# Disturbance Observer-Based Sliding Mode Control for MEMS with Prescribed Performance

Authors: Yuru Wang; Zhankui Song; Jie Dong; Zhenxiang Pan
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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 saty or yt/‖yt‖+δ with yt being the sliding mode variable and δ 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 V˙t≤−μ⋅γ⋅Vα1t−1−μ⋅γ⋅Vα2t+δ, in which Vt is a Lyapunov function; α1, α2, and γ are the design parameters; 0<μ<1 is a uncertain parameter, and δ is unknown but bounded function. Obviously, if the parameters 0<μ<1 and δ 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]:(1)q¨+Nq˙+Cq=τ+Dq˙+H,where q:=x¯t,y¯tT∈R2 is the displacement of the gyroscope’s reference point; τ:=τ1,τ2T∈R2 is the control input vector; H=h1,h2T∈R2 is the lumped disturbance; q˙ and q¨ are the velocity and acceleration of the gyroscope’s reference point; N, C, and D denote the system parameter matrixes expressed as follows:(2)N=nxxnxynxynyy,C=cxxcxycxycyy,D=02Ωz−2Ωz0,where Ωz denote the angular velocity about the z-axis; nxx, nyy, nxy, and nyx are the bounded damping ratios; and cxx, cyy, cxy, and cyx are the bounded spring parameters. In practical engineering, the damping ratios and the spring constants are difficult to obtain accurately, and therefore, N and C with these bounded parameters are treated as the uncertain system matrixes in this study. The MEMS gyroscope system structure is shown in Figure 1.Figure 1
Schematic diagram of the MEMS gyroscope.To complete the control system design, the following assumptions are need.Assumption 1.
The desired command signalsx¯d and y¯d are second-order differentiable and bounded.Assumption 2.
The each element of the lumped disturbance vectorH and H˙ are bounded, and therefore, there exists two unknown constants η1 and η2, such that H≤η1 and H˙≤η2.Assumption 3.
The system initial states (x¯0 and y¯0) are bounded, and there exist positive constant η3 satisfying(3)x¯0−x¯d02+y¯0−y¯d02≤η3.


### 2.2. Preliminaries

For analyzing the stability of the closed-loop system, the following lemmas are given.Lemma 1 (see [25]).
Consider the variableS defined as(4)S=z˙+c⋅z,where c is a positive design parameter and z is an error variable. If S∈L∞, then |z˙|∈L∞ and z∈L∞.Lemma 2 (see [26]).
Consider the systemς˙=fς,u, for positive definite function Vς, if there exist scalars κ,λ>0, 0<l<1, and 0<b<∞, which make(5)V˙ς≤−κVς−λVlς+b.Then, the trajectory of systemς˙=fς,u is practical finite-time stable, i.e.,(6)ς∈Vlς≤b1−ελ,0<ε<1,and the finite time TS can be estimated as(7)TS=1κ1−llnκV1−lς0+ελελ.


### 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 as(8)FP≔=t,e∈ℝ×ℝn|φt⋅et<1,where et is the system error, and φt is an arbitrarily bounded and continuous function satisfying φt>0 and supt≥0φt<∞. According to [29], if φt can be selected reasonably, the funnel control can force the system error to be constrained inside the funnel formed by the boundaries related with φt. It is worth mentioning that the funnel variable in traditional funnel control is defined as et/FP−‖et‖. Unfortunately, such funnel variable is not differentiable at the point et=0; 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.(9)ξ1=e1φ2−e1Te1,where e1 is a system error vector and φ is a performance function (φ will be given later). In the light of (9), the time derivative of ξ1 can be calculated as(10)ξ˙1=e˙1φ2−e1Te1−e1φ2−e1Te13/2⋅φφ˙−e1Te˙1=e˙1φ2−e1Te1φ2−e1Te13/2−e1φ2−e1Te13/2⋅φφ˙−e1Te˙1=e˙1φ2φ2−e1Te13/2−e1φφ˙φ2−e1Te13/2=ϑ⋅φ2e˙1−φ⋅φ˙⋅e1,in which ϑ=1/φ2−e1Te13. Based on (10), we can obtain(11)ξ¨1=−3φ2−e1Te1−5/2⋅φφ˙−e1Te1⋅φ2e˙1−φ⋅φ˙⋅e1+ϑ⋅φ2e¨1+φφ˙e˙1−φ˙2e1−φφ¨e1.It is worth noting that nonlinear functionφ in (9) is called performance function [29], and φ is chosen as follows:(12)φt=θ1−tTF⋅e1−TF/TF−t+θ2,if0≤t<TF,θ2,ift≥TF,where the related design parameters satisfy θ1>0, θ2>0, and TF>0. It is obvious that the function φ has the following properties:(1)
φ0=θ1+θ2 and φt>0(2)
limt⟶TF−φt=limt⟶TF+φt=θ2, limt⟶TF−φ˙t=limt⟶TF+φ˙t=0, and limt⟶TF−φ¨t=limt⟶TF+φ¨t=0(3)
φt, φ˙t, and φ¨t are boundedThe 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.


## 2.1. System Model

Consider the dynamics of the MEMS gyroscope described by the Cartesian coordinate system [23, 24]:(1)q¨+Nq˙+Cq=τ+Dq˙+H,where q:=x¯t,y¯tT∈R2 is the displacement of the gyroscope’s reference point; τ:=τ1,τ2T∈R2 is the control input vector; H=h1,h2T∈R2 is the lumped disturbance; q˙ and q¨ are the velocity and acceleration of the gyroscope’s reference point; N, C, and D denote the system parameter matrixes expressed as follows:(2)N=nxxnxynxynyy,C=cxxcxycxycyy,D=02Ωz−2Ωz0,where Ωz denote the angular velocity about the z-axis; nxx, nyy, nxy, and nyx are the bounded damping ratios; and cxx, cyy, cxy, and cyx are the bounded spring parameters. In practical engineering, the damping ratios and the spring constants are difficult to obtain accurately, and therefore, N and C with these bounded parameters are treated as the uncertain system matrixes in this study. The MEMS gyroscope system structure is shown in Figure 1.Figure 1
Schematic diagram of the MEMS gyroscope.To complete the control system design, the following assumptions are need.Assumption 1.
The desired command signalsx¯d and y¯d are second-order differentiable and bounded.Assumption 2.
The each element of the lumped disturbance vectorH and H˙ are bounded, and therefore, there exists two unknown constants η1 and η2, such that H≤η1 and H˙≤η2.Assumption 3.
The system initial states (x¯0 and y¯0) are bounded, and there exist positive constant η3 satisfying(3)x¯0−x¯d02+y¯0−y¯d02≤η3.


## 2.2. Preliminaries

For analyzing the stability of the closed-loop system, the following lemmas are given.Lemma 1 (see [25]).
Consider the variableS defined as(4)S=z˙+c⋅z,where c is a positive design parameter and z is an error variable. If S∈L∞, then |z˙|∈L∞ and z∈L∞.Lemma 2 (see [26]).
Consider the systemς˙=fς,u, for positive definite function Vς, if there exist scalars κ,λ>0, 0<l<1, and 0<b<∞, which make(5)V˙ς≤−κVς−λVlς+b.Then, the trajectory of systemς˙=fς,u is practical finite-time stable, i.e.,(6)ς∈Vlς≤b1−ελ,0<ε<1,and the finite time TS can be estimated as(7)TS=1κ1−llnκV1−lς0+ελελ.


## 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 as(8)FP≔=t,e∈ℝ×ℝn|φt⋅et<1,where et is the system error, and φt is an arbitrarily bounded and continuous function satisfying φt>0 and supt≥0φt<∞. According to [29], if φt can be selected reasonably, the funnel control can force the system error to be constrained inside the funnel formed by the boundaries related with φt. It is worth mentioning that the funnel variable in traditional funnel control is defined as et/FP−‖et‖. Unfortunately, such funnel variable is not differentiable at the point et=0; 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.(9)ξ1=e1φ2−e1Te1,where e1 is a system error vector and φ is a performance function (φ will be given later). In the light of (9), the time derivative of ξ1 can be calculated as(10)ξ˙1=e˙1φ2−e1Te1−e1φ2−e1Te13/2⋅φφ˙−e1Te˙1=e˙1φ2−e1Te1φ2−e1Te13/2−e1φ2−e1Te13/2⋅φφ˙−e1Te˙1=e˙1φ2φ2−e1Te13/2−e1φφ˙φ2−e1Te13/2=ϑ⋅φ2e˙1−φ⋅φ˙⋅e1,in which ϑ=1/φ2−e1Te13. Based on (10), we can obtain(11)ξ¨1=−3φ2−e1Te1−5/2⋅φφ˙−e1Te1⋅φ2e˙1−φ⋅φ˙⋅e1+ϑ⋅φ2e¨1+φφ˙e˙1−φ˙2e1−φφ¨e1.It is worth noting that nonlinear functionφ in (9) is called performance function [29], and φ is chosen as follows:(12)φt=θ1−tTF⋅e1−TF/TF−t+θ2,if0≤t<TF,θ2,ift≥TF,where the related design parameters satisfy θ1>0, θ2>0, and TF>0. It is obvious that the function φ has the following properties:(1)
φ0=θ1+θ2 and φt>0(2)
limt⟶TF−φt=limt⟶TF+φt=θ2, limt⟶TF−φ˙t=limt⟶TF+φ˙t=0, and limt⟶TF−φ¨t=limt⟶TF+φ¨t=0(3)
φt, φ˙t, and φ¨t are boundedThe 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 asx1=q and x2=q˙, and let Ξ=−H+Nq˙+Cq; then, (1) can be rewritten as(13)x˙1=x2,x˙2=Dx2+τ−Ξ.Letxd=x¯d,y¯dT, and the error transformation is defined as(14)e1=e11,e12T=x1−xd,(15)ξ1=ξ11,ξ12T=e1φ2−e1Te1.Combining (14) and (15), a sliding mode surface vector S=S1,S2T∈ℝ2 is introduced by(16)S=ξ˙1+cξ1,where c is a design parameter. Furthermore, the derivative of S is(17)S˙=−3φ2−e1Te1−52⋅φφ˙−e1Te1⋅φ2e˙1−φ⋅φ˙⋅e1+ϑφφ˙e˙1−φ˙2e1−φφ¨e1+cϑφ2e˙1−φφ˙e1+ϑφ2x˙2−x¨d=ω+ϑφ2τ,in which ω=ω1,ω2T=−3φ2−e1Te1−5/2φφ˙−e1Te1φ2e˙1−φφ˙e1+ϑφφ˙e˙1−φ˙2e1−φφ¨e1+cϑφ2e˙1−φφ˙e1−ϑφ2x¨d+ϑφ2Dx2−ϑφ2ΞFrom (17), the corresponding form of Sii=1,2 can be expressed as(18)S˙i=ωi+ϑφ2τi.For subsequent analysis, a compact set is defined as(19)Ωe=e1Te1+STS<ε,where ε is a positive constant.Remark 1.
To ensure initial conditions satisfyinge1T0e10<φ20, the design parameters θii=1,2 in (12) can be configured by(20)θ1+θ2>η3.Remark 2.
From Assumptions1 and 2 and the boundedness of φ, φ˙, and φ¨, it is inferred that ω˙ii=1,2 are bounded in compact set Ωe. Hence, there exist positive constants b such that maxω˙1+ω˙2<b.
In (18), the nonlinear functions ωii=1,2 contain uncertain parameters and external disturbance terms. In this study, a lumped disturbance observer (21) is constructed to estimate ωi, i.e.,(21)ω^i=−βi+κ1⋅Si,β˙i=κ1ϑφ2τi+ω^i−γi,where βi0=0 and γii=1,2 can be obtained by(22)γi=κ2⋅signSi,ifSi≥δ,κ2⋅Si2⋅signSi,ifSi<δ,with positive constant δ.
Next, the control lawsτi, i=1,2 are designed as(23)τi=−ϑ−1φ−2⋅κ1⋅|Si|0.5⋅signSi+κ2Si+ω^iifSi≥δ,−ϑ−1φ−2⋅κ1⋅|Si|1.5⋅signSi+κ2Si+ω^iifSi<δ,where κ1 and κ2 are the design parameters.
Letω˜i=ωi−ω^i, and the expression of ω˜˙i and S˙i for two cases is as follows.Case 1.Si≥δ.Differentiatingω˜i with respect to time, it yields(24)ω˜˙i=ω˙i−ω^˙i=ω˙i−κ1ϑφ2τi−ω^i−κ2⋅signSi+κ1S˙i=ω˙i−κ1⋅ω˜i−κ2⋅signSi.Meanwhile, by substituting (23) into (18), one has(25)S˙i=−κ1⋅Si0.5⋅signSi−κ2⋅Si+ω˜i.Case 2.Si<δ.Similar to the mathematical operation for caseSi≥δ, we obtain that(26)ω˜˙i=ω˙i−κ1⋅ω˜i−κ2⋅Si2⋅signSi,S˙i=−κ1⋅Si1.5⋅signSi−κ2⋅Si+ω˜i.


### 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 Ωe, if the control strategy (23) is applied, the control parameters κi, i=1,2 and c are appropriately selected as(27)c2−κ1c+2κ2>0κ2−c>1κ1−c>0.5.Then, the following control objectives are achieved.(1)
All signals in the closed-loop control system are ultimately uniformly bounded(2)
The errorse1i can converge to a predefined residual set in a predefined timeProof.
The following two cases need to be further considered in order to obtain the theoretical results.Case 1.
Si≥δ.
Letσ1=σ11,σ12T=|S1|0.5signS1,ω˜1T and σ2=σ21,σ22T=|S2|0.5⋅signS2,ω˜2T. Based on (24) and (25), we obtain(28)σ˙11=S1−0.5−0.5κ1σ11+0.5σ12−0.5κ2σ11,σ˙12=−S1−0.5κ2σ11−κ1σ12+ω˙1,σ˙21=S2−0.5−0.5κ1σ21+0.5σ22−0.5κ2σ21,σ˙22=−S2−0.5κ2σ21−κ1σ22+ω˙2.
From (28), it is obvious that(29)σ˙1=−S1−0.5⋅L1σ1−0.5c⋅S1−0.5⋅σ1−L2σ1−0.5+0.5c⋅σ1+ν1,σ˙2=−S2−0.5⋅L3σ2−0.5c⋅S2−0.5⋅σ2−L4σ2−0.5+0.5c⋅σ2+ν2,where ν1=0ω˙1, ν2=0ω˙2, L1=L3=0.5κ1−0.5c−0.5κ2−0.5c, and L2=L4=0.5κ2−0.5c−0.500κ1−0.5c−0.5
Choose the following Lyapunov function candidate:(30)VG=12σ1Tσ1+12σ2Tσ2.
It follows from taking (29) into account, differentiating VG respect to time, one has(31)V˙G=σ1Tσ˙1+σ2Tσ˙2=−S1−0.5σ1TL1σ1−0.5cS1−0.5σ1Tσ1−σ1TL2σ1−0.5c+1⋅σ1Tσ1+σ1Tν1−S2−0.5σ2TL3σ2−0.5cS2−0.5σ2Tσ2−σ2TL4σ2−0.5c+1⋅σ2Tσ2+σ2Tν1.
Note that the design parameters (α, c, κ1, and κ2) are selected by (27), the positive definiteness of the matrices L1, L3, L3, and L4 is guaranteed; moreover, with the help of Young’s inequality and Remark 2, the following relationships can be established, i.e.,(32)σ1Tν1≤0.5σ1Tσ1+0.5ν1Tν1≤0.5σ1Tσ1+0.5b2,σ2Tν2≤0.5σ2Tσ2+0.5ν2Tν2≤0.5σ1Tσ1+0.5b2.
Using (32), we have(33)V˙G≤−cS1−0.5σ1Tσ12−cσ1Tσ12−cS2−0.5σ2Tσ22−cσ2Tσ22+b22.
Due to(34)2VG≥σ1Tσ1≥S1,2VG≥σ2Tσ2≥S2,then for i=1,2, it follows that(35)2VG0.5≥Si0.5⇒2VG−0.5≤Si−0.5.
Combining with (35) and (33), it can be rewritten as(36)V˙G≤−c2VG−0.50.5σ1Tσ1−c2VG−0.50.5σ2Tσ2−cVG+b22≤−c2VG−0.5VG−cVG+b22=−2−0.5c⋅VG0.5−c⋅VG+b22.
From Lemma2, it is seen that ||σ1||22 and ||σ2||22 are finite-time stable. It means that the sliding mode variables Si, i=1,2 are bounded. According to Lemma 1 and the definition of Si, it implies that ξi is bounded, and therefore, there exists positive constants ρi, such that(37)ξi=eiφ2−e1Te1≤ρi,i=1,2,and we further have(38)1+ρ12e12≤ρ12φ2−e22≤ρ12φ2,1+ρ12e22≤ρ22φ2−e12≤ρ22φ2,namely,(39)e12≤ρ12φ21+ρ12<φ2,e22≤ρ22φ21+ρ22<φ2.Case 2.
Si<δ.
Letσ¯1=σ¯11,σ¯12T=S11.5signS1ω˜1T and σ¯2=σ¯21,σ¯22T=S21.5⋅signS2ω˜2T. One gets(40)σ¯˙11=S10.5−1.5κ1σ¯11+1.5σ¯12−1.5κ2σ¯11,σ¯˙12=−S10.5κ2σ¯11−κ1σ¯12+ω˙1,σ¯˙21=S2−0.5−1.5κ1σ¯21+1.5σ¯22−1.5κ2σ¯21,σ¯˙22=−S20.5κ2σ¯21−κ1σ¯22+ω˙2..
From (40), we can derive that(41)σ¯˙1=−S10.5⋅L¯1σ¯1−L¯2σ¯1−0.51+cσ¯1+ν1σ¯˙2=−S20.5⋅L¯3σ¯2−L¯4σ¯2−0.51+cσ¯2+ν2,where L¯2=L¯4=1.5κ2−0.5c−0.500κ1−0.5c−0.5 and L¯1=L¯3=1.5κ11.5κ20.
Consider the Lyapunov function as follows:(42)V¯G=12σ¯1Tσ¯1+12σ¯2Tσ¯2.
It follows from (41) that the time derivative of V¯G can be expressed as(43)V¯˙G≤−S10.5⋅σ¯1TL¯1σ¯1−σ¯1TL¯2σ¯1−cσ¯1Tσ¯12−S10.5⋅σ¯2TL¯3σ¯2−σ¯2TL¯4σ¯2−cσ¯2Tσ¯22+b22.
Based on the condition (27), it follows that L¯1>0, L¯2>0, L¯3>0, and L¯4>0, and we can rewrite (43) as(44)V¯˙G≤−cσ¯1Tσ¯12−cσ¯2Tσ¯22+b22=−c2⋅V¯G+b22,which implies that ||σ1||22, ||σ2||22, S1, and S2 are bounded; by virtue of Lemma 1 and the definition of Si, the result (38) can be obtained; it results in(45)e12≤ρ12φ21+ρ12<φ2e22≤ρ22φ21+ρ22<φ2.
Combining with (37) and (45), it follows from the definition of φt that(46)ei<θ1−tTF⋅e1−TF/TF−t+θ2,if0≤t<TFei<θ2,ift≥TF.
The above formula implies that the system error converges to a prescribed compact setΩ∗=ei|ei<θ2 within fixed time TF. All these complete the proof.Remark 3.
In view of (46), the size of system errors depends on the parameters θ2 and TF. These two parameters ensure that the control system can reach the preset accuracy θ2 within a fixed time TF. Nevertheless, TF cannot be infinitesimal, which should be greater than the sampling time of the control system.


## 3.1. The Prescribed Performance Controller Design

Before designing the fixed-time controller, the following new variables are defined asx1=q and x2=q˙, and let Ξ=−H+Nq˙+Cq; then, (1) can be rewritten as(13)x˙1=x2,x˙2=Dx2+τ−Ξ.Letxd=x¯d,y¯dT, and the error transformation is defined as(14)e1=e11,e12T=x1−xd,(15)ξ1=ξ11,ξ12T=e1φ2−e1Te1.Combining (14) and (15), a sliding mode surface vector S=S1,S2T∈ℝ2 is introduced by(16)S=ξ˙1+cξ1,where c is a design parameter. Furthermore, the derivative of S is(17)S˙=−3φ2−e1Te1−52⋅φφ˙−e1Te1⋅φ2e˙1−φ⋅φ˙⋅e1+ϑφφ˙e˙1−φ˙2e1−φφ¨e1+cϑφ2e˙1−φφ˙e1+ϑφ2x˙2−x¨d=ω+ϑφ2τ,in which ω=ω1,ω2T=−3φ2−e1Te1−5/2φφ˙−e1Te1φ2e˙1−φφ˙e1+ϑφφ˙e˙1−φ˙2e1−φφ¨e1+cϑφ2e˙1−φφ˙e1−ϑφ2x¨d+ϑφ2Dx2−ϑφ2ΞFrom (17), the corresponding form of Sii=1,2 can be expressed as(18)S˙i=ωi+ϑφ2τi.For subsequent analysis, a compact set is defined as(19)Ωe=e1Te1+STS<ε,where ε is a positive constant.Remark 1.
To ensure initial conditions satisfyinge1T0e10<φ20, the design parameters θii=1,2 in (12) can be configured by(20)θ1+θ2>η3.Remark 2.
From Assumptions1 and 2 and the boundedness of φ, φ˙, and φ¨, it is inferred that ω˙ii=1,2 are bounded in compact set Ωe. Hence, there exist positive constants b such that maxω˙1+ω˙2<b.
In (18), the nonlinear functions ωii=1,2 contain uncertain parameters and external disturbance terms. In this study, a lumped disturbance observer (21) is constructed to estimate ωi, i.e.,(21)ω^i=−βi+κ1⋅Si,β˙i=κ1ϑφ2τi+ω^i−γi,where βi0=0 and γii=1,2 can be obtained by(22)γi=κ2⋅signSi,ifSi≥δ,κ2⋅Si2⋅signSi,ifSi<δ,with positive constant δ.
Next, the control lawsτi, i=1,2 are designed as(23)τi=−ϑ−1φ−2⋅κ1⋅|Si|0.5⋅signSi+κ2Si+ω^iifSi≥δ,−ϑ−1φ−2⋅κ1⋅|Si|1.5⋅signSi+κ2Si+ω^iifSi<δ,where κ1 and κ2 are the design parameters.
Letω˜i=ωi−ω^i, and the expression of ω˜˙i and S˙i for two cases is as follows.Case 1.Si≥δ.Differentiatingω˜i with respect to time, it yields(24)ω˜˙i=ω˙i−ω^˙i=ω˙i−κ1ϑφ2τi−ω^i−κ2⋅signSi+κ1S˙i=ω˙i−κ1⋅ω˜i−κ2⋅signSi.Meanwhile, by substituting (23) into (18), one has(25)S˙i=−κ1⋅Si0.5⋅signSi−κ2⋅Si+ω˜i.Case 2.Si<δ.Similar to the mathematical operation for caseSi≥δ, we obtain that(26)ω˜˙i=ω˙i−κ1⋅ω˜i−κ2⋅Si2⋅signSi,S˙i=−κ1⋅Si1.5⋅signSi−κ2⋅Si+ω˜i.


## 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 Ωe, if the control strategy (23) is applied, the control parameters κi, i=1,2 and c are appropriately selected as(27)c2−κ1c+2κ2>0κ2−c>1κ1−c>0.5.Then, the following control objectives are achieved.(1)
All signals in the closed-loop control system are ultimately uniformly bounded(2)
The errorse1i can converge to a predefined residual set in a predefined timeProof.
The following two cases need to be further considered in order to obtain the theoretical results.Case 1.
Si≥δ.
Letσ1=σ11,σ12T=|S1|0.5signS1,ω˜1T and σ2=σ21,σ22T=|S2|0.5⋅signS2,ω˜2T. Based on (24) and (25), we obtain(28)σ˙11=S1−0.5−0.5κ1σ11+0.5σ12−0.5κ2σ11,σ˙12=−S1−0.5κ2σ11−κ1σ12+ω˙1,σ˙21=S2−0.5−0.5κ1σ21+0.5σ22−0.5κ2σ21,σ˙22=−S2−0.5κ2σ21−κ1σ22+ω˙2.
From (28), it is obvious that(29)σ˙1=−S1−0.5⋅L1σ1−0.5c⋅S1−0.5⋅σ1−L2σ1−0.5+0.5c⋅σ1+ν1,σ˙2=−S2−0.5⋅L3σ2−0.5c⋅S2−0.5⋅σ2−L4σ2−0.5+0.5c⋅σ2+ν2,where ν1=0ω˙1, ν2=0ω˙2, L1=L3=0.5κ1−0.5c−0.5κ2−0.5c, and L2=L4=0.5κ2−0.5c−0.500κ1−0.5c−0.5
Choose the following Lyapunov function candidate:(30)VG=12σ1Tσ1+12σ2Tσ2.
It follows from taking (29) into account, differentiating VG respect to time, one has(31)V˙G=σ1Tσ˙1+σ2Tσ˙2=−S1−0.5σ1TL1σ1−0.5cS1−0.5σ1Tσ1−σ1TL2σ1−0.5c+1⋅σ1Tσ1+σ1Tν1−S2−0.5σ2TL3σ2−0.5cS2−0.5σ2Tσ2−σ2TL4σ2−0.5c+1⋅σ2Tσ2+σ2Tν1.
Note that the design parameters (α, c, κ1, and κ2) are selected by (27), the positive definiteness of the matrices L1, L3, L3, and L4 is guaranteed; moreover, with the help of Young’s inequality and Remark 2, the following relationships can be established, i.e.,(32)σ1Tν1≤0.5σ1Tσ1+0.5ν1Tν1≤0.5σ1Tσ1+0.5b2,σ2Tν2≤0.5σ2Tσ2+0.5ν2Tν2≤0.5σ1Tσ1+0.5b2.
Using (32), we have(33)V˙G≤−cS1−0.5σ1Tσ12−cσ1Tσ12−cS2−0.5σ2Tσ22−cσ2Tσ22+b22.
Due to(34)2VG≥σ1Tσ1≥S1,2VG≥σ2Tσ2≥S2,then for i=1,2, it follows that(35)2VG0.5≥Si0.5⇒2VG−0.5≤Si−0.5.
Combining with (35) and (33), it can be rewritten as(36)V˙G≤−c2VG−0.50.5σ1Tσ1−c2VG−0.50.5σ2Tσ2−cVG+b22≤−c2VG−0.5VG−cVG+b22=−2−0.5c⋅VG0.5−c⋅VG+b22.
From Lemma2, it is seen that ||σ1||22 and ||σ2||22 are finite-time stable. It means that the sliding mode variables Si, i=1,2 are bounded. According to Lemma 1 and the definition of Si, it implies that ξi is bounded, and therefore, there exists positive constants ρi, such that(37)ξi=eiφ2−e1Te1≤ρi,i=1,2,and we further have(38)1+ρ12e12≤ρ12φ2−e22≤ρ12φ2,1+ρ12e22≤ρ22φ2−e12≤ρ22φ2,namely,(39)e12≤ρ12φ21+ρ12<φ2,e22≤ρ22φ21+ρ22<φ2.Case 2.
Si<δ.
Letσ¯1=σ¯11,σ¯12T=S11.5signS1ω˜1T and σ¯2=σ¯21,σ¯22T=S21.5⋅signS2ω˜2T. One gets(40)σ¯˙11=S10.5−1.5κ1σ¯11+1.5σ¯12−1.5κ2σ¯11,σ¯˙12=−S10.5κ2σ¯11−κ1σ¯12+ω˙1,σ¯˙21=S2−0.5−1.5κ1σ¯21+1.5σ¯22−1.5κ2σ¯21,σ¯˙22=−S20.5κ2σ¯21−κ1σ¯22+ω˙2..
From (40), we can derive that(41)σ¯˙1=−S10.5⋅L¯1σ¯1−L¯2σ¯1−0.51+cσ¯1+ν1σ¯˙2=−S20.5⋅L¯3σ¯2−L¯4σ¯2−0.51+cσ¯2+ν2,where L¯2=L¯4=1.5κ2−0.5c−0.500κ1−0.5c−0.5 and L¯1=L¯3=1.5κ11.5κ20.
Consider the Lyapunov function as follows:(42)V¯G=12σ¯1Tσ¯1+12σ¯2Tσ¯2.
It follows from (41) that the time derivative of V¯G can be expressed as(43)V¯˙G≤−S10.5⋅σ¯1TL¯1σ¯1−σ¯1TL¯2σ¯1−cσ¯1Tσ¯12−S10.5⋅σ¯2TL¯3σ¯2−σ¯2TL¯4σ¯2−cσ¯2Tσ¯22+b22.
Based on the condition (27), it follows that L¯1>0, L¯2>0, L¯3>0, and L¯4>0, and we can rewrite (43) as(44)V¯˙G≤−cσ¯1Tσ¯12−cσ¯2Tσ¯22+b22=−c2⋅V¯G+b22,which implies that ||σ1||22, ||σ2||22, S1, and S2 are bounded; by virtue of Lemma 1 and the definition of Si, the result (38) can be obtained; it results in(45)e12≤ρ12φ21+ρ12<φ2e22≤ρ22φ21+ρ22<φ2.
Combining with (37) and (45), it follows from the definition of φt that(46)ei<θ1−tTF⋅e1−TF/TF−t+θ2,if0≤t<TFei<θ2,ift≥TF.
The above formula implies that the system error converges to a prescribed compact setΩ∗=ei|ei<θ2 within fixed time TF. All these complete the proof.Remark 3.
In view of (46), the size of system errors depends on the parameters θ2 and TF. These two parameters ensure that the control system can reach the preset accuracy θ2 within a fixed time TF. Nevertheless, TF 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.,(47)nxx=n¯xx+Δnxx,nxy=n¯xy+Δnxy,nyy=n¯yy+Δnyy,cxx=c¯xx+Δcxx,cxy=c¯xy+Δcxy,cyy=c¯yy+Δcyy.The nominal parts are given as follows:(48)n¯xx=4.3×10−7Nsm,n¯xy=4.3×10−8Nsm,n¯yy=6.8×10−7Nsm,c¯xx=14.2,c¯xy=3,c¯yy=12.56.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.
(49)Δn¯xx=n¯xx×0.15,Δn¯xy=n¯xy×0.2sinπt,Δn¯yy=n¯yy×0.2cos2t,Δc¯xx=c¯xx×0.25sin2t,Δc¯xy=c¯xy×0.2sinπt,Δcyy=0.2c¯yy×cos2t,H=0.15sinπtcost,0.15cos2tsintT.Case 2.
(50)Δn¯xx=n¯xx×0.5,Δn¯xy=n¯xy×0.5sinπt,Δn¯yy=n¯yy×0.55cos2t,Δc¯xx=c¯xx×0.5sin2t,Δc¯xy=c¯xy×0.5sinπt,Δcyy=0.5c¯yy×cos2t,H=0.4sinπtcost,0.4cos2tsintT.
The design parameters and the initial condition are chosen asc=5, κ1=κ2=8, θ1=3, θ2=0.01, η1=3, η2=5, η3=1.5, δ=0.5, βi0=0, TF=1.2, x¯0=0.6, and y¯0=0.4. The desired command signals are given as xd=cosπt,sin2tT.
The tracking performance for statesx¯ and y¯ 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 (e11≤10−2 and e12≤10−2 for t≥1.2s). 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 TF (e11≤10−2 and e12≤10−2 for t≥1.2s).Figure 2
The trajectory trackingx¯ and x¯d with case 1.Figure 3
The trajectory trackingy¯ and y¯d with case 1.Figure 4
The system errore11 with case 1.Figure 5
The system errore12 with case 1.Figure 6
Sliding mode variablesS1 and S2 with case 1.Figure 7
Control signals with case 1.Figure 8
The trajectory trackingx¯ and x¯d with case 2.Figure 9
The trajectory trackingx¯ and x¯d with case 2.Figure 10
The system errore11 with case 2.Figure 11
The system errore12 with case 2.Figure 12
Sliding mode variablesS1 and S2 with case 2.Figure 13
Control signals with case 2.


## 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.



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*Source: 4880505-2022-08-28.xml*