output feedback nn control for two classes of discrete time systems with unknown control directions in a unified approach in this paper output feedback adaptive neural network nn controls are investigated for two classes of nonlinear discrete time systems with unknown control directions 1 nonlinear pure feedback systems and 2 nonlinear autoregressive moving average with exogenous inputs narmax systems to overcome the noncausal problem which has been known to be a major obstacle in the discrete time control design both systems are transformed to a predictor for output feedback control design implicit function theorem is used to overcome the difficulty of the nonaffine appearance of the control input the problem of lacking a priori knowledge on the control directions is solved by using discrete nussbaum gain the high order neural network honn is employed to approximate the unknown control the closed loop system achieves semiglobal uniformly ultimately bounded sguub stability and the output tracking error is made within a neighborhood around zero simulation results are presented to demonstrate the effectiveness of the proposed control

a monte carlo method for solving unsteady adjoint equations traditionally solving the adjoint equation for unsteady problems involves solving a large structured linear system this paper presents a variation on this technique and uses a monte carlo linear solver the monte carlo solver yields a forward time algorithm for solving unsteady adjoint equations when applied to computing the adjoint associated with burgers equation the monte carlo approach is faster for a large class of problems while preserving sufficient accuracy

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

sparse kernel regression modeling using combined locally regularized orthogonal least squares and d optimality experimental design the note proposes an efficient nonlinear identification algorithm by combining a locally regularized orthogonal least squares lrols model selection with a d optimality experimental design the proposed algorithm aims to achieve maximized model robustness and sparsity via two effective and complementary approaches the lrols method alone is capable of producing a very parsimonious model with excellent generalization performance the d optimality design criterion further enhances the model efficiency and robustness an added advantage is that the user only needs to specify a weighting for the d optimality cost in the combined model selecting criterion and the entire model construction procedure becomes automatic the value of this weighting does not influence the model selection procedure critically and it can be chosen with ease from a wide range of values

an adaptive multi scale basis method for modelling discrete time nonlinear dynamical systems based on locally applicable autoregressive moving average with exogenous armax models this paper proposes two new adaptive modelling schemes for discrete time non linear dynamic systems based on the concept of a multiscale which originates in approximation theory a general result on the convergence of modelling based on the multiscale basis functions with a least square estimated is derived with the advantage of the multiresolution nature of the multiscale basis the new modelling scheme can be used to capture both the global and local characteristics of a non linear system over distinct scales the algorithms also provide a trade off between the model structure model size and the modelling error one of which emphasizes the modelling error with variable model structure whilst the other stresses a desired model size with better modelling error data based modelling examples are used to demonstrate the effectiveness of the multiscale modelling schemes with different multiscale basis

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

lie bracket approximation of extremum seeking systems extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map it often requires advanced system theoretic tools to understand the qualitative behavior of extremum seeking systems in this paper a novel interpretation of extremum seeking is introduced we show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain lie brackets of the vector fields of the extremum seeking system it turns out that the lie bracket system directly reveals the optimizing behavior of the extremum seeking system furthermore we establish a theoretical foundation and prove that uniform asymptotic stability of the lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system we use the established results in order to prove local and semi global practical uniform asymptotic stability of the extrema of a certain map for multi agent extremum seeking systems

estimation based approach for real time optimisation of uncertain nonlinear systems abstractin this paper the minimisation of an unknown but measurable cost function with uncertain dynamics is considered the drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time varying parameters a novel estimation scheme based on almost invariant manifolds is proposed to estimate the time varying parameters a direct gradient based adaptive extremum seeking controller is designed to solve the uncertain optimisation problem this approach is shown to improve the transient performance of real time optimisation control systems

singularly perturbed lie bracket approximation we consider the interconnection of two dynamical systems where one has an input affine vector field by employing a singular perturbation and a lie bracket analysis technique we show how the trajectories can be approximated by two decoupled systems from this trajectory approximation result and the stability properties of the decoupled systems we derive stability properties of the overall system

error estimates for transport problems with high peclet number using a continuous dependence assumption in this paper we discuss the behavior of stabilized finite element methods for the transient advection diffusion problem with dominant advection and rough data we show that provided a certain continuous dependence result holds for the quantity of interest independent of the peclet number this quantity may be computed using a stabilized finite element method in all flow regimes as an example of a stable quantity we consider the parameterized weak norm introduced in burman 2014 the same results may not be obtained using a standard galerkin method we consider the following stabilized methods continuous interior penalty cip and streamline upwind petrov galerkin supg the theoretical results are illustrated by computations on a scalar transport equation with no diffusion term rough data and strongly varying velocity field

recursive hybrid algorithm for non linear system identification using radial basis function networks recursive identification of non linear systems is investigated using radial basis function networks a novel approach is adopted which employs a hybrid clustering and least squares algorithm the recursive clustering algorithm adjusts the centres of the radial basis function network while the recursive least squares algorithm estimates the connection weights of the network because these two recursive learning rules are both linear rapid convergence is guaranteed and this hybrid algorithm significantly enhances the real time or adaptive capability of radial basis function models the application to simulated real data are included to demonstrate the effectiveness of this hybrid approach

towards a parameter free method for high reynolds number turbulent flow simulation based on adaptive finite element approximation we present work towards a parameter free method for turbulent flow simulation based on adaptive finite element approximation of the navier stokes equations at high reynolds numbers in this model viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations and skin friction at solid walls is assumed to be negligible compared to inertial effects the result is a computational model without empirical data where the only parameter is the local size of the finite element mesh under adaptive refinement of the mesh based on a posteriori error estimation output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution and thus the resulting method has no adjustable parameters no ad hoc design of the mesh is needed instead the mesh is optimised based on solution features in particular no bounder layer mesh is needed we connect the computational method to the mathematical concept of a dissipative weak solution of the euler equations as a model of high reynolds number turbulent flow and we highlight a number of benchmark problems for which the method is validated

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

nonlinear difference equations in the numerical integration of nonlinear differential equations it is often possible to get phantom solutions which are qualitatively quite different from the true solutions whether the behaviour of a numerical solution is a true reflection of a nonlinear phenomenon being modelled by the differential equation or whether it is a mere consequence of the particular discretization method used in solving the equation is clearly a question of considerable importance this question is highlighted by recent analyses of difference equation approximations to one of the simplest nonlinear differential equations the verhulst equation

goal oriented error estimation and adaptivity for free boundary problems the domain map linearization approach in free boundary problems the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation we develop duality based a posteriori error estimates for functional outputs of solutions of free boundary problems that include both sources of error the derivation of an appropriate dual problem linearized adjoint is however nonobvious for free boundary problems to derive an appropriate dual problem we present the domain map linearization approach in this approach the free boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map we show for a bernoulli type free boundary problem that this dual problem corresponds to a poisson problem with a nonlocal robin type boundary condition furthermore we present numerical experiments that demonstrate the effectivity of the dual based error estimate and its usefulness in goal oriented adaptive mesh refinement

subgrid modeling for convection diffusion reaction in one space dimension using a haar multiresolution analysis in this paper we propose and study a subgrid model for linear convection diffusion reaction equations with fractal rough coefficients the subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference we show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h 4 we also present error estimates for the modeling error in terms of modeling residuals

extremum seeking control with an adaptive dither signal we present an output feedback law that steers a fully actuated control affine system with general drift vector field to a minimum of the output function the proposed control law is based on a suitable approximation of descent directions by lie brackets in the high frequency limit a novelty of our approach is an adaptive choice of the frequency parameter in this way the problem of determining a sufficiently large frequency parameter becomes obsolete the adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift the proposed control law does not only lead to convergence into a neighborhood of the set of minima but leads to exact convergence for the case of an output function with a global minimum and no other stationary points we prove global convergence

non linear systems stability and rationality the problem of when a non linear system can be represented as a quotient of two stable non linear systems is considered attention is mainly directed toward nonlinear discrete time recursive systems where recursive means that the relationship between an input sequence and the corresponding output sequence can be expressed in terms of a finite number of recursive equations necessary and sufficient conditions are derived for the existence of a fraction representation of a recursive system where the numerator and the denominator are stable recursive systems the explicit construction of such a fraction representation is described

an adaptive finite element splitting method for the incompressible navier stokes equations we present an adaptive finite element method for the incompressible navier stokes equations based on a standard splitting scheme the incremental pressure correction scheme the presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity an a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization time discretization and a term that measures the deviation of the splitting scheme from a pure galerkin scheme the computational error numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices it is further demonstrated that the computational error of the navier stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step

a new approach to computational turbulence modeling we present a new approach to computational fluid dynamics cfd using adaptive stabilized galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest we address the basic question of computability in cfd for a given flow what quantity is computable to what tole rance to what cost we focus on incompressible newtonian flow with medium to large rey nolds numbers involving both laminar and turbulent flow features we estimate a poste riori the output of the computed solution with the output based on the exact solution to the navier stokes equations thus circumventing introducing and modeling reynolds stresses in averaged navier stokes equations our basic tool is a representation formula for th e error in the quantity of interest in terms of a space time integral of the residual of a computed solution multiplied by weights related to derivatives of the solution of an associa ted dual problem with data connected to the output we use the error representation formula to derive an a posterori error estimate combining residuals with computed dual weights which is used for mesh adaptivity in space time with the objective of satisfying a give n error tolerance with minimal computational effort we show in concrete examples that outputs such as mean values in time of drag and lift of turbulent flow around a bluff body are c omputable on a pc with a tolerance of a few percent using a few hundred thousand mesh points in space we refer to our methodology as adaptive dns les where automatically by adaptivity certain features of the flow are resolved in a direct numerical simulation dns while certain other small scale turbulent features are left unresolved in a large eddy simulation les the stabilization of the galerkin method giving a weighted least square control of the residual acts as the subgrid model in the les the a posteriori error estimate takes into account both the error from discretization and the error from the subgrid model we pay particular attention to the stability of the dual solution from i perturbations repla cing the exact convection velocity by a computed velocity and ii computational solution of the dual problem which are the crucial aspects entering by avoiding using averaged navier stokes equations including reynolds stresses a crucial observation is that the contri bution from subgrid modeling in the a posteriori error estimation is small making it possib le to simulate aspects of turbulent flow without accurate modeling of reynolds stresses

on duality based a posteriori error estimation in various norms and linear functionals for large eddy simulation we derive a posteriori error estimates for the filtered velocity field in a large eddy simulation in various norms and linear functionals the a posteriori error estimates take the form of an integral in space time of a discretization residual and a modeling residual times a dual weight the discretization residual is directly computable and the modeling residual is estimated by a scale similarity model we approximate the dual weight by solving an associated linearized dual problem numerically computational examples from transition to turbulence in couette flow are presented

practical stability and stabilization presents a practical stability result for dynamical systems depending on a small parameter this result is applied to a practical stability analysis of fast time varying systems studied in averaging theory and of highly oscillatory systems studied by sussmann and liu 1999 furthermore the problem of practically stabilizing control affine systems with drift is discussed

extremum seeking for dynamic maps using lie brackets and singular perturbations abstract we introduce a framework for the analysis of extremum seeking systems for dynamic maps based on lie bracket approximations and singular perturbation theory using the introduced framework we provide an extremum seeking regulator for a unicycle and prove its stability properties

minimum seeking for clfs universal semiglobally stabilizing feedback under unknown control directions employing extremum seeking es for seeking minima of control lyapunov function clf candidates we develop 1 the first systematic design of es controllers for unstable plants 2 a simple non model based universal feedback law that emulates in an average sense the l g v controllers for stabilization with inverse optimality and 3 a new strategy for stabilization of systems with unknown control directions as an alternative to nussbaum gain controllers that lack exponential stability lack transient performance guarantees and lack robustness to changes in the control direction the stability analysis that underlies our designs is inspired by an analysis approach synthesized in a recent work by durr stankovic and johansson which combines a lie bracket averaging result of gurvits and li with a semiglobal practical stability result under small parametric perturbations by moreau and aeyels

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

stability of the dual navier stokes equations and efficient computation of mean output in turbulent flow using adaptive dns les we discuss aspects of adaptive dns les where adaptive finite element methods are used to accurately compute chosen output from a turbulent flow with the computational power of a pc the key to this break through is i application of the general approach to adaptive error control in galerkin methods based on duality coupled with ii crucial properties of turbulent flow allowing accurate computation of mean value quantities such as drag and lift without full resolution of all scales

a suboptimal bootstrap method for structure detection of non linear output error models with application to human ankle dynamics identification of non linear systems involves estimating unknown parameters and structure detection selection of a subset of candidate terms that best describe the observed output this is a necessary procedure to compute an efficient system description which may afford greater insight into the functionality of the system or a simpler controller design for non linear systems simple output additive noise can generate multiplicative terms between the input output and noise the terms associated with noise need to be modelled to obtain unbiased parameter estimates significantly increasing the number of candidate terms to be estimated and considered in special cases it may be possible to use an output error oe model structure and the instrumental variable iv estimator to obtain unbiased parameters without modelling the noise this significantly reduces the dimensionality of the structure computation problem therefore in this paper we propose a suboptimal bootstrap structure detection sobsd algo

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

frequency domain arx model and multi harmonic frf estimators for non linear dynamic systems abstract non linear dynamic systems respond at frequencies other than the excitation frequency however standard frequency response function estimators for linear systems do not accommodate this harmonic distortion a new multi harmonic frequency response function estimator that utilizes discrete frequency models for non linear systems is introduced here the multi harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain overdetermined least mean squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators the resulting multi harmonic frequency response function models are non parametric e g vary with amplitude when linear functions are used and parametric when non linear functions are used a new sensitive indicator for experimentally characterizing non linearity is introduced

adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation in this paper we first review our recent work on a new framework for adaptive turbulence simulation we model turbulence by weak solutions to the navier stokes equations that are wellposed with respect to mean value output in the form of functionals and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output we then derive a local energy estimate for a particular finite element method which we connect to related work on dissipative weak euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions the ideas are illustrated by numerical results where we observe a law of finite dissipation with respect to a decreasing mesh size

identification of nonlinear systems using polynomial nonlinear state space models in this paper we propose a method to model nonlinear systems using polynomial nonlinear state space equations obtaining good initial estimates is a major problem in nonlinear modelling it is solved here by identifying first the best linear approximation of the system under test the proposed identification procedure is successfully applied to measurements of two physical systems

prediction error estimation algorithm for non linear output affine systems a prediction error estimation algorithm is developed for non linear discrete time systems which can be represented by the output affine difference equation model the theory of hypothesis testing is employed to select a model with the correct structure problems relating to the use of output affine models in the identification of non linear systems are discussed and a comparison with the narmax nonlinear armax model is given

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples