eigenvalue assignment by minimal state feedback gain in lti multivariable systems in this article an improved method for eigenvalue assignment via state feedback in the linear time invariant multivariable systems is proposed this method is based on elementary similarity operations and involves mainly utilisation of vector companion forms and thus is very simple and easy to implement on a digital computer in addition to the controllable systems the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs moreover two types of state feedback gain matrices can be achieved by this method 1 the numerical one which is unique and 2 the parametric one in which its parameters are determined in order to achieve a gain matrix with minimum frobenius norm the numerical examples are presented to demonstrate the advantages of the proposed method

pole assignment by output feedback using cylindrical algebraic decomposition recently a new polynomial method for pole assignment by static output feedback has been developed in this paper we show how this method can be generalized to partial pole clustering some poles are exactly assigned the others being clustered in a specified region of the complex plane a constructive and systematic algorithm for the numerical resolution of both problems the collins cylindrical algebraic decomposition algorithm is presented and applied to an illustrative example

estimating the matrixp norm the holderp norm of anm n matrix has no explicit representation unlessp 1 2 or it is shown here that thep norm can be estimated reliably ino mn operations a generalization of the power method is used with a starting vector determined by a technique with a condition estimation flavour the algorithm nearly always computes ap norm estimate correct to the specified accuracy and the estimate is always within a factorn 1 1 p of a p as a by product a new way is obtained to estimate the 2 norm of a rectangular matrix this method is more general and produces better estimates in practice than a similar technique of cline conn and van loan

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

a closed form solution to the discrete time kalman filter and its applications abstract this paper presents a closed form solution to the discrete time kalman filter and its applications we first represent the kalman filter in terms of model parameters without using the riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises replacing the initial time with the fixed lag time to achieve the finite memory with respect to inputs and outputs and choosing the proper initial covariances on the recent finite horizon we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm as another application a closed form solution to the difference riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a luenberger type filter as in ackermann s formula

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

stability robustness of almost linear state equations sufficient conditions for stability robustness of finite dimensional autonomous systems are discussed the system comprises a stable linear nominal part and different types of unstructured norm bounded perturbations quantitative sufficient conditions for stability robustness are introduced for the case in which the perturbations are almost linear in the sense that both the size and the derivative of the perturbations are bounded these conditions describe a tradeoff between the size of the perturbations and their distance from linearity the arbitrary nonlinear case and a special linear case are shown to be limiting cases of the almost linear type as an illustration the same quantitative sufficient conditions for stability are applied to a system with slowly varying linear perturbations

loose eigenstructure assignment via rank one lmi approach with application to transient response shaping in h control this article proposes a new regional eigenstructure assignment via rank one lmi approach a gain parameter condition for the regional eigenvalue eigenstructure assignment is newly derived this assignment condition is easily combined with h design by means of enhanced lmi characterisation in the present approach the desired assignment of closed loop eigenvalues i e poles are not previously fixed but constrained into individual assignment regions to bring us more design freedom than classical exact assignment the regional assignment discussed in this article never falls into an ordinary lmi root clustering because the union of assignment regions is disjoint in general each closed loop eigenvector is also constrained into individual alignment cone to show the practical use of the extra design freedom brought by regional assignment a transient response shaping in h design is also discussed a useful design algorithm based on linearisation algorithm is proposed

approximate pole assignment a novel approach and some new algorithms for the pole assignment problem of a controllable time invariant linear multivariable system with linear state or output feedback are presented the main theme of the approach and the algorithms is to satisfy some constraints in the problem approximately various rank 1 update algorithms can then be derived depending on different extents to which the constraints are approximated functionals have to be minimized when eigenvectors of the closed loop system are chosen a perturbation analysis on the effect of the approximation and several numerical examples will be presented

a robust periodic pole assignment algorithm in this note a robust periodic pole assignment algorithm is proposed for linear time invariant discrete time systems the condition numbers of the eigenvector matrices of the closed loop system are assumed as a robustness measure and a periodic state feedback law is deduced by the minimization of the condition numbers associated to the eigenvectors of the monodromy matrix of the closed loop system the proposed periodic pole assignment algorithm has been tested on a number of examples giving satisfactory results

new approach to robust observer design this paper shows that based on the recent development of observer design solution an observer pole selection method can be formulated to minimize the observer gain to the system input it is proved that this method is a deterministic approach to the recovery of the loop transfer function and robustness of direct state feedback systems

reduced order observers a new algorithm and proof this paper gives a new recursive algorithm to construct a reduced order observer with prescribed spectrum of a given observable system the method is based on the staircase form and implicitly gives a new proof for the existence of such reduced order observers

a new approach for computing the state feedback gains of multivariable systems this note presents some new results in linear systems first the relationship between the polynomial matrix description and the state space representation of multivariable systems is clarified then we show that once such a relationship is determined the coprime matrix fraction description can be easily computed and we can further develop a closed form formula to solve the pole assignment problem of multivariable systems such formula can be thought of as an extension of the ackermann s formula for mimo systems thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

robust partial eigenvalue assignment problem for the second order system in this paper we study the partial eigenvalue assignment problem for the second order system where only a small part of eigenvalues of the open loop system is to be reassigned and the rest are required to remain unchanged it is desirable that the feedback controller not only assigns specific eigenvalues to the second order closed loop system but also that the system is robust or insensitive to perturbations we propose a numerical method such that the condition number of the matrix of the eigenvectors of the closed loop system is minimized in the method we only need the knowledge of the eigenvalues to be altered and the corresponding eigenvectors while we do not need the knowledge of the eigenstructures that are required to remain unchanged and are often unknown numerical examples show that the present method often leads to better conditioned closed loop system

comments with reply on computational algorithms for pole assignment by n k nichols some points of the paper by n k nichols see ibid vol ac 31 p 643 5 1986 concerning the robust pole assignment of linear multiinput systems are clarified it is stressed that the minimization of the condition number of the closed loop eigenvector matrix does not necessarily lead to robustness of the pole assignment it is shown why the computational method which nichols claims is robust is in fact numerically unstable with respect to the determination of the gain matrix in replying nichols presents arguments to support the choice of the conditioning of the closed loop poles as a measure of robustness and to show that the methods of j kautsky n k nichols and p vandooren 1985 are stable in the sense that they produce accurate solutions to well conditioned problems

pole placement by parametric output feedback abstract this note presents a new analytical solution to the problem of pole placement via constant output feedback under the condition m p n where n m and p are the number of states inputs and outputs respectively the approach is based upon parametric eigenstructure assignment of linear time invariant multivariable systems in combination with a special explicit formulation of the pole assignment equations thus the resulting analytical solution explicitly offers all remaining m p n degrees of freedom beyond eigenvalue assignment which can be used for additional design goals such as response shaping minimizing the norm of the feedback matrix and robust control respectively

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

algebraic characterization of controllability subspaces in the current base and its practical computation from the concepts of input chain and controllability chain introduced by heymann and in view of the generalized eigenvector assignment a new algebraic characterization of controllability subspaces css is developed compared with the previous ones based on matrix pencil theory it presents both the advantages of being stated in the current base and of being derived without resorting to use of canonical forms a practical algebraic algorithm is provided in order to compute the controllable structure of a given pair a b this approach is then applied to characterize css contained in a given subspace and span css via eigenvector assignment

least squares pole assignment by memoryless output feedback in this paper a pole assignment problem for a linear time invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed the cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum this approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters the relation between the modified cost functions and the original pole assignment problem is revealed the proposed approach is compared with other existing approaches and illustrated by numerical results

two sided estimation of diameters reduction rate for the longest edge n section of triangles with n 4 in this work we study the diameters reduction rate for the iterative application of the longest edge le n section of triangles for n p 4 the maximum diameter d n k of all triangles generated at the kth iteration of the le n section is closely connected with the properties of the triangular mesh generated by this refinement scheme the upper and the lower bounds for d 2 k were proved by kearfott in 9 and for d

output deadbeat control discrete time multivariate systems an algorithm is presented to compute output deadbeat controls for linear multivariable systems the algorithm based on quadratic optimization with no cost on control cheap control is numerically stable and covers the most general cases the resulting controls are internally stable and are shown to drive the outputs of the system to zero in minimum time using state feedback the central part of the algorithm is the construction of the nonunique state cost matrix two alternatives are presented having different complexities and resulting in solutions having different properties

a new robust stability measure for state feedback systems abstract this brief paper proposes the use of sensitivities of each individual system pole to measure robust stability of state feedback systems compared with the existing measures of this kind m 1 and m 2 this measure is much easier to work with than m 1 while at least as accurate and is definitely more accurate than m 2 the upper and lower bounds of this measure are also derived and proved to be the same as those of m 1 this measure can be maximized numerically using existing explicit and systematic procedures some analytical guidelines which relate explicitly and directly the state feedback system s individual pole selection performance and decoupling to this new measure are also available and presented

an iterative scheme for variational inequalities in this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities this iterative scheme not only contains as special cases the projection linear approximation and relaxation methods but also induces new algorithms then we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in en induced by symmetric positive definite matrices gm thus in contrast to the above mentioned methods this technique allows the possibility of adjusting the norm at each step of the algorithm this flexibility will generally yield convergence under weaker assumptions

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

a new proof of the degeneracy property of the longest edge n section refinement scheme for triangular meshes in this note by using complex variable functions we present a new simpler proof of the degeneracy property of the longest edge n section of triangles for n p 4 this means that the longest edge n section of triangles for n p 4 produces a sequence of triangles with minimum interior angle converging to zero

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

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

a matrix inverse eigenvalue problem and its application the problem of generating a matrix a with specified eigenvalues which maps a given set of vectors of another given set is presented an existence theorem is given and proved a stable algorithm for producing the matrix a is discussed the relation between this problem and the pole assignment problem in control theory is investigated the application of this problem in the design of neural networks is discussed

effects of using observers on stabilization of uncertain linear systems the effects of observers on robust linear feedback controllers are studied sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback the effects of the resulting high gain observers on the disturbance rejection bounds are also studied it is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection however if there is only additive plant disturbance and no measurement disturbance this degradation can be prevented

algorithms for refining triangular grids suitable for adaptive and multigrid techniques two general algorithms for refining triangular computational meshes based on the bisection of triangles by the longest side are presented and discussed the algorithms can be applied globally or locally for selective refinement of any conforming triangulation and always generate a new conforming triangulation after a finite number of interactions even when locally used the algorithms also ensure that all angles in subsequent refined triangulations are greater than or equal to half the smallest angle in the original triangulation the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth in a natural way proofs of the above properties are presented the second algorithm is a simpler improved version of the first which retains most of the properties of the latter the algorithms can be used either for constructing irregular computational meshes or for locally refining any given triangulation in this sense they can be adequately combined with adaptive and or multigrid techniques for solving finite element systems examples of the application of the algorithms are given and two possible generalizations are pointed out

a unified closed loop stability measure for finite precision digital controller realizations implemented in different representation schemes a computationally tractable unified finite word length closed loop stability measure is derived which is applicable to fixed point floating point and block floating point representation schemes both the dynamic range and precision of an arithmetic scheme are considered in this new unified measure for each arithmetic scheme the optimal controller realization problem is defined and a numerical optimization approach is adopted to solve it numerical examples are used to illustrate the design procedure and to compare the optimal controller realizations in different representation schemes

maximal stability robustness for state equations a measure for stability robustness of a linear time invariant finite dimensional system state equations is introduced an upper bound for the measure is derived using the characteristic values of the system it is shown that the set of optimal systems namely systems for which the stability robustness measure attains the bound contains the normal set which has been considered as the set of optimal robustness

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

mesh refinement processes based on the generalized bisection of simplices mesh refinement processes based on the generalized bisection of simplices are discussed and characterized in the context of the general theory of rheinboldt siam j numer anal 17 1980 pp 766 778 then it is proved that such processes allow the construction of sequences of naturally smooth conforming and nested nonuniform meshes in fact it is shown that there exists a generalized mesh refinement operator that for any conforming triangulation delta and for any v subset delta produces a nested smooth conforming triangulation delta containing successors of all elements of v and such that the minimum cell size of delta is bounded from below by half the minimum cell size of delta two conforming mesh refinement algorithms that allow the selective refinement of computational triangulations are explicitly given

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

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

a neural network based on the generalized fischer burmeister function for nonlinear complementarity problems in this paper we consider a neural network model for solving the nonlinear complementarity problem ncp the neural network is derived from an equivalent unconstrained minimization reformulation of the ncp which is based on the generalized fischer burmeister function f p a b email protected a b p a b we establish the existence and the convergence of the trajectory of the neural network and study its lyapunov stability asymptotic stability as well as exponential stability it was found that a larger p leads to a better convergence rate of the trajectory numerical simulations verify the obtained theoretical results

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

a new merit function for nonlinear complementarity problems and a related algorithm we investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one assuming that the complementarity problem is defined by a p 0 function we prove that every stationary point of the unconstrained problem is a global solution furthermore if the complementarity problem is defined by a uniform p function the level sets of the merit function are bounded the properties of the new merit function are compared with those of mangasarian solodov s implicit lagrangian and fukushima s regularized gap function we also introduce a new simple active set local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function

an algorithm for the assignment of system zeros abstract this paper presents an algorithm for assigning the locations and number of zeros to a linear system through synthesis of the system input and output space couplings the approach imposes no restrictions on the state space model can assign both real and complex zeros using only real arithmetic and is computationally efficient numerical properties of the algorithm are discussed and examples are given to illustrate its performance

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

sensitivity properties of multirate feedback control systems based on eigenstructure assignment some sampled data systems e g fly by wire control schemes have a necessarily multirate structure various input and or outputs sampled at different rates when considering a multirate system which has parameter uncertainty it is important to examine ways in which the full freedom of the multivariable design can be utilized to minimize the sensitivity to parameter variations given the accompanying problems induced by intersample ripple disturbance this note examines the design capabilities of a class of multirate systems with multiple input and fixed state sampling rates mifs based on eigenstructure assignment although the use of eigenstructure assignment for continuous and single rate discrete systems is well understood the eigenstructure assignment for the design of multirate feedback systems is an open topic of research accepting that the problems of intersample ripple are often magnified through multirate control there are advantages in terms of increased freedom for minimizing sensitivity and optimizing robustness to parameter variations a special feature of the mifs class of multirate systems is the ability to introduce extra design freedom in the eigenproblem by a suitable choice of eigenstructure assignment and sample rates the criteria for the selection of minimum sample rates to produce this extra freedom and the implication that this has on the eigenstructure assignment problem are outlined the improved insensitivity properties are demonstrated using an example comparing the performance of multirate and corresponding single rate designs

nonlinear integral type sliding surface for both matched and unmatched uncertain systems this note presents a new nonlinear integral type sliding surface which incorporates a virtual nonlinear nominal control to achieve prescribed specifications first the plant with matched uncertainties is considered the resultant closed loop system during ideal sliding mode behaves exactly like the nominal plant under the nonlinear nominal control which completely nullifies the matched uncertainties and consequently satisfies the prescribed specifications second the stability analysis of the proposed sliding mode for the systems with unmatched uncertainties is performed to exploit the stability conditions numerical results demonstrate the validity of the proposed concept

robustness in partial pole placement the robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined the projection procedure gives a reduced order pole assignment problem it is shown that the sensitivities of the assigned poles in the complete closed loop system are bounded in terms of the sensitivities of the assigned reduced order poles and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open loop poles if the assigned poles are well separated from the unaltered poles these bounds are expected to be tight the projection procedure is described in 3 and techniques for finding robust or insensitive solutions to the reduced order problem are given in 1 2

bounded flatness in q triangulated regular n simplexes bounded flatness of simplexes is crucial for simplicial algorithms to provide solutions of satisfactory accuracy by using combinatorial arguments we show that iterative q refinement with arbitrary mesh of euclidean simplexes a crucial step in the implementation of simplicial algorithms yields subsimplexes of bounded flatness the flatness is bounded by n 1 email protected n 2 f n r where f n r is the flatness of the regular unit n simplex

a two time scale neurodynamic approach to constrained minimax optimization this paper presents a two time scale neurodynamic approach to constrained minimax optimization using two coupled neural networks one of the recurrent neural networks is used for minimizing the objective function and another is used for maximization it is shown that the coupled neurodynamic systems operating in two different time scales work well for minimax optimization the effectiveness and characteristics of the proposed approach are illustrated using several examples furthermore the proposed approach is applied for h infty model predictive control

stabilization of projection based reduced order models for linear time invariant systems via optimization based eigenvalue reassignment abstract a new approach for stabilizing unstable reduced order models roms for linear time invariant lti systems through an a posteriori post processing step applied to the algebraic rom system is developed the key idea is to modify the unstable eigenvalues of the rom system by moving these eigenvalues into the stable half of the complex plane it is demonstrated that this modification to the rom system eigenvalues can be accomplished using full state feedback a k a pole placement algorithms from control theory this approach ensures that the modified rom is stable provided the system s unstable poles are controllable and observable however the accuracy of the stabilized rom is not guaranteed to remedy this difficulty and guarantee an accurate stabilized rom a constrained nonlinear least squares optimization problem for the stabilized rom eigenvalues in which the error in the rom output is minimized is formulated this optimization problem is small and therefore computationally inexpensive to solve performance of the proposed algorithms is evaluated on two test cases for which roms constructed via the proper orthogonal decomposition pod galerkin method suffer from instabilities

eigenvalues assignment by a particular lyapunov type equation this paper proposes a particular lyapunov type equation and uses its solution to synthesize a state feedback control such that the eigenvalues of the closed loop multiple inputs system can be assigned to the left half plane with arbitrary stability degree

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

robust pole assignment for synthesizing feedback control systems using recurrent neural networks this paper presents a neurodynamic optimization approach to robust pole assignment for synthesizing linear control systems via state and output feedback the problem is formulated as a pseudoconvex optimization problem with robustness measure i e the spectral condition number as the objective function and linear matrix equality constraints for exact pole assignment two coupled recurrent neural networks are applied for solving the formulated problem in real time in contrast to existing approaches the exponential convergence of the proposed neurodynamics to global optimal solutions can be guaranteed even with lower model complexity in terms of the number of variables simulation results of the proposed neurodynamic approach for 11 benchmark problems are reported to demonstrate its superiority

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

the decay of stabilizability with reynolds number in a linear model of spatially developing flows this article characterizes the gradual decay of stabilizability with reynolds number in the linear complex ginzburg landau model of spatially developing ow systems when a single spatially localized actuator is used to apply the control forcing it is shown that technically the system considered is linearly stabilizable for all actuator locations at any reynolds number however as the reynolds number is increased and an increasing number of modes of the open loop system become unstable the control authority on some of these open loop unstable modes is found to be exponentially small using nite precision arithmetic and any given numerical method for computing the feedback gains an e ective upper bound on the reynolds number is reached above which it is not possible to compute a linearly stabilizing control

robust eigenstructure assignment for the extended medium range air to air missile abstract a new sufficient condition for the robust stability of a linear time invariant system subject to linear time varying structured state space uncertainty is presented this sufficient condition is a sum of terms each of which involves the ith right eigenvector the ith left eigenvector and the real part of the ith eigenvalue this new robustness result is less conservative than an earlier result which involves the modal matrix and the real part of the eigenvalue which is closest to the imaginary axis in the complex plane a robust design is computed for the extended medium range air to air missile and the new design is compared with an earlier orthogonal projection eigenstructure assignment design the new design method uses the matlabtm optimization toolbox to minimize the integral of the deviations of the integrated roll rate which is approximately the bank angle from zero the integral is approximated by a sum of discrete samples of the integrated roll rate while normalizing the integration width to unity constraints are placed on the real part of the dutch roll and roll mode eigenvalues the damping ratios of the dutch roll and roll mode the aileron and rudder deflection rates and the sufficient condition for robust stability the new design satisfies the robustness condition while also yielding an improved transient response

numerically robust pole assignment for second order systems we propose two new methods for solution of the eigenvalue assignment problem associated with the second order control system global hsize 30pc specifically the methods construct feedback matrices f 1 and f 2 such that the closed loop quadratic pencil has a desired set of eigenvalues and the associated eigenvectors are well conditioned method 1 is a modification of the singular value decomposition based method proposed by juang and maghami which is a second order adaptation of the well known robust eigenvalue assignment method by kautsky et al for first order systems method 2 is an extension of the recent non modal approach of datta and rincon for feedback stabilization of second order systems robustness to numerical round off errors is achieved by minimizing the condition numbers of the eigenvectors of the closed loop second order pencil control robustness to large plant uncertainty will not be explicitly considered in this paper numerical results for both the two methods are favourable a compara

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

brief a gradient flow approach to on line robust pole assignment for synthesizing output feedback control systems pole assignment is a basic design method for synthesis of feedback control systems in this paper a gradient flow approach is presented for robust pole assignment in synthesizing output feedback control systems the proposed approach is shown to be capable of synthesizing linear output feedback control systems via on line robust pole assignment convergence of the gradient flow can be guaranteed moreover with appropriate design parameters the gradient flow converges exponentially to an optimal solution to the robust pole assignment problem and the closed loop control system based on the gradient flow is globally exponentially stable these desired properties make it possible to apply the proposed approach to slowly time varying linear control systems simulation results are shown to demonstrate the effectiveness and advantages of the proposed approach

an innovation representation for nonlinear systems with application to parameter and state estimation abstract an innovation model is derived for a nonlinear stochastic system described by a state variable representation the problem of state and system parameter estimation is solved through identification of the innovation model a recursive prediction error rpe algorithm is derived for the joint system parameter and state estimation through minimization of the innovation variance miv the algorithm is robust against the use of an erroneous model convergence and stability properties of the algorithm are also analyzed in order to ensure stability the algorithm needs an on line stability check at each iteration

an r linearly convergent derivative free algorithm for nonlinear complementarity problems based on the generalized fischer burmeister merit function in the paper j s chen s pan a family of ncp functions and a descent method for the nonlinear complementarity problem computational optimization and applications 40 2008 389 404 the authors proposed a derivative free descent algorithm for nonlinear complementarity problems ncps by the generalized fischer burmeister merit function j p a b 12 a b p a b 2 and observed that the choice of the parameter p has a great influence on the numerical performance of the algorithm in this paper we analyze the phenomenon theoretically for a derivative free descent algorithm which is based on a penalized form of j p and uses a different direction from that of chen and pan more specifically we show that the algorithm proposed is globally convergent and has a locally r linear convergence rate and furthermore its convergence rate will become worse when the parameter p decreases numerical results are also reported for the test problems from mcplib which further verify the theoretical results obtained

geometric views of the generalized fischer burmeister function and its induced merit function abstract in this paper we study geometric properties of surfaces of the generalized fischer burmeister function and its induced merit function then a visualization is proposed to explain how the convergent behaviors are influenced by two descent directions in merit function approach based on the geometric properties and visualization we have more intuitive ideas about how the convergent behavior is affected by changing parameter furthermore geometric view indicates how to improve the algorithm to achieve our goal by setting proper value of the parameter in merit function approach

longest edge n section algorithms in this paper we survey all known including own recent results properties of the longest edge n section algorithms these algorithms in classical and recently designed conforming form are nowadays used in many applications including finite element simulations computer graphics etc as a reliable tool for controllable mesh generation in addition we present a list of open problems arising in and around this topic

robust eigenstructure assignment in geometric control theory in this paper we employ the rosenbrock system matrix pencil for the computation of output nulling subspaces of linear time invariant systems which appear in the solution of a large number of control and estimation problems we also consider the problem of finding friends of these output nulling subspaces i e the feedback matrices that render such subspaces invariant with respect to the closed loop map and output nulling with respect to the output map and which at the same time deliver a robust closed loop eigenstructure we show that the methods presented in this paper offer considerably more robust eigenstructure assignment than the other commonly used methods and algorithms

robust pole placement in lmi regions discusses analysis and synthesis techniques for robust pole placement in linear matrix inequality lmi regions a class of convex regions of the complex plane that embraces most practically useful stability regions the focus is on linear systems with static uncertainty on the state matrix for this class of uncertain systems the notion of quadratic stability and the related robustness analysis tests are generalized to arbitrary lmi regions the resulting tests for robust pole clustering are all numerically tractable because they involve solving linear matrix inequalities lmis and cover both unstructured and parameter uncertainty these analysis results are then applied to the synthesis of dynamic output feedback controllers that robustly assign the closed loop poles in a prescribed lmi region with some conservatism this problem is again tractable via lmi optimization in addition robust pole placement can be combined with other control objectives such as h sub 2 or h sub spl infin performance to capture realistic sets of design specifications physically motivated examples demonstrate the effectiveness of this robust pole clustering technique

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

brief augmented gradient flows for on line robust pole assignment via state and output feedback this paper is concerned with robust pole assignment in synthesis of linear control systems via state and output feedbacks first both the pole assignment and robustness requirements are appropriately formulated as two optimization problems then gradient flow models are developed for the on line computation of feedback gain matrices that result in robust pole assignment by solving these two optimization problems a technique is introduced to facilitate the real time matrix inverse involved for realizing the gradient flow models the resulting augmented gradient flows have desired convergence properties simulation results are included to show the effectiveness of the proposed approach

neurodynamics based robust pole assignment for high order descriptor systems in this paper a neurodynamic optimization approach is proposed for synthesizing high order descriptor linear systems with state feedback control via robust pole assignment with a new robustness measure serving as the objective function the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem a neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high order singular systems with guaranteed optimality and exact pole assignment two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach

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 new design approach to unknown input observers this paper proposes a fundamentally new unknown input observer uio which has uniquely r decoupled modes provides r linearly independent combinations of plant system states and has zero gain to all p unknown inputs the parameter r is adjustable if the plant system satisfies the conditions of the existing uio then r n plant system order and there is no difference between our uio and the existing regular uio however if the plant system satisfies either of the following two significantly more general conditions m p m is the number of plant output measurements and at least one stable transmission zero then 1 n and our uio can still generate some constrained state feedback control signals hence this new uio can completely unify the regular uio r n and the static output feedback r m the two basic feedback structures of modern control theory far from being unified before

some necessary and sufficient conditions for eigenstructure assignment abstract some necessary and sufficient conditions for closed loop eigenstructure assignment by output feedback in time invariant linear multivariable control systems are presented a simple condition on a square matrix necessary and sufficient for it to be the closed loop plant matrix of a given system with some output feedback is the basis of the paper some known results on entire eigenstructure assignment are deduced from this the concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors

perturbation theory for homogeneous polynomial eigenvalue problems we consider polynomial eigenvalue problems p a alpha beta x 0 in which the matrix polynomial is homogeneous in the eigenvalue alpha beta uc2 in this framework infinite eigenvalues are on the same footing as finite eigenvalues we view the problem in projective spaces to avoid normalization of the eigenpairs we show that a polynomial eigenvalue problem is well posed when its eigenvalues are simple we define the condition numbers of a simple eigenvalue alpha beta and a corresponding eigenvector x and show that the distance to the nearest ill posed problem is equal to the reciprocal of the condition number of the eigenvector x we describe a bihomogeneous newton method for the solution of the homogeneous polynomial eigenvalue problem homogeneous pep

robust eigenvalue assignment for generalized systems abstract in this paper we examine the problem of robust pole placement using state feedback in generalized systems we develop a robustness theory for the finite generalized spectrum of the system as a partial problem and for the infinite pole placement problem as a second partial problem where perfect conditioning is always achievable we also give bounds between the complete closed loop robust eigenvalue problem and the two partial ones the basic tool that is exploited in the presented theory is the concept of chordal metric in this paper we take advantage of this notion and present a compact theory for the robust eigenvalue assignment problem in generalized systems the proposed theory is easy to implement retrieves the results for the state variable case as a special case and takes advantage of well known computational results

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

approximate set valued observers for nonlinear systems a set valued observer svo produces a set of possible states based on output measurements and a priori models of exogenous disturbances and noises previous work considered linear time varying systems and unknown but bounded exogenous signals in this case the sets of possible state vectors take the form of polytopes whose centers are optimal state estimates these polytopic sets can be computed by solving several small linear programs an svo can be constructed conceptually for nonlinear systems however the set of possible state vectors no longer takes the form of polytopes which in turn inhibits their explicit computation this paper considers an extended svo as in the extended kalman filter the state equations are linearized about the state estimate and a linear svo is designed along the linearization trajectory under appropriate observability assumptions it is shown that the extended svo provides an exponentially convergent state estimate in the case of sufficiently small initial condition uncertainty and provides a nondivergent state estimate in the case of sufficiently small exogenous signals

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

on the non degeneracy property of the longest edge trisection of triangles the longest edge le trisection of a triangle t is obtained by joining the two equally spaced points of the longest edge of t with the opposite vertex in this paper we prove that for any given triangle t with smallest interior angle t 0 if the minimum interior angle of the three triangles obtained by the le trisection of t into three new triangles is denoted by t 1 then t 1 email protected c 1 where c email protected 3arctan 3 5 3 1403 moreover we show empirical evidence on the non degeneracy property of the triangular meshes obtained by iterative application of the le trisection of triangles if t n denotes the minimum angle of the triangles obtained after n iterative applications of the le trisection then t n t c where c is a positive constant independent of n an experimental estimate of c 6 7052025350 is provided

proving the non degeneracy of the longest edge trisection by a space of triangular shapes with hyperbolic metric from an initial triangle three triangles are obtained joining the two equally spaced points of the longest edge with the opposite vertex this construction is the base of the longest edge trisection method let be an arbitrary triangle with minimum angle let be any triangle generated in the iterated application of the longest edge trisection let be the minimum angle of thus c with c 3 arctan 3 11 is proved in this paper a region of the complex half plane endowed with the poincare hyperbolic metric is used as the space of triangular shapes the metric properties of the piecewise smooth complex dynamic defined by the longest edge trisection are studied this allows us to obtain the value c

on the bisection method for triangles let uvw be a triangle with vertices u v and w it is bisected as follows choose a longest edge say vw of uvw and let a be the midpoint of vw the uvw gives birth to two daughter triangles uva and uwa continue this bisection process forever we prove that the infinite family of triangles so obtained falls into finitely many similarity classes and we obtain sharp estimates for the longest jth generation edge