solving boundary value problems integral and integro differential equations using gegenbauer integration matrices we introduce a hybrid gegenbauer ultraspherical integration method hgim for solving boundary value problems bvps integral and integro differential equations the proposed approach recasts the original problems into their integral formulations which are then discretized into linear systems of algebraic equations using gegenbauer integration matrices gims the resulting linear systems are well conditioned and can be easily solved using standard linear system solvers a study on the error bounds of the proposed method is presented and the spectral convergence is proven for two point bvps tpbvps comparisons with other competitive methods in the recent literature are included the proposed method results in an efficient algorithm and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems the proposed method can be applied on a broad range of mathematical problems while producing highly accurate results the developed numerical scheme provides a viable alternative to other solution methods when high order approximations are required using only a relatively small number of solution nodes

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

transfer matrix approach to the structural block diagonal decoupling problem the block diagonal decoupling problem for structured systems is treated using the transfer matrix approach an associated graph which displays the internal structure of the system invariant and interconnection is used to establish necessary and sufficient conditions for the solution of the problem these conditions are expressed in the graph in terms of shortest input output paths which are related to the infinite structure of the system

feedback decoupling of structured systems the feedback decoupling of structured linear systems is considered using results gathered on decoupling and the graph characterization of the structure at infinity a necessary and sufficient decoupling condition for structured systems is developed this condition has a graphical interpretation in terms of shortest input output paths

feedforward control pid control laws and almost invariant subspaces we define feedforward control as a control policy in which the exogenous disturbances are known for all time at the moment when the control is applied it is shown that disturbance decoupling by feedforward control is possible iff it is possible by pid control or iff approximate disturbance decoupling by state feedback is possible

the linear programming bound for binary linear codes combining delsarte s 1973 linear programming bound with the information that certain weights cannot occur new upper bounds for d sub min n k the maximum possible minimum distance of a binary linear code with given word length n and dimension k are derived

lagrange interpolation to compute the numerical solutions of differential integral and integro differential equations this paper describes new two methods for the numerical solution of linear integral equations of fredholm type and volterra type also the methods have been extended to linear integro differential and differential equations the methods are tested by different examples

structured systems within the transfer matrix approach application to decoupling abstract we consider a new approach to structured systems within the transfer matrix framework we associate with such a structured system a graph which concentrates both knowledge on structural properties of independent subsystems and on their interconnections the analysis of this associated graph allows to conclude on the decouplability of structured systems

on the weight distributions of optimal cosets of the first order reed muller codes we study the weight distributions of cosets of the first order reed muller code r 1 m for odd m whose minimum weight is greater than or equal to the so called quadratic bound some general restrictions on the weight distribution of a coset of r 1 m are obtained by partitioning its words according to their weight divisibility most notably we show that there are exactly five weight distributions for optimal cosets of r 1 7 in r 5 7 and that these distributions are related to the degree of the function generating the coset moreover for any odd m spl ges 9 we exhibit optimal cubic cosets of r 1 m whose weights take on exactly five values

proper rational solution to 2 d two sided polynomial matrix equations necessary and sufficient conditions are established for the existence of proper rational solutions to a 2 d two sided polynomial matrix equation of the form axb c two procedures for finding a proper rational solution to the equation are presented the first procedure can be used if x is a finite impulse response the second procedure is a general one

some new results on the minimum length of binary linear codes of dimension nine let n k d be the smallest integer n for which a binary linear code of length n dimension k and minimum distance d exists using the residual code technique the macwilliams identities and the weight distribution of appropriate reed muller codes we prove that n 9 64 133 n 9 120 spl ges 244 n 9 124 252 and n 9 184 371 we also show that puncturing a known 322 9 160 code yields length optimal codes with parameters 319 9 158 315 9 156 and 312 9 154

survey generic properties and control of linear structured systems a survey in this survey paper we consider linear structured systems in state space form where a linear system is structured when each entry of its matrices like a b c and d is either a fixed zero or a free parameter the location of the fixed zeros in these matrices constitutes the structure of the system indeed a lot of man made physical systems which admit a linear model are structured a structured system is representative of a class of linear systems in the usual sense it is of interest to investigate properties of structured systems which are true for almost any value of the free parameters therefore also called generic properties interestingly a lot of classical properties of linear systems can be studied in terms of genericity moreover these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way we review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms by properties we mean here system specific properties like controllability the finite and infinite zero structure and so on as well as solvability issues of certain classical control problems like disturbance rejection input output decoupling and so on in this paper we do not try to be exhaustive but instead by a selection of results we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool we emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well known polynomially bounded combinatorial techniques for larger systems

integral fractional pseudospectral methods for solving fractional optimal control problems the main purpose of this work is to provide a unified framework and develop integral fractional pseudospectral methods for solving fractional optimal control problems as a generalization of conventional pseudospectral integration matrices fractional pseudospectral integration matrices fpims and their efficient and stable computation are the key to our new approach in order to achieve this goal we take a special and smart way to compute fpims the essential idea is to transform the fractional integral of lagrange interpolating polynomials through a change of variables into their jacobi weighted integral which can be calculated exactly using the jacobi gauss quadrature this together with the stable barycentric representation of lagrange interpolating polynomials and the explicit barycentric weights for the gauss flipped radau and radau type points corresponding to the jacobi polynomials leads to an exact efficient and stable scheme to compute fpims even at millions of jacobi type points numerical results on two benchmark optimal control problems demonstrate the performance of the proposed pseudospectral methods

graph theoretic approach to symbolic analysis of linear descriptor systems abstract continuous descriptor systems e x ax bu y cx where e is a possibly singular matrix are symbolically analyzed by means of digraphs starting with four different digraph characterizations of square matrices and determinants the author favors the cauchy coates interpretation then an appropriate digraph representation of the matrix pencil se a is given which is followed by a digraph interpretation of det se a and the transfer function matrix c se a 1 b next a graph theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det se a this is very important for large scale systems finally as an application of the derived results an electrical network is analyzed symbolically

the generic number of invariant zeros of a structured linear system in this paper we consider a structured linear system represented by means of a directed graph we present a graph theoretic method to compute the generic number of invariant zeros of the corresponding rosenbrock matrix the method is based on a fundamental decomposition of the directed graph representing the structured system the generic number of invariant zeros is important in generic versions of problems involving controllability and pole placement

restrictions on the weight distribution of binary linear codes imposed by the structure of reed muller codes the words of a binary linear spl lsqb n k spl rsqb code c whose weights belong to a given subset i spl sub spl lcub 0 1 n spl rcub constitute a word in a certain reed muller code spl rfr spl mfr r k appropriate choices of i result in low values of the order r and thus yield restrictions on the weight distribution of c

on the structure at infinity of a structured system abstract we develop a graph theoretic characterization of the generic structure at infinity of the transfer matrix of a structured system we show that the generic structure at infinity can be determined by means of algorithms from combinatorial optimization based on the max flow min cut theorem and on results concerning minimal cost flows as an application of the obtained characterization we propose a structural version of two well known disturbance decoupling problems and we derive graph theoretic necessary and sufficient conditions for the solvability of each of the two problems

structural invertibility of transfer functions in this note we study certain properties of a transfer function in terms of the structure of the state space representation a b c specifically the rank of the transfer function and the degree of a decoupled system which can be obtained by applying a left decoupler are characterized in terms of the structure of a b c

a graph theoretic method for computing a formal representation of the largest controlled invariant subspace of a structured system the paper discusses a method to compute a kernel representation of the largest controlled invariant subspace in the kernel of the output mapping of a structured system the graph of the structured system is used to make the relevant equations and then a formula manipulation package can be used to compute a condensed kernel representation or even an image representation of the largest controlled invariant subspace

paper structural synthesis of multivariable controllers in this paper the problem of achieving a desired transfer function matrix h d s between external inputs and controlled outputs in a linear multivariable system by connecting proper stabilizing controllers between measured outputs and control inputs is solved in both transfer function and state space settings the class h d of achievable transfer functions is directly and constructively characterized via the theory of transfer function valuations for each h d s eh d the class of synthesizing controllers is determined similar valuation conditions are given for the asymptotic tracking and disturbance rejection problem in which h d s is only partially specified these results extend and complement earlier results the state space geometric solution of the problem of achieving a desired h d s is obtained by formulating it as an equivalent output feedback disturbance rejection problem a constructive solvability condition in terms of a pair of measurement and control invariant subspaces is given this requires a nontrivial generalization of the notion of c a b pairs an h d s is shown to be admissible if and only if it induces an appropriate pair of invariant subspaces the signal flow structure and certain factorizability conditions for a robust synthesis of the output feedback disturbance rejection problem are also given which extend earlier results on robust state feedback disturbance rejection it is shown that every compensator corresponds to a state feedback control law implemented by an unknown input observer the results of this paper are expected to be useful in the development of parameter optimization or other computer aided design algorithms where response specifications are to be traded against other design criteria such as sensitivity or stability margins since they explicitly characterize the algebraic variety h d in which the response transfer function may lie

transfer function conditions for c a b pairs state space conditions for the solution of the output feedback disturbance rejection problem have been given in a recent paper 1 in terms of c a b pairs this paper develops constructive and simple solvability conditions for this problem in terms of the rational matrix equation g s x s n s h s

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

graph theoretic methods for the computation of disturbance decoupling feedback matrices for structured systems abstract structured systems are considered for which the disturbance decoupling problem is known to be generically solvable it is recalled how this generic solvability is in one one correspondence with certain properties of the graph representing the structure of the system further a graph oriented method is derived for the efficient computation of a feedback that actually solves the disturbance decoupling problem it is believed that the method is especially useful in case of sparse system matrices the method may also be advantageous in case some of the entries in the system matrices have an unknown value since the influence of the various entries is easy to establish

a common solution to a pair of linear matrix equations over a principal ideal domain abstract a necessary and sufficient condition for the existence of a common solution to a pair of linear matrix equations over a principal ideal domain is obtained the equations are of the type a i b i xc i for i 1 2 the solvability condition is that the equations each have a solution and a bilateral linear matrix equation made up of the matrices a i b i and c i has a solution

disturbance localization and pole shifting by dynamic compensation this paper concerns the problem of disturbance localization and pole shifting by dynamic compensation in multivariable linear systems with inaccessible state vector a necessary and sufficient condition for the solvability of the problem is given two synthesis procedures of the compensator are presented one is based on an observer and the other is based on a dual observer the observer designed by the proposed procedure acts as as unknown input observer for a linear function an improved procedure of designing a reduced order unknown input observer is also given the result may be applicable to reducing the compensator order

regulator synthesis using c a b pairs the concept of a c a b pair of subspaces was introduced in an earlier paper by the author to study problems of compensator design here it is shown that the same concept plays a basic role in the theory of tracking and regulation from a few fundamental properties the main results of regulator theory are derived in a relatively simple and straightforward manner another new feature is that we are able to allow the presence of an unknown input thereby unifying the fields of tracking and regulation and disturbance decoupling all results are fully constructive and an example is given to illustrate the theory

solving rational matrix equations in the state space with applications to computer aided control system design a method of solving a class of linear matrix equations over various rings is proposed using results from linear geometric control theory an algorithm successfully implemented is presented along with non trivial numerical examples applications of the method to the algebraic control system design methodology are discussed

a geometric approach for structured systems application to disturbance decoupling structured systems described by state space models are considered for these systems the entries of the state space model matrices are supposed to be either fixed zeros or free independent parameters for such systems one can study structural properties i e properties that are valid for almost any value of the parameters we revisit the classical geometric theory in the context of structured systems in particular the well known notions of a b invariant and c a invariant subspaces are redefined and analysed in this context we characterize fixed subspaces included or containing these invariant subspaces and show that they play a key role in solving control problems as an application of this geometric approach for structured systems we propose new solutions to the disturbance decoupling problems by state feedback and by output measurement feedback the solvability conditions are easily checked on the associated graph of the system

almost invariant subspaces transmission and infinite zeros a lattice interpretation in the first part of the paper it is shown that the most important almost controlled invariant and almost conditionally invariant subspaces introduced by j c willems 4 5 are elements of the free distributive lattice generated by the well known subspaces download download full size image and download download full size image in the second part it is shown how the transmission zeros and the infinite zero structure may be obtained by duality within a state feedback context or an output injection one

feedback realizations in linear multivariable systems model following with stability and disturbance rejection abstract this paper deals with the problems of constructing a compensator for linear multi variable systems that simultaneously ensures i model following with stability and ii disturbance rejection necessary and sufficient conditions for the solvability of the problems are given by using a geometric approach the solvability conditions are explicit and the constructive procedure for finding a solution does not require any additional conditions for the plant

numerical expansion methods for solving integral equations by interpolation and gauss quadrature rules in this paper we introduce a numerical method for solving linear integral equations the main idea based on interpolation for unknown function where is interpolated in the zeros of the chebyshev s polynomials next we use gauss quadrature rules as gauss chebyshev or clenshaw curtis the technique is very effective and simple specially for integral equations of first kind as fredholm s and volterra s types in the end for showing efficiency of this method we use numerical examples

transfer function conditions for output feedback disturbance rejection a recent paper 1 has derived state space conditions under which the disturbance transfer function in a linear multivariable system can be zeroed by a dynamic compensator forced by a prescribed set of measurements the present note derives necessary conditions for this problem in terms of the orders of the open loop control disturbance and measurement transfer functions these necessary conditions are shown to be generically sufficient for solvability moreover they provide additional insight into the geometric solvability conditions are simple to check and extend the corresponding results obtained for the state feedback case 2

numerical approximations using chebyshev polynomial expansions el gendi s method revisited we present numerical solutions for differential equations by expanding the unknown function in terms of chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema or zeros of the chebyshev polynomial of order n el gendi s method the solutions are exact at these points apart from round off computer errors and the convergence of other numerical methods used in solving the linear system of equations applications to initial value problems in time dependent quantum field theory and second order boundary value problems in fluid dynamics are presented

disturbance decoupling by measurement feedback for structured transfer matrix systems structured transfer matrix systems are linear systems given by transfer matrices of which the infinite zero order of each nonzero entry is known while the associated infinite gains are unknown and assumed mutually independent in this paper necessary and sufficient conditions are derived for the generic solvability of the well known disturbance decoupling problem by measurement feedback for structured transfer matrix systems generic solvability here means solvability for almost all possible values for the infinite gains of the nonzero transfer matrix entries the conditions will be stated in terms of a bipartite graph that can be easily associated with a structured transfer matrix system the advantage of this is that the conditions then can be verified by means of graph theoretic algorithms

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

generation of pseudospectral differentiation matrices i we present a simple method for computing n times n pseudospectral differentiation matrices of order p in cal o pn 2 operations for the case of quasi polynomial approximation the algorithm is based on fornberg s finite difference algorithm and is numerically stable a fortran implementation is included a necessary and sufficient condition for d p d p 1 is also given

disturbance decoupling problems via dynamic output feedback this paper considers the disturbance decoupling problems with or without internal stability and pole placement via dynamic output feedback using polynomial and rational matrix techniques we show that in all three problems considered the central solvability condition can be expressed as a two sided matching problem a bxc where a b and c are the polynomial system matrices of certain natural subsystems of the system model and x is to be determined over various subrings of the rational functions this matching problem can in turn be reduced to certain appropriate zero cancellation conditions on the polynomial system matrices a b and c

simultaneous decoupling and disturbance rejection a structural approach abstract the combined problem of disturbance rejection and decoupling for linear systems is considered new necessary and sufficient conditions are derived for the existence of a state feedback controller that rejects the disturbances and simultaneously decouples the closed loop system these conditions are simple intuitively appealing and are expressed in terms of system structural invariants such as infinite zero orders then we consider the disturbance rejection with simultaneous decoupling problem for structured linear systems necessary and sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

output feedback disturbance decoupling graph interpretation for structured systems abstract in this paper we consider first the output feedback disturbance decoupling problem for linear systems we derive a set of new necessary or sufficient conditions for the solvability of this problem these conditions are expressed in terms of system structural invariants such as infinite zero orders and essential orders then we consider the output feedback disturbance decoupling problem for structured linear systems necessary or sufficient conditions for the generic solvability of this problem are provided they are expressed in terms of input output path length comparisons on the system associated graph

generation of higher order pseudospectral integration matrices a new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of chebyshev polynomials applications to initial value problems boundary value problems linear integral and integro differential equations are presented the present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods

new decoupling invariants the essential orders the aim of this paper is to introduce a new list of integers which is invariant under the action of biproper compensation and which plays a key role in the solution of the row by row decoupling problem several equivalent characterizations are provided within both the geometric and the transfer matrix approaches

disturbance rejection for structured systems the disturbance rejection problem for structured systems is solved a graph is associated with such a structured system disturbance rejection is shown to be solvable by state feedback whenever the control inputs reach the outputs more quickly than the disturbances in the associated graph the infinite structure of a structured system is characterized in terms of input output paths in the associated graph

set of open loop block diagonalizers of transfer matrices abstract the set of open loop right block diagonalizers of a given p m transfer matrix z with specified block sizes is studied a characterization of this set is obtained in terms of a maximal open loop block diagonalizer and a minimal proper basis of the kernel of z the relevance of the results to some non interacting control problems is pointed out in particular the results obtained yield a minimum delay solution to the block decoupling problem using dynamic state feedback

efficient and stable generation of higher order pseudospectral integration matrices the main purpose of this work is to provide new higher order pseudospectral integration matrices hpims for the chebyshev type points and present an exact efficient and stable approach for computing the hpims the essential idea is to reduce the computation of hpims to that of higher order chebyshev integration matrices hcims and take a very simple and recursive way to compute the hcims efficiently and stably extensive numerical results show that the new approach for computing the hpims has better stability than that of the recently derived elgindy s approach for large number of chebyshev type points

decoupling of structured systems by parameter independent precompensation and state feedback in this paper structured systems described by state space models are considered for these systems the entries of the state space model matrices are assumed to be either fixed zeros or free independent parameters we study the dynamic decoupling of structured linear systems it turns out that a large class of systems which is not state feedback decouplable can be made decouplable by the addition of a simple structured precompensator such a precompensator then has the property to be independent of the system parameters we give a necessary and sufficient condition for a structured system to be made feedback decouplable using a parameter independent precompensator furthermore we prove that when such a solution exists the precompensator can be chosen to be diagonal in this case the precompensator is constructed via a simple combinatorial algorithm our approach is based on a graph representation of the structured system all the results of this paper are related to some particular shortest sets of input output paths in the graph associated with the structured system

the structure of robust disturbance rejection control this paper deals with the problem of disturbance rejection under state feedback in linear multivariable control systems this problem is ill posed under arbitrary state space parameter perturbations however it is shown that with respect to arbitrary perturbations of the plant transfer function coefficients a robust solvability condition and possibly a robust synthesis can be obtained these results are significant because the class of arbitrary state space parameter variations is too rich for practical interest whereas the class of arbitrary variations in transfer functions more closely represents the realistic problem of engineering concern

state space solution of the causal factorization problem given rational matrix functions g h and r a necessary and sufficient condition is given for the existence of a causal solution to the equation gxh r a formula is given for a solution of the equation when it exists and this particular solution is used to parameterize the set of all causal solutions of the equation the results are specialized to the case when the function g is identically equal to identity applications of the author s results include exact model matching and disturbance decoupling problems