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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The time-average method currently available is limited to analyzing the specific performance of the automatic gain control-proportional and integral (AGC-PI) based velocity-controlled closed-loop in a micro-electro-mechanical systems (MEMS) vibratory gyroscope, since it is hard to solve nonlinear functions in the time domain when the control loop reaches to 3rd order. In this paper, we propose a linearization design approach to overcome this limitation by establishing a 3rd order linear model of the control loop and transferring the analysis to the frequency domain. Order reduction is applied on the built linear model's transfer function by constructing a zero-pole doublet, and therefore mathematical expression of each control loop's performance specification is obtained. Then an optimization methodology is summarized, which reveals that a robust, stable and swift control loop can be achieved by carefully selecting the system parameters following a priority order. Closed-loop drive circuits are designed and implemented using 0.35 μm complementary metal oxide semiconductor (CMOS) process, and experiments carried out on a gyroscope prototype verify the optimization methodology that an optimized stability of the control loop can be achieved by constructing the zero-pole doublet, and disturbance rejection capability (D.R.C) of the control loop can be improved by increasing the integral term.

Due to their wide applications in inertial navigation, automotive stability control and robots, MEMS gyroscopes have drawn tremendous attention of researchers in both academia and industry [

The time-average method is widely applied in the stability analysis of control loops [

To overcome these two drawbacks of the time-average method mentioned above, a linearization design approach is presented in this paper. A fully linear system model of the 3rd order closed-loop is established and thereafter the stability and performance are analyzed in the frequency domain. Compared to the conventional time-average method, the proposed linearization design approach has the following advantages: first, the stability criterion of the closed-loop can be simply obtained by applying a zero-pole method on the linear model; second, the mathematical expression of each performance specification is obtained by applying order reduction on the 3rd order linear model, so that the dilemma in [

This paper is organized as follows: Section 2 presents the overall linearization design approach, including building the linear model, revealing the stability criterion, analyzing the control loop's performance through mathematical expressions and numerical simulation results, and summarizing an optimization methodology. In Section 3, the implemented AGC-PI based closed-loop drive circuits are introduced briefly and detailed experimental results on a gyroscope prototype are presented to verify the proposed optimization methodology. Finally, a conclusion is given in Section 4.

The topology of the AGC-PI based velocity-controlled closed-loop for the drive mode of MEMS gyroscope is shown in _{ref}_{ref}_{error}

The kinetic equation of the drive mode in gyroscope is described as:
_{x}_{x}_{ext}

The velocity-controlled closed-loop is a nonlinear system, which processes the amplitude information of the sinusoidal resonance signal. It is hard to analyze the behavior of a nonlinear system, especially when the system usually reaches a 3rd order. Fortunately, a 3rd order linear model of the closed-loop can be built by linearizing two nonlinear modules in the loop, as shown in _{C/X}_{F/V}^{2}_{V/C}_{VGA}_{rect}

The nonlinearity of the closed-loop generates arises from two modules: the primary resonator and the VGA. As the phase perturbation term is negligible compared with _{x}_{ext_amp}_{amp}_{ext}_{in_amp}_{ref}/K_{rect}

The control loop is a negative feedback system. As shown in _{ref}_{in}_{out}_{dc}_{lpf}_{P}_{I}

The built linear system model shown in

Specifications including loop gain, bandwidth and phase margin are commonly used to evaluate the performances of a negative feedback system in the frequency domain. Mapping these specifications to the provided control loop, their influence on the control loop's performances is illustrated as follows. The loop gain determines the disturbance rejection capability of the control loop against the parameter variance in both mechanical structures and circuits. The bandwidth determines the recovery speed of the control loop and the phase margin reflects the stability of the control loop. These specifications correspond with transient response specifications in the time domain in [

The expressions of the bandwidth and phase margin can be calculated by solving a third-order equation generated from _{I}_{P}_{x}/(2Q)

According to

The order of the consideration on the performance specification optimization is usually loop gain > phase margin > bandwidth. The reason can be illustrated as follows: because the mechanical parameter variations are dominant in the control loop and most parameters of the primary resonator, like quality factor

Numerical simulation results of both the 3rd order control loop linear model and corresponding nonlinear model are provided in the following paragraphs for three reasons: first, it is intended to reveal the validity of the order-reduction; second, it is used to verify the derived results in _{I}

The parameters of the control loop used in the simulations are listed in _{p}_{mid}_{p}_{mid}_{dc}_{x}

The numerical simulations are carried out in Matlab. The simulation results of the transfer function in _{P}_{lpf}_{I}

The validity of the order-reduction is proved in _{P}_{P}_{x}_{P}

_{I}_{I}

The step responses of the nonlinear model in _{ref}_{rect}

Combining the theoretical and simulation results with the specification optimization order presented above, an optimization methodology of system parameters selection can be summarized as follows: to achieve a robust, stable and swift control loop, it is better to increase _{I}_{VGA}_{P}

To verify the proposed optimization method on a practical gyroscope prototype, AGC-PI based closed-loop drive circuits with flexible system parameters adjustment are designed. The block diagram is presented in _{P}_{I}

The drive circuits are implemented in a 0.35 μm CMOS technology. The supply voltage is 5 V. A symmetrical doubly decoupled z-axis gyroscope is tested, which adopts an electrostatical drive and capacitive sense. The gyroscope is vacuum-packaged and mounted on a printed circuit board (PCB) with the implemented drive circuits chip and other necessary external resistances and capacitances, as shown in

To verify the proposed optimization methodology accurately, the velocity-controlled closed-loop should be tested working as a linear system during the experiments. Carefully investigating the linear model, the input of the VGA module is approximated as the final steady value _{ref}

In our experiments, a 10 mV step-up signal is applied on the target voltage _{ref}_{P}_{I}_{P}_{I}_{P}_{P}_{P}_{P}_{I}_{I}_{I}

Disturbance rejection capability (D.R.C.) is an important performance specification of the control loop, which is inversely proportional to the loop gain of the negative feedback control loop. To evaluate the D.R.C. of the control loop against system parameter variation, experiments are carried out by varying the bias voltage _{p}_{dc}_{V/C}_{p}_{p}_{dc}_{V/C}_{Dr_amp}_{TIA_amp}_{p}

Both the magnitude of the drive signal and the TIA's output are recorded in _{I}_{p}_{p}_{I}_{I}_{I}_{p}

This paper presents a linearization design approach of the 3rd order velocity-controlled closed-loop for MEMS vibration gyroscope. As benefits of the built linear model, tedious stability analysis in the time domain can be transferred to the frequency domain and the design process of the control loop is simplified. The stability criterion and the mathematical expression of each performance specification of the 3rd order control loop are presented in this paper. Moreover, an optimization methodology which reveals that a robust, stable and swift control loop can be achieved by carefully selecting the system parameters following a priority order is summarized. Experiments carried out on a gyroscope prototype verify the optimization methodology that an optimized stability of the control loop can be achieved by canceling the pole of the resonator with the zero in the PI controller, and disturbance rejection capability (D.R.C) of the control loop can be improved by increasing the integral term _{I}

The authors gratefully acknowledge National Natural Science Foundation of China (project 61106025) and the CAS/SAFEA International Partnership Program for Creative Research Teams for financial support.

The authors declare no conflicts of interest.

^{2}ASIC for a 1000 deg/s 2-Axis Capacitive Micro-Gyroscope

AGC-PI based velocity-controlled closed-loop for the drive mode of MEMS vibratory gyroscope.

3rd order linear system model of the velocity-controlled closed-loop.

Zero-pole map of the 3rd order velocity-controlled closed-loop.

(_{P}

(_{lpf}

(_{I}

(

(_{VGA}

Block diagram of the implemented AGC-PI based closed-loop drive circuits.

Test board of the gyroscope prototype.

Measured step-up waveforms of the LPF's output with different proportional terms _{P}

Measured step-up waveforms of the LPF's output with different the integral term _{I}

Normalized magnitude of drive signal and TIA's output with different _{I}_{p}

Performance specifications of the control loop versus primary system parameters.

_{P} |
↑ | ↑ | ↓ |

_{I} |
↑ | – | – |

_{lpf} |
→ | ↑ | ↑ |

↑ | – | – | |

_{total} |
↑ | ↑ | ↓ |

↑ Increase; ↓ Decrease; → no change; – unknown.

Parameters of the control loop simulated in Matlab.

_{F/V}^{2}(N^{2}) |
3.7e-8 | _{x}(rad |
8063 × 2π |

_{C/X}(F |
1.2e-8 | 29.6 | |

_{V/C}(V |
2.4e12 | 1,368 | |

_{VGA}( |
20 | _{lpf}(rad |
100 × 2π |

_{ref}(V) |
0.2 | _{P} |
5 |

_{p}(V) |
12 | _{I} |
200 |

_{mid}(V) |
2.5 | 1 |

Comparison of simulation and measurement rise times with different proportional term _{P}

_{P} |
_{P} |
_{P} |
_{P} | |
---|---|---|---|---|

Simulated rise time with linear model (ms) | 16.4 | 12.8 | 7.4 | 3.7 |

Simulated rise time with nonlinear model (ms) | 12.0 | 8.6 | 5.1 | 2.8 |

Tested rise time (ms) | 12.4 | 8.9 | 5.3 | 3.0 |

Comparison of simulation and measurement rise times with different proportional term _{I}

_{I} |
_{I} |
_{I} |
_{I} | |
---|---|---|---|---|

Simulated rise time with linear model (ms) | 7.3 | 9.7 | 12.8 | 17.0 |

Simulated rise time with nonlinear model (ms) | 5.7 | 7.1 | 8.6 | 10.2 |

Tested rise time (ms) | 5.2 | 7.0 | 8.9 | 10.4 |

Measured D.R.C of the velocity-controlled closed-loop with different _{I}_{p}

_{p}(V) |
||||||||||
---|---|---|---|---|---|---|---|---|---|---|

K_{I} = 100, K_{P} = 5 |
419.8 | 372.0 | 332.8 | 299.8 | 272.6 | 249.6 | 229.8 | 63.4 | 36.2 | |

60.4 | 60.5 | 60.6 | 60.7 | 60.8 | 60.9 | 61.0 | 0.99 | |||

K_{I} = 200, K_{P} = 5 |
416.0 | 368.0 | 329.0 | 296.8 | 269.6 | 246.2 | 227.0 | 63.7 | 42.1 | |

59.8 | 59.9 | 59.9 | 60.0 | 60.0 | 60.1 | 60.1 | 0.5 | |||

K_{I} = 400, K_{P} = 5 |
416.6 | 369.0 | 330.4 | 297.4 | 270.4 | 247.6 | 227.8 | 63.5 | 45.7 | |

59.9 | 60.0 | 60.0 | 60.0 | 60.1 | 60.1 | 60.1 | 0.33 |

V.P. Variation Percentage (set _{p}