Four rendezvous problems for n non-identical simple-integral plants

Four rendezvous problems for n non-identical simple-integral plants (K ₁/s, K₂/s,[tdot],K_n/s), with a common input x(t) = A_mi^m u(t), m = 0,1,2,[tdot], and initial output conditions of (ξ₁,ξ₂,[tdot], ξ_n), are studied in this paper. These rendezvous problems, generally speaking, are concerned with transferring the n plants outputs to meet one another in a finite time (at a. fixed target or on a non-stationary one).     

Nonlocal quasi-Newton method for solving convex variational inequalities

Conclusion In the optimization problem [f ₀(x)│h_i(x)<-0,i=1,…,l] relaxation of the functionf ₀(x)+Nh⁺(x) does not produce, as we know [6, 7], α_k=1 in Newton's method with the auxiliary problem (5), (6), whereF(x)=f _0′(x). For this reason, Newton type methods based on relaxation off ₀(x)+Nh⁺(x) are not superlinearly convergent (so-called Maratos effect). The results of this article indicate that if (F(x)=f _0′(x), then replacement of the initial optimization problem with a larger equivalent problem (7) eliminates the Maratos effect in the proposed quasi-Newton method. This result is mainly of theoretical interest, because Newton type optimization methods in the space of the variablesx ∈R ⁿ are less complex. However to the best of our knowledge, the difficulties with nonlocal convergence arising in these methods (choice of parameters, etc.) have not been fully resolved [10, 11]. The discussion of these difficulties and comparison with the proposed method fall outside the scope of the present article, which focuses on solution of variational inequalities (1), (2) for the general caseF′(x)≠F′ ^T(x). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 78–91, November–December, 1994.     

On the general problem of pole assignment

Given a linear, time-invariant, minimal and strictly proper system  set of monic polynomials φ _i (s), i = l, 2, [tdot],q, such that φ _i (s) divides φ _i?1 (s), i = 2, 3, [tdot],q a method for finding a proper feedback system  which makes the invariant polynomials of the closed-loop system equal to the φ _i (s) is established and a sufficient condition which ensures the existence of  is determined. This method can also be used to assign the poles of the compensated system provided that certain poles are multiple. The order of the required compensator is generally loss than that of the Luenberger or the Wolovich compensation schemes.     

Exact 3-satisfiability is decidable in time <Emphasis Type="Italic">O</Emphasis>(2<Superscript>0.16254<Emphasis Type="Italic">n</Emphasis></Superscript>)

Let F=C ₁C _m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C _i contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2^0.18674n ). Thereby we improve a result in [2,3] stating X3SATO(2^0.2072n ) and a bound of O(2^0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SATO(2^0.16254n ).     

New PCG based algorithms for the solution of Hermitian Toeplitz systems

In order to solve Toeplitz linear systems A_n(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed A_n(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=A _n ⁻¹ (g) A_n(f). In this paper we prove that the union of the spectra of all the G_n is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λ _l ⁿ of G_n to r (and of λ _n ⁿ to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n^1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2⁻¹] and takes the maximum value 2⁻¹ when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.     

Interpolazione con splines verificanti una opportuna condizione

Summary A partition of the interval [x ₀,x _n+1] inton+1 subintervals [x _i ,x _i+1] (i=0,1,...,n) is considered. A spline functionf(x)∈C ^m , which coincides with a polynomialp _i (x)[p _i (x _i )=y _i ,p _i (x _i+1)=y _i+1] of degreem+1 in [x _i ,x _i+1 ], is introduced. Such a spline depends onm arbitrary constants. These constants are determined minimizing the integral .      

Algorithms for dense graphs and networks on the random access computer

We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size = (logn) a maximal matching in ann-vertex bipartite graph in timeO(n ²+n ^2.5/)=O(n ^2.5/logn), how to compute the transitive closure of a digraph withn vertices andm edges in timeO(n ²+nm/), how to solve the uncapacitated transportation problem with integer costs in the range [O.C] and integer demands in the range [–U.U] in timeO ((n ³ (log log/logn)^1/2+n² logU) lognC), and how to solve the assignment problem with integer costs in the range [O.C] in timeO(n ^2.5 lognC/(logn/loglogn)^1/4).Assuming a suitably compressed input, we also show how to do depth-first and breadth-first search and how to compute strongly connected components and biconnected components in timeO(n+n ²/), and how to solve the single source shortest-path problem with integer costs in the range [O.C] in time0 (n ²(logC)/logn). For the transitive closure algorithm we also report on the experiences with an implementation.Most of this research was carried out while both authors worked at the Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany. The research was supported in part by ESPRIT Project No. 3075 ALCOM. The first author acknowledges support also from NSERC Grant No. OGPIN007.     

A transformation of non-linear dynamical systems with a single singular singularly perturbed differential equation†

Singularly perturbed state differential equations of the form [xdot] = f(x, z, t, ?), x(t₀, ?) = x₀(?); μ(?)? = g(x, z, t, ?), z(t₀ ?) = z₀(?) with lim μ(?) = 0; ?, μ > 0 are considered, where the nominal equation 0 = g(x, z, t, 0)^{? → ∞} does not have to be solvable for z. A fairly general transformation of the above system into a form [xdot]^* = f ^*(x^*, z, t; z(1),...,z^(d?1), ? ); μ^*(?)z^(d)= g^*(x^*. z(⁰),...z^(d?1), t; ?), with dim x^* = dim x ?(d ? 1), d ? 1 is proposed. The transformed system stands a better chance of being analysed by existing methods (especially by those proposed by Hoppensteadt (1971) and Hoppensteadt and Mi ranker (1976)) than the original singular singularly perturbed form. Informative examples are presented.     

General eigenvalue placement in linear control systems by output feedback

Consider the time-invariant system E[xdot] = Ax + Bu, y = Cx where E is a square matrix that may be singular. The problem is to find constant matrices K and L, such that the feedback law u = Ky+L[ydot] yields x = exp (λt)v_i (where v_i is some constant vector) for some preassigned λ_i (i=l, 2, [tdot], r). This problem is equivalent to that of finding K and L which makes a preassigned λ _i an eigenvalue corresponding to the general eigenvalue problem {λ(E ? BLC) ? (A + BKC)}v=0. Using matrix generalized inverses, a method is developed for the construction of a linear system of equations from which the elements of K and L may be computed.     

Boundary control,stabilization and zero-pole dynamics for a non-linear distributed parameter system

In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter boundary control system governed by a scalar viscous Burgers' equation. Namely, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero dynamics as the gain parameters are increased to infinity. We also prove a robust version of this result, valid for perturbations by an unknown disturbance with arbitrary L² norm. For the controlled Burgers' equation forced by a disturbance we prove that, for all initial data in L²(0, 1), the closed-loop trajectories converge in L²(0, 1), uniformly in t∈[0, T] and in H¹(0, 1), uniformly in t∈[t₀, T] for any t₀>0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional boundary controls are included in the closed-loop system. This provides a proof of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous Dirichlet boundary controlled Burgers' equation. As an application of our convergence of trajectories results, we demonstrate that our boundary feedback control scheme provides a semiglobal exponential stabilizing feedback law in L², H¹ and L^∞ for the open-loop system consisting of Burgers' equation with Neumann boundary conditions and zero forcing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is ‘practically semiglobally stabilizable’ in L²-norm. Copyright © 1999 John Wiley & Sons, Ltd.     

New Minimum Distance Bounds for Linear Codes over Small Fields

Let [n, k, d] _q -codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). In this paper we consider codes over GF(3), GF(5), GF(7), and GF(8). Over GF(3), three new linear codes are constructed. Over GF(5), eight new linear codes are constructed and the nonexistence of six codes is proved. Over GF(7), the existence of 33 new codes is proved. Over GF(8), the existence of ten new codes and the nonexistence of six codes is proved. All of these results improve the corresponding lower and upper bounds in Brouwer's table [www.win.tue.nl/aeb/voorlincod.html].     

Stratification associated with local b-functions

The local b$b$-function _bf,p(s)$b_{f,p}\left(s\right)$ of an n$n$-variate polynomial f∈C[x]$f\in \mathbb{C}\left[x\right]$ (x=(_x1,…,_xn)$x=(x_1,\dots ,x_n)$) at a point p∈Cⁿ$p\in {\mathbb{C}}^n$ is constant on each stratum of a stratification of Cⁿ${\mathbb{C}}^n$. We propose a new method for computing such a stratification and _bf,p(s)$b_{f,p}\left(s\right)$ on each stratum. In the existing method proposed in Oaku (1997b), a primary ideal decomposition of an ideal in C[x,s]$\mathbb{C}[x,s]$ is needed and our experiment shows that the primary decomposition can be a bottleneck for computing the stratification. In our new method, the computation can be done by just computing ideal quotients and examining inclusions of algebraic sets. The precise form of a stratum can be obtained by computing the decomposition of the radicals of the ideals in C[x]$\mathbb{C}\left[x\right]$ defining the stratum. We also introduce various techniques for improving the practical efficiency of the implementation and we show results of computations for some examples.     

High performance polymer actuator based on carbon nanotube-ionic liquid gel: Effect of ionic liquid

In this study, we investigated a dependence of anionic species of ionic liquids (ILs) (IL: perfluoroalkyltrifluoroborate anions ([C_nF_2n+1BF₃]⁻ (n = 0, 1, 2) and bis(perfluoroalkylsulfonyl)imide anions ([(C_mF_2m+1SO₂)(C_nF_2n+1SO₂)N]⁻ (m, n = 0, 1, 2)) on electrochemical and electromechanical properties. 1-Ethyl-3-methylimidazolium (EMI⁺) was selected as a cation for ILs. 1-Ethyl-3-methylimidazolium trifluoromethyltrifluoroborate (EMI[CF₃BF₃]), 1-ethyl-3-methylimidazolium pentafluoroethyltrifluoroborate (EMI[CF₃CF₂BF₃]), 1-ethyl-3-methylimidazolium fluorosulfonyl(trifluoromethylsulfonyl)imide (EMI[FTA]) and 1-ethyl-3-methylimidazolium pentafluoroethylsulfonyl(trifluoromethylsulfonyl)imide (EMI[C₁C₂]) were synthesized according to the literatures. The generated strains of the bucky-gel electrodes of the actuators containing EMI[CF₃BF₃] (in the high frequency range: 10-0.5 Hz) and EMI[CF₃CF₂BF₃] (in the high frequency range of 1-0.5 Hz) are larger than that containing EMI[BF₄] (that is to say the quick response). For low frequencies (0.1-0.005 Hz), the generated strain containing EMI[CF₃CF₂BF₃] was larger than those containing other ILs (EMI[C_nF_2n+1BF₃] (n = 0, 1) and EMI[(C_mF_2m+1SO₂)(C_nF_2n+1SO₂)N] (m, n = 0, 1, 2)). The Young's modulus of actuators containing EMI[CF₃BF₃] and EMI[CF₃CF₂BF₃] were 145 and 110 MPa, respectively. The melting points of EMI[CF₃BF₃] and EMI[CF₃CF₂BF₃] are lower than that of EMI[BF₄].Therefore, trifluoromethyltrifluoroborate ([CF₃BF₃]⁻) and pentafluoroethyltrifluoroborate ([CF₃CF₂BF₃]⁻) anions performed much better as the actuator using the polymer-supported bucky-gel electrode containing the IL. These results are considered to be the actuator enough to apply actual applications (e.g. tactile display).     

Stochastic suppression and stabilization of delay differential systems

In this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system ${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)In this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system ${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)$, where δ(t) is the variable delay and f satisfies the one‐sided polynomial growth condition. Since f may defy the linear growth condition or the one‐sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t)=f(x(t), x(t?δ(t)), t)dt+qx(t)dw₁(t)+σ|x(t)|^βx(t)dw₂(t) by introducing two independent Brownian motions w₁(t) and w₂(t). This paper shows that the Brownian motion w₂(t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of β under the condition σ≠0. Moreover, for sufficiently large q, the Brownian motion w₁(t) may exponentially stabilize this system. Copyright © 2010 John Wiley & Sons, Ltd.     

Exact 3-satisfiability is decidable in time O(20.16254n)

Let F = C ₁ C _m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C _i contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2^0.18674n ). Thereby we improve a result in [2,3] stating X3SAT O(2^0.2072n ) and a bound of O(2^0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT O(2^0.16254n ).An extended abstract of this paper was presented at the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002).     

Robot calibration by mobile camera systems

A measurement technique for kinematic calibration of robot manipulators, which uses a stereo hand-eye system with moving camera coordinates, is presented in this article. The calibration system consists of a pair of cameras rigidly mounted on the robot end-effector, a camera calibration board, and a robot calibration fixture. The stereo cameras are precalibrated using the camera calibration board so that the 3D coordinates of any object point seen by the stereo cameras can be computed with respect to the camera coordinate frame [C] defined by the calibration board. Because [C] is fixed with respect to the tool frame [T] of the robot, it moves with the robot hand from one calibration measurement configuration to another. On each face of the robot calibration fixture that defines the world coordinate frame [W], there are evenly spaced dot patterns of uniform shape. Each pattern defines a coordinate frame [E_i], whose pose is known in [W]. The dot pattern is designed in such a way that from a pair of images of the pattern, the pose of [E_i] can be estimated with respect to [C] in each robot calibration measurement. By that means the pose of [C] becomes known in [W] at each robot measurement configuration. For a sufficient number of measurement configurations, the homogeneous transformation from [W] to [C] (or equivalently to [T]), and thus the link parameters of the robot, can be identified using the least-squares techniques. Because the cameras perform local measurements only, the field-of-view of the camera system can be as small as 50 × 50 mm², resulting in an overall accuracy of the measurement system as high as 0.05 mm. This is at least 20 times better than the accuracy provided by vision-based measurement systems with a fixed camera coordinate frame using common off-the-shelf cameras. © 1994 John Wiley & Sons, Inc.     

Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).     

Discrete-time convolution feedback systems †

This paper considers multi-input multi-output discrete-time feedback systems characterized by y = G?e and e = u — y. Theorem I shows that if the closed-loop impulse response H is stable in the sense that H ? ? ¹ _n×n (p), then [Gtilde] (z) =Ñ(z) [ [Dtilde](z)] ^? where Ñ(z), ;[Dtilde](z) are in ? ¹ _n×n (p) Theorem III gives necessary and sufficient conditions for H ? ? ¹ _n×n (p) Finally Theorem III gives necessary and sufficient conditions for stability when [Gtilde] (z) has a finite number of multiple poles in ∣z∣ > p : the case where the leading term of the Laurent expansion at each of these poles is singular is treated in detail.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

Graded tableaux for Rational Pavelka Logic

In this article, a sound and complete tableau system for Rational Pavelka Logic (RPL) is introduced. Extended formulas are used as the counterpart of the graded formulas. In this calculus, if we want to show that the graded formula (x, r) is tableau provable (in the finite fuzzy theory F, respectively), we develop a tableau for the extended formula [r, x] (for the set of extended formulas {[r, x], [x₁, a₁],…, [x_n, a_n] }, respectively). If this tableau closes we claim that (x, r) is tableau provable (in the fuzzy theory F, respectively). We claim also that x is valid at the degree equal to the l.u.b. that allows the closure of the tableaux. Our tableaux are a first step toward efficient procedures of automated deduction in narrow fuzzy logic with truth constants. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1273–1285, 2005.