Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007
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A Novel Oscillation Controller for Vibrational MEMS Gyroscopes1 Lili Dong2, *, Member, IEEE, Qing Zheng2, Student member, IEEE, Zhiqiang Gao2, Member, IEEE
Abstract—This paper introduces a novel oscillation controller for the drive axis of a MEMS gyroscope. Assuming the resonant frequencies of the drive and sense axes are matched and no external control input for sense axis, we employ the controller to drive the drive axis to resonance, to regulate the output amplitude of drive axis to a desired value, and to compensate for the mechanical-thermal noise and Quadrature error term. The controller is simply a combination of a traditional PD controller and a linear Extended State Observer (ESO). Since the controller design does not require exact information of system parameters, it is very robust against structural uncertainties of the drive axis. We simulate the controller on a vibrational piezoelectric beam gyroscope. The effectiveness of the controller is verified by the simulation results and performance analysis.
I. INTRODUCTION
M
EMS (Micro-electro-mechanical Systems) gyroscope is a silicon-based inertial rate sensor. Because it is compact and cost effective with low power consumption, it has been developed and applied in GPS assisted navigation, consumer electronics, automotives and so on [1]. Almost all reported MEMS gyroscopes [2, 3] are vibrational. They are capable of vibrating along two orthogonal axes. The working principle of the MEMS gyroscopes is based on the energy transfer from one axis (referred to as drive axis) to another (referred to as sense axis) caused by Coriolis acceleration, which is arising from a rotating reference frame and proportional to its rotation rate. Thus we can determine the rotation rate through sensing the vibration of the sense axis. Since the Coriolis acceleration is also proportional to the output amplitude of drive axis, the primary task of the controller for the MEMS gyroscope is to assure constantamplitude oscillation. In order to obtain a large response, another task of controller is to force the drive axis to vibrate at resonance. Ideally the parameters such as natural frequency, the damping coefficient, and the mass of the MEMS gyroscope are fixed, the vibrations along two axes are mechanically uncoupled, the natural frequencies of two axes are matched, and the gyroscope is only sensitive to rotation rate but not noise. However, in reality, the fabrication imperfection and
1 The work is supported by Ohio ICE (Instrumental, Control, and Electronic) consortium under grant 0260-0652-10. 2 The authors are with the Department of Electrical and Computer Engineering at Cleveland State University, Cleveland, OH 44115.
*Corresponding author. Email:
[email protected].
1-4244-0989-6/07/$25.00 ©2007 IEEE.
environmental variations cause parameter changes, mechanical couplings between two axes, and mechanical thermal noises along the axes which will degrade the sensitivity of the gyroscope. Since the early 1990s, more and more researchers have been trying to improve the performance of the gyroscopes through control electronics. Both open-loop and closed-loop controls can be used to control the oscillation of the drive axis [4]. However, when the gyroscope is driven in an open loop, any changes in system parameters will cause errors in the oscillation amplitude, leading to a measurement error of rotation rate. Thus most recently reported gyroscopes use feedback control to achieve the desired oscillation of drive axis [4-10]. In [5, 6], a typical Phase Locked Loop (PLL) is used to adjust the input frequency till the output of drive axis is -900 out of phase with the input indicating the resonance, and an Automatic Gain Control (AGC) loop is used to regulate the output amplitude. In this system, the input frequency is dependent on the mechanics of device and changing with environmental variations. In [7], an adaptive controller is used to tune the closed-loop frequency of drive axis to a given frequency while the amplitude regulation is disregarded. In [8], an adaptive oscillation controller is introduced without requiring an external driving input. However, the controller assumed an ideal model of drive axis and did not consider the coupling terms between both axes. In [9], an adaptive controller is designed to drive the vibrations of two axes of the gyroscope operating in an adaptive mode where the movements of the mass along two axes are equal. However, most reported MEMS gyroscopes are operating in conventional mode [10] where the movement of the mass along the drive axis is relatively large and the movement along the sense axis is very small. In [11], an adaptive control system is designed for measuring timevarying rotation rates of MEMS gyroscopes where the mechanical thermal noise and parameter variations are neglected. All of the controllers introduced in current literature [4-11] are based on precisely known mathematical models without structural uncertainties. In this paper, we aim to design an oscillation controller for the drive axis of the conventional MEMS gyroscope where the coupling terms, the noise, and the parameter variations are considered. We will apply a recently reported Active Disturbance Rejection Controller (ADRC) [12] to the drive axis. The ADRC is simply a combination of a linear observer and a PD controller. The disturbance represents any discrepancies between the mathematical model and a real system, hence is also taken as uncertainties of the system. The controller is built on the observed disturbance and compensates for it in real time. As reported in [12, 13, 14], ADRC has been successfully applied to mechanical control
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systems such as motion control, aircraft flight control, and jet engine control etc because of its robustness against parameter variations and its few numbers of tuning parameters during implementation. In this paper, we will extend its use to MEMS gyroscope. This paper is organized as follows. The dynamics of MEMS gyroscope is described in Section II. The oscillation controller is presented in Section III. Simulation results are shown in Section IV. It is followed by the performance and robustness analysis. Finally some concluding remarks are given in Section VI.
sensor scale factors, and ud is the control input for drive axis. The Quadrature errors are constant unknown signals, which need to be canceled out in control effort ud. The rotation rate Ω is assumed to be a constant unknown signal. In (1), mechanical thermal noise on the sense axis is represented by the random force N(t). The drive axis displacement x(t) is usually sufficiently large that the effects of thermal noise on the drive axis are negligible and are ignored [15]. The concept of mechanical thermal noise is originally from the thermal noise (Johnson noise) of a resistor in an electrical circuit. As introduced in [15-18], the Power Spectral Density (PSD) of the thermal noise in a resistor can be represented by
II. THE DYNAMICS OF MEMS GYROSCOPE
Sn(f)=4KBTR N2s,
A MEMS gyroscope can be understood as a silicon proof mass attached to a rigid reference frame through springs and dampers. The mass-spring-damper structure of a MEMS gyroscope is shown as Fig.1. The proof mass is driven into resonance along the drive axis (X axis) in order to obtain the largest response and phase synchronization. As the rigid frame rotates about rotation axis (Z axis) with the rate of Ω, a Coriolis acceleration is produced along sense axis (Y axis), which is perpendicular to both drive and rotation axes. The Coriolis acceleration provides the information of rotation rate. So we can determine the rotation rate through sensing the vibration of sense axis.
(2) -23
-1
where KB is a Boltzman constant (KB=1.38×10 JK ), T is the absolute temperature of the resistor, and R is resistance. In real systems, any dissipative process that is coupled to a thermal reservoir results in thermal noise. In the mechanical model of the MEMS gyroscope shown in Fig. 1, the damper is equivalent to the resistor since both of them are energy dissipative elements. Thus the resistance is equal to 2ςωnm. The mechanical-thermal noises resulted from dampers are caused by viscoelastic effects in the suspension of the gyroscope [9]. Because the PSD is constant over all frequencies, the mechanical thermal noise is considered as white noise. It is represented by (0, Sn) as in [9]. In this paper, we suppose the sense axis is working under the open-loop operation. Our control objective is to force the drive axis to oscillate at specified amplitude and resonant frequency in the presence of parameter variations, mechanical coupling, and mechanical-thermal noise. III. OSCILLATION CONTROLLER Both drive and sense axes of MEMS gyroscopes can be taken as lightly damped second-order systems. They are governed by the Newtonian law of motion. Then we could rewrite the drive axis model as x = f ( x , x, d ) + bud
Ω
Fig. 1. Mass-spring-damper structure of vibrational MEMS gyroscopes Assuming the natural frequencies of both axes are same, the vibrational MEMS gyroscope is modeled as k ud m 1 y + 2ςωn y + ωn2 y + ω xy x + 2Ωx = N (t ) m
x + 2ςωn x + ωn2 x + ω xy y − 2Ωy =
where b is the coefficient of the controller (b=K/m), d represents an extraneous unknown input force (known as external disturbance in its traditional sense) [12], and f accounts for all the other forces excluding the control effort ud [12]. It is f = −2ςωn x − ωn2 x − ω xy y + 2Ωy .
(1)
where x and y are the displacement outputs of drive and sense axes respectively, 2Ωx and 2Ω y are Coriolis accelerations, Ω is the rotation rate, ωn is the natural frequency of drive and sense axes, ω xy y and ωxy x are Quadrature errors caused
(3)
(4)
If an observer is designed to estimate the f, we can take ud as
ud =
1 ˆ (− f ( x , x, d ) + u0 ), b
(5)
where fˆ is the estimated f, u0 is a controller to be determined. Then (3) becomes
by spring coupling terms between two axes, ς is damping coefficient typically, m is the mass of the MEMS gyroscope, K is the parameter including forward gain and actuator and 3205
x = f ( x , x, d ) − fˆ ( x , x, d ) + u0 ≈ u0 .
(6)
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We suppose a desired signal r has a resonant frequency (ω) and the maximum amplitude (A) which the drive axis could output. And r is represented by r = A sin(ωt ).
u0 = k p e + k d e.
(8)
If we take the partially unknown f as a generalized disturbance or the discrepancy between the real system and its nominal model, the controller will estimate it and compensate for it actively, hence the name ADRC. The effectiveness of ADRC is dependent on the accurate estimation of the f. Consequently an Extended State Observer (ESO) is developed to estimate the disturbance f in real time. This can be achieved by using the linear state space representation of drive axis model and augmenting the state variables to include f [12]. We suppose f is unknown and bounded. Let x1=x, x2= x , x3=f, and X=[x1, x2, x3] T, we have X = AX + Bud + Eh
(9)
z = CX
where ⎡0 1 0⎤ ⎡0 ⎤ ⎡0⎤ ⎢ ⎥ ⎢ ⎥ A = ⎢ 0 0 1 ⎥ , B = ⎢b ⎥ , C = [1 0 0], E = ⎢⎢ 0 ⎥⎥ , h = f . ⎢⎣ 0 0 0 ⎥⎦ ⎢⎣0 ⎥⎦ ⎢⎣1 ⎥⎦
Based on (9), a state observer is given by
u0 = ωc2 e + 2ωc e.
(14)
Since the velocity of the movement along drive axis is not measurable, the controller u0 built on the observed velocity is shown as
u 0 = ω c2 (r − xˆ1 ) + 2ω c (r − xˆ 2 )
(15)
Assuming accurate estimation of the states, the ideal closedloop transfer function of controller is G( s) =
ωc2 + 2ωc s x = . r s 2 + 2ωc s + ωc2
(16)
From (16), we can see that this is a typically second-order critical-damping control system, where ωc is the only one tuning parameter for the control input ud. The details about how to tune the parameters of ADRC are introduced in [13]. The ESO (represented by (10)) and the ADRC (given by (13) and (14)) constitute the oscillation controller for the drive axis of the MEMS gyroscope. Under the oscillation control, the output of drive axis x is expected to be driven to the reference r which is a desired sinusoidal signal with constant amplitude and resonant frequency. Since the disturbance f could be estimated in ESO, the parameter variations, the Quadrature error, and the mechanical-thermal noise included in f will be compensated in the control effort as shown in (5) and (13). Another advantage of this oscillation controller is the few number (only two totally) of tuning parameters (ωc and ωo), making it easy to implement in real system. For tuning simplicity, we let ωo=5ωc as introduced in [13].
(10)
where the estimated state vector is Xˆ = [ xˆ 1 , xˆ 2 , xˆ 3 ] , and the vector of observer gain is L= [α1, α2, α3]T. We need to notice that the key part of (10) is the third state of observer xˆ3 , which is used to approximate f. The characteristic polynomial of observer is represented by p( s ) = s3 + α1s 2 + α 2 s + α 3 .
Then (8) becomes
(7)
Then our control goal is to drive the output signal x to the signal r. We have tracking error e=r-x. We can employ a common Proportional Derivative (PD) controller for u0 to drive tracking error e to zero. The controller can be represented by
Xˆ = AXˆ + Bu d + L( z − zˆ ) zˆ = CXˆ
controler u0, the controller parameters are chosen as Kp=ωc2 and Kd=2ωc, where ωc>0.
IV. SIMULATION RESULTS We simulated the oscillation controller on the model of a piezoelectrically-driven vibrational beam gyroscope [7], as shown in the figure below.
(11)
If the observer gains are selected as α1 = 3ωo, α2 = 3 ωo2 , α 3 = ωo3 , and ωo>0, the characteristic polynomial becomes p( s ) = ( s + ωo )3 .
(12)
Therefore we can change the observer gains through tuning the unique parameter ω0, which is the bandwidth of observer. After the observer is designed, the oscillation control of drive axis given by (6) becomes 1 u d = ( − xˆ3 + u 0 ). b
(13)
In order to minimize the number of tuning parameters of the
Fig. 2. Photograph of a vibratory beam gyroscope [7] The natural frequency and damping coefficient of the gyroscope are ωn=63881.1rad/s and ς=0.0005. The desired frequency of drive axis is ω=65973.4 rad/s (f=10.17KHz),
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which is approximately equal to natural frequency. Normally the desired output amplitude of drive axis is A=10-6 m. We use A=1 in simulation units to represent this. So the reference signal r = sin (65973.4t). The PSD of mechanical-thermal noise is 4.22×10-26 N2sec. We assume the magnitude of ωxy is 0.1% of natural frequency according to [9]. We suppose the rotation rate is constant and Ω = 0.1rad / s. The coefficient b=K/m=271780942.56. For the ADRC and ESO, we choose a controller gain ωc as 5×104 and an observer gain ωo as 25×104. The oscillation controller is represented by 2ω ω 1 u d = − xˆ 3 + (r − x) + c (r − xˆ 2 ) b b b 2 c
Tracking error e 0.6 0.4 0.2 0 -0.2
0
0.2
0.4
0.6
0.8 1 1.2 Time (seconds) (a)
1.4
1.4 1.5 1.6 Time (seconds) (b)
1.7
1.6
1.8
2 -3
x 10
0.01 0.005
(17)
0
ωc2
are divided by coefficient b, reducing the where ωc and gains of controller. Fig. 3 (a) shows the real output x of drive axis and the reference signal r in the first 2ms. Fig. 3 (b) shows the output x and reference signal in one period which is around 0.1ms as we desired. From the figure, we can see that after initial deviation, the output x can track the reference signal very well. Fig. 4 shows the tracking error between the reference signal r and the real output of drive axis x in different time ranges. The stabilized peak error is around 0.7% of real amplitude. Fig. 5 shows the stabilized tracking error as there is 28% variation of ωn and the magnitude of ωxy is 10% of natural frequency, which is 100 times original magnitude of Quadrature error term. The stabilized peak error is around 1.7% of real amplitude. The error is only a bit bigger than the one without variations. Fig.6 shows the output of drive axis and the tracking error as damping coefficient is 10 times original one, i.e. ς=0.005. The stabilized peak error of tracking error is around 1.7% of the real amplitude of x. As we choose the rotation rate as a timevarying sine signal, and Ω = 0.1sin(31.4t )rad / s, the simulation result of output x is same as Fig. 3. Our simulation results show the robustness of the oscillation controller against parameter variations, mechanical noise, and Quadrature error term.
-0.005 -0.01
1
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1.3
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2 -3
x 10
Fig. 4. The tracking error e without parameter variations Tracking error e 0.6 0.4 0.2 0 -0.2 -0.4
0
0.2
0.4
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0.8 1 1.2 Time (seconds) (a)
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1.4 1.5 1.6 Time (seconds) (b)
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2 -3
x 10
0.02 0.01 0 -0.01 -0.02
1
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2 -3
x 10
Fig. 5. The tracking error e with the variations of natural frequency and Quadrature error magnitude Drive axisoutout output Drive axis
The real output of drive axis and reference signal 1
1
0
0
-1
-1 0
0.1
0.2
0.3
0.4 0.5 0.6 Time (seconds) (a)
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2 -3
-3
x 10
Tracking error
x 10
(s)
0.02 0.01
1 0.5
0
0
-0.01
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-0.02
1
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1.2
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2 -3
x 10
-1 0.035
0.035
0.035 0.035 0.035 0.0351 0.0351 0.0351 0.0351 Time (seconds) (b)
(s)
Fig. 6. The output of drive axis and stabilized tracking error e with the variation of damping coefficient
Fig. 3. The output of drive axis and the reference signal without parameter variations
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The transfer function of the closed-loop system is
V. PERFORMANCE AND ROBUSTNESS ANALYSIS In this section, the steady-state performance of the closedloop control system is discussed. In addition, the robustness of ADRC against parameter variations is analyzed through frequency response [19]. The transfer function between the control input ud and the output x in (1) can be given by Gp (s) =
Gcl ( s ) =
The Laplace transform of ESO given by (10) is: sXˆ ( s) = ( A − LC ) Xˆ ( s) + LZ ( s) + BU d ( s).
(19)
The Laplace transform of the controller represented by (13) and (14) is: U d ( s) =
⎡ R(s) ⎤ 1 1 ⎡ k p kd ⎤⎦ ⎢ ⎥ − ⎡⎣ k p b⎣ ⎣ sR(s)⎦ b
kd
(20)
ˆ 1⎤⎦ X(s)
Replacing the Ud(s) in (19) with (20), we have s 3 + α1s 2 + α 2 s + α 3
U d (s) =
1
3
s + (α1 + kd ) s 2 + (α 2 + k p + kd α1 ) s b −
1 + Gc ( s )G p ( s )
(25)
.
As in (7), we have the reference signal r = A sin(ωt ) . The steady state output xss of the drive axis is
s2 + 2ςωn s + ωn2 X (s) k = 4 3 2 2 Ud (s) m s + 4ςωn s + (4Ω + 2ωn2 + 4ς 2ωn2 )s2 + 4ςωn3s + ωn4 − ωxy
(18)
G f ( s)Gc ( s )G p ( s )
xss = A Gcl ( jω ) sin (ωt + φ ) , Where the phase shift φ = ∠Gcl ( jω ) = tan −1
(26) Im [Gcl ( jω ) ]
. Re [Gcl ( jω ) ] Fig.8 shows the steady state magnitude error and the phase shift between the output and reference signal will converge to zero with the increase of the controller bandwidth ωc . As ωc
is 5 × 105 rad / sec as chosen in this paper, the steady-state magnitude error is about 1% of the reference magnitude and the phase shift is −0.006rad . Fig. 9 shows the frequency responses of the open-loop system in (24) with the variance of natural frequency ωn .
( k p + kd s ) R(s)
2 1 (k pα1 + kd α 2 + α 3 ) s + ( k pα 2 + kd α 3 ) s + k pα 3 Z ( s ). b s 3 + (α1 + kd ) s 2 + (α 2 + k p + kd α1 ) s
(21) Let Gc ( s ) = G f ( s) =
2 1 (k pα1 + kd α 2 + α 3 ) s + (k pα 2 + kd α 3 ) s + k pα 3 , b s 3 + (α1 + kd ) s 2 + (α 2 + k p + kd α1 ) s
( s 3 + α1s 2 + α 2 s + α 3 )(k p + kd s )
. (k pα1 + kd α 2 + α 3 ) s 2 + (k pα 2 + kd α 3 ) s + k pα 3 (22)
The equation (21) can be rewritten as
Fig. 8. The steady state magnitude error and phase shift
U d ( s ) = G f ( s)Gc ( s ) R( s ) − Gc ( s) Z ( s).
(23) 150
The block diagram of the closed-loop system is shown as below.
G f ( s)
Gc ( s)
ud
G p ( s)
wn2 wn wn1
100 Magnitude (dB)
r
The frequency responses of the open-loop system with different ωn
x
50
wn1=2wn wn=63881.1 wn2=0.5wn
0 -50 -100
Fig. 7. The block diagram of the closed-loop system This is a unit feedback system with a pre-filter G f ( s ) and a controller Gc ( s) . The transfer function of the open-loop system is Go ( s) = Gc ( s)G p ( s).
(24)
Phase (deg)
-150 -45 -90
wn2
-135
wn wn1
-180 -225 -270 3 10
4
10
5
10
6
7
10 10 Frequency (rad/sec)
8
10
10
9
Fig. 9. Frequency responses of the open-loop system with different natural frequency ωn 3208
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The stability margins with different ωn are shown in Table I. It can be observed that the stability margins keep unchanged as the natural frequency is ranging from 0.5ωn through 2 ωn. The frequency responses of the open-loop system with different damping ratio ς are given in Fig.9. In the figure, the frequency responses overlap with three different damping ratios. Fig.9 and Fig.10 show the strong robustness of ADRC against the variations of system parameters. TABLE I
mechanical thermal noise, and Quadrature errors, therefore, verified the effectiveness of the oscillation controller. The performance and robustness analysis further demonstrated the stability of the oscillation controller with the chosen controller and observer gains. It also confirmed the robustness of the controller against parameter variations. The implementation of the oscillation controller is currently being conducted on the vibrational beam gyroscope shown in Fig.2. The future experimental results are expected to match the simulation results of the paper.
STABILITY MARGINS WITH DIFFERENT ωn (rad / sec)
ωn = 63881.1 ωn1 = 2ωn ωn 2 = 0.5ωn
Gain Margin (dB) 14.2 14.2
Phase Margin (deg) 41.2 41.2
14.2
41.2
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[4]
The frequency responses of the open-loop system with different damping ration ζ 150 ζ 1=4ζ
Magnitude (dB)
100
[5]
ζ =0.0005 ζ 2=0.25ζ
50 0
[6]
-50 -100
[7]
-150 -45
Phase (deg)
-90
[8]
-135 -180
[9]
-225 -270 3 10
10
4
5
10
6
7
10 10 Frequency (rad/sec)
8
10
[10]
9
10
Fig. 10. The frequency responses of the open-loop system with different damping ratio ς
[11]
[12]
VI. CONCLUDING REMARKS We developed a control strategy known as ADRC for controlling the oscillation of the drive axis of a MEMS gyroscope. The controller is novel in two aspects. One is that the ADRC generalizes the notion of disturbance to represent any discrepancies between the mathematical description of the gyroscope and the real gyroscope, and can actively compensate for it in real time. The other is that we can design the bandwidth of the MEMS gyroscope through tuning the controller and observer bandwidths which are the only two parameters to be adjusted in the control system. The twoparameter tuning feature also makes ADRC very practical and easy to implement in the real world situation where the small size of the MEMS gyroscope makes it impossible to accommodate a complexity of tuning parameters. The oscillation controller was successfully simulated on a vibrational beam gyroscope model. The simulation results showed an accurate tracking of drive axis output to the desired signal with resonant frequency and specified amplitude in the presence of parameter variations,
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