stabilization of nonlinear systems via forwarding mod l sub g v forwarding is a tool for constructing stabilizers for nonlinear systems a key step in this design technique is to find an explicit solution to a partial differential equation pde which may be hard to find actually the pde may even not be solvable at all in this paper we show that it is possible to provide an additional degree of freedom for the solution of the aforementioned pde hence effectively extending the realm of application of the forwarding methodology our contribution is illustrated with the example of an inverted pendulum with a disk inertia

global stabilization for a class of switched nonlinear feedforward systems abstract the problem of global stabilization for a class of switched nonlinear feedforward systems under arbitrary switchings is investigated in this paper based on the integrator forwarding technique and the common lyapunov function method we design bounded state feedback controllers of individual subsystems to guarantee asymptotic stability of the closed loop system a common coordinate transformation of all subsystems is exploited to avoid individual coordinate transformations for subsystems that are required when applying the forwarding recursive design scheme an example is provided to demonstrate the effectiveness of the proposed design method

nonlinear control of feedforward systems with bounded signals the stabilization problem for a class of nonlinear feedforward systems is solved using bounded control it is shown that when the lower subsystem of the cascade is input to state stable and the upper subsystem not exponentially unstable global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the upper part of the state only this is made possible by invoking the bounded real lemma and a generalization of the small gain theorem thus stabilization is achieved with typical saturation functions saturations of constant sign or quantized control moreover the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback finally a new class of stabilizing control laws for a chain of integrators with input saturation is proposed some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems

stabilization of uncertain feedforward nonlinear systems with application to underactuated systems this note presents saturated controllers to stabilize uncertain feedforward nonlinear systems whose nominal dynamics contains uncertain gains and is subject to linear perturbations we assign a class of saturated controllers within which multiplicative coefficients appear before both states and saturation functions and the convergence analysis is conducted in the following way to verify the reduction of saturated terms we calculate the time derivative of a boundary surface in a small domain when the states reach the boundary surface to prove the asymptotic stability of the corresponding reduced system we use an m matrix based comparison principle in this way the suggested algorithm does not depend on the small gain theory and is suitable for dealing with strongly nonlinear underactuated systems including the cart pendulum system the overhead crane system and the uncertain vertical takeoff and landing vehicle

doubly asymptotic approximations for transient poroelastodynamics a doubly asymptotic approximation daa is an approximate temporal impedance relation at the boundary of a continuous medium here first and second order daas are formulated for an infinite external poroelastic medium described by biot s equations as with their acoustic elastodynamic and electromagnetic predecessors the poroelastodynamic daas approach exactness in both the early time high frequency and late time low frequency limits effecting a smooth transition between they also lend themselves to straightforward boundary element discretization producing matrix ordinary differential equations in time that are readily solved by numerical integration an initial examination of poroelastodynamic daa accuracy is presented for two problems with spherical symmetry

further facts about input to state stabilization previously published results about input to state stabilizability are shown to hold even for systems which are not linear in controls provided that a more general type of feedback is allowed applications to certain stabilization problems and coprime factorizations as well as comparisons to other results on input to state stability are briefly discussed

the gauge field copies and backlund transformation the authors show explicitly that the famous auto backlund transformation and non auto kruskal dodd bullough transformation for the sine gordon equation can be interpreted as the gauge field copies for the strength tensor associated with the lax pair for the sine gordon equation in the same way one interprets the auto and non auto backlund transformation for the liouville equation

doubly asymptotic approximations for vibration analysis of submerged structures doubly asymptotic approximations daa s are differential equations for boundary element analysis of the dynamic interaction between a complex structure and a surrounding infinite medium t l geers j acoust soc am 64 1500 1508 1978 p g underwood and t l geers int j solids struct 17 687 697 1981 in this paper the use of first and second order daa s for vibration analysis of submerged structures is examined first the governing discrete element equations for the general problem are set down and discussed then the accuracy of three daa forms is studied through the generation of numerical results for a submerged spherical shell although the first order daa is found to be inadequate the two second order forms show considerable promise work supported by the david w taylor naval ship r d center

new relationships between input output and lyapunov stability in this note we define a concept called small signal l p stability and show its relationship to lyapunov stability

swinging control of nonlinear oscillations the speed gradient method of control design used previously for problems of regulation and tracking is extended to oscillating systems with energy based objective functions the concept of swinging control is introduced meaning achievement of arbitrary large level of the objective function by arbitrary small control level the existence of swinging control for hamiltonian systems is established simulation results for pendulum swinging problem are demonstrated

stabilization of a class of underactuated mechanical systems via interconnection and damping assignment we consider the application of a formulation of passivity based control pbc known as interconnection and damping assignment ida to the problem of stabilization of underactuated mechanical systems which requires the modification of both the potential and the kinetic energies our main contribution is the characterization of a class of systems for which ida pbc yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction the class is given in terms of solvability of certain partial differential equations one important feature of ida pbc stemming from its hamiltonian formulation is that it provides new degrees of freedom for the solution of these equations using this additional freedom we are able to show that the method of controlled lagrangians in its original formulation may be viewed as a special case of our approach as illustrations we design asymptotically stabilizing ida pbcs for the classical ball and beam system and a novel inertia wheel pendulum

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

a relaxed condition for stability of nonlinear observer based controllers we study certainty equivalence design of observer based controllers and present a new stability condition that exploits state dependent convergence properties of observers this result eliminates the conservatism of treating the observer error as an exogeneous disturbance as an application we show that the reduced order variant of the class of observers in automatica 37 2001 9231 preserves global asymptotic stability in a certainty equivalence implementation in another application we study a nonminimum phase system which cannot be stabilized with the existing output feedback designs

semi global stabilization of partially linear composite systems via feedback of the state of the linear part abstract we consider the problem of semi global stabilization of a class of partially linear composite systems we show by explicit construction of the control laws that a cascade of linear stabilizable and nonlinear asymptotically stable subsystems is semi globally stabilizable by a dynamic feedback of the state of the linear subsystem if a the linear subsystem is right invertible and has all its invariant zeros in the closed left half s plane and b the only linear variables entering the nonlinear subsystem are the output of the linear subsystem our work generalizes previous results by c i byrnes and a isidori 1991 h j sussmann and p v kokotovic 1991 and a r teel 1992

integrator forwarding a new recursive nonlinear robust design we consider the global stabilization of nonlinear systems in strict feedforward form we show that these systems while not feedback linearizable can be globally transformed by feedback and diffeomorphism into lower triangular form the transformation is explicit and recursive we employ this transformation to design a globally stabilizing controller with optimality properties and an input to state stability property with respect to matched uncertainties c 1997 elsevier science ltd

dynamic lyapunov functions lyapunov functions are a fundamental tool to investigate stability properties of equilibrium points of linear and nonlinear systems the existence of lyapunov functions for asymptotically stable equilibrium points is guaranteed by converse lyapunov theorems nevertheless the actual computation of the analytic expression of the function may be difficult herein we propose an approach to avoid the computation of an explicit solution of the lyapunov partial differential inequality introducing the concept of dynamic lyapunov function these functions allow to study stability properties of equilibrium points similarly to standard lyapunov functions in the former however a positive definite function is combined with a dynamical system that render dynamic lyapunov functions easier to construct than lyapunov functions moreover families of standard lyapunov functions can be obtained from the knowledge of a dynamic lyapunov function by rendering invariant a desired submanifold of the extended state space the invariance condition is given in terms of a system of partial differential equations similar to the lyapunov pde differently from the latter however in the former no constraint is imposed on the sign of the solution or on the sign of the term on the right hand side of the equation several applications and examples conclude the paper

homogeneous approximation recursive observer design and output feedback we introduce two new tools that can be useful in nonlinear observer and output feedback design the first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity homogeneity in the bi limit exploiting this extension we give several results concerning stability and robustness for a homogeneous in the bi limit vector field the second tool is a new recursive observer design procedure for a chain of integrator combining these two tools we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

semi global stabilization of minimum phase nonlinear systems in special normal forms abstract we semi globally stabilize certain minimum phase nonlinear systems which are in a normal form where the nonlinear subsystem is driven by an output of a linear system that possesses possibly nonzero peaking exponents we eliminate the peaking phenomenon by stabilizing part of the linear system with a high gain linear control and part of the linear system with a small bounded control the interpretation of this approach will be that we are redefining the outputs to add asymptotically stable nonlinear zeros to the system in a manner that allows the new composite zero dynamics to be asymptotically stable on arbitrarily large compact sets

guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators we study the robustness of optimal regulators for nonlinear deterministic and stochastic multiinput dynamical systems under the assumption that all state variables can be measured we show that under mild assumptions such nonlinear regulators have a guaranteed infinite gain margin moreover they have a guaranteed 50 percent gain reduction margin and a 60 degree phase margin in each feedback channel provided that the system is linear in the control and the penalty to the control is quadratic thus extending the well known properties of lq regulators to nonlinear optimal designs these results are also valid for infinite horizon average cost stochastic optimal control problems

adding integral action for open loop exponentially stable semigroups and application to boundary control of pde systems the article deals with the output feedback regulation of exponentially stable systems by an integral controller we propose appropriate lyapunov functionals to prove exponential stability of the closed loop system an example of parabolic partial differential equation pde systems and an example of hyperbolic systems are worked out to show how exponentially stabilizing integral controllers are designed the proof is based on a novel lyapunov functional construction that employs the forwarding techniques

counterexamples to some results on the existence of field copies several criteria are known for determining which connectionsa are determined uniquely by their curvaturef or byf and its covariant derivatives on a principal bundle with semi simple gauge groupg over a 4 manifoldm a sufficient condition forf to determinea uniquely is that the linear mapb f b from lie algebra valued 1 forms to 3 forms pulled back tom via a local gauge be invertible on an open dense set inm recently f a doria has claimed that this condition is also necessary we present counterexamples to this claim and also to his assertion thatf determinesa uniquely if the restriction of the bundle to every open subset ofm has holonomy group equal tog andf is not degenerate as a 2 form over spacetime

locally optimal controllers and globally inverse optimal controllers in this paper we consider the problem of global asymptotic stabilization with prescribed local behavior we show that this problem can be formulated in terms of control lyapunov functions moreover we show that if the local control law has been synthesized employing an lq approach then the associated lyapunov function can be seen as the value function of an optimal problem with some specific local properties we illustrate these results on two specific classes of systems backstepping and feedforward systems finally we show how this framework can be employed when considering the orbital transfer problem

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

nonlinear and locally optimal controllers design for input affine locally controllable systems given a global nonlinear state feedback which globally stabilises an equilibrium the aim of this article is to modify the local behaviour of the trajectories in order to get local optimality with respect to a given quadratic cost a sufficient condition is given in terms of linear matrix inequalities lmis to design a locally optimal and globally stabilising control law this approach is illustrated on an academic inverted pendulum model in order to stabilise its upper equilibrium point an extension of the main result is then given to address the problematic cases moreover the cases in which the previous lmi condition failed to be satisfied is addressed and a new sufficient condition is then given which is not anymore linear

backlund transformation of partial differential equations from the painleve gambier classification ii tzitzeica equation from the existing methods of singularity analysis only we derive the two equations which define the backlund transformation of the tzitzeica equation this is achieved by defining a truncation in the spirit of the approach of weiss et al so as to preserve the lorentz invariance of the tzitzeica equation if one assumes a third order scattering problem this truncation admits a unique solution thus leading to a matrix lax pair and a darboux transformation in order to obtain the backlund transformation bt which is the main new result of this paper one represents the lax pair by an equivalent two component riccati pseudopotential this yields two different bts the first one is a bt for the hirota satsuma equation while the second one is a bt for the tzitzeica equation one of the two equations defining the bt is the fifth ordinary differential equation of gambier

doubly asymptotic approximations for transient elastodynamics abstract a doubly asymptotic approximation daa is an approximate temporal impedance relation at the boundary of a continuous medium it approaches exactness at both early and late times effecting a smooth transition between here first and second order daas are derived for a uniform isotropic elastic medium of either infinite or semi infinite extent the derivations proceed from pertinent singly asymptotic approximations and employ the method of operator matching previously used for acoustic domains a simple problem with spherical symmetry is considered that illustrates the characteristics of the singly and doubly asymptotic approximations

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

reinforcement learning for adaptive optimal control of unknown continuous time nonlinear systems with input constraints in this paper an adaptive reinforcement learning based solution is developed for the infinite horizon optimal control problem of constrained input continuous time nonlinear systems in the presence of nonlinearities with unknown structures two different types of neural networks nns are employed to approximate the hamilton jacobi bellman equation that is an recurrent nn is constructed to identify the unknown dynamical system and two feedforward nns are used as the actor and the critic to approximate the optimal control and the optimal cost respectively based on this framework the action nn and the critic nn are tuned simultaneously without the requirement for the knowledge of system drift dynamics moreover by using lyapunov s direct method the weights of the action nn and the critic nn are guaranteed to be uniformly ultimately bounded while keeping the closed loop system stable to demonstrate the effectiveness of the present approach simulation results are illustrated

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

robustness of nonlinear state feedback a survey a survey of robustness of nonlinear state feedback is given for series perturbations there are fairly complete results showing that under mild restrictions an optimal controller can tolerate an infinite increase in gain for gain reductions there are some results for systems linear in the control in particular there is a 50 reduction tolerance if the control penalty is quadratic usually the optimal controller cannot be computed exactly there are some results showing the effects of truncation on the robustness essentially robustness is maintained but in a reduced computable part of the state space

the geometry of gauge field copies we show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime provided that the field is not degenerate as a 2 form over spacetime in other words there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime we then show that the ambiguous potentials for those gauge fields are partitioned into gauge equivalence classes modulo certain homotopy classes as a consequence of the nontrivial connectivity of spacetime these homotopy classes depend on the gauge group on the holonomy group and on this last group s centralizer in the gauge group

global asymptotic stabilisation of non linear systems in this article the problem of global asymptotic stabilisation of non linear systems is considered first of all the global asymptotic stability of a class of non linear systems is considered it is proved that the class of non linear systems is globally asymptotically stable under certain appropriate conditions secondly based on the result obtained the global asymptotic stabilisation of a class of non linear systems is investigated and the feedback control laws which globally stabilise the class of systems are designed thirdly by the coordinate change and the results obtained the global stabilisation of the affine non linear systems are studied and then some sufficient conditions of stabilisation are presented finally the global asymptotic stabilisation of partially linear cascade non linear system is considered the sufficient conditions of global stabilisation are obtained and the corresponding feedback control laws which stabilise the non linear systems are designed in addition some

backstepping design with local optimality matching in nonlinear h sup spl infin optimal control design for strict feedback nonlinear systems our objective is to construct globally stabilizing control laws to match the optimal control law up to any desired order and to be inverse optimal with respect to some computable cost functional our recursive construction of a cost functional and the corresponding solution to the hamilton jacobi isaacs equation employs a new concept of nonlinear cholesky factorization when the value function for the system has a nonlinear cholesky factorization we show that the backstepping design procedure can be tuned to yield the optimal control law

on the stabilization of feedforward systems with bounded control abstract for the global asymptotic stabilization of nonlinear controllable in the first approximation cascades consisting of a globally asymptotically locally exponentially stable system driving a stable system a lyapunov like design is proposed this yields bounded control laws where the control magnitude can be chosen arbitrarily large this result provides an alternative to classical forwarding

uncertainty observer based robust tracking control for high order nonlinear systems using forwarding technique based on uncertainty observers we develop a robust tracking control approach for high order nonlinear systems subject to uncertainties satisfying mismatched condition in this study by combining

global stabilization of nonlinear cascade systems abstract we present sufficient conditions for the global stabilizability of two cascade connected nonlinear systems these are based on general results concerning global asymptotic stability of triangular systems which are proved in the last section for polynomial systems in particular the stabilizing feedback is given explicitly

a nonlinear small gain theorem for the analysis of control systems with saturation a nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation the theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear feedforward form this result in turn is applied to the control of 1 linear systems stable and unstable with inputs subject to magnitude and rate saturation and time delays 2 the cascade of globally asymptotically stable nonlinear systems with certain linear systems those that are stabilizable right invertible and such that all of their invariant zeros have nonpositive real part 3 the inverted pendulum on a cart and 4 the planar vertical takeoff and landing aircraft

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

global stabilization of partially linear composite systems a linear stabilizable nonlinear asymptotically stable cascade system is globally stabilizable by smooth dynamic state feedback if a the linear subsystem is right invertible and weakly minimum phase and b the only linear variables entering the nonlinear subsystem are the output and the zero dynamics corresponding to this output both of these conditions are coordinate free and there is freedom of choice for the linear output variable this result generalizes several earlier sufficient conditions for stabilizability moreover the weak minimum phase condition for the linear subsystem cannot be relaxed unless a growth restriction is imposed on the nonlinear subsystem

gd aided iol input output linearisation controller for handling affine form nonlinear system with loose condition on relative degree abstractinput output linearisation iol may encounter a singularity problem when applied to the tracking control of affine form nonlinear system afns which may not have a well defined relative degree the singularity problem has occurred in the area of control for decades in this paper we incorporate the gradient dynamics gd into iol which leads to the gd aided iol method to solve the singularity problem with the proposition of the loose condition on relative degree moreover detailed theoretical analyses on tracking error bound and convergence performance of the corresponding gd aided iol controller are presented simulations and comparisons substantiate that the proposed gd aided iol method is capable of completing the tracking control task and conquering the singularity encountered in the afns

robust disturbance attenuation of nonlinear systems using output feedback and state dependent scaling this paper presents an approach to output feedback stabilization with l 2 disturbance attenuation for nonlinear systems in the presence of dynamic uncertainties a new formulation of state dependent scaling is introduced into the output feedback design which unifies treatment of nonlinear and linear gains the effect of disturbance on the controlled output which is allowed to be any function of output measurements can be always attenuated to an arbitrarily small level with global asymptotic stability if the plant belongs to a wide class of triangular systems whose uncertainties do not necessarily have finite linear gains the uncertain dynamics is not limited to input to state stable systems either the approach is not only a natural extension of popular approaches in robust linear control but also advantageous to numerical computation which is applicable to non triangular systems as well as triangular systems

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

limitations on the stabilizability of globally minimum phase systems the author gives examples showing that in general it is possible for a globally minimum phase system in normal form to have states that cannot be driven asymptotically to the origin by means of any open loop control in particular this provides counterexamples to a number of recently published stabilization theorems it is established by means of examples that a minimum phase system in normal form need not be semiglobally stabilizable or small input semiglobally bibo stabilizable even if the zero dynamics is exponentially stable and the completeness condition holds

on compensating long actuator delays in nonlinear control we are interested in finite escape open loop unstable plants that are globally stabilizable in the absence of actuator delay but require controller redesign in the presence of delay the simplest such plant is z t z t 2 u t d where d is actuator delay of arbitrary length for this system we present a control law that compensates the delay and achieves feedback linearization of the entire ode delay infinite dimensional cascade however even though exponential stability is achieved the result is not global because the plant can have a finite escape with an initial condition z 0 ges 1 d before the feedback control reaches it at t d we prove a stability result whose region of attraction is essentially z 0 1 d and for which we derive an asymptotic stability bound in terms of the system norm z t 2 intt t d u thetas 2dthetas

forwarding design with prescribed local behavior among the non linear control techniques some lyapunov design methods forwarding backstepping take advantage of the structure of the system feedforward form feedback form to formulate a continuous control law which stabilizes globally and asymptotically the equilibrium in addition to stabilization we focus on the local behaviour of the closed loop system providing conditions under which we can predetermine the behaviour around the origin for feedforward systems

decentralized robust stabilization for large scale feedforward non linear systems in this paper we consider the decentralized robust stabilization problem for large scale feedforward non linear systems which possess an upper triangular structure while in the decentralized control area the commonly studied large scale feedback non linear systems possess a lower triangular structure our decentralized robust stabilizer for each subsystem takes a nested saturation feedback with the saturation levels satisfying a set of equations and inequalities global asymptotic stability of the closed loop large scale system has been proved and a simulation example is given