extending the class of solvable potentials i the infinite potential well with a sinusoidal bottom this is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems this is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal expanding the wave function in this basis makes the wave equation equivalent to a three term recursion relation for the expansion coefficients consequently finding solutions of the recursion relation is equivalent to solving the original problem doing so results in a larger class of solvable potentials the usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials giving the well known energy spectra and the corresponding wave functions moreover some of the new solvable problems show evidence of a klauder like phenomenon in the present work we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape

near optimum steady state regulators for stochastic linear weakly coupled systems abstract this paper presents an approach to the decomposition and approximation of the linear quadratic gaussian estimation and control problems for weakly coupled systems the global kalman filter is decomposed into separate reduced order local filters via the use of a decoupling transformation a near optimal control law is derived by approximating the coefficients of the truly optimal control law the order of approximation of the optimal performance is o en where n is the order of approximation of the coefficients a real world power system example demonstrates the failure of o e2 and o e4 approximations and the necessity for the existence of the o en theory the proposed method produces the reduction in both off line and on line computational requirements and leads to convergence under mild assumptions in addition only low order systems are involved in algebraic calculations and no analyticity requirement a standard assumption for the power series method is imposed on system coefficients

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

guaranteed level γ ℋ control in uncertain linear systems via linear matrix inequalities in this paper we solve guaranteed level control problems in uncertain linear systems in time varying systems we provide a time varying nonlinear differential equation for these problems which can be solved via differential equation solvers in time invariant systems we present linear matrix inequalities lmis which allow us to overcome the limitations of the existing algebraic riccati equation are formulations where one can hardly find their solutions since lmi problems are always convex one can solve them via convex optimization techniques such as interior point methods and can easily solve them even if additional constraints are considered as a result we directly handle the optimal guaranteed level control problem as well as a guaranteed level control problem with the maximal stability margin finally we present size reduced lmis

bounds on the joint and generalized spectral radius of the hadamard geometric mean of bounded sets of positive kernel operators abstract let 1 m be bounded sets of positive kernel operators on a banach function space l we prove that for the generalized spectral radius and the joint spectral radius the inequalities 1 1 m m 1 m 1 2 m 1 m 1 1 m m 1 m 1 2 m 1 m hold where 1 1 m m 1 m denotes the hadamard schur geometric mean of the sets 1 m

linear control of nearly singularly perturbed hydropower plants abstract in this paper we consider a special class of linear control systems represented by the standard singularly perturbed system matrix and with the control input matrix having three different nonstandard forms many real systems such as hydropower plants systems with only few actuators possess the control structure studied in this paper the obtained results are quite simplified comparing to the standard singularly perturbed control systems and in one case the optimal solution of the algebraic riccati equation is completely determined in terms of the reduced order algebraic lyapunov equations the proposed method is successfully applied to the reduced order design of optimal controllers for the real hydropower plant of the serbian power system

a simple criterion for global stabilizability of a class of nonlinear singularly perturbed systems in this paper a simple criterion is proposed for the global stabilizability of a class of nonlinear singularly perturbed systems our approach is based on the global nonlinear controller canonical transformation and the integral manifold method a globally stabilizing feedback control is also proposed such that the pre specified design manifold becomes an exact slow integral manifold and the trajectories of the closed loop system starting from any initial states are steered along the integral manifold to the origin

estimating the plant state from the compensator state for the system dot w sw where w x over z this note states and proves necessary and sufficient conditions under which a z is an estimate of a linear function of x i e z estimates tx or b a linear function of z is an estimate of x i e vz estimates x the mapping t has been used previously by the authors to compute the estimator controller form of a compensator whose state vector z had the same dimension as the plant state vector x

on the relation between additive and multiplicative decompositions of rational matrix functions in this paper a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented various extensions of the positive real lemma as well as of other classical factorization results both in the continuous and discrete time cases are established in the case of square factorizations the map between solutions to an asymmetric algebraic riccati equation and pairs of factors is shown to be a homeomorphism in this framework we also derive a geometric characterization of non square factorizations

the recursive reduced order numerical solution of the singularly perturbed matrix differential riccati equation under stability observability conditions imposed on a singularly perturbed system an efficient numerical method for solving the corresponding matrix differential riccati equation is obtained in terms of the reduced order problems the order reduction is achieved via the use of the chang transformation applied to the hamiltonian matrix of a singularly perturbed linear quadratic control problem an efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the chang transformation

parallel optimal control for weakly coupled nonlinear systems using successive galerkin approximation this technical note presents a new algorithm for the closed loop parallel optimal control of weakly coupled nonlinear systems with respect to performance criteria using the successive galerkin approximation sga by using the weak coupling theory the optimal control law can be obtained from two reduced order optimal control problems in parallel but the resulting problem is difficult to solve for nonlinear systems in order to overcome the difficulties inherent in the nonlinear optimal control problem the parallel optimal control laws are constructed in terms of the approximated solutions to two independent hamilton jacobi bellman equations using the sga method one of the purposes of this note is to design the closed loop parallel optimal control law for the weakly coupled nonlinear systems using the sga method the second is to reduce the computational complexity when the sga method is applied to the high order weakly coupled systems

reduced order solution to the finite time optimal control problems of linear weakly coupled systems the optimal solution to the finite time optimal control problem of weakly coupled linear systems is found in terms of completely decoupled reduced order differential equations for both closed loop and open loop control this is achieved through the use of the decoupling transformation that block diagonalizes the hamiltonian matrix of the weakly coupled linear quadratic control problem the convergence to the optimal solution is pretty rapid the proposed technique is very well suited for parallel computations

optimal control of weakly coupled bilinear systems abstract the optimization of the time invariant bilinear weakly coupled system with a quadratic performance criterion is considered a sequence of linear state and costate equations is constructed such that the open loop solution of the optimization problem is obtained in terms of the reduced order subsystems this leads to a reduction in the size of the required computations and allows parallel processing of information the near optimal closed loop control is obtained in the form of a linear feedback law with the feedback gains calculated from two reduced order independent time varying linear quadratic optimal control problems

convergence of the time invariant riccati differential equation and lq problem mechanisms of attraction the nature of the attraction of the solution of the time invariant matrix riccati differential equation towards the stabilizing solution of the algebraic riccati equation is studied this is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis various aspects of this convergence are analysed by displaying explicit mechanisms of attraction and connections are made with the literature the analysis ultimately shows the exponential nature of the convergence of the solution of the riccati differential equation and of the related finite horizon lq optimal state and control trajectories as the horizon recedes computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite horizon lq problem by the solution of an infinite horizon lq problem

computing the estimator controller form of a compensator a technique is presented for finding the estimator and controller gains to implement a desired compensator in estimator controller form the technique is derived by mapping the states of the compensator to those of the estimator controller via a linear transformation using the technique involves finding the real invertible solutions of a non symmetric steady state matrix riccati equation the ricatti equation may have zero two or more real invertible solutions from each solution a set of estimator and controller gains can be computed multiple solutions correspond to the freedom to choose which closed loop system poles are associated with the controller and which are associated with the estimator this technique can be applied to multi input multi output systems in an example the technique is applied to the despin control system of a dual spin satellite

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

singular perturbation modelling of large scale systems with multi time scale property the problem is investigated of singular perturbation modelling which plays an important role in cases of the application of the singular perturbation method to the design and analysis of large scale systems in both continuous and discrete time versions modelling concepts based on structural properties are studied and computer oriented algorithms are developed the results of two time scale systems are extended to multi time scale cases furthermore in order to verify the validity of these studies some numerical examples are given

near optimum regulators for stochastic linear singularly perturbed systems this paper presents a new approach to the decomposition and approximation of linear quadratic gaussian estimation and control problems for singularly perturbed systems the kalman filter is decomposed into separate slow mode and fast mode filters via the use of a decoupling transformation a near optimal control law is derived by approximating the coefficients of the optimal control law the order of approximation of the optimal performance is 0 mu n where n is the order of approximation of the coefficients

decoupling transformation for weakly coupled linear systems the non singular transformation that completely decouples a weakly coupled linear system under non restrictive conditions is defined the transformation matrices are obtained from two algebraic matrix equations algorithms that efficiently generate solutions to these equations are derived the proposed transformation also completely decouples the corresponding lyapunov matrix differential equation

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

on the bilinear transformation of lqg balanced realizations abstract this contribution shows that some useful properties of lqg balanced realizations of continuous time systems are preserved by applying bilinear transformations this fact suggests that lqg balancing of discrete systems and its approximation can be performed by using the typical schemes successfully adopted for continuous time systems in the paper the approximation of discrete linear time invariant systems by using the singular perturbation approximation approach for lqg balanced realizations is analyzed showing its usefulness in approximating discrete models including unstable ones for closed loop aims

discrete time optimal regulator with closed loop poles in a prescribed region this paper is concerned with the problem of designing discrete time optimal control systems with closed loop poles in a prescribed region of stability first by utilizing the property of the riccati equation with q being zero we develop a method for allocating poles in a disc with centre at the origin of the complex plane and with radius less than 1 secondly we deal with pole placement in a disc which is in the unit disc and also contacts the point 1 j0 of the complex plane to this end a bilinear transformation and continuous time regulator results are employed in each case the radius of the disc can be specified as a design parameter and the weighting matrices of the performance index are obtained to fulfil the desired pole allocations the design procedures are also illustrated by numerical examples

simple method for solving the constant gains of kalman filters with single output a simple method is described for the calculation of the optimal gains for steady state kalman filters in continuous and discrete time systems with single output the procedure consists of comparing the optimal closed loop characteristic polynomial with a determinant expansion containing the unknown constant gains it is then shown that for the case when a computer program is not used for finding the roots of a polynomial the optimal closed loop characteristic polynomial in the discretetime problem is not as readily obtainable as that in the continuous time problem even though the system under consideration is of lower order this drawback however is shown to be overcome by invoking the bilinear transformation

numerical fixed point solution for near optimum regulators of linear quadratic gaussian control problems for singularly perturbed systems the fixed point method is developed for obtaining an efficient numerical solution of the linear quadratic gaussian problem for singularly perturbed systems it is shown that each iteration step improves the accuracy by an order of magnitude that is the accuracy 0 ek can be obtained by performing only k 1 iterations in addition only low order systems are involved in algebraic manipulations and no analyticity requirements are imposed on the system coefficients

the polynomial approach to the lq non gaussian regulator problem a new approach for the solution of the regulator problem for linear discrete time dynamical systems with non gaussian disturbances is proposed this approach generalizes a previous result concerning the definition of the quadratic optimal regulator it consists of the definition of the polynomial optimal algorithm of an order spl nu for the solution of the linear quadratic non gaussian stochastic regulator problem for systems with partial state information the validity of the separation principle has also been proved in this case numerical simulations show the high performance of the proposed method with respect to the classical linear regulation techniques

exact decomposition of the algebraic riccati equation of deterministic multimodeling optimal control problems in this paper we show how to exactly decompose the algebraic riccati equations of deterministic multimodeling in terms of one pure slow and two pure fast algebraic riccati equations the algebraic riccati equations obtained are of reduced order and nonsymmetric however their o spl epsiv perturbations where spl epsiv spl par sub spl epsiv 2 sup spl epsiv 1 spl par and spl epsiv sub 1 spl epsiv sub 2 are small positive singular perturbation parameters are symmetric the newton method is perfectly suited for solving the nonsymmetric reduced order pure slow and pure fast algebraic riccati equations since excellent initial guesses are available from their o spl epsiv perturbed reduced order symmetric algebraic riccati equations that can be solved rather easily the proposed decomposition scheme might facilitates new approaches to multimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used

new method for optimal control and filtering of weakly coupled linear discrete stochastic systems abstract the algebraic regulator and filter riccati equations of weakly coupled discrete time stochastic linear control systems are completely and exactly decomposed into reduced order continuous time algebraic riccati equations corresponding to the subsystems that is the exact solution of the global discrete algebraic riccati equation is found in terms of the reduced order subsystem nonsymmetric continuous time algebraic riccati equations in addition the optimal global kaiman filter is decomposed into local optimal filters both driven by the system measurements and the system optimal control inputs as a result the optimal linear quadratic gaussian control problem for weakly coupled linear discrete systems takes decomposition and parallelism between subsystem filters and controllers

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

some spectral norm inequalities on hadamard products of nonnegative matrices abstract let a and b be nonnegative square matrices of the same order denote by and the spectral norm and the spectral radius respectively we prove the following inequalities a b a a 1 2 b b 1 2 a b 1 2 a t b b t a 1 2 a t b a t b a t b where denotes the hadamard product this interpolates the inequality a b a t b due to huang some spectral norm inequalities for an arbitrarily finite number of nonnegative square matrices are also obtained which refine some other results of huang

optimal reduced order solution of the weakly coupled discrete riccati equation the optimal solution of the weakly coupled algebraic discrete riccati equation is obtained in terms of a reduced order continuous type algebraic riccati equation via the use of a bilinear transformation the proposed method has a rate of convergence of o epsilon sup 2 where epsilon represents a small coupling parameter a real world physical example a chemical plant model demonstrates the efficiency of the proposed method simulation results obtained using a package for a computer aided control system are presented for this specific real world example the algorithm perfectly matches the presented theory since convergence with an accuracy of 10 sup 4 is achieved after nine iterations i e 0 68 sup 18 10 sup 4

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

geometric aspects of the riccati difference equation in the nonsymmetric case abstract this paper provides a collection of geometric results concerning the nonsymmetric riccati difference equation it gives representation formulas which express all the solutions in terms of a suitable number of known ones further results include geometric properties of the difference of two solutions and the generalization of some of the author s earlier results to the nonsymmetric case

invariant manifolds and projective combinations of solutions of the riccati differential equation abstract in this paper we show how families of solutions of the general riccati differential equation rde can be generated via projective combinations of a given number of reference solutions our approach is based upon the extension of the domain of the equation to the grassmannian manifold and the application of the radon lemma in this context we briefly discuss the relevance of our results to the study of the invariant manifolds of the equation and compare them to existing results concerning representation formulas for solutions of rde the results of the paper have been motivated by the recent characterization of solutions of the rde given in m pavon d d alessandro families of solution of matrix riccati differential equations siam j control optim 35 1 1997 194 204 which extends the classical results on the algebraic riccati equation due to willems coppel and shayman

robust control with pole shifting via performance index modification robust controllers that ensure closed loop eigenvalue placement are designed for linear systems with parametric uncertainties by using an appropriate state weighting matrix choice and a relative stability degree an lmi based algorithm is proposed to provide guaranteed cost controllers that ensure i the closed loop nominal eigenvalues shifting by a pre specified amount ii a prescribed degree of stability for all system uncertainties of a given class a certain level of closed loop performance is also ensured by minimizing the associated guaranteed cost

balanced canonical forms for minimal systems a normalized coprime factor approach abstract canonical forms are derived for the set of minimal systems of given order from a canonical form for a class of coinner transfer functions one of these canonical forms is in terms of so called riccati balanced coordinates the application of this work to model reduction is discussed

convex invertible cones of matrices a unified framework for the equations of sylvester lyapunov and riccati abstract convex cones of matrices which are closed under matrix inversion were defined and various connections with the algebraic lyapunov equation of general inertia were studied in n cohen i lewkowicz linear algebra and its appl 250 1997 105 131 here in the price of doubling the size of the matrices involved we introduce a unified framework for the equations of sylvester lyapunov and riccati this enables one to extend the convex invertible cone structure to all these three equations and explore related properties in particular the use of the matrix sign function for solving these equations is examined

supersymmetry of tridiagonal hamiltonians a positive semi definite hamiltonian h that has a tridiagonal matrix representation in a basis set allows a definition of forward and backward shift operators that can be used to define the matrix representation of its supersymmetric partner hamiltonian h with respect to the same basis we find explicit relationships connecting the matrix elements of both hamiltonians we present a method to obtain the orthogonal polynomials in the eigenstate expansion problem attached to h starting from those polynomials arising in the same problem for h this connection is established by using the notion of kernel polynomials we apply the obtained results to two known solvable models with different kinds of spectrum

optimal quadratic solution for the non gaussian finite horizon regulator problem in this paper the problem of the optimal quadratic regulator for non gaussian discrete time stochastic systems with a quadratic cost function is considered the main result here obtained is that such optimal control can be derived from the classical lqg solution by substituting the linear filtering part with a quadratic optimal filter numerical results show high performance of this method

bounds on the generalized and the joint spectral radius of hadamard products of bounded sets of positive operators on sequence spaces abstract recently audenaert 2010 2 horn and zhang 2010 15 huang 2011 16 and schep 2011 22 23 proved inequalities between the spectral radius of hadamard product denoted by of finite and infinite non negative matrices that define operators on sequence spaces and the spectral radius of their ordinary matrix product we extend these results to the generalized and the joint spectral radius of bounded sets of such operators moreover we prove new inequalities even in the case of the usual spectral radius of non negative matrices in particular we prove that a b 1 2 a a b b ab ab 1 4 ba ba 1 4 ab and a b 1 2 ab ba ab ab 1 4 ba ba 1 4 ab we also obtain related results in max algebra

parallel reduced order controllers for stochastic linear singularly perturbed discrete systems an approach to the decomposition and approximation of linear quadratic gaussian control problems for singularly perturbed discrete systems at steady state is presented the global kalman filter is decomposed into separate reduced order local filters through the use of a decoupling transformation a near optimal control law is derived by approximating coefficients of the optimal control law the proposed method allows parallel processing of information and reduces offline and online computational requirements a real world example demonstrates the efficiency of the proposed method

geometric properties and invariant manifolds of the riccati equation this short paper considers the general riccati matrix differential equation it reviews and extends results on the characterization and existence of equilibrium solutions establishes that the riccati equation has at most one stable equilibrium solution as t rightarrow infty or t rightarrow infty and confines the region of attraction to this unique stable equilibrium solution by identifying and utilizing linear and nonlinear invariant manifolds of the riccati equation

spectral radius of hadamard product versus conventional product for non negative matrices abstract we prove an inequality for the spectral radius of products of non negative matrices conjectured by x zhan we show that for all n n non negative matrices a and b a b a a b b 1 2 ab in which represents the hadamard product

on the bilinear transformation of riccati equations an interpretation of the bilinear transformation of the riccati equation is presented the bilinear transformation of a continuous type algebraic riccati equation are to a discrete type are and its inverse transformation have been used for solving are numerically however its physical meaning has not been clear and the problem formulation was complicated this note gives a physical interpretation and derivation of the bilinear transformation of the are finally cheap control for continuous time linear systems is considered from the view of the bilinearly transformed system

output feedback sliding mode control for sampled data systems we address the output feedback sliding mode control problem for a sampled data linear system with external disturbances of the matching type by taking into account the disturbance compensation a deadbeat high gain output feedback control strategy with additional dynamics is able to attenuate the disturbances in the framework of singular perturbation analysis it is shown that the closed loop system exhibits good robustness against exogenous disturbances

the exact slow fast decomposition of the algebraic ricatti equation of singularly perturbed systems the algebraic riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced order algebraic riccati equations corresponding to the slow and fast time scales the pure slow and pure fast algebraic riccati equations are asymmetric ones but their o epsilon perturbations are symmetric it is shown that the newton method is very efficient for solving the obtained asymmetric algebraic riccati equations the method presented is very suitable for parallel computations due to the complete and exact decomposition of the riccati equation this procedure might produce new insight into the two time scale optimal filtering and control problems

intertwining technique for a system of difference schrodinger equations and new exactly solvable multichannel potentials the intertwining operator technique is applied to difference schrodinger equations with operator valued coefficients it is shown that these equations appear naturally when a discrete basis is used for solving a multichannel schrodinger equation new families of exactly solvable multichannel hamiltonians are found

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

positive and negative solution of dual riccati equations by matrix sign function iteration abstract for a matrix algebraic riccati equation it is shown that the positive definite stabilizing solution the negative definite anti stabilizing solution and the positive and negative solutions of the corresponding dual equation may all be computed from a single matrix sign function iteration this is potentially far more efficient than four separate schur vector solutions moreover necessary and sufficient conditions are given for the existence of unique stabilizing or antistabilizing solutions of an algebraic riccati equation in terms of the matrix sign of an associated hamiltonian matrix

on the spectral radius and the spectral norm of hadamard products of nonnegative matrices abstract we prove the spectral radius inequality a 1 a 2 a k a 1 a 2 a k for nonnegative matrices using the ideas of horn and zhang we obtain the inequality a b a t b for nonnegative matrices which improves schur s classical inequality a b a b where denotes the spectral norm we also give counterexamples to two conjectures about the hadamard product

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

optimal control for a new class of singularly perturbed linear systems singularly perturbed linear control systems have been traditionally studied with one vector input that controls both the slow and fast state variables motivated by examples of real physical systems in this paper we extend the study to corresponding two vector input systems we have identified several classes of real physical singularly perturbed systems that can be more efficiently controlled using two vector inputs in such systems one vector input is used to control the slow state space variables and the other vector input is used to control the fast state variables this control structure offers more flexibility controlling given systems and appears to be more effective we study the linearquadratic regulator lqr problem for this new class of singularly perturbed systems and develop a numerical algorithm based on fixed point iterations to obtain its solution the latter is obtained in terms of reduced order sub problems which avoid possible numerical stiffness issues and are computationally advantageous for large scale systems

recursive algorithm for the optimal static output feedback control problem of linear weakly coupled systems a recursive algorithm is developed for solving the algebraic equations comprising the solution of the optimal static output feedback control problem of linear weakly coupled systems the proposed algorithm is very efficient from the numerical point of view since only low order systems are involved in algebraic calculations and the required solution can easily be obtained up to an arbitrary order of accuracy that is 0 e2k where e is a small weak coupling parameter the real world example demonstrates the effectiveness of the proposed reduced order algorithm and its advantages over the global full order algorithm

decentralized disturbance accommodation with limited plant model information the design of optimal disturbance accommodation and servomechanism controllers with limited plant model information is studied in this paper we consider discrete time linear time invariant systems that are fully actuated and composed of scalar subsystems each of which is controlled separately and influenced by a scalar disturbance each disturbance is assumed to be generated by a system with known dynamics and unknown initial conditions we restrict ourselves to control design methods that produce structured dynamic state feedback controllers where each subcontroller at least has access to the state measurements of those subsystems that can affect its corresponding subsystem the performance of such control design methods is compared using a metric called the competitive ratio which is the worst case ratio of the cost of a given control design strategy to the cost of the optimal control design with full model information we find an explicit minimizer of the competitive ratio and show that it is undo

stabilization of generalized linear systems via the algebraic riccati equation abstract this paper presents results from a study to examine the stabilization properties of linear time delay systems and systems depending on a parameter using techniques associated with the algebraic riccati equation are these classes of systems can be viewed as members of a broader class of systems called generalized linear systems gls techniques of spectral factorization for the gls play an essential role in the methods for performing this stabilization necessary and sufficient conditions for the existence of a polynomial solution to the generalized are are derived numerical examples are given in order to illustrate and clarify the results

block diagonally dominant positive definite approximate filters and smoothers abstract we examine stochastic dynamical systems where the transition matrix and the system noise q t covariance are nearly block diagonal when htr 1h is also nearly block diagonal where r is the observation noise covariance and h is the observation matrix our suboptimal filter smoothers are always positive semidefinite and have improved numerical properties applications for distributed dynamical systems with time dependent pixel imaging are discussed

nested e decompositions and clustering of complex systems abstract a graph theoretic method of extreme simplicity is proposed for decomposition and clustering of complex systems into interconnected subsystems with prescribed interaction strength the method is suitable for stability analysis via vector lyapunov functions and decentralized control and estimation of large scale systems the new decomposition scheme can be used in clustering aggregation of models in economics queuing networks and computer systems as well as decompositions in block iterative methods for solutions of linear algebraic equations

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

two new solvable potentials we succeeded in finding two new solvable potentials by working in a complete square integrable basis that carries a tridiagonal jacobi matrix representation for the wave operator the conventional methods do not lead to exact solutions for these potentials hence a new approach has been adopted for these potentials first we obtain the potential parameter spectrum the set of values of the potential parameters that lead to an exact solution at a given energy then the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum for a given potential parameter this procedure has been applied in 3d to obtain the energy spectrum for a special screened coulomb potential with a barrier and in 1d for a single wave potential

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

study of the discrete singularly perturbed linear quadratic control problem by a bilinear transformation this paper presents a new approach in the study of the linear quadratic control problem of singularly perturbed discrete systems by applying a bilinear transformation the algebraic discrete riccati equation is converted into a continuous one which can be solved by using the reduced order recursive method already documented in the control literature this method produces the reduced order near optimal solution up to an arbitrary order of accuracy and reduces the size of required computations the method is very suitable for parallel programming a real world example an f 8 aircraft demonstrates the efficiency of the proposed method