System Dynamics and Adaptive Control for MEMS Gyroscope Sensor

Abstract

This paper presents an adaptive control approach for Micro-Electro-Mechanical Systems (MEMS) z-axis gyroscope sensor. The dynamical model of MEMS gyroscope sensor is derived and adaptive state tracking control for MEMS gyroscope is developed. The proposed adaptive control approaches can estimate the angular velocity and the damping and stiffness coefficients including the coupling terms due to the fabrication imperfection. The stability of the closed-loop systems is established with the proposed adaptive control strategy. Numerical simulation is investigated to verify the effectiveness of the proposed control scheme.

1. Introduction

Gyroscopes are commonly used sensors for measuring angular velocity in many areas of applications such as navigation, homing, and control stabilization. Vibratory gyroscopes are the devices that transfer energy from one axis to other axis through Coriolis forces. The conventional mode of operation drives one of the modes of the gyroscope into a known oscillatory motion and then detects the Coriolis acceleration coupling along the sense mode of vibration, which is orthogonal to the driven mode. The response of the sense mode provides information about the applied angular velocity. Fabrication imperfections result in some cross stiffness and cross damping effects that may hinder the measurement of angular velocity of MEMS gyroscope. Therefore the angular velocity measurement and minimization of the cross coupling between two axes are challenging problems in gyroscopes that need to be solved using advanced control methods.

Adaptive control is an effective approach to handle parameter variations. In the presence of model uncertainties and external disturbances, sliding mode control is necessary to be incorporated into the adaptive control to improve the robust performance of control system. Sliding mode control is a robust control technique which has many attractive features such as robustness to parameter variations and insensitivity to disturbances. Adaptive sliding mode control has the advantages of combining the robustness of variable structure methods with the tracking capability of adaptive control. In the last few years, many applications have been developed using sliding mode control and adaptive control. Utkin [1] showed that variable structure control is insensitive to parameters perturbations and external disturbances. Ioannou and Sun [2] described the model reference adaptive control. Chou et al. [3] proposed an integral sliding surface and derived an adaptive law to estimate the upper bound of uncertainties. Some control algorithms have been proposed to control the MEMS gyroscope. Batur et.al. [4] developed a sliding mode control for a MEMS gyroscope system. Leland [5] presented an adaptive force balanced controller for tuning the natural frequency of the drive axis of a vibratory gyroscope. Novel robust adaptive controllers are proposed in [6–7] to control the vibration of MEMS gyroscope. Sun et al. [8] developed a phase-domain design approach to study the mode-matched control of MEMS vibratory gyroscope. Antonelli et al. [9] used extremum-seeking control to automatically match the vibration mode in MEMS vibrating gyroscopes. Feng et al. [10] presented an adaptive estimator-based technique to estimate the angular motion by providing the Coriolis force as the input to the adaptive estimator and to improve the bandwidth of microgyroscope. Tasi et al. [11] proposed integrated model reference adaptive control and time-varying angular rate estimation algorithm for micro-machined gyroscopes. Raman et al. [12] developed a closed-loop digitally controlled MEMS gyroscope using unconstrained sigma-delta force balanced feedback control. Some robust adaptive controllers for vibratory gyroscope system have been proposed in [13–14]. Park et al. [15] presented an adaptive controller for a MEMS gyroscope which drives both axes of vibration and controls the entire operation of the gyroscope. In this paper, the proposed adaptive control is different from the adaptive controller [15] in that an addition controller is incorporated into the state feedback controller to give more freedom in designing the adaptive controller. Therefore the error dynamics is determined by the reference model dynamics and addition control.

This paper investigates the adaptive control approach to identify the angular velocity of MEMS gyroscope using state tracking controllers. The contribution of this paper is that novel adaptive control is proposed to control the MEMS gyroscope and to estimate the angular velocity and all unknown gyroscope parameters. The paper is organized as follows. In section II, the dynamics of MEMS gyroscope is described. The adaptive controller is derived in section III. Simulation results are presented in section IV. The Conclusion is provided in section V.

2. Dynamics of Mems Vibrational Gyroscope

The inertial frame and rotating frame are shown in Fig. 1. The typical MEMS vibratory gyroscope includes a proof mass suspended by springs, an electrostatic actuation and sensing mechanisms for forcing an oscillatory motion and sensing the position and velocity of the proof mass.

Fig. 1.

Inertial frame and rotating frame

In the Fig. 1, r _A is position vector of an arbitrary point A as measured against the axes of the inertial frame. r _A/B is an arbitrary point A relative to the origin of the rotating axes, r _B is position vector of the origin of the rotating frame relative to the origin of the inertial frame B. Their relation can be expressed as

$$\begin{array}{rl}{\mathit{r}}_A& \hspace{0.17em}={\mathit{r}}_B+{\mathit{r}}_{{}^A/{}_B}\\ & \hspace{0.17em}={\mathit{r}}_B+xi+yj+zk.\end{array}$$

(1)

The velocity vector of an arbitrary point A as measured against the axes of the inertial frame can be derived as

$${\mathit{v}}_A={\mathit{v}}_B+\frac{dx}{dt}i+\frac{dy}{dt}j+\frac{dz}{dt}k+x\frac{di}{dt}+y\frac{dj}{dt}+z\frac{dz}{dt}.$$

(2)

Since di/dt = ω _z × i, dj/dt = Ω _z xj, dk/dt = Ω _z × k, substituting these properties into (2) yields

$$\begin{array}{rl}{\mathit{v}}_A& \hspace{0.17em}={\mathit{v}}_B+\frac{dx}{dt}i+\frac{dy}{dt}j+\frac{dz}{dt}k+x({\mathrm{\Omega }}_z\times i)+y({\mathrm{\Omega }}_z\times j)+k({\mathrm{\Omega }}_z\times k)\\ & \hspace{0.17em}={\mathit{v}}_B+v_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B}\end{array}$$

(3)

where v _A and v _B are the velocities with respect to the inertial frame, v _A/B = dx/dt i + dy/dt j + dz/dt k is the relative velocity vector of an arbitrary point A as measured against the axes of the rotating system.

The velocity of A relative to B is therefore made up of two terms - the velocity measured against the rotating axes and a component that results from the rotation of the axes, and is thus invisible to the observer in the rotating frame.

The property d(Ω _z × r _A/B )/dt = dΩ _z /dt × r _A/B + Ω _z × v _A/B + ω _z × Ω _z × r _A/B can be proved as

$$\begin{array}{rl}& \frac{d({\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B})}{dt}=\frac{d{\mathrm{\Omega }}_z}{dt}\times {\mathit{r}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times ({\mathit{v}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B})\\ & \hspace{0.17em}=\frac{d{\mathrm{\Omega }}_z}{dt}\times {\mathit{r}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times {\mathit{v}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times {\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B}.\end{array}$$

(4)

The property d v _A/B /dt = a _A/B + Ω _z × v _A/B can be proved as

$$\begin{array}{rl}& \frac{d{v}_{{}^A/{}_B}}{dt}=\frac{d(\frac{dx}{dt}i+\frac{dy}{dt}j+\frac{dz}{dt}k)}{dt}\\ & \hspace{0.17em}=\frac{d^{2}x}{dt}i+\frac{d^{2}y}{dt}j+\frac{d^{2}z}{dt}k+\frac{dx}{dt}\cdot \frac{di}{dt}+\frac{dy}{dt}\cdot \frac{dj}{dt}+\frac{dz}{dt}\cdot \frac{dk}{dt}\\ & \hspace{0.17em}=\frac{d^{2}x}{dt}i+\frac{d^{2}y}{dt}j+\frac{d^{2}z}{dt}k+\frac{dx}{dt}\cdot ({\mathrm{\Omega }}_z\times i)+\frac{dy}{dt}\cdot ({\mathrm{\Omega }}_z\times j)+\frac{dz}{dt}\cdot ({\mathrm{\Omega }}_z\times k)\\ & \hspace{0.17em}=\frac{d^{2}x}{dt}i+\frac{d^{2}y}{dt}j+\frac{d^{2}z}{dt}k+{\mathrm{\Omega }}_z\times \frac{dx}{td}i+{\mathrm{\Omega }}_z\times \frac{dy}{td}j+{\mathrm{\Omega }}_z\times \frac{dz}{td}k\\ & \hspace{0.17em}={\mathit{a}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times {\mathit{v}}_{{}^A/{}_B}.\end{array}$$

(5)

Differentiating (3) and using (4) and (5) yields

$${\mathit{a}}_A={\mathit{a}}_B+{\mathit{a}}_{{}^A/{}_B}+2{\mathrm{\Omega }}_z\times {\mathit{v}}_{{}^A/{}_B}+\frac{d{\mathrm{\Omega }}_z}{dt}\times {\mathit{r}}_{{}^A/{}_B}+{\mathrm{\Omega }}_z\times ({\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B})$$

(6)

where a _A and a _B are the accelerations with respect to the inertial frame, a _A/B = d²x/dt² i + d²y/dt² j + d²z/dt² k is the acceleration vector of an arbitrary point A as measured against the axes of the rotating system.

The acceleration of A relative to B is therefore made up of four terms - the acceleration measured against the rotating axes and three components that result from the rotation of the axes, and are thus invisible to the observer in the rotating frame. These are the Euler (tangential), Centripetal and Coriolis accelerations respectively.

Multiplying (6) by mass m gives

$$m{a}_A=m{a}_B+{ma}_{{}^A/{}_B}+2m{\mathrm{\Omega }}_z\times {\mathit{v}}_{{}^A/{}_B}+m(\frac{d{\mathrm{\Omega }}_z}{dt}\times {\mathit{r}}_{{}^A/{}_B})+m{\mathrm{\Omega }}_z\times ({\mathrm{\Omega }}_z\times {\mathit{r}}_{{}^A/{}_B})$$

(7)

where 2mΩ _z × v _A/B is the Coriolis force and mΩ _z × (Ω _z × r _A/B ) is the Centrifugal force.

The Coriolis force acting on the proof mass along x direction is derived as

$$F_{coriolis-x}=2m{\mathrm{\Omega }}_{z}k\times \dot{yj}=2m{\mathrm{\Omega }}_z\dot{yj}$$

(8)

The Coriolis force acting on the proof mass along y direction is derived as

$$F_{coriolis-y}=2m{\mathrm{\Omega }}_{z}k\times \dot{xi}=2m{\mathrm{\Omega }}_z\dot{xj}$$

(9)

By using the property of k × (k × i) = -i, the Centripetal force acting on the proof mass along x direction can be derived as

$$F_{centripetal-x}=m{\mathrm{\Omega }}_{z}k\times ({\mathrm{\Omega }}_{z}k\times xi)=-m{{\mathrm{\Omega }}_z}^{2}xi$$

(10)

.

By using the property of k × (k × j) = -j, the Centripetal forces acting on the proof mass along y direction can be derived as

$$F_{centripetal-y}=m{\mathrm{\Omega }}_{z}k\times ({\mathrm{\Omega }}_{z}k\times yj)=-m{{\mathrm{\Omega }}_z}^{2}yj.$$

(11)

We assume that the table where the proof mass is mounted is moving with a constant velocity; the gyroscope is rotating at a constant angular velocity Ω _z over a sufficiently long time interval; the centripetal forces mΩ _z ²x and mΩ _z ²y are assumed to be negligible; gyroscope undergoes rotation about the z axis only, and thereby Coriolis force is generated in a direction perpendicular to the drive and rotational axes. A z-axis MEMS gyroscope is depicted in Fig. 2.

Fig. 2.

A simple model of a MEMS Z axis gyroscope

With the assumptions the dynamics of gyroscope become

$$m\ddot{x}+d_{xx}\dot{x}+d_{xy}\dot{y}+k_{xx}x+k_{xy}y=u_x+2m{\mathrm{\Omega }}_z\dot{y}$$

(12)

$$m\ddot{y}+d_{xy}\dot{y}+d_{yy}\dot{y}+k_{xy}x+k_{yy}y=u_y-2m{\mathrm{\Omega }}_z\dot{x}$$

(13)

Fabrication imperfections contribute mainly to the symmetric spring and damping terms, k _xy and d _xy .

The x and y axes spring and damping terms k _xx , k _yy , d _xx and d _yy are mostly known, but have small unknown variations from their nominal values. The mass of proof mass can be determined very accurately, and u _x , u _y are the control forces in the x and y direction.

Dividing (12) and (13) by the reference mass and rewriting the gyroscope dynamics in vector forms result in

$$\ddot{q}+\frac{D}{m}q+\frac{k_a}{m}q=\frac{u}{m}-2\mathrm{\Omega }\dot{q}$$

(14)

where $\mathit{q}=\left[\begin{array}{c}x\\ y\end{array}\right],\mathit{u}\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],\mathbf{\Omega }=\left[\begin{array}{cc}0& -{\mathrm{\Omega }}_z\\ {\mathrm{\Omega }}_z& 0\end{array}\right],\mathit{D}=\left[\begin{array}{cc}{d}_{xx}& d_{xy}\\ d_{xy}& d_{yy}\end{array}\right],{\mathit{K}}_a=\left[\begin{array}{cc}{k}_{xx}& k_{xy}\\ k_{xy}& k_{yy}\end{array}\right]$ .

Using non-dimensional time t* = w₀t, and dividing both sides of (14) by w₀² and reference length q₀ give the final form of the non-dimensional equation of motion as

$$\frac{\ddot{q}}{q_0}+\frac{D}{mw0}\frac{\dot{q}}{q_0}+\frac{k}{m{w}_0^2}\frac{q}{q0}=\frac{u}{m{w}_0^2{q}_0}-2\frac{\mathrm{\Omega }}{w_0}\frac{\dot{q}}{q_0}.$$

(15)

Defining a set of new parameters as follows:

$$q^{\ast }=\frac{q}{q_0},D^{\ast }=\frac{D}{m{w}_0},{{\mathrm{\Omega }}_z}^{\ast }=\frac{{\mathrm{\Omega }}_z}{w_0},$$

(16)

$$\begin{gathered} u^{\ast }=\frac{u}{{m{w}_0}^2{q}_0},w_x=\sqrt{\frac{k_{xx}}{{m{w}_0}^2}},w_y=\sqrt{\frac{k_{yy}}{{m{w}_0}^2}}, \\ {w}_{xy}=\frac{k_{xy}}{{m{w}_0}^2}. \end{gathered}$$

(17)

Ignoring the superscript (*) for notational clarity, the nondimensional representation of (12) and (13) is

$$\ddot{q}+D\dot{q}+k_{b}q=u-2\mathrm{\Omega }\dot{q}$$

(18)

where $K_b=\left[\begin{array}{cc}{w_x}^2& w_{xy}\\ w_{xy}& {w_y}^2\end{array}\right].$ .

3. Adaptive Control Design

In this section an adaptive controller is proposed to identify the angular velocity and all unknown gyroscope parameters. The block diagram is shown in Fig. 3. En the adaptive control design, we consider the equation (9) as the system model.

Fig. 3.

Block diagram of adaptive control for MEMS gyroscope

Rewriting the gyroscope model (18) in state space form as

$$\dot{X}-AX+Bu$$

(19)

Where

$$\begin{array}{rl}\mathit{A}& \hspace{0.17em}=\left[\begin{array}{cccc}0& 01& 0& 0\\ -{w_x}^2& -d_{xx}& -w_{xy}& -(d_{xy}-2{\mathrm{\Omega }}_z)\\ 0& 0& 0& 1\\ -w_{xy}& -(d_{xy}+2\mathrm{\Omega }z)& -{w_y}^2& -d_{yy}\end{array}\right]\\ & \mathit{B}={\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]}^T,\mathit{u}=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],\mathit{X}=\left[\begin{array}{c}x\\ \dot{x}\\ y\\ \dot{y}\end{array}\right].\end{array}$$

(20)

The reference model x _m = A₁ sin(w₁t), y _m = A₂ sin(w₂t) is defined as

$${\ddot{q}}_m+K_m{q}_m=0$$

(21)

where K _m = diag{w₁² w₂²}.

Similar to (19), the reference model can be written as

$${\dot{X}}_m\left[\begin{array}{cccc}0& 1& 0& 0\\ -{w_1}^2& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -{w_2}^2& 0\end{array}\right]X_m\equiv A_m{X}_m$$

(22)

where A _m is a known constant matrix.

We make the following assumptions.

Assumption. There exists a constant matrix K * such that the following matching condition A + BK * ^T = A _m can always be satisfied.

The control target for MEMS gyroscope is (i) to design an adaptive controller so that the trajectory of X can track the state of reference model X _m ; (ii) to estimate the angular velocity of MEMS gyroscope and all unknown gyroscope parameters.

The tracking error and its derivative are

$$e(t)=X(t)-X_m(t)$$

(23)

$$\dot{e}=A_{m}e+(A-A_m)X+Bu.$$

(24)

The adaptive controller is proposed as

$$u(t)=K^T(t)X(t)+K_{f}e$$

(25)

where K (t) is an estimate of K *, the constant matrix K _f satisfies the condition that ( A _m + BK _f ) is Hurwitz.

We define the estimation error as $\tilde{\mathit{K}}(t)=\mathit{K}(t)-{\mathit{K}}^{\ast }$ and substitute this estimation error and (25) into (19)

yields

$$\dot{X}(t)=A_{m}X(t)+B{\tilde{K}}^T(t)X(t)$$

(26)

Then, we have the tracking error equation

$$\dot{e}(t)=(A_m+B{K}_f)e+B{\tilde{K}}^T(t)X(t)$$

(27)

Define a Lyapunov function

$$V=\frac{1}{2}{e}^{T}Pe+\frac{1}{2}tr[\tilde{K}{M}^{-1}{\tilde{K}}^T]$$

(28)

where P and M = diag{m₁ m₂} are positive definite matrix.

Differentiating V with respect to time yields

$$\begin{array}{rl}& \dot{V}=e^{T}Qe+tr\left[\tilde{K}{M}^{-1}{\dot{\stackrel{~}{K}}}^T\right]\\ & \hspace{0.17em}=-e^{T}Qe+e^{T}PB{\tilde{K}}^{T}X+tr\left[\tilde{K}{M}^{-1}{\dot{\stackrel{~}{K}}}^T\right]\end{array}$$

(29)

where P(A _m + BK _f ) + (A _m + BK _f ) ^T P = -Q, Q is positive definite matrix.

To make $\dot{V}\le 0$ , we choose the adaptive law as

$${\dot{\stackrel{~}{K}}}^T(t)={\dot{K}}^T(t)=-M{B}^T{P}^{T}e{X}^T(t)$$

(30)

with K(0) being arbitrary. This adaptive law yields

$$\dot{V}=-e^{T}Qe\le -{\lambda }_{min}(Q)\parallel e\parallel \le 0$$

(31)

The inequality $\dot{V}=\le -{\lambda }_{min}(Q)\parallel e\parallel$ implies that e is integrable as ${\int }_0^t\parallel \mathit{e}\parallel dt\le \frac{1}{{\lambda }_{min}(\mathit{Q})}[\mathit{V}(0)-\mathit{V}(t)]$ . Since V (0) is bounded and V (t) is nonincreasing and bounded, it can be concluded that $\underset{t\to \mathrm{\infty }}{lim}{\int }_0^t\parallel \mathit{e}\parallel dt$ is bounded. Since $\underset{t\to \mathrm{\infty }}{lim}{\int }_0^t\parallel \mathit{e}\parallel dt$ is bounded and $\dot{\mathit{e}}$ is also bounded, according to Barbalat's lemma, e will asymptotically converge to zero, $\underset{t\to \mathrm{\infty }}{lim}\mathit{e}(t)=0$ .

It can be shown that if f the persistent excitation can be satisfied, ie. w₁ ≠ w₂ controller parameter converges to its true values, $\tilde{\mathit{K}}\to 0$ . In other words, excitation of proof mass should be persistently exciting. Since $\tilde{\mathit{K}}\to 0$ , then the unknown angular velocity as well as all other unknown parameters can be determined from A + BK ^T = A _m and we obtain Ω _z = 0.25(k₂₂ - k₄₁).

Conclusion: if persistently exciting drive signals, x _m = A₁ sin(w₁t) and y _m = A₂ sin(w₂t) are used, then $\tilde{\mathit{K}}$ and e (t) all converge to zero asymptotically. Consequently the unknown angular velocity can be determined as lim_t→∞ Ω _z (t) = Ω _z . However it is difficult to establish the convergence rate.

4. Simulation Example

We will evaluate the proposed adaptive control on the lumped MEMS gyroscope model [7] using MATLAB/SIMULINK. The control objective is to design adaptive state tracking controller so that a consistent estimate of Ω _z can be obtained.

In the simulation, we allowed ± 5% parameter variations for the spring and damping coefficients with respect to their nominal values. We further assumed ± 5% magnitude changes in the coupling terms i.e. d _xy and ω _xy , again with respect to their nominal values. The external disturbance is a random variable signal with zero mean and unity variance. Parameters of the MEMS gyroscope are as follows: m = 0.57e − 8 kg, d _xx = 0.429e − 6 N s/m, d _xy = 0.0429e − 6 N s/m, d _yy = 0.687e − 36 N s/m, k _xx = 80.98 N/m, k _xy = 5 N/m, k _yy = 71.62 N/m, w₀ = 1kHz, q₀ = 10⁻⁶m.

The unknown angular velocity is assumed Ω _z = 5.0 rad/s and the initial condition on K matrix is K (0) = 0.95 K *. The desired motion trajectories are x _m = sin(w₁t) and y _m = 1.2sin(w₂t), where w₁ = 6.17kHz and w₂ = 5.11kHz. The adaptive gain of (34) is M = diag{20 20}. The K _f in (29) is chosen as ${\mathit{K}}_f=\left[\begin{array}{cccc}-10000& -10000& 1000& 20000\\ -1000& -1000& -1000& -1000\end{array}\right]$ . The adaptive gain of (41) is M = diag{20 20}

Fig. 4 depicts the tracking errors. It is observed that the tracking errors converge to zero asymptotically. Figs. 5 and 6 draw the adaptation of the angular velocity and controller parameters. It is shown that the estimates of angular velocity and controller parameters converge to their values. Fig.7 plots the control input using adaptive control.

Fig. 4.

The tracking error using adaptive control

The estimate of angular velocity using adaptive control has larger overshoot at the beginning but much smaller rise time. The model uncertainties and external disturbances are difficult to compensate in the adaptive controller, because there is no disturbance term in the derivation whereas the disturbance term can be dealt with well in the adaptive sliding mode control, therefore, adaptive sliding mode control is better than adaptive control in the presence of model uncertainties and external disturbance.

Simulations demonstrate that with the control laws (29), and the parameter adaptation laws (34), if the gyroscope is controlled to follow the mode-unmatched reference model, the persistent excitation condition is satisfied, i.e. w₁ ≠ w₂, and all unknown gyroscope parameters, including the angular velocity converge to their true values, and tracking error is going to zero asymptotically as time go on.

Fig. 5.

Adaptation of angular velocity using adaptive control

Fig. 6.

Adaptation of control parameters using adaptive control

Fig. 7.

Adaptive control input

5. Conclusion

This paper investigates the design of adaptive control for MEMS gyroscope. The dynamics model of the MEMS gyroscope is developed and nondimensionized. Novel adaptive approach is proposed and stability condition is established. Simulation results demonstrate that the effectiveness of the proposed adaptive control techniques in identifying the gyroscope parameters and angular velocity.