Key Engineering Materials Vols. 326-328 (2006) pp 249-252 online at http://www.scientific.net © (2006) Trans Tech Publications, Switzerland Online available since 2006/Dec/01
Design and performance test of digital rebalance loop for MEMS gyroscope Byung Su Chang1, a, Jang Gyu Lee1, b, Taesam Kang2,c 1
School of Electrical Engineering, Seoul National University, Sillim-dong, Gwanak-gu, Seoul, 151-742, Korea
2
Department of Aerospace Engineering, Konkuk University, Hwayang-dong, Gwangjin-gu, Seoul, 143-701, Korea a
[email protected],
[email protected],
[email protected]
Keywords: Gyroscope, MEMS, digital rebalance loop, DSP
Abstract. In this paper, a digital rebalance loop for MEMS gyroscope is designed and its performance test is performed. First, the system model of MEMS gyroscope is established by dynamic analysis. Then, the digital rebalance loop is designed using modern control technique. The performance of the digital rebalance loop is compared with that of conventional PID rebalance loop. Through frequency response analysis using MATLAB and experiments using a real MEMS gyroscope and digital controller, which is realized using digital signal processor (DSP), it is confirmed that the controller improves the performance of the gyroscope. Introduction A gyroscope is a basic inertial sensor, which can measure an external angular rate. The MEMS gyroscope is an inertial angular rate sensor fabricated using MEMS technology. When an external angular rate is applied to the MEMS gyroscope, the proof mass vibrating at resonant frequency is forced to vibrate in orthogonal direction due to the Coriolis force. The angular rate can be estimated by measuring the amplitude of the orthogonal oscillation. However, such operation, with open loop operation, has small bandwidth and narrow dynamic range. Furthermore, the system nonlinearity becomes larger as the amplitude of the orthogonal oscillation does. To overcome these disadvantages, a closed loop controller named rebalance loop can be used. The rebalance loop is a kind of feedback controller that keeps the orthogonal oscillation small. The magnitude of the feedback signal is proportional to the Coriolis force. Therefore, the control input is used for the estimation of external angular rate inputs. Furthermore, the bandwidth of the gyroscope can be made large by using suitable compensator [1, 3]. The rebalance loops are classified into two categories according to their torquing method. One is an analog rebalance loop that uses an analog torquing method, and the other is a digital rebalance loop that uses a digital torquing method. The former is simple in controller structure and relatively easy to achieve wide bandwidth. However, it is composed of analog circuits, which makes it difficult to realize complex control system designed using modern multivariable control theory such as H∞ controller. On the contrary, the latter is somewhat complex in its controller structure, but it can easily accommodate complex multivariable controller by just computer programming. Furthermore, an additional A/D converter that digitalizes the gyroscope output is not necessary since the gyroscope output is modulated in PWM (Pulse Width Modulation) form [4]. In the following, the principle of operation of the MEMS gyroscope is introduced, and the model of the gyroscope is explained. Next, the multivariable H∞ controller is designed and analyzed. Finally, designed controller is implemented using electronic components and DSP.
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Principle of Operation An operation of micro gyroscope is based on the motion of vibrating oscillation. The proof mass is driven along the driving axis (x-axis) at the resonant frequency of the driving mode as shown in Fig. 1. When an angular rate input along z-axis is applied to the sensor, the oscillation of the mass along the sensing axis (y-axis) is induced due to the Coriolis force, which is modulated by the oscillation along the driving axis. The angular rate can be estimated by measuring the amplitude of the orthogonal oscillation.
Fig. 1 Simple model of MEMS vibrating gyroscope Model Equations of System The governing equations of the vibratory gyroscope can be expressed as follows. The equation of the micro gyroscope can be simply expressed as a second order mass-spring-damper system. The dynamic model equation of sensing axis is given as Eq. 1: G (s) =
SΩF , Ms + C y s + K y
(1)
2
where M is the mass of the moving plate, Cy, the damping coefficient of the air damper and Ky the spring constant along the sensing axis. SΩF is scale factor between angular rate and Coriolis force. In this study, the rebalance loop for MEMS gyroscope is designed using multivariable H∞ control theory. To use H∞ control technique, Eq. 1 should be transformed into two-port system model as shown in Fig. 2. The two-port realization including scale factor models is given by Eq. 2: x = [ x1
x2 ] = [ x
z = [ z1
z2 ] ,
T
T
x& ]
T
x& = Ap x + Bp1 w + B p 2 u z = C p1 x + D p11 w + D p12 u y = C p 2 x + D p 21 w + D p 22 u
z1 = WΩ x1 , z2 = Wu u
y = Pg x1 w = [d
n]
Ap G p = C p1 C p 2
T
B p1 D p11 D p 21
0 −K y Bp 2 M D p12 = WΩ D p 22 0 Pg
1
0
0
−C y M 0
1 S ΩF M 0
0
0
0
0
0
R
0 0
1 SVT M . 0 Wu 0 0
(2)
Key Engineering Materials Vols. 326-328
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The x, z, u, w, and y denote the state vector, controlled output vector, control input vector, exogenous input vector (i.e., applied angular rates and measurement noises), and measurement output vector (i.e., displacement) respectively. SVT is a scale factor from control voltage to torquing force and Pg is a pick-off gain, that is, a scale factor from the displacement to sensing voltage. WΩ, Wu, and R denote the weighting factor for output voltage, sensing voltage, and noise covariance respectively.
Fig. 2 Block diagram of two-port feedback system Controller Design In this section, the H based controller for micro gyroscope is explained. The H∞ control method is the prevailing tool for the MIMO control system design. The H∞ method is usually known as robust to model uncertainties and external noises. Since MEMS gyroscope model is inaccurate due to the manufacturing error, the H∞ methodology is chosen for the design of the rebalance loop of them. The H control problem is to find a controller K(s) that minimizes H norm of the transfer matrix Tzw from w to z [2]. The H norm of Tzw is defined as Eq. 3. In addition, the controller can be enhanced using frequency weighting method that rejects noises in out-of-band frequency range. Tzw
∞
= sup σ (Tzw ( jω ) )
(3)
ω
Using the two-port system model expressed by Eq. 2, the H controller is designed. Fig. 3 shows a bode plots of closed loop feedback systems using the PID and the designed H controller. It shows that the H controller has wider bandwidth and rejects the high frequency noises better than the PID controller. B o d e D ia g ra m s 50
H -in f c o n t ro l P ID c o n t ro l
-50 -100 -150 200
P ID c o n t ro l
0 To: Y(1)
Phase (deg); Magnitude (dB)
0
-200 -400
H -in f c o n t ro l
-600 -800
104
105
F re q u e n c y (ra d / s e c )
Fig. 3 Bode plot of closed loop feedback system
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Experimental Mechanics in Nano and Biotechnology
Implementation of Digital Rebalance Loop The digital rebalance loop is composed of displacement sensing circuit and DSP. We choose TI TMS320F2812 DSP, which is high-performance, control-optimized and peripheral-included (i.e., embedded A/D converter and PWM event manager). Therefore, additional A/D converters and PWM signal generation parts are not needed. The digital rebalance loop is the torquing method, whose torquing signal has PWM form. Existing technique that generates PWM signal is the method that compares analog controller output with triangular wave signal. In this case, instability of triangular wave signal can lead to the noise. To overcome this disadvantage, The PWM signal is directly generated from DSP. It is possible because DSP has PWM event manager. Because signal generation is controlled by internal timer in DSP, we can obtain more accurate PWM signal. Experimental results To verify the performance of the designed controller, experiments are accomplished. Especially, its performance is compared with analog PID controller. Fig. 4 shows the results of experiment. From the result (a), the H∞ controller has better step response. Moreover, it has lower rms noise as shown in (b), (c) of Fig.4. 1.4
1.2
PID control
Angular rate (deg/sec)
1 H-inf control 0.8
0.6
0.4
0.2
0
0
0.005
0.01
0.015
0.02 0.025 0.03 Time (sec)
0.035
0.04
0.045
0.05
(a) Step response (b) PID (c) H∞ Fig. 4 Experimental results of closed loop MEMS gyroscope Summary In this paper, a digital rebalance loop for a MEMS gyroscope is proposed. The H∞ method is used to design the controller, which is a more systematic method than classical. In addition, the high-performance, control-optimized, and peripheral-included DSP is used to construct the designed rebalance loop. The simulation and experimental results demonstrate that the H∞ based digital rebalance loop has low noise and fast step responses than a conventional PID controller. References [1] W.T. Sung, J.G. Lee, J.W. Song and T. Kang: H∞ Controller design of MEMS gyroscope and its performance test, Proc. IEEE PLANS 2004 (2004), p. 63-69. [2] K. Zhou, J. C. Doyle, and K. Glover: Robust and Optimal Control, Prentice Hall, New Jersey (1996) [3] J. W. Song: State Weighted Model Reduction Based on Frequency Weighted Gramians and Its Application to Micro Inertial Sensors, PH. D. Thesis, Seoul National University (2002). [4] B. S. Chang, J. G. Lee and C. G. Park: Digital Rebalance Loop Design for Dynamically Tuned Gyroscope Using Digital Signal Processor, KSAS-JSASS Joint Symposium on Aerospace Engineering (2004), p.341-344.
