affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

the ii p method for estimating multivariate functions from noisy data the method for estimating an underlying smooth function of m variables x l xm using noisy data is based on approximating it by a sum of products of the form m m x m the problem is then reduced to estimating the univariate functions in the products a convergent algorithm is described the method keeps tight control on the degrees of freedom used in the fit many examples are given the quality of fit given by the method is excellent usually only a few products are enough to fit even fairly complicated functions the coding into products of univariate functions allows a relatively understandable interpretation of the multivariate fit

applying experimental design and regression splines to high dimensional continuous state stochastic dynamic programming in stochastic dynamic programming sdp with continuous state and decision variables the future value function is computed at discrete points in the state spac e interpolation can be used to approximate the values of the future value function between these discrete points however for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large statistical methods of experimental design and function estimation may be employed to overcome this curse of dimensionality in this paper we describe a method for estimating the future value function by multivariate adaptive regression splines mars fit over a discretization scheme based on orthogonal array oa experimental designs because orthogonal arrays only grow polynomially in the state space dimension our oa mars method is accurately able to solve higher dimensional sdp problems than previously possible to our knowledge the most efficient method published prior to this work employs tensor product cubic splines to approximate the future value function johnson et al 1993 the computational advantages of oa ma rs are demonstrated in comparisons with the method using tensor product cubic splines for applications of an inventory forecasting sdp with up to nine state variables computed on a small workstation in particular the storage of an adequate tensor product cubic spline for six dimensions exceeds the memory of our workstation and the run time for an accurate oa mars sdp solution would be at least an order of magnitude faster than using tensor product cubic splines for higher than six dimensions

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

fitting multivariate regression functions by interaction spline models in nonparametric multivariate regression analysis we seek methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality the original additive models approximate a regression function by additive univariate functions which overcome successfully the difficulty caused by the curse of dimensionality and have a clear interpretation but since they ignore any interactions between regressors they are not sufficiently flexible to cope with the case where interactions are of concern

a generalization of the structured singular value and its application to model validation the structured singular value spl mu gives a means of assessing the stability and performance of a system under a class of norm bounded structured perturbations this paper introduces a generalization of spl mu which guarantees stability with respect to perturbations where some elements of the perturbation structure are bounded from above and some are bounded from below this generalization is used to solve the frequency domain robust control model validation problem model validation determines whether or not there exist a model in the model set and a bounded noise signal that are consistent with the experimental input output measurements a computable convex upper bound to the generalized spl mu problem is presented and the relationship between the generalized spl mu problem and the standard spl mu and model validation problems is discussed

numerical solution of continuous state dynamic programs using linear and spline interpolation this paper demonstrates that the computational effort required to develop numerical solutions to continuous state dynamic programs can be reduced significantly when cubic piecewise polynomial functions rather than tensor product linear interpolants are used to approximate the value function tensor product cubic splines represented in either piecewise polynomial or b spline form and multivariate hermite polynomials are considered computational savings are possible because of the improved accuracy of higher order functions and because the smoothness of higher order functions allows efficient quasi newton methods to be used to compute optimal decisions the use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of hermite polynomials for the test problem and easier to program in comparison to linear interpolation use of splines in piecewise polynomial form reduced the cpu time to obtain results of equivalent accuracy by a factor of 250 330 for a stochastic

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

robust region of attraction estimation we propose a method to compute invariant subsets of the region of attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty parameter independent lyapunov functions are used to characterize invariant subsets of the robust region of attraction a branch and bound type refinement procedure reduces the conservatism we demonstrate the method on an example from the literature and uncertain controlled short period aircraft dynamics

robust performance of linear parametrically varying systems using parametrically dependent linear feedback abstract in this paper a parameter dependent control problem for linear parametrically varying lpv systems is presented sufficient conditions are given that guarantee an lpv system is exponentially stable and achieves an induced l 2 norm performance objective from the disturbance to error signals the usefulness of parameter dependent controllers is motivated from a gain scheduling viewpoint the resulting synthesis problem is reformulated into a convex optimization problem which can be solved using efficient new algorithms

gain scheduling via linear fractional transformations abstract a linear finite dimensional plant with rational state space parameter dependence is controlled using a parameter dependent controller the parameters are known to take on values in a unit ball and are known in real time the goal of control is to stabilize the parameter dependent closed loop system and provide disturbance error attenuation as measured in induced l 2 norms the approach taken uses the optimally scaled small gain theorem and solves the control synthesis problem by reformulating the existence conditions into a finite dimensional convex optimization

design of controllers for linear parameter varying systems by the gain scheduling technique abstract design of controllers for linear multi input multi output systems whose dynamics depend on a time varying parameter is studied design algorithms for the state feedback control law based on the gain scheduling technique are proposed state feedback control laws are designed so that one of the following goals is achieved i the poles of the closed loop system for all fixed parameter values are placed in a small neighborhood of a set of desired closed loop poles ii the stability of the closed loop system for all fixed parameter values guarantees that the of the system with the time varying parameter independent of the rate of change of the parameter rigorously justified and computationally appealing design algorithms are developed

on the performance of nonlinear dynamical systems under parameter perturbation we present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear odes as initial conditions and parameters are varied within compact sets over which stability is guaranteed this deviation is taken to be the integral over time of a user specified positive definite function of the difference between the trajectories for instance the l 2 norm we use sum of squares programming to obtain two polynomials which take as inputs the possibly differing initial conditions and parameters of the two families of odes and output upper and lower bounds to this transient deviation equality can be achieved using symbolic methods in a special case involving linear time invariant parameter dependent systems we demonstrate the utility of the proposed methods in the problems of model discrimination and location of worst case parameter perturbation for a single parameterised family of ode models

on efficient solutions to the continuous sensitivity equation using automatic differentiation shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation pde these shape sensitivities are described by a continuous sensitivity equation cse automatic differentiation ad can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation the approximate solution of the pde uses a spatial discretization mesh that often depends on the shape parameters therefore the straightforward application of ad introduces derivatives of the mesh there are two drawbacks to this approach first extra computational effort especially memory is used in these calculations due to mesh sensitivities second this mesh sensitivity information needs to be computed in order to obtain accurate results in this work we provide a methodology that avoids mesh sensitivities and their drawbacks by defining a modified pde on a fixed domain i e independent of the shape parameter such that ad provides the desired approximation of the cse using two examples we demonstrate significant improvement in the computational effort both in terms of floating point operations and memory requirements we explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code these changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation

a converse sum of squares lyapunov result with a degree bound although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems several fundamental questions remain unanswered in this paper we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a lyapunov function which is a sum of squares of polynomials in particular the main result states that if a system is exponentially stable on a bounded nonempty set then there exists a sum of squares lyapunov function which is exponentially decreasing on that bounded set furthermore we derive a bound on the degree of this converse lyapunov function as a function of the continuity and stability properties of the vector field the proof is constructive and uses the picard iteration our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

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

algebraic certificates of semi definiteness for polynomials over fields containing the rationals sum of squares sos decompositions for positive semidefinite polynomials are usually computed numerically using convex optimization solvers the precision of the decompositions can be improved by increasing the number of digits used in the computations but when the number of variables is greater than the length i e the minimum number of squares needed for the decomposition of the polynomial it is difficult to obtain an exact sos decomposition with the existing methods a new algorithm which works well in almost all such cases is proposed here the results of randomly generated experiments are reported to compare the proposed algorithm with those based on convex optimization

a linear algebra method to decompose forms whose length is lower than the number of variables into weighted sum of squares abstractin this paper an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables

delineating parameter unidentifiabilities in complex models scientists use mathematical modeling as a tool for understanding and predicting the properties of complex physical systems in highly parametrized models there often exist relationships between parameters over which model predictions are identical or nearly identical these are known as structural or practical unidentifiabilities respectively they are hard to diagnose and make reliable parameter estimation from data impossible they furthermore imply the existence of an underlying model simplification we describe a scalable method for detecting unidentifiabilities as well as the functional relations defining them for generic models this allows for model simplification and appreciation of which parameters or functions thereof cannot be estimated from data our algorithm can identify features such as redundant mechanisms and fast time scale subsystems as well as the regimes in parameter space over which such approximations are valid we base our algorithm on a quantification of regional parametric sensitivity that we call multiscale sloppiness traditionally the link between parametric sensitivity and the conditioning of the parameter estimation problem is made locally through the fisher information matrix this is valid in the regime of infinitesimal measurement uncertainty we demonstrate the duality between multiscale sloppiness and the geometry of confidence regions surrounding parameter estimates made where measurement uncertainty is non negligible further theoretical relationships are provided linking multiscale sloppiness to the likelihood ratio test from this we show that a local sensitivity analysis as typically done is insufficient for determining the reliability of parameter estimation even with simple non linear systems our algorithm can provide a tractable alternative we finally apply our methods to a large scale benchmark systems biology model of necrosis factor nf b uncovering unidentifiabilities

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

equilibrium independent passivity we extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium independent passivity a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point we provide a necessary input output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems an example from network stability analysis is presented which demonstrates the utility of this new definition we then proceed to numerical certification of equilibrium independent passivity using sum of squares programming finally through numerical examples we show that equilibrium independent passivity is less restrictive than incremental passivity

an existence result for polynomial solutions of parameter dependent lmis we show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set admits a branch of solutions polynomial with respect to the parameters this result is useful for studying e g parametric robustness or gain scheduling issues

sos based solution approach to polynomial lpv system analysis and synthesis problems based on sum of squares sos decomposition we propose a new solution approach for polynomial linear parameter varying lpv system analysis and control synthesis problems instead of solving matrix variables over positive cone the sos approach tries to find a suitable decomposition to verify the positiveness of given polynomials the complexity of the sos based numerical method is polynomial of the problem size and is computationally attractive this approach also leads to more accurate solutions to lpv systems than most existing relaxation methods several examples have been used to demonstrate benefits of the sos based solution approach

robust performance analysis with lmi based methods for real parametric uncertainty via parameter dependent lyapunov functions robust performance analysis for linear time invariant systems with linear fractional transformation real parametric uncertainty is considered new conditions of robust stability performance based on parameter dependent lyapunov functions are proposed the robust stability performance measures are robust pole location robust h sub spl infin performance and robust h sub 2 performance linear matrix inequality lmi based sufficient conditions for the existence of parameter dependent lyapunov functions are derived by using the quadratic separation concept the performances of the proposed conditions are compared with existing tests