strong convergence theorem for nonexpansive mappings on star shaped sets in hilbert spaces abstract in this paper using the concept of attractive points of a nonlinear mapping we obtain a strong convergence theorem of halpern s type 7 for nonexpansive mappings in a star shaped subset of a hilbert space using this result we obtain a new fixed point theorem and a strong convergence theorem which generalizes wittmann s theorem 26

comparison between h sub 2 and h sub infinity controllers the h sub infinity performances of a controller designed by h sub 2 methodology and a controller designed by h sub infinity methodology are quantitatively compared it is shown that the ratio of the h sub infinity performance of the two is no greater than 2k for the one block problem where k is the total number of right half plane transmission zeros for two subsystems namely the subsystem between the disturbance and the measurement and the subsystem between the control signal and the controlled output it is also shown that there are systems which can give a ratio arbitrarily close to the upper bound however those systems are of a pathological type some possible generalizations to two and four block problems are discussed

optimal interpolation for linear stochastic systems let y satisfying dy cxdt ddw y 0 0 represent the noisy observation of the gauss markov process x generated by dx axdt bdw x 0 x 0 on the finite interval 0 t in this paper we solve the following interpolation problem determine the best least squares estimate of x t t in 0 t given the increments of y on the intervals 0 t 1 and t 2 t where 0 t 1 t 2 t and the statistics of x 0 several alternative expressions for the optimal interpolator are derived some of which are phrased in terms of two kalman bucy estimates as a by product we also solve the interpolation problem for the missing increments of y

stochastic h2 h control with x u v dependent noise finite horizon case in this paper the finite horizon mixed h 2 h control problem is studied for the systems governed by ito type nonlinear stochastic differential equations with state control and external disturbance dependent noise it is shown that the mixed h 2 h control under consideration is associated with the four cross coupled hamilton jacobi equations a discrete approximation algorithm is presented for solving the coupled generalized differential riccati equations arising from linear stochastic h 2 h control a necessary condition and a sufficient condition for the existence of the finite horizon stochastic h 2 h control are derived in particular some previous results on deterministic and stochastic h 2 h control can be viewed as corollaries of our main theorems

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

an improved path following method for mixed h _ 2 h _ infty controller design among the few methods available to solve bilinear matrix inequalities bmis occurring in control design the path following method published some years ago appears to be one of the best ones as far as linearization methods are concerned however few details are given in the literature about its implementation and limits in this technical note this method is applied to the design of mixed h 2 h infin controllers with full details of the algorithm and some improvements over the one which has been published a few years ago the results obtained with the numerical example given in that same publication as well as with some other examples are compared with those given by other methods including a direct bmi solving program

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

h 2 optimal control with an h constraint the state feedback case in this paper we consider a mixed hz optimal control problem it is assumed that the plant as well as the feedback controller are finite dimensional and linear time invariant and that the plant state is available for feedback more specifically among all the state feedback controllers that minimize the he norm of a closed loop transfer matrix we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed h norm bound on some other closed loop transfer matrix when these conditions are met the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an he norm performance measure subject to an h norm constraint we also give state space formulae for computing the solutions some easily checkable sufficient conditions for the existence of solutions are given finally we give an example in which all solutions to the constrained optimization problem are necessarily dynamic i e there is no static gain solution even though plant state is available for feedback

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

mixed h 2 h autopilot design of bank to turn missiles using fuzzy basis function networks this paper presents a novel mixed h2 h autopilot design for bank to turn missiles based on sliding control theory the proposed autopilot utilizes a fuzzy basis function network to approximate the unknown nonlinear functions of bank to turn missile dynamics enabling a mixed h2 h controller with weight updating laws to be applied to attenuate the effects of approximation errors and external disturbances moreover the mixed h2 h autopilot design can minimize h2 performance criteria under a desired h disturbance rejection constraint unlike general mixed h2 h control problems the proposed h2 h autopilot with self tuning fuzzy basis function network can avoid solving coupled riccati equations by adequately selecting diagonal weighting matrices in the h2 performance criterion and h constraint therefore the proposed h2 h control approach can be implemented efficiently computer simulation results are presented to illustrate the effectiveness of the proposed mixed h2 h autopilot for btt missiles

robust controller design for stochastic nonlinear systems via convex optimization this paper presents a new nonlinear control design framework called convex optimization based stochastic steady state tracking error minimization cvstem for a class of ito stochastic nonlinear systems including stochastic lagrangian systems this state feedback controller design involves the use of multiple state dependent coefficient sdc models of a nonlinear system equation and computes its controller parameters by solving a convex optimization problem in order to minimize an upper bound of the steady state tracking error this problem is shown to be equivalent to the original nonlinear optimization problem of minimizing the upper bound the exponential stability and robustness properties due to the state dependent riccati inequality are proven using stochastic contraction analysis discrete time stochastic incremental contraction analysis with respect to a state and time dependent metric is also presented along with its explicit connection to continuous time contraction analysis a result of simulation is used to validate the superiority of the proposed controller compared to a known nonlinear controller a pid controller and an h infinity controller

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

synthesis of a robust broadband duct anc system using convex programming approach a robust active controller using spatially feedforward structure is proposed for broadband attenuation of noise in ducts to meet the requirements of performance and robust stability in the presence of plant uncertainties an h2 cost function and an h constrain are employed in the synthesis of the controller the design is then converted into a convex programming problem using q parametrization and frequency discretization an optimal controller that satisfies the quadratic cost functions and linear inequality constraints can be found by sequential quadratic programming the optimal controller was implemented via a digital signal processor dsp and verified by experiments experiment results showed that the system attained 16 5 db maximal attenuation and 5 9 db total attenuation in the frequency band 200 600 hz

h guaranteed cost control for uncertain continuous time linear systems abstract this paper deals with the h guaranteed cost control problems for continuous time uncertain systems it consist of the determination of a stabilizing state feedback gain which imposes on all possible closed loop models an h norm upper bound 0 assuming that the uncertain domain is convex bounded and the uncertain system is quadratic stabilizable with disturbance attenuation it is shown how to determine by means of a convex programming problem the global minimum of as a particular and important case for precisely known linear systems the last problem reduces to the classical h optimal control problem the results follow from the definition of a special parameter space on which the above mentioned problems are convex

approximating common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space abstract in this paper we obtain some fundamental properties for nonspreading mappings in a hilbert space further we study the approximation of common fixed points of nonexpansive mappings and nonspreading mappings in a hilbert space

two sided linear prediction and reconstruction of a stationary time series the problem of two sided linear prediction and reconstruction of a stationary process y on a finite interval is solved

static output feedback controllers stability and convexity the main objective of this paper is to solve the following stabilizing output feedback control problem given matrices a b sub 2 c sub 2 with appropriate dimensions find if one exists a static output feedback gain l such that the closed loop matrix a b sub 2 lc sub 2 is asymptotically stable it is known that the existence of l is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs hence this paper provides a new sufficient but not necessary condition for the solvability of the above stabilizing output feedback control problem as a natural extension we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index some examples borrowed from the literature are solved to illustrate the theoretical results

weak and strong convergence theorems for nonspreading mappings in hilbert spaces abstract in this paper we first obtain a weak mean convergence theorem of baillon s type for nonspreading mappings in a hilbert space further using an idea of mean convergence we prove a strong convergence theorem for nonspreading mappings in a hilbert space

probability one homotopy algorithms for solving the coupled lyapunov equations arising in reduced order h2 h modeling estimation and control optimal reduced order modeling estimation and control with respect to combined h 2 h criteria give rise to coupled lyapunov and riccati equations to develop reliable numerical algorithms for these problems this paper focuses on the coupled lyapunov equations which appear as a subset of the synthesis equations in particular this paper systematically examines the requirements of probability one homotopy algorithms to guarantee global convergence homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family the main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled lyapunov equations and variations based upon canonical forms these results are essential to the probability one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms

a new algorithm for two finite families of demicontractive mappings and equilibrium problems in this paper we introduce a new algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two finite families of demicontractive mappings the strong convergence theorem of the proposed algorithm is established under some suitable control conditions in a real hilbert space our result generalizes several recent results in the current literature

a duality based approach to the multiobjective h 2 h optimisation problem in this article a duality approach to multiobjective h 2 h problems is pursued in which real rational para hermitian multipliers and real valued ones are associated to h and as usual h 2 constraints respectively it is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective h 2 h problem to compute lower bounds on the latter and the corresponding approximate solutions to the original problem the maximisation of the dual functional over linearly parameterised finite dimensional classes of real rational multipliers is shown to be equivalent to semi definite linear programming problems once the optimal multipliers in such a class are obtained the corresponding approximate solutions can be computed from an unconstrained h 2 problem iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem this is done on the basis of locally incr

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

brief trading robustness with optimality in nonlinear control the mixed h 2 h control problem is studied for a class of nonlinear systems a procedure providing a sub optimal solution to this problem is discussed and analyzed in detail the sub optimal state feedback control law can be obtained solving two hamilton jacobi equations and a simple one parameter static minimization problem a simple one dimensional example is discussed in detail and applications to the optimal robust stabilization of the angular velocities of a rigid body are included

multiobjective robust control of lti systems subject to unstructured perturbations abstract we consider a multiobjective robust controller synthesis problem for an lti system subject to unstructured perturbations our design specifications include robust stability robust performance h 2 norm bounds and time domain bounds output and command input peak we derive sufficient conditions based on a single quadratic lyapunov function for the existence of an lti controller such that the closed loop system satisfies all specifications simultaneously these conditions can be numerically checked using finite dimensional convex optimization over lmis associated with a two dimensional search when considering only a subset of the specifications we recover previous results from the literature such as those obtained in mixed h 2 h control

optimal controller synthesis with d stability abstract in this paper we consider the problem of finding controllers which place the eigenvalues of the closed loop system matrix in a prespecified circular region in the left half plane and minimize an associated quadratic cost function we give solutions to both state feedback and output feedback synthesis problems

mixed h sub 2 h sub spl infin control of discrete time markovian jump linear systems in this paper we consider the mixed h sub 2 h sub spl infin control problem for the class of discrete time linear systems with parameters subject to markovian jump linear systems it is assumed that both the state variable and the jump variable are available to the controller the transition probability matrix may not be exactly known but belongs to an appropriate convex set for this controlled discrete time markovian jump linear system the problem of interest can be stated in the following way find a robust with respect to the uncertainty on the transition markov probability matrix mean square stabilizing state and jump feedback controller that minimizes an upper bound for the h sub spl infin norm under the restriction that the h sub spl infin norm is less than a prespecified value spl delta the problem of the determination of the smallest h sub spl infin norm is also addressed we present an approximate version of these problems via linear matrix inequality optimization

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

robust h 2 control based on dynamic multipliers in this article upper bounds on the worst case h 2 performance index relative to structured feedback perturbations are considered which are based on the minimisation of dual lagrangean functionals over linearly parametrised finite dimensional classes of dynamic multipliers it is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and linear matrix inequality lmi constraints an iterative scheme is suggested to generate linearly parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained as illustrated by two simple numerical examples finally with a view to synthesis procedures based on d k iterations relative to multipliers and controllers it is shown that the minimisation of the upper bounds corresponding to given multipliers with respect to linearly parametrised classes of youla parameters can also be cast as linear cost lmi constraint problems

synthesis of positive real spl hscr sub 2 controllers this paper describes a synthesis procedure for strictly positive real observer based controllers in particular we address the standard spl hscr sub 2 optimal control problem with the additional constraint on the strict positive realness of the controller a suboptimal version of this problem is shown to be a convex optimization problem and a linear matrix inequality based solution is provided one example illustrates the main results

suboptimal robust mixed h2h controller design for uncertain descriptor systems with distributed delays abstract this paper is concerned with the design problem of the optimal robust mixed h 2 h controllers for descriptor systems with time varying uncertainties and distributed delays in our method no decomposition of the coefficient matrix is needed the designed controller can guarantee regularity nonimpulsiveness and internal stability of the closed loop system and simultaneously minimizes an upper bound of the h2 performance measure satisfying an h norm bound constraint the control design problem is transformed into the solvability of a set of strict lmis which can be effectively solved by using interior point algorithms

derivation of the maximum entropy h controller and a state space formula for its entropy the problem of maximizing the entropy of a stabilized closed loop system is solved the solution exploits the parametrization of all closed loop systems that meet an h norm bound the central solution of this set is shown to maximize the entropy at infinity and for this case a particularly simple formula is derived for the maximum value of the entropy

brief on the strong stabilization and stable h controller design problems for mimo systems a state space based design method is developed to construct strongly stabilizing controllers for a class of mimo plants the problem is reduced to solving an algebraic riccati equation are which appears in the solution of a two block h problem strongly stabilizing controllers designed here are of the same order as the plant two sets of finite dimensional strongly stabilizing controllers are parameterized in terms of linear fractional transformations further optimization guidelines are given to find stable h controllers from these parameterizations stability robustness margins with respect to coprime factor uncertainties are determined for the controllers designed from the proposed method

mixed h sub 2 h sub infinity control a convex optimization approach the problem of finding an internally stabilizing controller that minimizes a mixed h sub 2 h sub infinity performance measure subject to an inequality constraint on the h sub infinity norm of another closed loop transfer function is considered this problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint both the state feedback and output feedback problems are considered it is shown that in the state feedback case one can come arbitrarily close to the optimal even over full information controllers mixed h sub 2 h sub infinity performance measure using constant gain state feedback moreover the state feedback problem can be converted into a convex optimization problem over a bounded subset of n n and n q where n and q are respectively the state and input dimensions real matrices using the central h sub infinity estimator it is shown that the output feedback problem can be reduced to a state feedback problem in this case the dimension of the resulting controller does not exceed the dimension of the generalized plant

a generalization in mixed h 2 h control with state feedback abstract in this note we consider a mixed h 2 h control problem with a performance cost which combines the h 2 norm cost and the so called auxiliary cost the main result is to give a formula for the optimal cost and a nearly optimal controller it is assumed that the state of the plant is available for feedback

ergodic theorems for hybrid sequences in a hilbert space with applications abstract in this paper we introduce the notion of generalized hybrid sequences extending the notion of nonexpansive sequences introduced and studied in our previous work djafari rouhani 1981 1990 1990 1997 2002 2004 2002 2 3 4 5 6 7 8 and prove ergodic and convergence theorems for such sequences in a hilbert space h subsequently we apply our results to prove new fixed point theorems for generalized hybrid mappings first introduced in kocourek et al 2010 14 takahashi and takeuchi 2011 20 defined on arbitrary nonempty subsets of h

families of solutions of matrix riccati differential equations the j c willems coppel shayman geometric characterization of solutions of the algebraic riccati equation are is extended to asymmetric riccati differential equations with time varying coefficients the coefficients do not need to satisfy any definiteness periodicity or system theoretic condition more precisely given any two solutions x 1 t and x 2 t of such equation on a given interval t 0 t 1 we show how to construct a family of solutions of the same equation of the form x t i pi t x 1 t pi t x 2 t where pi is a suitable matrix valued function even when specialized to the case of x 1 and x 2 equilibrium solutions of a symmetric equation with constant coefficients our results considerably extend the classical ones as no further assumption is made on the pair x 1 x 2 and on the coefficient matrices

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

stable controller synthesis for linear time invariant systems this paper is concerned with the problem of suboptimal stable mixed h 2 h x control for linear time invariant systems the designed controller with the same order as that of the considered system is required to satisfy a prescribed h x performance bound or a prescribed degree of stability by reducing the stable controller synthesis problem to a multiobjective state feedback control problem for two different state models sufficient conditions for the solvability of the considered problem are given in terms of solutions to algebraic riccati equations and matrix inequalities an lmi based iterative algorithm is developed to solve the stable controller synthesis problem the proposed algorithm is shown to be convergent examples are given to illustrate the proposed methods

weak and strong convergence theorems for nonspreading type mappings in hilbert spaces abstract weak and strong convergence theorems are proved in real hilbert spaces for a new class of nonspreading type mappings more general than the class studied recently in kurokawa and takahashi y kurokawa w takahashi weak and strong convergence theorems for nonspreading mappings in hilbert spaces nonlinear anal 73 2010 1562 1568 we explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of halpern s type for our class of mappings and hence resolved in the affirmative an open problem posed by kurokawa and takahashi in their final remark for the case where the mapping t is averaged

strong convergence theorems for 2 generalized hybrid mappings in hilbert spaces abstract in this paper we prove a strong convergence theorem of halpern s type for 2 generalized hybrid mappings in a hilbert space we also deal with strong convergence theorems by hybrid methods for these nonlinear mappings in a hilbert space

existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces a class of nonlinear operators in banach spaces is proposed we call each operator in this class a firmly nonexpansive type mapping this class contains the classes of firmly nonexpansive mappings in hilbert spaces and resolvents of maximal monotone operators in banach spaces we study the existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

relations between maximum entropy h control and combined h lqg control abstract the maximum entropy h control problem and a combined h lqg control problem are presented in a unified way and shown to be equivalent in the case of a full order controller and equalized h lqg weights the result links two separate approaches which have distinctly different methods of solution only two decoupled algebraic riccati equations are needed to state the controller maximum value of the entropy and minimum value of the auxiliary cost

the algorithms for studying the trade offs between h attenuation and uncertainty tolerance uncertainty tolerance and h attenuations are two important concerns in feedback control design it is therefore important to develop effective methods to compute the trade off between these two objectives using a quasiconvex optimization approach this paper develops algorithms for two problems in this regard 1 given the uncertainty size of the system compute the minimum h norm bound of the closed loop system reachable by state feedback control and 2 given the required maximum acceptable h norm bound of the closed loop system compute the maximum uncertainty size which can be tolerated when state feedback control is applied a numerical example is included to show the trade off between uncertainty tolerance and h attenuations and the effectiveness of the algorithms