Study of a MEMS Vibratory Gyroscope Using Adaptive Iterative Learning Control

Abstract

This paper proposes a framework, namely adaptive iterative learning control (AILC), which is used in the control of a microelectromechanical system (MEMS) gyroscope, to realize high-precision trajectory tracking control. According to the characteristics of the MEMS gyroscope's model, the proposed AILC algorithm includes an adaptive law of parametric estimation and an iteration control law, which is updated in the iterative domain without any prior knowledge of MEMS gyroscopes. The convergence of the method is proven by a Lyapunov-like approach, which shows that the designed controller can guarantee the stability of the system and make the output tracking errors to converge completely to zero while the iteration index tends to infinity. By comparing AILC and traditional PD-ILC, the simulation results demonstrate the effectiveness of AILC and its robustness against external random disturbance.

1. Introduction

MEMS gyroscopes could provide high performance angular-rate measurements leading to a wide range of applications, including navigation and guidance systems, automotive safety systems and consumer electronics. Extensive research efforts towards vibratory MEMS gyroscope -, which utilize vibrating elements to induce and detect Coriolis forces − have led to several innovative gyroscope fabrication and integration approaches and detection techniques. However, achieving robustness against fabrication variations and environmental fluctuations remains one of the greatest challenges in the commercialization and high-volume production of MEMS vibratory gyroscopes [1]. Moreover, MEMS gyroscopes belong to MIMO systems, which have parameter uncertainties and are vulnerable to external environmental influences. As a consequence, it is necessary to solve these problems by utilizing advanced control technologies, such as adaptive control, sliding mode variable control, adaptive sliding mode control, adaptive force-balancing control and fuzzy control. Paper [2] developed a sliding mode control for a MEMS gyroscope system. An adaptive controller was presented to drive both axes of vibration and control the entire operation of the gyroscope [3]. In reference [4], the SMC strategy was incorporated into an adaptive control for the vibration of a MEMS gyroscope. Paper [5] extended the method mentioned in [3] to three-axis MEMS gyroscopes. The adaptive force-balancing control proposed in [6] can identify major fabrication imperfections. In [7], an adaptive fuzzy compensator was designed to compensate dead-zone non-linearities for MEMS gyroscopes.

In 1984, Arimoto et al. [8] put forward the explicit formulation of an iterative learning control (ILC) algorithm with rigorous convergence analysis based on the previous research work of Uchiyama. After more than three decades of development, the ILC algorithm has become a branch of intelligent control with a well-established mathematical description and stability proof which can handle nonlinear and strong coupling systems in a very simple way and at low cost [9,10]. Currently, the ILC has made tremendous progress in terms of the learning law, convergence, robustness, learning speed and application, and it has been used in industrial robots, rotary systems, heat treatment, and chemical production processes, etc. [9]. Adaptive ILC, integrating the merits of adaptive control and ILC based on Lyapunov stability theory, is very popular and frequently attracts the attention of researchers. Paper [11] proposed an adaptive learning scheme for uncertain robot systems on the basis of parameter linearization; paper [12] provided not only an AILC method for industrial robots but also the results of experiments; in paper [13,14], the authors used standard Lyapunov stability theory to design an AILC algorithm, while [15–18] exploited a Lyapunov-like theory to prove the convergence of AILC. In paper [17], three AILC schemes were put forward for rigid robot manipulators, and the subsequent study of AILC was presented in paper [18], describing a unified AILC framework for uncertain nonlinear systems and interpreting the advantages and disadvantages of pure time-domain adaptation, pure iteration-domain adaptation and 2-D adaptation. In the last five years, there have been two main trends in AILC. The first one is AILC for discrete-time systems, namely the fusion of discrete-time adaptive control theory and ILC [19–21], which extends the application range of AILC; the second trend is the combination of AILC with other intelligent control theories [22,23] (fuzzy control, neural network control, etc.), which may assist in designing better controllers in achieving the desired performance, such as the relaxation of resetting conditions or alignment conditions, and robustness against external disturbances.

However, ILC has not been investigated for a MEMS gyroscope in the literature. Hence, an ILC strategy for MEMS vibratory gyroscopes is proposed in this paper. The research motivation of ILC is that some systems which run repetitive tasks in fixed time-intervals can improve their performance from previous operational experience - that is to say that systems can reduce tracking errors by incorporating past control information, such as tracking errors and control input signals. Due to the inherent characteristics of AILC, the approach requires less prior knowledge of process or systems at the stage of controller design than other methods, so it can deal with problems of unknown parameters or parametric uncertainties, especially for nonlinear, strong coupling dynamic systems. The main characteristics of the proposed AILC with their application to a MEMS gyroscope are highlighted as follows:

• 1)

  An AILC algorithm, including an adaptive law of parametric estimation and an iteration control law, is designed to improve the robustness of the control system. It integrates the advantages of adaptive control and ILC, and guarantees that the tracking error can converge to zero. The AILC design approach is used to drive the trajectory tracking errors to converge to zero rapidly.

• 2)

  An AILC is incorporated into the MEMS gyroscope to improve the tracking performance and strengthen the robustness of the control system. The convergence of the proposed algorithm using a Lyapunov-like theory can be established. This control scheme consists of conventional PD feedback control, unknown parameters and tracking errors of iteration.

• 3)

  Simulation studies verify the robustness of the algorithm using random disturbance signals instead of repetitive disturbance signals in every iterative control. By comparing AILC and traditional PD-ILC, the simulation results show the effectiveness of AILC and its robustness against external random disturbance

This paper is organized as follows. In Section 2, the dynamics of the MEMS gyroscope are described. In Section 3, the adaptive iterative learning controller is derived and the stability of the designed control systems is proved by Lyapunov's stability theory. Simulation results are presented in Section 4. Finally, conclusions are provided in Section 5.

2. Dynamics of the MEMS gyroscope

A typical MEMS vibratory gyroscope includes a proof mass suspended by spring beams, electrostatic actuations and sensing mechanisms for forcing an oscillatory motion and sensing the position and velocity of the proof mass. A z-axis MEMS gyroscope is depicted in Figure 1. The dynamics of a MEMS gyroscope are derived from Newton's law in the rotating frame.

Figure 1.

A simplified model of a MEMS z-axis gyroscope

As we know, Newton's law in the rotating frame becomes:

$F_r=F_{phy}+F_{centri}+F_{Colis}+F_{Eular}=m{a}_r$ (1)

where F_r is the total applied force to the proof mass in the gyro frame, F_phy is the total physical force to the proof mass in the inertial frame, F_centri is the centrifugal force, F_Colis is the Coriolis force, F_Euler is the Euler force, and a_r is the acceleration of the proof mass with respect to the gyro frame.

Assuming that the gyroscope is almost rotating at a constant angular velocity Ω over a sufficiently long time interval, that the stiffness of the spring in the z direction is much larger than that in the x,y directions, and the motion of the proof mass is constrained only along the x − y plane, we have Ω _x = Ω = 0.

In Equation (1), the expression for the Euler force reduces to $F_{Eular}=-m\frac{d\mathrm{\Omega }}{dt}\times r_r$ , where r_r is the position vector relative to the rotating gyroscope frame. Since the angular velocity Ω is a constant, we can obtain F_Eular = 0.

For another, the expression for centrifugal force reduces to F_centri = −mΩ×(Ω×r_r). Under a typical assumption Ω _x = Ω _y = 0, and the centrifugal forces along the x-axis and the y-axis can be expressed as F_centri–x = −mΩ _z ²x and F_centri–y = −mΩ _z ²y, where x, y are displacements of the proof mass along the x-axis and the y-axis. Considering the fact that F_centri–x and F_centri–y are very small - usually no more than one thousandth of other forces − as such they can be ignored.

Similarly, the expression for the Coriolis force reduces to F_Colis = −2mΩ × v_r, where v_r is the velocity vector relative to the rotating gyroscope frame. According to the principle of Coriolis force, the Coriolis forces along the x-axis and the y-axis can be expressed as F_Colis–x = 2mΩ _z ẏ and F_Colis–y = −2mΩ _z ẋ, where ẋ, ẏ are velocities of the proof mass along the x-axis and the y-axis.

The total physical force to the proof mass in the inertial frame F_phy mainly consists of the driving force, the spring force and the damping force. Here, we make the following statements: u_x, u_y are the control forces in the x,y directions; and k_xx, k_yy and d_xx, d_yy are the spring and damping coefficients of the x and y axes; k_xy and d_xy are the asymmetric spring and damping terms, which are mainly contributed by fabrication imperfections. Further, the total physical force in the inertial frame along the x-axis and the y-axis can be expressed as F_phy–x = u_x − k_xxx − k_xyy − d_xxẋ − d_xyẏ and F_phy–y = u_y − k_yyy − k_xyx − d_yyẏ − d_xyẋ.

Based on the above analysis, the dynamics of the gyroscope are simplified as follows:

$\begin{array}{l}m\stackrel{..}{x}+d_{xx}\stackrel{.}{x}+d_{xy}\stackrel{.}{y}+k_{xx}x+k_{xy}y=u_x+2m{\mathrm{\Omega }}_z\stackrel{.}{y}\\ m\stackrel{..}{y}+d_{xy}\stackrel{.}{x}+d_{yy}\stackrel{.}{y}+k_{xy}x+k_{yy}y=u_y-2m{\mathrm{\Omega }}_z\stackrel{.}{x}\end{array}$ (2)

Using non-dimensional time t* = ω₀t, where ω₀ is a natural resonance frequency about 1 KHz, then we have:

$\stackrel{.}{x}={\omega }_0{\stackrel{.}{x}}^{*},\stackrel{..}{x}={\omega }_0{}^2{\stackrel{..}{x}}^{*},\stackrel{.}{y}={\omega }_0{\stackrel{.}{y}}^{*},\stackrel{..}{y}={\omega }_0{}^2{\stackrel{..}{y}}^{*}$ (3)

Substituting Eq. (3) into Eq. (2) and dividing both sides of (3) by m, q₀, ω₀² (which are a reference mass, length and natural resonance frequency, respectively), we get the form of the non-dimensional equation of motion as:

$\begin{array}{l}\frac{{\stackrel{..}{x}}^{*}}{q_0}+\frac{d_{xx}}{m{\omega }_0}\frac{{\stackrel{.}{x}}^{*}}{q_0}+\frac{d_{xy}}{m{\omega }_0}\frac{{\stackrel{.}{y}}^{*}}{q_0}+\frac{k_{xx}}{m{\omega }_0{}^2}\frac{x^{*}}{q_0}+\frac{k_{xy}}{m{\omega }_0{}^2}\frac{y^{*}}{q_0}\\ =\frac{u_x}{m{\omega }_0{}^2{q}_0}+2\frac{{\mathrm{\Omega }}_z}{{\omega }_0}\frac{{\stackrel{.}{y}}^{*}}{q_0}\\ \frac{{\stackrel{..}{y}}^{*}}{q_0}+\frac{d_{xy}}{m{\omega }_0}\frac{{\stackrel{.}{x}}^{*}}{q_0}+\frac{d_{yy}}{m{\omega }_0}\frac{{\stackrel{.}{y}}^{*}}{q_0}+\frac{k_{xy}}{m{\omega }_0{}^2}\frac{x^{*}}{q_0}+\frac{k_{yy}}{m{\omega }_0{}^2}\frac{y^{*}}{q_0}\\ =\frac{u_y}{m{\omega }_0{}^2{q}_0}-2\frac{{\mathrm{\Omega }}_z}{{\omega }_0}\frac{{\stackrel{.}{x}}^{*}}{q_0}\end{array}$ (4)

The vector-form of the MEMS gyroscope dynamics model can be written as:

$\stackrel{..}{q}+D\stackrel{.}{q}+Kq=u-2\mathrm{\Omega }\stackrel{.}{q}$ (5)

where: $q=\left[\begin{array}{c}\frac{x^{*}}{q_0}\\ \frac{y^{*}}{q_0}\end{array}\right]$ , $D=\left[\begin{array}{cc}\frac{d_{xx}}{m{\omega }_0}& \frac{d_{xy}}{m{\omega }_0}\\ \frac{d_{xy}}{m{\omega }_0}& \frac{d_{yy}}{m{\omega }_0}\end{array}\right]$ , $K=\left[\begin{array}{cc}\frac{k_{xx}}{m{\omega }_0{}^2}& \frac{k_{xy}}{m{\omega }_0{}^2}\\ \frac{k_{xy}}{m{\omega }_0{}^2}& \frac{k_{yy}}{m{\omega }_0{}^2}\end{array}\right],u=\left[\begin{array}{c}\frac{u_x}{m{\omega }_0{}^2{q}_0}\\ \frac{u_y}{m{\omega }_0{}^2{q}_0}\end{array}\right],\mathrm{\Omega }=\left[\begin{array}{cc}0& -\frac{{\mathrm{\Omega }}_z}{{\omega }_0}\\ \frac{{\mathrm{\Omega }}_z}{{\omega }_0}& 0\end{array}\right]$ .

For concision, we define a set of new parameters as follows:

$\frac{x^{*}}{q_0}\to x,\frac{y^{*}}{q_0}\to y,\frac{d_{xx}}{m{\omega }_0}\to d_{xx},\frac{d_{xy}}{m{\omega }_0}\to d_{xy},\frac{d_{yy}}{m{\omega }_0}\to d_{yy},\frac{k_{xx}}{m{\omega }_0{}^2}\to {\omega }_x^2,\frac{k_{xy}}{m{\omega }_0{}^2}\to {\omega }_{xy},\frac{k_{yy}}{m{\omega }_0{}^2}\to {\omega }_y^2,\frac{u_x}{m{\omega }_0{}^2{q}_0}\to u_x,\frac{u_y}{m{\omega }_0{}^2{q}_0}\to u_y,\frac{{\mathrm{\Omega }}_z}{{\omega }_0}\to {\mathrm{\Omega }}_z$ . Then we have: $q=\left[\begin{array}{c}x\\ y\end{array}\right],D=\left[\begin{array}{cc}{d}_{xx}& d_{xy}\\ d_{xy}& d_{yy}\end{array}\right],K=\left[\begin{array}{cc}{\omega }_x^2& {\omega }_{xy}\\ {\omega }_{xy}& {\omega }_y^2\end{array}\right],u=\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right],\mathrm{\Omega }=\left[\begin{array}{cc}0& -{\mathrm{\Omega }}_z\\ {\mathrm{\Omega }}_z& 0\end{array}\right]$ .

Considering the system with external disturbance, the dynamics of the MEMS gyroscope can be expressed as:

$\stackrel{..}{q}+(D+2\mathrm{\Omega })\stackrel{.}{q}+Kq=u+d$ (6)

where d is an uncertain extraneous disturbance.

3. Adaptive iterative learning controller design

3.1 Control Algorithm Design

Based on the study of iterative learning control, the dynamics of the MEMS gyroscope may be expressed by:

${\stackrel{..}{q}}_k+(D+2\mathrm{\Omega }){\stackrel{.}{q}}_k+K{q}_k=u_k+d_k$ (7)

where the nonnegative integer $k\in \mathbb{Z}+$ denotes the operation or iteration number. The signals q_k ∈ Rⁿ, q̇ _k ∈ Rⁿ and q̈ _k ∈ Rⁿ are the position, velocity and acceleration vectors, respectively, and the iteration k · u_k ∈ Rⁿ is the control input vector. d_k ∈ Rⁿ is the vector containing the unmodelled dynamics and other unknown external disturbances.

We make the following assumptions prior to further discussion.

Assumption 1. The reference trajectory and its first and second time-derivative, namely q_d(t), q̇ _d (t) and q̈ _d (t), as well as the disturbance d_k(t), are bounded ∀t ∈ [0,T] and $\forall k\in \mathbb{Z}+$ .

Assumption 2. The resetting condition is satisfied, i.e., $q_d(0)-q_k(0)={\stackrel{.}{q}}_d(0)-{\stackrel{.}{q}}_k(0)=0,\forall k\in \mathbb{Z}+$ .

Moreover, we will also need the following properties, which are common to dynamics of MEMS gyroscope.

• i1)

  K_qk(t) + (D + 2Ω)q̇ _k (t) = Ψ(q_k, q̇ _k )ξ, where $\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)\in {ℝ}^{2\times 7}$ and $\xi \in {ℝ}^7$ . Since the signals q_k and q̇ _k are the position and velocity vectors of proof mass that can be detected, we can obtain that Ψ(q_k, q̇ _k ) is a known matrix. ξ is an unknown continuous vector, consisting of system structure parameters.

• i2)

  ‖q̈ _d − d_k‖ < β, where β is an unknown positive parameter.

• i3)

  θ(t) = [ξ ^T (t) β] ^T , where θ consists of system structure parameters and β.

The tracking control problem of a MEMS gyroscope is what this paper considers, so the control target is to design a suitable control law to make the system output q_k track the state of the desired trajectory q_d asymptotically. The block diagram of AILC for a MEMS gyroscope is shown in Figure 2; the tracking error between the reference state and the gyroscope state comes to the adaptive iterative learning controller.

Figure 2.

Block diagram of AILC for a MEMS gyroscope

Now, we can state the following result:

Theorem 1. If the control law (8), with the properties (i1–i3) and the parameter adaptation law (9), is applied to the nonlinear uncertain system defined by (7), then the system's tracking error can converge to zero, and all the closed-loop signals can be guaranteed bounded. Namely:

$u_k(t)=K_P{e}_k(t)+K_D{\stackrel{.}{e}}_k(t)+\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\widehat{\theta }}_k(t)$ (8)

with:

${\widehat{\theta }}_k(t)={\widehat{\theta }}_{k-1}(t)+\mathrm{\Gamma }{\phi }^T(q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\stackrel{.}{e}}_k(t)$ (9)

where ${\widehat{\theta }}_k(t)$ is the estimated value of an unknown parameter vector θ, ${\widehat{\theta }}_{-1}(t)=0,e_k(t)=q_d(t)-q_k(t)$ and ė _k (t) = q̇ _d (t) − q̇ _k (t). The matrix $\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k)\in {ℝ}^{2\times 8}$ is defined as $\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k)\stackrel{def}{=}[\begin{array}{cc}\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)& \mathrm{sgn}({\stackrel{.}{e}}_k)\end{array}]$ , where $\mathrm{sgn}({\stackrel{.}{e}}_k)\in {ℝ}^2$ is the vector obtained by applying the signum function to all the elements of ė _k . The matrices $K_P\in {ℝ}^{2\times 2},K_D\in {ℝ}^{2\times 2}$ and $\mathrm{\Gamma }\in {ℝ}^{8\times 8}$ are symmetric positive definite. Let assumption 1–2 be satisfied; then e_k(t), ė _k (t) are bounded for all $k\in \mathbb{Z}+$ and $\underset{k\to \infty }{\lim }{e}_k(t)=\underset{k\to \infty }{\lim }{\stackrel{.}{e}}_k(t)=0,\forall t\in [0,T]$ .

3.2 Control Algorithm Design

The proof of the above theorem is in the next three steps. The first step is to take a positive definite Lyapunov-like composite energy function, namely W_k, and show that this sequence is non-increasing with respect to k, and hence bounded if W₀ is bounded. In the second step, we show that W₀(t) is bounded for all t ∈ [0,T]. Finally, we show that $\underset{k\to \infty }{lim}{e}_k(t)=\underset{k\to \infty }{lim}{\stackrel{.}{e}}_k(t)=0,\forall t\in [0,T]$ .

Proof. Let us consider the following positive definite function as a Lyapunov function candidate:

$\begin{array}{l}{W}_k(e_k(t),{\stackrel{.}{e}}_k(t),{\tilde{\theta }}_k(t))=V_k(e_k(t),{\stackrel{.}{e}}_k(t))\\ +\frac{1}{2}{\displaystyle {\int }_0^t{\tilde{\theta }}^T{}_k(t)}{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k(t)d\tau \end{array}$ (10)

where $V_k(e_k(t),{\stackrel{.}{e}}_k(t))=\frac{1}{2}(e_k^T{K}_p{e}_k+{\stackrel{.}{e}}_k^T{\stackrel{.}{e}}_k),{\tilde{\theta }}_k(t)=\theta (t)-{\widehat{\theta }}_k(t)$ is the estimation error between the actual value θ _k (t) = [ξ ^T _k (t) β _k (t)] ^T and its estimated value ${\widehat{\theta }}_k(t)={[\begin{array}{cc}{\widehat{\xi }}_k^T(t)& {\widehat{\beta }}_k(t)\end{array}]}^T$ .

Differentiating W_k with respect to the iteration yields, we have:

$\begin{array}{l}\mathrm{\Delta }{W}_k=W_k-W_{k-1}=V_k-V_{k-1}\\ +\frac{1}{2}{\displaystyle {\int }_0^t({\tilde{\theta }}^T{}_k(t)}{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k(t)-{\tilde{\theta }}^T{}_{k-1}(t){\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_{k-1}(t))d\tau \end{array}$ (11)

We define ${\bar{\theta }}_k={\widehat{\theta }}_k(t)-{\widehat{\theta }}_{k-1}(t)$ ; then, we obtain ${\bar{\theta }}_k=-{\tilde{\theta }}_k(t)+{\tilde{\theta }}_{k-1}(t)$ and ${\tilde{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k-{\tilde{\theta }}^T{}_{k-1}{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_{k-1}=-{\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\bar{\theta }}_k-2{\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k$ .

Substituting the above equation into (11) generates:

$\mathrm{\Delta }{W}_k=V_k-V_{k-1}-\frac{1}{2}{\displaystyle {\int }_0^t({\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\bar{\theta }}_k+2{\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k)d\tau }$ (12)

Note that ${\displaystyle {\int }_0^t{\stackrel{.}{V}}_k(t)d\tau }=V_k(t)-V_k(0)$ , i.e., $V_k(t)=V_k(0)+{\displaystyle {\int }_0^t{\stackrel{.}{V}}_k(t)d\tau }$ and $\begin{array}{l}{\stackrel{.}{V}}_k(t)={\stackrel{.}{e}}_k^T{\stackrel{..}{e}}_k+{\stackrel{.}{e}}_k^T{K}_P{e}_k={\stackrel{.}{e}}_k^T({\stackrel{..}{e}}_k+K_P{e}_k)={\stackrel{.}{e}}_k^T({\stackrel{..}{q}}_d-{\stackrel{..}{q}}_k+K_P{e}_k)\\ ={\stackrel{.}{e}}_k^T({\stackrel{..}{q}}_d-d_k+(D+2\mathrm{\Omega }){\stackrel{.}{q}}_k+K{q}_k-K_D{\stackrel{.}{e}}_k-\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\widehat{\theta }}_k)\end{array}$ .

Hence:

$\begin{array}{l}{V}_k(t)=V_k(0)+{\displaystyle {\int }_0^t{\stackrel{.}{V}}_k(t)d\tau }=V_k({\stackrel{.}{e}}_k(0),e_k(0))\\ +{\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T({\stackrel{..}{q}}_d-d_k+(D+2\mathrm{\Omega }){\stackrel{.}{q}}_k+K{q}_k-K_D{\stackrel{.}{e}}_k-\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\widehat{\theta }}_k)d\tau }\end{array}$ (13)

where ${\stackrel{.}{e}}_k^T({\stackrel{..}{q}}_d-d_k)\le \Vert {\stackrel{.}{e}}_k^T\Vert \beta ={\stackrel{.}{e}}_k^T\beta \mathrm{sgn}({\stackrel{.}{e}}_k)$ , $K{q}_k(t)+(D+2\mathrm{\Omega }){\stackrel{.}{q}}_k(t)=\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)\xi$ , $\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)={\left(\begin{array}{ccccccc}{\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& 0& -2{\stackrel{.}{q}}_2& q_1& q_2& 0\\ 0& {\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& -2{\stackrel{.}{q}}_1& 0& q_1& q_2\end{array}\right)}_k$ , $\xi ={\left[\begin{array}{ccccccc}{d}_{xx}& d_{xy}& d_{yy}& {\mathrm{\Omega }}_z& {\omega }_x^2& {\omega }_{xy}& {\omega }_y^2\end{array}\right]}^T$ .

Further, we have

$\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)\xi +\beta \mathrm{sgn}({\stackrel{.}{e}}_k)=\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k)\theta$ , $\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k)={\left(\begin{array}{cccccccc}{\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& 0& -2{\stackrel{.}{q}}_2& q_1& q_2& 0& e_1\\ 0& {\stackrel{.}{q}}_1& {\stackrel{.}{q}}_2& -2{\stackrel{.}{q}}_1& 0& q_1& q_2& e_2\end{array}\right)}_k$ , $\theta ={\left[\begin{array}{cccccccc}{d}_{xx}& d_{xy}& d_{yy}& {\mathrm{\Omega }}_z& {\omega }_x^2& {\omega }_{xy}& {\omega }_y^2& \beta \end{array}\right]}^T$ . Substituting these equations into (13) yields:

$\begin{array}{l}{V}_k(t)\le V_k(0)+\\ {\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T(\beta \mathrm{sgn}({\stackrel{.}{e}}_k)+\mathrm{\Psi }(q_k,{\stackrel{.}{q}}_k)\xi -K_D{\stackrel{.}{e}}_k-\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\widehat{\theta }}_{k}d\tau }\\ =V_k({\stackrel{.}{e}}_k(0),e_k(0))+{\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T(\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\tilde{\theta }}_k-K_D{\stackrel{.}{e}}_k)d\tau }\end{array}$ (14)

According to Assumption 2, e_k(0) = 0 and ė _k (0) = 0, such that V_k(ė _k (0), e_k(0)). Thus:

$V_k(t)\le {\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T(\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\tilde{\theta }}_k-K_D{\stackrel{.}{e}}_k)d\tau }$ (15)

Considering the adaptive law ${\widehat{\theta }}_k(t)-{\widehat{\theta }}_{k-1}(t)=\mathrm{\Gamma }{\phi }^T(q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\stackrel{.}{e}}_k(t)={\bar{\theta }}_k(t)$ , we can obtain ${\bar{\theta }}^T{}_k(t)={\stackrel{.}{e}}_k^T(t)\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k)\mathrm{\Gamma }$ and it leads to:

${\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\bar{\theta }}_k={\stackrel{.}{e}}_k^T\phi \mathrm{\Gamma }{\mathrm{\Gamma }}^{-1}\mathrm{\Gamma }{\phi }^T{\stackrel{.}{e}}_k={\stackrel{.}{e}}_k^T\phi \mathrm{\Gamma }{\phi }^T{\stackrel{.}{e}}_k$ (16) $2{\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k=2{\stackrel{.}{e}}_k^T\phi \mathrm{\Gamma }{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k=2{\stackrel{.}{e}}_k^T\phi {\tilde{\theta }}_k$ (17)

Substituting (15–17) into (12) generates:

$\begin{array}{l}\mathrm{\Delta }{W}_k=V_k-V_{k-1}-\frac{1}{2}{\displaystyle {\int }_0^t({\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\bar{\theta }}_k+2{\bar{\theta }}^T{}_k{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_k)d\tau }\\ \le -V_{k-1}+{\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T(\phi {\tilde{\theta }}_k-K_D{\stackrel{.}{e}}_k)d\tau }\\ -\frac{1}{2}{\displaystyle {\int }_0^t({\stackrel{.}{e}}_k^T\phi \mathrm{\Gamma }{\phi }^T{\stackrel{.}{e}}_k+2{\stackrel{.}{e}}_k^T\phi {\tilde{\theta }}_k)d\tau }\\ =-V_{k-1}-\frac{1}{2}{\displaystyle {\int }_0^t{\stackrel{.}{e}}_k^T(\phi \mathrm{\Gamma }{\phi }^T+K_D){\stackrel{.}{e}}_{k}d\tau }\le 0\end{array}$ (18)

Hence, W_k is a non-increasing sequence. Thus, if W₀(t) is bounded, one can conclude that W_k is bounded.

In the next step, we will show that W₀(t) is bounded for all t ∈ [0,T].

From the definition of W₀(t) in (10), we have:

${\stackrel{.}{W}}_0({\stackrel{.}{e}}_0(t),e_0(t),{\tilde{\theta }}_0(t))={\stackrel{.}{V}}_0(e_0(t),{\stackrel{.}{e}}_0(t))+\frac{1}{2}{\tilde{\theta }}^T{}_0(t){\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0(t)$ (19)

In fact, differentiating (15) with k = 0, we have ${\stackrel{.}{V}}_k(t)\le {\stackrel{.}{e}}_k^T(\phi (q_k,{\stackrel{.}{q}}_k,{\stackrel{.}{e}}_k){\tilde{\theta }}_k-K_D{\stackrel{.}{e}}_k)$ . Thus, the time-derivative of W₀(t) in (19) can be bounded as follows:

${\stackrel{.}{W}}_0\le {\stackrel{.}{e}}_0^T(\phi {\tilde{\theta }}_0-K_D{\stackrel{.}{e}}_0)+\frac{1}{2}{\tilde{\theta }}^T{}_0{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0$ (20)

Since ${\widehat{\theta }}_{-1}(t)=0$ , we have ${\widehat{\theta }}_0(t)={\bar{\theta }}_0(t)=\mathrm{\Gamma }{\phi }^T(q_0,{\stackrel{.}{q}}_0,{\stackrel{.}{e}}_0){\stackrel{.}{e}}_0(t)$ such that ${\stackrel{.}{e}}_0^T\phi ={\widehat{\theta }}^T{}_0{\mathrm{\Gamma }}^{-1}$ . Hence:

$\begin{array}{l}{\stackrel{.}{W}}_0\le -{\stackrel{.}{e}}_0^T{K}_D{\stackrel{.}{e}}_0+({\widehat{\theta }}^T{}_0+\frac{1}{2}{\tilde{\theta }}^T{}_0){\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\\ =-{\stackrel{.}{e}}_0^T{K}_D{\stackrel{.}{e}}_0+({\theta }_0^T-{\tilde{\theta }}_0^T+\frac{1}{2}{\tilde{\theta }}^T{}_0){\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\\ =-{\stackrel{.}{e}}_0^T{K}_D{\stackrel{.}{e}}_0-\frac{1}{2}{\tilde{\theta }}^T{}_0{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0+{\theta }_0^T{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\end{array}$ (21)

Using Young's inequality, i.e., a² + b² ≥ 2ab, we have ${\theta }_0^T{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\le K_1{\Vert {\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\Vert }^2+\frac{1}{4K_1}{\Vert \theta \Vert }^2$ for any K₁ > 0, since K_D is a positive definite matrix and its eigenvalues satisfying λ_d2 ≥ λ_d1 ≥ 0 such that $-{\stackrel{.}{e}}_0^T{K}_D{\stackrel{.}{e}}_0\le -{\lambda }_{d1}{\Vert {\stackrel{.}{e}}_0\Vert }^2$ . Similarly, we have $-\frac{1}{2}{\tilde{\theta }}^T{}_0{\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\le -\frac{1}{2}{\lambda }_1{\Vert {\tilde{\theta }}_0\Vert }^2$ and $K_1{\Vert {\mathrm{\Gamma }}^{-1}{\tilde{\theta }}_0\Vert }^2\le K_1{\lambda }_2^2{\Vert {\tilde{\theta }}_0\Vert }^2$ where λ₁ and λ₂ denote the minimum and maximum eigenvalues of Γ⁻¹ respectively.

Hence:

$\begin{array}{l}{\stackrel{.}{W}}_0\le -{\lambda }_{d1}{\Vert {\stackrel{.}{e}}_0\Vert }^2-\frac{1}{2}{\lambda }_1{\Vert {\tilde{\theta }}_0\Vert }^2+K_1{\lambda }_2^2{\Vert {\tilde{\theta }}_0\Vert }^2+\frac{1}{4K_1}{\Vert \theta \Vert }^2\\ =-{\lambda }_{d1}{\Vert {\stackrel{.}{e}}_0\Vert }^2-(\frac{1}{2}{\lambda }_1-K_1{\lambda }_2^2){\Vert {\tilde{\theta }}_0\Vert }^2+\frac{1}{4K_1}{\Vert \theta \Vert }^2\end{array}$ (22)

Taking $K_1\le \frac{{\lambda }_1}{2{\lambda }_2^2}$ , we obtain $\frac{1}{2}{\lambda }_1-K_1{\lambda }_2^2\ge 0$ . Eq. (22) can be expressed as ${\stackrel{.}{W}}_0\le \frac{{\theta }_{\max }^2}{4K_1}$ , which implies that W₀(t) is uniformly continuous and thus bounded over [0,T].

Since W_k(t) is a Lyapunov-like composite energy function and its sequence is non-increasing, we have 0 ≤ W_k(t) ≤ W₀, ∀k ≥ 1. On the other hand, because W₀(t) is bounded over [0,T], we can conclude that W_k(t) is bounded $\forall k\in \mathbb{Z}+,\forall t\in [0,T]$ .

Note that W_k(t) can be written as $W_k=W_0+{\displaystyle \sum _{j=1}^k\mathrm{\Delta }{W}_j}$ . Hence, using (18), we have:

$W_k\le W_0-{\displaystyle \sum _{j=1}^k{V}_{j-1}}\le W_0-\frac{1}{2}{\displaystyle \sum _{j=1}^k({\stackrel{.}{e}}_{j-1}^T{\stackrel{.}{e}}_{j-1}+e_{j-1}^T{K}_P{e}_{j-1})}$ (23)

which leads to:

${\displaystyle \sum _{j=1}^k({\stackrel{.}{e}}_{j-1}^T{\stackrel{.}{e}}_{j-1}+e_{j-1}^T{K}_P{e}_{j-1})}\le 2(W_0-W_k)\le 2W_0$ (24)

Eq. (24) implies that $\underset{k\to \infty }{lim}{e}_k(t)=\underset{k\to \infty }{lim}{\stackrel{.}{e}}_k(t)=0,\forall t\in [0,T]$ . When the iterations tend towards infinity, the designed controller can guarantee the stability of the system and make the output tracking error converge to zero completely.

4. Simulation example

A simulation example with a z-axis vibratory MEMS gyroscope is implemented in this section for the purpose of evaluating the performance of the AILC scheme. The parameters of the adopted MEMS gyroscope are as:

$\begin{array}{l}m=1.8\times {10}^{-7},k_{xx}=63.955N∕m,k_{yy}=95.92N∕m,\\ k_{xy}=12.779N∕m,d_{xx}=1.8\times {10}^{-6}N\cdot s∕m,\\ d_{yy}=1.8\times {10}^{-6}N\cdot s∕m,d_{xy}=3.6\times {10}^{-7}N\cdot s∕m.\end{array}$

Since the general displacement range of the MEMS gyroscope in each axis is at the sub-micrometre level, it is reasonable to choose 1μm as the reference length q₀. Given that the usual natural frequency of each axis of a MEMS gyroscope is within the KHz range, we thus choose ω₀ as 1 KHz. The unknown angular velocity is assumed to be Ω _z = 100rad / s. As such, the non-dimensional values of the MEMS gyroscope parameters are listed as follows:

ω² _x = 355.3, ω² _y = 532.9, ω _xy = 70.99, d_xx = 0.01, d_yy = 0.01, d_xy = 0.002, Ω _z = 0.1. Because all the gyroscope parameters are unknown or uncertain, they are invisible to our controller.

In this simulation example, the desired motion trajectories are described by $q_d=\left[\begin{array}{c}{x}_d\\ y_d\end{array}\right]=\left[\begin{array}{c}\sin (6.17t)\\ 1.2\sin (5.11t)\end{array}\right]$ and the initial values of the system are selected as q_k(0) = q_d(0), q̇ _k (0) = q̇ _d (0). Moreover, the coefficients of the adaptive iterative learning controller are given as $K_P=\left[\begin{array}{cc}70& 0\\ 0& 70\end{array}\right],K_D=\left[\begin{array}{cc}70& 0\\ 0& 70\end{array}\right]$ and the adaptive gain is chosen as Γ = diag(90,90,90,90,90,90,90,90). The disturbances are chosen as a random signal $d_k=\left[\begin{array}{c}rand(1)\sin (t)\\ rand(1)\sin (t)\end{array}\right]$ .

In order to demonstrate the effectiveness of the control algorithm, a comparable investigation is accomplished between the proposed AILC and PD-type iterative learning controller (PD-ILC). The PD iterative learning controller is defined as u_k(t) = u_k−1(t) + K_pek(t) + K_Dek(t), where u₋₁(t) = 0 and the other parts, such as the definitions of the variables and parameters' values, are the same as the AILC algorithm. In the simulation, the iterations of these two iterative algorithms are set as five-times over the time interval [0,2s]. The simulation results are shown in Figures 3∼5.

Figure 3.

Comparison of the position tracking of the x-axis and the y-axis between AILC and PD-ILC

Figure 4.

Maximum absolute value of the x-axis and y-axis position errors for the two methods

Figure 5.

Comparison of the velocity tracking of the x-axis and the y-axis between the AILC and the PD-ILC

Figure 3 depicts the MEMS gyroscope along the x-axis and the y-axis position tracking trajectories between the AILC algorithm and the PD-ILC algorithm in the fifth iteration cycle. It can be observed intuitively that the actual motion trajectory of the MEMS gyroscope is consistent with the desired reference trajectory using the AILC algorithm, demonstrating that the adaptive tracking performance is satisfactory. However, the actual motion trajectory has an obvious interval from the reference trajectory when utilizing the PD-ILC algorithm.

Figure 4 describes that the maximum absolute value of the x-axis and the y-axis position errors in two methods of five iterations, where e_x = max(|x_d(t) − x_k(t)|), e_y = max(|y_d(t) − y_k(t)|), ∀t∈ [0,T]. From Figure 4, it can be seen that the maximum absolute value of the x-axis and the y-axis corresponding to the AILC decreases obviously in contrast with the PD-ILC.

Figure 5 and Figure 6 show the MEMS gyroscope velocity tracking trajectories and the maximum absolute value of the velocity tracking errors along the x-axis and the y-axis between two methods. The maximum absolute value of the velocity tracking errors along the x-axis and the y-axis are defined as ${\stackrel{.}{e}}_x=\max (\mid {\stackrel{.}{x}}_d(t)-{\stackrel{.}{x}}_k(t)\mid ),{\stackrel{.}{e}}_y=\max (\mid {\stackrel{.}{y}}_d(t)-{\stackrel{.}{y}}_k(t)\mid )$ . It can be clearly observed from Figures 5–6 that the velocity tracking performance using the AILC method is much better than that utilizing the PD-ILC method.

Figure 6.

Maximum absolute value of the x-axis and y-axis velocity errors for the two methods

5. Conclusion

In this paper, we have proposed the design of a new AILC for MEMS vibratory gyroscopes under unknown model uncertainties and external disturbances. First, the dynamic model of a two-axis MEMS vibratory gyroscope is briefly introduced. Next, the AILC approach is used to drive the trajectory tracking errors to rapidly converge to zero. This control scheme consists of conventional PD feedback control, unknown parameters and the tracking error of iteration. Based on Lyapunov's stability theory, we construct a Lyapunov definite function candidate, demonstrating that the position and velocity tracking errors all converge to zero. Finally, comparative simulation results between the AILC algorithm and a PD-ILC algorithm are implemented to verify the effectiveness of the proposed AILC strategies.