MLP-Based Neural Guaranteed Performance Control for MEMS Gyroscope With Logarithmic Quantizer

In this paper, a minimum-learning-parameter (MLP) based neural control method is proposed for micro-electro-mechanical system (MEMS) gyroscope with prescribed performance and input quantization. For the first time, a logarithmic quantizer (LQ) is employed to generate smooth input control signal for MEMS gyroscope, which greatly reduces the communication data size as well as actuator bandwidth. To improve the performance of MEMS gyroscope in the presence of quantization error, a prescribed performance control scheme consisting of preselected performance boundaries and an error transformation is utilized, such that preselected transient and steady-state properties can be assured. In contrast to the neural control strategies subject to the issue of learning explosion, a MLP-based neural network (NN) is introduced to estimate the unknown uncertainties using the norm of neural weight. To eliminate the effect of quantization error induced by LQ, a robust quantized control is designed to further ensure the closed-loop system suffering from discontinuous dynamics with prescribed ultimately uniformly bounded (UUB) performance. In the end, a series of simulations are presented to validate the superiority of the proposed control methodology.


I. INTRODUCTION
As an efficient angular rate sensor, micro-electro-mechanical system (MEMS) gyroscope is extensively utilized in inertial navigation, consumer electronics, automobile and even national defense industry by virtue of its remarkable advantages over traditional ones, especially its tiny size, easy integration, higher energy efficiency and availability of mass production [1]- [3]. Commonly, to obtain the precise information of MEMS gyroscope in terms of rotation rate, fast and accurate tracking of the reference trajectory is of high priority. However, in industrial application, diverse operating situations of MEMS gyroscope inevitably induce unknown external disturbances, which typically degrades the control The associate editor coordinating the review of this manuscript and approving it for publication was Nasim Ullah .
performance. What is more, the parametric uncertainties and couplings between drive and sense modes resulting from fabrication flaws further render the MEMS gyroscope having uncertain dynamics and as a consequence, the measuring accuracy of angular rate is unavoidably decreased. Hence, it is of great significance to design an elegant control method for MEMS gyroscope in pursuit of higher measuring accuracy.
Focusing on raising the measuring accuracy of MEMS gyroscope, tremendous advanced control schemes have been proposed, such as sliding-mode control [4]- [6], fuzzy logic control [5], [7], robust and adaptive control [7], [8], as well as neural networks (NNs) [6], [9]. As an example, in [7], the backstepping control, where the controller design contains several steps and, in each step, the virtual control law is recursively designed such that the ultimately uniformly bounded (UUB) properties of MEMS gyroscope can be achieved. In [8], to deal with the issue of term explosion arising from multiple differentiating, the dynamic surface control (DSC) is explored, where a first-order low-pass filter is applied to approximate the time derivative of virtual control signal and meanwhile, NNs are employed to online identify the unknown dynamics by virtue of their universal approximation, such that, the robustness against uncertainties of MEMS gyroscope is greatly improved. Despite the fact that available works [4]- [9] for MEMS gyroscope are of great superiority, it is worth pointing out that almost all of them are subject to the following two serious problems. One is concerned with bandwidth restriction of actuator, i.e., the existing results for MEMS gyroscope are almost achieved under the unpractical assumption that the actuator of MEMS gyroscope can perform with arbitrarily high precision and the capacity of communication channel from control module to actuator is large enough to transmit continuous signal with infinite accuracy, which is in contradiction with the physical truth of MEMS gyroscope. Another problem is that although previous works can realize remarkable closed-loop stability with UUB properties, the upper bound of final tracking error cannot get rid of the heavy dependence on unknown disturbances and as a consequence, in pursuit of guaranteed performance in terms of tracking accuracy, controller parameters are always required to be repeatedly tuned and some design conservatism is introduced. Therefore, how to design an advanced control methodology for MEMS gyroscope considering input quantization and predetermined performance is still an open issue and deserves deep study.
Quantization, a ubiquitous process in modern digital communication, refers to the conversion of continuous signal to the one with finite-precision via mapping from infinite sets to finite sets. With the aid of quantization, the control signal can be transmitted over a constrained communication channel, performed with a limited digital microprocessor while the normal operation of system will not be influenced. Enlightened by the quantization in communication systems, researchers are more and more focused on quantized control of uncertain systems subject to input quantization. Logarithmic quantizer (LQ) [10], [11], one of the most employed quantizers in digital communication systems, can largely reduce the communication data size and meanwhile, it is capable of generating quantized signals with constant quantization signal-to-noise ratio (SNR) owing to its exponentially transformed quantization levels compared with uniform quantizer (UQ) [12], such that the robustness of system will not change along with the variation of control signal magnitude. Nevertheless, the introduction of input quantization typically induces unexpected quantization error, which will inevitably influence the tracking accuracy of MEMS gyroscope, while to the best of authors' knowledge, how to make a compromise between the tracking performance and bandwidth restriction is rarely studied in existing results [4]- [9]. Thus, it is an urgent and tough issue to eliminate the influence of quantization error and guarantee high-accuracy tracking performance for MEMS gyroscope in the existence of quantized input.
To further improve the tracking performance in both transient and steady-state phases and remove the design conservatism universally existing in previous control methods for MEMS gyroscope, which can only ensure the closed-loop stability with UUB, prescribed performance control (PPC) proposed in [13]- [15] is an efficient strategy. Nowadays, PPC has been commonly used in uncertain nonlinear systems such as air-breathing hypersonic vehicle (AHV) [16], [17] and [35], robotic manipulator [13] and underwater vehicles [18]. With the aid of a proper equivalent error transformation function, PPC can assure the tracking error within predetermined boundaries, such that the prescribed performance can be realized without the tedious tuning of design parameters. For instance, within the framework of the controller design for backstepping, the PPC methodology in [19] is capable of realizing the tracking of both velocity and altitude with prescribed performance for AHV. However, the analytical differentiating of virtual control law inevitably leads to the issue of differentiating explosion, which may further affect the normal operation of controller. To eliminate this obstacle, dynamic surface control (DSC) [20], [21] is usually combined with PPC, where a first-order low-pass filter is employed to approximate the time derivative of virtual control signal. However, to the best of authors' knowledge, the available PPC studies are typically designed for actuator with arbitrarily high performing accuracy, which is not feasible for algorithm implementation in digital microprocessor of MEMS gyroscope.
In light of the manufacturing defects and diverse working conditions of MEMS gyroscope, its measuring accuracy is typically affected by uncertainties containing external disturbances, parametric uncertainties and coupling between two operation modes. Hence, the accurate identification of lumped disturbances directly determines the performance of MEMS gyroscope. Under such circumstances, neural networks (NNs) are widely employed to online identify unknown disturbances for uncertain systems [22]- [24] by virtue of their universal approximation. However, the complex structure of NN unavoidably induces heavy computational load, such that an unexpected time delay may be introduced to controller, which can further result in poor transient performance especially under some fast time-varying measuring conditions. Thus, how to avoid the problem of learning explosion without degrading the identification accuracy is of great urgency and necessity.
To solve the issues mentioned above, a MLP-based neural guaranteed performance control for MEMS gyroscope with logarithmic quantizer is proposed and its superiorities over available studies are collected in TABLE 4 in APPENDIX, which can be summarized as follows: Different from the available works [4]- [9] under the unpractical hypotheses that the actuator can perform with arbitrarily high accuracy and data transmission VOLUME 8, 2020 from control module to actuator can be realized via unlimited communication channel, LQ is firstly employed to generate finite-precision input control signal for MEMS gyroscope that possible digital controller can be promoted. In addition, different from existing quantization error decomposition [25], [35] relying on the control input to be bounded, which is hard to ensure before the controller design, a novel quantization decomposition approach is investigated for MEMS gyroscope to relax such condition and guarantee closed-loop stability. Furthermore, PPC technique is explored in controller design in the existence of input quantization to assure both transient and steady-state tracking performance within preselected performance boundaries. Hence, a compromise between the tracking performance of MEMS gyroscope and bandwidth restriction can be obtained.
Although most existing works [22]- [24] can realize high-accuracy identification of unknown disturbances by means of universal approximation of NNs, it is worth pointing out that almost all of them suffer from the problem of learning explosion. Inspired by the minimum-learning-parameter (MLP) technique in [26], [27] and [36], we employ MLP to reconstruct the lumped disturbances consisting of external disturbances, parametric uncertainties and coupling between two operation modes, which greatly reduces learning dimension, such that the computational load is remarkably eased without decreasing the identification accuracy. What's more, the application of DSC technique effectively eliminates the problem of term explosion caused by analytical differentiation, such that the feasibility of algorithm is notably improved.
With the aid of Lyapunov stability analysis, the prescribed UUB properties of closed-loop system can be ensured. Extensive simulation results are presented to further validate the effectiveness as well as superiority of the proposed control method. This rest paper is arranged as follows. The problem formulation and preliminaries will be introduced in Section II. Section III presents the procedure of controller design and section IV proposes the performance analysis. Section V provides a series of simulation results and finally section VI concludes this paper.

II. PROBLEM STATEMENT AND PRELIMINARIES A. MODELING OF MEMS GYROSCOPE
The schematic diagram of a z-axis vibrational MEMS gyroscope is depicted in Fig. 1 and conventionally, the nondimension model of MEMS gyroscope, as borrowed form [4], [5], [8], [9], [28] can be formulated as: where ] T denote the displacements and velocities of drive and sense mode, respectively. F = [F 1 , F 2 ] T represents the lumped disturbances, which can be described as where u i for i = 1, 2 denotes the input signal for MEMS gyroscope and m represents the mass of gyroscope. ω 0 is the reference angular frequency. D xx , D yy and K xx , K yy respectively refer to the uncertain damping coefficients and spring coefficients. K xy , D xy are unknown coupling parameters and ξ is the external disturbances. Remark 1: It is worth pointing out that the model of MEMS gyroscope mentioned in [4], [5], [8], [9], [28] commonly assume that the input control signal u can be performed with arbitrary accuracy. While in fact, the widespread use of digital microprocessor leads to the fact that most electrostatic actuators of MEMS gyroscope cannot meet such requirement. To break through this restriction, we explore a modified model of MEMS gyroscope as follows.
where Q(u) denotes the quantized control signal generated by LQ, which is formulated as where u ri = δ 1−r i u mi with integer r = 1, 2, . . . , N , u mi is the smallest quantization level and δ i = (1 − η i )/(1 + η i ) is a positive design parameter, which determines the quantization density. Generally speaking, the bigger δ i is, the more quantization levels will Q(u i ) has and the smoother will quantization performance be. With the aid of LQ, the input control signal is limited in the finite set of U = {±u ri ,0}, such that the issue mentioned in Remark 1 is eliminated.
The quantized signal Q(u i ) can be considered as the composition of a continuous part and a discontinuous one, which can be described as where Lemma 1 [26]: For ι(u i ) and T (u i ) in (6), there always exist the following inequalities.
Assumption 1 [30]: The reference signals x pdi , its time derivativeẋ pdi , as well as its second derivative with regard to timeẍ pdi for i = 1, 2 are commonly considered to be smooth, continuous and bounded functions. Accordingly, we assume a positive constant B 0 belonging to a compact set 0 defined as Control Objective: Under assumption1, for the MEMS gyroscope modeled as (4), our objective is to design a MLP-based neural guaranteed performance control with input quantization to steer position state x p of MEMS gyroscope to follow the given trajectory. Under the proposed control scheme, the prescribed UUB stability of closed-loop system can be assured and high tracking accuracy are expected.

B. RADIAL BASIS FUNCTION NEURAL NETWORK
In control field, radial basis function neural networks (RBFNNs) are commonly applied to approximate arbitrary unknown continuous function over a compact set x m ∈ ⊂ m by virtue of its universal approximation [31], [32] and the output of RBFNN can be formulated as where ω = [ω 1 , ω 2 , . . . , ω n ] T ∈ n ,x m = [x 1 , x 2 ,. . . ,x m ] T ∈ m respectively denote the weight and input vector. n is the number of neural codes and m represents the input number. h(x m ) = [h 1 (x m ), h 2 (x m ),. . . ,h n (x m )] T ∈ n denotes the basis function of RBFNN and it is typically selected as the following Gaussian function formed as with c j = [c j1 , c j2 ,. . . ,c jm ] T ∈ m and b being the center and width parameter of basis function h j (x m ).
Remark 2: It is notable that the elements of hidden layer will exponentially grow with the increase of learning dimension and based on universal approximation theorem, high-accuracy approximation requires the learning dimension to be large enough, such that high computational burden will be induced to the controller, which will further lead to some severe problems to MEMS gyroscope, especially in some fast time-varying operation environments.

III. CONTROLLER DESIGN
In this section, the proposed control scheme will be elaborated. Its architecture is depicted in Fig. 2 and it is clear that with the aid of proposed PPC, DSC, LQ and MLPNN techniques, fast and high-accuracy tracking of reference trajectory with slight computational burden and limited actuator bandwidth can be realized.

A. PRESCRIBED PERFORMANCE CONTROL
Given the dimensionless reference trajectory x pd , the tracking error e p = [e p1 , e p2 ] T can be defined as To guarantee the tracking result with prescribed properties, PPC is an efficient approach to improve the tracking performance in both transient and steady-state phases, which can be described as with i _,¯ i being positive constants, ρ i (t) for i = 1, 2 being the preselected performance function and it is typically chosen as the following exponential decaying one [13]- [15].
where ρ i0 , ρ i∞ , l i are all positive design parameters and ρ i0 = ρ i (0), ρ i∞ = ρ i (∞), such that ρ i0 limits the transient overshoots, ρ i∞ ensures the maximum steady-state tracking error and l i decides convergence rate of performance function ρ i (t). Note that it is hard to design control law based on inequality (13), some equivalent error transformation is required to eliminate the time-varying constraints exerted on system states. To facilitate the controller design, the error transformation function (z i (t)) is introduced to convert constrained error into an unconstrained one and it is formulated as with where z i (t) is the transformed error. Obviously, (z i (t)) is a smooth, strictly increasing function and one has (17) Following (15), z i (t) can be subsequently obtained as with ζ i (t) = e pi (t)/ρ i (t) being the normalized function. Lemma 2 [13]: As long as the transformed error z i is bounded, the MEMS gyroscope tracking error e pi is within the preselected boundaries, i.e., − i ρ i (t) < e pi (t) <¯ i ρ i (t) always holds.

B. MLPNN
Considering the unknown time-varying disturbances imposed on the system, fast and accurate online disturbances identification is crucial for the performance of MEMS gyroscope. Thus, we employ RBFNN to approximate the lumped disturbances.
where x i = [x pi , x vi ] T represents the input matrix, ε i is the approximation error and based on universal approximation theorem [32], [34], as long as the number of neural codes is large enough, there always exists an ideal value ω * i such that ε i can be arbitrarily small and one always has whereε i is a positive constant. As mentioned in Remark 1, the node number is required to be large enough to realize high-accuracy identification and consequently, the elements requiring to be adaptively updated will exponentially grow with node number, which will cause the issue of learning explosion. To reduce the learning dimension, a MLPNN technique is applied to reconstruct the lumped disturbances F i , which is formulated as where W * i = ||ω * i || 2 is defined as the norm of weight vector ω * i . Subsequently, we can easily obtain the approximation value of F i , which is defined aŝ whereŴ is the estimation value of W * , which can be produced via the following adaptive update law: with e vi denoting the tracking error of velocity subsystem, i being the adaptive gain, κ i being a positive design parameter. Remark 3: Using MLP technique, we only need to online updateŴ i rather than each element ofω i like the works [19]- [21], such that the learning dimension is remarkably reduced, which greatly eases the computational burden without decreasing the identification accuracy.

C. QUANTIZEDCONTROL DESIGN
Inspired by the recursive controller design of backstepping control, MEMS gyroscope system is split up into two subsystems and accordingly, the controller design contains the following two parts. STEP 1: Recollecting the transformed error z i (t) defined in (18), taking its time derivation produceṡ To guarantee the tracking performance with prescribed UUB properties, the virtual control law is chosen as where k 1i is the controller gain of position subsystem.
To avoid the issue of ''complexity explosion'' caused by analytical differentiation of α i (t), we explore a first-order low-pass filter to obtain the time derivation of virtual control signal, which is formulated as with τ i being the time delay constant of filter. Define the filtering error σ i as Then, taking time derivative along (28) results iṅ and obviously 38600 VOLUME 8, 2020 where b i (ẍ pdi ,ż i , e pi ,ė pi , ρ i ,ρ i ,ρ i ) is a continuous and smooth function satisfying |b i | ≤ B i . Now, consider the following Lyapunov function candidate Considering (29) and (30), we can obtain the time Based on Young's inequality, the following inequality holds.
Substituting (33) into (32) produceṡ Simultaneously, considering (24), (26) and (28),ż i can be rewritten in the form oḟ Subsequently, consider the Lyapunov function candidate Combined with (35), its time derivative is formulated aṡ With the aid of Young's inequality, we have Substituting (38) into (37),V z i can be rewritten aṡ STEP 2: In this step, the actual control law is elaborated to eliminate the quantization error as well as ensure the UUB closed-loop stability of system. Firstly, we define the tracking error of MEMS gyroscope velocity subsystem as e vi = x vi −ᾱ i . Next, using (4) and (6), its time derivative can be written aṡ Then, select the following Lyapunov function candidate Based on (21) and (40), taking time derivative along (41) produceṡ Define the auxiliary control signal υ i as with k 2i being the controller gain of velocity subsystem. To make up for the quantization error induced by LQ, adding and subtracting e vi υ i , (42) can be rewritten in the following forṁ withW i = W * i −Ŵ i being the updating error. Now, select the actual control law as where β 1i , β 2i are positive design parameters. With the aid of Lemma 1, we can easily obtain Meanwhile, the following inequalities apparently hold Then (46) can be further rewritten in the following concise form Substituting (48) into (44),V e vi can be finally written aṡ

IV. PERFORMANCE ANALYSIS
Theorem 1: Consider the MEMS gyroscope model (4), the control signals (45), the virtual control law (26), the adaptive update law (23) and DSC (27). For any bounded initial conditions satisfying − i _ρ i (0) < e pi (0) <¯ i ρ i (0), i = 1, 2, there always exist constants defined by (54), such that all the involved signals in MEMS gyroscope system converge to small residual sets and the prescribed tracking performance, i.e., − i _ρ i (t) < e pi (t) <¯ i ρ i (t) can be satisfied. Proof: Choose the overall Lyapunov function candidate as: With the aid of (34), (39) and (49), taking time derivative along with (50) produceṡ Recalling the definition of updating errorW i , using Young's inequality, the following inequalities can be obtained With the aid of (23) and (52), (50) can be further rewritten asV Then, from (53), we can easily obtain that if the following inequalities hold, it can be derived thaṫ with C being a positive constant defined as The initial conditions are all bounded, so we can make a feasible assumption that V (0) = A. Subsequently, as long as K > C/A holds, we can draw the conclusion that ∀t > 0, V (t) ≤ 0, in other word, V (t) ≤ A always holds. Hence, the signals involved in closed-loop system are all UUB and meanwhile, with the aid of Lemma 2, it can be discovered that tracking error e pi always falls into the prescribed boundary, such that Theorem 1 is eventually proved.

V. SIMULATION RESULTS
To validate the effectiveness of our proposed control method, several simulations are performed via MATLAB/ SIMULINK. Firstly, the physical parameters of MEMS gyroscope, borrowed from [4], are chosen as The reference trajectory of displacements is chosen as Choose the external disturbances ξ as      that guaranteed tracking performance can be obtained via appropriately selecting performance functions. Furthermore, the convergence rate is very fast, as can be seen from Fig. 3, only after 0.08s, both tracking error of position and velocity subsystem coincide with the reference trajectory perfectly. Meanwhile, the quantization performance is clearly reflected from Fig. 6, where the continuous input signal is converted to the one taking values in finite sets, such that the issue of actuator limitation is eliminated and data size requiring to be transmitted is remarkably lessened. Although the precision of input signal is notably reduced, high-accuracy tracking performance as well as identification of uncertainties can still be available, which is mainly attributed to the effective error compensation methodology and PPC technique.

B. COMPARATIVE SIMULATIONS
To further demonstrate the superiorities of our proposed control method over existing ones, some comparative simulations are performed and presented in this section. Firstly, the simulations among proposed control scheme, backstepping control [7] are given in Fig. 7. In light of the fairness, we apply these algorithms with the same control parameters.    The results clearly indicate that both transient and steadystate tracking performance of the proposed algorithm are much better than that of backstepping control. In addition, the proposed method performs faster convergence rate of tracking error by virtue of performance boundaries and effective error transformation mechanism.
In addition, to further clarify the remarkable quantization performance, a statistic with respect to data size of input signal as well as tracking accuracy is presented in TABLE 2, from which we can discover that with the aid of LQ as well as effective error compensation mechanism, the data amount of control input required to be transmitted is notably reduced while the steady tracking accuracy is almost the same as the case without LQ, such that the compromise between tracking performance and actuator bandwidth can be finally obtained.
Furthermore, aiming at testifying the superiorities of MLPNN in terms of computational burden and tracking accuracy, we conduct the comparative simulations between MLPNN and RBFNN [34], the simulation results are demonstrated in TABLE 3, where the average computation time denotes the average time of updating of each neural weight. It can be easily discovered that the heavy computation burden of RBFNN appears due to the learning process of each element of neural weight vectors. Evidently, the computation time of RBFNN is much longer than MLPNN as the nodes number further improves. While with the aid of MLPNN, we can achieve almost the same accurate identification results with much slighter computation load.

VI. CONCLUSION
In this paper, we propose a MLP-based neural control for MEMS gyroscope with guaranteed performance and quantized input. Firstly, we employ a LQ to generate input signal for MEMS gyroscope, the issues of actuator bandwidth and communication resource constraints are effectively eliminated. Considering the quantization error imposing on MEMS gyroscope system, a PPC technique is employed to realize guaranteed tracking performance in the existence of uncertainties. In addition, MLPNN is utilized to realize highaccuracy identification of lumped disturbances with slight computational burden. In the near future, we will focus on an event-triggered fuzzy control scheme with full-state constraints in pursuit of reliable operation of MEMS gyroscope with resource limitation. APPENDIX See Table 4.