On the NP-Hardness of Checking Matrix Polytope Stability and Continuous-Time Switching Stability

Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of $k$ matrices in $R^{n times n}$ are stable. In particular, we are interested whether there exist algorithms which can solve this problem in time polynomial in $n$ and $k$. We show that if $k= lceil n^{d} rceil $ for any fixed real $d>0$, then the problem is NP-hard, meaning that no polynomial-time algorithm in $n$ exists provided that $P ne NP$, a widely believed conjecture in computer science. On the other hand, when $k$ is a constant independent of $n$ , then it is known that the problem may be solved in polynomial time in $n$. Using these results and the method of measurable switching rules, we prove our main statement: verifying the absolute asymptotic stability of a continuous-time switched linear system with more than $n^{d}$ matrices $A_{i} in R^{n times n}$ satisfying $0 succeq A_{i} + A_{i}^{T}$ is NP-hard.      

Post-CMOS-Compatible Aluminum Nitride Resonant MEMS Accelerometers

This paper describes the development of aluminum nitride (AlN) resonant accelerometers that can be integrated directly over foundry CMOS circuitry. Acceleration is measured by a change in resonant frequency of AlN double-ended tuning-fork (DETF) resonators. The DETF resonators and an attached proof mass are composed of a 1- $muhbox{m}$ -thick piezoelectric AlN layer. Utilizing piezoelectric coupling for the resonator drive and sense, DETFs at 890 kHz have been realized with quality factors $(Q)$ of 5090 and a maximum power handling of 1 $muhbox{W}$. The linear drive of the piezoelectric coupling reduces upconversion of $1/f$ amplifier noise into $1/f^{3}$ phase noise close to the oscillator carrier. This results in lower oscillator phase noise, $-$96 dBc/Hz at 100-Hz offset from the carrier, and improved sensor resolution when the DETF resonators are oscillated by the readout electronics. Attached to a 110-ng proof mass, the accelerometer microsystem has a measured sensitivity of 3.4 Hz/G and a resolution of 0.9 $hbox{mG}/surdhbox{Hz}$ from 10 to 200 Hz, where the accelerometer bandwidth is limited by the measurement setup. Theoretical calculations predict an upper limit on the accelerometer bandwidth of 1.4 kHz.$hfill$ [2008-0190]      

$ {cal L}_{2}$ Gain of Periodic Linear Switched Systems: Fast Switching Behavior

We investigate the $ {cal L}_{2}$ gain of periodic linear switched systems under fast switching. For systems that possess a suitable notion of a time-average system, we characterize the relationship between the ${cal L}_{2}$ gain of the switched system and the ${cal L}_{2}$ gain of its induced time-average system when the switching rate is sufficiently fast. We show that the switched system ${cal L}_{2}$ gain is in general different from the average system ${cal L}_{2}$ gain if the input or output coefficient matrix switches. If only the state coefficient matrix switches, the input-output energy gain for a fixed ${cal L}_{2}$ input signal is bounded by the ${cal L}_{2}$ gain of the average system as the switching rate grows large. Additionally, for a fixed ${cal L}_{2}$ input, the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows.      

A Fast Liquid-Metal Droplet Microswitch Using EWOD-Driven Contact-Line Sliding

Liquid-metal (LM) droplet-based MEMS switches have mostly been restricted to slow applications until now due to the following reasons: 1) a relatively large switching gap (distance) needed to accommodate imprecise volumes and locations of droplets on the device and 2) lack of high-speed actuation to move the droplets quickly across the switching gap. To combat these problems, we explore switching by sliding the solid–LM–gas triple contact line rather than the entire droplet. This new approach allows us to use a microframe, which not only consistently positions the LM droplet but also makes the switching gap less sensitive to the errors in the deposited-droplet volume, allowing us to design microswitches with very small switching gaps (e.g., 10 $muhbox{m}$ for 600 $muhbox{m}$-diameter droplets). Furthermore, a study of electrowetting-on-dielectric identifies a regime of fast contact-line sliding at the onset of droplet spreading. By moving the contact line fast across a small switching distance, we demonstrate a low-latency LM switch with 60 $muhbox{s}$ switch-on latency ( $sim$20 times better than other LM-switch technologies) and better than 5 $muhbox{s}$ signal rise/fall time, while boasting no contact bounce, as expected from an LM switch. High power-handling capability and long-term reliability are also discussed. $hfill$[2008-0135]      

How to Make $T$-Transitive a Proximity Relation

Three ways to approximate a proximity relation $R$ (i.e., a reflexive and symmetric fuzzy relation) by a $T$ -transitive one where $T$ is a continuous Archimedean $t$-norm are given. The first one aggregates the transitive closure $overline{R}$ of $R$ with a (maximal) $T$-transitive relation $B$ contained in $R$ . The second one computes the closest homotecy of $overline{R}$ or $B$ to better fit their entries with the ones of $R$. The third method uses nonlinear programming techniques to obtain the best approximation with respect to the Euclidean distance for $T$ the $Lstrok$ukasiewicz or the product $t$-norm. The previous methods do not apply for the minimum $t$-norm. An algorithm to approximate a given proximity relation by a ${rm Min}$-transitive relation (a similarity) is given in the last section of the paper.      

A Laterally Movable Gate Field Effect Transistor

A laterally movable gate field effect transistor (LMGFET) device that directly couples lateral mechanical gate motion to drain current of a FET is presented in this paper. Lateral motion of the FET gate results in a change in channel width, keeping the channel length and the gap between the gate and the oxide layer constant. This results in a change in channel current that, in principle, is linearly proportional to mechanical motion. The operating principle of an LMGFET, along with details of the fabrication process for a depletion-type LMGFET device, is described. Fabricated LMGFET shows an average drain current sensitivity to gate motion of $-5.8 muhbox{A}/muhbox{m}$ at $V_{rm DS} = 20 hbox{V}$ and $V_{rm GS} = 0 hbox{V}$ for 60-$muhbox{m}$ gate motion. A model for the fabricated LMGFET is developed based on electrical measurements. The device shows promise both as a sensor and as an actuator in MEMS and other related applications.$hfill$ [2008-0147]      

Simultaneous $H^{infty}$ Control for Nonlinear Systems

This note develops a novel method for designing simultaneous $H^{infty}$ state feedback controllers for a collection of single-input nonlinear systems. Based on the Kalman—Yakubovich—Popov Lemma, necessary and sufficient conditions for the existence of simultaneous $H^{infty}$ controllers are derived by the control storage function approach. A universal formula for constructing continuous, time-invariant, simultaneous $H^{infty}$ state feedback controllers is presented.      

Reflection Coefficients of Polynomials and Stable Polytopes

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Two linear Schur invariant transformations with a free parameter in the polynomial coefficient space are introduced. The first transformation ${cal R}^{n}times{cal R}rightarrow{cal R}^{n}$ maps an arbitrary stable polytope into another stable polytope. The second transformation ${cal R}^{n}times{cal R}rightarrow{cal R}^{n+1}$ maps a stable tilted $n$-dimensional hyperrectangle defined by the discrete Kharitonov theorem into a stable $(n+1)$- dimensional polytope.      

Visual Servo Control Achieving Nanometer Resolution in $X$ –$Y$–$Z$

This paper presents a three-axis vision motion sensor and its applications to visual servo control. The vision sensor is integrated with a three-axis piezo stage to form a visual servo control system that achieves nanometer resolution in all three $x$– $y$–$z$ motion axes. Motion measurement is achieved using a single interferometer-equipped optical microscope. A real-time image-processing algorithm that processes interference fringe patterns and that achieves nanometer out-of-plane resolution is presented. Furthermore, a feedback-control scheme is introduced to control the sensor plane using an Objective-Z-Positioner to enable automatic tracking of moving objects. It expands the out-of-plane measurement range of the vision sensor beyond its inherent depth of field of several micrometers to 100 $mu$m and beyond. An integrated visual servo system is implemented and experimental results are shown.      

Reducing Anchor Loss in MEMS Resonators Using Mesa Isolation

In microelectromechanical systems resonators, dissipation of energy through anchor points into the substrate adds to resonator energy loss, contributing to low values of $Q$. A design for improving $Q$ based on the reflection of anchor-generated surface acoustic waves is presented here. In this design, the resonator is surrounded by a trench, or a mesa, that partially reflects the wave energy back to the resonator. Depending on the distance from the resonator to the mesa, the reflected wave interferes either constructively or destructively with the resonator, increasing or decreasing $Q$. The proposed design is experimentally tested using a dome-shaped flexural mode resonator for a range of distances of the mesa from the resonator. Improvements in $Q$ of up to 400% are observed. The resonator/mesa system is modeled using a commercially available finite-element code. Experiments and simulations compare well, suggesting that a finite-element-method analysis can be used in the preliminary design of mesas for the optimization of $Q$. The concept of using mesas to improve $Q$ is simulated for both flexural and in-plane modes of vibration.$hfill$[2008-0149]      

Analytical Model of Valveless Micropumps

The flow driven by a valveless micropump with a single cylindrical pump chamber and two diffuser/nozzle elements is studied theoretically using a 1-D model. The pump cavity is driven at an angular frequency $omega$ so that its volume oscillates with an amplitude $V_{rm m}$. The presence of diffuser/nozzle elements with pressure-drop coefficients $zeta_{+}$, $zeta_{-}( ≫ zeta_{+})$ and throat cross-sectional area $A_{1}$ creates a rectified mean flow. In the absence of frictional forces the maximum mean volume flux (with zero pressure head) is $Q_{0}$ where $Q_{0}/V_{rm m}omega = (zeta_{-} -break zeta_{+})pi/16(zeta_{-}+zeta_{+})$, while the maximum pressure that can be overcome is $Delta P_{max}$ where $ Delta P_{max}A_{1}^{2}/V_{rm m}^{2} omega^{2} !=! (zeta_{-} -break zeta_{+})/16$. These analytical results agree with numerical calculations for the coupled system of equations and compare well with the experimental results of Stemme and Stemme.$hfill$ [2008-0244]      

Design, Modeling, and Fabrication of MEMS-Based Multicapillary Gas Chromatographic Columns

This paper describes different approaches to achieve high-performance microfabricated silicon-glass separation columns for microgas chromatography systems. The capillary width effect on the separation performance has been studied by characterization of 250-, 125-, 50-, and 25-$muhbox{m}$ -wide single-capillary columns (SCCs) fabricated on a $10 times 8 hbox{mm}^{2}$ die. The highest plate number (12 500/m), reported to date for MEMS-based silicon-glass columns, has been achieved by 25-$muhbox{m}$-wide columns coated by a thin layer of polydimethylsiloxane stationary phase using static coating technique. To address the low sample capacity of these narrow columns, this paper presents the first generation of MEMS-based “multicapillary” columns (MCCs) consisting of a bundle of narrow-width rectangular capillaries working in parallel. The theoretical model for the height-equivalent-to-a-theoretical-plate $(HETP)$ of rectangular MCCs has been developed, which relates the $HETP$ to the discrepancies of the widths and depths of the capillaries in the bundle. Two-, four-, and eight-capillary MCCs have been designed and fabricated to justify the separation ability of these columns. These MCCs capable of multicomponent gas separation provide a sample capacity as large as 200 ng compared to 5.5 ng for 25-$muhbox{m}$-wide SCCs.$hfillhbox{[2007-0309]}$      

Fuzzy Subsethood for Fuzzy Sets of Type-2 and Generalized Type- ${n}$

In this paper, we use Zadeh's extension principle to extend Kosko's definition of the fuzzy subsethood measure $S(G,H)$ to type-2 fuzzy sets defined on any set $X$ equipped with a measure. Subsethood is itself a fuzzy set that is a crisp interval when $G$ and $H$ are interval type-2 sets. We show how to compute this interval and then use the result to compute subsethood for general type-2 fuzzy sets. A definition of subsethood for arbitrary fuzzy sets of type- $n ≫ 2$ is then developed. This subsethood is a type-( $n-1$) fuzzy set, and we provide a procedure to compute subsethood of interval type-3 fuzzy sets.      

Cryogenic Etching of Silicon: An Alternative Method for Fabrication of Vertical Microcantilever Master Molds

This paper examines the use of deep reactive ion etching of silicon with fluorine high-density plasmas at cryogenic temperatures to produce silicon master molds for vertical microcantilever arrays used for controlling substrate stiffness for culturing living cells. The resultant profiles achieved depend on the rate of deposition and etching of an $hbox{SiO}_{x}hbox{F}_{y}$ polymer, which serves as a passivation layer on the sidewalls of the etched structures in relation to areas that have not been passivated with the polymer. We look at how optimal tuning of two parameters, the $ hbox{O}_{2}$ flow rate and the capacitively coupled plasma power, determine the etch profile. All other pertinent parameters are kept constant. We examine the etch profiles produced using electron-beam resist as the main etch mask, with holes having diameters of 750 nm, 1 $muhbox{m}$ , and 2 $muhbox{m}$. $hfill$[2008-0317]      

On $Delta$-Transforms

Any set of two legs in a Gough–Stewart platform sharing an attachment is defined as a $Delta$component. This component links a point in the platform (base) to a line in the base (platform). Thus, if the two legs, which are involved in a $Delta$ component, are rearranged without altering the location of the line and the point in their base and platform local reference frames, the singularity locus of the Gough–Stewart platform remains the same, provided that no architectural singularities are introduced. Such leg rearrangements are defined as $Delta$-transforms, and they can be applied sequentially and simultaneously. Although it may seem counterintuitive at first glance, the rearrangement of legs using simultaneous $Delta$-transforms does not necessarily lead to leg configurations containing a $Delta$component. As a consequence, the application of $Delta$-transforms reveals itself as a simple, yet powerful, technique for the kinematic analysis of large families of Gough–Stewart platforms. It is also shown that these transforms shed new light on the characterization of architectural singularities and their associated self-motions.      

Relaxed Conditions for the Exponential Stability of a Class of Linear Time-Varying Systems

This technical note states new sufficient conditions for the exponential stability of linear time-varying (LTV) systems of the form $mathdot{x}(cdot)=A(t)x(t)$. The approach proposed derives and uses the notion of perturbed frozen time (PFT) form that can be associated to any LTV system. Exploiting the Bellman-Gronwall lemma, relaxed stability conditions are then stated in terms of “average” parameter variations. Salient features of the approach are: pointwise stability of $A(cdot)$ is not required, $Vertmathdot{A}(cdot)Vert$ may not be bounded, the stability conditions also apply to uncertain systems. The approach is illustrated by numerical examples.      

A Necessary and Sufficient Frequency Domain Criterion for the Passivity of SISO Sampled-Data Systems

We present a frequency domain solution to the sampled-data passivity problem. Our analysis is exact in the sense that we take into account the intersample behavior of the system. We use frequency response (FR) operators to first obtain necessary conditions on the sampling rate $T$ and the relative degree of the open-loop transfer function $G_{11}$ for achieving a passive continuous-time closed-loop system. Then, assuming passivity of $G_{11}$ and closed-loop stability, we derive a necessary and sufficient condition for discrete-time controllers that render a passive closed-loop system. We apply the obtained results to the problem of stability of haptic systems.      

Exponential Stability on Stochastic Neural Networks With Discrete Interval and Distributed Delays

This brief addresses the stability analysis problem for stochastic neural networks (SNNs) with discrete interval and distributed time-varying delays. The interval time-varying delay is assumed to satisfy $0≪d_{1}leq d(t) leq d_{2}$ and is described as $d(t)= d_{1}+h(t)$ with $0leq h(t) leq d_{2}-d_{1}$. Based on the idea of partitioning the lower bound $d_{1}$, new delay-dependent stability criteria are presented by constructing a novel Lyapunov–Krasovskii functional, which can guarantee the new stability conditions to be less conservative than those in the literature. The obtained results are formulated in the form of linear matrix inequalities (LMIs). Numerical examples are provided to illustrate the effectiveness and less conservatism of the developed results.      

Performance Recovery of Feedback-Linearization-Based Designs

We consider a tracking problem for a partially feedback linearizable nonlinear system with stable zero dynamics. The system is uncertain and only the output is measured. We use an extended high-gain observer of dimension $n+1$, where $n$ is the relative degree. The observer estimates $n$ derivatives of the tracking error, of which the first $(n-1)$ derivatives are states of the plant in the normal form and the $n$th derivative estimates the perturbation due to model uncertainty and disturbance. The controller cancels the perturbation estimate and implements a feedback control law, designed for the nominal linear model that would have been obtained by feedback linearization had all the nonlinearities been known and the signals been available. We prove that the closed-loop system under the observer-based controller recovers the performance of the nominal linear model as the observer gain becomes sufficiently high. Moreover, we prove that the controller has an integral action property in that it ensures regulation of the tracking error to zero in the presence of constant nonvanishing perturbation.      

Tungsten-Based SOI Microhotplates for Smart Gas Sensors

This paper is concerned with the design, fabrication, and characterization of novel high-temperature silicon on insulator (SOI) microhotplates employing tungsten resistive heaters. Tungsten has a high operating temperature and good mechanical strength and is used as an interconnect in high temperature SOI-CMOS processes. These devices have been fabricated using a commercial SOI-CMOS process followed by a deep reactive ion etching (DRIE) back-etch step, offering low cost and circuit integration. In this paper, we report on the design of microhotplates with different diameters (560 and 300 $muhbox{m}$) together with 3-D electrothermal simulation in ANSYS, electrothermal characterization, and analytical analysis. Results show that these devices can operate at high temperatures (600 $^{circ}hbox{C}$ ) well beyond the typical junction temperatures of high temperature SOI ICs (225 $^{circ}hbox{C}$), have ultralow dc power consumption (12 mW at 600 $^{circ}hbox{C}$), fast transient time (as low as 2-ms rise time to 600 $^{circ}hbox{C}$), good thermal stability, and, more importantly, a high reproducibility both within a wafer and from wafer to wafer. We also report initial tests on the long-term stability of the tungsten heaters. We believe that this type of SOI microhotplate could be exploited commercially in fully integrated microcalorimetric or resistive gas sensors. $hfill$[2007-0275]