control of an unstable reaction diffusion pde with long input delay abstract a smith predictor like design for compensation of arbitrarily long input delays is available for general controllable possibly unstable lti finite dimensional systems such a design has not been proposed previously for problems where the plant is a pde we present a design and stability analysis for a prototype problem where the plant is a reaction diffusion parabolic pde with boundary control the plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay with an unbounded input operator the predictor based feedback design extends fairly routinely within the framework of infinite dimensional backstepping however the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ode the unbounded character of the input operator requires that the stability be characterized in terms of the h 1 rather than the usual l 2 norm of the actuator state the analysis involves an interesting structure of interconnected pdes of parabolic and first order hyperbolic types where the feedback gain kernel for the undelayed problem becomes an initial condition in a pde arising in the compensator design for the problem with input delay space and time variables swap their roles in an interesting manner throughout the analysis

robust adaptive output feedback tracking for a class of nonlinear time delayed plants within the model reference adaptive control mrac framework a continuous adaptive output feedback control scheme is developed for a class of nonlinear siso dynamic systems with time delays which is robust with respect to unknown time varying plant delays and to an external disturbance with unknown bounds a special form of the lyapunov krasovskii functional with a virtual adaptation gain vector is introduced to prove stability

adaptive control scheme for uncertain time delay systems the paper exposes some of the potential of a recently introduced backstepping transformation for linear uncertain time delay systems to address the classic problems of equilibrium regulation under partial measurements disturbance rejection parameter or delay adaptation for each of these problems an implementable control strategy is proposed it is analyzed through a convergence analysis of infinite dimensional dynamics based on a transport partial differential equation representation of the estimated input delay and the mentioned backstepping transformation the considered controllers contain this representation of the system under the form of a distributed parameter system and use it to determine relevant adaptation strategies an illustration example is proposed

predictor based control for an uncertain euler lagrange system with input delay controlling a nonlinear system with actuator delay is a challenging problem because of the need to develop some form of prediction of the nonlinear dynamics developing a predictor based controller for an uncertain system is especially challenging in this paper tracking controllers are developed for an euler lagrange system with time delayed actuation parametric uncertainty and additive bounded disturbances the developed controllers represent the first input delayed controllers developed for uncertain nonlinear systems that use a predictor to compensate for the delay the results are obtained through the development of a novel predictor like method to address the time delay in the control input lyapunov krasovskii functionals are used within a lyapunov based stability analysis to prove semi globally uniformly ultimately bounded tracking experimental results illustrate the performance and robustness of the developed control methods

ground states for schrodinger type equations with nonlocal nonlinearity abstract the problem of the existence of stable solitary wave solutions for nonlinear schrodinger type equations with a generalized cubic nonlinearity is considered these types of equations have recently arisen in the context of optical communications as averaging approximations to nonlinear dispersive equations with widely separated time scales in this paper it is shown that under general conditions on the kernel of the nonlocal term stable standing wave solutions exist for these equations

brief paper lyapunov tools for predictor feedbacks for delay systems inverse optimality and robustness to delay mismatch we consider lti finite dimensional completely controllable but possibly open loop unstable plants with arbitrarily long actuator delay and the corresponding predictor based feedback for delay compensation we study the problem of inverse optimal re design of the predictor based feedback law we obtain a simple modification of the basic predictor based controller which employs a low pass filter and has been proposed previously by mondie and michiels for achieving robustness to discretization of the integral term in the predictor feedback law the key element in our work is the employment of an infinite dimensional backstepping transformation and the resulting lyapunov function for the infinite dimensional systems consisting of the state of the ode plant and the delay state the lyapunov function allows us to quantify the lyapunov stability properties under the modified feedback the inverse optimality of the feedback and its disturbance attenuation properties for the basic predictor feedback the availability of the lyapunov function also allows us to prove robustness to small delay mismatch in both positive and negative directions

positive solutions for 2p order and 2q order nonlinear ordinary differential systems abstract in this paper by using krasnosel skii fixed point theorem and under suitable conditions we present the existence of single and multiple positive solutions to the following systems 1 p u 2 p a t f u t v t t 0 1 1 q v 2 q b t g u t v t t 0 1 u 2 i 0 u 2 i 1 0 0 i p 1 v 2 j 0 v 2 j 1 0 0 j q 1 where 0 0 p q n we derive two explicit intervals of and such that for any and in the two intervals respectively the existence of at least one solution to the systems is guaranteed and the existence of at least two solutions for and in appropriate intervals is also discussed

brief paper stabilization of linear strict feedback systems with delayed integrators the problem of compensation of input delays for unstable linear systems was solved in the late 1970s systems with simultaneous input and state delay have remained a challenge although exponential stabilization has been solved for systems that are not exponentially unstable such as chains of delayed integrators and systems in the feedforward form we consider a general system in strict feedback form with delayed integrators which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays and design a predictor feedback controller for this class of systems exponential stability is proven with the aid of a lyapunov krasovskii functional that we construct using the pde backstepping approach

adaptive backstepping control of uncertain linear systems under unknown actuator delay trajectory tracking of uncertain time delay linear systems by output feedback control is of theoretical importance and practical value in this paper concentrating on a class of linear plants whose relative degree equals to system dimension we develop a lyapunov based control scheme to achieve trajectory tracking despite some classic difficulties including unmeasurable system state unknown plant parameters and unknown input time delay a comprehensive approach combining adaptive backstepping output feedback with prediction based boundary control is employed in the design the stability analysis exhibits the global boundedness of all closed loop system signals and the tracking performance is also guaranteed

an adaptive smith controller for time delay systems with relative degree n 2 abstract this paper addresses the control of time delay systems whose relative degree does not exceed two an adaptive smith controller together with an adaptive law similar to the delay free case is proposed by using lyapunov krasovskii functionals for an appropriate model transformation of the original system semiglobal stability of the closed loop system and asymptotic convergence of the output error is established strict positive realness together with the low relative degree of the plant is exploited to establish the stability properties robustness properties of the adaptive controller are briefly discussed

robust output feedback model reference adaptive control of siso plants with multiple uncertain time varying state delays within the model reference adaptive control framework a simple adaptive output feedback control scheme is developed for a class of linear single input single output dynamic systems with multiple state delays which is robust with respect to multiple unknown time varying plant delays and to an external disturbance with unknown bounds a special form of the lyapunov krasovskii functional with ldquovirtualrdquo adaptation gain is introduced to prove stability

globally stable adaptive controller for systems with delay we propose a globally stable finite spectrum assignment adaptive controller for systems with delay it is assumed that the process is minimum phase though not necessarily stable and the delay is known it is shown that the plant output asymptotically tracks a reference model output with all signals remaining bounded

pulse confinement in optical fibers with random dispersion short range correlated uniform noise in the dispersion coefficient inherent in many types of optical fibers broadens and eventually destroys all initially ultra short pulses however under the constraint that the integral of the random component of the dispersion coefficient is set to zero pinned periodically or quasi periodically along the fiber the dynamics of the pulse propagation changes dramatically for the case that randomness is present in addition to constant positive dispersion the pinning restriction significantly reduces average pulse broadening if the randomness is present in addition to piece wise constant periodic dispersion with positive residual value the pinning may even provide probability distributions of pulse parameters that are numerically indistinguishable from the statistically steady case the pinning method can be used to both manufacture better fibers and upgrade existing fiber links

adaptive sinusoidal disturbance cancellation for unknown lti systems despite input delay solutions exist for the problems of canceling sinusoidal disturbances and compensating input delays in this paper two problems are considered simultaneously and an adaptive controller is designed to cancel unknown sinusoidal disturbances forcing an unknown linear time invariant system in controllable canonical form despite input delay the design is based on three steps 1 parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters 2 representation of the delay as a transport pde 3 design of the adaptive controller by using the backstepping boundary control technique for pdes it is proven that the equilibrium of the closed loop system is stable and the state of the considered system converges to zero as t with perfect disturbance estimation the effectiveness of the controller is illustrated with a simulation example of a second order system

adaptive control of delay system the result of exact model matching for delay systems developed by the author is extended to the adaptive control of delay systems likewise using results from the adaptive control of systems without delay the principle of constructing adaptive systems for delay plants is also very simple and clear

stabilizing effects of dispersion management a cubic nonlinear schrodinger equation nls with periodically varying dispersion coefficient as it arises in the context of fiber optics communication is considered for sufficiently strong variation corresponding to the so called strong dispersion management regime the equation possesses pulse like solutions which evolve nearly periodically this phenomenon is explained by constructing ground states for the averaged variational principle and justifying the averaging procedure furthermore it is shown that in certain critical cases e g quintic nonlinearity in one dimension and cubic nonlinearity in two dimensions the dispersion management technique stabilizes the pulses which otherwise would be unstable this observation seems to be new and is reminiscent of the well known kapitza s effect of stabilizing the inverted pendulum by rapidly moving its pivot 2001 published by elsevier science b v

compensating the distributed effect of diffusion and counter convection in multi input and multi output lti systems compensation of infinite dimensional input or sensor dynamics in siso lti systems is achieved using the backstepping method for mimo lti systems with distributed input or sensor dynamics governed by diffusion with counter convection we develop a methodology for constructing control laws and observers that compensate the infinite dimensional actuator or sensor dynamics the explicit construction of the compensators are based on novel transformations which can be considered of backstepping forwarding type of the finite dimensional state of the plant and of the infinite dimensional actuator or sensor states based on these transformations we construct explicit lyapunov functionals which prove exponential stability of the closed loop system or convergence of the estimation error in the case of observer design finally we illustrate the effectiveness of our controller with a numerical example

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

stability and robustness properties of a simple adaptive controller a low order adaptive tracking controller is proposed for linear time invariant plants with a relative degree not exceeding two the order of the plant is not required to be known a priori the algorithm is robustly stable with respect to small linear time invariant plant perturbations and bounded disturbances the robustness is achieved by using a projection of the parameter estimates in the control law an additional a priori information needed to design the controller is bounds on the plant parameters in the absence of plant perturbations and disturbances perfect tracking is obtained

predictor based stabilization of neutral type systems with input delay in this contribution we present an extension of the prediction scheme proposed in manitius and olbrot 1979 for the compensation of the input delay to the case of linear neutral type systems with input delay for simplicity of the presentation we treat the case of systems with one state delay