# Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement

Authors: Alexander Jimenez-Triana; Guchuan Zhu; Lahcen Saydy
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The present work addresses the problem of chaos control in an electrostatic MEMS resonator by using an output-feedback control scheme. One of the unstable orbits immersed in the chaotic attractor is stabilized in order to produce a sustained oscillation of the movable plate composing the microstructure. The orbit is carefully chosen so as to produce a high amplitude oscillation. This approach allows the enhancement of oscillation amplitude of the resonator at a reduced control effort, since the unstable orbit already exists in the system and it is not necessary to spend energy to create it. Realistic operational conditions of the MEMS are considered including parametric uncertainties in the model and constraints due to the difficulty in measuring the speed of the plates of the microstructure. A control law is constructed recursively by using the technique of backstepping. Finally, numerical simulations are carried out to confirm the validity of the developed control scheme and to demonstrate the effect of controlling orbits immersed in the chaotic attractor.


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## Body

## 1. Introduction

This paper addresses the problem of chaos control in an electrostatic MEMS resonator with parametric uncertainties and constraints in the measure of its state. The considered device is a structure with dual drive using differential actuation, as shown in Figure1.Figure 1
Schematic of the electrostatically actuated MEMS resonator.Many MEMS devices exhibit chaotic behaviour in certain operational conditions [1–4]. It is observed in recent results concerning the improvement of the output energy in oscillators using MEMS [5] that controlling MEMS in chaotic mode can be useful to increase the amplitude of the periodic oscillations of these systems. Therefore, controlling chaos in MEMS can enhance the performance of different systems in which an accurate control of amplitude of oscillation is mandatory.Chaos control has been extensively studied in the literature. Different methods have been applied to stabilize equilibrium points and unstable orbits immersed in the chaotic attractor or to track orbits that can be chaotic. The considered approaches cover the application of a wide range of control techniques, from the use of Lyapunov theory to stabilize chaotic oscillators designed via local linearization or linear feedback [6, 7], to optimal control [8], robust control [9], and adaptive control [10], to name a few. Chaos control in MEMS is reported recently in [1, 11]. In [1] a fuzzy control algorithm is applied to control the chaotic motion and track a sinusoidal orbit. In [11] an adaptive control is used to control an uncertain model of a MEMS device and to synchronize it with a class of uncertain chaotic oscillators. Nevertheless, the enhancement of oscillation amplitude for MEMS resonators by appropriate control techniques remains an interesting and open research subject.The oscillation amplitude of electrostatic MEMS resonators is restricted by the so-called pull-in phenomena related to a bifurcation arising in electrostatic actuation [12]. The stable traveling range is limited to one-third of its full gap due to the instability beyond that point. The pull-in phenomena can be used to accelerate the reaction speed for switching structures. However, it limits the stable operational range of the device, which is mandatory for applications requiring a sustained oscillation of the plates.In order to operate electrostatic MEMS devices beyond the pull-in, one can use closed-loop control schemes (see, e.g., [13, 14]). However, it may require a considerable control effort to produce an arbitrary trajectory. In order to increase the amplitude of periodic oscillations of the device with a low energy consumption, we consider in the present work controlling the resonator in chaotic mode to stabilize an orbit which is part of the chaotic attractor.There are different ways to select this orbit. One can pick an unstable periodic orbit embedded in the chaotic attractor, take an orbit close to an unstable periodic one, or even choose a part of a chaotic trajectory that has an almost periodic behavior.In the present work, we tackle the amplitude enhancement problem by stabilizing one of these orbits to produce a high amplitude oscillation.There are some constraints that must be considered in a practical implementation of a chaos control scheme applied to MEMS. In particular, it is very difficult to sense the velocity of the plates during the operation of the device, which motivates the use of output-feedback control techniques. In this work, the output is chosen as the position of the plates, and, in order to implement the control, the velocity must be estimated by constructing an observer. We deal with the uncertainties in the model and the constraints on sensing the velocity by designing a controller by means of an adaptive observer backstepping scheme.The organization of the rest of the paper is as follows. In Section2, the dynamics of a chaotic MEMS actuator are briefly explained. Then, the design of an adaptive observer control scheme is presented in Section 3. Simulations are carried out and the results are presented in Section 4. Finally, concluding remarks are given in Section 5.


## 2.  Dynamics of a Chaotic MEMS Resonator

### 2.1. Model of the Chaotic MEMS

The MEMS device considered in this study is presented in Figure1, which is based on a model given in [1, 15]. This device is modeled by a nonlinear mass-spring-damper system with external electrostatic actuation, expressed as
(1)mz¨+bz˙+c1z+c3z3=F,
where m is the mass of the movable structure, b is the damping constant, c1 and c3 are the linear and cubic stiffness coefficients, respectively, and F is the electrostatic actuation force given by
(2)F=12C0d(d-z)2(VDC+u)2-12C0d(d+z)2VDC2.In (2), C0 is the capacitance of the actuator at rest, which has an initial gap d between the movable and fixed plates. The input voltage Vi=(VDC+u) is applied as shown in Figure 1 and contains two components: a DC voltage VDC and a forcing voltage u, which may be a control signal that varies according to control objectives.Note that, in the present work, we ignore the dynamics of the driving circuit by assuming that they are much faster than those of the mechanical structure. Moreover, as the MEMS are used as a resonator, we assume that the system operates in such a way that the moveable and fixed plates do not come into contact. Hence, we do not take into account contact dynamics. Nevertheless, for those applications, where the contact may occur, a different model should be considered to capture the dynamics of impacts and the control strategy should be modified adequately [16].In order to facilitate control system design and analysis, it is convenient to work in normalized coordinates and redefine the time variable. For this purpose, we define the new variablesψ=z/d, τ=ω0t, and u~=uA and introduce the following dimensionless constants:
(3)σ=c1mω02,η=c3d2mω02,δ=C0VDC22mω02d2,A=2δ1VDC,μ=bmω0,
where ω0 is the natural frequency defined by
(4)ω0=c1m.
In this way, by assuming that the DC voltage is sufficiently higher than the amplitude of the forcing voltage u, the dimensionless equation of motion (1) may be fairly approximated by
(5)ψ¨+μψ˙+σψ+ηψ3=δ(1(1-ψ)2-1(1+ψ)2)+u~(1-ψ)2.It can be shown that system (5) behaves chaotically by selecting u=VAC⁡sin(ω~t) and VAC⁡A=0.04, ω~=0.5, δ=0.338, μ=0.01, σ=1, and η=12, as indicated in the Lyapunov exponents diagram in Figure 2. The Lyapunov exponents are calculated by using the algorithm proposed in [17]. The existence of one positive Lyapunov exponent confirms the sensitive dependence on initial conditions of the system.Figure 2
Lyapunov exponents for the system (5) with the set of parameters: δ=0.338, A=0.04, ω=0.5, σ=1, μ=0.01, and η=12.System (5) can be written in state space form as
(6)ψ˙1=ψ2,ψ˙2=-μψ˙2-σψ1-ηψ13+f(ψ1)+u~(1-ψ1)2,y=ψ1,
where
(7)f(ψ1)=δ(1(1-ψ1)2-1(1+ψ1)2)
and y is the output.


### 2.2.  Unstable Orbits in the Strange Attractor

The main idea to obtain a sustained oscillation of the MEMS with a low energy consumption is not to create a new orbit different from those already present in the attractor. The orbit could be an unstable periodic orbit embedded in the chaotic attractor, an orbit close to an unstable periodic one, or even part of a chaotic trajectory which has an almost periodic behavior.In order to find one of such orbits, one may measure the phase space dynamics of the system and obtain a set of points from which one can extract the orbit by applying the close return method [18]. Note that other approaches, such as the shooting method [19], can also be used for this purpose. With the aim of simplifying the discussion, we will only consider systems, where the state x=(x1,x2)∈R2 and an external periodic force is applied. Under these conditions, a Poincaré map can be easily constructed from stroboscopically logging the value of the variable x at multiples of the period T of the forcing. Time can be seen as one of the states, and the problem of finding periodic orbits reduces to find fixed points in R2 on the Poincaré surface.The Poincaré map may be defined as follows. Given the differential equation(8)x˙=f(x,t),x∈R2,
with periodic phase space of period T such that
(9)f(x,t)=f(x,t+T),
then the Poincaré map is defined as
(10)P:Σ⟶Σ,x⟼φ(T,x),
where φ(t,·) denotes the flow generated by (8) and Σ is a Poincaré section as shown in Figure 3.Figure 3
Poincaré sectionΣ and Poincaré map P for the periodic vector field defined by (8) and (9).The procedure for finding orbits with almost periodic behaviour is similar to the one of finding periodic orbits. In both cases, the image under the Poincaré map of a point close to the orbit, which intersects the Poincaré surface, is close to the point. Finding such point is the aim of the close return method.For the purpose of applying the method, one records discretely sampled time seriesx(0),x(Ts),…,x((N-1)Ts) containing N points, where Ts is the sample time. Then, for a k given, one takes a point x(kTs) and its iteration m on the map x(kTs+mT) and evaluates the distance d(x(kTs),x(kTs+mT)) between the points, by using, for example, the Euclidean norm. If the distance is less than a small ϵs, then this is a m-period orbit that could be used.In order to show the application of the close return method, we use three different time series from which we extract unstable orbits of different periods. Each of these series comprises a set ofN=120000 points with sample time of Ts=0.01, which corresponds to a logging time of approximately 100 times the period of the forcing term. The orbits were obtained by using ϵs=0.005 and can be seen in Figures 4, 5, and 6. The figures suggest that the bifurcations occurred when the period increases.Unstable orbits found using close return method. The period of the orbit is identified withn.
(a)
(b)
(c)
(d)
(e)
(f)Unstable orbits found using close return method for a second set of data. The period of the orbit isn.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Unstable orbits found using close return method for a third set of data.n identifies the period of the orbit.
(a)
(b)
(c)
(d)
(e)
(f)
(g)In order to enhance the oscillation amplitude of the plates, we chose an unstable orbit surrounding the chaotic attractor, which produces a high amplitude periodic output. This orbit, shown in Figure7, was used as the reference trajectory in the simulation given subsequently. It can be approximated by its truncated Fourier series up to five terms, given by
(11)yr=a0+a1cos⁡(w1t)+b1sin(w1t)+a2cos⁡(2w1t)+b2sin⁡(2w1t)+a3cos⁡⁡(3w1t)+b3sin(3w1t)+a4cos⁡⁡(4w1t)+b4sin⁡(4w1t)+a5cos⁡(5w1t)+b5sin(5w1t),
where
(12)a0=0.0175400,a1=-0.4498000,b1=-0.000563,a2=0.0112600,b2=0.0008906,a3=-0.029140,b3=0.0000376,a4=0.0006330,b4=0.000231,a5=-0.001090,b5=0.0000852.Figure 7
Unstable orbit surrounding the chaotic attractor.By taking the time derivative of (11), we obtain
(13)y˙r=a1′cos⁡(w1t)+b1′sin(w1t)+a2′cos⁡(2w1t)+b2′sin⁡(2w1t)+a3′cos⁡⁡(3w1t)+b3′sin(3w1t)+a4′cos⁡⁡(4w1t)+b4′sin⁡(4w1t)+a5′cos⁡(5w1t)+b5′sin(5w1t),
where
(14)a1′=-0.00052,b1′=0.4163,a2′=0.0016,b2′=-0.0208,a3′=0.00010,b3′=0.0809,a4′=0.00085,b4′=-0.0023,a5′=0.00039,b5′=0.00500.The second derivative ofyr is obtained by taking the derivative of (13).In what follows, we use the following general assumption.Assumption 1.
The reference signalyr(t) and its first two derivatives are known and bounded. In addition, yr(p)(t), ∀p≥3, is piecewise continuous.


## 2.1. Model of the Chaotic MEMS

The MEMS device considered in this study is presented in Figure1, which is based on a model given in [1, 15]. This device is modeled by a nonlinear mass-spring-damper system with external electrostatic actuation, expressed as
(1)mz¨+bz˙+c1z+c3z3=F,
where m is the mass of the movable structure, b is the damping constant, c1 and c3 are the linear and cubic stiffness coefficients, respectively, and F is the electrostatic actuation force given by
(2)F=12C0d(d-z)2(VDC+u)2-12C0d(d+z)2VDC2.In (2), C0 is the capacitance of the actuator at rest, which has an initial gap d between the movable and fixed plates. The input voltage Vi=(VDC+u) is applied as shown in Figure 1 and contains two components: a DC voltage VDC and a forcing voltage u, which may be a control signal that varies according to control objectives.Note that, in the present work, we ignore the dynamics of the driving circuit by assuming that they are much faster than those of the mechanical structure. Moreover, as the MEMS are used as a resonator, we assume that the system operates in such a way that the moveable and fixed plates do not come into contact. Hence, we do not take into account contact dynamics. Nevertheless, for those applications, where the contact may occur, a different model should be considered to capture the dynamics of impacts and the control strategy should be modified adequately [16].In order to facilitate control system design and analysis, it is convenient to work in normalized coordinates and redefine the time variable. For this purpose, we define the new variablesψ=z/d, τ=ω0t, and u~=uA and introduce the following dimensionless constants:
(3)σ=c1mω02,η=c3d2mω02,δ=C0VDC22mω02d2,A=2δ1VDC,μ=bmω0,
where ω0 is the natural frequency defined by
(4)ω0=c1m.
In this way, by assuming that the DC voltage is sufficiently higher than the amplitude of the forcing voltage u, the dimensionless equation of motion (1) may be fairly approximated by
(5)ψ¨+μψ˙+σψ+ηψ3=δ(1(1-ψ)2-1(1+ψ)2)+u~(1-ψ)2.It can be shown that system (5) behaves chaotically by selecting u=VAC⁡sin(ω~t) and VAC⁡A=0.04, ω~=0.5, δ=0.338, μ=0.01, σ=1, and η=12, as indicated in the Lyapunov exponents diagram in Figure 2. The Lyapunov exponents are calculated by using the algorithm proposed in [17]. The existence of one positive Lyapunov exponent confirms the sensitive dependence on initial conditions of the system.Figure 2
Lyapunov exponents for the system (5) with the set of parameters: δ=0.338, A=0.04, ω=0.5, σ=1, μ=0.01, and η=12.System (5) can be written in state space form as
(6)ψ˙1=ψ2,ψ˙2=-μψ˙2-σψ1-ηψ13+f(ψ1)+u~(1-ψ1)2,y=ψ1,
where
(7)f(ψ1)=δ(1(1-ψ1)2-1(1+ψ1)2)
and y is the output.


## 2.2.  Unstable Orbits in the Strange Attractor

The main idea to obtain a sustained oscillation of the MEMS with a low energy consumption is not to create a new orbit different from those already present in the attractor. The orbit could be an unstable periodic orbit embedded in the chaotic attractor, an orbit close to an unstable periodic one, or even part of a chaotic trajectory which has an almost periodic behavior.In order to find one of such orbits, one may measure the phase space dynamics of the system and obtain a set of points from which one can extract the orbit by applying the close return method [18]. Note that other approaches, such as the shooting method [19], can also be used for this purpose. With the aim of simplifying the discussion, we will only consider systems, where the state x=(x1,x2)∈R2 and an external periodic force is applied. Under these conditions, a Poincaré map can be easily constructed from stroboscopically logging the value of the variable x at multiples of the period T of the forcing. Time can be seen as one of the states, and the problem of finding periodic orbits reduces to find fixed points in R2 on the Poincaré surface.The Poincaré map may be defined as follows. Given the differential equation(8)x˙=f(x,t),x∈R2,
with periodic phase space of period T such that
(9)f(x,t)=f(x,t+T),
then the Poincaré map is defined as
(10)P:Σ⟶Σ,x⟼φ(T,x),
where φ(t,·) denotes the flow generated by (8) and Σ is a Poincaré section as shown in Figure 3.Figure 3
Poincaré sectionΣ and Poincaré map P for the periodic vector field defined by (8) and (9).The procedure for finding orbits with almost periodic behaviour is similar to the one of finding periodic orbits. In both cases, the image under the Poincaré map of a point close to the orbit, which intersects the Poincaré surface, is close to the point. Finding such point is the aim of the close return method.For the purpose of applying the method, one records discretely sampled time seriesx(0),x(Ts),…,x((N-1)Ts) containing N points, where Ts is the sample time. Then, for a k given, one takes a point x(kTs) and its iteration m on the map x(kTs+mT) and evaluates the distance d(x(kTs),x(kTs+mT)) between the points, by using, for example, the Euclidean norm. If the distance is less than a small ϵs, then this is a m-period orbit that could be used.In order to show the application of the close return method, we use three different time series from which we extract unstable orbits of different periods. Each of these series comprises a set ofN=120000 points with sample time of Ts=0.01, which corresponds to a logging time of approximately 100 times the period of the forcing term. The orbits were obtained by using ϵs=0.005 and can be seen in Figures 4, 5, and 6. The figures suggest that the bifurcations occurred when the period increases.Unstable orbits found using close return method. The period of the orbit is identified withn.
(a)
(b)
(c)
(d)
(e)
(f)Unstable orbits found using close return method for a second set of data. The period of the orbit isn.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Unstable orbits found using close return method for a third set of data.n identifies the period of the orbit.
(a)
(b)
(c)
(d)
(e)
(f)
(g)In order to enhance the oscillation amplitude of the plates, we chose an unstable orbit surrounding the chaotic attractor, which produces a high amplitude periodic output. This orbit, shown in Figure7, was used as the reference trajectory in the simulation given subsequently. It can be approximated by its truncated Fourier series up to five terms, given by
(11)yr=a0+a1cos⁡(w1t)+b1sin(w1t)+a2cos⁡(2w1t)+b2sin⁡(2w1t)+a3cos⁡⁡(3w1t)+b3sin(3w1t)+a4cos⁡⁡(4w1t)+b4sin⁡(4w1t)+a5cos⁡(5w1t)+b5sin(5w1t),
where
(12)a0=0.0175400,a1=-0.4498000,b1=-0.000563,a2=0.0112600,b2=0.0008906,a3=-0.029140,b3=0.0000376,a4=0.0006330,b4=0.000231,a5=-0.001090,b5=0.0000852.Figure 7
Unstable orbit surrounding the chaotic attractor.By taking the time derivative of (11), we obtain
(13)y˙r=a1′cos⁡(w1t)+b1′sin(w1t)+a2′cos⁡(2w1t)+b2′sin⁡(2w1t)+a3′cos⁡⁡(3w1t)+b3′sin(3w1t)+a4′cos⁡⁡(4w1t)+b4′sin⁡(4w1t)+a5′cos⁡(5w1t)+b5′sin(5w1t),
where
(14)a1′=-0.00052,b1′=0.4163,a2′=0.0016,b2′=-0.0208,a3′=0.00010,b3′=0.0809,a4′=0.00085,b4′=-0.0023,a5′=0.00039,b5′=0.00500.The second derivative ofyr is obtained by taking the derivative of (13).In what follows, we use the following general assumption.Assumption 1.
The reference signalyr(t) and its first two derivatives are known and bounded. In addition, yr(p)(t), ∀p≥3, is piecewise continuous.


## 3. Output-Feedback Control of Chaotic MEMS

The objective of the control is to stabilize the unstable orbit (11) of the chaotic uncertain system (6). This objective can be achieved by forcing the tracking error to tend to zero:
(15)lim⁡t→∞[y(t)-yr(t)]=0.We use adaptive observer backstepping to deal with the parametric uncertainties in the model and also consider more realistic designs where only the output is available for measurement.


### 3.1. Adaptive Observer Backstepping Control

In order to apply a backstepping design procedure to system (6), we transform it into the parametric output-feedback form by using the following change of variables [20]:
(16)x1=ψ1,x2=ψ2+μψ1.System (6) becomes
(17)x˙1=x2-θ1x1,x˙2=-θ2x1-θ3x13+θ4f(x1)+β(y)u,y=x1,
where θ=(θ1θ2θ3θ4)=(μσηδ) is a vector of uncertain parameters and
(18)β(y)=1(1-y)2.Equation (17) can be written in a more compact form
(19)x˙=Ax+ky+bu+φθ,y=cTx,
where x=(x1x2)T is the state vector, y is the output, u is the input, and k=(k1k2)T is a vector with k1,k2∈Z+, defined in such a way that the matrix A, given by
(20)A=[-k11-k20],
is Hurwitz, b=(0β)T, φ is a nonlinear matrix defined by
(21)φ=[-x10000-x1-x13f(x1)]=(φ1φ2φ3φ4),
and c=[10]T.We will reconstruct the state of the system through the use of the filtersξi,i=0,1,…,4. The state is reconstructed as
(22)x=ξ0+θ1ξ1+θ2ξ2+θ3ξ3+θ4ξ4+ϵ,
where ϵ is the error between the state and its estimation. The virtual estimate of x depends on the uncertain parameter θ, but all we need in the design is the vector (ξ0ξ1ξ2ξ3ξ4). In order to guarantee that the error ϵ tends exponentially to be zero, we define the filters as
(23)ξ˙0=[-k11-k20][ξ01ξ02]+[k1k2]y+[0β]u,ξ˙1=[-k11-k20][ξ11ξ12]+[-y0],ξ˙2=[-k11-k20][ξ21ξ22]+[0-y],ξ˙3=[-k11-k20][ξ31ξ32]+[0-y3],ξ˙4=[-k11-k20][ξ41ξ42]+[0f(y)]
or in a compact way as
(24)ξ˙0=Aξ0+ky+bu,ξ˙j=Aξj+φj(y),j=1,2,3,4.From (22) we can obtain the error dynamic equation
(25)ϵ˙=x˙-ξ˙0-θ1ξ˙1-θ2ξ˙2-θ3ξ˙3-θ4ξ˙4,
and by using (19) and (23) we obtain
(26)ϵ˙=(Ax+ky+bu+φθ)-(Aξ0+ky+bu)-θ1(Aξ1+φ1)-θ2(Aξ2+φ2)-θ3(Aξ3+φ3)-θ4(Aξ4+φ4)=A(x-ξ0-θ1ξ1-θ2ξ2-θ3ξ3-θ4ξ4)=Aϵ.
Since A is Hurwitz, ϵ converges to zero exponentially.The adaptive output-feedback controller is designed recursively applying backstepping as follows.Step 1.
We define the first error variable as the tracking error:(27)z1=y-yr,
where yr is the reference signal which satisfies Assumption 1.
Its dynamic is given by(28)z˙1=y˙-y˙r=x˙1-y˙r=x2+θ1(-x1)-y˙r;
since we cannot measure x2, it cannot be used as a virtual control. We replace it with its virtual estimation and the corresponding error:
(29)x2=ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+ϵ2;
hence (28) becomes
(30)z˙1=ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+θ1(-x1)-y˙r+ϵ2.
By choosing ξ02 as the virtual control and defining the second error variable as
(31)z2=ξ02-α1-y˙r
(30) becomes
(32)z˙1=(z2+α1+y˙r)+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+θ1(-x1)-y˙r+ϵ2.
Since θ is an uncertain vector, we must use its first estimated value ϑ1=(ϑ11ϑ12ϑ13ϑ14)T. By defining the first stabilizing function α1 as
(33)α1=-c1z1-d1z1-ϑ1ω,
where ω=(ξ12-x1ξ22ξ32ξ42)T, then (32) becomes
(34)z˙1=-c1z1+z2-d1z1+(θ-ϑ1)ω+ϵ2.
If we use the Lyapunov function candidate
(35)V1=12z12+12γ1(θ-ϑ1)(θ-ϑ1)T+1d1ϵTP0ϵ,
where γ1>0 is the adaptation gain and P0=P0T>0 satisfies P0A+ATP0=-I, then
(36)V˙1=z1z˙1-1γϑ˙1(θ-ϑ1)T-1d1ϵTϵ=z1(-c1z1+z2-d1z1+(θ-ϑ1)ω+ϵ2)-1γ(θ-ϑ1)ϑ˙1T-1d1ϵTϵ=-c1z12+z1z2-d1z12+(θ-ϑ1)(z1ω-1γϑ˙T)+z1ϵ2-1d1ϵTϵ=-c1z12+z1z2-d1(z1-12d1ϵ2)2+14d1ϵ22-1d1ϵTϵ+(θ-ϑ1)(z1ω-1γϑ˙1T)⩽z1z2-c1z12+(θ-ϑ1)(z1ω-1γϑ˙1T)-34d1ϵTϵ.
It is possible to eliminate the term (θ-ϑ1) by the update law
(37)ϑ˙1T=γz1ω.Step 2.
The derivative ofz2 is
(38)z˙2=ξ˙02-α˙1-y¨r=βu+k2(x1-ξ01)-y¨r-α˙1=βu+k2(x1-ξ01)-∂α1∂y×(ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+ϵ2+θ1(-x1))-∂α1∂ξ12(-k2ξ11)-∂α1∂ξ22(-k2ξ21-x1)-∂α1∂ξ32(-k2ξ31-x13)-∂α1∂ξ42(-k2ξ41+f1)-∂α1∂yry˙r-∂α1∂ϑ1(γz1ω)-y¨r.
The control signal is chosen as
(39)βu=-k2(x1-ξ01)+∂α1∂ξ12(-k2ξ11)+∂α1∂ξ22(-k2ξ21-x1)+∂α1∂ξ32(-k2ξ31-x13)+∂α1∂ξ42(-k2ξ41+f1)+∂α1∂yry˙r+∂α1∂ϑ1(γz1ω)+y¨r+∂α1∂y(ξ02+ϑ2ω)-d2(∂α1∂y)2z2-c2z2-z1,
where ϑ2=(ϑ21ϑ22ϑ23ϑ24)T is a second estimate of ϑ. The dynamic of the error z2 is
(40)z˙2=-c2z2-z1-d2(∂α1∂y)2z2-∂α1∂yϵ2-∂α1∂y(θ-ϑ2)ω.
Defining the Lyapunov function
(41)V2=V1+12z22+12γ(θ-ϑ2)(θ-ϑ2)T+1d2ϵTP0ϵ,
its derivative is computed as
(42)V˙2=V˙1+z2z˙2-1γϑ˙2(θ-ϑ2)T-1d2ϵTϵ≤z1z2-c1z12-34d1ϵTϵ-c2z22-z1z2-d2(∂α1∂y)2z22-∂α1∂yϵ2z2-(∂α1∂yωz2+1γϑ˙2)(θ-ϑ2)-1d2ϵTϵ≤-c1z12-c2z22-(34d1+1d2)ϵTϵ-(∂α1∂yωz2+1γϑ˙2)(θ-ϑ2).
To eliminate the term (θ-ϑ2) we use the update law
(43)ϑ˙2=-γ∂α1∂yωz2,
and we finally obtain
(44)V˙2≤-c1z12-c2z22-(34d1+1d2)ϵTϵ.By applying the LaSalle-Yoshizawa theorem [20], we can conclude that z1, z2, ϑ1, and ϑ2 are bounded and z→0 as t→∞. Therefore, the output y(t) tracks yr(t) globally and asymptotically.


## 3.1. Adaptive Observer Backstepping Control

In order to apply a backstepping design procedure to system (6), we transform it into the parametric output-feedback form by using the following change of variables [20]:
(16)x1=ψ1,x2=ψ2+μψ1.System (6) becomes
(17)x˙1=x2-θ1x1,x˙2=-θ2x1-θ3x13+θ4f(x1)+β(y)u,y=x1,
where θ=(θ1θ2θ3θ4)=(μσηδ) is a vector of uncertain parameters and
(18)β(y)=1(1-y)2.Equation (17) can be written in a more compact form
(19)x˙=Ax+ky+bu+φθ,y=cTx,
where x=(x1x2)T is the state vector, y is the output, u is the input, and k=(k1k2)T is a vector with k1,k2∈Z+, defined in such a way that the matrix A, given by
(20)A=[-k11-k20],
is Hurwitz, b=(0β)T, φ is a nonlinear matrix defined by
(21)φ=[-x10000-x1-x13f(x1)]=(φ1φ2φ3φ4),
and c=[10]T.We will reconstruct the state of the system through the use of the filtersξi,i=0,1,…,4. The state is reconstructed as
(22)x=ξ0+θ1ξ1+θ2ξ2+θ3ξ3+θ4ξ4+ϵ,
where ϵ is the error between the state and its estimation. The virtual estimate of x depends on the uncertain parameter θ, but all we need in the design is the vector (ξ0ξ1ξ2ξ3ξ4). In order to guarantee that the error ϵ tends exponentially to be zero, we define the filters as
(23)ξ˙0=[-k11-k20][ξ01ξ02]+[k1k2]y+[0β]u,ξ˙1=[-k11-k20][ξ11ξ12]+[-y0],ξ˙2=[-k11-k20][ξ21ξ22]+[0-y],ξ˙3=[-k11-k20][ξ31ξ32]+[0-y3],ξ˙4=[-k11-k20][ξ41ξ42]+[0f(y)]
or in a compact way as
(24)ξ˙0=Aξ0+ky+bu,ξ˙j=Aξj+φj(y),j=1,2,3,4.From (22) we can obtain the error dynamic equation
(25)ϵ˙=x˙-ξ˙0-θ1ξ˙1-θ2ξ˙2-θ3ξ˙3-θ4ξ˙4,
and by using (19) and (23) we obtain
(26)ϵ˙=(Ax+ky+bu+φθ)-(Aξ0+ky+bu)-θ1(Aξ1+φ1)-θ2(Aξ2+φ2)-θ3(Aξ3+φ3)-θ4(Aξ4+φ4)=A(x-ξ0-θ1ξ1-θ2ξ2-θ3ξ3-θ4ξ4)=Aϵ.
Since A is Hurwitz, ϵ converges to zero exponentially.The adaptive output-feedback controller is designed recursively applying backstepping as follows.Step 1.
We define the first error variable as the tracking error:(27)z1=y-yr,
where yr is the reference signal which satisfies Assumption 1.
Its dynamic is given by(28)z˙1=y˙-y˙r=x˙1-y˙r=x2+θ1(-x1)-y˙r;
since we cannot measure x2, it cannot be used as a virtual control. We replace it with its virtual estimation and the corresponding error:
(29)x2=ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+ϵ2;
hence (28) becomes
(30)z˙1=ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+θ1(-x1)-y˙r+ϵ2.
By choosing ξ02 as the virtual control and defining the second error variable as
(31)z2=ξ02-α1-y˙r
(30) becomes
(32)z˙1=(z2+α1+y˙r)+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+θ1(-x1)-y˙r+ϵ2.
Since θ is an uncertain vector, we must use its first estimated value ϑ1=(ϑ11ϑ12ϑ13ϑ14)T. By defining the first stabilizing function α1 as
(33)α1=-c1z1-d1z1-ϑ1ω,
where ω=(ξ12-x1ξ22ξ32ξ42)T, then (32) becomes
(34)z˙1=-c1z1+z2-d1z1+(θ-ϑ1)ω+ϵ2.
If we use the Lyapunov function candidate
(35)V1=12z12+12γ1(θ-ϑ1)(θ-ϑ1)T+1d1ϵTP0ϵ,
where γ1>0 is the adaptation gain and P0=P0T>0 satisfies P0A+ATP0=-I, then
(36)V˙1=z1z˙1-1γϑ˙1(θ-ϑ1)T-1d1ϵTϵ=z1(-c1z1+z2-d1z1+(θ-ϑ1)ω+ϵ2)-1γ(θ-ϑ1)ϑ˙1T-1d1ϵTϵ=-c1z12+z1z2-d1z12+(θ-ϑ1)(z1ω-1γϑ˙T)+z1ϵ2-1d1ϵTϵ=-c1z12+z1z2-d1(z1-12d1ϵ2)2+14d1ϵ22-1d1ϵTϵ+(θ-ϑ1)(z1ω-1γϑ˙1T)⩽z1z2-c1z12+(θ-ϑ1)(z1ω-1γϑ˙1T)-34d1ϵTϵ.
It is possible to eliminate the term (θ-ϑ1) by the update law
(37)ϑ˙1T=γz1ω.Step 2.
The derivative ofz2 is
(38)z˙2=ξ˙02-α˙1-y¨r=βu+k2(x1-ξ01)-y¨r-α˙1=βu+k2(x1-ξ01)-∂α1∂y×(ξ02+θ1ξ12+θ2ξ22+θ3ξ32+θ4ξ42+ϵ2+θ1(-x1))-∂α1∂ξ12(-k2ξ11)-∂α1∂ξ22(-k2ξ21-x1)-∂α1∂ξ32(-k2ξ31-x13)-∂α1∂ξ42(-k2ξ41+f1)-∂α1∂yry˙r-∂α1∂ϑ1(γz1ω)-y¨r.
The control signal is chosen as
(39)βu=-k2(x1-ξ01)+∂α1∂ξ12(-k2ξ11)+∂α1∂ξ22(-k2ξ21-x1)+∂α1∂ξ32(-k2ξ31-x13)+∂α1∂ξ42(-k2ξ41+f1)+∂α1∂yry˙r+∂α1∂ϑ1(γz1ω)+y¨r+∂α1∂y(ξ02+ϑ2ω)-d2(∂α1∂y)2z2-c2z2-z1,
where ϑ2=(ϑ21ϑ22ϑ23ϑ24)T is a second estimate of ϑ. The dynamic of the error z2 is
(40)z˙2=-c2z2-z1-d2(∂α1∂y)2z2-∂α1∂yϵ2-∂α1∂y(θ-ϑ2)ω.
Defining the Lyapunov function
(41)V2=V1+12z22+12γ(θ-ϑ2)(θ-ϑ2)T+1d2ϵTP0ϵ,
its derivative is computed as
(42)V˙2=V˙1+z2z˙2-1γϑ˙2(θ-ϑ2)T-1d2ϵTϵ≤z1z2-c1z12-34d1ϵTϵ-c2z22-z1z2-d2(∂α1∂y)2z22-∂α1∂yϵ2z2-(∂α1∂yωz2+1γϑ˙2)(θ-ϑ2)-1d2ϵTϵ≤-c1z12-c2z22-(34d1+1d2)ϵTϵ-(∂α1∂yωz2+1γϑ˙2)(θ-ϑ2).
To eliminate the term (θ-ϑ2) we use the update law
(43)ϑ˙2=-γ∂α1∂yωz2,
and we finally obtain
(44)V˙2≤-c1z12-c2z22-(34d1+1d2)ϵTϵ.By applying the LaSalle-Yoshizawa theorem [20], we can conclude that z1, z2, ϑ1, and ϑ2 are bounded and z→0 as t→∞. Therefore, the output y(t) tracks yr(t) globally and asymptotically.


## 4.  Simulation Results

Numerical simulations have been carried out to validate the design made in Section3. Two orbits are stabilized using the control strategy developed in the previous section with the aim of comparing the corresponding control effort.First we will stabilize the unstable orbit (11) of the chaotic system (6), and then we stabilize the orbit:
(45)yr1=a0+1.15a1cos⁡(w1t)+a2cos⁡(2w1t),y˙r1=-1.15a1w1sin(w1t)-2w1a2sin(2w1t),
where
(46)a0=0.01754,a1=-0.4,a2=0.01126.
These orbits have the same fundamental frequency w1, and (45) is a perturbation of (11). The orbits are shown in Figure 8 and their states are compared in Figure 9. Although these two orbits are similar, one is immersed in the chaotic MEMS attractor while the other is not.Figure 8
Unstable orbit immersed in the chaotic attractor and orbit not immersed.Reference signal and its derivative: (a)yr; (b) y˙r.
(a)
(b)Figure10(a) shows the output of the system when the orbit immersed in the chaotic attractor is stabilized by the algorithm developed in Section 3. In Figure 10(b) the output of the system is shown when the nonimmersed orbit is stabilized by using the same algorithm. The agreement between the curves and the effect of the controller can be seen.Simulation results for the controller: (a) output for the immersed orbit; (b) output for the nonimmersed orbit; (c) tracking errors; (d) control signals.
(a)
(b)
(c)
(d)The tracking errors are compared in Figure10(c) from which we can observe a rapid decay as expected. A comparison between the control signals for stabilizing the two orbits is given in Figure 10(d). It clearly shows that, in steady-state, an affordable control effort is required for tracking the orbit immersed in the chaotic attractor, which is significantly smaller than that for the nonimmersed orbit.Finally, the evolution of the estimates of the parameters given by the vectorθ is shown in Figure 11 for the immersed orbit and in Figure 12 for the nonimmersed orbit. We can verify that the estimated values of the parameters are bounded as expected.Estimated parameterϑ for the immersed orbit: (a) ϑ11; (b) ϑ12; (c) ϑ13; (d) ϑ14.
(a)
(b)
(c)
(d)Estimated parameterϑ for the nonimmersed orbit: (a) ϑ21; (b) ϑ22; (c) ϑ23; (d) ϑ24.
(a)
(b)
(c)
(d)


## 5. Conclusions

This work presented a solution for the enhancement of the oscillation amplitude of a MEMS resonator based on the chaotic properties of the system. In order to enhance the amplitude of the oscillation at a reduced control effort, a high amplitude unstable orbit immersed in the chaotic attractor was chosen as the reference trajectory. The method of close returns was used to compute the orbit, which was then approximated by a five-term truncated Fourier series. With the aim of stabilizing this orbit, a control law was designed by using output-feedback and considering parametric uncertainties in the model. It has been shown through adaptive observer backstepping that this control law allows the system output to globally and asymptotically track the desired orbits. Numerical simulations were carried out and the results show that the unstable orbits immersed in the chaotic attractor are stabilized with an affordable control effort compared to those required for stabilizing a nonimmersed orbit. This demonstrated the advantage of the proposed approach.Finally, it is worth noting that the scheme developed in the present work is realistic in the sense that it is based on output-feedback and considers parametric uncertainties in the model. Variables as the speed of some components of the MEMS are very difficult to be measured, and a precise knowledge of the parameters of the system is hard due to the fabrication technology. In the approach proposed, the speed of the plates of the device was estimated by means of a virtual observer at the price of increasing the differential equations needed to construct the controller. Reducing the dynamic equations is beyond the scope of this paper and will be considered in a future work.



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*Source: 745963-2014-09-09.xml*