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.      

${cal H}_{infty}$ 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.      

$H_{infty}$ Controller Synthesis via Switched PDC Scheme for Discrete-Time T--S Fuzzy Systems

This paper is concerned with the problem of designing switched state feedback $H_{infty}$ controllers for discrete-time Takagi--Sugeno (T--S) fuzzy systems. New types of state feedback controllers, namely, switched parallel distributed compensation (PDC) controllers, are proposed, which are switched based on the values of membership functions. Switched quadratic Lyapunov functions are exploited to derive a new method for designing switched PDC controllers to guarantee the stability and $H_{infty}$ performances of closed-loop nonlinear systems. The design conditions are given in terms of solvability of a set of linear matrix inequalities. It is shown that the new method provides better or at least the same results of the existing design methods via the pure PDC scheme with a quadratic Lyapunov function or switched constant controller gain scheme. Numerical examples are given to illustrate the effectiveness of the proposed method.      

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.      

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.      

A Parameter-Dependent Approach to Robust $H_{infty }$ Filtering for Time-Delay Systems

This paper is concerned with the problem of robust $H_{infty }$ filtering for linear continuous-time systems with polytopic parameter uncertainties and time-varying delay in the state. We utilize the polynomially parameter-dependent idea to solve the robust $H_{infty }$ filtering problem, with new linear matrix inequality conditions obtained for the existence of admissible filters. These conditions are developed based on homogeneous polynomially parameter-dependent matrices of arbitrary degree. The delay-dependence and polynomial parameter-dependence guarantee the proposed approach to be potentially less conservative, which is shown via a numerical example.      

$ {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.      

Delay-Dependent $hbox{H}_{infty }$ Filter Design for Discrete-Time Fuzzy Systems With Time-Varying Delays

This paper investigates delay-dependent $hbox{H}_{bminfty }$ filter design problems for discrete-time fuzzy systems with time-varying delays. First, a novel delay-dependent piecewise Lyapunov–Krasovskii functional (DDPLKF) is proposed in which both the upper bound of delays and the delay interval are considered. Based on this DDPLKF, the delay-dependent stability criteria for discrete-time systems with constant or time-varying delays are obtained, respectively. Then, delay-dependent full-order and reduced-order $hbox{H}_{bminfty }$ filter design approaches are proposed. The filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs). Simulation examples are also given to illustrate the performance of the proposed approaches. It is shown that our approaches are less conservative and that the corresponding $hbox{H}_{bminfty }$ filters can achieve better performance than the existing approaches.      

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.      

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.      

6H-SiC JFETs for 450 $^{circ}hbox{C}$ Differential Sensing Applications

N-channel 6H-SiC depletion-mode junction field-effect transistors (JFETs) have been fabricated, and characterized for use in high-temperature differential sensing. Electrical characteristics of the JFETs have been measured and are in good agreement with predictions of an abrupt-junction long-channel JFET model. The electrical characteristics were measured across a 2-in wafer for temperatures from 25 $^{ circ}hbox{C}$ to 450 $^{circ}hbox{C}$, and the extracted pinchoff voltage has a mean of 11.3 V and a standard deviation of about 1.0 V at room temperature, whereas pinchoff current has a mean of 0.41 mA with standard deviation of about 0.1 mA. The change in pinchoff voltage is minimal across the measured temperature range, whereas pinchoff current at 450 $^{circ}hbox{C}$ is about half its value at room temperature, consistent with the expected change in the $nmu_{n}$ product. The characterization of differential pairs and hybrid amplifiers constructed using these differential pairs is also reported. A three-stage amplifier with passive loads has a differential voltage gain of 50 dB, and a unity-gain frequency of 200 kHz at 450 $^{circ}hbox{C}$, limited by test parasitics. A two-stage amplifier with active loads has reduced sensitivity to off-chip parasitics and exhibits a differential voltage gain of 69 dB with a unity-gain frequency of 1.3 MHz at 450 $^{circ}hbox{C}$.$hfill$[2009-0029]      

A Solution to the Continuous-Time ${rm H}_{infty}$ Fixed-Interval Smoother Problem

The minimum-variance fixed-interval smoother is a state-space realization of the Wiener solution generalized for time-varying problems. It involves forward and adjoint Wiener-Hopf factor inverses in which the gains are obtained by solving a Riccati equation. This technical note introduces a continuous-time ${rm H}_{infty}$ smoother having the structure of the minimum-variance version, in which the gains are obtained by solving a Riccati equation that possesses an indefinite quadratic term. It is shown that the smoother exhibits an increase in mean-square-error, the error is bounded, and the upper error bound is greater than that for the ${rm H}_{infty}$ filter.      

${cal H}_{infty}$ Filtering of Discrete-Time Markov Jump Linear Systems Through Linear Matrix Inequalities

This technical note addresses the discrete-time Markov jump linear systems ${cal H}_{infty}$ filtering design problem. First, under the assumption that the Markov parameter is measurable, the main contribution is the linear matrix inequality (LMI) characterization of all linear filters such that the estimation error remains bounded by a given ${cal H}_{infty}$ norm level, yielding the complete solution of the mode-dependent filtering design problem. Based on this result, a robust filter design able to deal with polytopic uncertainty is considered. Second, from the same LMI characterization, a design procedure for mode-independent filtering is proposed. Some examples are solved for illustration and comparisons.      

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]      

A ${bm lambda}$-Cut and Goal-Programming-Based Algorithm for Fuzzy-Linear Multiple-Objective Bilevel Optimization

Bilevel-programming techniques are developed to handle decentralized problems with two-level decision makers, which are leaders and followers, who may have more than one objective to achieve. This paper proposes a ${lambda}$-cut and goal-programming-based algorithm to solve fuzzy-linear multiple-objective bilevel (FLMOB) decision problems. First, based on the definition of a distance measure between two fuzzy vectors using ${lambda}$-cut, a fuzzy-linear bilevel goal (FLBG) model is formatted, and related theorems are proved. Then, using a ${lambda}$-cut for fuzzy coefficients and a goal-programming strategy for multiple objectives, a ${lambda}$-cut and goal-programming-based algorithm to solve FLMOB decision problems is presented. A case study for a newsboy problem is adopted to illustrate the application and executing procedure of this algorithm. Finally, experiments are carried out to discuss and analyze the performance of this algorithm.      

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.      

Neurodynamic Programming and Zero-Sum Games for Constrained Control Systems

In this paper, neural networks are used along with two-player policy iterations to solve for the feedback strategies of a continuous-time zero-sum game that appears in $L_{2}$ -gain optimal control, suboptimal $H_{infty}$ control, of nonlinear systems affine in input with the control policy having saturation constraints. The result is a closed-form representation, on a prescribed compact set chosen a priori, of the feedback strategies and the value function that solves the associated Hamilton–Jacobi–Isaacs (HJI) equation. The closed-loop stability, $L_{2}$-gain disturbance attenuation of the neural network saturated control feedback strategy, and uniform convergence results are proven. Finally, this approach is applied to the rotational/translational actuator (RTAC) nonlinear benchmark problem under actuator saturation, offering guaranteed stability and disturbance attenuation.      

A Programmable Array for Contact-Free Manipulation of Floating Droplets on Featureless Substrates by the Modulation of Surface Tension

This paper presents a contactless droplet manipulation system that relies on thermally generated Marangoni flows. Programmable 2-D control of aqueous microdroplets suspended in an oil film on a plain featureless glass substrate is achieved using a 128-pixel heater array suspended 100–500 $mu hbox{m}$ above the oil layer. The heaters generate surface temperature perturbations $(≪ 25 ^{circ}hbox{C})$, resulting in local Marangoni flows that can move droplets in either a push or a pull mode. Programmed movement is achieved by the sequential activation of the heaters, with digital control circuitry and a graphical interface providing addressable control of each heater. Droplets with diameters of 300–1000 $muhbox{m}$ are manipulated and merged at speeds up to 140 $muhbox{m/s}$. Evaporation rates can be reduced by almost two orders of magnitude by using a two-layer-oil medium, and the choice of an optimum carrier fluid can achieve fluid velocities over 17 000 $mu hbox{m/s}$. The system provides a contactless platform for parallel droplet-based assays. As such, it circumvents the challenges of sample contamination and loss that occur when a droplet interacts with a solid surface.$hfill$[2008-0272]      

Discharge-Based Pressure Sensors for High-Temperature Applications Using Three-Dimensional and Planar Microstructures

Two versions of microdischarge-based pressure sensors, which operate by measuring the change, with pressure, in the spatial current distribution of pulsed dc microdischarges, are reported. The inherently high temperatures of the ions and electrons in the microdischarges make these devices amenable to high-temperature operation. The first sensor type uses 3-D arrays of horizontal bulk metal electrodes embedded in quartz substrates with electrode diameters of 1–2 mm and 50–100-$muhbox{m}$ interelectrode spacing. These devices were operated in nitrogen over a range of 10–2000 torr, at temperatures as high as 1000 $^{circ}hbox{C}$. The maximum measured sensitivity was 5420 ppm/torr at the low end of the dynamic range and 500 ppm/torr at the high end, while the temperature coefficient of sensitivity ranged from $-$925 to $-$550 ppm/K. Sensors of the second type use planar electrodes and have active areas as small as 0.13 $hbox{mm}^{2}$. These devices, when tested in a chemical sensing system flowing helium as a carrier gas, had a maximum sensitivity of 9800 ppm/torr, a dynamic range of 25–200 torr, and a temperature coefficient of sensitivity of approximately $-$1412 ppm/K.$hfill$ [2008-0262]      

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.