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.      

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.      

$L_2$--$E$ FIR Filters for Deterministic Continuous-Time-State Space Signal Models

In this paper, a new $L_2$-- $E$ performance criterion is introduced, which is represented as a gain between the disturbance during the recent time interval and the current estimation error. Based on the $L_2$--$E$ performance criterion, the $L_2$-- $E$ finite impulse response (FIR) filter (LEFF) is proposed for deterministic continuous-time-state space signal models without requiring the stochastic information such as variances and means. The LEFF is designed to minimize the maximum value of the $L_2$-- $E$ performance criterion together with prior constraints such as linearity, unbiased property in the deterministic sense, and FIR structure, simultaneously. Via simulation, the LEFF is compared with other deterministic infinite impulse response (IIR) filters such as the $H_infty$ and $L_2$-- $L_infty$ filters.      

$H_{infty}$ Filtering for Fuzzy Singularly Perturbed Systems

This paper considers the problem of designing $H_{infty}$ filters for fuzzy singularly perturbed systems with the consideration of improving the bound of singular-perturbation parameter $epsilon$. First, a linear-matrix-inequality (LMI)-based approach is presented for simultaneously designing the bound of the singularly perturbed parameter $epsilon$, and $H_{infty}$ filters for a fuzzy singularly perturbed system. When the bound of singularly perturbed parameter $epsilon$ is not under consideration, the result reduces to an LMI-based design method for $H_{infty}$ filtering of fuzzy singularly perturbed systems. Furthermore, a method is given for evaluating the upper bound of singularly perturbed parameter subject to the constraint that the considered system is to be with a prescribed $H_{infty}$ performance bound, and the upper bound can be obtained by solving a generalized eigenvalue problem. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.      

Polymeric Thermal Microactuator With Embedded Silicon Skeleton: Part II—Fabrication, Characterization, and Application for 2-DOF Microgripper

This paper presents the fabrication, characterization, and application of a novel silicon-polymer laterally stacked electrothermal microactuator. The actuator consists of a deep silicon skeleton structure with a thin-film aluminum heater on top and filled polymer in the trenches among the vertical silicon parts. The fabrication is based on deep reactive ion etching, aluminum sputtering, SU8 filling, and KOH etching. The actuator is 360 $muhbox{m}$ long, 125 $mu hbox{m}$ wide, and 30 $muhbox{m}$ thick. It generates a large in-plane forward motion up to 9 $muhbox{m}$ at a driving voltage of 2.5 V using low power consumption and low operating temperature. A novel 2-D microgripper based on four such forward actuators is introduced. The microgripper jaws can be moved along both the $x$- and $y$ -axes up to 17 and 11 $muhbox{m}$, respectively. The microgripper can grasp a microobject with a diameter from 6 to 40 $muhbox{m}$ . In addition, the proposed design is suitable for rotation of the clamped object both clockwise and counterclockwise. $hfill$[2007-0192]      

Two-Axis Microstage System

We report on the design, fabrication, and testing of a two-axis stage that was used to move a cantilever arm. We utilized Sandia National Laboratories' MEMS foundry process that incorporates five levels of polysilicon and four sacrificial layers of silicon dioxide. The $X$– $Y$ stage was actuated with the full 110 $muhbox{m}$ of travel in both dimensions. To demonstrate the travel of the stage, a cantilevered arm was attached to the stage and extended off the edge of the chip. Rotational motion from torsional ratcheting actuators is converted to linear motion through an intricate mechanical system. The rotational engines drive two independent spiral cams that provide the linkage system with the angular to linear translation needed to move the stage arms. A pin and the slotted $X$ –$Y$ stage arms allow motion in both the $X$ - and $Y$ -directions. This initial device has been successfully operated and demonstrates a useful design for producing 2-D motion using a MEMS. The unidirectional rotational drives used in these experiments did not allow the retraction of the stage, but there appears to be a clear path to back-and-forth motion using bidirectional rotational drives or linear actuators.$hfill$ [2007-0273]      

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]      

$ {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.      

Fracture Toughness, Fracture Strength, and Stress Corrosion Cracking of Silicon Dioxide Thin Films

The fracture toughness, fracture strength, and stress corrosion cracking behavior of thin-film amorphous silicon dioxide $(hbox{SiO}_{2})$ deposited on silicon wafers via plasma-enhanced chemical vapor deposition have been measured using specimens with length scales comparable to micromachined devices. Clamped–clamped microtensile specimens were fabricated using standard micromachining techniques. These devices exploit residual tensile stresses in the film to create stress intensity factors at precrack tips and stress concentrations at notches, in order to measure fracture toughness and fracture strength, respectively. The fracture toughness of thin-film $hbox{SiO}_{2}$ was $0.77 pm 0.15 hbox{MPa}cdot hbox{m}^{1/2}$, and the fracture strength was $0.81 pm 0.06 hbox{GPa}$. Stress corrosion cracking (slow crack growth) was also measured in the $hbox{SiO}_{2}$ devices with sharp precracks subjected to residual tensile stresses. These data are used to predict lifetimes for a $hbox{SiO}_{2}$-based microdevice. $hfill$[2007-0304]      

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]      

Seal Formation in Silicon Planar Patch-Clamp Microstructures

This paper presents a microfabricated planar patch-clamp electrode design and looks at the impact of several physical characteristics on seal formation. The device consists of a patch aperture, 1.5–2.5 $muhbox{m}$ in diameter and 7–12 $muhbox{m}$ in depth, with a reverse-side deep-etched 80- $muhbox{m}$ well. The patch aperture was coated with either thermal oxide or plasma-enhanced chemical vapor deposited (PECVD) $ hbox{SiO}_{2}$. Some of the thermal oxide devices were converted into protruding nozzle structures, and some were boron-doped. Seal formation was tested with cultured N2a neuroblastoma cells. The PECVD oxide devices produced an average seal resistance of 34 $hbox{M}Omega (n = 24)$ , and the thermal oxide devices produced an average seal resistance of 96 $hbox{M}Omega (n = 59)$. Seal resistance was found to positively correlate with patch aperture depth. Whole-cell recordings were obtained from 14% of cells tested with the thermal oxide devices, including a single recording where a gigaohm seal was obtained.$hfill$ [2007-0159]      

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.      

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.      

The Global Kernel $k$-Means Algorithm for Clustering in Feature Space

Kernel $k$-means is an extension of the standard $k$ -means clustering algorithm that identifies nonlinearly separable clusters. In order to overcome the cluster initialization problem associated with this method, we propose the global kernel $k$-means algorithm, a deterministic and incremental approach to kernel-based clustering. Our method adds one cluster at each stage, through a global search procedure consisting of several executions of kernel $k$-means from suitable initializations. This algorithm does not depend on cluster initialization, identifies nonlinearly separable clusters, and, due to its incremental nature and search procedure, locates near-optimal solutions avoiding poor local minima. Furthermore, two modifications are developed to reduce the computational cost that do not significantly affect the solution quality. The proposed methods are extended to handle weighted data points, which enables their application to graph partitioning. We experiment with several data sets and the proposed approach compares favorably to kernel $k$ -means with random restarts.      

Distance-Dependent Head-Related Transfer Functions Measured With High Spatial Resolution Using a Spark Gap

A measurement of head-related transfer functions (HRTFs) with high spatial resolution was carried out in this study. HRTF measurement is difficult in the proximal region because of the lack of an appropriate acoustic point source. In this paper, a modified spark gap was used as the acoustic sound source. Our evaluation experiments showed that the spark gap was more like an acoustic point source than others previously used from the viewpoints of frequency response, directivity, power attenuation, and stability. Using this spark gap, high spatial resolution HRTFs were measured at 6344 spatial points, with distances from 20 to 160 cm, elevations from $-$40$^circ$ to 90 $^circ$, and azimuths from 0$^circ$ to 360$^circ$. Based on these measurements, an HRTF database was obtained and its reliability was confirmed by both objective and subjective evaluations.      

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.      

On Compensating Long Actuator Delays in Nonlinear Control

We are interested in finite-escape open-loop unstable plants that are globally stabilizable in the absence of actuator delay but require controller redesign in the presence of delay. The simplest such plant is $ {mathdot Z} (t)= Z(t)^{2} + U(t-D)$, where $D$ is actuator delay of arbitrary length. For this system we present a control law that compensates the delay and achieves feedback linearization (of the entire ODE+delay infinite-dimensional cascade). However, even though exponential stability is achieved, the result is not global because the plant can have a finite escape with an initial condition $Z(0)geq 1/D$ before the feedback control “reaches” it at $t=D$ . We prove a stability result whose region of attraction is essentially $Z(0)≪ 1/D$ and for which we derive an asymptotic stability bound in terms of the system norm $Z(t)^{2} + int _{t-D}^{t} U(theta )^{2} dtheta $.      

Decentralized $H_{infty }$ Filter Design for Discrete-Time Interconnected Fuzzy Systems

This paper describes a decentralized $H_{infty }$ filter design for discrete-time interconnected fuzzy systems based on piecewise-quadratic Lyapunov functions. The systems consist of $J$discrete-time interconnected Takagi–Sugeno (T–S) fuzzy subsystems, and a decentralized $H_infty$ filter is designed for each subsystem. It is shown that the stability of the overall filtering-error system with $H_{infty }$ performance can be established if a piecewise-quadratic Lyapunov function can be constructed. Moreover, the parameters of filters can be obtained by solving a set of linear matrix inequalities that are numerically feasible. Two simulation examples are given to show the effectiveness of the proposed approach.      

Real-Time Temperature Compensation of MEMS Oscillators Using an Integrated Micro-Oven and a Phase-Locked Loop

We present a new temperature compensation system for microresonator-based frequency references. It consists of a phase-locked loop (PLL) whose inputs are derived from two microresonators with different temperature coefficients of frequency. The resonators are suspended within an encapsulated cavity and are heated to a constant temperature by the PLL controller, thereby achieving active temperature compensation. We show repeated real-time measurements of three 1.2-MHz prototypes that achieve a frequency stability of $pm$ 1 ppm from $-20 ^{circ}hbox{C}$ to $+80 ^{circ}hbox{C}$, as well as a technique to reduce steady-state frequency errors to $pm$0.05 ppm using multipoint calibration.$hfill$[2009-0074]      

General Transformation for Block Diagonalization of Multitime-Scale Singularly Perturbed Linear Systems

In this technical note, a general transformation for block-diagonalization (decoupling) of multitime-scale singularly perturbed linear systems composed of ${mbi N}$ subsystems is presented. The resulting decoupling corresponds to subsystem time-scales and produces ${mbi N}$ completely independent subsystems operating in different time scales.