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.      

Stability Analysis and $H_{infty}$ Controller Design of Discrete-Time Fuzzy Large-Scale Systems Based on Piecewise Lyapunov Functions

This paper is concerned with stability analysis and $H_{infty}$ decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of $J$ interconnected discrete-time Takagi–Sugeno (T–S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The $H_{infty}$ controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.      

Relaxed ${cal H}_{infty }$ Stabilization Conditions for Discrete-Time Fuzzy Systems With Interval Time-Varying Delays

To derive less-conservative delay- and range-dependent ${cal H}_{infty }$ stabilization conditions for discrete-time Takagi–Sugeno (T–S) fuzzy systems with interval time-varying delays, the use of a fuzzy-weighting-dependent Lyapunov–Krasovskii functional (FWLKF), in which all variables are set to be affinely or quadratically dependent on fuzzy weighting functions, is proposed. Subsequently, parameterized linear matrix inequality (PLMI)-based ${cal H}_infty$ stabilization conditions are derived by following the free-weighting matrix approach. To fully exploit the convexity of fuzzy weighting functions, the derived PLMIs are sequentially replaced by a finite set of LMIs by considering all possible conditions associated with fuzzy weighting functions.      

${cal H}_{infty}$ State-Feedback Control Design for Fuzzy Systems Using Lyapunov Functions With Quadratic Dependence on Fuzzy Weighting Functions

This paper proposes a method for designing an ${cal H}_{infty}$ state-feedback fuzzy controller for discrete-time Takagi–Sugeno (T-S) fuzzy systems. To derive less conservative ${cal H}_{infty}$ stabilization conditions, this paper enhances the interactions among the fuzzy subsystems using a multiple Lyapunov function with quadratic dependence on fuzzy weighting functions. Besides, for more allocation of the nonlinearity to the fuzzy control system, this paper introduces a slack variable that is quadratically dependent on the one-step-past fuzzy weighting functions as well as the current ones. In the derivation, the ${cal H}_{infty}$ stabilization conditions are formulated in terms of parameterized linear matrix inequalities (PLMIs), which are reconverted into LMI conditions with the help of an efficient relaxation technique.      

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.      

On Constructing Parsimonious Type-2 Fuzzy Logic Systems via Influential Rule Selection

Type-2 fuzzy systems are increasing in popularity, and there are many examples of successful applications. While many techniques have been proposed for creating parsimonious type-1 fuzzy systems, there is a lack of such techniques for type-2 systems. The essential problem is to reduce the number of rules, while maintaining the system's approximation performance. In this paper, four novel indexes for ranking the relative contribution of type-2 fuzzy rules are proposed, which are termed $R$-values, $c$-values, $omega _1 $-values, and $omega _2 $-values. The $R$-values of type-2 fuzzy rules are obtained by applying a QR decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model. The $c$-values rank rules based on the effects of rule consequents, while the $omega _1 $-values and $omega _2 $-values consider both the rule-base structure (via firing strength matrices) and the output contribution of fuzzy rule consequents. Two procedures for utilizing these indexes in fuzzy rule selection (termed “forward selection” and “backward elimination”) are described. Experiments are presented which demonstrate that by using the proposed methodology, the most influential type-2 fuzzy rules can be effectively retained in order to construct parsimonious type-2 fuzzy models.      

$H_{infty}$ 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.      

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 Novel Approach to $H_{bm infty}$ Decentralized Fuzzy-Observer-Based Fuzzy Control Design for Nonlinear Interconnected Systems

In general, due to the interactions among subsystems, it is difficult to design an $H_{infty }$-decentralized output-feedback controller for nonlinear interconnected systems. This study introduces $H_{infty }$-decentralized fuzzy-observer-based fuzzy control design, where the premise variables depend on the state variables estimated by a fuzzy observer, for nonlinear interconnected systems via T--S fuzzy models. The fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables depend on the state variables only. A novel decoupled method is proposed in this study to transform the non-linear matrix inequality (non-LMI) conditions into some LMI forms. By the proposed decoupled method, the problem of $H_{infty }$-decentralized fuzzy-observer-based fuzzy control design for nonlinear interconnected systems is characterized in terms of solving an eigenvalue problem (EVP) with five prespecified scalars for each subsystem. In general, it is a difficult task to solve the EVP with five prespecified scalars. Fortunately, this special EVP can be easily solved by using a genetic algorithm and an LMI-based optimization method. Finally, a simulation example is given to illustrate the design procedure and robust performance of the proposed methods.      

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.      

An Interactive Satisfying Method Based on Alternative Tolerance for Multiple Objective Optimization With Fuzzy Parameters

An interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters. Using the $alpha $ -level sets of the fuzzy numbers, all the objectives are modeled as the fuzzy goals, and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution. Via a specific attainable point programming model, the membership functions can be modified, and then, a lexicographic two-phase programming procedure is constructed correspondingly to find the final solution. In a special case, the objective constraint is added instead of changing the membership functions; therefore, the dissatisfying objectives for the decision maker can be improved step by step. The presented method not only acquires the $alpha $ -Pareto optimal or weak $alpha $-Pareto optimal solution of the fuzzy multiple objective optimization, but also satisfies the progressive preference of the decision maker. A numerical example shows its power.      

$L_2$--$E$ 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.      

Large-Deflection Spiral-Shaped Micromirror Actuator

We present a new two-axis spiral-shaped micromirror manipulator developed for free-space optical switching. The actuator is an electrostatic actuator, which is composed of two different parts that are fabricated using conventional surface-micromachining processes and are then assembled. Instead of conventional monolithic self-assembling design approaches, these two parts are fabricated on two different chips and assembled using a spatial–mechanical approach. The design utilizes the increased flexibility of the spiral-shaped electrode and the zipping-effect technique in order to increase the maximum rotation angle. The footprint of the assembled device is $600 muhbox{m} times 600 muhbox{m}$, and the height of the micropyramid is 200 $muhbox{m}$. The switch is simulated using an energy method and a coupled electromechanical model. Its performance is measured statically using a reflection measurement approach. A continuous rotational actuation of 17$^{ circ}$ has been achieved with an actuation voltage of 235 V.$ hfill$[2008-0319]      

Robust Output Regulation of T--S Fuzzy Systems With Multiple Time-Varying State and Input Delays

This paper proposes a robust output regulator design for Takagi–Sugeno (T–S) fuzzy systems with uncertain time delays. For generalization of the control problem, we assume that there exist affine dynamics, unknown multiple time-varying delays in state and input, uncertain parameters, disturbance, and partial state feedback. Both continuous-time and discrete-time cases are considered in a unified theoretical derivation. Here, we introduce a memoryless fuzzy observer and a fuzzy integral compensator. In comparison with previous studies, the proposed scheme drops the need for coordinate transformation, calculation of the desired operational point, which is known state and input delays, and exact fuzzy modeling. The control result is asymptotic output regulation for systems without disturbance and guaranteed $H^{infty}$ performance for systems with disturbance. Moreover, a piecewise constant output regulation is achieved without controller redesign. In addition, the developed fuzzy regulator can be applied to both affine and nonaffine uncertain time-delay systems. Finally, numerical simulations are performed by using two typical time-delay systems to validate the expected performance.      

Piecewise $H_{infty}$ Controller Design of Uncertain Discrete-Time Fuzzy Systems With Time Delays

This paper considers the robust $H_{infty}$ control of uncertain discrete-time fuzzy systems with time delays based on piecewise Lyapunov--Krasovskii functionals. It is shown that the stability with $H_{infty}$ disturbance attenuation performance can be established for the closed-loop fuzzy control systems if there exists a piecewise Lyapunov--Krasovskii functional, and moreover, the functional and the corresponding controller can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A numerical example is given to demonstrate the efficiency and the advantage of the proposed method.      

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]      

Radioisotope Thin-Film Fueled Microfabricated Reciprocating Electromechanical Power Generator

A radioisotope power generator with a potential lifetime of decades is demonstrated by employing a 100.3-year half-lifetime $^{63}hbox{Ni}$ radioisotope thin-film source to electrostatically actuate and cause reciprocation in a microfabricated piezoelectric unimorph cantilever. The radioisotope direct-charged electrostatic actuation of the piezoelectric unimorph cantilever results in the conversion of radiation energy into mechanical energy stored in the strained unimorph cantilever. The gradual accumulation of the actuation charges leads to the pull-in of the unimorph cantilever into the radioisotope thin-film, and the resulting discharge leads to vibrations in the unimorph cantilever. During the vibrations, the stored mechanical energy is converted into electrical energy by the piezoelectric thin-film. The generator was realized by using both microfabricated lead zirconate titanate oxide–silicon (PZT–Si) and aluminum nitride–silicon (AlN–Si) unimorph cantilevers. The radioisotope direct-charged electrostatic actuation of the AlN–Si unimorph cantilevers by a 2.9-mCi $^{63}hbox{Ni}$ thin-film radiating 0.3 $muhbox{W}$ led to charge–discharge–vibrate cycles that resulted in the generation of 0.25% duty cycle 12.95- $muhbox{W}$ power pulses (across an optimal load impedance of 521 $hbox{k}Omega$) at an overall energy conversion efficiency of 3.97%. These electrical power pulses can potentially be useful for periodically sampling sensor microsystems. $hfill$[2008-0009]      

A Fuzzy Qualitative Framework for Connecting Robot Qualitative and Quantitative Representations

This paper proposes a novel framework for describing articulated robot kinematics motion with the goal of providing a unified representation by combining symbolic or qualitative functions and numerical sensing and control tasks in the context of intelligent robotics. First, fuzzy qualitative robot kinematics that provides theoretical preliminaries for the proposed robot motion representation is revisited. Second, a fuzzy qualitative framework based on clustering techniques is presented to connect numerical and symbolic robot representations. Built on the $k-{{bb AGOP}}$ operator (an extension of the ordered weighted aggregation operators), k-means and Gaussian functions are adapted to model a multimodal density of fuzzy qualitative kinematics parameters of a robot in both Cartesian and joint spaces; on the other hand, a mixture regressor and interpolation method are employed to convert Gaussian symbols into numerical values. Finally, simulation results in a PUMA 560 robot demonstrated that the proposed method effectively provides a two-way connection for robot representations used for both numerical and symbolic robotic tasks.      

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.