The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

A study of associative evidential reasoning

Associative evidential reasoning is the mechanism of combining evidence and evaluating hypotheses, which is the core of many computational systems. It is shown that under the generalized symmetry condition, that is, f(a,b)=neg (f(neg(a), neg(b))), where f is the combination operator satisfying common requirements like associativity and monotonicity, and neg maps positive elements to negative ones and vice versa, f is uniquely determined by a one-place mapping from the positive region to the set of positive reals. Furthermore, such combination formulas cannot be made robust, and quantizing the region will cause the loss of associativity or other inconsistencies. The implications on evidential reasoning system are: there exists often only one kind of formula for combining evidence; the quest for robust combination is often infeasible; and the attempt of converting numerical degrees of belief to linguistic quantifiers and vice versa is destined to be fruitless     

Decomposition of 2-D linear systems into 1-D systems in thefrequency domain

A method is presented for the decomposition of the frequency domain of 2-D linear systems into two equivalent 1-D systems having dynamics in different directions and connected by a feedback system. It is shown that under some assumptions the decomposition problem can be reduced to finding a realizable solution to the matrix polynomial equation X(z₁)P(z₂ )+Q(z₁)Y(z₂ )=D(z₁, z₂). A procedure for finding a realizable solution X(z₁ ), Y(z₂) to the equation is given     

Strict aperiodic property of polynomials with perturbedcoefficients

Let a family of polynomials be P(s)=t ₀sⁿ+t₁s ⁿ±¹ + . . . + t_n where 0<a_j⩽t_j⩽b _j. V.L. Kharitonov (1978) derived a necessary and sufficient condition for the above equation to have only zeros in the open left-half plane. The present authors derive some similar results for the equation to be strictly aperiodic (distinct real roots)     

On the imaging of fractal surfaces

An analysis is presented of the imaging of surfaces modeled by fractal Brownian elevation functions of the sort used in computer graphics. It is shown that, if Lambertian reflectance modest surface slopes and the absence of occlusions and self shadowing are assumed, a fractal surface with Fourier power spectrum proportional to f ^β produces an image with power spectrum proportional to f^2-β; here, f is the spatial frequency and β is related to the fractional dimension value. This allows one to use the spectral falloff of the images to predict the fractal dimension of the surface     

Order-determination theorems and system identification

Theorems for order-determination without a priori knowledge of upper bounds on the order in MIMO dynamic systems are developed. Also, deterministic procedures are introduced to determine orders and estimate parameters simultaneously by recursively computing the order-determining quantity S_n(a,b,k), which plays a crucial role in order-determination procedures, and the least-squares estimate &thetas;_n(a,b) of &thetas;(p,q), with p and q denoting the true orders     

Decomposing the Laplacian

A correction is made and an elementary derivation is given of Clark's (ibid., vol.11, p.43-57, 1989) decomposition of the Laplacian Δ²f(x,y) into a second directional derivative in the gradient direction of f, and a product of gradient magnitude by curvature of the level curve through f(x,y)     

Optimal algorithms on the pipelined hypercube and related networks

Parallel algorithms for several important combinatorial problems such as the all nearest smaller values problem, triangulating a monotone polygon, and line packing are presented. These algorithms achieve linear speedups on the pipelined hypercube, and provably optimal speedups on the shuffle-exchange and the cube-connected-cycles for any number p of processors satisfying 1⩽p⩽n/((log³n)(loglog n)²), where n is the input size. The lower bound results are established under no restriction on how the input is mapped into the local memories of the different processors     

Comments on `Conditions for stable zeros of sampled systems' by M.Ishitobi

The commenter argues that the result of the above-titled work (see ibid., vol.37, no.10, p.1558-1561, Oct. 1992) is incorrect. It is pointed out that when sampling a continuous-time system G(s ) using zero-order hold, the zeros of the resulting discrete-time system H(z) become complicated functions of the sampling interval T. The system G(s) has unstable continuous-time zeros, s=0.1±i. The zeros of the corresponding sampled system start for small T from a double zero at z=1 as exp(T(0.1±i )), i.e., on the unstable side. For T>1.067 . . . the zeros become stable. The criterion function of the above-titled work, F(T)=G*(jω_s/2)= H(-1)T/2, is, however, positive for all T, indicating only stable zeros. The zero-locus crosses the unit circle at complex values     

Computational aspects of the bilinear transformation basedalgorithm for S-plane to Z-plane mapping

The author analyzes the computational complexity of an algorithm by F.D. Groutage et al. (ibid., vol.AC-32, no.7, p.635-7, July 1987) for performing the transformation of a continuous transfer function to a discrete equivalent by a bilinear transformation. Groutage et al. defend their method by noting that their technique is not limited to the bilinear transformation. Rather, it can be extended to any higher-order integration rule (Simpson, Runge-Kutta, etc.), or to any higher-order expansion of the ln function. In general, using the method, s can be any appropriate mapping function s=f (z)     

Fault tree based diagnostics using fuzzy logic

Fuzzy set theory is investigated as a tool for the diagnostics of systems described by means of a fault tree. The objective is to diagnose component failures from the observation of fuzzy symptoms using the information contained in a fault tree. A two-step procedure is used to solve the problem. In this first step, causal reasoning is used to diagnose failure modes, consisting of minimal cut-sets of basic events, from the observation of triggered gates treated as symptoms. In the second step, the authors identify the particular components which have failed based on the diagnosed failure modes. To perform this second step, the solution of a fuzzy relational equation a=∧(S ^T αx) connecting failure mode a to basic events x is derived. With this method, the diagnostics equations can be symmetrically generated and solved in terms of the tree's basic events. The systematic nature with which a diagnosis can be generated from a fault tree lends this method to potential application of object-based programming techniques     

A new realization theory

A realization theory that is based on the Kalman system theory and motivated by the Fuhrmann realization theory is presented. In the Fuhrmann realization theory, the realization state-space is defined by a natural polynomial module K_Q, which is related to a nonsingular polynomial matrix Q. The module K_Q is formed of all f where Q^-1f is strictly proper. In the realization theory proposed, the realization state-space is defined by a different module M_Q, which is formed of properly truncated, strictly proper formal power series of Q^-1f. The realization theory can be used to prove the dual-realization scheme induced by the Fuhrmann realization theory     

Two-dimensional convolution on a pyramid computer

An algorithm for convolving a k×k window of weighting coefficients with an n×n image matrix on a pyramid computer of O(n²) processors in time O(logn+k²), excluding the time to load the image matrix, is presented. If k=Ω (√log n), which is typical in practice, the algorithm has a processor-time product O(n ² k²) which is optimal with respect to the usual sequential algorithm. A feature of the algorithm is that the mechanism for controlling the transmission and distribution of data in each processor is finite state, independent of the values of n and k. Thus, for convolving two {0, 1}-valued matrices using Boolean operations rather than the typical sum and product operations, the processors of the pyramid computer are finite-state     

On the number of digital straight line segments

A closed-form expression has been reported in the literature for L_N, the number of digital line segments of length N that correspond to lines of the form y=ax+β, O⩽α, β<1. The authors prove an asymptotic estimate for L_N that might prove useful for many applications, namely, L_N=N ³/π²+O(N² log N). An application to an image registration problem is given     

Algorithms and bounds for shortest paths and diameter in faultyhypercubes

In an n-dimensional hypercube Qn, with the fault set |F|<2_n-2, assuming S and D are not isolated, it is shown that there exists a path of length equal to at most their Hamming distance plus 4. An algorithm with complexity O (|F|logn) is given to find such a path. A bound for the diameter of the faulty hypercube Qn-F, when |F|<2_n-2, as n+2 is obtained. This improves the previously known bound of n+6 obtained by A.-H. Esfahanian (1989). Worst case scenarios are constructed to show that these bounds for shortest paths and diameter are tight. It is also shown that when |F|<2n-2, the diameter bound is reduced to n+1 if every node has at least 2 nonfaulty neighbors and reduced to n if every node has at least 3 nonfaulty neighbors     

Consensus with dual failure modes

The problem of achieving consensus in a distributed system is discussed. Systems are treated in which either or both of two types of faults may occur: dormant (essentially omission and timing faults) and arbitrary (exhibiting arbitrary behavior, commonly referred to as Byzantine). Previous results showed that are number of dormant faults may be tolerated when there are no arbitrary faults and that, at most, [n-1/3] arbitrary faults may be tolerated when there are no dormant faults (n is the number of processors). A continuum is established between the previous results: an algorithm exists iff n >f_max+2m_max and c >f_max+m_max (where c is the system connectivity), when faults are constrained so that there are at most f_max and at most m_max of these that are arbitrary. An algorithm is given and compared to known algorithms. A method is given to establish virtual links so that the communications graph appears completely connected     

Damping ratio of polynomials with perturbed coefficients

Let a family of polynomials be P(s)=t ₀Sⁿ+t₁s ^n-1 . . .+t_n where O<α _j⩽t_j⩽β. Recently, C.B. Soh and C.S. Berger have shown that a necessary and sufficient condition for this equation to have a damping ratio of φ is that the 2ⁿ⁺¹ polynomials in it which have t_k=α_k or t_k=β_k have a damping ratio of φ. The authors derive a more powerful result requiring only eight polynomials to be Hurwitz for the equation to have a damping ratio of φ using Kharitonov's theorem for complex polynomials     

Simultaneous stabilization of single-input single-output discretetime systems

The problem of finding one compensator which simultaneously stabilizes a family of single-input-single-output (SISO) discrete-time plants is considered. The family of plants is described by the transfer functions {Pz, q): q∈Q}, which are generated via a z-transformation. A number of assumptions describing the set of allowable plants are then given. These assumptions include some regularity conditions on the plants and a minimum-phase requirement. The satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator C(z) for simultaneous stabilization. An iterative computation method is provided for control design     

True 3D animation for displaying VLSI modeling data

The authors report on a flexible system that provides visual feedback to VLSI designers with a novel display method. The basic problem is to display simultaneously two or more functions f ₁, f₂, . . ., each of which depends on the two spatial variables x and y. The method is based on results from visual perception experiments, which indicated that the human visual system can view simultaneously two or more images that are separated in depth, even if they are combined (superimposed) over a common spatial domain. Two display modes have been implemented: a gray-level mode and a contour mode in which only the edges separating adjacent regions are displayed in an effort to make things easier for the user. This depth-separation technique allows the viewer to register spatially the x,y distribution of multiple variables in an animation sequence. The technique can be applied to many other situations in which a single-plane episode involves several variables     

Further remarks on potentially global stabilizability

Given a control system with a nonlinear drift term, it is proved that a linear-like criterion for almost-global stabilizability can be stated, at least for control systems of the form x'=f( x)+Bu, where f is a polynomial but not necessarily homogeneous vector field. These stability criteria also allow for the control of the size of the region of attraction