h controller synthesis by j lossless coprime factorization this paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the mathcal h infty norm of the closed loop is strictly less than a given bound by the use of coprime factorizations it is shown that the mathcal h infty control problem is fundamentally related to the so called analytic systems considered by helton et al regional conference series in mathematics 68 1987 such problems admit a solution if and only if a certain j lossless factorization exists interpreted in the mathcal h infty control context this means that for a controller of the requisite type to exist the plant must admit a certain j lossless coprime factorization the full mathcal h infty synthesis problem requires that two nested j lossless factorizations exist the results are independent of whether discrete time or continuous time systems are being considered it is then shown that j lossless factorization is equivalent to the existen

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

on the perfect regulation of optimal regulators an alternative proof is provided for an earlier result of francis 1 on the perfect regulation of optimal regulators our proof shown in the present paper 1 is an interesting application of anderson s result on the spectral factorization and may be simpler and more transparent than the one in 1

j inner outer factorization j spectral factorization and robust control for nonlinear systems the problem of expressing a given nonlinear state space system as the cascade connection of a lossless system and a stable minimum phase system inner outer factorization is solved for the case of a stable system having state space equations affine in the inputs the solution is given in terms of the stabilizing solution of a certain hamilton jacobi equation the stable minimum phase factor is obtained as the solution of an associated nonlinear spectral factorization problem as an application one can arrive at the solution of the nonlinear h sub spl infin control problem for the disturbance feedforward case

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

on representing and generating kernels by fuzzy equivalence relations kernels are two placed functions that can be interpreted as inner products in some hilbert space it is this property which makes kernels predestinated to carry linear models of learning optimization or classification strategies over to non linear variants following this idea various kernel based methods like support vector machines or kernel principal component analysis have been conceived which prove to be successful for machine learning data mining and computer vision applications when applying a kernel based method a central question is the choice and the design of the kernel function this paper provides a novel view on kernels based on fuzzy logical concepts which allows to incorporate prior knowledge in the design process it is demonstrated that kernels mapping to the unit interval with constant one in its diagonal can be represented by a commonly used fuzzy logical formula for representing fuzzy rule bases this means that a great class of kernels can be represented by fuzzy logical concepts apart from this result which only guarantees the existence of such a representation constructive examples are presented and the relation to unlabeled learning is pointed out

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

on cancellation in h optimal controllers abstract this note clarifies the mechanism of pole zero cancellations that appeared in the derivation of the h optimal controller via a solution of the corresponding model matching problem the cancellable factor is explicitly expressed using chain scattering matrix descriptions a simple state space formula of the sub optimal controller is also presented based on the chain scattering forms and we show that there is a mixed sensitivity problem that can be solved by calculating a solution of only one riccati equation in each iteration

the role of the dissipation matrix in singular optimal control the optimal cost in the regular stationary linear quadratic optimal control problem is given by the maximal solution of an algebraic riccati equation this solution minimizes the rank of a certain matrix which we call the dissipation matrix the rank minimization problem is also meaningful in the singular case does it provide the optimal cost for the singular control problem this question was posed by j c willems in 1971 the answer is yes

a normal form approach to approximate input output linearization for maximum phase nonlinear siso systems a control synthesis scheme is presented for nonlinear single input single output systems which have completely unstable antistable zero dynamics the approach is similar in spirit to linear approaches for nonminimum phase systems and involves the derivation of an input output linearizing controller for a suitably defined nonlinear minimum phase approximation to the original system the linearizing controller achieves an approximately linear input output response and internal stability

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

a j spectral factorization approach to control necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with mathcal h infty control j control are derived using j spectral factorization theory the existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled j spectral factorization problems with the second factor providing a parametrization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of nonnegative definite stabilizing solutions to two indefinite algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given a virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework and no additional auxiliary riccati equations are required

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

computation of inner outer factorization of rational matrices we propose a new numerically reliable computational approach to determine the inner outer factorization of a rational transfer matrix g of a linear descriptor system in contrast to existing computationally involved one shot methods which require the solution of riccati or generalized riccati equations the new approach relies on an efficient recursive zeros dislocation technique the resulting inner and outer factors always have minimal order descriptor representations the approach proposed is completely general being applicable whenever g is proper strictly proper or not or of full column row rank or not

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

factorization and feedback stabilization for nonlinear systems we establish the equivalence of internal input out stability for two feedback configurations of a nonlinear time varying plant p for which a related plant g is assumed to have a factorization g r with both r and r 1 incrementally stable this extends a factorization principle for stabilizability previously given only for the linear time invariant case as an application of a special case we recover a version of the youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature we use degree theory to parametrize a collection of solutions of the h control problem for the case of a 1 gain stable or lossless plant in the case of a plant g having a j inner outer factorization this last result combined with the above mentioned factorization principle leads to results on the h control problem for p

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

spectral and inner outer factorizations for discrete time systems the spectral and inner outer factorization problems are solved for discrete time systems with transfer matrices of arbitrary rank simple and explicit state space formulas expressed in terms of the stabilizing solution to the discrete time algebraic riccati system are derived the results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered