leader follower swarm tracking for networked lagrange systems abstract in this paper swarm tracking problems with group dispersion and cohesion behaviors are discussed for a group of lagrange systems the agent group is separated into two subgroups one is called the leader group whose members are encapsulated with the desired generalized coordinates and generalized coordinate derivatives the other one referred to as the follower group is guided by the leader group the objective is to guarantee distributed tracking of generalized coordinate derivatives for the followers and to drive the generalized coordinates of the followers close to the convex hull formed by those of the leaders both the case of constant leaders generalized coordinate derivatives and the case of time varying leaders generalized coordinate derivatives are considered the proposed control algorithms are shown to achieve velocity matching connectivity maintenance and collision avoidance in addition the sum of the steady state distances between the followers and the convex hull formed by the leaders is shown to be bounded and the bound is explicitly given simulation results are presented to validate the effectiveness of theoretical conclusions

characterizations of input to state stability for infinite dimensional systems we prove characterizations of input to state stability iss for a large class of infinite dimensional control systems including some classes of evolution equations over banach spaces time delay systems ordinary differential equations ode and switched systems these characterizations generalize well known criteria of iss proved by sontag and wang for ode systems for the special case of differential equations in banach spaces we prove even broader criteria for iss and apply these results to show that under some mild restrictions the existence of a noncoercive iss lyapunov functions implies iss we introduce the new notion of strong iss siss that is equivalent to iss in the ode case but is strictly weaker than iss in the infinite dimensional setting and prove several criteria for the siss property at the same time we show by means of counterexamples that many characterizations which are valid in the ode case are not true for general infinite dimensional systems

the iiss property for globally asymptotically stable and passive nonlinear systems this paper investigates the integral input to state stability iiss property for passive nonlinear systems we show that under mild technical assumptions a passive nonlinear system which is globally asymptotically stable is also iiss moreover the integral term from the definition of the iiss property has a very simple form like an l 1 norm we illustrate the result by proving that the drive train of a wind turbine with quadratic torque control is iiss

combining iiss and iss with respect to small inputs the strong iiss property this paper studies the notion of strong iiss which is defined as the combination of input to state stability iss with respect to small inputs and integral input to state stability iiss this notion char acterizes the robustness property that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small but may diverge for stronger disturbances we provide several lyapunov based sufficient conditions for strong iiss one of them relies on iiss lyapunov functions admitting a radially non vanishing class k dissipation rate although such dissipation inequality appears natural in view of the existing lyapunov characterization of iiss and iss we show through a counter example that it is not a necessary condition for strong iiss less conservative conditions are then provided as well as lyapunov tools to estimate the tolerated input magnitude that preserves solutions boundedness

on stabilizability and sampling for infinite dimensional systems some questions either motivated by or related to time sampling regarding the stabilizability of infinite dimensional or distributed parameter systems are considered in particular these include the characteristics of those systems which are finite rank stabilizable and the preservation of stabilizability under sampling a pathological class of systems which when sampled may be rendered unstabilizable by arbitrarily small perturbations in the sampling rate is identified the approach taken in establishing these results is based upon a spectral decomposition of the infinitesimal generator of the underlying open loop semigroup the results obtained apply to control systems with dynamics described by broad classes of partial and functional differential equations that arise frequently in applications

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

nonlinear small gain theorems for discrete time feedback systems and applications we derive in this work a local nonlinear small gain theorem in the framework of input to state stability for discrete time systems our primary objective is to show that as in the continuous time context these discrete time nonlinear small gain theorems are very effective in stability analysis and synthesis for various classes of discrete time control systems two converse lyapunov theorems for discrete exponential stability are developed to assist these applications new results in stability and stabilization presented in this paper are significant extensions of previous work by other authors ieee trans automat control 38 1993 1398 39 1994 2340 33 1988 1082

infinite dimensional lur e systems input to state stability and convergence properties we consider forced lur e systems in which the linear dynamic component is an infinite dimensional well posed system numerous physically motivated delay and partial differential equations are know

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

strong iiss for a class of systems under saturated feedback this paper proposes sufficient conditions under which nonlinear input affine systems can be made strongly iiss in the presence of actuator saturation strong iiss was recently proposed as a compromise between the strength of input to state stability iss and the generality of integral input to state stability iiss it ensures in particular that solutions are bounded provided that the disturbance magnitude is below a certain threshold and that they tend to the origin in response to any vanishing disturbance we propose a growth rate condition under which the bounded feedback law proposed by lin and sontag for disturbance free nonlinear systems ensures strong iiss in the presence of perturbations we illustrate our findings on the angular velocity control of a spacecraft with limited power thrusters in the specific case of linear time invariant systems with neutrally stable internal dynamics we provide a simple static state feedback that ensures strong iiss in presence of actuator saturations this second result is illustrated by the robust stabilization of the harmonic oscillator in both cases we provide an estimate of the maximum disturbance amplitude that can be tolerated without compromising solutions boundedness

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

convergent systems vs incremental stability two similar stability notions are considered one is the long established notion of convergent systems the other is the younger notion of incremental stability both notions require that any two solutions of a system converge to each other yet these stability concepts are different in the sense that none implies the other as is shown in this paper using two examples it is shown under what additional assumptions one property indeed implies the other furthermore this paper contains necessary and sufficient characterizations of both properties in terms of lyapunov functions

stability of higher order discrete time lur e systems abstract we consider discrete time lur e systems obtained by applying nonlinear feedback to a system of higher order difference equations arma models the arma model relates the inputs and outputs of the linear system and does not involve any internal or state variables a stability theory subsuming results of circle criterion type is developed including criteria for input to output stability a concept which is very much reminiscent of input to state stability

a circle criterion for strong integral input to state stability abstract we present sufficient conditions for integral input to state stability iiss and strong iiss of the zero equilibrium pair of continuous time forced lur e systems where by strong iiss we mean the conjunction of iiss and small signal iss our main results are reminiscent of the complex aizerman conjecture and the well known circle criterion we derive a number of corollaries including a result on stabilisation by static feedback in the presence of input saturation in particular we identify classes of forced lur e systems with saturating nonlinearities which are strongly iiss but not iss

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

invariance of the generalised berry phase under unitary transformations application to the time dependent generalised harmonic oscillator a generalisation of the berry phase which reduces to the usual berry phase in the adiabatic and cyclic case is derived in a gauge invariant way this generalised berry phase unlike the berry phase is invariant under unitary transformations a time dependent generalised harmonic oscillator with a cross term in the hamiltonian is taken as an example the hamiltonian of this system is not in general the energy an energy the time derivative of which is the power is obtained from the equation of motion when the system is quantised the generalised berry phase is zero and remains invariant under unitary transformations on the other hand if the energy is chosen to be the hamiltonian a non zero generalised berry phase is obtained but the power is given incorrectly in this case the total phase which includes the dynamical phase and the generalised berry phase is equal in the adiabatic limit to the correct total phase

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

integral control of linear systems with actuator nonlinearities lower bounds for the maximal regulating gain closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller is known to ensure asymptotic tracking of constant reference signals provided that 1 the steady state gain of the linear part of the plant is positive 2 the positive integrator gain is sufficiently small and 3 the reference value is feasible in a very natural sense here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step response and second order systems with a time delay in the input or output the lower bounds are given in terms of open loop frequency step response data and the lipschitz constant of the nonlinearity and are hence readily obtainable

on finite gain stabilizability of linear systems subject to input saturation this paper deals with global finite gain input output stabilization of linear systems with saturated controls for neutrally stable systems it is shown that the linear feedback law suggested by the passivity approach indeed provides stability with respect to every l p norm explicit bounds on closed loop gains are obtained and they are related to the norms for the respective systems without saturation these results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple size 1 jordan blocks contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state space sense this is shown in particular for the double integrator

stability of nonnegative lur e systems a stability instability trichotomy for a class of nonnegative continuous time lur e systems is derived asymptotic exponential and input to state stability concepts are considered the presented trichotomy rests on perron frobenius theory absolute stability theory and recent input to state stability results for lur e systems applications of the results derived arise in various fields including density dependent population dynamics and two examples are discussed in detail

semi global incremental input to state stability of discrete time lur e systems abstract we present sufficient conditions for semi global incremental input to state stability of a class of forced discrete time lur e systems the results derived are reminiscent of well known absolute stability criteria such as the small gain theorem and the circle criterion we derive a natural sufficient condition which guarantees that asymptotically almost periodic inputs generate asymptotically almost periodic state trajectories as a corollary we obtain sufficient conditions for the converging input converging state property to hold

gauge covariant properties of a linear nonautonomous quantum system time dependent even and odd coherent states a new method for finding the dynamical invariant for a general time dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion it is shown that the dynamical invariant for different gauged hamiltonians are connected by time dependent gauge transformations therefore the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system in this representation the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged hamiltonians the exact even and odd coherent states for a time dependent harmonic oscillator are constructed in terms of these gauge covariant bases the harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

coordination and collision avoidance for lagrangian systems with disturbances in this paper we address the problem of cooperation and collision avoidance for lagrangian systems with input disturbances we design control laws that guarantee cooperation as well as collision free maneuvers we show using a two step proof that the avoidance part of the control laws guarantees safety of the agents independently of the coordinating part then we establish an ultimate bound on the region to which all the agents converge to the obtained theoretical results are illustrated through numerical examples

low gain integral control for multi input multioutput linear systems with input nonlinearities we consider the inclusion of a static antiwindup component in a continuous time low gain integral controller in feedback with a multi input multi output stable linear system subject to an input nonlinearity from a class of functions that includes componentwise diagonal saturation we demonstrate that the output of the closed loop system asymptotically tracks every constant reference vector which is feasible in a natural sense provided that the integrator gain is sufficiently small robustness properties of the proposed control scheme are investigated and three examples are discussed in detail

invariance of the geometrical phase under time dependent unitary transformations the effects of time dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting a detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle hilbert space to space of states changes the measured relative phase geometrical phase and dynamical phase are all invariant under these transformations

input to state stability of discrete time lur e systems an input to state stability theory which subsumes results of circle criterion type is developed in the context of discrete time lur e systems the approach developed is inspired by the complexified aizerman conjecture

time evolution of quantum systems with time dependent hamiltonian and the invariant hermitian operator we study the time evolution of a class of exactly solvable time dependent quantum systems with a time dependent hamiltonian given by a linear combination of su 1 1 and su 2 generators with the help of the invariant hermitian operator the exact common solutions of the schrodinger equations for both the su 1 1 and su 2 systems are obtained in terms of eigenstates of the invariant operator the adiabatic and non adiabatic berry phases are calculated with the exact solutions moreover we derive an explicit time evolution operator which is used to investigate the time dependent two photon squeezing states and su 2 squeezing states the squeezing properties of the time dependent su 1 1 coherent states are also discussed

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

some spectral properties of operator valued positive real functions abstract we consider operator valued positive real functions h and show that the intersections of the point continuous and residual spectra of h s with the imaginary axis do not depend on s in particular if h is positive real and h z is invertible for some z in the open right half plane then h s is invertible for all s in the open right half plane furthermore we prove that the eigenspace of h s corresponding to an imaginary eigenvalue does not depend on s it is also shown that the intersection of the numerical range of h s with the imaginary axis is independent of s finally we prove that under suitable assumptions application of a sufficiently positive real static output feedback to a positive real transfer function leads to a strictly positive real closed loop system

input to state stability of differential inclusions with applications to hysteretic and quantized feedback systems input to state stability iss of a class of differential inclusions is proved every system in the class is of lur e type a feedback interconnection of a linear system and a set valued nonlinearity applications of the iss results in the context of feedback interconnections with a hysteresis operator or a quantization operator in the feedback path are developed

existence of unmixed solutions of the discrete time algebraic riccati equation and a nonstrictly bounded real lemma abstract the existence of a solution of the discrete time algebraic riccati equation is established assuming modulus controllability and positive semidefiniteness on the unit circle of the popov function as an application a nonstrictly bounded real lemma is obtained

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

fully distributed flocking with a moving leader for lagrange networks with parametric uncertainties this paper addresses the leader follower flocking problem with a moving leader for networked lagrange systems with parametric uncertainties under a proximity graph here a group of followers move cohesively with the moving leader to maintain connectivity and avoid collisions for all time and also eventually achieve velocity matching in the proximity graph the neighbor relationship is defined according to the relative distance between each pair of agents each follower is able to obtain information from only the neighbors in its proximity involving only local interaction we consider two cases i the leader moves with a constant velocity and ii the leader moves with a varying velocity in the first case a distributed continuous adaptive control algorithm accounting for unknown parameters is proposed in combination with a distributed continuous estimator for each follower in the second case a distributed discontinuous adaptive control algorithm and estimator are proposed then the algorithm is extended to be fully distributed with the introduction of gain adaptation laws in all proposed algorithms only one hop neighbors information e g the relative position and velocity measurements between the neighbors and the absolute position and velocity measurements is required and flocking is achieved as long as the connectivity and collision avoidance are ensured at the initial time and the control gains are designed properly numerical simulations are presented to illustrate the theoretical results

stabilization of well posed infinite dimensional systems by dynamic sampled data feedback it is shown that a set of six natural conditions is necessary and sufficient for the existence of a finite dimensional stabilizing sampled data controller for a well posed infinite dimensional system the underlying stability concept for the sampled data system is reminiscent of the notion of input to state stability from nonlinear control theory

impulsive practical tracking synchronization of networked uncertain lagrangian systems without and with time delays abstract this paper presents a procedure for studying tracking synchronization of networked lagrangian systems via impulsive control where each agent is allowed to be nonidentical even uncertain dynamics some algebraic criteria for tracking synchronization without and with time delays are established respectively it turns out that these criteria can provide a novel impulsive control strategy to synchronize globally networked lagrangian systems to a given time varying target trajectory with a desired tracking error bound called as practical tracking synchronization a distinctive feature of the developed control strategy is fully to take into account the effects of impulsive constraints and thereby to yield impulsive synchronization motion of networked lagrangian systems provided that each agent instantaneously interacts with its neighbors only at some discrete moments as a direct application of the theoretical results practical tracking synchronization of nonidentical 3 dof mobile robots without and with time delays is discussed in detail simulation results illustrate and visualize the effectiveness and feasibility of the proposed control technique

low gain control of uncertain regular linear systems it is well known that closing the loop around an exponentially stable finite dimensional linear time invariant plant with square transfer function matrix bg s compensated by a controller of the form k s gamma 0 where k in bbb r and gamma 0 in bbb r mm will result in an exponentially stable closed loop system which achieves tracking of arbitrary constant reference signals provided that i all the eigenvalues of bg 0 gamma 0 have positive real parts and ii the gain parameter k is positive and sufficiently small in this paper we consider a rather general class of infinite dimensional linear systems called regular systems for which convenient representations are known to exist both in time and in frequency domain the purpose of the paper is twofold i we extend the above result to the class of exponentially stable regular systems and ii we show how the parameters k and gamma 0 can be tuned adaptively the resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms nor is the injection of probing signals required

brief input to state stability for a class of lurie systems we analyze input to state stability iss for the feedback interconnection of a linear block and a nonlinear element this study is of importance for establishing robustness against actuator nonlinearities and disturbances in the absolute stability framework we prove iss from a positive real property of the linear block by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity when this growth condition is violated examples show that the iss property is lost the result is used to give a simple proof of boundedness for negative resistance oscillators such as the van der pol oscillator in a separate application we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear