numerical algorithms for eigenvalue assignment by constant and dynamic output feedback algorithms are proposed for eigenvalue assignment eva by constant as well as dynamic output feedback the main algorithm is developed for single input multioutput systems and the results are then extended to multiinput multioutput systems in computing the feedback use is made of the fact that the closed loop eigenvalues can almost always be assigned arbitrarily close to the desired locations in the complex plane provided the system satisfies the condition m p n where m p and n are respectively the number of inputs outputs and states of the system the eva problem has been treated as a converse of the algebraic eigenvalue problem the proposed algorithms are based on the implicitly shifted qr algorithm for solving the algebraic eigenvalue problem the performance of the algorithms is illustrated by several numerical examples

partial eigenvalue assignment problem of high order control systems using orthogonality relations in this paper we present an explicit solution to the partial eigenvalue assignment problem of high order control system using orthogonality relations between eigenvectors of the matrix polynomial our solution can be implemented with only a partial knowledge of the spectrum and the corresponding left eigenvectors of the matrix polynomial we show that the number of eigenvalues and eigenvectors that need to remain unchanged will not affected by feedback a numerical example is given to illustrate the applicability and the practical usefulness of the proposed method

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

solving the maximum duo preservation string mapping problem with linear programming abstract in this paper we introduce the maximum duo preservation string mapping problem mpsm which is complementary to the minimum common string partition problem mcsp when each letter occurs at most k times in any input string the version of mpsm is called k mpsm in order to design approximation algorithms for mpsm we also introduce the constrained maximum induced subgraph problem cmis and the constrained minimum induced subgraph cnis problem we show that both cmis and cnis are np complete we also study the approximation algorithms for the restricted version of cmis which is called k cmis k 2 using linear programming method we propose an approximation algorithm for 2 cmis with approximation ratio 2 and an approximation algorithm for k cmis k 3 with approximation ratio k 2 based on approximation algorithms for k cmis we get approximation algorithms for k mpsm with the same approximation ratio

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

an algorithm for the multiinput eigenvalue assignment problem a very simple and inexpensive algorithm is presented for pole placement in the multiinput case the algorithm consists of orthogonal reduction to a block hessenberg form and a simple linear recursion it yields a matrix f such that a bf has any specified set of eigenvalues whenever the system is controllable it is extremely easy to program on a computer the algorithm is not a robust pole placement algorithm but appears to give comparable results in most well conditioned cases at a fraction of the cost it is a direct noniterative algorithm and no eigenvalues or singular values are computed the algorithm does not need any complex arithmetic even when complex conjugate eigenvalues need to be assigned

controllability analysis and eigenvalue assignment for generalized state space systems abstract an extended hessenberg triangular form is introduced as a tool for the controllability analysis of linear descriptor generalized state space systems the extended hessenberg triangular form may be applied to the multiple input eigenvalue placement via state feedback with the advantage that the number of eigenvalues that have to be assigned with each input may be decided only unitary transformations that lead to numerical stability are applied to the systems the assignment of the eigenvalues to both finite or infinite positions is considered

assignment of the characteristic polynomial of a hessenberg matrix an algorithm is described for determining the first row of an upper hessenberg matrix associated with a completely controllable linear system so as to produce a prespecified characteristic polynomial the scheme is faster and more flexible than two existing procedures and is competitive with regard to accuracy on a variety of sample problems

a parallel algorithm for the sylvester observer equation we present a new algorithm for solving the sylvester observer equation arising in the context of the luenberger observer the algorithm embodies two main computational phases the solution of several independent equation systems and a series of matrix matrix multiplications the algorithm is thus well suited for parallel and high performance computing by reducing the coefficient matrix a to lower hessenberg form one can implement the algorithm efficiently with few floating point operations and little workspace the algorithm has been successfully implemented on a cray c90 a comparison both theoretical and experimental has been made with the well known hessenberg schur algorithm which solves an arbitrary sylvester equation our theoretical analysis and experimental results confirm the superiority of the proposed algorithm both in efficiency and speed over the hessenberg schur algorithm

computation of transfer function matrices of generalized state space models a reliable computational algorithm is proposed for the evaluation of the transfer function matrices of linear multivariable systems described by generalized state space models the algorithm is based on a new efficient and numerically stable procedure for computing irreducible state space realizations of generalized state space models for each input output channel the proposed algorithm computes an irreducible realization using this procedure then the parameters of the corresponding transfer function gain poles zeros are determined from the resulting irreducible model

a multishift hessenberg method for pole assignment of single input systems a new algorithm is proposed for the pole assignment of single input linear time invariant systems the proposed algorithm belongs to the family of hessenberg methods and is based on an implicit multishift qr like technique the new method compares favourably in many respects speed memory usage with existing numerically stable methods its improved vectorizability guarantees good opportunities for parallel implementation on high performance computers

parallel and large scale matrix computations in control some ideas abstract the design and analysis of time invariant linear control systems give rise to a variety of interesting linear algebra problems numerically effective methods now exist for several of these problems however algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent in this paper we propose several efficient general purpose parallel algorithms for single input and multiinput eigenvalue assignment problems a desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix vector multiplication solution of a linear system and computations of the eigensystem and singular values of a symmetric matrix for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms the proposed algorithms thus have potential for implementations on some existing and future parallel processors we also propose a numerical method for the sylvester matrix equation arising in the construction of a luenberger observer the method does not need reduction to condensed forms and is thus suitable for large and sparse matrices the method also exhibits a certain parallelism

an algorithm for pole assignment in high order multivariable systems an efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions a given multiinput system is first transformed to an upper block hessenberg form by means of orthogonal state coordinate transformations the state feedback problem is then reformulated in terms of the sylvester equation the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively a distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast computational aspects of the algorithm are discussed and numerical examples are provided

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

eigenvalue assignment in a single input linear system a procedure for eigenvalue assignment in a single input linear system in hessenberg form is described the procedure uses real arithmetic throughout and includes a facility for estimating the robustness of the assigned eigenvalues the assigned eigenvalues may be real or complex simple or multiple

on the selection of poles in the single input pole placement problem abstract it is well known that the single input pole placement problem can be quite sensitive to perturbations in the data recent results by mehrmann and xu show how this sensitivity depends on the location of the poles in many applications it suffices to prescribe a set k in the complex plane that contains the poles mehrmann and xu formulated a minimization problem for allocating the poles in a given set k so that the sensitivity to perturbations is reduced the present paper uses methods of potential theory to derive simple algorithms that yield approximate solutions of this minimization problem

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

value iteration in a class of communicating markov decision chains with the average cost criterion markov decision processes with denumerable state space and discrete time parameter are considered the performance index of a control policy is the lim sup expected average cost criterion and the the main structural restrictions on the model are the following i under the action of any stationary policy the state space is a communicating class ii the cost function has an almost monotone or penalized structure v s borkar siam j control optim 21 1983 pp 652 666 22 1984 pp 965 978 and iii some stationary policy induces an ergodic chain with finite average cost in this context it is shown that the value iteration scheme can be used to construct convergent approximations of a solution to the optimality equation as well as a sequence of stationary policies whose limit points are optimal

are the coefficients of a polynomial well conditioned functions of its roots we show that the coefficients of a polynomial can be rather illconditioned functions of its given roots in certain cases however the coefficients are well conditioned e g if the roots lie symmetrically in a closed angular sector for an angle less than centered at the origin of the complex plane we introduce and study condition numbersb l k for the sensitivity of the polynomial coefficienta k to changes in the rootx l and develop an algorithm to compute the condition numbers and to decide whether a given set of roots will generate the polynomial coefficients in an ill way this algorithm is an essential preliminary for certain numerical pole placement algorithms in linear control theory

asymptotic orders of reachability in perturbed linear systems a framework for studying asymptotic orders of reachability in perturbed linear time invariant systems is developed the systems of interest are defined by matrices that have asymptotic expansions in powers of a perturbation parameter epsilon about the point 0 the reachability structure is exposed by means of the smith form of the reachability matrix this approach is used to provide insight into the kinds of inputs needed to reach weakly reachable target states into the structure of high gain feedback for pole placement and into the types of inputs that steer trajectories arbitrarily close to almost a b invariant subspaces and almost a b controllability subspaces

parameterized tractability of the maximum duo preservation string mapping problem in this paper we investigate the parameterized complexity of the maximum duo preservation string mapping problem the complementary of the minimum common string partition problem we show that this problem is fixed parameter tractable when parameterized by the number k of conserved duos by first giving a parameterized algorithm based on the color coding technique and then presenting a reduction to a kernel of size o k 6

an algorithm for the multiinput pole assignment problem abstract we present an algorithm for the multiinput pole assignment problem this algorithm is based on basic linear operations such as matrix vector multiplications resolution of upper triangular systems of smaller sizes than the original problem and resolution of systems with multiple right hand sides for which efficient and parallel algorithms have been developed furthermore those independent systems can be computed in parallel the accuracy of the algorithm has been checked on some numerical examples

an algorithm to assign eigenvalues in a hessenberg matrix single input case we propose a conceptually simple algorithm to assign eigenvalues in a hessenberg matrix the method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer numerical experiments compare favorably to those of other methods recently proposed for the problem

on the matrix equation xh hx and the associated controllability problem abstract in this paper a simple method for generating a nonsingular solution of the matrix equation xh hx where the matrix h is in unreduced lower hessenberg form is proposed the controllability of the matrix pair h b where b is the last column vector of solution matrix x is also addressed in addition a nonsingular symmetric solution of the matrix equation xh h t x is obtained and the matrix pair h b where b is the last row of the solution matrix of this equation is shown to be controllable

jordan pair assignment via state feedback this note considers the assignment of the eigenstructure of the closed loop system via state feedback this assignment is achieved by assigning the jordan pair of the denominator matrix polynomial of the controllable mimo system via state feedback it turns out that for proper assignment one must take into account the almost forgotten infinite eigenstructure of the system even though they are invariant under constant state feedback however the infinite eigenstructure affects the selection of the eigenvectors

on eigenvalue and canonical form assignments abstract we first present a constructive matrix procedure to assign an arbitrary nonderogatory matrix by state feedback specifically given a controllable pair m c t the procedure finds a vector f and a transforming matrix l such that the closed loop matrix m c t f t is similar to a preassigned arbitrary nonderogatory matrix such as the jordan companion schwarz or triangular the method is direct in the sense that it does not require knowledge of eigenvalues and eigenvectors of m or solution of any matrix equation or even computation of the characteristic polynomial we then propose an algorithm to assign an arbitrary normalized hessenberg matrix given a controllable pair m c t and a normalized upper hessenberg matrix b the algorithm computes an upper triangular matrix l l ij with l ij 1 and a vector f such that l m c t f l 1 b the algorithm seems to be computationally more effective than the former and in particular can be used to assign most of the important canonical forms above and a given set of eigenvalues as well

pole assignment in vibratory systems by multi input control the problem of reassigning some poles of a vibratory system while keeping the other poles unchanged is considered the problem may be solved uniquely by single input state feedback control a family of solutions to the partial pole assignment problem may be obtained by applying multi input control forces an algorithm for determining a multi input control which is small in some sense is presented the non iterative algorithm proposed defines a closed form solution to the partial pole assignment problem in its natural second order form and no first order realization is used the reduction in the control effort achieved by the proposed method is demonstrated by numerical examples

a note on the convergence rate of the value iteration scheme in controlled markov chains this work is concerned with controlled markov chains with bounded costs assuming that the transition probabilities satisfy a simultaneous doeblin condition it is shown that schweitzer s transformation on the transition law yields a strong ergodicity condition that implies that the solution to the average cost optimality equation can be approximated at a geometric rate via the value iteration scheme

pole assignment a new proof and algorithm in this paper a new proof of the pole assignment theorem is given this proof is a very straightforward one it is not based on canonical forms and also the reduction to the single input case heymann s lemma is not used furthermore an algorithm is given which allows to take into account numerical aspects with respect to the feedback construction for the multi input case furthermore the non uniqueness of the feedback matrix in the multi input case may be exploited in order to reduce the gains

feedback controls for pole subset assignment a pole assignment problem for multi input linear systems is discussed in conventional pole assignment problems an input is concerned with all poles simultaneously or it assigns only a specified subset of poles here we propose a design method which makes each input assign an arbitrary subset of poles thus when we replace a pole by using this method it is enough to reconstruct only the corresponding input

state feedback design using reduced order nonsquare descriptions a pole placement technique is proposed for computing state feedback in multi input linear systems the algorithm has good numerical behavior and utilizes nonsquare pencils and the notion of decomposability the fact that the method deals with reduced order pencils and exploits the schur hessenberg form improves the conditioning of the problem

linear and numerical linear algebra in control theory some research problems abstract this paper presents an informal survey of theoretical and computational aspects of three linear algebra problems arising in control theory namely the sylvester observer matrix equation the matrix eigenvalue assignment problem and the stability and stabilization of large second order systems difficulties are highlighted recent developments are mentioned some research problems in these areas which should be of interest to both linear algebraists and control theorists are described

on computational algorithms for pole assignment a number of computationally reliable direct methods for pole assignment by feedback have recently been developed these direct procedures do not necessarily produce robust solutions to the problem however in the sense that the assigned poles are insensitive to perturbalions in the closed loop system this difficulty is illustrated here with results from a recent algorithm presented in this transactions and its causes are examined a measure of robustness is described and techniques for testing and improving robustness are indicated

recursive eigenstructure assignment in linear systems in arbel and rath 1985 an iterative eigenvalue assignment algorithm was presented this algorithm has the shortcomings of clustering most of the closed loop eigenvalues on the real axis a general method is given here for recursive eigenstructure assignment in linear systems the method eliminates the shortcomings of the algorithm of arbel and rath it is shown that the right and left eigenvectors of the closed loop system matrix can be determined in terms of those of a small dimension matrix a c it is also shown that the results of arbel and tse 1980 are special cases of those of the proposed method moreover the arbitrary parameters beyond eigenvalue assignment are shown to be embedded in the choice of a certain arbitrary invertible matrix s furthermore computer oriented steps are outlined for recursive eigenstructure assignment in large scale systems numerical examples are worked out to illustrate the generality and feasibility of the proposed method

parallel algorithms for certain matrix computations abstract the complexity of performing matrix computations such as solving a linear system inverting a nonsingular matrix or computing its rank has received a lot of attention by both the theory and the scientific computing communities in this paper we address some nonclassical matrix problems that find extensive applications notably in control theory more precisely we study the matrix equations ax xa t c and ax xb c the inverse of the eigenvalue problem called pole assignment and the problem of testing whether the matrix b ab a n 1 b has full row rank for these problems we show two kinds of pram algorithms on one side very fast i e polylog time algorithms and on the other side almost linear time and processor efficient algorithms in the latter case the algorithms rely on basic matrix computations that can be performed efficiently also on realistic machine models

pole assignment via sylvester s equation it is shown that the pole assignment problem can be reduced to solving the linear matrix equations ax xa bg fx g successively for x and then f for almost any choice of g the result is a new pole assignment procedure and proof of the pole assignment theorem that should play an important role in both theoretical and practical applications

the matrix equation xa atx and an associated algorithm for solving the inertia and stability problems abstract we develop a constructive procedure for generating nonsingular solutions of the matrix equation xa a t x and establish an interesting relationship between a given solution x of the above equation and the associated matrix polynomial p a the latter is then used to develop an algorithm for computing the inertia of a matrix the algorithm is more efficient than the other common procedures

causal stabilization of forward models of discrete lti processes under constraints on the set of elementary divisors of the characteristic matrix the paper investigates the stabilization problem for discrete lti processes by realizable digital controllers the process is described in the form of forward models which are mostly used by engineers but proved to be unsuitable for causality considerations the set of all stabilizing causal controllers is constructed in a parametric form moreover the authors construct the set of all causal stabilizing controllers which satisfy additional constraints i e the set of elementary divisors of the characteristic matrix of the closed loop has a certain structure of practical relevance as an example the causal structural stabilization problem is solved for the discrete forward model of the double integrator

revisiting the parameterized complexity of maximum duo preservation string mapping abstract in the maximum duo preservation string mapping max duo psm problem the input consists of two related strings a and b of length n and a nonnegative integer k the objective is to determine whether there exists a mapping m from the set of positions of a to the set of positions of b that maps only to positions with the same character and preserves at least k duos which are pairs of adjacent positions we develop a randomized algorithm that solves max duo psm in 4 k n o 1 time and a deterministic algorithm that solves this problem in 6 855 k n o 1 time the previous best known deterministic algorithm for this problem has 8 e 2 k o k n o 1 running time beretta et al 2016 1 2 we also show that max duo psm admits a problem kernel of size o k 3 improving upon the previous best known problem kernel of size o k 6

an algorithm for the multi input complex eigenvalue assignment problem in this paper a simple efficient and accurate method for complex eigenvalue assignment in the multi input linear system is presented the method considers both cases all eigenvalues need to assign are complex with their complex conjugates the other case is the arbitrary eigenvalues need to assign are mixed some are real and the others are complex which are closed under complex conjugations the algorithm proposed in this method is a modification to the one proposed by arnold and datta ieee trans automat contr 35 10 1986 1149 which considers the case where all the eigenvalues need to assign are only real also it is a kind of extension to the algorithms suggested by ramadan in press and ramadan and el shazly in press where arbitrary preassigned complex and mixed eigenvalues are assigned for single input case respectively the method first transforms the problem to block hessenberg form of needed form using orthogonal reduction then a simple linear recursion is carried out to find the feedback matrix so that the closed loop matrix has the prescribed eigenvalues the main advantages of the algorithm besides being simple and has recursive nature it avoids complex arithmetic and easy to program on computer numerical examples are given to illustrate the reduction process the algorithm performance and the accuracy of the method

eigenvalue translation method for stabilizing an unsymmetric lanczos reduction process the unsymmetric lanczos reduction method has been recently developed to reduce the size of a large scale linear system which is the discretized form of a time dependent partial differential equation problem with a large physical domain this has been applied to solve the time dependent advection dispersion equation discretized by finite element or finite difference methods however the reduced system sometimes suffers time instability because of relocation of the approximate eigenvalues into the left half plane this paper develops a method for stabilizing the reduced system while preserving the accuracy of the solution the unstable eigenvalues are translated from the left half complex plane to the right half leaving eigenvalues in right half plane unchanged the results of numerical simulations of the synthetic and practical field contaminant transport problems show the efficiency and accuracy of this method 1998 john wiley sons ltd

computation of minimal order state space realizations and observability indices using orthogonal transformations in this paper an algorithm is presented for obtaining a minimal order state space realization of a strictly proper rational function matrix the algorithm can also be used to compute the observability indices of any given state space system a controllable but unobservable state space realization of the given rational function matrix is first obtained by inspection the algorithm then performs a sequence of simple coordinate transformations on the state vector of the system the coordinate transformation matrices are orthogonal and are easily constructed from the system matrices each coordinate transformation operates on submatrices of the state space system matrices of lower dimension than the preceding one the sequence of coordinate transformations terminates after at most v 1 1 transformations where v 1 is the maximal observability index of the state space model the observability indices of the system are also determined at this time the transformed system matrices are obtained in a form wh

projection and deflation method for partial pole assignment in linear state feedback two projection methods are proposed for partial pole placement in linear control systems these methods are of interest when the system is very large and only a few of its poles must be assigned the first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace the second method can be regarded as a variant of the weilandt deflation technique used in eigenvalue methods

pole placement preconditioning the partial pole placement problem has received considerable attention in control theory where it is applied to stabilize time invariant linear control systems this paper discusses application of the partial pole placement problem to the construction of preconditioners for linear systems of equations numerical examples show that these preconditioners can improve the rate of convergence of the restarted gmres methods significantly

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

a global minimum search algorithm for estimating the distance to uncontrollability abstract let a r n n and b r n m we suggest a new search algorithm for estimating the distance a b of a controllable pair a b to the set of uncontrollable pairs by estimating the global minimum of the function min a i b c where min denotes the smallest singular value of a matrix using simple properties of this function due to ralph byers one first observes that provided rank b n the minimization problem can be transformed to a minimization problem in the bounded region x z x a 2 z a 2 in the two dimensional real plane the algorithm then progressively partitions this region into simplexes and by determining whether their vertices x j z j satisfy z j min y r min a x j iy i b it computes after a finite number of steps upper and lower bounds for a b the difference between the two is small if a b is small while the lower bound is larger if a b is large thus ensuring safe decisions an error analysis numerical examples and an operation count are all presented only simple modifications of the search region are necessary to extend the applicability of the algorithm to the case when rank b n

efficient eigenvalue assignments for general linear mimo systems abstract this paper deals with the transformation of linear multi input multi output mimo systems into frobenius canonical form with the ultimate objective of developing a new computationally efficient methodology for a poleplacement procedure both time invariant and time varying systems are considered the conventional pole placement steps for time invariant siso single input single output systems are generalized for both classes this is a unique study of the expansion of the pole placement capability in particular for time varying mimo systems this depth of generalization has been neglected in the past due to its complex formulation the practical advantage of the proposed technique is that it does not require the coefficients of the characteristic polynomial the eigenvalues of the original system or the coefficients of the characteristic polynomial of the transformed system the repercussions of such a development are expected in nonlinear systems theory as well considering the fact that some recently developed techniques yield exact i o linearization with time varying coefficients

on minimum cost per unit time control of markov chains the minimum cost per unit time control problem is studied for a class of markov chains that though important in applications does not fit the conventional framework for this problem existence of optimal stationary strategies and necessary and sufficient conditions for optimality are established

approaches to robust pole assignment abstract robust pole assignment is a non linear optimization problem in many variables we describe numerical methods for determining robust or well conditioned so lutions to the problem of pole assignment by state feedback the solutions are chosen to minimize various objective functions based on the condition number of the eigenvector matrix careful choice of parametrization and objective function avoids singularities and artificial variable constraints explicit formulae for the gradient and hessian permit rigorous stopping criteria and rapid local convergence several computational examples are included

design of state feedback for large scale multivariable systems this note presents a general method which reduces the computational requirements in the state feedback design of large scale multivariable systems the given system is first transformed into a general block canonical form by using simple equivalent transformation the state feedback problem is then reformulated in terms of a sylvester equation finally the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively

feedback stabilization of a second order system a nonmodal approach a novel approach for feedback stabilization of a second order model is proposed specifically two nonmodal algorithms are described and it is shown mathematically that under some mild assumptions on the damping matrix the feedback matrix obtained by each algorithm indeed stabilizes a closed loop system the first algorithm requires the solution of a symmetric positive definite linear system and the inversion of a small matrix a remarkable feature of this algorithm is that it does not require knowledge of the stiffness and damping for implementation which makes it very feasible for practical application the second algorithm requires the solution of a small linear least squares problem and an estimate of the stability index which can be obtained just by computing the extremal eigenvalues of the data matrices which are symmetric for almost all practical applications these minimal computational requirements make the proposed algorithms suitable for practical implementations even for large and sparse systems using the state of the art techniques of matrix computations this is in sharp contrast with the traditional modal approach ordinarily used by practicing engineers which requires the solution of a quadratic eigenvalue problem or equivalently the solution of a 2n x 2n generalized eigenvalue problem and therefore may not be practical for very large and sparse problems besides the proposed algorithms the paper also contains some mathematical results on the bounds of the eigenvalues of a second order pencil which may be useful for investigating stability of second order systems and are of independent interest

the lanczos arnoldi algorithm and controllability we present methods for computing the controllable space of a linear dynamic system the methods are based on orthogonalizing krylov sequences and can use much less storage than other methods with little or no loss of numerical stability

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

an algorithm for computing state feedback in multiinput linear systems a new algorithm is proposed in this note to compute the state feedback for arbitrary pole placement in multiinput linear systems the algorithm has good numerical behavior it also allows the user to adjust the eigenstructure of the resulting feedback system and therefore improve the condition of the problem and computation

numerical linear algebra aspects of control design computations the interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems control and estimation theory is discussed the impact of finite precision finite range arithmetic including the implications of the proposed ieee floating point standard s on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning basic tools from numerical linear algebra such as linear equations linear least squares eigenproblems generalized eigenproblems and singular value decomposition are then outlined a selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials frequency response system balancing and matrix riccati equations the implementation of such algorithms as robust mathematical software is then discussed a number of issues are addressed including characteristics of reliable mathematical software availability and evaluation language implications fortran ada etc and the overall role of mathematical software as a component of computer aided control system design

a direct algorithm for pole assignment of time invariant multi input linear systems using state feedback abstract a direct algorithm is suggested for the computation of the linear state feedback for multi input systems such that the resultant closed loop system matrix has specified eigenvalues the extra freedom can be used in different ways for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed loop matrix the algorithm uses unitary transformations for numerical reliability and is based on ideas from the qr algorithm for solving the eigenproblem the stability of the algorithm is proven by doing a backward rounding error analysis and numerical examples are given as well

the matrix equation xa bx r and its applications abstract we study the well known sylvester equation xa bx r in the case when a and b are given and r is known up to its first n 1 rows we prove new results on the existence and uniqueness of x our results essentially state that in case a is a nonderogatory matrix there always exists a solution to this equation a solution is uniquely determined by its first row x 1 and there is an interesting relationship between x 1 and the rows of r we also give a complete characterization of the nonsingularity of x in this case as applications of our results we develop direct methods for constructing symmetrizers and commuting matrices computing the characteristic polynomial of a matrix and finding the numbers of common eigenvalues between a and b some well known important results on symmetrizers bezoutians and inertia are recovered as special cases

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

hessenberg and hessenberg triangular forms in linear system thcory abstract in this paper we distinguish between free hessenberg forms where only the system matrix is transformed into a special structure and system hessenberg forms where either an input or output matrix is simultaneously simplified we show how the freedom in the choice of the transformation matrix for the free hessenberg form can be used to obtain a unique system hessenberg form we discuss numerical aspects and outline some basic applications for both types of hessenberg forms in control and system theory we show that the system hessenberg form can be extremely sensitive to perturbations a property not necessarily shared by the free hessenberg form several examples are explored and a first order perturbation analysis is given analogous aspects are also discussed for the hessenberg triangular forms resulting from reduction of a matrix pencil the main emphasis of this paper is on systems with one input

robust exact pole placement via an lmi based algorithm this technical note deals with the robust exact pole placement problem pole placement algorithms that guarantee a small variation of the assigned poles against possible perturbations the solution to this problem is related to the solvability of a sylvester like equation thus the main issue is to compute a well conditioned solution to this equation also the best candidate solution must possess the minimal condition number to reduce sensitivity to perturbation this problem is cast as a global optimization under linear matrix inequality constraints for which a numerical convergent algorithm is provided and compared with the most attractive methods in the literature

arnoldi methods for large sylvester like observer matrix equations and an associated algorithm for partial spectrum assignment abstract we propose an arnoldi based numerical method for solving a sylvester type equation arising in the construction of the luenberger observer given an n n matrix a and an n m matrix g the method simultaneously constructs an m m hessenberg matrix h with a preassigned spectrum and an n m orthonormal matrix x such that ax xh g we consider the case when a is large and sparse so that the standard techniques such as the well known hessenberg schur method for solving a sylvester equation cannot be easily applied as a byproduct we propose an algorithm for the partial pole assignment problem for large matrices

on structural decomposition and stabilization of large scale control systems graph theoretic concepts are used to decompose a large control system into a block triangular interconnection of input reachable subsystems stabilization of the system is thereby reduced to stabilization of a number of smaller subsystems the special structural properties of the obtained subsystems can be exploited to further simplify the stabilization problem

strong 1 optimal stationary policies in denumerable markov decision processes abstract usual conditions for existence of stationary average optimal policies in denumerable mdps with general bounded rewards are shown to be also sufficient for strong 1 optimality moreover we prove that all limit points of discounted optimal stationary policies when the discount factor goes to 1 are strong 1 optimal

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

eigenvalue assignment in an upper hessenberg matrix a simple algorithm is described for the determination of the elements in the first row of a real unreduced upper hessenberg matrix so that this matrix has a specified set of eigenvalues the method is extended for application to descriptor systems

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems