degenerate hopf bifurcations in discontinuous planar systems abstract we study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities this is done for the most generic cases by computing some kind of lyapunov constants our computations are based on the so called r p q generalized polar coordinates introduced by lyapunov and they are essentially different from the ones used in the smooth case these lyapunov constants are also used to generate limit cycles for some concrete examples

necessary and sufficient condition for stabilizability of discrete time linear switched systems in this paper the stabilizability of discrete time linear switched systems is considered several sufficient conditions for stabilizability are proposed in the literature but no necessary and sufficient the main contributions are the necessary and sufficient conditions for stabilizability based on the set theory and the characterization of a universal class of lyapunov functions an algorithm for computing the lyapunov functions and a procedure to design the stabilizing switching control law are provided based on such conditions moreover a sufficient condition for non stabilizability for switched system is presented several academic examples are given to illustrate the efficiency of the proposed results in particular a lyapunov function is obtained for a system for which the lyapunov metzler condition for stabilizability does not hold

quadratic stabilizability of switched linear systems with polytopic uncertainties in this paper we consider quadratic stabilizability via state feedback for both continuous time and discrete time switched linear systems that are composed of polytopic uncertain subsystems by state feedback we mean that the switchings among subsystems are dependent on system states for continuous time switched linear systems we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices then the switched system is quadratically stabilizable via state feedback for discrete time switched linear systems we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non negative scalars and a common positive definite matrix for both continuous time and discrete time switched systems we propose the switching rules by using the obtained common positive definite matrix

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

the inverse integrating factor and the poincare map this work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit we show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit when the regular orbit is a limit cycle we can determine its associated poincare return map in terms of the inverse integrating factor in particular we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it we also apply this result to study the homoclinic loop bifurcation we only consider homoclinic loops whose critical point is a hyperbolic saddle and whose poincare return map is not the identity a local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable that is the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method

on the stability of limit cycles for planar differential systems abstract we consider a planar differential system x p x y y q x y where p and q are c 1 functions in some open set u r 2 and d dt let be a periodic orbit of the system in u let f x y u r 2 r be a c 1 function such that p x y f x x y q x y f y x y k x y f x y where k x y is a c 1 function in u and x y f x y 0 we assume that if p u is such that f p 0 and f p 0 then p is a singular point we prove that 0 t p x q y t dt 0 t k t dt where t 0 is the period of as an application we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

suboptimal switching control consistency analysis for switched linear systems this paper introduces the concept of consistency for continuous time switched linear systems having the switching function as a primary control signal to be designed a switching control strategy is strictly consistent whenever it improves performance compared to the ones of all isolated subsystems conditions under which a min type switching strategy is strictly consistent for the classes of h2 and h performance indexes are determined this property makes clear the importance of switching systems control design in both theoretical and practical application frameworks moreover with this property it is not necessary to assume that all the subsystems are not stable in order to make a switching strategy design problem well posed the theory is illustrated by means of several academic examples

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

on stability l2 gain and h control for switched systems this paper addresses the issues of stability l 2 gain analysis and h control for switched systems via multiple lyapunov function methods a concept of general lyapunov like functions is presented a necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized lyapunov like functions which enables derivation of improved stability tests an l 2 gain characterization and a design method for stabilizing switching laws a solution to the h control problem for switched systems is also provided

stability analysis of lur e type switched systems this technical note aims to introduce stability analysis of lur e type switched systems in frequency domain a new state input dependent switching function is proposed and it is the key issue to obtain a stability condition that generalizes the celebrated popov criterion to deal with this class of switched nonlinear systems likewise the case of time invariant systems we propose a frequency domain stability test that is expressed through a convex combination of the subsystems state space matrices this task is not trivial due to the time varying nature of the nonlinear systems under consideration the theory is illustrated by a simple example

stability and stabilization of switched linear systems with mode dependent average dwell time in this paper the stability and stabilization problems for a class of switched linear systems with mode dependent average dwell time mdadt are investigated in both continuous time and discrete time contexts the proposed switching law is more applicable in practice than the average dwell time adt switching in which each mode in the underlying system has its own adt the stability criteria for switched systems with mdadt in nonlinear setting are firstly derived by which the conditions for stability and stabilization for linear systems are also presented a numerical example is given to show the validity and potential of the developed techniques

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

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

on input to state stability of discrete time switched nonlinear time varying systems in this paper input to state stability iss for discrete time switched nonlinear time varying sntv systems is investigated starting with discrete time nonlinear time varying ntv systems some improved sufficient conditions are proposed to verify the iss of systems by using the weak implication form iss wi iss lyapunov function weak dissipative form iss wd iss lyapunov function and interval descent technique then the results obtained are extended to study the iss of discrete time sntv systems several relaxed conditions are given by using piecewise wi iss and wd iss lyapunov functions minimum dwell time and infinite switching methods respectively comparing with the existing results the obtained conditions release the requirement on negative definiteness of the differences of piecewise lyapunov functions moreover all subsystems are allowed to be unstable in the case of infinite switching finally a numerical example is given to illustrate the theoretical results

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

uniformly stabilizing sets of switching sequences for switched linear systems a set of switching sequences of a discrete time switched linear system is said to be uniformly stabilizing if the system is uniformly exponentially stable over all switching sequences that belong to the set based on the recent result that a switched linear system is uniformly stable if and only if an increasing countable union of linear matrix inequality conditions is satisfied we characterize all uniformly stabilizing sets of switching sequences a discrete time hybrid automaton of the automobile engine is considered to illustrate the result

brief paper stability of switched positive linear systems with average dwell time switching in this paper the stability analysis problem for a class of switched positive linear systems splss with average dwell time switching is investigated a multiple linear copositive lyapunov function mlclf is first introduced by which the sufficient stability criteria in terms of a set of linear matrix inequalities are given for the underlying systems in both continuous time and discrete time contexts the stability results for the splss under arbitrary switching which have been previously studied in the literature can be easily obtained by reducing mlclf to the common linear copositive lyapunov function used for the system under arbitrary switching those systems finally a numerical example is given to show the effectiveness and advantages of the proposed techniques

stability and stabilization of discrete time switched systems this paper addresses two strategies for stabilization of discrete time linear switched systems the first one is of open loop nature trajectory independent and is based on the determination of an upper bound of the 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 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 is expressed being non convex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

technical communique switched state feedback control for continuous time uncertain systems this paper is concerned to design a switched state feedback robust control for continuous time systems subject to norm bounded uncertainty as important features of the proposed design method we mention that it can handle a general lft structure for the uncertainty and it is based on stability conditions that can be numerically solved by means of lmis and a line search moreover the switching rule as well as the state feedback gains are determined from the minimization of a guaranteed cost function derived from a multi objective criterion the theoretical results are illustrated with an academic example

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

modal occupation measures and lmi relaxations for nonlinear switched systems control this paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence this is done by introducing modal occupation measures which allow to relax the problem as a primal linear programming lp problem its dual linear program of hamilton jacobi bellman inequalities is also characterized the lps are then solved numerically with a converging hierarchy of primal dual moment sum of squares sos linear matrix inequalities lmi because of the special structure of switched systems we obtain a much more efficient method than could be achieved by applying standard moment sos lmi hierarchies for general optimal control problems

stabilization of switched systems via composite quadratic functions this paper investigates stabilization of switched systems by using a lyapunov function composed of a family of continuously differentiable functions switching laws are constructed by using the directional derivatives along the trajectories of the subsystems conditions for stabilization are established with careful consideration of sliding modes and directional derivatives along sliding modes three types of composite quadratic lyapunov functions the max of quadratics the min of quadratics and the convex hull of quadratics are used for deriving matrix conditions of stabilization and for constructing switching laws dual stabilization results are established with respect to the pair of conjugate lyapunov functions the max of quadratics and the convex hull of quadratics conditions of stabilization are derived as bilinear matrix inequalities and solved with lmi based numerical tools relationship between the newly derived conditions and some existing conditions are investigated it is observed that the min of quadratics which is nondifferentiable and nonconvex may be a more convenient tool than the other two types of functions which are convex and or differentiable numerical examples are used to demonstrate the synthesis results and the advantage of the composite quadratic functions over existing multiple lyapunov functions in particular better results have been obtained when the number of quadratic functions is greater than the number of subsystems

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

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

switching rule design for affine switched systems with guaranteed cost and uncertain equilibrium condition this paper addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized the switching rule is determined by solving an lmi problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system the potential of the results is illustrated on a true refrigeration system namely a domestic refrigerator where the purpose is to control the temperature in the fresh food and the freezer compartments

multiplicity of limit cycles and analytic m solutions for planar differential systems abstract this work deals with limit cycles of real planar analytic vector fields it is well known that given any limit cycle of an analytic vector field it always exists a real analytic function f 0 x y defined in a neighborhood of and such that is contained in its zero level set in this work we introduce the notion of f 0 x y being an m solution which is a merely analytic concept our main result is that a limit cycle is of multiplicity m if and only if f 0 x y is an m solution of the vector field we apply it to study in some examples the stability and the bifurcation of periodic orbits from some non hyperbolic limit cycles

stability of planar switched systems the linear single input case we study the stability of the origin for the dynamical system dot x t u t ax t 1 u t bx t where a and b are two 2 2 real matrices with eigenvalues having strictly negative real part x in mbox bf r 2 and u 0 infty to 0 1 is a completely random measurable function more precisely we find a coordinates invariant necessary and sufficient condition on a and b for the origin to be asymptotically stable for each function u the result is obtained without looking for a common lyapunov function but studying the locus in which the two vector fields ax and bx are collinear there are only three relevant parameters the first depends only on the eigenvalues of a the second depends only on the eigenvalues of b and the third contains the interrelation among the two systems and it is the cross ratio of the four eigenvectors of a and b in the projective line cp1 in the space of these parameters the shape and the convexity of the region in which there is stability are studied this bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation

linear quadratic regulation of switched systems using informed policies the problem of designing a switching and control policy for regulating the state of a switched linear system to zero while minimizing a quadratic cost appears in numerous applications however obtaining the optimal policy is in general computationally intractable here we propose a class of suboptimal policies that exploit information in terms of upper or lower bounds on the optimal cost we analyze the performance of these novel policies obtaining new bounds on the optimal cost which are tighter than the initial ones the usefulness of these policies and performance bounds is illustrated in the context of resource aware 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

switching rule design for affine switched systems using a max type composition rule abstract this paper presents conditions for designing a switching rule that drives the state of the switched dynamic system to a desired equilibrium point the proposed method deals with the class of switched systems where each subsystem has an affine vector field and considers a switching rule using max composition the results guarantee global asymptotic stability of the tracking error dynamics even if sliding mode occurs at any switching surface of the system in addition the method does not require a hurwitz convex combination of the dynamic matrices of the subsystems two numerical examples are used to illustrate the results

local smoothness maps a new method for solving inverse problems with the accurate recovery of sharp gradients we describe a novel bayesian approach to solving inverse problems by simultaneously estimating the reconstructed signal and the local smoothness map lsm which is a generalization of the global smoothness parameter that is often used to stabilize inverse problems the greater flexibility afforded by the introduction of the local smoothness map makes the new method very effective on inverse problems that involve discontinuities or other regions with sharp gradients we demonstrate the lsm method on the problem of reducing noise in one dimensional 1 d signals

dynamic output feedback control of switched linear systems this paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete time domains a particular class of matrix inequalities the so called lyapunov metzler inequalities provides conditions for open loop stability analysis and closed loop switching control using state and output feedback switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the hamilton jacobi bellman inequality an upper bound on the optimal cost associated with a closed loop stabilizing switching strategy is provided as well the solution of the output feedback problem is based on the construction of a full order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule throughout the theoretical results are illustrated by means of academic examples a realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported

multiple lyapunov functions based small gain theorems for switched interconnected nonlinear systems multiple lyapunov functions mlfs based small gain theorems are presented for switched interconnected nonlinear systems with unstable subsystems which extend the small gain technique from its original non switched nonlinear version to a switched nonlinear version each low dimensional subsystem does not necessarily have the input to state stability iss property in the whole state space and it only has individual iss property in some subregions of the state space the novelty of this paper is that integral type mlfs and small gain techniques are utilized to establish some mlfs based small gain theorems for switched interconnected nonlinear systems which derive various stability results under some novel switching laws designed and construct integral type mlfs the small gain theorems proposed cover several recent results as special cases which also permit removal of a common restriction in which all low dimensional subsystems in switched interconnected systems are iss or only some are iss and others are not finally two illustrative examples are presented to demonstrate the effectiveness of the results provided

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

a necessary and sufficient condition for robust asymptotic stability of time variant discrete systems a necessary and sufficient condition for the stability of time variant interval matrices is presented this condition allows stability to be tested by checking only products of vertex extreme matrices the implementation of the test in the form of an algorithm and two illustrative examples are provided

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

stabilization and robust h control for sector bounded switched nonlinear systems this paper presents stability analysis and robust h control for a particular class of switched systems characterized by nonlinear functions that belong to sector sets with arbitrary boundaries the sector boundaries can have positive and or negative slopes and therefore we cover the most general case in our approach using the special structure of the system but without making additional assumptions e g on the derivative of the nonlinear functions and by proposing new multiple lyapunov function candidates we formulate stability conditions and a control design procedure in the form of matrix inequalities the proposed lyapunov functions are more general than the quadratic functions previously proposed in the literature as they incorporate the nonlinearities of the system and hence lead to less conservative stability conditions the stabilizing switching controllers are designed through a bi level optimization problem that can be efficiently solved using a combination of a convex optimization algorithm and a line search method the proposed optimization problem is achieved using a special loop transformation to normalize the arbitrary sector bounds and by other linear matrix inequalities lmi techniques

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

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

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

hybrid dynamical systems robust control synthesis problems the paper presents a new approach to robust control synthesis problems for hybrid dynamical systems the hybrid system under consideration is a composite of a continuous plant and a discrete event controller state and output feedback problems are considered the main results are given in terms of the existence of suitable solutions to a dynamic programming equation and a riccati differential equation of the h filtering type these results show a connection between the theories of hybrid dynamical systems and robust and nonlinear control

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

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

stabilizability of switched linear systems does not imply the existence of convex lyapunov functions counterexamples are given which show that a linear switched system with controlled switching that can be stabilized by means of a suitable switching law does not necessarily admit a convex lyapunov function both continuous and discrete time cases are considered this fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched system in particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a lyapunov function

design of stabilizing switching laws for mixed switched affine systems this technical note presents stability analysis and stabilization for a general class of switched systems characterized by nonlinear functions the proposed approach is composed of approximating the switched nonlinear system with a switched affine system that has a mixture of controlled and autonomous switching behavior utilizing a joint polyhedral partitioning approach a stabilizing switching law based on quadratic lyapunov functions and with considering the autonomous switching between polyhedral regions is proposed to ensure the decrease of the overall lyapunov function two approaches are proposed 1 guarantee continuity of the lyapunov function over boundaries of polyhedral regions and 2 relax the continuity requirement by using additional matrix inequalities the second approach is less conservative but with more variables and matrix inequalities than in the first method with fixing one scalar variable the stabilization conditions will have the form of linear matrix inequalities lmis further the sufficient conditions for stabilizing the original switched nonlinear system using the proposed switching schemes are presented finally through two examples the performance of the proposed stabilization methods is demonstrated

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note