implies that and . Let , and therefore, we obtain

##### 3.3. Velocity-Free Fixed-Time Prescribed Performance Control Strategy with Input Constraint

The problem of control input saturation exists in a variety of practical control systems. When a system suffers from an adverse factor, the controller produces a larger output value to offset the effect of the external adverse factor on the system. In this case, the closed-loop system becomes more sensitive to input saturation. Thus, the research for the control input constraints is more meaningful. In this section, input constraints are considered, and for this purpose, in (2) is defined as

Here, and describe the nonlinear saturation characteristic of the actuator, which are considered aswhere and is a known bound. To further soften control signal, is approximated by using the following smooth function:

In the light of (30) and (31), one haswith , , and satisfying .

Based on (29)–(32), the controlled plant (2) is written asin which . In view of the intervention of , a velocity observation error vector and an observation system are constructed, respectively, aswhere , , , and .

Lemma 2. *For the nonlinear system (36b), if the parameter is chosen such that , then the variable supported by (36b) can estimate system state , and moreover, can converge to zero in a finite time.*

*Proof. *From (33) and (36b), one hasThen, consider the following Lyapunov function:By using (35), the time derivative of is calculated asBased on the condition , we haveSimilar to the analysis of Lemma 1, we can also obtain that the reconstruction error converge to zero after a finite time. The proof is completed.

To achieve the control objective, the following transform of coordinate is made:where and are the virtual control laws, and they are designed aswith parameter updating lawThe following design processes are used in designing the proposed control law. Step 1: with the help of (36a), (40), (41), and (43), the derivative of is(i)Substituting (44) into (47) yields Step 2: by applying (41), (42), and (45), the time derivative of can expressed as where . Step 3: taking time derivative of yieldsIn (51), the details of is presented as follows:To continuous, the updating law of is designed aswhere and is updated byDefine a compact set as follows:where is a positive constant.

Notice that and are bounded in ; therefore, there exists positive constants and , such that

Theorem 2. *Consider a feedback control loop containing controlled plant (33), the velocity observation system (36b), and the input updating law (51); then, for any initial condition in , if the design parameters , , , , and are chosen such thatthen the following goals hold:*(1)

*The constraints of system error variable are not violated, i.e., for*(2)

*The errors converge to a prescribed compact set for*

*Proof. *Consider the following Lyapunov function:where and . By applying (46) and (47) and (49)–(53), the time derivative of in (54) can expressed aswhere and . Multiplying on both its sides by and integrating it over , we haveFrom the above equation, it is clear that and . Let ; similar to the analysis of Theorem 1, we can obtain the following results:All these complete the proof. The control structure is presented in Figure 2.

*Remark 2. *From (56), the size of the system error depend on the parameters and when . The small and not only ensure small tracking errors but also shorten the time for errors to reach specified accuracy. Notwithstanding, these two parameters cannot be infinitesimal, which should be selected according to the control requirements of the actual controlled system. It is important to note that the other design parameters only need to satisfy the design parameter conditions in Theorem 2.

*Remark 3. *According to Theorem 2, the strong robustness of the proposed method stems from the performance function and the virtual control laws (44) and (45). When system errors are far away from the origin, in (44) and (45) to uncertainties plays an important role for enforcing the system error to enter the preassigned error convergence set in .

#### 4. Numerical Simulations

This section presents the numerical simulation analysis of the proposed method for the MEMS subjected to the uncertain parameters. In the simulation, the nominal part of the MEMS parameters are taken as , , and , , and , , and , , and

To show the effectiveness of the proposed control strategy, uncertain system parameters are introduced, the details of which are

, , , , , , , , and .

In addition, the initial state value is , the desired command signals are given as as well as , and the related control system parameters are choose as , , , , , , , , , and .

Figures 3 and 4 show the time responses of the velocity observation errors, where is the observation error of and is the observation error of . From the plots of and , we can conclude that and can be accurately estimated by the observer system (36b). Moreover, the observer error converge to a very small region with when . The outputs of the state tracking are presented in Figures 5 and 6, which show that the system states are indeed bounded by the proposed method, as claimed in Theorem 1. More specifically, the system tracking errors and strictly evolve within the predefined performance envelope and converge to when , and meanwhile, and converge to when , as shown in Figures 5 and 6. The control effect of control signals are depicted byFigures 7 and 8. The amplitude of the control is relatively large in the first 0.5 seconds, which is mainly due to the fact that the control system needs to output more energy to compensate for the impact of the uncertainty of the system parameters. After that, control signals change smoothly and slightly with steady and small oscillations.

#### 5. Conclusion

In this study, an adaptive compensation problem for MEMS subject to system parameters’ uncertainty based on prescribed performance fixed-time control strategy has been investigated. Firstly, a finite-time state reconstruction system is exploited to estimate the velocity information of MEMS gyroscope. Secondly, a new input updating law control anti-saturation technique is utilized to ensure tracking errors converge to a preset compact set around zero in a preassigned time. Finally, numerical simulations illustrate the effectiveness of the proposed methods. It should be pointed out that other actuator constraints such as dead-zone or actuator faults have not been investigated in this research. Further results will be reported in the future works.

#### Data Availability

The data used to study the findings of the study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported in part by the Natural Science Foundation of Liaoning Province (China), under Grant 2019-ZD-0131.