Nonlinear dynamics of MEMS resonator in PLL-AGC self-oscillation loop

The work is devoted to the study of a MEMS resonator dynamics under the action of phase-locked and automatic gain control loops. Particular attention is directed to the study of the nonlinearity factor of the resonator elastic restoring force. It was found that the determination of control system parameters based on the stability analysis of the operating resonant mode, in the general case, does not provide the required phase adjustment and stabilization of the oscillation amplitude. Stable multifrequency modes of oscillations are found, and an analytical study of the mechanisms of their appearance and evolution is carried out under variation of the key parameters of the system. The real regions of the control system stable operation are determined (which do not coincide, as was found, with the regions of stability of the operating resonant mode, due to the presence of hidden attractors in the phase space of the system). A methodology has been developed for identifying such areas of stable operation. A significant complication of the structure of possible motions in the system with an increase in the Q-factor of the resonator is revealed.


Introduction
High requirements for the stability of the operation of resonant sensors (in particular, the sensitive elements of microelectromechanical vibration gyroscopes) lead to the need to use feedback control systems that ensure the constancy of the oscillation amplitude of the resonator and its frequency tuning to resonance. The first of these tasks, as a rule, is solved using the automatic gain control (AGC) system [1][2][3][4]. To solve the second problem, phase-locked loop (PLL) systems [5][6][7] are widely used.
Mathematical modeling of the dynamics of MEMS resonators considering operation of closed-loop control systems is of significant theoretical and applied interest.
A large number of scientific works are devoted to various aspects of this problem. In [8], an analytical study of stationary oscillatory regimes for a mechanically linear resonator was carried out taking into account the action of the AGC system. In the same work, the dynamics of a system of two weakly coupled oscillators in a control loop is considered. In [9], the dynamics of the resonator under the action of the PLL is studied. The region of stability of the required resonant oscillation mode in the parameter space of the control system is found. Calculations of the resonator dynamics are performed by a direct numerical method, taking into account the nonlinearity of the elastic restoring force. A qualitative study of the dynamics of a mechanically linear resonator with the combined action of the AGC and PLL systems was carried out in [10]. This work also considers the issues of tuning the natural frequency of the resonator, the generation of primary oscillations for a two-mass vibration gyroscope and the adjustment of the frequencies of its primary and secondary oscillations. The mathematical apparatus of these works is applied methods of nonlinear oscillations theory (in most cases, the averaging method).
A separate area of research is the study of essentially nonlinear effects characteristic to the considered control algorithms and controlled systems in a whole. In a number of works, the nonlinear dynamics and bifurcations of the operating modes of the PLL circuit [11,12] are studied. The existence of hidden attractors in the dynamics of the PLL is revealed. The difficulties associated with this in the numerical simulation of the processes of interest are noted and overcome [13]. There are also known works devoted to the study of the nonlinear dynamics of MEMS resonators in control loops that implement the amplitude and phase stabilization of oscillations [14][15][16]. The relevance of the latter direction is especially significant in the context of the tasks of designing architectures and ensuring stable dynamic modes of operation of high-quality sensitive elements of micromechanical inertial navigation systems (vibration gyroscopes, resonant modal-localized accelerometers), where the stability of reaching the required amplitudes of stationary oscillations in the presence of inevitable mechanical and other nonlinearities directly affects the accuracy of the sensor. In recent work [17], a novel control strategy for a micro-electro-mechanical gyroscope with a drive mode excited through parametric resonance was presented using PLL and AGC loops. The analysis of the drive mode control loops was conducted using the multiple scales method, and the robustness of the suggested control loops to parameters perturbation was demonstrated.
Despite active research in the field of nonlinear dynamics of controlled MEMS devices, many qualitative issues related to the complexity of possible modes of operation of MEMS elements in the closed control loops, their resistance to variations in parameters and external disturbances in harsh environments, energy and time efficiency of the used control algorithms remain insufficiently developed, being at the same time critical from the point of view of the practical implementation of such systems in real-world applications.
The present paper is devoted to a qualitative (parametric) study of the nonlinear dynamics of a MEMS resonator with a nonlinear restoring force under the combined action of the AGC and PLL systems. The novelty of the work consists in the construction and analytical study of the properties of a high-order dynamic system in slow variables, obtained from the original model of the "mechanical resonator-AGC-PLL" system using the averaging method. The use of numerical methods of the continuation theory made it possible to identify and explain the nature of the appearance of fundamentally different modes of resonator oscillations, which are not predicted by the analysis of the linearized equations of motion. The issues of stability of stationary oscillatory modes, their evolution and branching are considered depending on the key mechanical parameters of the resonator and parameters of the control system.

Mathematical model
Let us investigate the dynamics of a micromechanical resonator under the action of a control system designed for resonant frequency tuning and stabilization of the vibration amplitude. Tuning the frequency of the electrostatic comb-drive to resonance is provided by a phase-locked loop system. Stabilization of the amplitude of the primary oscillations is achieved by the operation of the automatic gain control loop.
The PLL circuit contains a phase detector, a low-pass filter (LPF), a controller and a voltage-controlled oscillator (VCO) [7,18]. The circuit is a closed-loop control system whose task is to provide the required phase difference between the output signal of the capacitance-voltage converter of the resonator displacement sensor and the VCO reference signal.
The amplitude channel, which implements the AGC circuit, consists of an amplitude detector and a PI controller.
The mathematical model of the resonator in the PLL-AGC system is described by the following system of equations [9,10]: where x-displacement of the resonator, ω n -natural frequency of the resonator, c = 2ω n ξ = ω n /Q-dissipation parameter (ξ -coefficient of relative attenuation, Q-quality factor of the resonator), β-coefficient for the cubic term in the restoring force, θ -phase of the VCO (ω =θ -instantaneous VCO frequency), z-VCO control signal, y -output signal of the phase detector, ω 0 -natural (free) frequency of the VCO, K V C O -proportional gain of the VCO, K P L L I -integral gain of the controller, K G , λ P L Ltransmission coefficient and frequency of the PLL lowpass filter, A-output signal of the AGC circuit, Bcontrol signal of the AGC, r -estimate of the current oscillation amplitude, K p , K AGC I -proportional and integral coefficients of the controller of the AGC circuit, λ AGC -frequency of the pole of the AGC low-pass filter, X 0 -required amplitude of stationary oscillations.
It is convenient to associate the parameter of the geometric nonlinearity of the resonator β with the ratio κ of the linear and cubic components of the elastic restoring force at the displacement amplitude X 0 : It should be noted that a qualitative study of the dynamics of a controlled MEMS resonator and the stability of its stationary oscillatory modes for a mechanically linear case under the action of only the PLL system (without automatic gain control) was carried out in the seminal article [9]. The cited work also presents some results of the numerical integration of the equations of motion taking into account mechanical nonlinearity. It is noted there that when the β parameter reaches a certain limit value, "the control scheme fails to track the resonant frequency." The problem of determining the areas of stable operation of the resonant tuning system and classifying possible modes of motion, taking into account nonlinearity, was not posed in the cited article. The present work aims to investigate this group of questions. As follows from the formulation of the problem (1), the joint action of the AGC and PLL systems will be taken into account.

Qualitative analysis of the system nonlinear dynamics
For a qualitative (parametric) study of the dynamics of the system, we will apply the averaging method [19]. We represent the sought solution for the mechanical degree of freedom in the form where a(t)-amplitude, θ(t) + φ(t)-instantaneous phase of oscillations. Instantaneous VCO frequency is θ ; thus, φ is phase difference between the resonator and the VCO. Differentiating (3), we geṫ Assume thaṫ Note that the VCO phase dynamics is described by simple equation: The variable θ is "fast" and can be thought of as normalized time. Following the procedure of the averaging method, substitute (3), (4) and (6) into (1), express from the first equation (1) and (5)ȧ,φ and average the right parts of the system over the explicitly incoming "time" θ . Thus, we come to an autonomous system of differential equations describing the evolution of slow variablesã,φ,z,ỹ,B,r : Possible stationary periodic regimes of the system (1) correspond to fixed points (equilibrium positions) of the dynamical system (8): The stationary valuesz are determined from the equation Finally, The main interest is a solution for whichφ 0 = − π 2 andz 0 has the sign of ω n − ω 0 .

Stability analysis of stationary regimes
The stability analysis of the considered stationary regime is performed by calculating the eigenvalues of the Jacobian of the right-hand side of the system (8) at a given fixed point. For a linear resonator (κ = 0), the area of stability in the parameter space is characterized by the following conditions [10]: For the general case κ = 0, the derived conditions characterizing the stability of the system are quite cumbersome and are not explicitly given here.
Among the parameters of the system (8), let us single out the groups of fixed and variable. The values of the fixed parameters are given in Table 1.

Case of mechanically linear resonator
Let us present the results of numerical integration of the initial (1) and averaged (8) As can be seen from the figures, when the stability conditions of the control system are satisfied, the resonator reaches the resonant oscillation mode with the required amplitude. At the same time, equations in slow variables allow one to simulate the dynamics of a system accurately and much more efficiently from a computational point of view, in comparison with direct integration.
The following Figs. 3, 4 show the calculation results for the values of the control system parameters corresponding to the instability region of the AGC loop.
As can be seen from the figures, going out of the stability region in terms of the parameters of the AGC circuit leads to oscillations of the resonator with variable amplitude, i.e., to incorrect operation of the control system. Leaving the stability region in terms of the parameters of the PLL circuit also leads to oscillations of the resonator with variable amplitude. As in the previous case, at the end of the transient process, the system enters a two-frequency stationary oscillation mode in the vicinity of the required resonant mode.
Everywhere below, we take K AGC

Taking into account the nonlinearity of the restoring force
Further stages of the study will be devoted to the analysis of the influence of the nonlinearity factor of the resonator κ and the peculiarities of the system dynamics for various values of the quality factor Q.
Let us investigate the stability of the required resonant operation mode with respect to changes in the parameter κ. Figure 7 shows the dependence of the maximum real part among all eigenvalues for the inves-tigated fixed point of the system (8) depending on κ for Q = 10, 100, 1000.
As can be seen from the figure, for Q = 10 at κ ≈ 3% there is a loss of stability of the stationary solution.
For larger values of the quality factor in the considered range of κ, the stability of the control system tuned according to the linear model is not violated.
As can be seen from Fig. 8a, when choosing the initial conditions inside a small neighborhood of the stationary solution, the system actually reaches the required resonant oscillation mode. According to Fig.  8b, in spite of the fact that the steady state mode is stable for the selected parameters values, under arbitrary (in this case, zero) initial conditions, a continuous two-frequency oscillatory process is observed without reaching the required resonance mode. Hence, it can be concluded that there exist complex stable multifrequency periodic oscillation modes outside the basin of attraction of the investigated single-frequency resonant mode-hidden attractors [11,13].
As can be seen from Fig. 9a, the dynamics of the system outside the stability region of the stationary regime has the expected character of exponential growth from the initial state. According to Fig. 9b, under zero initial conditions for the considered values of the nonlinearity parameter, there are no qualitative differences between the behavior of the system inside and outside the stability region of the required resonant mode; in both cases, we have two-frequency oscillatory mode. Thus, it can be concluded that the adjustment of the parameters of the PLL-AGC system for a nonlinear resonator based on the stability conditions for the stationary mode is not justified and, in the general case, does not lead to the required stabilization of the amplitude and phase of the resonator oscillations.
Let us now investigate the dynamics of higher-Q systems. Based on Fig. 7, we see that for Q = 100 in the range of values κ to 10%, the stability of the stationary mode is preserved. Hereinafter, the calculations will be performed for zero initial conditions (i.e., outside, as will be shown below, a very narrow basin of attraction of single-frequency motion). Figures 10, 11 and 12, 13 show the simulation results for the values κ equal to 2 and 2.5%, respectively.
As can be seen from the figures, at κ = 2% the control system provides stabilization of the oscillation amplitude of the resonator and its resonance tuning.
From the figures above, it can be seen that at κ = 2.5% the system enters a complex multifrequency mode of motion despite the predictions of stability analysis. Physically, the solution obtained corresponds to the hysteresis motion of the representing point in the "amplitude-frequency" space according to the resonance (amplitude-frequency) characteristic of the mechanical resonator-due to nonlinearity, the skeletal curve has a slope to the right, which leads to the possibility of the phenomenon of "breakdowns" of oscillations when passing through the corresponding limit points.

Evolution and branching of steady-state oscillatory regimes
From a practical point of view, it is important to determine the areas in the parameter space of the mechanical structure and control system, corresponding to the possibility of maintaining the required resonant oscillation mode for a sufficiently wide range of initial conditions and amplitudes of external disturbing influences (temperature instability, vibrations, shocks and acoustic noise).
Such an analysis can be carried out using applied methods of nonlinear dynamics and the theory of bifurcations [20][21][22]. Let us consider the specific task of determining the minimum value of the nonlinearity parameter κ, for which the described above undesirable oscillatory modes are possible. The calculations will be carried out for different values of the quality factor Q. We assume other parameters of the system to be fixed.

Factor of mechanical nonlinearity
As can be seen from Fig. 8b, for Q = 10 the twofrequency oscillatory mode is observed at κ = 2.5%. This solution is a stable (attractive) limit cycle. For such objects of the phase space of a dynamical system, numerical-analytical methods have been developed that allow the continuation of the solution by the parameters of interest. The numerical implementation of these algorithms is performed in the MATCONT [23] software package. Let us graphically represent the evolution of the limit cycle when κ changes in the subspace ã,φ (Fig. 14).
We see that at κ ≈ 2.5% the fold bifurcation of the limit cycle occurs-at values κ < 2.5%, the undesirable oscillatory mode does not exist; as κ grows, the nature of the two-frequency mode becomes more complicated, and the amplitude of oscillations around the required value ã 0 ,φ 0 increases. This is illustrated in Figures 15a-15b, where the deviations of the amplitude and phase of the oscillations from the corresponding resonance (required) values are shown.
Note that in accordance with Fig. 7, for κ = 3% the Andronov-Hopf bifurcation is observed in the system-from a stable stationary resonant regime with an amplitudeã 0 , an unstable limit cycle branches off into the region of smaller κ, becoming stable through the aforementioned fold bifurcation; the stationary regime itself at κ > 3% is unstable. These features of the system explain the numerically observed fact presented above that undesirable oscillatory modes can also arise when the condition for the stability of the required resonant mode is met-the reason is the narrowness of the basin of attraction of this solution and the presence of an adjacent stable multifrequency motion.
A similar calculation was carried out for Q = 30. Figure 16 shows the evolution of the limit cycle when κ changes in the subspace ã,φ .
As can be seen from Fig. 16, with an increase in the Q-factor of the resonator, the structure of undesirable dynamic modes in the system becomes much more complicated; there is a multiplicity of possible multifrequency modes with one value of the nonlinear- We see that with an increase in the quality factor, the lower κ boundary for the region of possible multifrequency modes shifts to the left: in the case under consideration, κ min = 2%. Unlike the case Q = 10, here the Andronov-Hopf bifurcation is supercriticalthe branching of the stable limit cycle is directed to the region of larger κ.

Q-factor influence
Of key practical interest is the continuation of the above bifurcation of the limit cycle in other parameters (first of all, in the resonator Q-factor), what would characterize the boundary of the region of existence of parasitic oscillation modes. The MATCONT software package allows us to solve such problems-to perform continuation of bifurcations of codimension 1. The expected result here is a curve on the (κ, Q) plane, which characterizes the above-mentioned boundary of the stable operation of the PLL-AGC loop. In the present work, this problem was not completely solved due to significant numerical difficulties. Below are the results concerning the dependence of multifrequency oscillatory modes on the parameter Q at a fixed value of the nonlinearity parameter κ. Figures 18-19 show the evolution in the subspace ã,φ of the limit cycle when Q changes for κ = 2.5%. Figures 20a-20b show the deviations of the amplitude and phase of the oscillations from the required resonance values.
As can be seen from Figures 20a-20b,both the lower and the upper Q boundaries of the existence of parasitic oscillatory modes are observed. It should be noted that this, generally speaking, does not imply the stability of the control loop operation outside the found zones. This is due to the fact that the numerical continuation of the limit cycle is performed only up to certain values of the active parameter; the structure of the phase space of the  The results presented above confirm the complexity of the dynamics of the system at high values of the quality factor and the instability of the adopted settings of the PLL-AGC loop.
It is noteworthy that the dynamics of the system has a similar character at Q = 140 (at the value for which, according to the Figures 20a-20b, twofrequency oscillations around the required resonance mode were found). This indicates the multiplicity of possible dynamic modes for the selected parameter values (among which the κ parameter is of great importance). Note that by choosing the initial conditions close to the resonant mode, it is possible to obtain the motions shown in the above figures.

Analysis of the influence of the PLL and AGC loops settings
In the course of the research, it was found that the values of the poles frequencies of the low-pass filters λ AGC , λ P L L have a significant effect on the operation of oscillations control system. In this regard, the dynamics of the system was considered in detail when these parameters were being changed. It is assumed in the calculation that λ AGC = λ P L L = λ. As noted earlier, f n /10 was taken as the base frequency value λ. Following Figs. 23-24b show the results of calculating the evolution of the limit cycle for Q = 10, κ = 2.5% depending on n λ , where λ = f n n λ .
As can be seen from the figures, the found branch of periodic solutions (limit cycles) is closed. System dynamics when n λ is out of range ≈ [5,13] is subject to special consideration. Figure 25 shows the results of numerical simulation for various values n λ .
It is noteworthy that with a decrease in the filter frequency λ (with an increase in n λ to a value of 13), the control system does not provide stable resonator dynamics; on the contrary, the stationary mode with an amplitude close to X 0 is achieved by increasing λ (n λ = 4). It is also important to note that an increase in the frequency of the low-pass filter leads to a gradual violation of the initial assumption about the slowness of the phase detector variable y compared to the phase variable θ , which in turn leads to a decrease in the accuracy of the results obtained when studying the averaged system (8).

Conclusions
This work presents the results of a qualitative study of the dynamics of a micromechanical resonator taking into account the action of a control system with PLL and AGC loops. Particular attention was directed to studying the features associated with the presence of a nonlinear component of the elastic restoring force of the resonator. It has been found that the choice of control system parameters based on the analysis of the stability of the operating resonance mode, in general case, does not provide the required stabilization of the amplitude and relative phase of oscillations. Numerical simulation revealed attractive (stable) complex multi-frequency oscillation modes with variable amplitude and phase. An analytical study of the mechanisms of the appearance of these modes and their evolution when changing the key parameters of the system: the degree of mechanical nonlinearity, the Q-factor of the resonator and the frequency of the poles of the low-pass filter of the PLL and AGC circuits is carried out. For the considered computational cases, the real regions of stable operation of the control system are determined (which, as was found, do not coincide with the regions of stability of the operating resonance mode, due to the presence of hidden attractors in the phase space of the system). A technique has been developed for determining such areas of stable operation for specific parameters of the considered real systems. Significant difficulties associated with the qualitative analysis of the nonlinear dynamics of high-Q controlled resonators have been revealed. However, for this case, the models presented above give accurate numerical results that are of immediate practical importance.

Conflict of interest
The authors declare that they have no conflict of interest.