# Velocity-Free Adaptation Compensation Control of MEMS with Prescribed Performance

Authors: Jie Dong; Yanzhao Zhang; Xiaotian Liu; Zhankui Song
Journal: Mathematical Problems in Engineering
 (2022)

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## 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.


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## Body

## 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](1)mω02ζ0⋅x¨+c11ω0ζ0⋅x˙+ω0ζ0⋅c12−2mΩz∗y˙,+ζ0b11−mΩz∗2x+b12ζ0y+b13ζ03x3=u1,mω02ζ0⋅y¨+c22ω0ζ0⋅y˙+ω0ζ0⋅c21+2mΩz∗x˙,+ζ0b22−mΩz∗2y+b21ζ0x+b23ζ03y3=u2,where m represents the weight of internal moving mass, ω0 denotes the reference frequency, ζ0 is the reference length, c11,c12,c21,c22, b11, b12,b21, and b22 are system parameters, Ωz is angular velocity, x and y are the displacements of drive axis and sensitive axis, and u1 and u2 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.Figure 1
Schematic diagram of MEMS gyroscope.Letζ1=x,yT, ζ˙1=ζ2, u=ux,uyT=u1ω0−2ξ0−1,u2ω0−2ξ0−1T, and Ξ=Ξ1,Ξ2T withΞ1=c11⋅x˙/mω0+c12−2mΩz∗y˙/mω0+b11−mΩz∗2x/mω02+b12y+b13ζ02x3/mω02,Ξ2=c22⋅y˙/mω0+c21−2mΩz∗x˙/mω0+b22−mΩz∗2y/mω02+b12x+b23ζ02y3/mω02.Then, the controlled plant (1) can be written as(2)ζ˙1=ζ2,ζ˙2=−Ξ+um.Throughout this study, the following definition and assumptions are needed.Definition 1 (see [25]).
The nonlinear function(2) is a speed function:(3)φt=11−ψfϑ−1+ψf,where 0<ψf<1 is the design parameter and ϑ can be described as(4)ϑ=TS/TS−t4,for0≤t<TS,∞,fort≥TS,with initial condition ϑ0=1 and preset-time parameter TS, and moreover, function ϑ has some attractive features:(1)
ϑ is positive and strictly increasing for 0≤t<TS, and thus, ϑ−1 is strictly decreasing in some time interval(2)
ϑ=∞ and ϑ−1=0, for t≥TS(3)
ϑ is n-order smooth function, and its n-order n⟶∞ derivative is bounded for 0≤t<TS
Obviously, some important properties can be obtained fromϑ:(i) (P1)
φ˙t≥0, and φt is strictly increasing with time and positive for 0≤t<TS; moreover, φ0=1 and φt=1/ψf, for t≥TS(P2)
φ˙t is continuously differentiable bounded functionAssumption 1.
The desired command signalsxd and yd are bounded, and their derivatives x˙d and y˙d are also bounded.Assumption 2.
For the system initial states, there exists a positive constantc satisfying(5)x0−xd02+y0−yd02<c2.
This 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].


## 2.1. System Model and Preliminaries

The vibratory model of MEMS gyroscope is described by [11](1)mω02ζ0⋅x¨+c11ω0ζ0⋅x˙+ω0ζ0⋅c12−2mΩz∗y˙,+ζ0b11−mΩz∗2x+b12ζ0y+b13ζ03x3=u1,mω02ζ0⋅y¨+c22ω0ζ0⋅y˙+ω0ζ0⋅c21+2mΩz∗x˙,+ζ0b22−mΩz∗2y+b21ζ0x+b23ζ03y3=u2,where m represents the weight of internal moving mass, ω0 denotes the reference frequency, ζ0 is the reference length, c11,c12,c21,c22, b11, b12,b21, and b22 are system parameters, Ωz is angular velocity, x and y are the displacements of drive axis and sensitive axis, and u1 and u2 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.Figure 1
Schematic diagram of MEMS gyroscope.Letζ1=x,yT, ζ˙1=ζ2, u=ux,uyT=u1ω0−2ξ0−1,u2ω0−2ξ0−1T, and Ξ=Ξ1,Ξ2T withΞ1=c11⋅x˙/mω0+c12−2mΩz∗y˙/mω0+b11−mΩz∗2x/mω02+b12y+b13ζ02x3/mω02,Ξ2=c22⋅y˙/mω0+c21−2mΩz∗x˙/mω0+b22−mΩz∗2y/mω02+b12x+b23ζ02y3/mω02.Then, the controlled plant (1) can be written as(2)ζ˙1=ζ2,ζ˙2=−Ξ+um.Throughout this study, the following definition and assumptions are needed.Definition 1 (see [25]).
The nonlinear function(2) is a speed function:(3)φt=11−ψfϑ−1+ψf,where 0<ψf<1 is the design parameter and ϑ can be described as(4)ϑ=TS/TS−t4,for0≤t<TS,∞,fort≥TS,with initial condition ϑ0=1 and preset-time parameter TS, and moreover, function ϑ has some attractive features:(1)
ϑ is positive and strictly increasing for 0≤t<TS, and thus, ϑ−1 is strictly decreasing in some time interval(2)
ϑ=∞ and ϑ−1=0, for t≥TS(3)
ϑ is n-order smooth function, and its n-order n⟶∞ derivative is bounded for 0≤t<TS
Obviously, some important properties can be obtained fromϑ:(i) (P1)
φ˙t≥0, and φt is strictly increasing with time and positive for 0≤t<TS; moreover, φ0=1 and φt=1/ψf, for t≥TS(P2)
φ˙t is continuously differentiable bounded functionAssumption 1.
The desired command signalsxd and yd are bounded, and their derivatives x˙d and y˙d are also bounded.Assumption 2.
For the system initial states, there exists a positive constantc satisfying(5)x0−xd02+y0−yd02<c2.
This 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 stateζ2. Then, a new fixed-time prescribed performance controller is developed.


### 3.1. Velocity Observation System Design

Firstly, we define a velocity observation error vector as(6)Z=z1,z2T=ζ2−S,where S≜s1,s2T is updated by the following nonlinear system:(7)S˙=μ1Z+μ2signZ+μ3ZZ22+um,where Z22=ZTZ, signZ=signz1,signz2T, and μ1, μ2, and μ3 are positive design parameters. Notice that μ3Z/Z22=0 if Z=0.Lemma 1.
For the controlled plant (1), if the parameter μ2 is chosen such that μ2≥ϖ, then the variable S provided by (6) can estimate the system state ζ2; in particular, Z22 can converge to zero in a finite time.Proof.
With the help of (1), (5), and (6), we obtain(8)Z˙=ζ˙2−S˙.=−Ξ−μ1Z−μ2signZ−μ3Z/Z22.
Consider the following Lyapunov function candidate:(9)VO=0.5⋅ZTZ.
Taking the tine derivative of (9) along (8), it follows that(10)V˙O=ZT−Ξ−μ1Z−μ2signZ−μ3Z/Z22≤−μ1Z22+ϖ−μ2⋅Z−μ3,where Z=z1+z2. Furthermore, we have(11)V˙O≤−μ2<0.
Integrating (11) from t0 to t yields(12)∫t0tV˙Odτ=VOt−VOt0≤−∫t0tμ2dτ=−μ2t−t0.
From (8) and (11), we have VOt≡0, for t≥t0+VOt0/μ2. Hence, we can further obtain Z=0 due to VOt≡0, for t≥t0+VOt0/μ2. In view of Z=ζ2−S, it means ζ2−S=0 for t≥t0+VOt0/μ2. This shows that S is able to reconstruct the system state ζ2, 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 errorse1 and e2 as well as the auxiliary transformed error θ are introduced as(13)e1=e11,e12T=ζ1−ζd,(14)e2=e21,e22T=S−αd,(15)θ=θ1,θ2T=φt⋅e1.In (12) and (13), ζd=xd,ydT is the desired command vector and αd is a virtual control law which will be given later. Next, a compact set is defined as(16)Ωθ=θ|θTθ<κ2,where κ is a positive design parameter satisfying κ>c. The design process of control law is as follows:Step 1: from (1), (5), (12), and (13), the derivative of θ is(17)θ˙=φ˙t⋅e1+φt⋅ζ2−ζ˙d=φ˙t⋅e1+φt⋅Z+S−ζ˙d=φ˙t⋅e1+φt⋅Z+e2+αd−ζ˙d.Design the virtual control lawαd as follows:(18)αd=ζ˙d−φ−1t⋅φ˙t⋅e1+ηθ−12φt⋅θκ2−θTθ.where η is a positive design parameter. Substituting the virtual control law (17) into (16) yields(19)θ˙=−ηθ+φt⋅e2+φt⋅Z−12⋅φ2t⋅θκ2−θTθ.Remark 1.
To ensure that the initial valueθT0θ0 satisfies θT0θ0<κ2, κ>c need to be met. Since θT0θ0=φ20e1T0e10=ζ110−xd02+ζ120−yd02, if κ>c hold under the assumption 2, then θT0θ0<κ.Step 2: taking the tine derivative of (13), it follows that(20)e˙2=S˙−α˙d=um+δ,where δ=μ1Z+μ2signZ+μ3Z/Z22−α˙d. It is worth noting that the virtual law αd contains some relevant measured signals which are likely to be mixed with interfering noise, so the differential operations of αd may bring harmful and limited spike pulse. In this work, the derivative term of αd and the function μ1Z+μ2signZ+μ3Z/Z22 are combined as a hybrid nonlinear term, and we will utilize the adaptive compensation method to estimate the upper bound of δ22 instead of calculating δ directly, thus avoiding calculation for α˙d.
Now, the control lawu is designed as(21)u=m⋅−ηe2−φtκ2−θTθ⋅θ−W^⋅e2222,with the parameter updating law(22)W^˙=γ1⋅e2222−γ2⋅W^,where γ1 and γ2 are positive design parameters.
Before the main result of stability is given, the following compact set is defined below, i.e.,(23)ΩG1=θ,e2,Z:12logκ2κ2−θTθ+e2Te22+ZTZ2≤kc1,where kc1 is a positive constant.
Note thatδ is bounded in ΩG1, and therefore, there exists a positive constant W such that(24)δ22≤W.
In the follow-up content, we will useW^ to estimate W, and W˜≜W−W^.
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 μ2≥ϖ, and the control law (20) with parameter updating law as specified in equations (21) is used; then, for any initial states in ΩG1, there exist design parameters η>0,γ1>0,γ2>0,andκ>c>0 such that all signals in the closed-loop control system are ultimately uniformly bounded and that the error e122 can converge to a predefined residual set in a predefined time.Proof.
Consider the following Lyapunov function:(25)VG1=12logκ2κ2−θTθ+12e2Te2+12γ1W˜2+12ZTZ,where W˜=W−W^. Taking the derivative of VG1 with respect to time and using (9), (18), and (19), we have(26)V˙G1=θTθ˙κ2−θTθ+e2Te˙2−1γ1W˜W^˙+ZTZ˙≤−η⋅θTθκ2−θTθ+φt⋅θTe2κ2−θTθ+φt⋅θTZκ2−θTθ−12⋅φ2t⋅θTθκ2−θTθ2+e2T⋅um+δ−1γ1W˜W^˙−μ1Z22.
Substituting (21) and (22) into (24), one has(27)V˙G1≤−η⋅θTθκ2−θTθ−ηe22+e2Tδ+φt⋅θTZκ2−θTθ−μ1Z22−12⋅φ2t⋅θTθκ2−θTθ2−W⋅e2222+γ2W˜W^.
It is noted that the following inequalities hold:(28)γ2W˜W^≤−12γ2W˜2+12γ2W2,e2T⋅δ≤W⋅e2222+12,φt⋅θTZκ2−θTθ≤φ2t⋅θTθ2⋅κ2−θTθ2+12Z22,−θTθκ2−θTθ≤−logκ2κ2−θTθ.
With the help of (26), then (25) can be written as(29)V˙G1≤−2η2logκ2κ2−θTθ−2η2e22−μ1−12⋅Z22,−12γ2W˜2+γ22W2+12,≤−ηG1∗⋅VG1+ΔG1,where ηG1∗=min2η,2μ1−1,γ1γ2 and ΔG1=γ2W2+1/2. It follows from (27) that limt⟶∞W˜2≤2ΔG1/ηG1∗ and limt⟶∞e222≤2ΔG1/ηG1∗, and moreover, θTθ<κ2 holds. From the definition of θ and φ, it implies that x−xd<φ−1κ and y−yd<φ−1κ. Let ζx,y=maxx−xd,y−yd, and therefore, we obtain(30)ζx,y<1−ψf⋅TS−tTS4κ+ψfκ,for0≤t<TS,ζx,y<ψfκ,fort≥TS.


### 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,u in (2) is defined as(31)u=ux,uyT=satv=satvx,satvyT.Here,satvx and satvy describe the nonlinear saturation characteristic of the actuator, which are considered as(32)satvj=v¯max⋅signvjifvj≥v¯max,vj,ifvj≥v¯max,where j=x,y and v¯max is a known bound. To further soften control signal, satvj is approximated by using the following smooth function:(33)Γvj=v¯max⋅tanhvjv¯max=v¯max⋅evj/v¯max−e−vj/v¯maxevj/v¯max+e−vj/v¯max.In the light of (30) and (31), one has(34)satv=satvxsatvy=Γv+ε=Γvx+εxΓvy+εy,with Γv=Γvx,ΓvyT, ε=εx,εyT, and εj=satvj−Γvjj=x,y satisfying εj≤v¯max⋅1−tanh1.Based on (29)–(32), the controlled plant (2) is written as(35)ζ˙1=ζ2,ζ˙2=−Ξ∗+Γv/m,in which Ξ∗=Ξ−ε/m. In view of the intervention of Γv, a velocity observation error vector Z∗ and an observation system are constructed, respectively, as(36a)Z∗=ζ2−S∗,(36b)S˙∗=μ1Z∗+μ2signZ∗+μ3Z∗/Z∗22+Γv/m,where Z∗=z1∗,z2∗T, signZ∗=signz1∗,signz2∗T, S∗=s1∗,s2∗T, and Z∗22=Z∗TZ∗.Lemma 2.
For the nonlinear system (36b), if the parameter μ2 is chosen such that μ2≥ϖ+1−tanh1/m, then the variable S∗ supported by (36b) can estimate system state ζ2, and moreover, Z∗22 can converge to zero in a finite time.Proof.
From (33) and (36b), one has(37)Z˙∗=ζ˙2−S˙∗=−Ξ+ε/m−μ1Z∗−μ2signZ∗−μ3Z∗/Z∗22.
Then, consider the following Lyapunov function:(38)VO∗=0.5⋅Z∗TZ∗.
By using (35), the time derivative of VO∗ is calculated as(39)V˙O∗=Z∗T−Ξ+ε/m−μ1Z∗−μ2signZ∗−μ3Z∗/Z∗22≤−μ1Z∗22+ϖ−μ2+1−tanh1/m⋅Z∗−μ3.
Based on the conditionμ2≥ϖ+1−tanh1/m, we have(40)V˙O∗≤−μ2<0.
Similar to the analysis of Lemma1, we can also obtain that the reconstruction error S∗−ζ2 converge to zero after a finite time. The proof is completed.
To achieve the control objective, the following transform of coordinate is made:(41)e1∗=e11∗,e12∗T=ζ1−ζd,(42)e2∗=e21∗,e22∗T=S∗−αd∗,(43)e3∗=e31∗,e32∗T=Γv−βd,(44)θ∗=θ1∗,θ2∗T=φt⋅e1∗,where αd∗ and βd are the virtual control laws, and they are designed as(45)αd∗=ζ˙d−φ−1t⋅φ˙t⋅e1∗+ηθ∗−12φt⋅θ∗κ2−θ∗Tθ∗,(46)βd=m⋅−ηe2∗−φtκ2−θ∗Tθ∗⋅θ∗−W^1∗⋅e2∗2,with parameter updating law(47)W^˙1∗=γ1⋅e2∗22/2−γ2W^1∗.
The 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 θ∗ is(48)θ˙∗=φ˙te1∗+φte˙1∗=φ˙te1∗+φtζ2−ζ˙d,=φ˙te1∗+φt⋅Z∗+S∗−ζ˙d,=φ˙te1∗+φt⋅Z∗+e2∗+αd∗−ζ˙d.(i)
Substituting (44) into (47) yields(49)θ˙∗=−ηθ∗+φt⋅e2∗+φt⋅Z∗−12⋅φ2t⋅θ∗κ2−θ∗Tθ∗.Step 2: by applying (41), (42), and (45), the time derivative of e2∗ can expressed as(50)e˙2∗=S˙∗−α˙d∗=Γvm+δ∗=βdm+e3∗m+δ∗=−ηe2∗−φtκ2−θ∗Tθ∗⋅θ∗−W^1∗⋅e2∗2+e3∗m+δ∗,whereδ∗=μ1Z∗+μ2signZ∗+μ3Z∗/Z∗22−α˙d∗.Step 3
: taking time derivative ofe3∗ yields(51)e˙3∗=∂Γv∂v⋅v˙−β˙d.
In (51), the details of ∂Γv/∂v is presented as follows:(52)∂Γv∂v=diag∂Γvx∂vx,∂Γvy∂vy,=diag4evx/v¯max+e−vx/v¯max2,4evy/v¯max+e−vy/v¯max2>0.
To continuous, the updating law ofv is designed as(53)v˙=Φ⋅−η⋅e3∗−e2∗m−W^2∗e3∗2,v0=0.1,where Φ=diag∂Γvx/∂vx−1,∂Γvy/∂vy−1 and W^2∗ is updated by(54)W^˙2∗=γ1e3∗Te3∗2−γ2W^2∗.
Define a compact set as follows:(55)ΩG2=θ∗,e2∗,e3∗,Z∗:12logκ2κ2−θ∗Tθ∗+e2∗Te2∗2+e3∗Te3∗2+Z∗TZ∗2≤kc2,where kc2 is a positive constant.
Notice thatδ∗22 and β˙d22 are bounded in ΩG2; therefore, there exists positive constants W1∗ and W2∗, such that(56)δ∗22≤W1∗,β˙d22≤W2∗.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 ΩG2, if the design parameters η, γ1, γ2, κ, and μ2 are chosen such that(57)η>0,γ1>0,γ2>0,κ>c>0,μ2≥ϖ+1−tanh1/m,then the following goals hold:(1)
The constraints of system error variableθ∗Ttθ∗t are not violated, i.e., θ∗Ttθ∗t<κ2 for ∀t≥0(2)
The errorse1∗ converge to a prescribed compact set ΩZ=e1∗|e1∗<ψfκ for t≥TProof.
Consider the following Lyapunov function:(58)VG2=12logκ2κ2−θ∗Tθ∗+12e2∗Te2∗+12e3∗Te3∗+12γ1W˜1∗2+W˜2∗2+12Z∗TZ∗,where W˜1∗=W1∗−W^1∗ and W˜2∗=W2∗−W^2∗. By applying (46) and (47) and (49)–(53), the time derivative of VG2 in (54) can expressed as(59)V˙G2=θ∗Tθ˙∗κ2−θ∗Tθ∗+e2∗Te˙2∗+e3∗Te˙3∗−1γ1W˜1∗W^˙1∗+W˜2∗W^˙2∗+Z∗TZ˙∗≤−η⋅θ∗Tθ∗κ2−θ∗Tθ∗+φt⋅θ∗TZ∗κ2−θ∗Tθ∗−12⋅φ2t⋅θ∗Tθ∗κ2−θ∗Tθ∗2−ηe2∗Te2∗−e2∗Te2∗⋅W^1∗2+e2∗Te2∗⋅W1∗2+12−1γ1W˜1∗W^˙1∗−η⋅e3∗Te3∗−e3∗Te3∗⋅W^2∗2+e3∗Te3∗⋅W2∗2+12−1γ1W˜2∗W^˙2∗−μ1Z∗22≤−η⋅θ∗Tθ∗κ2−θ∗Tθ∗−η⋅e2∗Te2∗−η⋅e3∗Te3∗−12γ2⋅W˜1∗2+W˜2∗2−μ1−12⋅Z∗22+γ22⋅W1∗2+W2∗2+12,(60)≤−ηG2∗⋅VG2+ΔG2,where ηG2∗=min2η,2μ1−1,γ1γ2 and ΔG2=0.5γ2W1∗2+W2∗2+1. Multiplying on both its sides by eηG2∗t and integrating it over t0,t, we have(61)VG2t≤eηG2∗t0−tVG2t0−ΔG2ηG2∗+ΔG2ηG2∗<eηG2∗t0VG2t0+ΔG2ηG2∗.
From the above equation, it is clear thatx−xd<φ−1κ and y−yd<φ−1κ. Let ζx,y=maxx−xd,y−yd; similar to the analysis of Theorem 1, we can obtain the following results:(62)ζx,y<1−ψf⋅TS−tTS4κ+ψfκ,for0≤t<TS,ζx,y<ψfκ,fort≥TS.
All these complete the proof. The control structure is presented in Figure2.Figure 2
The control system structure.Remark 2.
From (56), the size of the system error ζx,y depend on the parameters ψf and TS when t≥TS. The small ψf and TS 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 Theorem2, the strong robustness of the proposed method stems from the performance function φt and the virtual control laws (44) and (45). When system errors are far away from the origin, φt/κ2−θ∗Tθ∗ in (44) and (45) to uncertainties plays an important role for enforcing the system error to enter the preassigned error convergence set in t≥TS.


## 3.1. Velocity Observation System Design

Firstly, we define a velocity observation error vector as(6)Z=z1,z2T=ζ2−S,where S≜s1,s2T is updated by the following nonlinear system:(7)S˙=μ1Z+μ2signZ+μ3ZZ22+um,where Z22=ZTZ, signZ=signz1,signz2T, and μ1, μ2, and μ3 are positive design parameters. Notice that μ3Z/Z22=0 if Z=0.Lemma 1.
For the controlled plant (1), if the parameter μ2 is chosen such that μ2≥ϖ, then the variable S provided by (6) can estimate the system state ζ2; in particular, Z22 can converge to zero in a finite time.Proof.
With the help of (1), (5), and (6), we obtain(8)Z˙=ζ˙2−S˙.=−Ξ−μ1Z−μ2signZ−μ3Z/Z22.
Consider the following Lyapunov function candidate:(9)VO=0.5⋅ZTZ.
Taking the tine derivative of (9) along (8), it follows that(10)V˙O=ZT−Ξ−μ1Z−μ2signZ−μ3Z/Z22≤−μ1Z22+ϖ−μ2⋅Z−μ3,where Z=z1+z2. Furthermore, we have(11)V˙O≤−μ2<0.
Integrating (11) from t0 to t yields(12)∫t0tV˙Odτ=VOt−VOt0≤−∫t0tμ2dτ=−μ2t−t0.
From (8) and (11), we have VOt≡0, for t≥t0+VOt0/μ2. Hence, we can further obtain Z=0 due to VOt≡0, for t≥t0+VOt0/μ2. In view of Z=ζ2−S, it means ζ2−S=0 for t≥t0+VOt0/μ2. This shows that S is able to reconstruct the system state ζ2, 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 errorse1 and e2 as well as the auxiliary transformed error θ are introduced as(13)e1=e11,e12T=ζ1−ζd,(14)e2=e21,e22T=S−αd,(15)θ=θ1,θ2T=φt⋅e1.In (12) and (13), ζd=xd,ydT is the desired command vector and αd is a virtual control law which will be given later. Next, a compact set is defined as(16)Ωθ=θ|θTθ<κ2,where κ is a positive design parameter satisfying κ>c. The design process of control law is as follows:Step 1: from (1), (5), (12), and (13), the derivative of θ is(17)θ˙=φ˙t⋅e1+φt⋅ζ2−ζ˙d=φ˙t⋅e1+φt⋅Z+S−ζ˙d=φ˙t⋅e1+φt⋅Z+e2+αd−ζ˙d.Design the virtual control lawαd as follows:(18)αd=ζ˙d−φ−1t⋅φ˙t⋅e1+ηθ−12φt⋅θκ2−θTθ.where η is a positive design parameter. Substituting the virtual control law (17) into (16) yields(19)θ˙=−ηθ+φt⋅e2+φt⋅Z−12⋅φ2t⋅θκ2−θTθ.Remark 1.
To ensure that the initial valueθT0θ0 satisfies θT0θ0<κ2, κ>c need to be met. Since θT0θ0=φ20e1T0e10=ζ110−xd02+ζ120−yd02, if κ>c hold under the assumption 2, then θT0θ0<κ.Step 2: taking the tine derivative of (13), it follows that(20)e˙2=S˙−α˙d=um+δ,where δ=μ1Z+μ2signZ+μ3Z/Z22−α˙d. It is worth noting that the virtual law αd contains some relevant measured signals which are likely to be mixed with interfering noise, so the differential operations of αd may bring harmful and limited spike pulse. In this work, the derivative term of αd and the function μ1Z+μ2signZ+μ3Z/Z22 are combined as a hybrid nonlinear term, and we will utilize the adaptive compensation method to estimate the upper bound of δ22 instead of calculating δ directly, thus avoiding calculation for α˙d.
Now, the control lawu is designed as(21)u=m⋅−ηe2−φtκ2−θTθ⋅θ−W^⋅e2222,with the parameter updating law(22)W^˙=γ1⋅e2222−γ2⋅W^,where γ1 and γ2 are positive design parameters.
Before the main result of stability is given, the following compact set is defined below, i.e.,(23)ΩG1=θ,e2,Z:12logκ2κ2−θTθ+e2Te22+ZTZ2≤kc1,where kc1 is a positive constant.
Note thatδ is bounded in ΩG1, and therefore, there exists a positive constant W such that(24)δ22≤W.
In the follow-up content, we will useW^ to estimate W, and W˜≜W−W^.
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 μ2≥ϖ, and the control law (20) with parameter updating law as specified in equations (21) is used; then, for any initial states in ΩG1, there exist design parameters η>0,γ1>0,γ2>0,andκ>c>0 such that all signals in the closed-loop control system are ultimately uniformly bounded and that the error e122 can converge to a predefined residual set in a predefined time.Proof.
Consider the following Lyapunov function:(25)VG1=12logκ2κ2−θTθ+12e2Te2+12γ1W˜2+12ZTZ,where W˜=W−W^. Taking the derivative of VG1 with respect to time and using (9), (18), and (19), we have(26)V˙G1=θTθ˙κ2−θTθ+e2Te˙2−1γ1W˜W^˙+ZTZ˙≤−η⋅θTθκ2−θTθ+φt⋅θTe2κ2−θTθ+φt⋅θTZκ2−θTθ−12⋅φ2t⋅θTθκ2−θTθ2+e2T⋅um+δ−1γ1W˜W^˙−μ1Z22.
Substituting (21) and (22) into (24), one has(27)V˙G1≤−η⋅θTθκ2−θTθ−ηe22+e2Tδ+φt⋅θTZκ2−θTθ−μ1Z22−12⋅φ2t⋅θTθκ2−θTθ2−W⋅e2222+γ2W˜W^.
It is noted that the following inequalities hold:(28)γ2W˜W^≤−12γ2W˜2+12γ2W2,e2T⋅δ≤W⋅e2222+12,φt⋅θTZκ2−θTθ≤φ2t⋅θTθ2⋅κ2−θTθ2+12Z22,−θTθκ2−θTθ≤−logκ2κ2−θTθ.
With the help of (26), then (25) can be written as(29)V˙G1≤−2η2logκ2κ2−θTθ−2η2e22−μ1−12⋅Z22,−12γ2W˜2+γ22W2+12,≤−ηG1∗⋅VG1+ΔG1,where ηG1∗=min2η,2μ1−1,γ1γ2 and ΔG1=γ2W2+1/2. It follows from (27) that limt⟶∞W˜2≤2ΔG1/ηG1∗ and limt⟶∞e222≤2ΔG1/ηG1∗, and moreover, θTθ<κ2 holds. From the definition of θ and φ, it implies that x−xd<φ−1κ and y−yd<φ−1κ. Let ζx,y=maxx−xd,y−yd, and therefore, we obtain(30)ζx,y<1−ψf⋅TS−tTS4κ+ψfκ,for0≤t<TS,ζx,y<ψfκ,fort≥TS.


## 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,u in (2) is defined as(31)u=ux,uyT=satv=satvx,satvyT.Here,satvx and satvy describe the nonlinear saturation characteristic of the actuator, which are considered as(32)satvj=v¯max⋅signvjifvj≥v¯max,vj,ifvj≥v¯max,where j=x,y and v¯max is a known bound. To further soften control signal, satvj is approximated by using the following smooth function:(33)Γvj=v¯max⋅tanhvjv¯max=v¯max⋅evj/v¯max−e−vj/v¯maxevj/v¯max+e−vj/v¯max.In the light of (30) and (31), one has(34)satv=satvxsatvy=Γv+ε=Γvx+εxΓvy+εy,with Γv=Γvx,ΓvyT, ε=εx,εyT, and εj=satvj−Γvjj=x,y satisfying εj≤v¯max⋅1−tanh1.Based on (29)–(32), the controlled plant (2) is written as(35)ζ˙1=ζ2,ζ˙2=−Ξ∗+Γv/m,in which Ξ∗=Ξ−ε/m. In view of the intervention of Γv, a velocity observation error vector Z∗ and an observation system are constructed, respectively, as(36a)Z∗=ζ2−S∗,(36b)S˙∗=μ1Z∗+μ2signZ∗+μ3Z∗/Z∗22+Γv/m,where Z∗=z1∗,z2∗T, signZ∗=signz1∗,signz2∗T, S∗=s1∗,s2∗T, and Z∗22=Z∗TZ∗.Lemma 2.
For the nonlinear system (36b), if the parameter μ2 is chosen such that μ2≥ϖ+1−tanh1/m, then the variable S∗ supported by (36b) can estimate system state ζ2, and moreover, Z∗22 can converge to zero in a finite time.Proof.
From (33) and (36b), one has(37)Z˙∗=ζ˙2−S˙∗=−Ξ+ε/m−μ1Z∗−μ2signZ∗−μ3Z∗/Z∗22.
Then, consider the following Lyapunov function:(38)VO∗=0.5⋅Z∗TZ∗.
By using (35), the time derivative of VO∗ is calculated as(39)V˙O∗=Z∗T−Ξ+ε/m−μ1Z∗−μ2signZ∗−μ3Z∗/Z∗22≤−μ1Z∗22+ϖ−μ2+1−tanh1/m⋅Z∗−μ3.
Based on the conditionμ2≥ϖ+1−tanh1/m, we have(40)V˙O∗≤−μ2<0.
Similar to the analysis of Lemma1, we can also obtain that the reconstruction error S∗−ζ2 converge to zero after a finite time. The proof is completed.
To achieve the control objective, the following transform of coordinate is made:(41)e1∗=e11∗,e12∗T=ζ1−ζd,(42)e2∗=e21∗,e22∗T=S∗−αd∗,(43)e3∗=e31∗,e32∗T=Γv−βd,(44)θ∗=θ1∗,θ2∗T=φt⋅e1∗,where αd∗ and βd are the virtual control laws, and they are designed as(45)αd∗=ζ˙d−φ−1t⋅φ˙t⋅e1∗+ηθ∗−12φt⋅θ∗κ2−θ∗Tθ∗,(46)βd=m⋅−ηe2∗−φtκ2−θ∗Tθ∗⋅θ∗−W^1∗⋅e2∗2,with parameter updating law(47)W^˙1∗=γ1⋅e2∗22/2−γ2W^1∗.
The 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 θ∗ is(48)θ˙∗=φ˙te1∗+φte˙1∗=φ˙te1∗+φtζ2−ζ˙d,=φ˙te1∗+φt⋅Z∗+S∗−ζ˙d,=φ˙te1∗+φt⋅Z∗+e2∗+αd∗−ζ˙d.(i)
Substituting (44) into (47) yields(49)θ˙∗=−ηθ∗+φt⋅e2∗+φt⋅Z∗−12⋅φ2t⋅θ∗κ2−θ∗Tθ∗.Step 2: by applying (41), (42), and (45), the time derivative of e2∗ can expressed as(50)e˙2∗=S˙∗−α˙d∗=Γvm+δ∗=βdm+e3∗m+δ∗=−ηe2∗−φtκ2−θ∗Tθ∗⋅θ∗−W^1∗⋅e2∗2+e3∗m+δ∗,whereδ∗=μ1Z∗+μ2signZ∗+μ3Z∗/Z∗22−α˙d∗.Step 3
: taking time derivative ofe3∗ yields(51)e˙3∗=∂Γv∂v⋅v˙−β˙d.
In (51), the details of ∂Γv/∂v is presented as follows:(52)∂Γv∂v=diag∂Γvx∂vx,∂Γvy∂vy,=diag4evx/v¯max+e−vx/v¯max2,4evy/v¯max+e−vy/v¯max2>0.
To continuous, the updating law ofv is designed as(53)v˙=Φ⋅−η⋅e3∗−e2∗m−W^2∗e3∗2,v0=0.1,where Φ=diag∂Γvx/∂vx−1,∂Γvy/∂vy−1 and W^2∗ is updated by(54)W^˙2∗=γ1e3∗Te3∗2−γ2W^2∗.
Define a compact set as follows:(55)ΩG2=θ∗,e2∗,e3∗,Z∗:12logκ2κ2−θ∗Tθ∗+e2∗Te2∗2+e3∗Te3∗2+Z∗TZ∗2≤kc2,where kc2 is a positive constant.
Notice thatδ∗22 and β˙d22 are bounded in ΩG2; therefore, there exists positive constants W1∗ and W2∗, such that(56)δ∗22≤W1∗,β˙d22≤W2∗.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 ΩG2, if the design parameters η, γ1, γ2, κ, and μ2 are chosen such that(57)η>0,γ1>0,γ2>0,κ>c>0,μ2≥ϖ+1−tanh1/m,then the following goals hold:(1)
The constraints of system error variableθ∗Ttθ∗t are not violated, i.e., θ∗Ttθ∗t<κ2 for ∀t≥0(2)
The errorse1∗ converge to a prescribed compact set ΩZ=e1∗|e1∗<ψfκ for t≥TProof.
Consider the following Lyapunov function:(58)VG2=12logκ2κ2−θ∗Tθ∗+12e2∗Te2∗+12e3∗Te3∗+12γ1W˜1∗2+W˜2∗2+12Z∗TZ∗,where W˜1∗=W1∗−W^1∗ and W˜2∗=W2∗−W^2∗. By applying (46) and (47) and (49)–(53), the time derivative of VG2 in (54) can expressed as(59)V˙G2=θ∗Tθ˙∗κ2−θ∗Tθ∗+e2∗Te˙2∗+e3∗Te˙3∗−1γ1W˜1∗W^˙1∗+W˜2∗W^˙2∗+Z∗TZ˙∗≤−η⋅θ∗Tθ∗κ2−θ∗Tθ∗+φt⋅θ∗TZ∗κ2−θ∗Tθ∗−12⋅φ2t⋅θ∗Tθ∗κ2−θ∗Tθ∗2−ηe2∗Te2∗−e2∗Te2∗⋅W^1∗2+e2∗Te2∗⋅W1∗2+12−1γ1W˜1∗W^˙1∗−η⋅e3∗Te3∗−e3∗Te3∗⋅W^2∗2+e3∗Te3∗⋅W2∗2+12−1γ1W˜2∗W^˙2∗−μ1Z∗22≤−η⋅θ∗Tθ∗κ2−θ∗Tθ∗−η⋅e2∗Te2∗−η⋅e3∗Te3∗−12γ2⋅W˜1∗2+W˜2∗2−μ1−12⋅Z∗22+γ22⋅W1∗2+W2∗2+12,(60)≤−ηG2∗⋅VG2+ΔG2,where ηG2∗=min2η,2μ1−1,γ1γ2 and ΔG2=0.5γ2W1∗2+W2∗2+1. Multiplying on both its sides by eηG2∗t and integrating it over t0,t, we have(61)VG2t≤eηG2∗t0−tVG2t0−ΔG2ηG2∗+ΔG2ηG2∗<eηG2∗t0VG2t0+ΔG2ηG2∗.
From the above equation, it is clear thatx−xd<φ−1κ and y−yd<φ−1κ. Let ζx,y=maxx−xd,y−yd; similar to the analysis of Theorem 1, we can obtain the following results:(62)ζx,y<1−ψf⋅TS−tTS4κ+ψfκ,for0≤t<TS,ζx,y<ψfκ,fort≥TS.
All these complete the proof. The control structure is presented in Figure2.Figure 2
The control system structure.Remark 2.
From (56), the size of the system error ζx,y depend on the parameters ψf and TS when t≥TS. The small ψf and TS 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 Theorem2, the strong robustness of the proposed method stems from the performance function φt and the virtual control laws (44) and (45). When system errors are far away from the origin, φt/κ2−θ∗Tθ∗ in (44) and (45) to uncertainties plays an important role for enforcing the system error to enter the preassigned error convergence set in t≥TS.


## 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 asm=0.57×10−8kg, c¯11=4.29×10−7Ns/m, and c¯22=4.29×10−8Ns/mc¯12=4.29×10−8Ns/m, c¯21=4.29×10−8Ns/m, and b¯11=80.98N/mb¯22=71.62N/m, b¯12=b¯21=5N/m, and b¯13=b¯23=3.56×10−6N/mΩ¯z∗=5rad/s, ζ0=0.1×10−4m, and ω0=1kHzTo show the effectiveness of the proposed control strategy, uncertain system parameters are introduced, the details of which arec11=c¯11×0.5sinπt, c22=c¯22×0.45cost, c12=c¯12×0.6sin2.5t, c21=c¯21×0.5sin1.5t, b11=b¯11×0.6sin2t, b22=b¯22×0.5sint, b12=b21=b¯21×0.5sin2t, b13=b23=b¯23×0.6cost, and Ωz∗=Ω¯z∗×0.62.In addition, the initial state value isζ0=0.2,0.2T, the desired command signals are given as xd=sin2t as well as yd=0.8sin3t, and the related control system parameters are choose as μ1=5, μ2=15, μ3=η=7, ψf=0.005, TS=1, κ=2.5, γ1=20, γ2=0.01, v¯max=300, and W^1∗0=W^1∗0=0.1.Figures3 and 4 show the time responses of the velocity observation errors, where z1 is the observation error of x˙−s1 and z2 is the observation error of y˙−s2. From the plots of z1 and z2, we can conclude that x˙ and y˙ can be accurately estimated by the observer system (36b). Moreover, the observer error converge to a very small region with maxz1,z2≤2×10−4 when t≥0.5s. 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 e11 and e12 strictly evolve within the predefined performance envelope and converge to ±4.5×10−3 when t≥1s, and meanwhile, s1−x˙d and s2−y˙d converge to ±8×10−3 when t≥1s, 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.Figure 3
The observation estimation of system statex˙.Figure 4
The observation estimation of system statey˙.Figure 5
The tracking response ofx and x˙.Figure 6
The tracking response ofy and y˙.Figure 7
The updating value of parameterW^1 and W^2.Figure 8
The control inputux and uy.


## 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.



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*Source: 1499685-2022-06-16.xml*