l_ infty and l_ 2 low gain feedback their properties characterizations and applications in constrained control low gain feedback has found several applications in constrained control robust control and nonlinear control in this paper we first generalize the existing low gain design methods by introducing the notion of l vanishment and by providing a full characterization of feedback gains that achieve such a property we observe that l low gain feedback can lead to energy peaking namely the control energy required by l low gain feedback increases toward infinity as the low gain parameter decreases to zero motivated by this observation we consider the notion of l2 vanishment and establish several of its characterizations based on which a new design approach referred to as the l2 low gain feedback approach for linear systems is developed different from the l low gain feedback the l2 low gain feedback is instrumental in the control of systems with control energy constraints as an application of l2 low gain feedback the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper the notions of l and l2 vanishment also allow us to establish a systematic approach to the design of l and l2 low gain feedback the advantage of this new design approach is that it results in a family of control laws including those resulting from the existing design methods

interaction of a solitary wave with an external force abstract using the forced korteweg de vries equation as a model we investigate the interaction of a solitary wave with an isolated moving external force of small amplitude we use both analysis and numerical calculations our analysis is based on a multi scale perturbation asymptotic analysis and is carried through to either the first or second order depending on whether the first or more accurate second order expression for the solitary wave speed is used the perturbation theory leads to a pair of autonomous equations for the solitary wave amplitude and position and these are analyzed in detail for the two extreme cases when the lengthscale of the external force is either much greater or much less than that of the solitary wave the theory predicts that the main regimes are passage when the solitary wave passes through the external force with some amplitude modulation repulsion when the solitary wave is reflected by the external force with a significant amplitude change and trapping when the solitary wave remains either partially or totally trapped at the location of the external force these theoretical predictions are confirmed in our numerical simulations of the forced korteweg de vries equation

brief paper a new concept of invariance for saturated systems in this paper the notions of sns invariance and sns domain of attraction are introduced the sns domain of attraction serves as an estimation of the domain of attraction of a saturated system it is shown that in the case of single input saturated systems any contractive set is contained in the sns domain of attraction another important characteristic of the sns domain of attraction is that it contains any estimation obtained by means of a linear difference inclusion of the saturated system a simple algorithm that converges to the sns domain of attraction is presented an illustrative example is given

designing stabilizing controllers for uncertain systems using the riccati equation approach a useful technique for determining a linear feedback control law stabilizes an uncertain system is the riccati equation approach of i r petersen and c v hollot 1986 they consider systems with time varying uncertainty in the system matrix and obtain the constant feedback gains for the linear stabilizing controller in terms of the solutions of a riccati equation the technique is extended to include problems with time varying uncertainty in the input connection matrix several examples are included to demonstrate the efficacy of this result

anti disturbance control for nonlinear systems subject to input saturation via disturbance observer this paper is concerned with the problem of the disturbance observer based control for a class of continuous time uncertain systems subject to input saturation and nonlinearity the input of the system includes two parts the control input and the disturbance input the nonlinearity of the system which satisfies a global lipschitz condition is considered as two cases of known nonlinearity and unknown nonlinearity by virtue of the technique of the disturbance observer based controller the anti disturbance controllers are designed respectively with both the polytopic and dead zone representations of the saturation function which ensure that the resulting closed loop systems are asymptotically stable with an estimation of the domain of attraction described by the level set of the lyapunov function further an iterative optimization method is used to obtain the maximum estimation of the domain of the set of initial states an example of application design for a flight control system illustrates the effectiveness of the proposed results

an nth step set invariance approach to the analysis and design of discrete time linear systems subject to actuator saturation abstract for a discrete time linear system subject to actuator saturation an analysis and design approach based on n th step set invariance is proposed an n th step invariant set is a set from which all trajectories will enter a contractively invariant set in n steps and thus can be used as an estimate of the domain of attraction such an estimate is less conservative than an estimate obtained on the basis of contractive invariance of ellipsoids which excludes initial conditions from which the trajectories overshoot outside the ellipsoid before converging to the origin in comparison with the existing approach based on conventional set invariance the proposed approach results in a larger domain of attraction and better performance both in the analysis of the closed loop system under a given feedback gain and in the design of feedback gain in comparison with the saturated and nonsaturated sns invariant set approach which is an analysis approach the proposed approach can be used for control design even in the presence of plant uncertainties

analysis and design of discrete time linear systems with nested actuator saturations abstract this paper is concerned with the analysis and design of discrete time linear systems subject to nested saturation functions by utilizing a new compact convex hull representation of the saturation nonlinearity a linear matrix inequalities lmis based condition is obtained for testing the local and global stability of the considered nonlinear system the estimation of the domain of attraction and the design of feedback gains such that the estimation of the domain of attraction for the resulting closed loop system is maximized are then converted into some lmis based optimization problems compared with the existing results on the same problems the proposed solutions are less conservative as more slack variables are introduced into the conditions a couple of numerical examples are worked out to validate the effectiveness of the proposed approach

on the prediction of the number of solitons excited by an arbitrary potential an observation from inverse scattering abstract a heuristic estimate for the soliton production rate by a pulse is verified for the korteweg de vries equation using inverse scattering an observation from this result which is shown to hold for some other nonlinear equations and for the case of the forced nonlinear schrolinger equation is that production is determined by quantities that are invariant under rescaling of the original nonlinear equations we speculate that this result may be useful to the development of an inverse scattering theory for forced nonlinear systems

local stabilization of linear systems under amplitude and rate saturating actuators this note addresses the problem of local stabilization of linear systems subject to control amplitude and rate saturation considering the actuator represented by a first order system subject to input and state saturation a condition for the stabilization of an a priori given set of admissible initial states is formulated from certain saturation nonlinearities representation and quadratic stability results from this condition an algorithm based on the iterative solution of linear matrix inequalities based problems is proposed in order to compute the control law

semiglobal stabilization of linear discrete time systems subject to input saturation via linear feedback an are based approach we revisit the problem of semiglobal stabilization of linear discrete time systems subject to input saturation and give an algebraic riccati equation are based approach to the proof of a fact established earlier lin and saberi 1995 i e a linear discrete time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open loop poles are located inside or on the unit circle moreover we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work

quadratic stability for hybrid systems with nested saturations the problems of characterizing quadratic stability and computing an estimation of the domain of attraction for saturated hybrid systems are addressed hybrid systems presenting saturations and nested saturations on signals involved in both the continuous time and the discrete time dynamics are considered geometrical characterizations of local and global quadratic stability are provided computation oriented conditions for quadratic stability are given in form of convex constraints

on maximizing the convergence rate for linear systems with input saturation in this note we consider a few important issues related to the maximization of the convergence rate inside a given ellipsoid for linear systems with input saturation for continuous time systems the control that maximizes the convergence rate is simply a bang bang control through studying the system under the maximal convergence control we reveal several fundamental results on set invariance an important consequence of maximizing the convergence rate is that the maximal invariant ellipsoid is produced we provide a simple method for finding the maximal invariant ellipsoid and we also study the dependence of the maximal convergence rate on the lyapunov function

anti windup design with guaranteed regions of stability for discrete time linear systems abstract the purpose of this paper is to study the determination of stability regions for discrete time linear systems with saturating controls through anti windup schemes considering that a linear dynamic output feedback has been designed to stabilize the linear discrete time system without saturation a method is proposed for designing an anti windup gain that maximizes an estimate of the basin of attraction of the closed loop system in the presence of saturation it is shown that the closed loop system obtained from the controller plus the anti windup gain can be locally modeled by a linear system with a deadzone nonlinearity then based on the use of a new sector condition and quadratic lyapunov functions stability conditions in an lmi form are stated these conditions are then considered in a convex optimization problem in order to compute an anti windup gain that maximizes an estimate of the basin of attraction of the closed loop system moreover considering asymptotically stable open loop systems it is shown that the conditions can be slightly modified in order to determine an anti windup gain that ensures global stability an extension of the proposed results to the case of dynamic anti windup synthesis is also presented in the paper

on slater s criterion for the breakup of invariant curves abstract we numerically explore slater s theorem in the context of dynamical systems to study the breakup of invariant curves slater s theorem states that an irrational translation over a circle returns to an arbitrary interval in at most three different recurrence times expressible by the continued fraction expansion of the related irrational number the hypothesis considered in this paper is that slater s theorem can be also verified in the dynamics of invariant curves hence we use slater s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area preserving maps

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

adaptation coordination and local interactions via distributed approachability abstract this paper investigates the relation between cooperation competition and local interactions in large distributed multi agent systems the main contribution is the game theoretic problem formulation and solution approach based on the new framework of distributed approachability and the study of the convergence properties of the resulting game model approachability theory is the theory of two player repeated games with vector payoffs and distributed approachability is here presented for the first time as an extension to the case where we have a team of agents cooperating against a team of adversaries under local information and interaction structure the game model turns into a nonlinear differential inclusion which after a proper design of the control and disturbance policies presents a consensus term and an exogenous adversarial input local interactions enter into the model through a graph topology and the corresponding graph laplacian matrix given the above model we turn the original questions on cooperation competition and local interactions into convergence properties of the differential inclusion in particular we prove convergence and exponential convergence conditions around zero under general markovian strategies we illustrate our results in the case of decentralized organizations with multiple decision makers

nonlinear regulation in constrained input discrete time linear systems the use of composite linear and non linear feedback laws for the control of constrained input discrete time linear systems is re examined by making use of the delta operator formulation of a discrete time system an apparent restriction on the magnitude of the non linear control law is removed and the similarities between the continuous and discrete time solutions to the problem are elucidated in order to develop the results unconstrained systems are treated initially but it is shown that locally the sufficient conditions for the stabilization of such systems are actually equivalent to those for the stabilization of the corresponding constrained systems

interaction of a moving dipole with a soliton in the kdv equation the korteweg de vries equation with the perturbing term e x vt a point like dipole which models disturbances produced by a small body moving in a liquid layer is considered in the case v 0 a non emitting dipole a soliton may be pinned by the moving dipole in the adiabatic approximation the pinned state is stable provided e 0 in this case a pair of solitons may also be pinned by the dipole but that pinned state is unstable other types of solitary pinned profiles and their stability are discussed oscillations of a soliton near the adiabatically stable pinned state are accompanied by emission of quasi linear waves the emission intensity is calculated in a general form and it is demonstrated that the oscillation are subject to radiative instability due to the fact that the energy of the system is not positive definite the same model is considered with the burgers dissipative term the dissipation may compensate the radiative instability and render the pinned state of a soliton completely stable besides it is demonstrated that the burgers term gives rise to multisoliton pinned profiles a maximum possible number of solitons in the profile is found

brief paper stability and stabilization of discrete time periodic linear systems with actuator saturation this paper is concerned with the problems of stability and stabilization for discrete time periodic linear systems subject to input saturation both local results and global results are obtained for local stability and stabilization the so called periodic invariant set is used to estimate the domain of attraction the conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities lmis which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers the periodic enhancement technique is introduced to reduce the conservatism in the methods as a by product less conservative results for controller analysis and design for discrete time time invariant systems with input saturation are obtained for global stability by utilizing the special properties of the saturation function a saturation dependent periodic lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system the corresponding conditions are expressed in the form of lmis and can be efficiently solved several numerical and practical examples are given to illustrate the theoretical results proposed in the paper

robust control of multi input periodic discrete time systems with saturating actuators this paper proposes the use of a time varying sliding surface for robust transient shaping of periodic linear discrete time plants subject to saturating actuators in the presence of bounded matched uncertainties a constructive procedure is presented and ultimate boundedness of state trajectories is proved

barriers to transport and mixing in volume preserving maps with nonzero flux abstract we identify some geometric structures secondary tori that restrict transport and prevent mixing in perturbations of integrable volume preserving systems with nonzero net flux unlike the customary kam tori secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction we also note that secondary tori persist under the addition of a net flux which destroys all customary kam tori we introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori we then implement the algorithm and after running it present assertions regarding the distribution of the escape times of the unbounded orbits the abundance of secondary tori the size of the resonant regions and the robustness of the tori under the addition of a mean flux

brief stability analysis of discrete time systems with actuator saturation by a saturation dependent lyapunov function in this paper stability of discrete time linear systems subject to actuator saturation is analyzed using a saturation dependent lyapunov function this saturation dependent lyapunov function captures the real time information on the severity of actuator saturation and leads to less conservative estimate of the domain of attraction which is based on the solution of an lmi optimization problem numerical examples are presented to show the effectiveness of the proposed method

a complete characterization of the maximal contractively invariant ellipsoids of linear systems under saturated linear feedback as level sets of quadratic lyapunov functions ellipsoids have been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback for a linear system with a single input subject to actuator saturation based on a convex hull representation of saturation functions a necessary and sufficient condition for an ellipsoid to be contractively invariant was previously established which through the solution of an lmi problem leads to the maximal ellipsoidal invariant set in this technical note for a linear system with multiple inputs subject to actuator saturation we develop a complete characterization of the maximal ellipsoidal invariant set by algebraic computation simulation results demonstrate the effectiveness of our results

multiple supercritical solitary wave solutions of the stationary forced korteweg de vries equation and their stability the first order approximation of long nonlinear surface waves in a channel flow of an inviscid incompressible fluid over a bump results in a forced korteweg de vries equation fkdv eta t lambda eta x 2 alpha eta eta x beta eta xxx f x x quad infty 0 the forcing represented by the function f x in the fkdv equation is due to the bump on the bottom of the channel in this paper the solitary wave solutions of the stationary fkdv equation sfkdv are studied the supercritical solitary wave solutions of the sfkdv equation exist only when the upstream flow velocity c is greater than a crucial value u c sqrt gh or equivalently lambda lambda c 0 the existence of supercritical positive solitary wave solutions spsws of the sfkdv equation is proved some ordered properties and extreme properties of spsws are discussed there may exist more than two spsws for a nonlocal forcing an analytic expression of the spsws is found w

the perturbed cubic schrodinger equation selection mechanism resonant limits and spatial chaos separable solutions of the perturbed cubic schrodinger equation that are sinusoidal in time with frequency e are considered two processes associated with the integrable limit e 0 are demonstrated the first will be called a selection mechanism and is demonstrated for ginzburg landau type perturbations the second is a resonant limit process and will be demonstrated for perturbations containing spatial driving terms of wave number q spatially chaotic behavior is studied in the limits e 0 q 0 and e q such that q e const

siso linear system control via saturating actuator l2 tracking performance in the presence of arbitrary shaped inputs most studies on linear system control in the presence of actuator saturation have focused on closed loop asymptotic stabilisation available tracking results are generally limited to constraint compatible inputs references disturbances with known exogenous input models the problem of output reference tracking in the presence of arbitrary shaped not necessarily constraint compatible inputs has yet to be investigated in this study we show that an tracking performance is achievable for siso linear systems using a given class of saturated controllers accordingly the tracking error energy resp power is bounded by the energy resp power of a well defined mismatch input depending on the disturbance and reference compatibility in the disturbance free case this entails perfect asymptotic tracking of arbitrary shaped constraint compatible reference signals otherwise the average tracking quality is proportional to the mismatch input power in the case of periodic inputs constraint compatible or not

exact characterization of invariant ellipsoids for single input linear systems subject to actuator saturation we present a necessary and sufficient condition for an ellipsoid to be an invariant set of a linear system under a saturated linear feedback the condition is given in terms of linear matrix inequalities lmis and can be easily used for optimization based analysis and design

greene s residue criterion for the breakup of invariant tori of volume preserving maps abstract invariant tori play a fundamental role in the dynamics of symplectic and volume preserving maps codimension one tori are particularly important as they form barriers to transport such tori foliate the phase space of integrable volume preserving maps with one action and d angles for the area preserving case greene s residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits even though kam theory applies to the three dimensional case the robustness of tori in such systems is still poorly understood we study a three dimensional reversible volume preserving analogue of chirikov s standard map with one action and two angles we investigate the preservation and destruction of tori under perturbation by computing the residue of nearby periodic orbits we find tori with diophantine rotation vectors in the spiral mean cubic algebraic field the residue is used to generate the critical function of the map and find a candidate for the most robust torus

comments on the inverse scattering transform for the forced toda lattice abstract it is demonstrated that the standard procedure for treating initial value problems with the ist almost works also for boundary value problems the only remaining difficulty preventing one from solving integrable systems with boundary values such as the forced toda lattice is the determination of one set of scattering data from the boundary value once this would be known then the forced toda lattice would become completely integrable

stability of some stationary solutions for the forced kdv equation abstract the forced kdv fkdv equation has been established by recent studies as a simple mathematical model capable of describing the physics of a shallow layer of fluid in response to external forcing for a particular one parameter family of forcings characterized by a wave amplitude parameter for a certain supercritical forcing distribution the exact stationary solutions are known we study the stability of these solutions as the parameter varies the linear stability analysis is first carried out by exploring the structure of the spectrum of the associated eigenvalue problem using a perturbation technique applied to a neighbourhood of isolated parameter values where eigenfunctions corresponding to eigenvalue zero can be expressed in closed form the results identify a set of intervals in the parameter space corresponding to different types of manifestation of instability in the region of the parameter space where the linear stability analysis fails to provide an answer we discuss a nonlinear analysis that provides a sufficient condition for stability

taking into account period variations and actuators saturation in sampled data systems this paper deals with the problem of stability and stabilization of sampled data systems under asynchronous samplings and actuators saturation the method is based on the first hand on the use of a novel class of lyapunov functionals whose derivative is negative along the trajectories of the continuous time model of the sampled data system it is shown that this fact guarantees that a quadratic lyapunov function is strictly decreasing for the discrete time asynchronous system on the other side the control saturation is taken into account from the use of a modified sector condition these ingredients lead to the formulation of improved lmi conditions that can be cast in optimization problems aiming at enlarging estimates of the region of attraction of the closed loop system or maximizing the bounds on the sampling period jitter for which stability and stabilization are ensured

linear discrete time global and regional anti windup an lmi approach in this article we address linear anti windup design for linear discrete time control systems guaranteeing regional and global stability and performance the techniques that we develop are the discrete time counterpart of existing techniques for anti windup augmentation which lead to convex constructions by way of linear matrix inequalities lmis when adopting static and plant order anti windup augmentation interesting system theoretic interpretations of the performance bounds for the non linear closed loop can also be given we show here that parallel results apply to the discrete time case we derive the corresponding conditions and prove their effectiveness by adapting the continuous time approaches to the discrete time case

a quantum mechanics analogy for the nonlinear schrodinger equation in the finite line abstract a quantum mechanics analogy is used to determine the forces acting on and the energies of solitons governed by the nonlinear schrodinger equation in finite intervals with periodic and with homogeneous dirichlet neumann and robin boundary conditions it is shown that the energy densities remain nearly constant for periodic while they undergo large variations for homogeneous boundary conditions the largest variations in the force and energy densities occur for the neumann boundary conditions but for all the boundary conditions considered in this paper the magnitudes of these forces and energies recover their values prior to the interaction of the soliton with the boundary after the soliton rebound process is completed it is also shown that the quantum momentum changes sign but recovers its original value after the collision of the soliton with the boundaries the asymmetry of the robin boundary conditions shows different dynamic behaviour at the left and right boundaries of the finite interval

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

h sub spl infin design with pole placement constraints an lmi approach this paper addresses the design of state or output feedback h sub spl infin controllers that satisfy additional constraints on the closed loop pole location sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane these conditions are expressed in terms of linear matrix inequalities lmis and the authors formulation is therefore numerically tractable via lmi optimization in the state feedback case mixed h sub 2 h sub spl infin synthesis with regional pole placement is also discussed finally the validity and applicability of this approach are illustrated by a benchmark example

breakup of invariant tori for the four dimensional semi standard map abstract we compute the domain of existence of two dimensional invariant tori with fixed frequency vectors for a four dimensional complex symplectic map the map is a generalization of the semi standard map studied by greene and percival it has three parameters a 1 and a 2 representing the strength of the kicks in each degree of freedom and the coupling the domain of existence of a torus in a 1 a 2 is shown to be complete and log convex for fixed k a 1 a 2 explicit bounds on the domain for fixed k are obtained numerical results show that quadratic irrationals can be more robust than the cubic irrational the spiral mean

brief paper an improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation this paper is concerned with the treatment of saturation nonlinearity existing commonly in practical control systems an improved treatment of such nonlinearity is proposed which includes the well known linear differential inclusion method for handling saturation nonlinearity as a special case to show the effectiveness of this new treatment the problem of analysis and design of linear systems subject to nested saturation function is revisited the proposed results not only complement the existing work on this problem but also are less conservative than the existing methods four numerical examples are worked out to illustrate the applicability of the established approaches

a locally conservative eulerian lagrangian finite difference method for the forced kdv equation abstract we propose a locally conservative eulerian lagrangian finite difference method fdlcelm for approximating the solution of the forced korteweg de vries equation the new numerical method employs an operator splitting scheme that solves the nonlinear transport equation and the linear dispersive equation sequentially in order to conserve mass in the transport fractional step we trace back each grid cell in time along the integral curve the dispersive fractional step will be solved using a cell centered finite difference method numerical examples are provided to confirm and illustrate the accuracy and mass conservation property of the new method we also study the numerical stability and time evolution of various stationary solitary wave solutions in the presence of one or two bumps

antiwindup for stable linear systems with input saturation an lmi based synthesis this paper considers closed loop quadratic stability and l sub 2 performance properties of linear control systems subject to input saturation more specifically these properties are examined within the context of the popular linear antiwindup augmentation paradigm linear antiwindup augmentation refers to designing a linear filter to augment a linear control system subject to a local specification called the unconstrained closed loop behavior building on known results on h sub spl infin and lpv synthesis the fixed order linear antiwindup synthesis feasibility problem is cast as a nonconvex matrix optimization problem which has an attractive system theoretic interpretation the lower bound on the achievable l sub 2 performance is the maximum of the open and unconstrained closed loop l sub 2 gains in the special cases of zero order static and plant order antiwindup compensation the feasibility conditions become convex linear matrix inequalities it is shown that if and only if the plant is asymptotically stable plant order linear antiwindup compensation is always feasible for large enough l sub 2 gain and that static antiwindup compensation is feasible provided a quasi common lyapunov function between the open loop and unconstrained closed loop exists using the solutions to the matrix feasibility problems the synthesis of the antiwindup augmentation achieving the desired level of l sub 2 performance is then accomplished by solving an additional lmi

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

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

robust control of uncertain multi inventory systems via linear matrix inequality we consider a continuous time linear multi inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes we address the problem of e stabilising the inventory since this implies some reduction of the inventory costs the main results are certain conditions under which e stabilisability is possible through a saturated linear state feedback control all the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations numerical simulations are provided

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

a magnitude and rate saturation model and its use in the solution of a static anti windup problem in this paper we address the anti windup design problem for linear control systems with strictly proper controllers in the presence of input magnitude and rate saturation using generalized sector condition we provide an lmi based procedure for the construction of a linear anti windup gain acting on the controller state equation such that regional closed loop stability is guaranteed and suitable performance measures are optimized the approach is successfully illustrated on a simulation example

the soliton birth rate in the forced toda lattice the soliton birth rate in the semi infinite toda lattice is studied the lattice is forced by driving the zeroth particle with a constant velocity into the remainder of the lattice an approximate solution for the soliton birth rate is derived and it is shown to compare quite favorably with the actual birth rate

antiwindup design with guaranteed regions of stability an lmi based approach this note addresses the design of antiwindup gains for obtaining larger regions of stability for linear systems with saturating inputs considering that a linear dynamic output feedback has been designed to stabilize the linear system without saturation a method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed loop system it is shown that the closed loop system obtained from the controller plus the antiwindup gain can be modeled by a linear system with a deadzone nonlinearity a modified sector condition is then used to obtain stability conditions based on quadratic lyapunov functions differently from previous works these conditions are directly in linear matrix inequality form some numerical examples illustrate the effectiveness of the proposed design technique when compared with the previous ones

birkhoff averages and rotational invariant circles for area preserving maps abstract rotational invariant circles of area preserving maps are an important and well studied example of kam tori john greene conjectured that the locally most robust rotational circles have rotation numbers that are noble i e have continued fractions with a tail of ones and that of these circles the most robust has golden mean rotation number the accurate numerical confirmation of these conjectures relies on the map having a time reversal symmetry and such high accuracy has not been obtained in more general maps in this paper we develop a method based on a weighted birkhoff average for identifying chaotic orbits island chains and rotational invariant circles that does not rely on these symmetries we use chirikov s standard map as our test case and also demonstrate that our methods apply to three other well studied cases

output feedback stabilization of linear systems with actuator saturation the note presents a method for designing an output feedback law that stabilizes a linear system subject to actuator saturation with a large domain of attraction this method applies to general linear systems including strictly unstable ones a nonlinear output feedback controller is first expressed in the form of a quasi lpv system conditions under which the closed loop system is locally asymptotically stable are then established in terms of the coefficient matrices of the controller the design of the controller coefficient matrices that maximizes an estimate of the domain of attraction is then formulated and solved as an optimization problem with lmi constraints

local stabilization of discrete time linear systems with saturating controls an lmi based approach deals with the problem of local stabilization of linear discrete time systems subject to control saturation a linear matrix inequalities based framework is proposed in order to compute a saturating state feedback that stabilizes the system with respect to a given set of admissible initial states and in addition guarantees some dynamical performances when the system operates in the zone of linear behavior i e when the controls are not saturated

improvements to the linear differential inclusion approach to stability analysis of linear systems with saturated linear feedback ellipsoids as level sets of quadratic lyapunov functions and the convex hull of ellipsoids as a level set of a certain composite quadratic lyapunov function have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback by expressing the saturated linear feedback law on the convex hull of a group of linear feedback laws which in turn expresses the linear system under this saturated linear feedback in a linear differential inclusion conditions have been established under which an ellipsoid or the convex hull of a group of ellipsoids are contractively invariant sets and are thus estimates of the domain of attraction these conditions are usually less conservative for single input systems than for multiple input systems in this paper we consider multiple input systems and establish conditions for contractive invariance of the convex hull of ellipsoids that are less conservative than the existing conditions

the maximal contractively invariant ellipsoids for discrete time linear systems under saturated linear feedback abstract in this paper we consider the problem of determining the maximal contractively invariant ellipsoids for discrete time linear systems with multiple inputs under saturated linear feedback we propose an algebraic computational approach to determining such maximal contractively invariant ellipsoids we divide the state space into several regions according to the saturation status of each input and compute the possible maximal contractively invariant ellipsoids on each region except the region where none of inputs saturate and on their intersections the minimal one among these possible maximal contractively invariant ellipsoids is the maximal contractively invariant ellipsoids of the system simulation results demonstrate the effectiveness of our methods

the forced toda lattice an example of an almost integrable system a method for solving forced integrable systems is presented the method requires the knowledge of at least one piece of information about the solution once this is known one may then construct the remainder of the solution in this sense these systems are almost integrable the forced semi infinite toda lattice is used as an example and to illustrate the method

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

dynamic output feedback for discrete time systems under amplitude and rate actuator constraints this work proposes a technique for the design of stabilizing dynamic output feedback controllers for discrete time linear systems with rate and amplitude saturating actuators the nonlinear effects introduced by the saturations in the closed loop system are taken into account by using a generalized sector condition which leads to theoretical conditions for solving the problem directly in the form of linear matrix inequalities

robust controller design of uncertain discrete time delay systems with input saturation and disturbances this technical note deals with the problem of robust controller design of uncertain discrete time delay systems subject to control input saturation and bounded external disturbances a sufficient condition is obtained which guarantees the existence of state feedback controllers and an admissible initial condition domain such that the resulting closed loop system is uniformly exponentially convergent to a ball with certain exponential decay rate for every initial condition emanating from the admissible domain an iterative linear matrix inequality relaxation scheme can be used to construct desired state feedback controllers a simulation example is provided to demonstrate the effectiveness of the proposed method

control of linear systems subject to time domain constraints with polynomial pole placement and lmis this note focuses on the control of continuous time linear systems subject to time domain constraints input amplitude limitation output overshoot on closed loop signals using recent results on positive polynomials it is shown that finding a youla kuc spl breve era polynomial parameter of fixed degree hence a controller of fixed order such that time domain constraints are satisfied amounts to solving a convex linear matrix inequality lmi optimization problem as soon as distinct strictly negative closed loop poles are assigned by pole placement proceeding this way time domain constraints are handled by an appropriate choice of the closed loop zeros

global stabilization with low computational cost of the discrete time chain of integrators by means of bounded controls this note proposes a bounded nonlinear control law composed of saturation functions for the discrete time chain of integrators a dynamical adaptation rule of the saturation levels involved in the control law is proposed to improve the closed loop performances the note unifies the original work of yang 1997 with static saturation level and convergence improvements that recently appeared in the continuous time case the possible ranges for the controller s parameters are extended with respect to existing results

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

discrete time l and l2 norm vanishment and low gain feedback with their applications in constrained control the existing low gain feedback which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero has been designed in very specific ways in this paper by recognizing the l and l 2 slow peaking phenomenon that exists in discrete time systems under low gain feedback more general notions of l and l 2 norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals low gain feedback that does not lead to l and l 2 slow peaking in the control input are respectively referred to as l and l 2 low gain feedback based on the notions of l and l 2 vanishment not only can the existing low gain feedback be recognized as an l low gain feedback but also a new design approach referred to as the l 2 low gain feedback approach is developed for discrete time linear systems parallel to the effectiveness of l low gain feedback in magnitude constrained control the l 2 low gain feedback is instrumental in the control of discrete time systems with control energy constraints the notions of l and l 2 vanishment also result in a systematic approach to the design of l and l 2 low gain feedback by providing a family of solutions including those resulting from the existing design methods

on improving transient performance in global control of multiple integrators system by bounded feedback abstract the global stabilization problem for multiple integrators systems by bounded control is considered two classes of nonlinear feedback laws are proposed the first one consisting of nested saturation functions is a modification and generalization of that in a r teel global stabilization and restricted tracking for multiple integrators with bounded controls systems control letters 18 3 1992 165 171 and the other one consists of parallel connections of saturation functions both these two types of nonlinear feedback laws need only n n n 2 if n is even and n n 1 2 if n is odd where n is the length of the multiple integrators saturation elements furthermore the poles of the closed loop system can be placed on any location of the left real axis when none of the saturation elements in the control laws is saturated both of them exhibit simpler structure can significantly improve the transient performance of the closed loop system and are very superior to the other existing methods simulation of a fourth order system is used to illustrate the effectiveness of the proposed methods

the forced nonlinear schrodinger equation abstract in many applications the nonlinear schrodinger equation nls occurs with a nonlinear boundary value being specified in these cases one may consider the semi infinite nls as being forced and driven by the boundary value here we numerically study one such case where the nls field at the origin q x 0 t is specified as a boundary value we find that solitons may be created by the forcing we devise a model to predict these numerical results and do so with reasonable quantitative results and excellent qualitative results the model is able to predict the threshold for soliton production fairly accurately

semi global exponential stabilization of linear discrete time systems subject to input saturation via linear feedbacks abstract in this paper we show that a linear discrete time system subject to input saturation is semi globally exponentially stabilizable via linear state and or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable and has all its poles located inside or on the unit circle furthermore the semi globally stabilizing feedback laws are explicitly constructed the results presented here are parallel to our earlier results on the continuous time counterpart lin and saberi 1993

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

brief paper output feedback design for saturated linear plants using deadzone loops in this paper we present an lmi based synthesis approach on output feedback design for input saturated linear systems by using deadzone loops algorithms are developed for minimizing the upper bound on the regional l 2 gain for exogenous inputs with l 2 norm bounded by a given value and for minimizing this upper bound with a guaranteed reachable set or domain of attraction the proposed synthesis approach will always lead to regionally stabilizing controllers if the plant is exponentially unstable to semi global results if the plant is non exponentially unstable and to global results if the plant is already exponentially stable where the only requirement on the linear plant is detectability and stabilizability the effectiveness of the proposed techniques is illustrated with one example