stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

controllability and linearized regulation a nonlinear controllable plant under mild technical conditions admits a precompensator with the following property along control trajectories joining pairs of states the composite system precompensator plus plant is up to first order isomorphic to a parallel connection of integrators

a new methodology for observer design and implementation observers are usually formulated as explicit systems of differential equations and implemented using standard ode solvers in this paper we show that there can be advantages in formulating the observer as a differential algebraic equation dae we propose a canonical dae observer and show that it can be implemented using standard dae solvers the dae implementation of the observer can be considered regardless of the design philosophy used we present a simple design strategy for this canonical dae observer based on the extended linearization method

nonlinear control via uniform system approximation abstract we introduce the notion of uniform system approximation and use it to solve the following problem given a single input nonlinear system f g x f x g x u that is linearly controllable on a manifold e of equilibrium points find vector fields f g such that f g and f g agree to first order on e and f g is input to state linearizable on a neighborhood of e the solution proceeds by constructing a linearizable uniform system approximation and yields as a by product a feedback control law that can be used to guarantee stable operation of the given system in a neighborhood of e this includes for example smooth transitions between operating points that are not necessarily close to each other a simple example is given to illustrate the approach

higher order approximate feedback linearization about a manifold for multi input systems considers the problem of approximately feedback linearizing a multi input nonlinear system around the equilibrium manifold spl epsi while making the error terms be of highest order on spl epsi necessary and sufficient conditions are given for approximately feedback linearizing the system around spl epsi with error terms of order spl rho on spl epsi a simple multi input example is given to illustrate the higher order approximate feedback linearization procedure

feedback linearization families for nonlinear systems characterizations are developed for parameterized linear dynamic feedback control laws that can arise when linearizing a nonlinear dynamic feedback control law for a specified nonlinear system about a family of constant operating points such characterizations are important in applying the recently developed extended linearization design approach to various types of control problems to illustrate the input output decoupling problem is considered

feedback control of analytic nonlinear systems by extended linearization for mufti input mufti output analytic nonlinear systems a design method based on the family of linearizations of the system parameterized by the family of constant operating points is considered using the cauchy kowalewski theorem it is shown that nonlinear state feedback control laws and output observer feedback control laws exist such that the eigenvalues of the family of closed loop linearizations have specified analytically scheduled values with respect to the family of closed loop operating points

robust sampled data fuzzy control for nonlinear systems and its applications free weight matrix method in this paper sampled data control is proposed to stabilize the nonlinear system which is expressed as a takagi sugeno t s fuzzy submodels based on suitable lyapunov krasovskii functional lkf along with new weighted integral inequalities the sufficient conditions are derived in terms of linear matrix inequalities which ensure the exponential stability of the proposed closed loop t s fuzzy system the peculiarity of this paper is as novel integral inequalities and lkf are proposed less conservative results are provided when compared to existing results besides that wind energy conversion systems chaotic systems and an inverted pendulum are validated with derived sufficient conditions the corresponding simulation results show the merit of the proposed results over the existing works

a unified view of some numerical methods for fractional diffusion abstract in recent years a number of numerical methods for the solution of fractional laplace and more generally fractional diffusion problems have been proposed the approaches are quite diverse and include among others the use of best uniform rational approximations quadrature for dunford taylor like integrals finite element approaches for a localized elliptic extension into a space of increased dimensions and time stepping methods for a parabolic reformulation of the fractional differential equation a systematic comparison both theoretical and experimental of these approaches has thus far been lacking a main contribution of the present work is the observation that all approaches mentioned above can in fact be interpreted as realizing different rational approximations of a univariate function over the spectrum of the original non fractional diffusion operator while this is obvious for some of the methods it is a new result in particular for extension based and time stepping approaches this observation allows us to cast all described methods into a unified theoretical and computational framework which has a number of benefits theoretically it enables us to develop new convergence proofs for several of the studied methods clarifies similarities and differences between the approaches suggests how to design new and improved methods and allows a direct comparison of the relative performance of the various methods practically it provides a single simple to implement efficient and fully parallel algorithm for the realization of all studied methods for instance this immediately yields a fast and memory efficient way of realizing all tensor product extension methods and lets us parallelize the otherwise inherently sequential time stepping approach finally we present a detailed numerical study comparing all investigated methods for various fractional exponents and draw conclusions from the results the comparison is made fair by the central insight that the computational effort of all these methods depends only on a single parameter the degree of the underlying rational approximation as a point of comparison we also test a simple rational approximation method based on a black box direct rational approximation algorithm which performs very well in practice

brief analysis and design of gain scheduled sampled data control systems the existence of gain scheduled sampled data controllers that meet an important linearization requirement about a family of equilibria is investigated

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

a new method for linearizing non linear systems the pseudolinearization consider the single input non linear system x f x u it has been shown that under mild conditions there exist transformations z t x v s x u such that in the z space the linear tangent model is independent of the operating point a constructive procedure to obtain the above transformations is described this new method is computationally attractive and effective in many practical cases

a stability property in this note we obtain a stability property not for a point of equilibrium but for a certain family of asymptotically stable equilibria of a dynamical system with an input we show that this property depends not only on the initial condition but also on the time derivative of the input function we suggest possible applications of this result to some control problems

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

bayesian inference in models with euclidean structures abstract when a parameter space is endowed with a euclidean structure there is a euclidean measure called the structural or geometric prior which is an obvious candidate to represent ignorance concerning an unknown parameter the rationale for using this prior is strengthened when the unknown parameter is the canonical parameter of an exponential family then the structural prior is the uniform prior for the canonical parameter it is shown also that euclidean structures can be combined in different ways with group structures the structural or geometric prior is then the product of the invariant prior for the group structure and the structural or geometric prior for the euclidean structure

iterative low rank approximation solvers for the extension method for fractional diffusion abstract we consider the numerical method for fractional diffusion problems which is based on an extension to a mixed boundary value problem for a local operator in a higher dimensional space we observe that when this problem is discretized using tensor product spaces as is commonly done the solution can be very well approximated by low rank tensors we provide some analysis to support this observation this fact motivates us to apply iterative low rank approximation algorithms in order to efficiently solve this extended problem in particular we employ a recently proposed greedy tucker approximation method as well as a more classical greedy rank one update method throughout all objects of interest are kept in suitable low rank approximations which dramatically reduces the required amount of memory compared to the full formulation of the extended problem our approach can be used for general non structured space discretizations if the space discretization itself has tensor product structure we can further decompose the problem in order to deal with even lower dimensional objects we also note that the approach can be directly applied to higher order discretizations both in space and the extended variable a further contribution of our work is a rank one version of a diagonal preconditioner which can mitigate the severe ill conditioning of the extended problem arising due to mesh grading and singularity of the coefficient this preconditioner can be realized using a simple pre and postprocessing step and does not affect the computational effort of our algorithm in several numerical examples we demonstrate the convergence behavior of the proposed methods in particular the tucker approximation approach requires only a few iterations in order to reach the discretization error in all tested settings

on dynamic input output linearization of discrete time nonlinear systems this paper studies the problem of linearizing the input output map of an analytic discrete time nonlinear system locally around a given trajectory necessary and sufficient conditions are given for the existence of a regular dynamic state feedback control law under which the input dependent part of the response of a nonlinear system becomes linear in the input and independent of the initial state the proposed conditions are less restrictive than those obtained by lee and marcus for linearizing the input output map via a static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system firstly the solvability conditions are expressed in terms of the inversion algorithm secondly the proof of the existence and construction of the dynamic state feedback compensator relies on this algorithm

inner statistical inference ii for any linear model there is according to an invariance argument a uniquely defined prior representing ignorance which will be called the inner prior it is shown that the corresponding posterior probabilities are weighted averages of conditional confidence levels this frequency interpretation compares favorably with the usual interpretation of confidence levels as unconditional probabilities of coverage of particular interest to users of statistical methods is that inner posterior intervals are always shorter than the corresponding confidence intervals and sometimes dramatically so

studies in the history of probability and statistics xlviii the bayesian contributions of ernest lhoste summary the contributions of ernest lhoste are largely unknown outside france and even within that country are not well known his important contributions were in the two areas of the development of prior distributions that represent little or no information and a sophisticated posterior analysis for normal and binomial populations his results are similar to those of haldane 1948 and jeffreys 1961 but they appeared much earlier in 1923 he gave a great deal of thought to how to represent vague prior knowledge and his results represent a significant and unique contribution to bayesian ideas in the early part of the 20th century

exact and approximate feedback linearization without the linear controllability assumption the feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable in this paper the assumption of linear controllability is removed necessary and sufficient conditions are given through lie orbital symmetries thus giving a geometric characterization of the problem in the analytic case both the exact and the approximate linearization problems are considered in the analytic case

state estimation for non linear uncertain systems a design based on properties related to the uncertainty bound observers for non linear uncertain dynamical systems are proposed the non linearity and uncertainty in the systems are assumed to be bounded the bound is however unknown the observer contains an adaptive scheme designed to accommodate the uncertainty the design of the observer is only based on certain functional properties of the bound no statistical information about the uncertainty needs to be assumed or utilized

approximate noninteracting control with stability for nonlinear systems a notion of approximate noninteracting control with stability for nonlinear systems is introduced and the design of both static and dynamic state feedback control laws that achieve this property for a given plant is considered this approach to the problem is based on the notion of extended linearization and begins with an extension of the linearization approach pointwise to all constant operating points in an open neighborhood of the nominal under natural assumptions it is shown that obstructions to extract noninteracting control with internal stability or input state output stability can be circumvented in the context of approximate noninteraction

design of p i controllers for dc to dc power supplies via extended linearization in this article the method of extended linearization is used for designing stabilizing non linear proportional integral p i controllers which regulate to a constant value either the average output voltage or the average input current of pwm controlled dc to dc power converters such as the boost the buck boost and the cuk converters the design is carried out on the basis of the ziegler nichols p i controller design method applied to a family of linearized transfer function models of converters parametrized by constant operating equilibrium points of the average pwm controlled circuit

further results on the observer design problem abstract in this paper the observer design problem for nonlinear systems is considered sufficient lyapunov like conditions are presented for the existence of a nonlinear observer the theory we develop considerably improves and extends the results of our recent work 14

an extended linearization approach to nonlinear system inversion the inversion problem for nonlinear systems is considered from both the internal viewpoint state equation representations and the external viewpoint volterra series representations in order to circumvent difficulties in the known exact inversion theory a method for obtaining what will be called a partial inverse system is proposed for a given nonlinear system a cascade system typically containing differentiators is found such that the overall nonlinear system has a family of constant operating points with inputs and outputs identical furthermore the linearization of the overall system about each of these constant operating points is the identity system

input output linearization approach to state observer design for nonlinear system presents a state observer for a class of nonlinear systems based on the input output linearization while the previous result presented state observers for nonlinear systems of full relative degree we proposed a procedure fur the design of nonlinear state observers which do not require the hypothesis of full relative degree assuming that there exists a global state observer for internal dynamics and that some functions are globally lipschitz we can design a globally convergent state observer it is also shown that if the zero dynamics are locally exponentially stable then there exists a local state observer an example is given to illustrate the proposed design of nonlinear state observers

on stability properties of nonlinear systems with slowly varying inputs systems with slowly varying inputs are discussed as a special class of two time scale systems and singular perturbation results are seen as convenient tools to analyze their properties a lemma by f c hoppensteadt trans amer math soc vol 123 p 521 35 1966 for a parameterized family of systems is restated and applied to the analysis of systems with slowly varying inputs

stability analysis of the simplest takagi sugeno fuzzy control system using circle criterion in this paper we present two sufficient conditions in the frequency domain for the global stability analysis of the simplest takagi sugeno t s fuzzy control system based on the circle criterion and its graphical interpretation these conditions are significant in graphically designing the simplest t s fuzzy controller in the frequency domain four numerical examples are provided to demonstrate the efficiency of our two frequency domain based conditions furthermore this t s fuzzy controller and the linear proportional controllers are also compared it is concluded that the simplest t s fuzzy controller can outperform the linear proportional controllers in a noisy environment with external disturbances

dynamic gain scheduled control and extended linearisation extensions explicit formulae and stability a controller designed for linearisations at various trim operating points of a nonlinear system using linear approaches is not necessarily performing well or stable once scheduled with a state under dynamic conditions the key idea of using this scheduled control law design is to retain states close to the current usually dynamically varying operating point dynamic gain scheduling dgs is a technique aimed to resolve this controller scheduling issue for rapidly changing dynamics and states it entails scheduling the control law gains with a fast varying state variable rather than with a slowly varying state it has been successfully applied to the aircraft system models allowing also for nested loops the aim of this paper is to extend dgs to a more general setting allowing for more complex multi input dynamics incorporating the more simple static scheduling within the same controller hence given a linear design suitable transformations are provided which allow fast scheduling of multi variable co

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

two level schemes of cauchy problem method for solving fractional powers of elliptic operators abstract in this paper we develop and investigate numerical algorithms for solving equations of the first order for the fractional powers of discrete elliptic operators a y 0 1 for v h with v h a finite element or discrete approximation space the main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the cauchy problem method vabishchevich 2015 the first approach is based on a solution of the related time dependent pde and the second approach is based on diagonal pade approximations to 1 x when the relation of the solution on two times levels is used the second and fourth order approximations are constructed by both approaches in order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for t close to 0 results of numerical experiments are presented they agree well with the theoretical results

hp finite elements for fractional diffusion the purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional laplacian the approach is based on a reformul

on the pseudo linearization problem for nonlinear systems abstract based on well known results for linear systems an alternate treatment of tthe pseudo linearization problem of reboulet and champetier is given sufficient conditions for pseudo linearization are obtained in the general case and these are shown to be satisfied under mild hypothesis by systems with one or two inputs advantages of the new approach include the simplicity of the derivation and a more explicit representation for a pseudo linearizing transformation

brief stability preserving interpolation methods for the synthesis of gain scheduled controllers synthesis of gain scheduled controllers for nonlinear plants often requires that a parameter varying controller be generated from a finite set of linear time invariant controllers we propose interpolation methods for this task with the property that stability of the linearized closed loop system is preserved for all values of the scheduling parameter in addition slow variation arguments are presented that establish local stability of the nonlinear closed loop system with gain scheduled controller

stabilization of nonlinear systems under variable sampling a fuzzy control approach this paper investigates the problem of stabilization for a takagi sugeno t s fuzzy system with nonuniform uncertain sampling the sampling is not required to be periodic and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound by using the input delay approach the t s fuzzy system with variable uncertain sampling is transformed into a continuous time t s fuzzy system with a delay in the state though the resulting closed loop state delayed t s fuzzy system takes a standard form the existing results on delay t s fuzzy systems cannot be used for our purpose due to their restrictive assumptions on the derivative of state delay a new condition guaranteeing asymptotic stability of the closed loop sampled data system is derived by a lyapunov approach plus the free weighting matrix technique based on this stability condition two procedures for designing state feedback control laws are given one casts the controller design into a convex optimization by introducing some over design and the other utilizes the cone complementarity linearization idea to cast the controller design into a sequential minimization problem subject to linear matrix inequality constraints which can be readily solved using standard numerical software an illustrative example is provided to show the applicability and effectiveness of the proposed controller design methodology

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

an enhanced velocity based algorithm for safe implementations of gain scheduled controllers abstractthis paper presents an enhanced velocity based algorithm to implement gain scheduled controllers for nonlinear and parameter dependent systems a new scheme including pre and post filtering is proposed with the assumption that the time derivative of the controller inputs is not available for feedback control it is shown that the proposed control structure can preserve the input output properties of the linearised closed loop system in the neighbourhood of each equilibrium point avoiding the emergence of the so called hidden coupling terms moreover it is guaranteed that this implementation will not introduce unobservable or uncontrollable unstable modes and hence the internal stability will not be affected a case study dealing with the design of a pitch axis missile autopilot is carried out and the numerical simulation results confirm the validity of the proposed approach

observer based control law for a class of non linear systems control and observation laws are considered for a class of non linear systems an algebraic procedure is proposed to design approximate observer based controllers this procedure is similar to that used for linear time invariant systems the paper is completed by an example based on elastically jointed robots

identification of non linear parametrically varying models using separable least squares in this paper a novel identification algorithm for a class of non linear possibly parameter varying models is proposed the algorithm is based on separable least squares ideas these models are given in the form of a linear fractional transformation lft where the forward part is represented by a conventional linear regression and the feedback part is given by a non linear map which can take into account scheduling variables available for measurement the non linear part of the model can be parameterized according to various paradigms like e g neural network nn or general nonlinear autoregressive exogenous narx models the estimation algorithm exploits the separability of the criterion used to estimate the parameters when using a nn it is possible the explicit computation of the frechet derivative needed to implement a separable least square algorithm

brief an observer for systems with nonlinear output map a nonlinear observer with the feedback gain weighted by the sensitivity of the output with respect to the state is developed for systems with nonlinear output map the observer can be obtained from the extended kalman filter by a special choice of time varying weighting matrices it is shown that the estimation error dynamics are asymptotically stable and a region of attraction is derived the observer is applied to the top blown converter process for estimating the content of impurities in the liquid metal using plant data from the converter at ssab oxelosund ab the observer is shown to provide accurate estimates of the carbon content

on the state observation and output feedback problems for nonlinear uncertain dynamic systems abstract in this paper we examine the problems of state observation and state trajectory control by output feedback for the class of non linear systems in 3 11 we begin by modifying a known discontinuous variable structure type observer into a continuous type observer that guarantees the observation error is globally exponentially stable ges we then modify a known discontinuous variable structure type output feedback controller into continuous type output feedback controller that forces the state to the origin in the ges sense specific time varying bounds on the observation error and the state trajectory are also developed revealing how the corresponding observer or control parameters can be adjusted to improve the system performance

feedback control of multiinput nonlinear systems by extended linearization control of a nonlinear system is addressed by considering its family of linearizations given that a family of linear state feedback controllers has been designed to meet some linear control objective it is shown how these controllers can be pieced together into an overall nonlinear state feedback controller the resulting closed loop system has the property that its linearizations are equal to the desired linearly controlled systems a similar approach can be used to design nonlinear state estimators the approach is illustrated by applying it to the control of a planar two link manipulator

parametric stabilization is easy a polynomially parametrized family of continuous time controllable linear systems is always stabilizabte by polynomially parametrized feedback

a neural approach for control of nonlinear systems with feedback linearization several schemes for feedback linearization using neural networks have been investigated and compared then an approach to design a neurocontroller in the sense of feedback linearization is introduced the contents include 1 full input output linearization when a system has relative degree n 2 partial input output linearization when a system has relative degree r r n and 3 approximate linearization when the involutivity condition does not hold corresponding programs and examples are given to illustrate the proposed methodology

brief paper a velocity algorithm for the implementation of gain scheduled controllers a new method is proposed to implement gain scheduled controllers for nonlinear plants given a family of linear feedback controllers designed for linearizations of a nonlinear plant about constant operating points a nonlinear gain scheduled controller is derived that preserves the input output properties of the linear closed loop systems locally about each equilibrium point the key procedures in the proposed method are to provide integral action at the inputs to the plant and differentiate some of the measured outputs before they are fed back to the scheduled controller for a fairly general class of systems the nonlinear gain scheduled controllers are easy to obtain and their structure is similar to that of the original linear controllers

control problems in super articulated mechanical systems control of super articulated mechanical systems sams controlled mechanical system in which the dimension of the configuration space exceeds the dimension of the control input space is studied as a starting point a graphical characterization of general sams is developed in which a so called control flow diagram cfd is constructed to represent the interaction forces among the degrees of freedom of the system three types of structure namely the chain structure the tree structure and isolated points are identified degree of complexity of the system is defined based on the structure of the system a systematic backstepping type controller design is proposed for systems with a chain structure to achieve global asymptotic stability and trajectory tracking such a controller design itself illustrates the sufficient conditions for the existence of smooth feedback control such that the systems with a chain structure can be globally asymptotically stabilized the connection of the authors control algorithm with existing results is discussed

survey approximate linearization via feedback an overview fostered by a growing interest in nonlinear control theory and catalyzed by the discovery in the early 1980s of the exact conditions under which a nonlinear plant can be linearized by static state feedback and coordinate transformation in the last decades there has been a rapid increase of interest in the search for approximate solutions to the problem of linearizing nonlinear systems by state or output feedback main reason for that is the limited applicability of the rigorous methods and the complexity sensitivity and design difficulties of the exact linearizing compensators if any in the present paper the literature on the subject is reviewed and organized in what is believed to be a new and consistent perspective recent works especially in the area of data based techniques are in fact described and related whenever possible to fundamental results previously obtained by model based differential geometric methods this is expected to bring modern system linearization methods closer to the needs of practicing control engineers and to stimulate further research eventually able to fill visible gaps in this direction