brief paper a duality based convex optimization approach to l2 gain control of piecewise affine slab differential inclusions this paper introduces the dual parameter set of a piecewise affine pwa system this is a key concept to enable a convex formulation of pwa controller synthesis for pwa slab differential inclusions using a new convex relaxation another important contribution of the paper is to present pwa l 2 gain analysis and synthesis results for pwa systems whose output is also a pwa function of the state as opposed to a piecewise linear pwl function unlike other results existing in the literature the sufficient lmi conditions in this paper are valid for synthesis even when the pwa systems include sliding modes a numerical example with sliding modes illustrates the new approach

generalized and hypergeneralized projectors abstract generalizations of the class of projectors are considered within the classes of normal and ep matrices some properties are derived and results on the sum difference and product of generalized and hypergeneralized projectors are given

local stabilization of switched affine systems this technical note considers the local stabilization problem for the class of switched affine systems the main idea is to use an alternative representation of the switched affine system as a nonlinear system with input constraints switching laws can be derived by emulating locally classical controllers it is shown that by restricting to local stabilization the classical constraint on the existence of constant stable convex combinations may be easily avoided the approach may be interpreted as a generalization where convex combinations defined as functions of the system state are being used constructive methods for deriving switching laws are proposed

brief paper fault tolerant control for a class of nonlinear sampled data systems via a euler approximate observer in this paper a fault tolerant control ftc scheme is developed for a class of nonlinear sampled data systems first a euler approximate discrete model is used to describe the plant under the sampling under this model an observer based fault estimation method is proposed to guarantee the accuracy of both the state and fault estimation values the conditions to make the approximate model consistent with the exact model are presented then an active fault tolerant controller which has a constraint condition for the sampling time is designed to make the faulty system stable finally an inverted pendulum is used to show the efficiency of the proposed method

the exact number of squares in fibonacci words all our words sequences are binary a square is a subword of the form uu concatenation two squares are distinct if they are of different shape not just translates of each other otherwise they are repeated fibonacci words are defined by f sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub j 0 7962

polynomial lpv approach to robust h control of nonlinear sampled data systems this paper addresses l2 gain analysis and robust h control of nonlinear sampled data systems described by a polynomial linear parameter varying plpv model first conditions for l2 gain analysis of sampled data plpv systems are derived using a modified krasovskii functional in which the distance between the real and measured parameters are considered as uncertainties the resulting conditions are formulated in terms of parameterised linear matrix inequalities plmi second a solution approach for satisfying plmi conditions over the set of whole parameters and uncertainties is proposed using the sum of squares decomposition method third conditions for robust h sampled data controller synthesis for nonlinear systems described by plpv models are derived in terms of parameterised bilinear matrix inequalities pbmi with the maximum allowable sampling period as the parameter to make the resulting bilinear matrix inequality bmi problems tractable an iterative algorithm is developed and utilised to s

a note on k generalized projections abstract in this note we investigate characterizations for k generalized projections i e a k a on hilbert spaces the obtained results generalize those for generalized projections on hilbert spaces in hong ke du yuan li the spectral characterization of generalized projections linear algebra appl 400 2005 313 318 and those for matrices in j benitez n thome characterizations and linear combinations of k generalized projectors linear algebra appl 410 2005 150 159

stability analysis and control design of discrete time switched affine systems this technical note focuses on stability analysis and control design of switched affine systems in discrete time domain the stability conditions are obtained by taking into account that the system trajectories governed by a certain switching function converge to a set of attraction boldsymbol mathcal v containing a desired equilibrium point these conditions follow from the adoption of a general quadratic lyapunov function whose time variation is bounded above by a concave convex function with center determined by minimax theory our main contribution is to provide a stabilizing state feedback switching function and an invariant set of attraction boldsymbol mathcal v with minimum volume as far as this class of quadratic lyapunov functions is adopted the results are applied to sampled data control of continuous time switched affine systems with chattering avoidance the speed control of a dc motor is presented

optimal off line detection of repetitions in a string an algorithm is presented to detect within optimal time o n log n and space o n off line on a ram all of the distinct repetitions in a given textstring on a finite alphabet the proposed strategy is self contained as it depends more heavily on algorithmic design considerations than on the combinatorial properties of the output it is based on a new data structure the leaf tree which is particularly suited to exploit simple properties of the suffix tree associated with the string to be analyzed

dynamic output feedback h control of continuous time switched affine systems this paper deals with output feedback switching function control design for continuous time switched affine systems assuring global asymptotic stability of a desired equilibrium point the set of all attainable equilibrium points is provided more specifically the main purpose is to design a full order switched affine filter together with a switching rule assuring not only global asymptotic stability but also a guaranteed h performance level the design conditions do not require that the open loop subsystems be stable and outperform other techniques available in the literature to date the results are compared with recent ones based on a quadratic lyapunov function and on a max type lyapunov function numerical examples illustrate the theoretical results and are used for comparisons

further results on generalized and hypergeneralized projectors abstract the notions of generalized and hypergeneralized projectors introduced by gros and trenkler j gro s j trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 are revisited on the one hand the present paper provides several new characterizations of these sets and on the other the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered moreover the paper demonstrates the usefulness in studying the properties of generalized and hypergeneralized projectors of the representation of complex matrices given in corollary 6 by hartwig and spindelbock r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256

absolute stability with a generalized sector condition this paper generalizes the linear sector in the classical absolute stability theory to a sector bounded by concave convex functions this generalization allows more flexible or more specific description of the nonlinearity and will thus reduce the conservatism in the estimation of the domain of attraction we introduce the notions of generalized sector and absolute contractive invariance for estimating the domain of attraction of the origin necessary and sufficient conditions are identified under which an ellipsoid is absolutely contractively invariant in the case that the sector is bounded by piecewise linear concave convex functions these conditions can be exactly stated as linear matrix inequalities moreover if we have a set of absolutely contractively invariant aci ellipsoids then their convex hull is also aci and inside the domain of attraction we also present optimization technique to maximize the absolutely contractively invariant ellipsoids for the purpose of estimating the domain of attraction the effectiveness of the proposed method is illustrated with examples

stability analysis of a class of pwm systems this note considers stability analysis of a class of pulsewidth modulated pwm systems that incorporates several different switched mode dc dc converters the systems of the class typically have periodic solutions a sampled data model is developed and used to prove stability of these solutions conditions for global and local exponential stability are derived using quadratic and piecewise quadratic lyapunov functions the state space is partitioned and the stability conditions are verified by checking a set of coupled linear matrix inequalities lmis

brief paper stability of sampled data piecewise affine systems under state feedback this paper addresses stability of sampled data piecewise affine pwa systems consisting of a continuous time plant and a discrete time emulation of a continuous time state feedback controller the paper presents conditions under which the trajectories of the sampled data closed loop system will exponentially converge to a neighborhood of the origin moreover the size of this neighborhood will be related to bounds on perturbation parameters related to the sampling procedure in particular related to the sampling period finally it will be shown that when the sampling period converges to zero the performance of the stabilizing continuous time pwa state feedback controller can be recovered by the emulated controller

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

asymptotic stability of sampled data piecewise affine slab systems this paper addresses stability analysis of closed loop sampled data piecewise affine pwa slab systems in particular we study the case in which a pwa plant is in feedback with a discrete time emulation of a pwa controller we consider the sampled data system as a continuous time system with a variable time delay the contributions of this work are threefold first we present a modified lyapunov krasovskii functional lkf for studying pwa systems with time delays that is less conservative when compared to previously suggested alternatives second based on the new lkf sufficient conditions are provided for asymptotic stability of sampled data pwa slab systems to the origin these conditions become linear matrix inequalities lmis in the case of a piecewise linear pwl controller finally we present an algorithm for finding a lower bound on the maximum delay that preserves asymptotic stability therefore the output of the algorithm provides an upper bound on the minimum sampling frequency that guarantees asymptotic stability of the sampled data system the new results are successfully applied to a unicycle example

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

stability of sampled data piecewise affine systems a time delay approach this paper addresses the stability analysis of sampled data piecewise affine pwa systems consisting of a continuous time plant in feedback connection with a discrete time emulation of a continuous time state feedback controller the sampled data system is first considered as a continuous time system with a variable time delay conditions under which the trajectories of the sampled data closed loop system will converge to an attracting invariant set are then presented it is also shown that when the sampling period converges to zero these conditions coincide with sufficient conditions for non fragility of the stabilizing continuous time pwa state feedback controller the results are successfully applied to a helicopter example

a note on generalized and hypergeneralized projectors abstract gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 have introduced two generalizations of orthogonal projectors called generalized projectors and hypergeneralized projectors in this note we characterize these generalizations by their spectral decompositions

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

a characterization of the squares in a fibonacci string a finite fibonacci stringfn is defined as follows f0 b f1 a for every integer n 2 fn fn 1fn 2 for n 1 the length of fn is denoted by the infinite fibonacci stringf is the string which contains every fn n 1 as a prefix apart from their general theoretical importance fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the abelian squares in a given string in this paper we provide a characterization of all the squares in f hence in every prefix fn this characterization naturally gives rise to a algorithm which specifies all the squares of fn in an appropriate encoding this encoding is made possible by the fact that the squares of fn occur consecutively in runs the number of which is by contrast the known general algorithms for the computation of the repetitions in an arbitrary string require time and produce outputs when applied to a fibonacci string fn

piecewise affine state feedback for piecewise affine slab systems using convex optimization abstract this paper shows that lyapunov based state feedback controller synthesis for piecewise affine pwa slab systems can be cast as an optimization problem subject to a set of linear matrix inequalities lmis analytically parameterized by a vector furthermore it is shown that continuity of the control inputs at the switchings can be guaranteed by adding equality constraints to the problem without affecting its parameterization structure finally it is shown that piecewise affine state feedback controller synthesis for piecewise affine slab systems to maximize the decay rate of a quadratic control lyapunov function can be cast as a set of quasi concave optimization problems analytically parameterized by a vector before casting the synthesis in the format presented in this paper lyapunov based piecewise affine state feedback controller synthesis could only be formulated as a bi convex optimization program which is very expensive to solve globally thus the fundamental importance of the contributions of the paper relies on the fact that for the first time the piecewise affine state feedback synthesis problem has been formulated as a convex problem with a parameterized set of lmis that can be relaxed to a finite set of lmis and solved efficiently to a point near the global optimum using available software furthermore it is shown for the first time that in some situations the global can be exactly found by solving only one concave problem

characterizations of k s 1 potent matrices and applications abstract recently situations where a matrix coincides with some of its powers have been studied this kind of matrices is related to the generalized inverse matrices on the other hand it is possible to introduce another class of matrices that involve an involutory matrix generalizing the well known idempotent matrix widely useful in many applications in this paper we introduce a new kind of matrices called k s 1 potent as an extension of the aforementioned ones first different properties of k s 1 potent matrices have been developed later the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view in terms of powers of the matrix by means of the group inverse and via a block representation of a matrix of index 1 finally an application of the above results to study linear combinations of k s 1 potent matrices is derived

properties of the combinations of commutative idempotents this paper deals with idempotent operators i e p2 p and n potent operators i e pn p we characterize the invertibility the group invertibility and idempotency of the linear combinations of idempotent operators or n potent operators under certain commutativity property imposed on them

switched affine systems using sampled data controllers robust and guaranteed stabilization the problem of robust and guaranteed stabilization is addressed for switched affine systems using sampled state feedback controllers based on the existence of a control lyapunov function for a relaxed system we propose three sampled data controls global attracting sets computed by solving a sequence of optimization problems guarantee practical and global asymptotic stabilization for the whole system trajectories in addition robust margins with respect to parameters uncertainties and non uniform sampling are provided using input to state stability finally a buck boost converter is considered to illustrate the effectiveness of the proposed approaches

observer based control of piecewise affine systems this paper presents a new synthesis method for both state and dynamic output feedback control of a class of hybrid systems called piecewise affine pwa systems the synthesis procedure delivers stabilizing controllers that can be proven to give either asymptotic or exponential convergence rates the synthesis method builds on existing pwa stability analysis tools by transforming the design into a closed loop analysis problem wherein the controller parameters are unknown more specifically the proposed technique formulates the search for a piecewise quadratic control lyapunov function and a piecewise affine control law as an optimization problem subject to linear constraints and a bilinear matrix inequality the linear constraints in the synthesis guarantee that sliding modes are not generated at the switching the resulting optimization problem is known to be hard but suboptimal solutions can be obtained using the three iterative algorithms presented in the paper the new synthesis technique allows con

sampled data piecewise affine differential inclusions this paper addresses exponential stability and stabilization of sampled data piecewise affine differential inclusions pwadi with an unknown nonuniform sampling rate the contributions of this paper are threefold first given a controller sufficient conditions for exponential stability of closed loop sampled data pwadi are presented using a piecewise smooth krasovskii functional second assuming the controller is piecewise linear and the krasovskii functional is piecewise quadratic the stability conditions are cast as linear matrix inequalities lmis third sufficient conditions for sampled data controller synthesis for pwadi are formulated in terms of lmis with the maximum allowable sampling period as a parameter the effectiveness of the approach is shown via a pwadi example that is motivated by a nonlinear system

intersecting periodic words let a a 1 2v be a word which is a square and for which a 1 2u is also a square and a k 1 min k 2w 2v has period w where u

the runs conjecture the runs conjecture proposed by kolpakov and kucherov 1999 7 states that the number of occurrences of maximal repetitions runs in a string of length n runs n is at most n we almost solve the conjecture by proving that runs n 1 029n this bound is obtained using a combination of theory and computer verification

practical optimal state feedback control law for continuous time switched affine systems with cyclic steady state in this article a method for computing an optimal state feedback control law for continuous time switched affine systems exhibiting cyclic behaviour in steady state is presented the hybrid solutions are deduced from the fillipov solutions it is shown that the optimal trajectory synthesis implies to determine singular arcs algebraic conditions are given to obtain these particular arcs of the trajectory a numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two point boundary value problem occurring in optimal control synthesis the interpolation of the solutions set through a neural network yields a state feedback control law several examples in the power converters field show the feasibility and the efficiency of the method

how many runs can a string contain given a string x x 1 n a repetition of period p in x is a substring u r x i 1 i rp p u r 2 where neither u x i 1 i p nor x i 1 i r 1 p 1 is a repetition the maximum number of repetitions in any string x is well known to be q nlogn a run or maximal periodicity of period p in x is a substring u rt x i 1 i rp t of x where u r is a repetition t a proper prefix of u and no repetition of period p begins at position i of x or ends at position i rp t 1 in 2000 kolpakov and kucherov showed that the maximum number r n of runs in any string x 1 n is o n but their proof was nonconstructive and provided no specific constant of proportionality at the same time they presented experimental data to prompt the conjecture r n

matrices a such that ra as 1r when rk i abstract this paper examines matrices a c n n such that ra a s 1 r where r k i the identity matrix and where s and k are nonnegative integers with k 2 spectral theory is used to characterize these matrices the cases s 0 and s 1 are considered separately since they are analyzed by different techniques

three overlapping squares the three squares lemma 9 famously explored the consequences of supposing that three squares occur at the same position in a string essentially it showed that this phenomenon could not occur unless the longest of the three squares was at least the sum of the lengths of the other two more recently several papers 10 30 21 13 have greatly extended this result to a new periodicity lemma npl by supposing that only two of the squares occur at the same position with a third occurring in a neighbourhood to the right in these cases also similar restrictions apply in this paper an alternative strategy is proposed the consequences of having only two squares at neighbouring positions are carefully analyzed and then the observation is made that the analysis applies in a straightforward way though perhaps with complicated details to the three neighbouring squares problem in its full generality we then apply these new insights first to proofs of the final two remaining unproved subcases out of a total of 14 of the npl 10 then to an instance of the more general problem

on linear combinations of generalized projectors abstract baksalary and baksalary linear algebra appl 321 2000 3 established a complete solution to the problem of when a linear combination of two different projectors is also a projector by listing all situations in which nonzero complex numbers c 1 c 2 and nonzero complex matrices p 1 p 2 p 1 p 2 satisfying p i 2 p i i 1 2 form a matrix p c 1 p 1 c 2 p 2 such that p 2 p in the present paper the same problem is considered for generalized projectors g 1 and g 2 defined by gros and trenkler linear algebra appl 264 1997 463 as matrices satisfying g i 2 g i i 1 2 their results concerning the sum g 1 g 2 and difference g 1 g 2 appear to be very special cases of the general solution established herein

path connectivity of k generalized projectors abstract let b h be the set of all bounded linear operators on a hilbert space h an operator a b h is said to be a k generalized projector if a k a where k 2 is an integer and a denotes the adjoint of a denote by b h k gp the set of all k generalized projectors in b h in this paper we show that any two homotopic k generalized projectors are path connected and that there does not exist a segment p q b h k gp when p and q are two different k generalized projectors

open loop nash equilibrium in polynomial differential games via state dependent riccati equation this paper studies finite as well as infinite time horizon nonzero sum polynomial differential games in both cases we explore the so called state dependent riccati equations to find a set of strategies that guarantee an open loop nash equilibrium for this particular class of nonlinear games such a method presents advantages in simplicity of the design of equilibrium strategies and yields computationally effective solution algorithms we demonstrate that this solution leads the game to an e or quasi equilibrium and provide an upper bound for this e quantity the proposed solution is given as a set of n coupled polynomial riccati like state dependent differential equations where each equation includes a p linear form tensor representation for its polynomial part we provide an algorithm for finding the solution of the state dependent algebraic equation in the infinite time case based on a hamiltonian approach and give conditions on the existence of such stabilizing solutions for a third order polynomial a numerical example is presented to illustrate effectiveness of the approach

spectral study of r s 1 k and r s 1 k potent matrices abstract the r s 1 k and r s 1 k potent matrices have been studied in several recent papers we continue these investigations from a spectral point of view specifically a spectral study of r s 1 k potent matrices is developed using characterizations involving an associated matrix pencil a r the corresponding spectral study for r s 1 k potent matrices involves the pencil a r in order to present some properties the relevance of the projector i a a where a is the group inverse of a is highlighted in addition some applications and numerical examples are given particularly involving pauli matrices and the quaternions

an alternative characterization of generalized projectors abstract one of the results of gros and trenkler linear algebra appl 264 1997 463 asserts that a square complex matrix k is a generalized projector if and only if it is i quadripotent ii normal and iii partial isometry the authors supplemented this statement by proving that condition iii in the above characterization can actually be deleted the purpose of the present note is to show that also an alternative reduction of the set of conditions i iii is possible as a consequence of establishing the redundancy of ii under the presence of i and iii

generalized projections in banach algebras in this note we investigate generalized projections in banach algebras our results generalize results obtained for bounded linear operators on hilbert spaces