Computation of minimal-order realizations of generalized state-space systems

In this paper we consider the problem of obtaining minimal-order representations of generalized state-space systems described by equations of the formE x(t)=A x(t)+B u(t),y(t)=C x(t)+D u(t) withE singular and det(sE–A)0. The underlying principle is that of removal of impulsive and exponential uncontrollable and unobservable modes. When this is followed by the removal of the remaining impulsive modes, we get a minimal-order generalized or standard state-space representation. Simple reduction procedures and numerical algorithms based on these principles are developed and illustrated by means of two numerical examples.P. Misra's research was supported in part by AFWAL, Wright Patterson Air Force Base under Grant F33615-88-C-3605. R. V. Patel's research was supported by NSERC of Canada under Grant A1345.     

Integrally Small Perturbations of Linear Nonautonomous Systems

Let A(t) and B(t) (t≥0) be variable n×n-matrices. Assuming that the system [(x)\dot]=A(t)x\dot{x}=A(t)x is exponentially stable and the matrix norm of the integral ò₀^t (B(s)-A(s)) ds\int_{0}^{t} (B(s)-A(s))\,ds is sufficiently small, for the system [(x)\dot]=B(t)x\dot{x}=B(t)x we derive explicit stability conditions, which improve the well-known ones in appropriate situations. The results are illustrated by a numerical example.     

New results on the inverse-system/deconvolution problem for linear dynamical systems

A new theory and realization methodology for the inverse-system/deconvolution problem associated with dynamical systems is developed. This new theory, based on the assumption of a finite-dimensional, linearized dynamical system and uncertain input signalss(t) that have a feature calledwaveform structure, effectively overcomes the longstanding difficulties of realizing inverse systems for linear systems that have more poles than zeros and/or have zeros in the right half-plane. The results obtained here are derived for both continuous and discrete-time linear systems, using either analog or digital signal processing, and includetime-varying systems with uncertainvector input signalss(t)=(s ₁ (t)...sr(t)) and/orvector output measurementsy(t)=(y ₁ (t)...y _m (t)).     

On the stability of real exponential polynomials with interval-valued delays

We consider the family ? of exponential polynomials. $$f_T (z): = f_0 (z) + f_1 (z)e^{ - zT_1 } + f_2 (z)e^{ - T_2 } + \cdots + f_n (z)e^{ - zT_n } ,T : = T_1 \times T_2 \times \cdots \times T_n ,$$ where thef _k(z),k=0(1)n, are real polynomials under the degree restriction deg(f ₀)>deg(f _k),k=1(1)n, and theT _k,k=1(1)n, are intervals inR ₊. The functions in ? are characteristic functions of linear, retarded dynamical systems with constant coefficients and finitely many interval-valued discrete delays. A stability criterion for ? is expounded; ? is stable if (a) ? contains a stable member and (b) a certain functionalT:S(y) →; {0, 1} vanishes fory in a compact interval inR ₊. Here,S(y) is the boundary of a circular are regionS(y) in the complex plane derived fromf _T. Tools needed for a computer implementation are compiled.     

Some properties of order-Statistics filters

LetX ₁,X ₂,... be a stationary sequence of random variables with Pr{X _t, x}=F(x),t=1, 2,... Also let _i ^n,(t) ,i=1,...,n, denote the ith order statistic (OS) in the moving sample (X _t–N ,...,X _t,...,X _t+N) of odd sizen=2N+1. ThenY _t=a _i X _i ^n(t) with a _i=1 is an order-statistics filter. In practicea _i0,i=1,...,n. Fort>N, the sequence {Y _t} is also stationary. IfX ₁ X ₂, ... are independent, the autocorrelation function (r)=corr(Y _t,Y _t+r) is zero forr >n – 1 and forr n – 1 can be evaluated directly in terms of the means, variances, and covariances of the OS in random samples of sizen +r fromF(x).In special cases several authors have observed that the spectral density functionf() of {Y _t} is initially decreasing for > 0. This result is made more precise and shown to hold generally under white noise. The effect of outliers (impulses) is also discussed.This research was supported by the U.S. Army Research Office.     

Critical Layer Thickness in Exponentially Graded Heteroepitaxial Layers

Exponentially graded semiconductor layers are of interest for use as buffers in heteroepitaxial devices because of their tapered dislocation density and strain profiles. Here we have calculated the critical layer thickness for the onset of lattice relaxation in exponentially graded In _x Ga_1?x As/GaAs (001) heteroepitaxial layers. Upwardly convex grading with \( x = x_{\infty } \left( {1 - {\rm e}^{ - \gamma /y} } \right) \) was considered, where y is the distance from the GaAs interface, γ is a grading length constant, and x _∞ is the limiting mole fraction of In. For these structures the critical layer thickness was determined by an energy-minimization approach and also by consideration of force balance on grown-in dislocations. The force balance calculations underestimate the critical layer thickness unless one accounts for the fact that the first misfit dislocations are introduced at a finite distance above the interface. The critical layer thickness determined by energy minimization, or by a detailed force balance model, is approximately \( h_{\rm{c}} \approx <Exponentially graded semiconductor layers are of interest for use as buffers in heteroepitaxial devices because of their tapered dislocation density and strain profiles. Here we have calculated the critical layer thickness for the onset of lattice relaxation in exponentially graded In _x Ga_1−x As/GaAs (001) heteroepitaxial layers. Upwardly convex grading with x = x_￥ ( 1 - e ^{- g/y} ) x = x_{\infty } \left( {1 - {\rm e}^{ - \gamma /y} } \right) was considered, where y is the distance from the GaAs interface, γ is a grading length constant, and x _∞ is the limiting mole fraction of In. For these structures the critical layer thickness was determined by an energy-minimization approach and also by consideration of force balance on grown-in dislocations. The force balance calculations underestimate the critical layer thickness unless one accounts for the fact that the first misfit dislocations are introduced at a finite distance above the interface. The critical layer thickness determined by energy minimization, or by a detailed force balance model, is approximately h_c ? < h_{\rm{c}} \approx < Although these results were developed for exponentially graded In _x Ga_1−x As/GaAs (001), they may be generalized to other material systems for application to the design of exponentially graded buffer layers in metamorphic device structures such as modulation-doped field-effect transistors and light-emitting diodes.     

Estimation of Energy Aliasing Error for Nonbandlimited Signals

In this paper, we investigate the problem of approximating a given (not necessarily bandlimited) signal, x(t), by a (bandlimited) interpolation or sampling series of the form:

A method for robust stabilization

LetP(y,M) be a family of polynomials, depending on a controller parameter vector y in ⁿ⁺¹, defined as follows: a family of interval matrices, and y in ⁿ⁺¹, set Given an initial stabilizing controller y⁰, this paper provides a simple method to robustify y⁰, i.e., to obtain a controller parameter vector y¹ which is more robust than y⁰. Furthermore we define the stability factort for a given robustly stable controller y, which serves as a measure of the robustness of y.     

Composition and parameters of domains resulting from spinodal decomposition of quaternary alloys in epitaxial GaInP/Ga x In1 ? x As y P1 ? y /GaInP/GaAs(001) heterostructures

Structural and optical properties of two- and three-layer epitaxial heterostructures containing GaInP/Ga _x In_{1 − x} As _y P_{1 − y} quaternary alloy layers were studied. Domain formation due to spinodal decomposition of the quaternary alloy was detected in three-layer heterostructures. As a result, an additional long-wavelength band appears in the photoluminescence spectra, and an additional doublet of the line appears in X-ray diffraction patterns of the (006) line. The domain composition was determined on the basis of Vegard’s law and the Kouphal equation. Original Russian Text ? E.P. Domashevskaya, N.N. Gordienko, N.A. Rumyantseva, B.L. Agapov, P.V. Seredin, L.A. Bityutskaya, I.N. Arsent’ev, L.S. Vavilova, I.S. Tarasov, 2008, published in Fizika i Tekhnika Poluprovodnikov, 2008, Vol. 42, No. 9, pp. 1086–1093.     

Model-based analytical FOE determination

An analytical procedure for the estimation of the focus of expansion (FOE) location is introduced. The method is applied to a vector field which has been obtained from the analysis of the relative translational movement of a rigid body with respect to the acquiring camera (monocular system). A polynomial model is fitted to the vector field; the order and shape of the model are determined by the kinetic analysis of the Euclidean camera centered coordinate system in which the camera is moving, and some hypothesis concerning the approximation of the inverse of the depth function 1/Z(x, y). Two basic properties of complex differentiation theory are applied, assuming that each vector of the optic flow is a real-valued complex function of the form v(x, y)=v_x(x, y)+iv_y(x, y). These properties provide the set of points for which both components of the vector field v_x(x, y) and v_y(x, y) are harmonic and, as a subset, the set of points for which both components are zero. The use of perspective projection states that the FOE is the locus where v_x(x, y)=0=v_y(x, y) and, therefore, where the conditions stated by both properties meet.     

Photoresponse asymmetry of CdZnTe crystals

It is shown that transformation of photoelectric characteristics of sensors based on Cd_{1 − x} Zn _x Te (x = 0.05–0.15) crystals into parametric spectral I(λ)-dI/dλ, kinetic I(t)-dI/dt, and dynamic U-I(Δy) _{f, λ} signatures (I(λ) is the photocurrent, U is the voltage, f is the frequency, t is the time, and y is the coordinate) makes it possible to reveal integrative photoresponse features caused by the photoresponse asymmetry and nonlinearity. Indices of asymmetry and balance of dynamic and energy photoinduced states are suggested; these indices represent systematically the effect of multiscale fields on photoelectronic processes. Using these indices, the ranges of external effects at which systematic features of photoresponse of sensors are minimal or maximal are determined, which makes it possible to increase the efficiency of purposeful selection and treatment of sensors.     

A canonical form for discrete-time systems defined overZ + *

It is shown that for each memberG of a large class of causal time invariant nonlinear input-output maps, with inputs and outputs defined on the nonnegative integers, there is a functionalA on the input set such that (Gs)(k) has the representationA(F _k s) for allk and each inputs, in whichF _k is a simple linear map that does not depend onG. More specifically, this holds—with anA that is unique in a certain important sense—for anyG that has approximately finite memory and meets a certain often-satisfied additional condition. Similar results are given for a corresponding continuous-time case in which inputs and outputs are defined on ₊. An example shows that the members of a large family of feedback systems have these A-map representations.     

Order selection statistical test for nonstationary AR models

A new statistical test for selecting the order of a nonstationary AR modelyk is presented based on the predictive least-squares principle. This test is of the same order as the accumulated cost function _n = _k=1 ⁿ ( _k ^* – _k )²;i.e., ^* wherey _k ^* is the predictive least-square estimate. It is constructed to show how many times the integrated AR processy _k is differenced in order to obtain a stationary AR process given that the exact order of the process is unknown.     

On the structure of the privacy hierarchy

An N argument function f(x ₁,...,x _N ) is called t-private if a protocol for computing f exists so that no coalition of at most t parties can infer any additional information from the execution, other than the value of the function. The motivation of this work is to understand what levels of privacy are attainable. So far, only two levels of privacy are known for N argument functions which are defined over finite domains: functions that are N-private and functions that are (N – 1)/2-private but not N/2-private.In this work we show that the privacy hierarchy for N-argument functions which are defined over finite domains, has exactly (N + 1)/2 levels. We prove this by constructing, for any N/2 t N – 2, an N-argument function which is t-private but not (t + 1)-private.This research was supported by US-Israel Binational Science Foundation Grant 88-00282.     

Qualitative analysis of motion properties of semistate models of large-scale systems

This paper studies the behavior of motions of large-scale (LS) semistate systems (SSS) governed byP _i (t)x _i =M _i (t,x _i )x _i +f _i (t)+h _i (t, x), i=1,2,...,s, =(x ₁ ^T x ₂ ^T x _s ^T )^T, where matricesP _i (t) are singular. Using Lyapunov's approach and the tools for LS system analysis, a variant of attractivity and ultimate boundedness of appropriate time-variable sets are investigated. The results are based on a specific choice of the aggregate functions. It is assumed that the reduction of equations to a normal form of lower order is inconvenient. The aggregation-decomposition approach used in this paper reduces the dimensionality of an aggregate matrix of the system to the number of its systems. Motion properties of LS systems are deduced from the properties of its isolated subsystems, the character of interconnections, and the conditions imposed on the system aggregate matrix. Sufficient algebraic conditions for the above-mentioned motion properties are developed.     

Approximations of a two-unit cold standby system's reliability behaviour

In this paper we study the reliability behaviour of a two-unit cold standby system with priority. At time t = 0, the priority unit (p) begins to work and the standby unit (s) is in cold standby. The p-unit has priority whether it is working or being repaired. A single repairman is available, and the repaired unit is as good as new. We assume that the p-unit's working time X₁ has a general life distribution F(x), and its repair time Y₁ has a general distribution G(x). The s-unit's working time X₂ has an exponential distribution with mean , and its repair time has an exponential distribution with mean . Using the Markov renewal process and stochastic comparison, we give bounds of the mean time to the first failure of the system, and the system's availability.     

Graph Coloring Applied to Secure Computation in Non-Abelian Groups

We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,?), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted. Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t<n/2, but it requires exponential communication complexity $O({\binom{2 t+1}{t}}^{2} \cdot N_{g})$ group elements and round complexity $O(\binom{2 t + 1}{t} \cdot N_{g})$ , for a G-circuit of size N _g . Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity $\mathcal{P}\mathit{oly}(n)\cdot N_{g}$ for any t∈O(n ^1?? ) where ? is any positive constant. For our third protocol, there is a probability δ (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t<n/2. It has communication complexity O(n ^5.056(n+log?δ ^?1)²?N _g ) and the number of rounds is O(n ^2.528?(n+log?δ ^?1)?N _g ).     

Growth and Characterization of Pulsed-Laser-Deposited Ilmenite–Hematite Thin Films

The ilmenite–hematite (1 − x) FeTiO₃ · xFe₂O₃ solid solution system is considered to be a novel material for spin-electronics, microelectronics, high-temperature electronics, and radhard electronics. This paper focuses on thin films of composition x = 0.33 grown on (100) MgO single-crystal substrates using pulsed-laser deposition (PLD) under different argon–oxygen mixtures. The surface of the MgO was found to possess MgO₂ crystals, yielding an orientation relationship, [001] MgO ∥ [011] MgO₂ and (00)MgO ∥ (10) MgO₂. The structural characterizations show that the films are crystalline and homogeneous without any secondary phase. The films show a weak and inclined (110) growth epitaxy. A bandgap of 3.4–3.7 eV was obtained for these films from optical measurements carried out in the UV–visible region. Electrical measurements confirmed the semiconducting behavior. However, the resistivity was found to increase substantially on the slightest addition of oxygen into the chamber.     

Algebraic nonlinearity and its applications to cryptography

The algebraic nonlinearity of an n-bit boolean function is defined as the degree of the polynomial f(X) Z ₂[x ₁, x ₂,..., x _n] that represents f. We prove that the average degree of an ANF polynomial for an n-bit function is n+o(1). Further, for a balanced n-bit function, any subfunction obtained by holding less than n-[log n]- 1 bits constant is also expected to be nonaffine. A function is partially linear if f(X) has some indeterminates that only occur in terms bounded by degree 1. Boolean functions which can be mapped to partially linear functions via a linear transformation are said to have a linear structure, and are a potentially weak class of functions for cryptography. We prove that the number of n-bit functions that have a linear structure is asymptotic .The author is presently employed by the Distributed System Technology Center, Brisbane, Australia.Project sponsored in part by NSERC operating Grant OGP0121648, and the National Security Agency under Grant Number MDA904-91-H-0012. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon.