nonlinear state feedback for l 1 optimal control this paper considers l1 optimal control problems with full state feedback in contrast to h optimal control previous work has shown that linear l1 optimal controllers can be dynamic and arbitrarily high order however this paper shows that continuous memoryless nonlinear state feedback perform as well as dynamic linear state feedback the derivation which is nonconstructive relies on concepts from viability theory

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

robust control through piecewise lyapunov functions for discrete time varying uncertain systems this paper is concerned with the robust stabilization by state feedback of a linear discrete time system with time varying uncertain parameters an optimization problem involving a set of linear matrix inequalities and scaling parameters provides both the robust feedback gain and the piecewise lyapunov function used to ensure the closed loop stability in the case of linear time varying systems involving the convex combination of two matrices only two scaling parameters constrained into the interval 0 1 are needed allowing a simple numerical solution as illustrated by means of examples

nonquadratic lyapunov functions for robust stability analysis of linear uncertain systems in this note we derive conditions of existence of a homogeneous polynomial lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box bounded uncertainty verification of these conditions is reduced to solving a convex minimization problem we produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic lyapunov functions

brief paper non conservative matrix inequality conditions for stability stabilizability of linear differential inclusions this paper shows that the matrix inequality conditions for stability stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative it is established that the existing stability stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions this implies that the composite quadratic functions are universal for robust possibly constrained stabilization problems of linear differential inclusions in particular a linear differential inclusion is stable stabilizable with without constraints iff it admits a lyapunov control lyapunov function in these classes examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions to confirm the stability stabilizability of the same system

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

convex synthesis of robust controllers for linear systems with polytopic time varying uncertainty the design of robust controllers for linear systems with structured time varying uncertainty is a fundamental problem in control systems this problem is still open because the available methods for robust analysis lead to nonconvex optimization whenever design variables are present this paper proposes a novel approach that overcomes this difficulty specifically continuous time linear systems with polytopic time varying uncertainty are considered the basic problem consists of designing a robust static output feedback controller ensuring robust asymptotical stability a novel approach is proposed through the introduction of robust stabilizability functions rsfs i e functions able to establish whether a controller is robustly asymptotically stabilizing in particular rsfs based on homogeneous polynomial lyapunov functions hplfs are searched for the proposed approach requires the solution of two convex optimization problems with constraints expressed as linear matrix inequalities lmis for any size of the lmis the proposed approach provides a sufficient condition for the existence of a sought controller moreover this condition is also necessary when the size of the lmis is large enough several extensions of the proposed methodology are presented which deal with the synthesis of controllers ensuring desired decay rate fixed order dynamic output controllers and controllers with the minimum norm

stability analysis of discrete time switched systems a switched homogeneous lyapunov function method abstractthis paper addresses the stability issue of discrete time switched systems with guaranteed dwell time the approach of switched homogeneous lyapunov function of higher order is formally proposed by means of this approach a necessary and sufficient condition is established to check the exponential stability of the considered system with the observation that switching signal is actually arbitrary if the dwell time is one sample time a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals using the augmented argument a necessary and sufficient exponential stability criterion is given for discrete time switched systems with delays a numerical example is provided to show the advantages of the theoretical results

criteria for robust absolute stability of time varying nonlinear continuous time systems the present paper establishes results for the robust absolute stability of a class of nonlinear continuous time systems with time varying matrix uncertainties of polyhedral type and multiple time varying sector nonlinearities by using the variational method and the lyapunov second method criteria for robust absolute stability are obtained in different forms for the given class of systems specifically the parametric classes of lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability the piecewise linear lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases two examples are presented as applications of the present results to a particular second order system and to a specific class of systems with time varying interval matrices in the linear part

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

common polynomial lyapunov functions for linear switched systems in this paper we consider linear switched systems dot x t a u t x t x in r n u in u a u u in u compact and the problem of asymptotic stability for arbitrary switching functions uniform with respect to switching uas given a uas system it is always possible to build a common polynomial lyapunov function our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems this result answers a question raised by dayawansa and martin a generalization to a class of piecewise polynomial lyapunov functions is given

robust stability of time varying uncertain systems with rational dependence on the uncertainty robust stability of time varying uncertain systems is a key problem in automatic control this note considers the case of linear systems with rational dependence on an uncertain time varying vector constrained in a polytope which is typically addressed in the literature by using the linear fractional representation lfr a novel sufficient condition for robust stability is derived in terms of a linear matrix inequality lmi feasibility test by exploiting homogeneous polynomial lyapunov functions the square matrix representation and an extended version of polya s theorem which considers structured matrix polynomials it is shown that this condition is also necessary for second order systems and that this condition is less conservative than existing lmi conditions based on the lfr for any order

brief paper robust stability of time varying polytopic systems via parameter dependent homogeneous lyapunov functions this paper deals with robust stability analysis of linear state space systems affected by time varying uncertainties with bounded variation rate a new class of parameter dependent lyapunov functions is introduced whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms this class of lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time invariant perturbations the main result of the paper is a sufficient condition to determine the sought lyapunov function which amounts to solving an lmi feasibility problem derived via a suitable parameterization of polynomial homogeneous forms moreover lower bounds on the maximum variation rate for which robust stability of the system is preserved are shown to be computable in terms of generalized eigenvalue problems numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

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

piecewise lyapunov functions for robust stability of linear time varying systems in this paper we investigate the use of two term piecewise quadratic lyapunov functions for robust stability of linear time varying systems by using the so called s procedure and a special variable reduction method we provide numerically efficient conditions for the robust asymptotic stability of the linear time varying systems involving the convex combinations of two matrices an example is included to demonstrate the usefulness of our 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

constrained stabilization of continuous time linear systems in this paper we consider the stabilization problem for linear continuous time systems under state and control constraints we show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain continuous feedback stabilizing control and we show how to use existing numeric procedures for discrete time systems to solve the continuous time problem we propose a new discontinuous stabilizing control law for scalar input systems which has the advantage of being successfully applicable to systems with quantized control

robust asymptotic stabilizability of petersen s counterexample via a linear static controller a stabilizing linear static controller is discovered for a linear time varying uncertain system of petersen we use the concept of a sandwich cone decomposition to show that petersen s system is stabilizable by linear static controllers we establish the surprising result that petersen s system is stabilizable against time varying uncertainties on any given compact subset of the uncertainty controllability space our work suggests the use of scalar quadratic lyapunov functions a special subclass of polyhedral lyapunov functions in establishing stability for other linear time varying uncertain systems and it reopens the study of the conjecture stabilizability via nonlinear control always implies stabilizability via linear control

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

robust stability analysis of nonlinear hybrid systems we present a methodology for robust stability analysis of nonlinear hybrid systems through the algorithmic construction of polynomial and piecewise polynomial lyapunov like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials several improvements compared to previous approaches are discussed such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems

persistent disturbance rejection via static state feedback in contrast with spl hscr sub spl infin and spl hscr sub 2 control theories the problem of persistent disturbance rejection l sup 1 optimal control leads to dynamic controllers even when the states of the plant are available for feedback using viability theory shamma showed 1993 in a nonconstructive way that in the state feedback case the same performance achieved by any dynamic linear time invariant controller can be achieved using memoryless nonlinear state feedback in this paper we give an alternative constructive proof of these results for discrete and continuous time systems the main result of the paper shows that in both cases the l sup 1 norm achieved by any stabilizing state feedback linear dynamic controller can be also achieved using a memoryless variable structure controller

parameter memorized lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper a novel parameter memorized lyapunov function is proposed for stability analysis of discrete time linear systems with time varying parametric uncertainties the parameter memorized lyapunov function depends on the uncertain parameters with a memory of a certain interval it is shown that the previous parameter dependent lyapunov function is a special memoryless case of the parameter memorized lyapunov function and as a result the parameter memorized lyapunov function approach leads to less conservative results furthermore if the length of the interval for parameter memory is sufficiently long a nonconservative stability analysis result can be achieved numerical examples are provided to evaluate the obtained theoretical results

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

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

stability analysis and control of a class of lpv systems with piecewise constant parameters stability criteria characterizing the asymptotic stability of a class of lpv systems with piecewise constant parameters under constant and minimum dwell time are derived it is shown that for such systems the conditions for the stability under minimum dwell time can be seen as a unifying stability concept lying in between quadratic and robust stability thereby including them as extremal cases the results are then extended to address the stabilization problem using a particular class of time dependent gain scheduled state feedback controllers several examples are given for illustration

boundedness of rough oscillatory singular integral on triebel lizorkin spaces we obtain appropriate sharp bounds on triebel lizorkin spaces for rough oscillatory integrals with polynomial phase by using these bounds and using an extrapolation argument we obtain some new and previously known results for oscillatory integrals under very weak size conditions on the kernel functions

robust state feedback control of ltv systems nonlinear is better than linear examples of linear control systems with fast time varying uncertain coefficients are given which can be stabilized by a nonlinear memoryless state feedback but cannot be stabilized by a linear time invariant dynamic state feedback by means of one of these examples the authors show that the closed loop quadratic stability margin may be infinitely smaller than the actual stability margin

haar based stability analysis of lpv systems a new gridding based algorithm for stability analysis of linear parameter varying lpv systems is presented the algorithm inherits the main features of classical gridding techniques it can handle a vast class of parametric dependencies as well as non convex parametric domains novelty of the proposed approach lies in the use of haar wavelet transform theory to guarantee constraint satisfaction over the entire parametric domain even for an arbitrarily sparse grid it represents a major improvement over traditional gridding approaches which fail to provide such a certificate without requiring a posteriori verification tests the resulting algorithms rely on semidefinite programing and are related to sufficient stability conditions whose degree of conservatism decreases as the grid density and the haar truncation level increase two numerical examples corroborate the validity of the proposed algorithms

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

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

establishing robust stability of discrete time systems with time varying uncertainty this paper addresses the problem of establishing robust asymptotical stability of discrete time linear systems polynomially affected by time varying uncertainty confined into a polytope a linear matrix inequality lmi condition for establishing robust asymptotical stability is proposed by introducing a novel approach for establishing the existence of a common homogeneous polynomial lyapunov function hplf this approach consists firstly of introducing a gram matrix built with respect to the state and parametrized by an arbitrary vector function of the uncertainty and secondly of requiring that a transformation of the introduced gram matrix is a sum of squares sos of matrix polynomials the approach hence is referred to as a gram sos approach it is shown that the proposed lmi condition is sufficient for any degree of the hplf candidate that includes quadratic robust stability as a special case and that is also necessary for a sufficiently large degree of the hplf candidate numerical examples also show that the proposed lmi condition can outperform alternative ones in terms of conservatism and computational burden

sufficient and necessary lmi conditions for robust stability of rationally time varying uncertain systems this technical note addresses robust stability of uncertain systems with rational dependence on unknown time varying parameters constrained in a polytope first the technical note proves that a sufficient linear matrix inequality lmi condition that we previously proposed based on homogeneous polynomial lyapunov functions hplfs and on the introduction of an extended version of polya s theorem is also necessary second the technical note proposes a new sufficient and necessary lmi condition by exploiting properties of the simplex and sum of squares sos parameter dependent polynomials lastly the technical note investigates relationships among these conditions and conditions based on the linear fractional representation lfr it is worth remarking that sufficient and necessary lmi conditions for this problem have not been proposed yet in the literature

affine parameter dependent lyapunov functions for lpv systems with affine dependence this paper deals with the certification problem for robust quadratic stability robust state convergence and robust quadratic performance of linear systems that exhibit bounded rates of variation in their parameters we consider both continuous time ct and discrete time dt parameter varying systems in this paper we provide a uniform method for this certification problem in both cases and we show that contrary to what was claimed previously the dt case requires a significantly different treatment compared to the existing ct results in the established uniform approach quadratic lyapunov functions which are affine in the parameter are used to certify robust stability robust convergence rates and robust performance in terms of linear matrix inequality feasibility tests to exemplify the procedure we solve the certification problem for mathscr l 2 gain performance both in the ct and the dt cases a numerical example is given to show that the proposed approach is less conservative than a method with slack variables

comprehensive lyapunov functions for linear switching systems abstract for a linear dynamical switching system we analyse maximal asymptotic growth of trajectories depending on the initial point both discrete and continuous time systems in r d are considered we prove the existence of a lyapunov norm in r d with the following property for every invariant linear subspace l r d of the system the restriction of the norm on l provides a tight upper bound for the growth of trajectories on l for this we introduce the concept of the spectral normal form of a family of matrices properties of the comprehensive lyapunov norms are analysed and methods of their construction are discussed

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper we consider discrete time systems with polytopic time varying uncertainty we look for a class of parameter dependent lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter we show that extending the new discrete time stability condition proposed by de oliveira et al systems control lett 36 1999 135 to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a lyapunov function this allows to check asymptotic stability of the system under study the obtained linear matrix inequalities condition can also be used to cope with the control synthesis problem