cartan decomposition subgroups of 2 for g sl 3 r and g so 2 n we give explicit practical conditions that determine whether or not a closed connected subgroup h of g has the property that there exists a compact subset c of g with chc g to do this we fix a cartan decomposition g ka k of g and then carry out an approximate calculation of khk n a for each closed connected subgroup h of g

a high order numerical method for the helmholtz equation with nonstandard boundary conditions we describe a high order accurate methodology for the numerical simulation of time harmonic waves governed by the helmholtz equation our approach combines compact finite difference schemes that provide an inexpensive venue toward high order accuracy with the method of difference potentials developed by ryaben kii the latter can be interpreted as a generalized discrete version of the method of calderon s operators in the theory of partial differential equations the method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing it introduces a universal framework for treating boundary conditions of any type a significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive in this paper we address various types of boundary conditions using the method of difference potentials we demonstrate th

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

on switching h controllers for a class of linear parameter varying systems abstract we consider switching h controllers for a class of linear parameter varying lpv systems scheduled along a measurable parameter trajectory the candidate controllers are selected from a given controller set according to the switching rules based on the scheduling variable we provide sufficient conditions to guarantee the stability of the switching lpv systems in terms of the dwell time and the average dwell time our results are illustrated with an example where switching between two robust controllers is performed for an lpv system

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

gain scheduled and nonlinear systems dynamic analysis by velocity based linearization families a family of velocity based linearizations is proposed for a nonlinear system in contrast to the conventional series expansion linearization a member of the family of velocity based linearizations is valid in the vicinity of any operating point not just an equilibrium operating point the velocity based linearizations facilitate dynamic analysis far from the equilibrium operating points and enable the transient behaviour of the nonlinear system to be investigated using velocity based linearizations stability conditions are derived for both smooth and non smooth nonlinear systems which avoid the restrictions to trajectories lying within an unnecessarily perhaps excessively small neighbourhood about the equilibrium operating points inherent in existing frozen input theory for systems where there is no restriction on the rate of variation the velocity based linearization analysis is global in nature the analysis techniques developed although quite general are motivated by the gain scheduling design

fast gain scheduling on tracking problems using derivative information current gain scheduling practice is necessarily limited to slow variation in scheduling variables this paper proposes a control law which reflects the exogenous signal and its time derivative with assumption of its availability the proposed controller achieves better performance than the existents for tracking fast scheduling variables while rejecting exogenous signal

linear quadratic bumpless transfer a method for bumpless transfer using ideas from lq theory is presented and shown to reduce to the hanus conditioning scheme under certain conditions

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

input output pseudolinearization for nonlinear systems an input output pseudolinearized system has the property that its family of linearizations about constant operating points has input output behavior that is independent of the operating point we develop a necessary and sufficient condition for existence of a state feedback and state coordinate change that transforms a given system into a pseudo normal form that is input output pseudolinearized these ideas are applied to obtain new approaches to approximate tracking and approximate inversion of nonlinear systems

a stability property of nonlinear sampled data systems with slowly varying inputs a stability analysis is presented that deals with the response of a nonlinear sampled data system to a slowly varying exogenous input signal the main result similar to existing results for purely continuous time and discrete time systems establishes that if the system possesses a manifold of exponentially stable constant operating points equilibria corresponding to constant values of the input signal then an initial state close to this manifold and a slowly varying input signal yield a trajectory that remains close to the manifold the analysis involves casting the sampled data system as a continuous time system with discrete jumps at the sampling instants

poincare series for non riemannian locally symmetric spaces abstract we initiate the spectral analysis of pseudo riemannian locally symmetric spaces g h beyond the classical cases where h is compact automorphic forms or is trivial analysis on symmetric spaces for any non riemannian reductive symmetric space x g h on which the discrete spectrum of the laplacian is nonempty and for any discrete group of isometries whose action on x is sufficiently proper we construct l 2 eigenfunctions of the laplacian on x x for an infinite set of eigenvalues these eigenfunctions are obtained as generalized poincare series i e as projections to x of sums over the orbits of eigenfunctions of the laplacian on x we prove that the poincare series we construct still converge and define nonzero l 2 functions after any small deformation of inside g for a large class of groups thus the infinite set of eigenvalues we construct is stable under small deformations this contrasts with the classical setting where the nonzero discrete spectrum varies on the teichmuller space of a compact riemann surface we actually construct joint l 2 eigenfunctions for the whole commutative algebra of invariant differential operators on x

feedback control of analytic nonlinear systems by extended linearization for mufti input mufti output analytic nonlinear systems a design method based on the family of linearizations of the system parameterized by the family of constant operating points is considered using the cauchy kowalewski theorem it is shown that nonlinear state feedback control laws and output observer feedback control laws exist such that the eigenvalues of the family of closed loop linearizations have specified analytically scheduled values with respect to the family of closed loop operating points

feedback linearization families for nonlinear systems characterizations are developed for parameterized linear dynamic feedback control laws that can arise when linearizing a nonlinear dynamic feedback control law for a specified nonlinear system about a family of constant operating points such characterizations are important in applying the recently developed extended linearization design approach to various types of control problems to illustrate the input output decoupling problem is considered

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

on a stability theorem for nonlinear systems with slowly varying inputs recently m kelemen ieee trans automat contr vol ac 31 766 768 aug 1986 presented a stability result that deals with the response of a nonlinear system to slowly varying input signals in this work a proof is given under a weaker hypothesis on the input signal by constructing a lyapunov function that is standard in the control literature this proof may be more accessible than the other approaches for completeness brief proofs of some intermediate results that are regarded as well known are included a simple example is given to show that this approach can yield explicit bounds in certain situations of interest

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

actions propres sur les espaces homogenes reductifs let g be a linear semisimple real lie group and h be a re ductive subgroup of g we give a necessary and sufficient condition for the existence of a nonabelian free discrete subgroup r of g acting properly on g h for instance such a group r does exist for sl 2n ir sl 2n 1 ir but does not for sl 2n 1 r sl 2n r with n 1

on surface radiation conditions for an ellipse we compare several on surface radiation boundary conditions in two dimensions for solving the helmholtz equation exterior to an ellipse we also introduce a new boundary condition for an ellipse based on a modal expansion in mathieu functions we compare the osrc to a finite difference method

proper action on a homogeneous space of reductive type an action of l on a homogeneous space g h is investigated where l h g are reductive lie groups a criterion of the properness of this action is obtained in terms of the little weyl group of g in particular r rankg r rankh iff calabi markus phenomenon occurs i e only finite subgroups of g can act properly discontinuously on g h then by using cohomological dimension theory of a discrete group l g h is proved compact iff d g d l d h where d g denotes the dimension of a riemannian symmetric space associated with g etc these results apply to the existence problem of lattice in g h several series of classical pseudo riemannian homogeneous spaces are found to admit non uniform lattice as well as uniform lattice while some necessary condition for the existence of uniform lattice is obtained when rankg rankh partially supported by grant in aid for scientific research no 63740074

brief analysis and design of gain scheduled sampled data control systems the existence of gain scheduled sampled data controllers that meet an important linearization requirement about a family of equilibria is investigated

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

a new method for linearizing non linear systems the pseudolinearization consider the single input non linear system x f x u it has been shown that under mild conditions there exist transformations z t x v s x u such that in the z space the linear tangent model is independent of the operating point a constructive procedure to obtain the above transformations is described this new method is computationally attractive and effective in many practical cases

improved accuracy for the helmholtz equation in unbounded domains based on properties of the helmholtz equation we derive a new equation for an auxiliary variable this reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original unknown computations confirm the improved accuracy of the new models in both two and three dimensions this also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k we examine the accuracy for both waveguide and scattering problems as a function of k h and the forcing mode l the use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k connections with parabolic approximations are analysed copyright 2004 john wiley sons ltd

a stability property in this note we obtain a stability property not for a point of equilibrium but for a certain family of asymptotically stable equilibria of a dynamical system with an input we show that this property depends not only on the initial condition but also on the time derivative of the input function we suggest possible applications of this result to some control problems

direct implementation of high order bgt artificial boundary conditions abstract local artificial boundary conditions abcs for the numerical simulation of waves have been successfully used for decades most notably the boundary conditions due to engquist majda bayliss gunzburger turkel and higdon the basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution the larger the number of terms canceled the higher the order of the boundary condition and in turn the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary in practice however the use of local abcs has been limited to low orders first and second because higher order boundary conditions involve higher order derivatives of the solution which may harm well posedness and cause numerical instabilities they are also difficult to implement especially in finite elements a prominent exception is the development of local high order abcs based on auxiliary variables in the current paper we implement high order bayliss turkel abcs directly with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either instead we represent the solution at the boundary as an expansion with respect to a continuous basis for the spherical artificial boundary the basis consists of eigenfunctions of the beltrami operator spherical harmonics which enable replacing the high order derivatives in the abcs with powers of the corresponding eigenvalues the continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials mdp it maintains high order accuracy even when the boundary is not aligned with the discretization grid

compact 2d and 3d sixth order schemes for the helmholtz equation with variable wave number several studies have presented compact fourth order accurate finite difference approximation for the helmholtz equation in two or three dimensions several of these formulae allow for the wave number k to be variable other papers have extended this further to include variable coefficients within the laplacian which models non homogeneous materials in electromagnetism later papers considered more accurate compact sixth order methods but these were restricted to constant k in this paper we extend these compact sixth order schemes to variable k in both two and three dimensions results on 2d and 3d problems with known analytic solutions verify the sixth order accuracy we demonstrate that for large wave numbers the second order scheme cannot produce comparable results with reasonable grid sizes

interpolation for gain scheduled control with guarantees here a methodology is presented which considers the interpolation of linear time invariant lti controllers designed for different operating points of a nonlinear system in order to produce a gain scheduled controller guarantees of closed loop quadratic stability and performance at intermediate interpolation points are presented in terms of a set of linear matrix inequalities lmis the proposed interpolation scheme can be applied in cases where the system must remain at the operating points most of the time and the transitions from one point to another rarely occur e g chemical processes satellites

multiple model adaptive mixing control the discrete time case most of the work in multiple model adaptive control with various forms of switching focused on continuous time systems the purpose of this technical note is to extend the results of one approach the adaptive mixing control amc to discrete time systems further the technical note solves the tracking problem which has not been addressed in most schemes of this class stability and robustness properties of the amc scheme for discrete time systems are analyzed it is shown that in the ideal case when no disturbances or unmodeled dynamics are present the tracking error converges to zero in the non ideal case the mean square tracking error is of the order of magnitude of the modeling error provided the unmodeled dynamics satisfy a norm bound condition while these robustness results are conceptually similar to those of traditional robust adaptive control the proposed scheme does not suffer from the drawback of stabilizability of the estimated plant and in addition performs much better in simulation studies furthermore it allows well developed results from robust control to be incorporated in the design

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

design of controllers for linear parameter varying systems by the gain scheduling technique abstract design of controllers for linear multi input multi output systems whose dynamics depend on a time varying parameter is studied design algorithms for the state feedback control law based on the gain scheduling technique are proposed state feedback control laws are designed so that one of the following goals is achieved i the poles of the closed loop system for all fixed parameter values are placed in a small neighborhood of a set of desired closed loop poles ii the stability of the closed loop system for all fixed parameter values guarantees that the of the system with the time varying parameter independent of the rate of change of the parameter rigorously justified and computationally appealing design algorithms are developed

input output linearization by velocity based gain scheduling the velocity based analysis framework provides direct support for divide and conquer design approaches such as the gain scheduling control design methodology whereby the design of a non linear system is decomposed into the design of an associated family of linear systems the velocity based gain scheduling approach is quite general and directly supports the design of feedback configurations for which the closed loop dynamics are non linear however the present paper concentrates on the velocity based design of controllers which when combined with a non linear plant attain linear dynamics the resulting approach is a direct generalization to non linear systems of classical frequency domain pole zero inversion which is in many ways complementary to the input output linearization approach in particular the former is dynamic and reduces to open loop inversion in the linear case whilst the latter is essentially static and utilizes full state feedback

deformation of proper actions on reductive homogeneous spaces let g be a real reductive lie group and h a closed reductive subgroup of g we investigate the deformation of standard compact quotients of g h that is of quotients of g h by discrete groups that are uniform lattices in some closed reductive subgroup l of g acting properly and cocompactly on g h for l of real rank 1 we prove that after a small deformation in g such a group keeps acting properly discontinuously and cocompactly on g h more generally we prove that the properness of the action of any convex cocompact subgroup of l on g h is preserved under small deformations and we extend this result to reductive homogeneous spaces g h over any local field as an application we obtain compact quotients of so 2n 2 u n 1 by zariski dense discrete subgroups of so 2n 2 acting properly discontinuously

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

the l2 l2 bumpless transfer problem for linear plants its definition and solution a novel characterization of bumpless transfer among alternative controllers both in continuous and discrete time is introduced the bumpless transfer problem is to design a compensation scheme guaranteeing an l 2 respectively l 2 bound on the mismatch after the switching time between the actual plant output and a particular target ideal response minimization of the gain from initial plant state mismatch to l 2 respectively l 2 plant output mismatch provides improved transients a solution to the bumpless transfer problem is given both for plants without input saturation and for plants with input saturation the solution guarantees anti windup features in the latter case

a stability property and its application to discrete time nonlinear system control an inequality on the solution of discrete time nonlinear systems is established a special case of this inequality leads to a stability result that is the counterpart of the kelemen theorem on continuous time systems application of this inequality gives some results on stability regulation and tracking of the discrete time nonlinear systems

analytic framework for blended multiple model systems using linear local models in this paper it is shown that the dynamics of a conventional type of blended multiple model system are only weakly related to the local models from which it is formed a novel class of velocity based blended multiple model systems is proposed for which the dynamics are directly related to the local models indeed the solution to the blended multiple model system locally to a specific operating point is approximated by the weighted linear combination of the solutions to the local models moreover in contrast to conventional blended multiple model systems the velocity based blended multiple model systems employs linear local models thereby providing a degree of continuity with established linear methods and consequently facilitating analysis and design

exact feedforward linearization based on differential flatness this article deals with the trajectory aspect of differential flatness from a feedforward point of view the notion of exact feedforward linearization based on differential flatness is introduced a differentially flat system to which a nominal feedforward deduced from flatness is applied is equivalent by change of coordinates to a linear multivariable brunovsk form if the initial condition is consistent with the one considered in the design of the nominal trajectory in its second part the new notion states that there exist unique solutions in the vicinity of the desired trajectory when applying a nominal feedforward to the corresponding flat system to the end of stabilizing the desired trajectory the information from the brunovsk form is used to design the combination of the nominal feedforward and an additional feedback part in the case of extended pid controls for the latter stability is proven using a theorem by kelemen thus the overall control structure turns out to be quite simple and e

local absorbing boundary conditions for elliptical shaped boundaries we compare several local absorbing boundary conditions for solving the helmholtz equation by a finite difference or finite element method exterior to a general scatterer these boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface in order to compare the computational solution with an analytical solution we consider as an example scattering about an ellipse we solve the helmholtz equation with both finite differences and finite elements we also introduce a new boundary condition for an ellipse based on a modal expansion

approximate state feedback linearization using spline functions abstract this paper presents two numerical methods for synthesizing an approximate feedback linearizing control for single input nonlinear systems both techniques use part of the pseudo linearizing coordinate transformation as an output function t1 the first synthesis uses a well known recursive algorithm to compute the control given t1 splines represent functions that cannot be computed in closed form the second method yields a simplified controller that can be computed regardless of the plant s complexity experimental results from a rotating inverted pendulum show an improved performance of the approximate feedback linearization compared with pseudo linearization while highlighting the practical numerical difficulties that arise in the first method simulations of an acrobot illustrate the second method which is also improved over pseudo linearization

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

on stability properties of nonlinear systems with slowly varying inputs systems with slowly varying inputs are discussed as a special class of two time scale systems and singular perturbation results are seen as convenient tools to analyze their properties a lemma by f c hoppensteadt trans amer math soc vol 123 p 521 35 1966 for a parameterized family of systems is restated and applied to the analysis of systems with slowly varying inputs

dynamic gain scheduled control and extended linearisation extensions explicit formulae and stability a controller designed for linearisations at various trim operating points of a nonlinear system using linear approaches is not necessarily performing well or stable once scheduled with a state under dynamic conditions the key idea of using this scheduled control law design is to retain states close to the current usually dynamically varying operating point dynamic gain scheduling dgs is a technique aimed to resolve this controller scheduling issue for rapidly changing dynamics and states it entails scheduling the control law gains with a fast varying state variable rather than with a slowly varying state it has been successfully applied to the aircraft system models allowing also for nested loops the aim of this paper is to extend dgs to a more general setting allowing for more complex multi input dynamics incorporating the more simple static scheduling within the same controller hence given a linear design suitable transformations are provided which allow fast scheduling of multi variable co

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

stabilization analysis of discrete nonlinear systems in this paper we set a new framework for the analysis of nonlinear discrete systems to carry out the stabilization theorem and to design stable controllers of the systems within that framework this framework includes many practically important systems described for example by lipschitz functions functions with arbitrary order bounded derivatives bilinear functions and all polynomial functions an important existence theorem for stabilizing has been proven for this framework the theorem basically states that under definite rather broad conditions the zero solution of the plants can be made stable by a nonlinear output feedback in addition this nonlinear feedback is from a larger class of functions than these classes given by current literature following these stabilization theorems a new design procedure for the output feedback control law is presented all results presented are easy to use and convenient to apply the application is illustrated by several examples including the design of stabilizing control for a bilinear control system

stabilization of nonlinear systems with moving equilibria this note provides a new method for the stabilization of nonlinear systems with parametric uncertainty unlike traditional techniques our approach does not assume that the equilibrium remains fixed for all parameter values the proposed method combines different optimization techniques to produce a robust control that accounts for uncertain parametric variations and the corresponding equilibrium shifts comparisons with analytical gain scheduling are provided

a computationally efficient lyapunov based scheduling procedure for control of nonlinear systems with stability guarantees we propose an alternative to gain scheduling for stabilization of nonlinear systems for a useful class of nonlinear systems the characterization of a region of stability based on a control lyapunov function is computationally tractable in the sense that computation times vary polynomially with the state dimension for a fixed number of scheduling variables using this fact we develop a procedure to expand the region of stability by constructing control lyapunov functions to various trim points of the system a lyapunov based control synthesis algorithm is used to construct a control law that guarantees closed loop stability for initial conditions in the expanded region of state space this control asymptotically recovers the optimal stability margin in the sense of a lyapunov derivative which in turn can be seen as a performance measure robustness to bounded disturbances and stabilization under bounded control are easily incorporated into this framework in the worst case the computational complexity of the analysis problem that develops in the new method is increased by an exponential in the disturbance dimension similarly we can handle control constraints with an increase in computational complexity of no more than an exponential in the control dimension we demonstrate the new control design procedure on an example

discrete groups and non riemannian homogeneous spaces a basic question in geometry is to understand compact locally homogeneous manifolds i e those compact manifolds that can be locally modelled on a homogeneous space j h of a finite dimensional lie group h this means that there is an atlas on a manifold m consisting of local diffeomorphisms with open sets in j h where the transition functions between these open sets are given by translations by elements of h a basic example is the case of m j h f where f c h is a discrete subgroup acting freely and properly discontinuously on j h with a compact quotient these examples can be abstractly characterized as the complete case see g we then call f a cocompact lattice on j h when j is compact j h has an h invariant riemannian metric so that m itself becomes a riemannian manifold in a natural way thus the class of locally homogeneous riemannian manifolds includes but is not exhausted by the locally symmetric spaces for h aff rn and j gl n r the spaces j h f are the complete affinely flat manifolds a non riemannian example which has been extensively studied when h is simple and j is not compact the situation is far from being well understood some special series of such homogeneous spaces admitting a cocompact lattice have been constructed by kulkarni ku and by t kobayashi k for example kulkarni shows the existence of such a discrete group f for the homogeneous spaces so 1 2n so 2 2n and kobayshi does the same for u 1 n so 2 2n on the other hand the constructions they give for these and other similar series are quite special and there has been the general suspicion that most homogeneous spaces j h with h simple and j not compact do not admit a cocompact lattice we recall that when j is compact i e the riemannian case such f always exist by a result of borel b one general situation in which it is known that no such f exists is the case in which j h is noncompact and of reductive type and r rank h r rank j this situation was studied by calabi markus cm wolf w and kobayashi k and the nonexistence of f in this situation is a special case of a more general result known as the calabi markus phenomenon kobayashi and ono ko use the hirzebruch proportionality principle to show that some other special series do

nonlinear overtaking optimal control sufficiency stability and approximation the authors consider infinite horizon nonlinear optimal control problems with unbounded performance indexes using notions of overtaking optimality in the framework of optimal control new sufficient conditions of caratheodory hamilton jacobi type for both weak overtaking and overtaking optimality are presented and compared with known results from the calculus of variations setting using local structural properties of a related control hamiltonian conclusions are drawn on aspects of existence of overtaking and weak overtaking optimal controls stability of optimal trajectories and approximation of weak overtaking optimal controls both similarities and differences among alternative approaches to overtaking weak overtaking and strong optimal control problems are discussed the formulation and results are illustrated in the classical linear quadratic tracking problem and in an optimization approach to gain scheduling

high order numerical simulation of the transmission and scattering of waves using the method of difference potentials the method of difference potentials generalizes the method of calderon s operators from pdes to arbitrary difference equations and systems it offers several key advantages such as the capability of handling boundaries interfaces that are not aligned with the discretization grid variable coefficients and nonstandard boundary conditions in doing so the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid previously we have applied the method of difference potentials to solving several variable coefficient interior helmholtz problems with fourth and sixth order accuracy we have employed compact finite difference schemes as a core discretization methodology those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself numerical experiments corroborate the high order accuracy of our method for variable coefficients regular grids and non conforming boundaries in the current paper we extend the previously developed methodology to exterior problems we present a complete theoretical analysis of the algorithm as well as the results of a series of numerical simulations specifically we study the scattering of time harmonic waves about smooth shapes subject to various boundary conditions we also solve the transmission scattering problems in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside in all the cases our methodology guarantees high order accuracy for regular grids and non conforming boundaries and interfaces

robustness analysis with respect to exogenous perturbations for flatness based exact feedforward linearization a methodology to analyze robustness with respect to exogenous perturbations for exact feedforward linearization based on differential flatness is presented the analysis takes into consideration the tracking error equation and makes thereafter use of a stability result by kelemen coupled with results issued from interval analysis this turns exact feedforward linearization based on differential flatness into a general control methodology for flat systems

phase reduction models for improving the accuracy of the finite element solution of time harmonic scattering problems i general approach and low order models this paper introduces a new formulation of high frequency time harmonic scattering problems in view of a numerical finite element solution it is well known that pollution error causes inaccuracies in the finite element solution of short wave problems to partially avoid this precision problem the strategy proposed here consists in firstly numerically computing at a low cost an approximate phase of the exact solution through asymptotic propagative models secondly using this approximate phase a slowly varying unknown envelope is introduced and is computed using coarser mesh grids the global procedure is called phase reduction in this first paper the general theoretical procedure is developed and low order propagative models are numerically investigated in detail improved solutions based on higher order models are discussed showing the potential of the method for further developments

deformation of compact clifford klein forms of indefinite riemannian homogeneous manifolds 1 1 let g be a lie group and h a closed subgroup of g if a discrete subgroup of g acts properly discontinuously and freely on g h then the double coset space g h carries naturally a manifold structure such that the quotient map g h g h is locally diffeomorphic the manifold g h is said to be a clifford klein form of g h if it is compact then is said to be a uniform lattice for g h a typical example is a compact riemann surface mg with genus g 2 which is biholomorphic to a compact clifford klein form of the poincare disk g h sl 2 r so 2 by the uniformization theorem it is important from geometric view point that a clifford klein form g h inherits any g invariant local geometric structure on g h such as indefinite riemannian metric complex structure symplectic structure causal structure and so on

brief stability preserving interpolation methods for the synthesis of gain scheduled controllers synthesis of gain scheduled controllers for nonlinear plants often requires that a parameter varying controller be generated from a finite set of linear time invariant controllers we propose interpolation methods for this task with the property that stability of the linearized closed loop system is preserved for all values of the scheduling parameter in addition slow variation arguments are presented that establish local stability of the nonlinear closed loop system with gain scheduled controller

continuous time non linear flatness based predictive control an exact feedforward linearisation setting with an induction drive example a general flatness based framework for non linear continuous time predictive control is presented it extends the results of fliess and marquez 2000 to the non linear case the mathematical setting which is valid for multivariable systems is provided by the theory of flatness based exact feedforward linearisation introduced by the authors hagenmeyer and delaleau 2003b thereby differential flatness does not only yield an easy calculation of the predicted trajectories considering the respective system constraints but allows to use simple linear feedback parts in a two degree of freedom control structure moreover this formalism permits one to handle non minimum phase systems and furthermore to deal with parameter uncertainties and exogenous perturbations respective robustness analysis tools are available finally an induction drive example is discussed in detail and experimental results for this fast electro mechanical system are presented

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

an enhanced velocity based algorithm for safe implementations of gain scheduled controllers abstractthis paper presents an enhanced velocity based algorithm to implement gain scheduled controllers for nonlinear and parameter dependent systems a new scheme including pre and post filtering is proposed with the assumption that the time derivative of the controller inputs is not available for feedback control it is shown that the proposed control structure can preserve the input output properties of the linearised closed loop system in the neighbourhood of each equilibrium point avoiding the emergence of the so called hidden coupling terms moreover it is guaranteed that this implementation will not introduce unobservable or uncontrollable unstable modes and hence the internal stability will not be affected a case study dealing with the design of a pitch axis missile autopilot is carried out and the numerical simulation results confirm the validity of the proposed approach

an lmi approach to control oriented identification and model in validation of lpv systems this note proposes a control oriented identification framework for a class of linear parameter varying systems that takes into account both the dependence of part of the model on time varying parameters as well as the possible existence of a nonparametric component the main results of the note show that the problems of obtaining and validating a model for these systems can be recast as linear matrix inequality feasibility problems moreover as the information is completed the algorithm is shown to converge in the l sub 2 induced topology to the actual plant additional results include deterministic bounds on the identification error these results are illustrated with a practical example arising in the context of active vision

multiple model adaptive control with mixing despite the remarkable theoretical accomplishments and successful applications of adaptive control the field is not sufficiently mature to solve challenging control problems where strict performance and robustness guarantees are required critical to the design of practical control systems for these challenging applications and currently lacking in parameter estimation based adaptive control schemes is an approach that explicitly accounts for robust performance and stability specifications towards this goal this paper describes a robust adaptive control approach called adaptive mixing control that makes available the full suite of powerful design tools from lti theory e g mixed synthesis the stability and robustness properties of adaptive mixing control are analyzed it is shown that the mean square regulation error is of the order of the modeling error provided the unmodeled dynamics satisfy a norm bound condition and when the parameter estimate converges to its true value which is guaranteed if a persistence of excitation condition is satisfied the adaptive closed loop system converges exponentially fast to a closed loop system comprising the plant and some lti controller that satisfies the control objective a benchmark example is presented which is used to compare the adaptive mixing controller with other adaptive schemes

lq bumpless transfer between two tracking controllers in this article a linear quadratic lq scheme is proposed for bumpless transfer between two tracking controllers the controllers have integral actions which are used to eliminate the steady state tracking errors a compensator is designed using the lq technique to minimise the differences between two sets of signals the differences between two control signals as well as the differences between the two input signals which drive the two controllers under such circumstances complete information of the system is taken into account and utilised for achieving satisfactory control it is shown that the ideal bumpless transfer is achievable with the proposed control scheme moreover the results are extended to discrete time systems finally simulation results are given to demonstrate the effectiveness of the proposed control scheme

some numerical aspects of approximate linearization of single input non linear systems we characterize approximate linearization to order m of a single input non linear system in terms of the existence of an integrating factor to order m 2 which is equivalent to the well known condition that a system of vector fields be order m 1 involutive we present a constructive method to obtain the required transformation for approximate linearization which results in substantial computational savings

identification of non linear parametrically varying models using separable least squares in this paper a novel identification algorithm for a class of non linear possibly parameter varying models is proposed the algorithm is based on separable least squares ideas these models are given in the form of a linear fractional transformation lft where the forward part is represented by a conventional linear regression and the feedback part is given by a non linear map which can take into account scheduling variables available for measurement the non linear part of the model can be parameterized according to various paradigms like e g neural network nn or general nonlinear autoregressive exogenous narx models the estimation algorithm exploits the separability of the criterion used to estimate the parameters when using a nn it is possible the explicit computation of the frechet derivative needed to implement a separable least square algorithm

parameter dependent state observer design for affine lpv systems in this paper the design of a parameter dependent state observer that allows to estimate the state of an affine linear parameter varying lpv system is investigated the observer has the property of being parameter dependent since the corresponding state space matrices are scheduled using an interpolation method moreover the stability of the estimation error is based on the existence of an affine parameter dependent lyapunov function the main contribution of this paper is that the problem of the observer design and the existence of such a lyapunov function is interpreted as a flexible lmi feasibility condition some illustrative examples are presented at the end of this paper

the gain scheduling and the robust state feedback stabilization problems we consider dynamical uncertain systems of the form x spl dot a x w b x w u where w t spl isin w is an unknown but bounded time varying parameter we consider the robust state feedback stabilization problem in which we consider a control of the form u spl phi x and the gain scheduling stabilization problem in which a control of the form u spl phi x w full information control is admitted we show that for convex processes those systems in which for fixed x the set of all a x w b x w is convex the two problems are equivalent i e if there exists a locally lipschitz gain scheduling stabilizing control then there exists a robustly stabilizing control continuous everywhere except possibly at the origin thus for convex process stabilization knowledge of w t is not an advantage for the compensator then we consider the special class of polytopic lpv systems and show that there is no loss of regularity as in the general case if we pass from a gain scheduling controller to a state feedback controller in particular no discontinuity at the origin may occur we show that the existence of a dynamic controller always implies the existence of a static one finally we show that we can always find a linear gain scheduling controller for a stabilizable system this means that a possible advantage of the online measurement of the parameter w t is that this always allows for linear compensators whose implementation can be easier than that of nonlinear ones

anosov representations domains of discontinuity and applications the notion of anosov representations has been introduced by labourie in his study of the hitchin component for sl n r subsequently anosov representations have been studied mainly for surface groups in particular in the context of higher teichmuller spaces and for lattices in so 1 n in this article we extend the notion of anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties in particular given an anosov representation g we explicitly construct open subsets of compact g spaces on which acts properly discontinuously and with compact quotient as a consequence we show that higher teichmuller spaces parametrize locally homogeneous geometric structures on compact manifolds we also obtain applications regarding non standard compact clifford klein forms and compactifications of locally symmetric spaces of infinite volume

nonlinear system stabilization via hierarchical switching control a nonlinear control system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed specifically using equilibria dependent lyapunov functions a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems the switching nonlinear controller architecture is designed based on a generalized lower semicontinuous lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria the proposed framework provides a rigorous alternative to designing gain scheduled feedback controllers and guarantees local and global closed loop system stability for general nonlinear systems

feedback control of multiinput nonlinear systems by extended linearization control of a nonlinear system is addressed by considering its family of linearizations given that a family of linear state feedback controllers has been designed to meet some linear control objective it is shown how these controllers can be pieced together into an overall nonlinear state feedback controller the resulting closed loop system has the property that its linearizations are equal to the desired linearly controlled systems a similar approach can be used to design nonlinear state estimators the approach is illustrated by applying it to the control of a planar two link manipulator

interpolation of observer state feedback controllers for gain scheduling the authors propose a method of interpolating linear time invariant controllers with observer state feedback structure in order to generate a continuously varying family of controllers that stabilizes a family of linear plants gain scheduling is a motivation for this work and the interpolation method yields guidelines for the design of gain scheduled controllers the method is illustrated with the design of a missile autopilot using loop shaping h infinity controllers

bumpless transfer between observer based gain scheduled controllers this paper deals with bumpless transfer between a number of observer based controllers in a gain scheduling architecture linear observer based controllers are designed for a number of linear approximations of a non linear system in a set of operating points and gain scheduling control can subsequently be achieved by interpolating between each controller the youla jabr bongiorno kucera yjbk parameterization is used to achieve a smooth scheduling between the controllers this approach produces a scheduled controller as a linear fractional transformation between a fixed controller and a scheduling parameter the approach is tested on a simple but highly non linear model of a fossil fuel power plant

brief paper a velocity algorithm for the implementation of gain scheduled controllers a new method is proposed to implement gain scheduled controllers for nonlinear plants given a family of linear feedback controllers designed for linearizations of a nonlinear plant about constant operating points a nonlinear gain scheduled controller is derived that preserves the input output properties of the linear closed loop systems locally about each equilibrium point the key procedures in the proposed method are to provide integral action at the inputs to the plant and differentiate some of the measured outputs before they are fed back to the scheduled controller for a fairly general class of systems the nonlinear gain scheduled controllers are easy to obtain and their structure is similar to that of the original linear controllers

gain scheduling dynamic linear controllers for a nonlinear plant abstract for a general type of nonlinear tracking problem we consider gain scheduling based on a family of linear dynamic controllers designed for a family of plant linearizations about constant operating points a necessary and sufficient condition for the existence of gain scheduled controllers satisfying appropriate requirements is presented using this condition the impact of linear controller configuration on the existence of a gain scheduled controller can be assessed and ad hoc approaches to scheduling can be analyzed

brief robustness analysis of exact feedforward linearization based on differential flatness a methodology to analyze robustness with respect to parametric uncertainty for exact feedforward linearization based on differential flatness is presented the analysis takes into consideration the tracking error equation and thereafter makes use of a stability result by kelemen coupled with results issued from interval analysis theory

a general approach for high order absorbing boundary conditions for the helmholtz equation when solving a scattering problem in an unbounded space one needs to implement the sommerfeld condition as a boundary condition at infinity to ensure no energy penetrates the system in practice solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary bayliss gunzburger and turkel bgt suggested an absorbing boundary condition abc as a sequence of operators aimed at annihilating elements from the solution s series representation their method was practical only up to a second order condition later hagstrom and hariharan hh suggested a method which used auxiliary functions and enabled implementation of higher order conditions we compare various absorbing boundary conditions abcs and introduce a new method to construct high order abcs generalizing the hh method we then derive from this general method abcs based on different series representations of the solution to the helmholtz equation in polar elliptical and spherical coordinates some of these abcs are generalizations of previously constructed abcs and some are new these new abcs produce accurate solutions to the helmholtz equation which are much less dependent on the various parameters of the problem such as the value of k or the eccentricity of the ellipse in addition to constructing new abcs our general method sheds light on the connection between various abcs computations are presented to verify the high accuracy of these new abcs

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly

gain scheduled controller design an analytic framework directly incorporating non equilibrium plant dynamics a velocity based linearization framework is employed to develop a novel rigorous approach to gain scheduling design the proposed approach enables knowledge concerning the plant dynamics at non equilibrium operating points to be incorporated directly into the controller design since the velocity based linearization framework supports the analysis of the transient response performance considerations can be accommodated the approach retains continuity with linear methods which is central to the existing conventional gain scheduling methodology and since a single type of linearization is employed throughout the design procedure is both straightforward and conceptually appealing

brief stability and l2 gain properties of lpv systems stability and l 2 gain properties of linear parameter varying systems are obtained under assumed bounds on either the maximum or average value of the parameter rate