Reduced-Order Modeling of Parametrically Excited Micro-Electro-Mechanical Systems (MEMS)

Abstract

Reduced-order modeling is a systematic way of constructing models with smaller number of states that can capture the “essential dynamics” of the large-scale systems, accurately. In this paper, reduced-order modeling and control techniques for parametrically excited MEMS are presented. The techniques proposed here use the Lyapunov-Floquet (L-F) transformation that makes the linear part of transformed equations time invariant. In this work, three model reduction techniques for MEMS are suggested. First method is simply an application of the well-known Guyan-like reduction method to nonlinear systems. The second technique is based on singular perturbation, where the transformed system dynamics is partitioned as fast and slow dynamics and the system of differential equations is converted into a differential algebraic (DAE) system. In the third technique, the concept of invariant manifold for time-periodic systems is used. The “time periodic invariant manifold” based technique yields “reducibility conditions”. This is an important result because it helps us to understand the various types of resonances present in the system. These resonances indicate a tight coupling between the system states, and in order to retain the dynamic characteristics, one has to preserve all these “resonant” states in the reduced-order model. Thus, if the “reducibility conditions” are satisfied, only then a nonlinear order reduction based on invariant manifold approach is possible. It is found that the invariant manifold approach yields the most accurate results followed by the nonlinear projection and linear technique. These methodologies are general, free from small parameter assumptions, and can be applied to a variety of MEM systems like resonators, sensors and filters. The reduced-order models can be used for parametric study, sensitivity analysis and/or controller design. The controller design is based on the reduced-order system. Thus, first the reduced-order model of the large-scale system is constructed that captures the essential dynamics. If a controller is designed to stabilize this reduced-order system, then it guarantees that the large-scale system is controlled. The theoretical framework to design linear and nonlinear controllers is also presented.

1. Introduction

Micro-Electro-Mechanical Systems (MEMS) are miniature systems capable of controlling mechanical structures. These systems are fabricated using I.C. processing technology. A lot of research initiatives are supported by the governments, automobile and space organizations to develop micro-sensors, micro-machined gyroscopes, accelerometers, switches, micro-engines, micro-pumps and so on. The numbers of MEM devices continue to grow exponentially and national attention is focused on MEMS technology and education [1].

Even though the MEMS technology has advanced a lot, enabling innovative designers to design and make new MEM devices (which were not possible a few years ago) and experimentalists to conduct various experiments for performance and efficiency evaluation, there seems to be a “gap” between the theoretical predictions and experimental results. This deviation is predominant especially when these miniature systems are operated at extreme, critical conditions. Many times these discrepancies are attributed to the flaws in manufacturing, experimental errors, human errors and scaling effects. However, if one critically evaluates the arguments presented in this regard, it is not difficult to observe that the fundamental reason behind this issue seems to be that the mathematical models developed for a particular miniature system may not be accurate enough to capture all of the dynamics. The assumptions regarding linearity, magnitude of forces, some parameters being small (correct at macro-level) may not be valid at micro-level. Thus, a MEM device based on an inadequate mathematical model or analyzed using the approximate techniques may not capture all of the dynamical effects, especially when it is used for very crucial applications like medical, defense, aerospace, safety sensors and operated under extreme conditions like large operating range, noise, as well as thermal and magnetic effects.

The mathematical models describing MEMS dynamics have lots of degrees of freedom (arising either from a complex model or descretizing the governing partial differential equations). These modes are highly nonlinear and difficult to analyze within the constraints of limited time, computational resources, efforts, and desired accuracy. One possible way to analyze these systems is by the means of “order reduction”. Order reduction is replacing the original large-scale system by an equivalent small-scale system, which is easier to analyze. The response of this small-scale system is similar to the original large-scale system and it preserves all stability and bifurcation characteristics. One can investigate the effect of change in the system parameters on the stability of this small system and extend the conclusions to the stability of large-scale system. If one designs a controller for this small-scale system to achieve asymptotic stability, a limit cycle (by means of bifurcation control) or marginal stability then the same behavior will be depicted by the large-scale system. Controlling a very large-scale system by means of reduced-order controller has lots of advantages like relatively easy controller design, less controller effort and suitability for real time implementation.

In this research effort, we concentrate upon order reduction and control issues related to time periodic MEM systems. These MEMS are described by linear or nonlinear mathematical models, where the periodic coefficients multiply the state vectors. These types of systems constitute a large class of MEMS like resonant oscillators [2–4], micro-cantilevers [5, 6], micro-flight mechanisms [7], parametrically excited MEMS based filters [8] and miniature rate gyroscopes in electrostatic force fields [9, 10]. These systems exhibit very rich and interesting dynamics and are generally described by coupled Mathieu type equations [11]. Even a single DOF system described by a linear Mathieu type equation is not easy to analyze. It has regions of stability and instability in certain parameter range. Conventionally, these types of problems are analyzed using the averaging or perturbation type approaches [11]. These methodologies depend upon assumptions that there exists a generating solution and the parametric excitation terms are small. For general MEMS, either or both the assumptions may not be valid implying that the analysis performed using these techniques may not yield accurate results and the subsequent experimental findings deviate from them.

The order reduction and bifurcation studies for nonlinear time periodic systems have not received considerable attention in the structural dynamics community and never attempted for parametrically excited nonlinear MEM system. On the other hand, there exists a vast amount of knowledge on order reduction of time invariant systems at macro-level [12]. Many MEMS researchers have attempted order reduction of MEMS using these techniques [13]. Typically, this order reduction problem is two pronged. The first one is the order reduction from structural aspect where the aim is to construct a reduced-order system that retains the structure of the original system either in the state space or in second order form. This reduced-order system is then used for response analysis [14], sensitivity analysis [15], optimization and so on. Another approach to view this order reduction problem is controller design, which is very active research area in control community [16]. Here the problem is formulated in the state space with a controller. The system order is reduced and finally the analyst comes up with a suitable reduced-order controller that can be used to control the original large-scale system.

From structural point of view, various order reduction techniques like classical Guyan reduction [17] and its extensions to nonlinear systems can be used. For nonlinear systems, the order reduction techniques based on “nonlinear normal modes” [18] (which are the nonlinear counterpart of linear normal modes) can be used. Shaw and Pierre [19] presented an alternate formulation for nonlinear normal modes by applying the center manifold theory. Afterwards Shaw et al. [20] presented a rigorous order reduction technique where the configuration subspace is confined to a high dimensional curved manifold, described by “master” degrees of freedom of the subset. Similar ideas are presented by the structural dynamics researchers [21–24] for order reduction in second order form. Gabby et al. [23] successfully implemented an approach based on basis function and presented an automated procedure for constructing reduced-order models for MEMS that captures the nonlinear effects. The same approach was modified to incorporate stress-stiffening effects [24]. Though these methodologies yield good results, they cannot be used directly for order reduction of time periodic systems.

From control point of view, quite a few order reduction techniques are available. For small linear systems, balanced truncation approximation [25], singular perturbation approximation [26] and Hankel norm approximation [27] can be used and are successfully implemented in software routines like MATLAB [28] and SILICOT [29]. These methods require solutions of the Lyapunov equations or Singular Value Decomposition, which can be a computationally demanding task. For large linear systems, order reduction algorithms based on Krylov spaces [30] are proposed. Krylov spaces, in most cases are good candidates for lower order subspace. The most popular algorithms based on Krylov space are Lancoz [31] and Arnoldi [32] methods. Another interesting approach of order reduction is Padé and Padé type approximations [33]. The objective here is to find out a rational transfer function of smaller dimension, which retains the essential behavior of the large dimensional transfer function. This is done by matching moments of the original and reduced-order transfer function around some fixed point [34]. All of these techniques are applicable to autonomous linear systems. For nonlinear systems order reduction and controller design techniques based on Proper Orthogonal Decomposition (POD) and Empirical eigenvectors [35] are proposed. The techniques based on POD are more suitable when experimental data is available.

Recently, Sinha et al. [36] presented a novel technique for order reduction of nonlinear time periodic systems based on “time periodic invariant manifold technique”. This technique is based on the Lyapunov-Floquet (L-F) transformation [37], which converts a linear time varying system into a time invariant one and at the same time preserves all stability and bifurcation properties. This approach also yields unique “reducibility” conditions, which answer very fundamental questions in order reduction. “Is order reduction really possible? and if so then which states should be retained and which should be eliminated?”. Some new techniques based on the L-F transformation and singular perturbation are discussed by Redkar, et al. [38]. From control point of view, Deshmukh et al. [39] proposed a method for order reduction of a large-scale linear system and implemented control algorithm successfully. So far, these techniques are applied to large-scale structural systems. However, preliminary results show that similar techniques can be used for order reduction and control of MEM devices. These approaches are free from “small parameter” assumption and take into account “strong” nonlinear effects. It is expected that the order reduction and control techniques based on the L-F transformation and “invariant manifold” techniques will show a new path in the research of parametrically excited nonlinear systems and reduce the “gap” between theoretical results and experimental findings. These tools will certainly help MEMS researchers to design, study, simulate, analyze, optimize and control complex parametrically excited MEMS (and possibly NEMS) systems in a better way.

2. Model Order Reduction Techniques

2.1. Order Reduction via Linear Projection

Consider a nonlinear time periodic MEM system described by

\dot{x} (t) = A (t) x (t) + f (x, t),

(1)

where x is an n vector of states, $A (t)$ is an $n \times n$ time periodic matrix and $f (x, t)$ is a nonlinear time periodic nvector such that $f (0, t) = 0$ .

Applying the L-F transformation [37] $x (t) = Q (t) \bar{y} (t)$ produces

\dot{\bar{y}} (t) = R \bar{y} (t) + Q^{- 1} (t) f (\bar{y}, t),

(2)

where $Q (t)$ is the L-F transformation matrix of dimension $n \times n$ .

After the modal transformation $\bar{y} (t) = M z (t),$ we have

\dot{z} (t) = J z (t) + M^{- 1} Q^{- 1} (t) f (z, t) \equiv J z (t) + w (z, t),

(3)

where J is the Jordan form of R and $w (z, t)$ represents an appropriately defined nonlinear time varying vector consisting of monomials of z_j.

The objective of order reduction is to replace the nonlinear time periodic system given by (3) by an equivalent system given by

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + w_{r} (z_{r}, t),

(4)

where z_r is an $r (r \geq n)$ vector of dominant states, J_r is an $r \times r$ Jordan block corresponding to dominant states and $w_{r} (z_{r}, t)$ is a nonlinear vector function of appropriate dimensions consisting only of dominant states. Equation (4) can be integrated numerically and using the transformation $x (t) = Q (t) M z_{r} (t)$ , all of the states in x can be recovered.

This linear projection technique assumes $z (t) \approx z_{r} (t)$ . It is simple and easy to implement. However, the selection of dominant states depends upon the judgment of analyst. It does not give a clear insight of system dynamics if the system dynamics is complex and involves internal and/or parametric resonance.

2.2. Order Reduction Using Nonlinear Projection

Once again, consider a nonlinear time periodic MEM system described by

\dot{x} (t) = A (t) x (t) + f (x, t),

(5)

where $x (t)$ , $A (t)$ , $f (x, t)$ are as defined before. Using the L-F and modal transformation equation (5) can be written as

\dot{z} (t) = J z (t) + w (z, t),

(6)

where $z (t)$ , J, $w (z, t)$ are as defined before.

The objective here again is to approximate (6) by a reduced-order system given by

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + w_{r} (z_{r}, ψ (z_{r}, t), t),

(7)

where $z_{r} (t)$ , J_r, $w_{r} (z_{r}, t)$ are as defined before and $ψ (z_{r}, t)$ is a nonlinear time periodic operator relating non-dominant states to dominant states as

z_{s} (t) = ψ (z_{r}, t) .

(8)

This transformation from (6) to (7) is achieved by means of a nonlinear projection given by (8). As before, (7) can be integrated numerically and using the L-F and modal transformation all of the states in x can be recovered.

Depending upon the complexity of large-scale system, availability of computational power and the required accuracy of reduced-order system, one may choose the method to compute the nonlinear projection $ψ (z_{r}, t)$ . For details, refer to Redkar et al. [38]. The nonlinear projection method yields “a non-flat” sub-manifold which itself is time periodic and this technique is expected to yield better results than its linear counterpart.

2.3. Order Reduction Using Time Periodic Invariant Manifold

This methodology is based on the “Time Periodic Invariant Manifold Theory”. According to this theory, there exists a time periodic relationship between the dominant and the non-dominant states of the system and it is possible to replace (under certain conditions) the non-dominant states by dominant states. Thus, the order of system can be reduced.

Consider again, the nonlinear time periodic MEM system,

\dot{x} (t) = A (t) x (t) + f (x, t)

(9)

with all of the terms appearing in (9) being defined as before. After the L-F and modal transformation equation (9) can be transformed into (6) and further partitioned as

\begin{matrix} {\begin{Bmatrix} {\dot{z}}_{r} \\ {\dot{z}}_{s} \end{Bmatrix}} = [\begin{bmatrix} J_{r} & 0 \\ 0 & J_{s} \end{bmatrix}] {\begin{Bmatrix} z_{r} \\ z_{s} \end{Bmatrix}} + {\begin{Bmatrix} w_{r} (z_{r}, z_{s}, t) \\ w_{s} (z_{r}, z_{s}, t) \end{Bmatrix}} & \begin{matrix} (a) \\ (b) \end{matrix} \end{matrix}

(10)

At this stage, we assume a nonlinear relationship between the dominant $(z_{r})$ and the non-dominant $(z_{s})$ states as

z_{s} = \sum_{i} h_{i} (z_{r}, t) \equiv H (z_{r}, t),

(11)

where

\begin{matrix} h_{i} = \sum_{\bar{m}} {\bar{h}}_{i} (t) z_{1}^{m_{1}} \dots z_{r}^{m_{r}}, \\ \bar{m} = {(m_{1}, \dots, m_{r})}^{T}, \\ m_{1} + \dots + m_{r} = i, i = 2,3, \dots, k . \end{matrix}

(12)

Here ${\bar{h}}_{i} (t)$ are the unknown periodic vector coefficients with period $2 T$ . Substitution of (11) into (10) ordering terms and expanding ${\bar{h}}_{i} (t)$ and $w_{s} (z_{r}, t)$ in Fourier series yields the following “reducibility condition”. (For details, see Sinha et al. [40])

\begin{gathered} \frac{\bar{i} ν π}{T} + \sum_{l = 1}^{r} (m_{l} λ_{l}) - {\bar{λ}}_{p} \neq 0, \\ \forall ν = 0, \pm 1, \pm 2 \dots, p = 1,2, \dots, s, \end{gathered}

(13)

where $λ_{1}, λ_{2}, \dots, λ_{r}$ are the eigenvalues of the Jordan matrix J_r and ${\bar{λ}}_{p}; p = 1,2, \dots, s$ are the eigenvalues of J_s.

It can be shown that if the “reducibility condition” is satisfied then vector $H (z_{r}, t)$ can be obtained and the non-dominant states can be expressed in terms of the dominant states [36]. However, when this “reducibility condition” is not satisfied, such a reduction is not possible and the non-dominant states cannot be expressed as functions of the dominant states.

Assuming the “reducibility condition” is satisfied, the reduced-order equation can be written as

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + w_{r} (z_{r}, H (z_{r}, t), t) .

(14)

As before, this reduced-order equation can be integrated numerically and using the L-F and modal transformation, all the states in x can be obtained.

3. Application: A MEM Torsional Mode Oscillator

In order to demonstrate the applications of the order reduction approaches, we consider a coupled oscillator MEM sensor-actuator system. Baskaran and Turner [3, 4] have analyzed this system to study electrostatic interaction and their effect on the system stability. The scanning electron micrograph (SEM) of the system is shown in the Figure 1(a). The system is composed of two torsional oscillators actuated using electrostatic forces as shown in Figure 1(b). For design, fabrication and parameter estimation details, we refer to the papers by Baskaran and Turner [3, 4]. The mechanical equivalent system is shown in Figure 1(c), where the springs (k_actuation) represent the electrostatic stiffness due to interdigitated comb fingers and the pendulums represent the MEM torsional beams.

MEM Torsional Oscillators [3].

Assuming the device is operated in vacuum and has negligible structural damping, the undamped equation of motion can be shown [3, 4] to be

\begin{gathered} I_{1} {\ddot{θ}}_{1} + k_{1} θ_{1} + k_{act 1} {(V_{1} - V_{2})}^{2} θ_{1} + k_{12} {(V_{2} - V_{3})}^{2} (θ_{1} - θ_{2}) = 0, \\ I_{2} {\ddot{θ}}_{2} + k_{2} θ_{2} + k_{act 2} {(V_{3} - V_{4})}^{2} θ_{2} + k_{12} {(V_{2} - V_{3})}^{2} (θ_{2} - θ_{1}) = 0, \end{gathered}

(15)

where $I_{1}, I_{2}$ are the mass moment of inertia of the oscillators, $θ_{1}, θ_{2}$ are angular displacement of the oscillators, $k_{1}, k_{2}$ are mechanical stiffness of the oscillators, $k_{act 1}, k_{act 2}, k_{12}$ are electrostatic stiffness of the comb-finger configurations, $V_{1}, V_{2}, V_{3}, V_{4}$ are external potentials applied to contact pads. These external potentials can be DC, harmonic wave, impulse, and so forth. In the original work of Baskaran and Turner [3, 4], they have used square-rooted sinusoidal signal with large amplitude range (20–70 V RMS) at approximately twice the natural frequencies and near the sum of the two natural frequencies of the torsional oscillators. Equation (15) are valid for small angular displacements. For moderate and large angular displacement, one has to consider the nonlinear effects arising from spring nonlinearity, that is, spring stiffness as a nonlinear function of angular displacement and geometric nonlinearity $(\sin θ \approx θ - θ^{3} / 6)$ and retain higher order terms in the Taylor series expansion. Thus, the nonlinear equations of motion approximated up to cubic order can be shown to be

\begin{matrix} I_{1} {\ddot{θ}}_{1} + [k_{1} + k_{act 1} {(V_{1} - V_{2})}^{2}] (θ_{1} - \frac{θ_{1}^{3}}{6}) + k_{12} {(V_{2} - V_{3})}^{2} \\ \times [A (θ_{1} - θ_{2}) - B {(θ_{1} - θ_{2})}^{2} - C {(θ_{1} - θ_{2})}^{3}] = 0, \\ I_{2} {\ddot{θ}}_{2} + [k_{2} + k_{act 2} {(V_{3} - V_{4})}^{2}] (θ_{2} - \frac{θ_{2}^{3}}{6}) + k_{12} {(V_{2} - V_{3})}^{2} \\ \times [A (θ_{2} - θ_{1}) + B {(θ_{1} - θ_{2})}^{2} + C {(θ_{1} - θ_{2})}^{3}] = 0, \end{matrix}

(16)

where A, B, C are coupling constants. Depending upon the type of external forcing voltages $(V_{1}, V_{2}, V_{3}, V_{4}),$ (16) and (18) can have time periodic and/or constant coefficients. For illustration purposes, we assume $V_{1} = 1 + \sin (ω t), V_{2} = 1, V_{3} = 2, V_{4} = 2 + \sin (ω t)$ . Noting that $si n^{2} (ω t) = (1 - \cos (2 ω t)) / 2$ , the equations can be simplified further as

\begin{matrix} {\ddot{θ}}_{1} + (ω_{n_{1}}^{2} - ε p_{1} \cos (2 ω t)) θ_{1} - (p_{11} - ε p_{1} \cos (2 ω t)) \frac{θ_{1}^{3}}{6} \\ - a θ_{2} - b {(θ_{1} - θ_{2})}^{2} - c {(θ_{1} - θ_{2})}^{3} = 0, (a), \\ {\ddot{θ}}_{2} + (ω_{n_{2}}^{2} - ε p_{2} \cos (2 ω t)) θ_{2} - (p_{22} - ε p_{2} \cos (2 ω t)) \frac{θ_{2}^{3}}{6} \\ - a θ_{1} + b {(θ_{1} - θ_{2})}^{2} + c {(θ_{1} - θ_{2})}^{3} = 0, (b) \end{matrix}

(17)

where

\begin{gathered} I_{1} = I_{2} = I, \\ ω_{n_{1}}^{2} = \frac{k_{1} + 0.5 k_{act 1} + k_{12} A}{I}, ω_{n_{2}}^{2} = \frac{k_{2} + 0.5 k_{act 2} + k_{12} A}{I}, \\ ε p_{1} = \frac{0.5 k_{act 1}}{I}, ε p_{2} = \frac{0.5 k_{act 2}}{I}, \\ p_{11} = \frac{k_{1} + 0.5 k_{act 1}}{I}, p_{22} = \frac{k_{2} + 0.5 k_{act 2}}{I}, \\ a = \frac{k_{12} A}{I}, b = \frac{k_{12} B}{I}, c = \frac{k_{12} C}{I} . \end{gathered}

(18)

Equation (17) can be written in the state space form as

\begin{matrix} {\begin{Bmatrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \\ {\dot{x}}_{3} \\ {\dot{x}}_{4} \end{Bmatrix}} \\ = [\begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - (ω_{n_{1}}^{2} - ε p_{1} \cos (2 ω t)) & a & 0 & 0 \\ a & - (ω_{n_{2}}^{2} - ε p_{2} \cos (2 ω t)) & 0 & 0 \end{bmatrix}] \\ \times {\begin{Bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{Bmatrix}} \\ + {\begin{Bmatrix} 0 \\ 0 \\ (p_{11} - ε p_{1} \cos (2 ω t)) \frac{x_{1}^{3}}{6} + b {(x_{1} - x_{2})}^{2} + c {(x_{1} - x_{2})}^{3} \\ (p_{22} - ε p_{2} \cos (2 ω t)) \frac{x_{2}^{3}}{6} - b {(x_{1} - x_{2})}^{2} - c {(x_{1} - x_{2})}^{3} \end{Bmatrix}} \end{matrix}

(19)

where

{\begin{Bmatrix} x_{1} & x_{2} & x_{3} & x_{4} \end{Bmatrix}}^{T} = {\begin{Bmatrix} θ_{1} & θ_{2} & {\dot{θ}}_{1} & {\dot{θ}}_{2} \end{Bmatrix}}^{T} .

(20)

Applying the L-F transformation, (19) can be written as

\dot{\bar{y}} = R \bar{y} + Q^{- 1} (t) f (\bar{y}, t),

(21)

where $\bar{y} = {{\bar{y}}_{1} {\bar{y}}_{2} {\bar{y}}_{3} {\bar{y}}_{4}}^{T}$ and R is a 4 × 4 time invariant matrix. By choosing a suitable set of values of coupling parameters $(a, b, c)$ , the system can be made to undergo “parametric resonance”, (3:1 and 2:1), “internal resonance”, “true internal resonance” or “true combination resonance”.

For the detailed discussion on these resonances, refer to Sinha et al. [36]. These resonances are discussed in the context of this example.

3.1. The Case of No Resonance

The system parameters are chosen such that the system does not exhibit any resonance, that is, the “reducibility condition” in (13) is satisfied. The parameter set for this case is given in Table 1.

Table 1:

Parameter set for the case when no parametric/internal resonance exists.

Parameter	$ω_{n_{1}}^{2}$	$ω_{n_{2}}^{2}$	$ε p_{1}$	$ε p_{2}$	p ₁₁	p ₂₂	ω	a	b	c

Value	4	5	2	2	3	4	π	1	0	3

For this set of parameters, the multipliers are $(- 0.28 \pm 0.95 i, - 0.73 \pm 0.67 i)$ . The L-F transformation matrix $Q (t)$ is computed from the linear part of (19) by the method suggested by Sinha et al. [37]. The Floquet exponents (eigenvalues of R matrix) are $(\pm 1.28 i, \pm 0.74 i)$ . It can be easily verified that the ratio of the angles of Floquet multipliers (or the ratio of Floquet exponents) for this set is 1.72, which implies they are not in “true internal resonance” and the “reducibility condition” given by (13) is satisfied.

Using the modal transformation, $\bar{y} = M z$ , where M is the 4 × 4 modal matrix of R and $z = {z_{1} z_{2} z_{3} z_{4}}^{T}$ , (21) can be written as

{\begin{Bmatrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \\ {\dot{z}}_{3} \\ {\dot{z}}_{4} \end{Bmatrix}} = [\begin{bmatrix} 0 & - 0.74 & 0 & 0 \\ 0.74 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1.28 \\ 0 & 0 & 1.28 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{1} \\ z_{2} \\ z_{3} \\ z_{4} \end{Bmatrix}} + {\begin{Bmatrix} w_{1} (z, t) \\ w_{2} (z, t) \\ w_{3} (z, t) \\ w_{4} (z, t) \end{Bmatrix}} .

(22)

where $w_{i} (z, t)$ are nonlinear functions with periodic coefficients comprising of all of the states, the long expressions for $w_{i} (z, t)$ are obtained using Mathematica and omitted here for brevity. At this stage, we reduce the order of (22) using the three methods described in Section 2.

3.1.1. Order Reduction Using the Linear Method

Equation (22) comprises of 4 states, ${z_{1} z_{2} z_{3} z_{4}}^{T}$ . We choose $z_{r} = {z_{1} z_{2}}^{T}$ as the dominant states, as they correspond to the smaller exponent of the system. Following the procedure outlined in Section 2.1, we neglect the contribution from the non-dominant states $z_{s} = {z_{3} z_{4}}^{T}$ and the system dynamics is approximated by

{\begin{Bmatrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \end{Bmatrix}} = [\begin{bmatrix} 0 & - 0.74 \\ 0.74 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{1} \\ z_{2} \end{Bmatrix}} + {\begin{Bmatrix} w_{1} (z_{1}, z_{2}, 0,0, t) \\ w_{2} (z_{1}, z_{2}, 0,0, t) \end{Bmatrix}}

(23)

Equation (23) is the reduced-order model of MEM system described by (22). This reduced-order system is integrated numerically with typical initial conditions and all the states in $x = {θ_{1} θ_{2} {\dot{θ}}_{1} {\dot{θ}}_{2}}^{T}$ are obtained using the L-F and modal transformation. In Figure 2(a), the time trace of θ₁ obtained using the above procedure is compared with the time trace of θ₁ obtained by integrating the original (19). It can be seen that these time traces do not match exactly. In order to show the long time behavior, we simulate both the reduced-order and the large-scale system for relatively longer period of time and plot the Poincaré maps (for both) sampled at the time period ( $T = 2 π / ω_{p}$ ) of parametric excitation. The Poincaré map of large-scale system is shown in Figure 3(a), while Figure 3(b) shows the Poincaré map of reduced system using the linear method. We observe that the Poincaré map of the large-scale system shows a “band” implying the existence of a complex quasiperiodic motion that is absent in the Poincaré map of linearly reduced system. However, the amplitudes are quite comparable. Thus, we may conclude that the reduced model based on the linear method captures the amplitude but does not capture the frequency content accurately for the given initial conditions.

Time trace comparison.

Comparison of Poincaré maps (no resonance case).

3.1.2. Order Reduction Using the Nonlinear Projection Based on Singular Perturbation Method

Once again, we take $z_{r} = {z_{1} z_{2}}^{T}$ as the dominant states to be retained. By following the procedure discussed in Section 2.2, we assume the derivatives of the non-dominant states to be small compared to the derivatives of the dominant states and rewrite (22) as

{\begin{Bmatrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \\ ε {\dot{z}}_{3} \\ ε {\dot{z}}_{4} \end{Bmatrix}} = [\begin{bmatrix} 0 & - 0.74 & 0 & 0 \\ 0.74 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1.28 \\ 0 & 0 & 1.28 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{1} \\ z_{2} \\ z_{3} \\ z_{4} \end{Bmatrix}} + {\begin{Bmatrix} w_{1} (z, t) \\ w_{2} (z, t) \\ w_{3} (z, t) \\ w_{4} (z, t) \end{Bmatrix}} .

(24)

We let ε = 0, and resorting to one fixed point iteration, as discussed earlier, z₃ and z₄ are approximated as

[\begin{bmatrix} 0 & - 1.28 \\ 1.28 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{3} \\ z_{4} \end{Bmatrix}} = - {\begin{Bmatrix} w_{3} (z_{1}, z_{2}, 0,0, t) \\ w_{4} (z_{1}, z_{2}, 0,0, t) \end{Bmatrix}} .

(25)

We substitute the single fixed-point iteration solution of ${z_{3}, z_{4}}$ given by (25) into the top half of (24) to obtain the reduced-order model as

{\begin{Bmatrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \end{Bmatrix}} = [\begin{bmatrix} 0 & - 0.74 \\ 0.74 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{1} \\ z_{2} \end{Bmatrix}} + {\begin{Bmatrix} {\bar{w}}_{1} (z_{1}, z_{2}, t) \\ {\bar{w}}_{2} (z_{1}, z_{2}, t) \end{Bmatrix}} .

(26)

Equation (26) is integrated numerically and using the L-F and the modal transformation x vector is constructed as before.

Once again, we compare the time trace of θ₁ obtained using the nonlinear projection technique with the time trace of θ₁ obtained by integrating the original (19). It can be seen that these time traces (given in Figure 2(b)) match better when compared to the time traces obtained by the linear method (Figure 2(a)).

When we observe the Poincaré map of reduced system (given in Figure 3(c)) via nonlinear projection, we find that it is very close to the Poincaré map of the large-scale system (given in Figure 3(a)), but does not show a “band”. However, comparing the time traces we conclude that the nonlinear projection yields a better-reduced model than the linear method.

3.1.3. Order Reduction Using an Invariant Manifold

As discussed in Section 2.3 we try to relate the non-dominant states to the dominant states by a time periodic nonlinear transformation. If the system does not exhibit “true internal resonance” (like the case under consideration) then the “reducibility condition” is satisfied and the system order can be reduced.

We start with (22) and select the same states [ $z_{r} = {z_{1} z_{2}}^{T}$ ] as the dominant states and try to find a nonlinear time periodic relationship of the form given in (11). For this particular example, the relationship between z_s and z_r are given by

z_{s} = \sum_{i} h_{i} (z_{1}, z_{2}, t) \equiv H (z_{1}, z_{2}, t), s = 3,4

(27)

where

h_{i} = \sum_{\bar{m}} {\bar{h}}_{i} (t) z_{1}^{m_{1}} \dots z_{2}^{m_{2}}, \bar{m} = {(m_{1}, m_{2})}^{T}, m_{1} + m_{2} = 3 .

(28)

Here ${\bar{h}}_{i} (t)$ are the unknown periodic vector coefficients with period $2 T$ . We substitute (27) into (22). After expanding ${\bar{h}}_{i} (t)$ and $w_{s} (z_{r}, t) (s = 3,4)$ in the Fourier series and neglecting the terms of higher order, we obtain the relationship between the dominant and the non-dominant states as

z_{3} = H_{1} (z_{1}, z_{2}, t), z_{4} = H_{2} (z_{1}, z_{2}, t) .

(29)

Equation (29) is substituted into top half of (22) to obtain the reduced-order model as

{\begin{Bmatrix} {\dot{z}}_{1} \\ {\dot{z}}_{2} \end{Bmatrix}} = [\begin{bmatrix} 0 & - 0.74 \\ 0.74 & 0 \end{bmatrix}] {\begin{Bmatrix} z_{1} \\ z_{2} \end{Bmatrix}} + {\begin{Bmatrix} {\underline{w}}_{1} (z_{1}, z_{2}, t) \\ {\underline{w}}_{2} (z_{1}, z_{2}, t) \end{Bmatrix}} .

(30)

This symbolic computation was performed using Mathematica. Numerical solution of (30) was carried out and the L-F and modal transformation was used to determine $θ_{1}, θ_{2}, {\dot{θ}}_{1},$ and ${\dot{θ}}_{2}$ .

Once again the time trace of the reduced-order system (θ₁) obtained by this method is compared with the time trace of ${\dot{θ}}_{1}$ (Figure 2(c)) obtained by numerical integration of the original system given by (22). As we can see, the difference between the two time traces is hardly distinguishable. Even the Poincaré map of the original system (Figure 3(a)) matches the Poincaré map of the reduced-order system (Figure 3(d)) quantitatively as well as qualitatively. Therefore, it can be concluded that the invariant manifold order reduction technique yields the most accurate reduced-order model when compared to the “linear” and the “nonlinear projection” methods.

3.2. The True Internal Resonance Case

For the time periodic systems, a “true internal resonance” is said to occur when some linear combination of λ_l (the Floquet exponents of dominant states) and ${\bar{λ}}_{p}$ (the Floquet exponents of non-dominant states) add up to zero for ν = 0 (cf., (20)). The types of nonlinearities present in the system determine the exact combination. In this case, the “reducibility condition” as given by (13) is not satisfied. For illustration, we select the system parameters as given in Table 2. For this set of parameters, the Floquet multipliers are $(- 0.76 \pm 0.64 i, 0.97 \pm 0.22 i)$ and the eigenvalues of R matrix (Floquet exponents) are $(\pm 0.230 i, \pm 0.703 i)$ . It is also to be observed that the H vector has the form

\begin{matrix} H_{j} = a_{1 j} (t) z_{1}^{3} z_{2}^{0} + a_{2 j} (t) z_{1}^{2} z_{2}^{1} + a_{3 j} (t) z_{1}^{1} z_{2}^{2} + a_{4 j} (t) z_{1}^{0} z_{2}^{3} \\ j = 1,2 . \end{matrix}

(31)

where $a_{i j} (t)$ are periodic functions of time.

Table 2:

Parameter values for the case when “True internal resonance” exists.

Parameter	$ω_{n_{1}}^{2}$	$ω_{n_{2}}^{2}$	$ε p_{1}$	$ε p_{2}$	p ₁₁	p ₂₂	ω	a	b	c

Value	5	6	2.5	0.5	4	5	π	1	0	3

In this case, the “reducibility condition” is not satisfied since for ν = 0, (13) yields $3 \times 0.230 i - 0.703 i \approx 0$ . Note that $m_{1} = 3, m_{2} = 0; λ_{1}, λ_{2} = \pm 0.230 i and {\bar{λ}}_{1}, {\bar{λ}}_{2} = \pm 0.703 i$ . Thus, the modes cannot be decoupled and order reduction is not possible using the invariant manifold approach.

However, linear or nonlinear projection based order reduction is certainly possible for small initial conditions when the nonlinear effects are not significant.Figure 4 shows comparison of error for linear and nonlinear projection based reduced-order models. In Figure 4, Error Norm $(e)$ is plotted against initial conditions (on log scale) where

e = \frac{\sum_{i = 1}^{n} {({θ_{1}}_{i_{Original}} - {θ_{1}}_{i_{Reduced}})}^{2}}{n} .

(32)

This norm was computed for n = 3000. It can be observed that for very small initial conditions, the error as defined by (32) between the responses of the linear reduced-order model and original system (shown as dash-dot line) is small. However, as the initial conditions increase and nonlinear effects dominate the response, the error becomes significant and grows without bounds. Similarly, for the reduced-order system via nonlinear projection the error norm defined by (32) is small for very small initial conditions but increases rapidly and continues to grow (shown by the dashed line) with a small change in the initial conditions. Thus, it can be concluded that we need to retain all of the states in the dynamical system and order reduction using linear or nonlinear methods does not yield acceptable results.

Figure 4

Comparison of Error in the case of true internal resonance.

4. Controller Design

The controller design for a MEM device is a very interesting problem. For a complex, nonlinear MEM system designing is full state controller is a very difficult task. However, one can make use of the order reduction methodologies and design a reduced-order controller, which will control the “dominant dynamics” in the system. This controller will be easier to design at the same time guarantee that the original system will meet the specifications. Application of the L-F transformation, order reduction methodologies and bifurcation studies enable researchers to design linear and nonlinear controllers. It is noted that one can design linear or nonlinear controllers using symbolic computation techniques [41, 42], Lyapunov type approach, Feedback linearization [43] and back stepping [44] to mention a few. All of these controller design approaches integrate very well to the order reduction methodology presented here. As an example, we present the theoretical framework to design a simple linear controller and nonlinear controller for the reduced-order system. There are various combinations possible like reduced-order system using linear approach along with linear or nonlinear controller, reduced-order system based on singular perturbation approach along with linear or nonlinear controller and invariant manifold based reduced-order system with linear or nonlinear approach. The detailed discussion on advantages and disadvantages of these combinations along with examples will be presented elsewhere.

4.1. Linear Controller Design

Consider a nonlinear MEM system described by

\dot{x} (t) = A (t) x (t) + f (x, t) + B (t) u,

(33)

where $x (t), A (t), f (x, t)$ are as defined before, $B (t)$ is a time periodic matrix of appropriate dimensions and u is a control vector and controllability condition is satisfied. This system may be unstable and the objective is to design a stabilizing control. After the L-F transformation the system can be transformed to

\dot{\bar{y}} (t) = R \bar{y} (t) + Q^{- 1} (t) f (\bar{y}, t) + Q^{- 1} (t) B (t) u .

(34)

The system described by (34) is also a large order system. At this stage the order reduction techniques can be applied and a reduced-order system of dimension $r (\geq n)$ can be constructed which retains the “dominant” (or unstable) dynamics to be controlled and given by

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + w_{r} (z_{r}, t) + B_{r} (t) u .

(35)

Now consider the linear part of (35):

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + B_{r} (t) u .

(36)

Assuming $u = - K z_{r} (t)$ (36) can be written as

{\dot{z}}_{r} (t) = [J_{r} - B_{r} (t) K] z_{r} (t)

(37)

with unknown gains K. It is possible to compute the Floquet Transition Matrix (FTM) of the system given by (37) symbolically [41] or numerically and place its eigenvalues inside unit circle (for asymptotic stability) by appropriate choice of K. The linear control will make the system asymptotically stable as long as the nonlinear terms are bounded in the mean [39]. By inverting the modal and L-F transformation, it is possible to find out control vector that would stabilize the system.

u = K T^{T} M^{- 1} Q^{- 1} (t),

(38)

where $T = {[I_{r \times r} : 0_{r \times (n - r)}]}^{T}$ . I and $0$ are the Identity and null matrices.

4.2. Nonlinear Controller Design

Theoretically, the necessity for nonlinear control for critical MEM systems arises only in the situation when the system has linearly uncontrollable critical modes, and all of the unstable modes are controllable. From a practical point of view, however, there may be reasons to choose nonlinear control even if the system is linearly controllable; among these reasons are better system performance and lower energy requirements that may be achieved with a nonlinear controller. It is known that sometimes performance can be significantly improved if the system is operated near a stability boundary. However, it may not be safe to do so, especially if a subcritical bifurcation occurs when the system crosses the boundary. On the other hand, if small vibrations can be allowed without causing system failure, then systems can be operated very close to the stability boundary if the bifurcation is of the supercritical type.

It should be noted that for critical systems (due to so called “resonance”) the normal form reduction cannot eliminate all of the nonlinear terms. Therefore, the resulting normal form will be not only nonlinear, but also time-dependent, in general. Thus, the nonlinear controller design to control bifurcations is an extension of bifurcation studies.

Once again, the objective is to design a nonlinear controller to control the bifurcations of the original MEM system described by large number of nonlinear differential equations with time periodic coefficients given by (34). The first step in the analysis is order reduction. Once the system order is reduced, the system dynamics is described by

{\dot{z}}_{r} (t) = J_{r} z_{r} (t) + w_{r} (z_{r}, t) + B_{r} (t) \bar{u} .

(39)

This equation is same as (35) except the control vector $\bar{u} = \bar{u} (z_{r}, t, β)$ is nonlinear and β are unknown control gains. As before, (39) can be partitioned as

[\begin{bmatrix} {\dot{z}}_{rs} \\ {\dot{z}}_{rc} \end{bmatrix}] = [\begin{bmatrix} J_{rs} & 0 \\ 0 & J_{rc} \end{bmatrix}] [\begin{bmatrix} z_{rs} \\ z_{rc} \end{bmatrix}] + [\begin{bmatrix} {\bar{f}}_{1} (z_{rs}, z_{rc}, t) \\ {\bar{f}}_{2} (z_{rs}, z_{rc}, t) \end{bmatrix}] + B_{r} (t) \bar{u}

(40)

where $z_{rc}$ are the critical (bifurcating) states to be controlled and $z_{rs}$ are the remaining states. In order to design the controller $\bar{u}$ the next step is to simplify (40) with the help of time periodic center manifold approach and time periodic normal form theory as discussed by David and Sinha [45]. In this case due to presence of unknown control gains β the resulting normal form is

\dot{y} = J_{rc} y + g (y, t, β) .

(41)

This is the simplest nonlinear form of (40), and for the codimension one bifurcations it is either time invariant or can be brought into a time invariant form by further simple coordinate transformations. For example, in case of a symmetry breaking bifurcations (when one of the Floquet multipliers is plus one and the system has a trivial equilibrium on both sides of the critical point) (41) takes the form

\dot{y} = (a + β_{1}) y^{2} + (b + β_{2}) y^{3}

(42)

which is a one dimensional time invariant equation [45]. The equilibrium of this equation (and thereby the equilibrium of (40)) can be stabilized by an appropriate choice of the control parameters $β_{1}, β_{2}$ .

5. Discussion and MEMS Application Areas

The reduced-order modeling techniques proposed in this work are general can be used for order reduction of macro as well as micro systems. They are free from small parameter assumption. We presented a MEMS case study to show how these methods can be used to simulate and study system dynamics. The example in previous section had 4 states and we reduced its dimension to 2 states. Even for a model comprising large number of states (say 100) the methodology remains the same. For systems with varying nonlinear parameters (e.g., tunable sensors and filters) this order reduction can be performed in symbolic form. It is possible to obtain a concise expression governing the dynamics of the system with symbolic nonlinear parameters. One can study the effect of change of these dominant nonlinear parameters in the operating range of the device or tune them to obtain the desired response. Typical practical applications of this approach are described below.

5.1. Parametric Resonance Based Sensors

In the case of linear oscillator with parametric excitation (when frequency of periodic term ω_p and the natural frequency ω_n satisfy certain integral relationship) response of the system is amplified in the form of exponentially growing solution. It is characterized by multipliers becoming either +1 or −1 resulting in zero Floquet exponents corresponding to the bifurcating states. In the presence of nonlinearity, the system undergoes bifurcation and depending upon the type of bifurcation may even become chaotic. The states undergoing resonance (and dominating the system response) corresponds to bifurcating states and must be retained in the reduced-order model. This reduced-order model is purely nonlinear and in some cases can be simplified further. By constructing the versal deformation and tuning the nonlinearity it is possible to control the post bifurcation behavior. This kind of analysis also yields useful information like effect of parameter dependence, sensitivity and robustness in the operating range.

5.2. Active Amplifiers and De-Amplifiers

Parametrically excited MEMS amplifiers and de-amplifiers have been used to increase the Quality factor of system and to suppress noise, respectively. Reduced-order modeling techniques described here can help design these systems more effectively and efficiently and provide a better understanding of dynamics. It is possible to tune the collective modes in MEMS arrays by means of controlling the bifurcation and achieve amplification or de-amplification. This approach can be used in switching where a particular MEM device can be switched on or off by simply controlling the nonlinearity.

5.3. Large-Scale Integration of MEM Devices

The ongoing efforts of developing parametrically excited MEM devices comprising of various subcomponents give rise to very complex models with very large number of states. The techniques presented here can be applied straightforwardly to reduce these complex models to simple models with less number of states. These reduced-order models can be used for experimental validation and optimization of the MEM device to obtain desired response.

6. Conclusions

In this paper, order reduction techniques for parametrically excited nonlinear MEMS are suggested that can be used to design controller. All order reduction techniques make use of the L-F transformation that produces equivalent systems with time invariant linear parts but the nonlinear parts remain time-periodic. These approaches do not restrict parametric excitation to be small.

The first procedure is a linear technique that can be considered as a state apace version ofthe “Guyan Reduction” where the entire non-dominant dynamics is neglected. As an improvement over the linear technique, a “nonlinear projection method”, based on singular perturbation, is suggested. The “invariant manifold technique” is a nonlinear order reduction methodology similar to the concept of “Time Periodic Center Manifold Theory”. It also yields a “reducibility condition” that determines whether (or not) an order reduction is possible. Reduced-order models constructed from this method capture the dynamics of original large-scale system very accurately in comparison with the other two approaches. As an extension to order reduction, linear and nonlinear controller design approaches are also suggested that integrate well with the order reduction methodologies presented here.

These reduced-order modeling techniques are applicable to general nonlinear time periodic MEM devices. They can provide computable, efficient procedures for order reduction, bifurcation studies and control and convenient for implementation on digital computers and real time application.

In conclusion, it can be stated that some possible techniques for order reduction of general nonlinear time periodic MEMS undergoing strong parametric excitation have been suggested. Some immediate applications in the active area of MEM sensors and amplifiers are also discussed. Several extensions, generalizations and applications are in progress.

Footnotes

Nomenclature

References

Bryzek

, “Impact of MEMS technology on society,” Sensors and Actuators A, vol. 56, no. 1–2, pp. 1–9, 1996.

Buks

and Roukes

M. L.

, “Electrically tunable collective response in a coupled micromechanical array,” Journal of Microelectromechanical Systems, vol. 11, no. 6, pp. 802–807, 2002.

Baskaran

and Turner

K. L.

, “Electrostatic interactions in coupled micro electro mechanical systems,” in Micro/MEMS Conference, Proceedings of SPIE, Adelaide, South Australia, December 2001.

Baskaran

and Turner

K. L.

, “Mechanical domain coupled mode parametric resonance and amplification in a torsional mode micro electro mechanical oscillator,” Journal of Micromechanics and Microengineering, vol. 13, no. 5, pp. 701–707, 2003.

Lifshitz

R. O. N.

and Cross

M. C.

, “Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays,” Physical Review B, vol. 67, no. 13, Article ID 134302, pp. 1–12, 2003.

Napoli

Baskaran

Turner

, and Bamieh

, “Understanding mechanical domain parametric resonance in micro-cantilevers,” in Proceedings of the IEEE Micro Electro Mechanical Systems (MEMS '03), pp. 169–172, Kyoto, Japan, 2003.

Miki

and Shimoyama

, “Dynamics of a microflight mechanism with magnetic rotational wings in an alternating magnetic field,” Journal of Microelectromechanical Systems, vol. 11, no. 5, pp. 584–591, 2002.

Nguyen

C. T.-C.

and Howe

R. T.

, “Design and performance of CMOS micromechanical resonator oscillators,” in Proceedings of IEEE International Frequency Control Symposium, pp. 127–134, 1994.

Bucher

Elka

, and Balmes

, “On the dynamics and modeling of a micro electro-mechanical structure,” preprint.

10.

Seshia

A. A.

Howe

R. T.

, and Montague

, “An integrated microelectromechanical resonant output gyroscope,” in Proceedings of the 15th IEEE Micro Electro Mechanical Systems (MEMS '02), pp. 722–726, Las Vegas, Nev, USA, January 2002.

11.

Neyfeh

A. H.

and Mook

D. T.

, Nonlinear Oscillations, John Wiley & Sons, New York, NY, USA, 1979.

12.

Mahmoud

M. S.

and Singh

M. G.

, Large Scale Systems Modeling, Pergamon Press, Oxford, UK, 1981.

13.

Rudnyi

and Korvink

, “Review: automatic model order reduction for transient simulation of MEMS-based devices,” Sensors Update, vol. 11, no. 1, pp. 3–33, 2002.

14.

Mukherjee

Fedder

Ramaswamy

, and White

, “Emerging simulation approaches for micromachined devices,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 19, no. 12, pp. 1572–1589, 2000.

15.

Westby

E. R.

and Fjeldly

T. A.

, “Nonlinear analytical reducedorder modeling of MEMS,” in Proceedings of the International Conference on Modeling and Simulation of Microsystems (MSM '02), vol. 1, pp. 150–153, San Juan, Puerto Rico, USA, 2002.

16.

Antoulus

, “Approximations of linear dynamical systems,” Encyclopedia of Electrical and Electronics Engineering, vol. 11, pp. 403–422, 1999.

17.

Guyan

R. J.

, “Reduction of stiffness and mass matrices,” AIAA Journal, vol. 3, p. 380, 1965.

18.

Vakakis

A. F.

, “Nonlinear normal modes (NNMs) and their applications in vibration theory: An overview,” Mechanical Systems and Signal Processing, vol. 11, no. 1, pp. 3–22, 1997.

19.

Shaw

S. W.

and Pierre

, “Nonlinear normal modes and invariant manifolds,” Journal of Sound and Vibration, vol. 150, no. 1, pp. 170–173, 1991.

20.

Shaw

S. W.

Pierre

, and Pesheck

, “Modal analysis-based reduced-order models for nonlinear structures-an invariant manifold approach,” Shock and Vibration Digest, vol. 31, no. 1, pp. 3–16, 1999.

21.

Burton

T. D.

and Young

M. E.

, “Model reduction and nonlinear normal modes in structural dynamics,” in Proceedings of the International Mechanical Engineering Congress and Exposition (IMECE '94), vol. 192, pp. 9–16, Chicago, Ill, USA, November 1994.

22.

Burton

T. D.

and Rhee

, “On the reduction of nonlinear structural dynamics models,” Journal of Vibration and Control, vol. 6, no. 4, pp. 531–556, 2000.

23.

Gabbay

L. D.

Mehner

J. A. N. E.

Senturia

S. D.

Gabbay

L. D.

Mehner

J. A. N. E.

, and Senturia

S. D.

, “Computer-aided generation of nonlinear reduced-order dynamic macromodels-I: non-stress-stiffened case,” Journal of Microelectromechanical Systems, vol. 9, no. 2, pp. 262–269, 2000.

24.

Mehner

Gabbay

, and Senturia

, “Computer-aided generation of nonlinear reduced-order dynamic macromodels-II: stress-stiffened case,” Journal of Microelectromechanical Systems, vol. 9, no. 2, pp. 270–278, 2000.

25.

Antoulus

and Sorensen

, “Approximation of large scale dynamical systems: an overview,” Technical Report, Rice University, Houston, Tex, USA, 1999.

26.

Kokotović

Khalil

, and O'Reilly

, Singular Perturbation Methods in Control Analysis and Design, SIAM Press, Philadelphia, Pa, USA, 1999.

27.

Antoulus

Soresen

, and Zhou

, “On the decay rate of the Hankel singular values and related issues,” Technical Report, Rice University, Houston, Tex, USA, 2001.

28.

http://www.mathworks.com/.

29.

Varga

, “Model reduction routines for SLICOT,” Niconet Report 1999-8, 1999.

30.

Freund

R. W.

, “Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation,” in Applied and Computational Control, Signals, and Circuits, Datta

B. N.

, Ed., vol. 1, pp. 435–498, Birkhäuser, Boston, UK, 1999.

31.

Grimme

Sorensen

, and Van Dooren

, “A rational Lanczos algorithm for model reduction,” Numerical Algorithms, vol. 12, no. 1–2, pp. 33–63, 1996.

32.

Silveira

L. M.

Kamon

Elfadel

, and White

, “A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits,” Computer Methods in Applied Mechanics and Engineering, vol. 169, no. 3–4, pp. 377–389, 1999.

33.

Bai

and Freund

R. W.

, “A partial Padé-via-Lanczos method for reduced-order modeling,” Linear Algebra and Its Applications, vol. 332–334, pp. 139–164, 2001.

34.

Gallivan

Grimme

, and Van Dooren

, “Multi-point Padé approximants of large scale system via two-sided rational Krylov algorithm,” in Proceedings of IEEE Conference on Decision and Control, Lake Buena Vista, Fla, USA, 1994.

35.

Newman

A. J.

, “Model reduction via Karhunen-Loéve expansion part I: an exposition,” Technical Report, Electrical Engineering Department, University of Maryland, College Park, Md, USA, 1996.

36.

Sinha

S. C.

Redkar

Butcher

E. A.

, and Deshmukh

, “Order reduction of nonlinear time periodic systems using invariant manifolds,” in Proceedings of the ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 5, pp. 1197–1207, Chicago, Ill, USA, September 2003.

37.

Sinha

S. C.

and Pandiyan

, “Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov-Floquet transformation,” International Journal of Nonlinear Mechanics, vol. 29, no. 5, pp. 687–702, 1994.

38.

Redkar

Sinha

S. C.

, and Butcher

E. A.

, “Some techniques for order reduction of nonlinear time periodic systems,” in Proceedings of ASME International Mechanical Engineering Congress, vol. 116, pp. 649–658, Washington, DC, USA, November 2003.

39.

Deshmukh

Sinha

S. C.

, and Joseph

, “Order reduction and control of parametrically excited dynamical systems,” Journal of Vibration and Control, vol. 6, no. 7, pp. 1017–1028, 2000.

40.

Sinha

S. C.

Redkar

, and Butcher

E. A.

, “Order reduction of nonlinear time periodic systems,” in Proceedings of the 10th International Congress on Sound and Vibration, pp. 2041–2048, Stockholm, Sweden, July 2003.

41.

Sinha

S. C.

and Butcher

E. A.

, “Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems,” Journal of Sound and Vibration, vol. 206, no. 1, pp. 61–85, 1997.

42.

David

and Sinha

S. C.

, “Bifurcation control of nonlinear systems with time-periodic coefficients,” Journal of Dynamic Systems, Measurement and Control, vol. 125, no. 4, pp. 541–548, 2003.

43.

Zhang

and Sinha

S. C.

, “Development of a feedback linearization technique for parametrically excited nonlinear systems via normal forms,” Journal of Computational and Nonlinear Dynamics, vol. 2, no. 2, pp. 124–131, 2007.

44.

Deshmukh

V. S.

and Sinha

S. C.

, “Control of dynamic systems with time-periodic coefficients via the Lyapunov-Floquet transformation and backstepping technique,” Journal of Vibration and Control, vol. 10, no. 10, pp. 1517–1533, 2004.

45.

David

and Sinha

S. C.

, “Versal deformation and local bifurcation analysis of time-periodic nonlinear systems,” Nonlinear Dynamics, vol. 21, no. 4, pp. 317–336, 2000.