lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

estimation of the domain of attraction for discrete time linear impulsive control systems with input saturation abstract this paper discusses the domain of attraction and stability analysis for discrete time linear impulsive systems with input saturation the first goal is to give a stable condition for the system the second is to present a strategy to find an inner estimate of the domain of attraction via establishing lyapunov functions and using other mathematical methods finally we present several examples to illustrate the validity of the conditions we have derived

brief paper estimating the region of attraction for large scale systems with uncertainties in this paper a new approach is proposed for estimating regions of attraction for large scale dynamic systems in designing control laws for such systems it is essential to incorporate the underlying information structure constraints while keeping the number of optimization variables at a minimum the proposed method successfully accomplishes both of these objectives it is computationally efficient and can produce decentralized control laws without imposing structural constraints on the lyapunov function a feature that can considerably improve the quality of the estimate the design algorithm is based on linear matrix inequalities and can easily accommodate various types of uncertainties in the system model an example with 300 states is provided to demonstrate the suitability of this approach for large sparse systems

control effort considerations in the stabilization of uncertain dynamical systems abstract this paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system we focus attention on the amplitude norm of the controller which is used to achieve this end the uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded a lyapunov function is used to establish the stability of the closed loop system the paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given lyapunov function furthermore a procedure is indicated for the construction of the desired feedback control law

semi invariants and their use for stability analysis of planar systems semi invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector eigenvalue pair for linear systems and are therefore very useful to depict the behaviour of the system in this article semi invariants are used to construct explicitly lyapunov functions useful for studying the stability of the origin for continuous time systems moreover using well known tools from differential geometry such as orbital symmetries it is shown how semi invariants can be found for several classes of systems important connections with centre manifold theory are pointed out by using the proposed general techniques a new proof of a result claimed to bendixson bendixson i 1901 sur les courbes definies par des equations differentielles acta mathematics 24 1 88 is given

composite quadratic lyapunov functions for constrained control systems a lyapunov function based on a set of quadratic functions is introduced in this paper we call this lyapunov function a composite quadratic function some important properties of this lyapunov function are revealed we show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids these results are used to study the set invariance properties of continuous time linear systems with input and state constraints we show that for a system under a given saturated linear feedback the convex hull of a set of invariant ellipsoids is also invariant if each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators then their convex hull can also be made invariant by the same actuators for a set of ellipsoids each invariant under a separate saturated linear feedback we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant

solving quadratic distance problems an lmi based approach the computation of the minimum distance of a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems the paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality lmi techniques exploiting a suitable representation of homogeneous forms a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one parameter family of lmi optimization problems several properties of the proposed technique are discussed in particular tightness of the lower bound is investigated providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori extensive numerical simulations are reported showing promising performances of the proposed method

vandermonde matrices on integer nodes the rectangular case we consider a rectangular vandermonde matrix v on integer nodes using combinatorial identities we give an explicit cholesky factorization of the matrix b v rm t v a factorization of the pseudo inverse of v and asymptotic estimates of the extremal eigenvalues of b we also discuss the numerical properties of the proposed formulas

arc length based maximal lyapunov functions and domains of attraction estimation for polynomial nonlinear systems in the phase space of a dynamical system containing an asymptotically stable equilibrium point the arc length function alf is defined as sum of length differential elements of phase trajectories starting from state points and ending at the equilibrium point it is shown that receding from the origin and verging on the domain of attraction doa boundary causes alf to be increased drastically according to the latter issue and regarding to other properties it is shown that alf is a maximal lyapunov function to this end a numerical method to approximate the alf as a polynomial function is proposed to ensure that the approximated function has positive value and negative derivative inside the desired region the homotopy continuation method is used thus the approximated alf presents an ensured lyapunov behavior to estimate doa the efficacy of the proposed method is demonstrated by several simulation examples

on the number of factors of sturmian words abstract we prove that for m 1 card a m 1 m i 1 m i 1 i where a m is the set of factors of length m of all the sturmian words and is the euler function this result was conjectured by dulucq and gouyou beauchamps 1987 who proved that this result implies that the language m 0 a m c is inherently ambiguous we also give a combinatorial version of the riemann hypothesis

properties of the composite quadratic lyapunov functions a composite quadratic lyapunov function introduced recently was shown to be very useful in the study of set invariance properties for linear systems with input and state constraints and for systems with a class of convex concave nonlinearities in this note more properties about this function are revealed in particular we study the continuity of the optimal parameter involved in this function this continuity is crucial in the construction of a continuous feedback law which makes the convex hull of a group of ellipsoids invariant

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

robust region of attraction estimation we propose a method to compute invariant subsets of the region of attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty parameter independent lyapunov functions are used to characterize invariant subsets of the robust region of attraction a branch and bound type refinement procedure reduces the conservatism we demonstrate the method on an example from the literature and uncertain controlled short period aircraft dynamics

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

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

computing output feedback controllers to enlarge the domain of attraction in polynomial systems the problem of computing controllers to enlarge the domain of attraction da of equilibrium points of polynomial systems is considered in order to deal with such a problem a technique for computing static nonlinear output feedback controllers which maximize the largest estimate of the da leda induced by a given polynomial lyapunov function is proposed the main contribution of the note is to show that a lower bound of the maximum achievable leda and a corresponding controller can be obtained through linear matrix inequality optimizations moreover a necessary condition for tightness of this lower bound is presented which is also a sufficient condition to establish the tightness of the lower bound of the leda for a given controller

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

some combinatorial properties of sturmian words abstract in this paper we give a characterization of finite sturmian words by palindrome words which generalizes a property of the fibonacci words we prove that the set st of finite sturmian words coincides with the set of the factors of all the words w such that w ab cxy with a b c palindromes x y a b and x y moreover using this result we prove that st is equal to the set of the factors of all words w having two periods p and q which are coprimes and such that w p q 2 several other combinatorial properties concerning special and bispecial elements of st are shown as a consequence we give a new and purely combinatorial proof of the enumeration formula of st

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

robust static output feedback controllers via robust stabilizability functions this technical note addresses the design of robust static output feedback controllers that minimize a polynomial cost and robustly stabilize a system with polynomial dependence on an uncertain vector constrained in a semialgebraic set the admissible controllers are those in a given hyper rectangle for which the system is well posed first the class of robust stabilizability functions is introduced i e the functions of the controller that are positive whenever the controller robustly stabilizes the system second the approximation of a robust stabilizability function with a controller dependent lower bound is proposed through a sums of squares sos program exploiting a technique developed in the estimation of the domain of attraction third the derivation of a robust stabilizing controller from the found controller dependent lower bound is addressed through a second sos program that provides an upper bound of the optimal cost the proposed method is asymptotically non conservative under mild assumptions

rational lyapunov functions for estimating and controlling the robust domain of attraction this paper addresses the estimation and control of the robust domain of attraction rda of equilibrium points through rational lyapunov functions lfs and sum of squares sos techniques specifically continuous time uncertain polynomial systems are considered where the uncertainty is represented by a vector that affects polynomially the system and is constrained into a semialgebraic set the estimation problem consists of computing the largest estimate of the rda lerda provided by a given rational lf the control problem consists of computing a polynomial static output controller of given degree for maximizing such a lerda in particular the paper shows that the computation of the best lower bound of the lerda for chosen degrees of the sos polynomials which requires the solution of a nonconvex optimization problem with bilinear matrix inequalities bmis can be reformulated as a quasi convex optimization problem under some conditions moreover the paper provides a necessary and sufficient condition for establishing tightness of this lower bound lastly the paper discusses the search for optimal rational lfs using the proposed strategy

a generalization of zubov s method to perturbed systems a generalization of zubov s theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point for the construction it is necessary to consider solutions in the viscosity sense as a consequence maximal robust lyapunov functions can be characterized as viscosity solutions

design of robust state feedback laws existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices the uncertainties are modelled as sector bounded non linearities and feedback laws ensuring global asymptotic closed loop stability are obtained using lyapunov e direct method the algorithms essentially amount to repeatedly solving a parameter dependent riccati equation until the maximal solution becomes positive definite

estimating the domain of attraction for uncertain polynomial systems estimating the domain of attraction da of equilibrium points is a problem of fundamental importance in systems engineering several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the da leda for a given lyapunov function lf however the problem of estimating the robust da rda that is the da guaranteed for all possible uncertainties in an uncertain system it is still an unsolved problem in this paper lmi methods are proposed for estimating the rda in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope specifically the issue of computing the robust leda rleda that is the intersection of all ledas is considered for common and parameter dependent lfs providing constant and parameter dependent lower bounds the computation of approximations with simple shape of the rleda in the case of parameter dependent lfs is also discussed

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

brief piecewise affine lyapunov functions for discrete time linear systems with saturating controls this paper is concerned with piecewise affine pwa functions as lyapunov function candidates for stability analysis of time invariant discrete time linear systems with saturating closed loop control inputs using a pwa model of saturating closed loop system new necessary and sufficient conditions for a pwa function be a lyapunov function are presented based on linear programming formulation of these conditions an effective algorithm is proposed for construction of such lyapunov functions for estimation of the region of local asymptotic stability compared to piecewise linear functions like minkowski functions pwa functions are more adequate to capture the dynamical effects of saturation nonlinearities giving strictly less conservative results the complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension being hence appropriate for problems with large state dimension but with few saturating inputs

vandermonde matrices on integer nodes this paper deals with vandermonde matrices widetilde v whose nodes are the first n integer numbers we give an analytic factorization of such matrices and explicit formulas for the entries of their inverses and explore their computational issues we also give asymptotic estimates of the frobenius norm of both widetilde v and its inverse

symmetric functions and the vandermonde matrix this work deduces the lower and the upper triangular factors of the inverse of the vandermonde matrix using symmetric functions and combinatorial identities the l and u matrices are in turn factored as bidiagonal matrices the elements of the upper triangular matrices in both the vandermonde matrix and its inverse are obtained recursively the particular value x i 1 q qi 1 in the indeterminates of the vandermonde matrix is investigated and it leads to q binomial and q stirling matrices it is also shown that q stirling matrices may be obtained from the pascal matrix

control sharing and merging control lyapunov functions given two control lyapunov functions clfs a merging is a new clf whose gradient is a positive combination of the gradients of the two parents clfs the merging function is an important trade off since this new function may for instance approximate one of the two parents functions close to the origin while being close to the other far away for nonlinear control affine systems some equivalence properties are shown between the control sharing property i e the existence of a single control law which makes simultaneously negative the lyapunov derivatives of the two given clfs and the existence of merging clfs it is shown that even for linear time invariant systems the control sharing property does not always hold with the remarkable exception of planar systems the class of linear differential inclusions is also discussed and similar equivalence results are presented for this class of systems linear matrix inequalities conditions are provided to guarantee the control sharing property finally a constructive procedure based on the recently considered r functions is defined to merge two smooth positively homogeneous clfs

region of attraction analysis of nonlinear stochastic systems using polynomial chaos expansion abstract a method is presented to estimate the region of attraction roa of stochastic systems with finite second moment and uncertainty dependent equilibria the approach employs polynomial chaos pc expansions to represent the stochastic system by a higher dimensional set of deterministic equations we first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty dependent equilibrium point of the stochastic system a connection between the boundedness of the moments of the stochastic system and the lyapunov stability of its pc expansion is then derived defining corresponding notions of a roa for both system representations we show how this connection can be leveraged to recover an estimate of the roa of the stochastic system from the roa of the pc expanded system two optimization programs obtained from sum of squares programming techniques are provided to compute inner estimates of the roa the first optimization program uses the lyapunov stability arguments to return an estimate of the roa of the pc expansion based on this result and user specifications on the moments for the initial conditions the second one employs the shown connection to provide the corresponding roa of the stochastic system the method is demonstrated by two examples

approximation and monotonicity of the maximal invariant ellipsoid for discrete time systems by bounded controls an analytic approximation of the maximal invariant ellipsoid for a discrete time linear system with bounded controls is derived the approximation is expressed explicitly in terms of the coefficient matrices of the system and the positive definite matrix that represents the shape of the invariant ellipsoid it is shown that this approximation is very close to the exact maximal invariant ellipsoid obtained by solving either an lmi based optimization problem or a nonlinear algebraic equation furthermore the necessary and sufficient condition for such an approximation to be equal to the exact maximal invariant ellipsoid is established on the other hand the monotonicity of the maximal invariant ellipsoid resulting from the minimal energy control with guaranteed convergence rate problem is established that shows a trade off between increasing the size of the invariant ellipsoid and increasing the convergence rate of the closed loop system under a bounded control two illustrative examples demonstrate of the effectiveness of the results

logical composition of lyapunov functions this article introduces the use of r functions to compose single lyapunov functions lfs via classic boolean operators with the aim to obtain a rich family of non conventional generally non convex functions the main benefit of the proposed composition is the nice geometric interpretation since it corresponds to intersection and union operations in the phase space region the composition of lfs is parameterised through a variable and classic compositions of lfs through min and max operations are recovered as a special case for a particular value of the proposed logical composition is applied to region of asymptotic stability ras estimation problems where the union of several lfs corresponds to the union of the ras estimates obtained from the separate use of each lf likewise the intersection of several lfs defined on independent subsets of the state space variables provides a single lf for the overall dynamical system sufficient conditions for the composition function to be an lf are provide

regional robust stabilisation and domain of attraction estimation for mimo uncertain nonlinear systems with input saturation abstractthe regional robust stabilisation by means of linear time invariant state feedback control for a class of uncertain mimo nonlinear systems with parametric uncertainties and control input saturation is investigated the nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain of attraction doa inside a given polytopic region in the state space a novel set of sufficient linear matrix inequality lmi conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger doas by considering the polytopic description of the saturated inputs a few examples are presented showing favourable comparisons with recently published similar control design methodologies

weak repetitions in sturmian strings in this paper we analyze the weak repetitions in sturmian strings and show that an optimally efficient algorithm to compute the weak repetitions in sturmian strings according to the output encodings defined in the literature is quadratic in the string length finally we present an encoding that leads to a linear time algorithm

fast qr factorization of vandermonde matrices abstract we introduce a fast algorithm for computing the qr factors of a complex column vandermonde matrix v the complexity of the algorithm is 5 mn 7 n 2 2 o m where m is the number of rows in v and n is the number of columns we assume that m n the matrices q and r may be computed independently if desired two special cases for the row vandermonde matrix with real elements or unit magnitude elements are also studied and similar results are obtained

parameter dependent lyapunov functions for linear systems with constant uncertainties robust stability of linear time invariant systems with respect to structured uncertainties is considered the small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition it is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic lyapunov function which proves robust stability with respect to all allowable time varying perturbations in this technical note we show that if the small gain condition is satisfied with frequency varying scalings then an explicit parameter dependent lyapunov function can be constructed to prove robust stability with respect to constant uncertainties this lyapunov function has a rational quadratic dependence on the uncertainties

safe nonlinear control design for input constrained polynomial systems using sum of squares programming we study the feedback stabilisation problem for input affine polynomial systems subject to polytopic input constraints first we characterise a subset of the state space which we refer to as the

inner approximating reachable sets for polynomial systems with time varying uncertainties in this paper we propose a convex programming based method to address a long standing problem of inner approximating backward reachable sets of state constrained polynomial systems subject to time varying uncertainties the backward reachable set is a set of states from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties it is equal to the zero sublevel set of the unique lipschitz viscosity solution to a hamilton jacobi partial differential equation hje we show that inner approximations of the backward reachable set can be formed by zero sublevel sets of its viscosity supersolutions consequently we reduce the inner approximation problem to a problem of synthesizing polynomial viscosity supersolutions to this hje such a polynomial solution in our method is synthesized by solving a single semidefinite program we also prove that polynomial solutions to the formulated semidefinite program exist and can produce a convergent sequence of inner approximations to the interior of the backward reachable set in measure under appropriate assumptions this is the main contribution of this paper several illustrative examples demonstrate the merits of our approach

rapidly exponentially stabilizing control lyapunov functions and hybrid zero dynamics this paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models systems with impulse effects through control lyapunov functions the periodic orbit is assumed to lie in a c1 submanifold z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics the associated restriction dynamics are termed the hybrid zero dynamics the orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics prior results on the stabilization of such periodic orbits with respect to the full order dynamics of the system with impulse effects have relied on input output linearization of the dynamics transverse to the zero dynamics manifold the principal result of this paper demonstrates that a variant of control lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface z can be used to achieve exponential stability of the periodic orbit in the full order dynamics thereby significantly extending the class of stabilizing controllers the main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control lyapunov functions

on enlarging the basin of attraction for linear systems under saturated linear feedback we consider the problem of enlarging the basin of attraction for a linear system under saturated linear feedback an lmi based approach to this problem is developed for discrete time system this approach is enhanced by the lifting technique which leads to further enlargement of the basin of attraction the low convergence rate inherent with the large invariant set hence the large basin of attraction is prevented by the construction of a sequence of invariant ellipsoids nested within the large one obtained

establishing convexity of polynomial lyapunov functions and their sublevel sets there has been renewed interest in the use of polynomial and homogeneous polynomial lyapunov functions for addressing stability and performance issues in systems and control theory however no condition is yet available to establish the convexity of these functions and the convexity of their sublevel sets properties that can be useful in many applications in this paper new conditions for establishing these convexity properties are proposed which can be handled through convex linear matrix inequalities optimizations by means of sum of squares relaxations

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

a decomposition technique for nonlinear dynamical system analysis a method for analyzing large scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented it is shown that the decomposition approach can be used to scale the sum of squares programming framework for nonlinear systems analysis the method constructs subsystem lyapunov functions which are used to form a composite lyapunov function for the whole system further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem lyapunov functions

brief paper estimating the domain of attraction for non polynomial systems via lmi optimizations this paper proposes a strategy for estimating the domain of attraction da for non polynomial systems via lyapunov functions lfs the idea consists of converting the non polynomial optimization arising for a chosen lf in a polynomial one which can be solved via lmi optimizations this is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst case remainders in truncated taylor expansions moreover a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen lf and another condition is provided for establishing whether such an estimate has been found the proposed strategy can readily be exploited with variable lfs in order to search for optimal estimates lastly it is worth remarking that no other method is available to estimate the da for non polynomial systems via lmis

estimating the domain of attraction via union of continuous families of lyapunov estimates this paper proposes a new approach to estimate the domain of attraction of equilibrium points of polynomial systems the idea consists of estimating the domain of attraction via the union of a continuous family of lyapunov estimates rather than via one lyapunov estimate only as done in existing methods this family is obtained through a convex lmi optimization by deriving a stability condition which takes simultaneously into account all considered lyapunov functions moreover inner approximations of the union of this family via a set with simple shape are also derived

rectangular vandermonde matrices on chebyshev nodes abstract we consider a rectangular vandermonde matrix v on chebyshev nodes using combinatorial identities we give an explicit factorization of the matrix v a factorization of thepseudo inverse of v and prove that the condition number 2 v does not depend onthe dimension of nodes set we also discuss the numerical properties of the proposed formulas

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

region of attraction analysis with integral quadratic constraints abstract a general framework is presented to estimate the region of attraction of attracting equilibrium points the system is described by a feedback connection of a nonlinear polynomial system and a bounded operator the input output behavior of the operator is characterized using an integral quadratic constraint this allows to analyze generic problems including for example hard nonlinearities and different classes of uncertainties adding to the state of practice in the field which is typically limited to polynomial vector fields the iqc description is also nonrestrictive with the main result given for both hard and soft factorizations optimization algorithms based on sum of squares techniques are then proposed with the aim to enlarge the inner estimates of the roa numerical examples are provided to show the applicability of the approaches these include a saturated plant where bounds on the states are exploited to refine the sector description and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft iqcs

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

polynomial level set method for polynomial system reachable set estimation in this paper we present a polynomial level set method for advecting a semi algebraic set for polynomial systems this method uses the sub level representation of sets the problem of flowing these sets under the advection map of a dynamical system is converted to a semi definite program which is then used to compute the coefficients of the polynomials the method presented in this paper does not require either the sets being positively invariant or star shaped hence the proposed algorithm can describe the behavior of system states both inside and outside the domain of attraction and can also be used to describe more complex shapes of sets we further address the related problems of constraining the degree of the polynomials various numerical examples are presented to show the effectiveness of advection approach

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

region of attraction estimation using invariant sets and rational lyapunov functions this work addresses the problem of estimating the region of attraction ra of equilibrium points of nonlinear dynamical systems the estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a lyapunov function moreover we present conditions for the existence of lyapunov functions linked to the positively invariant set formulation we propose connections to fundamental results on estimates of the ra are presented and support the search of lyapunov functions of a rational nature we then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the ra at each iteration

stability analysis of uncertain genetic sum regulatory networks this paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions specifically we first consider uncertain genetic networks where the regulation occurs at the transcriptional level and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity we hence show that this condition can be formulated as a convex optimization through polynomial lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation smr of polynomials which allows to establish whether a polynomial is a sum of squares sos via a linear matrix inequality lmi then we propose a method for computing a family of bounding sets by means of convex optimizations it is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter dependent nonlinear equations and is hence unknown lastly the proposed approach is extended to models where the regulation occurs at different levels and both mrna and protein dynamics are nonlinear

structural stabilization of uncertain systems necessity of the matching condition this paper investigates one aspect of the problem of stabilizing an uncertain linear system that is the systems under consideration contain uncertain parameters which are unknown but bounded the question arises as to whether such a system can be stabilized via feedback control in some of the previous papers in this area the system is assumed to satisfy a so called matching condition this type of assumption is used to assure that the uncertain system can be stabilized it is known however that the matching condition is not a necessary condition for stabilizability this paper introduces a strengthened notion of stabilizability referred to as structural stabilizability via a nominally determined quadratic lyapunov function it is then shown that if a system is to have this stronger property the matching condition must necessarily be satisfied

brief paper a parametric lyapunov equation approach to low gain feedback design for discrete time systems low gain feedback a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero has found several applications in constrained control systems robust control and nonlinear control in the continuous time setting there are currently three ways of constructing low gain feedback laws the eigenstructure assignment approach the parametric are based approach and the parametric lyapunov equation based approach the eigenstructure assignment approach leads to feedback gains explicitly parameterized in the low gain parameter the parametric are based approach results in a lyapunov function along with the feedback gain but requires the solution of an are for each value of the parameter the parametric lyapunov equation based approach possesses the advantages of the first two approaches and results in both an explicitly parameterized feedback gains and a lyapunov function the first two approaches have been extended to discrete time setting this paper develops the parametric lyapunov equation based approach to low gain feedback design for discrete time systems

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

on the normal matrix of the polynomial ls problem over the chebyshev points this note deals with the properties of the normal matrix of the polynomial ls problem over the chebyshev points and presents an explicit cholesky factorization of that matrix