Low-order self-tuner for fault-tolerant control of a class of unknown nonlinear stochastic sampled-data systems

Based on the modified state-space self-tuning control (STC) via the observer/Kalman filter identification (OKID) method, an effective low-order tuner for fault-tolerant control of a class of unknown nonlinear stochastic sampled-data systems is proposed in this paper. The OKID method is a time-domain technique that identifies a discrete input–output map by using known input–output sampled data in the general coordinate form, through an extension of the eigensystem realization algorithm (ERA). Then, the above identified model in a general coordinate form is transformed to an observer form to provide a computationally effective initialization for a low-order on-line “auto-regressive moving average process with exogenous (ARMAX) model”-based identification. Furthermore, the proposed approach uses a modified Kalman filter estimate algorithm and the current-output-based observer to repair the drawback of the system multiple failures. Thus, the fault-tolerant control (FTC) performance can be significantly improved. As a result, a low-order state-space self-tuning control (STC) is constructed. Finally, the method is applied for a three-tank system with various faults to demonstrate the effectiveness of the proposed methodology.     

Chaotic Homeomorphisms of Rn, Lifted from Torus Homeomorphisms

We establish the existence of self-homeomorphisms of Rⁿ, n 2, which are chaotic in the sense of Devaney, preserve volumeand are spatially periodic. Moreover, we show that in the spaceof volume-preserving homeomorphisms of the n-torus with meanrotation zero, those with chaotic lifts to Rⁿ are dense, withrespect to the uniform topology. An application is given forfixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–FranksTheorem). 1991 Mathematics Subject Classification 54H20.     

A low-order active fault-tolerant state space self-tuner for the unknown sampled-data nonlinear singular system using OKID and modified ARMAX model-based system identification

In this paper, we present two control schemes for the unknown sampled-data nonlinear singular system. One is an observer-based digital redesign tracker with the state-feedback gain and the feed-forward gain based on off-line observer/Kalman filter identification (OKID) method. The presented control scheme is able to make the unknown sampled-data nonlinear singular system to well track the desired reference signal. The other is an active fault tolerance state-space self-tuner using the OKID method and modified autoregressive moving average with exogenous inputs (ARMAX) model-based system identification for unknown sampled-data nonlinear singular system with input faults. First, one can apply the off-line OKID method to determine the appropriate (low-) order of the unknown system order and good initial parameters of the modified ARMAX model to improve the convergent speed of recursive extended-least-squares (RELS) method. Then, based on modified ARMAX-based system identification, a corresponding adaptive digital control scheme is presented for the unknown sampled-data nonlinear singular system with immeasurable system state. Moreover, in order to overcome the interference of input fault, one can use a fault-tolerant control scheme for unknown sampled-data nonlinear singular system by modifying the conventional self-tuner control (STC). The presented method can effectively cope with partially abrupt and/or gradual system input faults. Finally, some illustrative examples including a real circuit system are given to demonstrate the effectiveness of the presented design methodologies.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

Hypercyclic and Chaotic Convolution Operators

Every convolution operator on a space of ultradifferentiablefunctions of Beurling or Roumieu type and on the correspondingspace of ultradistributions is hypercyclic and chaotic whenit is not a multiple of the identity. The operator of differentiationis hypercyclic on the space A^–, but it need not be hypercyclicon radial weighted algebras of entire functions.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

Waves in the Presence of an Infinite Dock with Gap

After extending some completeness results of an earlier paper,the two-dimensional problem of the infinite dock with gap isconsidered. With the frequency and 2 the non-dimensional lengthof the gap, eigenvalues of K = ²/g are first computed and thentheir apparent asymptotic form is established not only to orderK^–1 but also up to terms in K^–2.     

Error estimates for stochastic differential games: the adverse stopping case

^** Email: frederic.bonnans{at}inria.fr^*** Email: stefania.maroso{at}inria.fr^**** Email: zidani{at}ensta.fr We obtain error bounds for monotone approximation schemes ofa particular Isaacs equation. This is an extension of the theoryfor estimating errors for the Hamilton–Jacobi–Bellmanequation. To obtain the upper error bound, we consider the ‘Krylovregularization’ of the Isaacs equation to build an approximatesub-solution of the scheme. To get the lower error bound, weextend the method of Barles & Jakobsen (2005, SIAM J. Numer.Anal.) which consists in introducing a switching system whosesolutions are local super-solutions of the Isaacs equation.     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

Numerical Investigations into the Stokes Phenomenon. II

The Stokes phenomenon associated with the differential equationsW " = WZ (z²–a²) and W" = w(z² –a²)(x²–b²)isconsidered. As an application to the method introduced in paper I, somenumerical and analytical results concerning the Stokes constantsof these equations are presented.     

A complete characterization of the feedback-linearized input-output response

The problem of characterizing the feedback-linearized input–outputresponse of a single-input single-output nonlinear system viastatic state feedback is completely solved. The proposed approachreduces the problem of determining the structure of the mostgeneral linearized input–output response, together withthe respective control law, to that of solving a nonhomogeneoussystem of first-order partial differential equations. Necessaryand sufficient conditions for this system of equations to havea solution are established which are of an algebraic natureand henceforth easily checkable. Based on these conditions,the linearized input–output response is completely characterized,prior to determining the desired control law. Furthermore, thegeneral solution of this system of equations is derived whichleads to the determination of the general analytical expressionfor the control law that satisfies the input–output linearizationproblem.     

Wavelet stabilization and preconditioning for domain decomposition

The aim of this paper is to propose a stabilization technique in order to circumvent the two inf–sup conditions neededfor stability of the discretization of the three fields formulationfor domain decomposition methods (following the work of Brezzi& Marini (1993) Optimisation et Controle 89–96). Realization of this technique in terms of wavelets is presented. Furthermore, the resulting discrete problem is shown to admitan asymptotically optimal preconditioner also based on wavelets. Received 24 February, 1999. Revised 7 October, 1999. ^* aivlis@ian.pv.cnr.it; http://www.ian.pv.cnr.it/~aivlis kunoth@iam.un-bonn.de; http://www.iam.uni-bonn.de/~kunoth     

The Fast Numerical Solution of Very Large Elliptic Difference Schemes

Let L_kv_k = g_k be a system of difference equations discretizingan elliptic boundary value problem. Assume the system to be"very large", that means that the number of unknowns exceedsthe capacity of storage. We present a method for solving theproblem with much less storage requirement. For two-dimensionalproblems the size of the needed storage decreases from O(h^–2)to (or even O(h^–5/4)). The computational work increasesonly by a factor about six. The technique can be generalizedto nonlinear problems. The algorithm is also useful for computerswith a small number of parallel processors.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

On a Comparison Principle and the Critical Fujita Exponents for Solutions of Semilinear Parabolic Inequalities

The paper examines blow-up phenomena for the inequality u_t–Lu–|u|^q–1u_t–L–||^q–1 (*) in the half-space x Rⁿ, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rⁿ. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.     

Bounds For Gradient Trajectories and Geodesic Diameter of Real Algebraic Sets

Let M Rⁿ be a connected component of an algebraic set ^–1(0),where is a polynomial of degree d. Assume that M is containedin a ball of radius r. We prove that the geodesic diameter ofM is bounded by 2rv(n)d(4d–5)^n–2, where v(n) =(1/2)((n+1)/2)(n/2)^–1.This estimate is based on the bound rv(n)d(4d–5)^n–2for the length of the gradient trajectories of a linear projectionrestricted to M. 2000 Mathematics Subject Classification 32Bxx,34Cxx (primary), 32Sxx, 14P10 (secondary).     

Spines and Embeddings of n-Manifolds

Every compact, connected PL manifold Mⁿ, with MⁿØ, collapsesto a codimension-one subpolyhedron Q^n–1, called a spineof Mⁿ. The purpose of this paper is to prove that, if Q^n–1is appropriately chosen, one can reconstruct Mⁿ from Q^n–1,after taking the Cartesian product with an interval I=[0, 1].     

A discrete-time reduced-order controller for robust stabilization of plant in the normalized coprime-factor description

Using the technique of singlar perturbation, a procedure fordesigning a discrete–time reduced–order controllerfor robust stabilization is proposed. The theoretical aspectsare derived directly from the results of the generalized Popov–Yakubovichtheory recently developed. ¹Vlad Ionescu, 3 Emile Zola, Bucharest, 71272, Romania. ²Cristian Oara, 34 Austrului, Bucharest, 73115, Romania.