interactive algorithms for multiple criteria nonlinear programming problems abstract in this paper several interactive procedures for solving multiple criteria nonlinear programming problems have been developed these are based on the generalized reduced gradient method for solving single objective nonlinear programming problems they can handle maximization problems with nonlinear concave objectives nonlinear convex constraints and an implicit quasi concave preference function of the decision maker the interactive procedures work with information of varying degrees of accuracy from the decision maker thereby extending and strengthening a number of existing methods

limit cycles in planar piecewise linear differential systems with nonregular separation line abstract in this paper we deal with planar piecewise linear differential systems defined in two zones we consider the case when the two linear zones are angular sectors of angles and 2 respectively for 0 we study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using melnikov functions these limit cycles appear studying higher order piecewise linear perturbations of a linear center it is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five moreover for these values of we prove the existence of systems with four limit cycles up to fifth order and for 2 we provide an explicit example with five up to sixth order in general the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line

design of multivariable feedback systems with stable plant this paper considers in a general algebraic framework the design of a unity feedback multivariable system with a stable plant the method is based on a simple parameterization of the four closed loop transfer functions in terms of p the plant transfer function and q h y 1 u 1 in particular the i o transfer function q h y 2 u 1 pq using the framework of rational transfer functions we show that the closed loop system will be exponentially stable if and only if q is exponentially stable furthermore if both p and q are strictly proper then the controller is also strictly proper the basic result is design theorem 2 an algorithm is given for obtaining strictly proper controllers such that the resulting i o map is decoupled all its poles can be chosen by the designer and the same holds for zeros except of course for the c zeros prescribed by the c zeros of the plant a discussion is included to temper these results by the constraints imposed by noise and plant saturation

decoupling of multivariable control systems over unique factorization domains necessary and sufficient conditions are established for the existence of a state variable feedback decoupling of an m input m output time invariant linear control system over a unique factorization domain an explicit computation is provided for the feedback and the feedforward gain matrix also necessary and sufficient conditions for the existence of a stability preserving state feedback decoupling are given the results are illustrated by some examples

open loop decoupling of mimo plants the complete set of decoupled forms of a mimo plant p which might be non square and may have unstable pole zero coincidences achievable using open loop precompensation is obtained in terms of certain multiplicities of the poles and zeros of p this approach which also yields the precompensator directly is much simpler than the methods available

decoupling controller design for linear multivariable plants decoupling controller design for linear time invariant square multiple input multiple output mimo plants under the unity feedback configuration is discussed for plants with no coincidences of unstable poles and zeros a simplified necessary and sufficient condition for closed loop stability is given the simplified condition leads to a simple parameterization of the set of all achievable decoupled input output i o maps and an algorithm which allows the design of decoupling controllers to achieve preassigned closed loop poles

feedback representation of precompensators application to decoupling linear multi input multi output plants in this paper we examine the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by two parameter compensation based on our earlier results boussaid and guerin 1996 we prove that any given plant p of full row rank can be decoupled with stability by the considered configuration computation of the decoupling compensators follows by easy constructions the decoupling problem with a minimum number of unstable zeros in the decoupled system is also formulated and solved in the generic case

nonlinear multiple objective optimization an algorithm and some theory this paper presents an algorithm and the supporting theory for solving a class of nonlinear multiple criteria optimization problems using zionts wallenius type of interaction the zionts wallenius method as extended in this paper can be used for solving multiple criteria problems with concave objective and implicit value functions and convex feasible regions modifications of the method to handle nonconvex feasible regions and general nonlinear objective functions are also discussed

limit cycles in 4 star symmetric planar piecewise linear systems abstract our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is x y 0 we consider the symmetric case that is one vector field defined in the odd quadrants and the other in the even ones we deal with equilibrium points of center focus type with matrices in real jordan form in each vector field when the infinity is monodromic in this case we provide the center classification at infinity we prove that the maximum order of a weak focus is five moreover we show the existence of systems exhibiting five limit cycles bifurcating from infinity

h 2 design of one degree of freedom decoupling controllers for square plants decoupling design of one degree of freedom controllers is treated for the generalised plant model within the h 2 framework the class of all realisable closed loop transfer matrices is parametrised under a mild assumption on non unity feedback sensors a set of compact assumptions is presented to guarantee the existence of the optimal h 2 controller together with the optimal one the class of all h 2 controllers that yield finite h 2 cost is obtained the optimal closed transfer matrix and its associated controller are shown to be strictly proper under the reasonable order assumptions on the generalised plant

integral constraints on sensitivity vectors for multivariable linear systems this paper develops integral constraints on sensitivity vectors for multivariable feedback systems due to either unstable poles or non minimum phase zeros of the plant the results presented cover the continuous rational or distributed case and are based on the use of directions corresponding to zeros or poles of the plant the integral constraints are in vector form and give new insights into the inherent trade offs in sensitivity reduction for multivariable systems and the cost of decoupling

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

a comparative study of interactive tradeoff cutting plane methods for momp abstract in this paper four interactive cutting plane methods developed for multiple objective mathematical programming problems are compared to one another using randomly selected problems tradeoff cutting plane methods converge to the best compromise solution by iteratively reducing the objective space by cutting planes the cutting planes are constructed using partial preference information obtained from the decision maker the performance of the four methods is evaluated and compared to one another by solving the test problems under i four classes of preference functions in addition to the original preference functions to simulate the nonexisting decision maker ii different precision levels in the decision maker s comparative accuracy iii decision space expanded or contracted and iv inconsistent responses by the decision maker based on the comparative study some guidelines for performing a comparative study of multicriteria interactive methods are presented

kkt transformation approach for multi objective multi level linear programming problems abstract the earlier karush kuhn tucker kkt transformation method has been applied to multi level decentralized programming problems ml d pps when the decision variable set was divided into subsets where each decision maker dm of the system controlled only a particular subset but had no control over any decision variables of some other subset in this paper we give the mathematical formulation and corresponding development of ml d pps by kkt transformation when dms have absolute control over certain decision variables but some variables may be shared and hence controlled by two or more dms

transfer function matrix approach to decoupling problem with stability a simple necessary and sufficient condition for solvability of decoupling with internal stability for unity output feedback for non singular plants is proposed

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

generalized hopf bifurcation emerged from a corner in general planar piecewise smooth systems abstract we investigate a generalized hopf bifurcation emerged from a corner located at the origin which is the intersection of n discontinuity boundaries in planar piecewise smooth dynamical systems with the jacobian matrix of each smooth subsystem having either two different nonzero real eigenvalues or a pair of complex conjugate eigenvalues we obtain a novel result that the generalized hopf bifurcation can occur even when the jacobian matrix of each smooth subsystem has two different nonzero real eigenvalues according to the eigenvalues of the jacobian matrices and the number of smooth subsystems we provide a general method and prove some generalized hopf bifurcation theorems by studying the associated poincare map

multivariable feedback sensitivity and decentralized control in this paper the problem of sensitivity reduction by feedback is studied and related to a problem of decentralized control a plant will be represented by an n times n matrix of frequency responses which may be unstable or irrational the object will be to find conditions on p s under which a diagonal feedback f s can make the sensitivity parallel i p s f s 1 parallel arbitrarily small over some specified frequency interval j omega 0 j omega 0 without violating a global sensitivity bound parallel i p s f s 1 parallel leq m m geq some const 1 for re s geq 0 it will be shown that such a diagonal feedback of the high gain type can be constructed whenever p 1 s is analytic in re s geq 0 p s satisfies an attenuation condition near s infty and p s approaches diagonal dominance at high frequencies it will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide band control scheme

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

uniqueness of limit cycles for sewing planar piecewise linear systems abstract this paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line and only one singularity of monodromic type the proofs are based in an extension of rolle s theorem for dynamical systems on the plane

internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation we consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation a sufficient condition is derived for the existence of a cascade proper compensator c s such that when employed in a unity feedback loop involving the proper transfer function matrix p o of a free of unstable hidden modes system sigma p o will not only internally stabilize the feedback closed loop system sigma p o c but will also give rise to a closed loop transfer function matrix h yr diag which is nonsingular diagonal and has desired poles based on this analysis an algorithmic procedure for the computation of such a compensator is presented

wiener hopf design of the general two degree of freedom controllers and the connection to state space formulas h 2 design of the two degree of freedom controller configuration is treated for the generalized plant in both the frequency and state space domains a frequency domain wiener hopf formula is derived first under a set of compact assumptions that gives the entire class of controllers for which the feedback control system is stable and a quadratic cost functional is finite the h 2 design method suggested in this paper allows the reference inputs to include the persistent signals such as step or ramp functions and the class of persistent reference inputs that can be tracked with zero steady state error is characterized explicitly to circumvent the computational difficulties of the wiener hopf formulas the corresponding state space domain formulas are derived by rationalizing the polynomial fractional descriptions it is shown that the well known standard time domain assumptions for the h 2 design problem have the corresponding equivalent frequency domain conditions and this equivalence relationship is show

decoupling linear multiinput multioutput plants by dynamic output feedback an algebraic theory this paper presents an algebraic theory for the design of a decoupling compensator for linear time invariant multiinput multioutput systems the design method uses a two input one output compensator which gives a convenient parametrization of all diagonal input output i o maps and all disturbance to output d o maps achievable by a stabilizing compensator for a given plant it is shown that this method has two degrees of freedom any achievable diagonal i o map and any achievable d o map can be realized simultaneously by a choice of an appropriate compensator the difference between all achievable diagonal and nondiagonal i o maps and the cost of decoupling is discussed for some particular algebraic settings

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

generalized hopf bifurcation emanated from a corner for piecewise smooth planar systems abstract in this paper we study the existence of periodic orbits bifurcating from a corner in a piecewise smooth planar dynamical system this phenomenon is interpreted as generalized hopf bifurcation philos trans r soc london ser a 359 1789 2001 2483 2496 hopf bifurcation for nonsmooth planar systems universitat zu koln 2000 northeast math j 17 4 2001 383 386 northeast math j 17 3 2001 261 264 generalized hopf bifurcation for non smooth planar dynamical systems university cologne 2003 in the case of smoothness hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis this characterization does not apply to a piecewise smooth system due to the lack of linearization in fact the generalized hopf bifurcation is determined by interactions between the geometrical structure of the corner and the eigenstructure of each smooth subsystem we combine a geometrical method and an analytical method to investigate the generalized hopf bifurcation the bifurcating periodic orbits are obtained by studying the fixed points of a return map

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm