an improved stability criterion for discrete time lur e systems with sector and slope restrictions this paper introduces an improved stability criterion for discrete time lur e systems with sector and slope restrictions for the stability criterion a lur e lyapunov functional candidate is constructed to include quadratic terms and integration terms with nonlinearities to handle the resulting integration terms in the lyapunov difference there is proposed a new bound lemma where the integration terms are relaxed into the second order terms by the geometric point of view in addition an appropriate weighting method uses slack variables from the sector and slope restrictions in the overall derivation the stabilization condition is formulated in terms of a parameterized linear matrix inequality plmi which is then converted into the linear matrix inequality an example illustrates that the proposed criterion presents a less conservative result than the previous criteria in the literature

system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

stability analysis of discrete time lur e systems a class of lyapunov functions is proposed for discrete time linear systems interconnected with a cone bounded nonlinearity using these functions we propose sufficient conditions for the global stability analysis in terms of linear matrix inequalities lmi only taking the bounded sector condition into account unlike frameworks based on the lur e type function the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary hence a wider range of cone bounded nonlinearities can be covered we also show that there is a link between global stability lmi conditions based on this new lyapunov function and a transfer function of an auxiliary system being strictly positive real in addition the novel function is considered in the local stability analysis problem of discrete time lur e systems subject to a saturating feedback a convex optimization problem based on sufficient lmi conditions is formulated to maximize an estimate of the basin of attraction another specificity of this new lyapunov function is the fact that the estimate is composed of disconnected sets numerical examples reveal the effectiveness of this new lyapunov function in providing a less conservative estimate with respect to the quadratic function

brief paper a revisited tsypkin criterion for discrete time nonlinear lur e systems with monotonic sector restrictions this paper revisits a well known tsypkin criterion for stability analysis of discrete time nonlinear luroe systems when nonlinearities are monotonic and sector restricted by 0 1 where 1 is positive de nite it is shown by kapila and haddad that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k ng z is strictly positive real wherek is nonnegative diagonal and g z represents a transfer function of the linear part of the luroe system with invertible or identically zero g 0 this paper extends this criterion when 1 is positive diagonal by choosing a new lyapunov function to obtain an lmi criterion from a frequency domain interpretation of this lmi criterion another su cient criterion is generated which establishes that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k 1 z k n g z is strictly positive real wherek andk are nonnegative diagonal and orthogonal to each other 1998 elsevier science ltd all rights reserved

lmi searches for anticausal and noncausal rational zames falb multipliers abstract given a linear time invariant plant the search for a suitable multiplier over the class of zames falb multipliers is a challenging problem which has been studied for several decades recently a new linear matrix inequality search has been proposed over rational and causal zames falb multipliers this letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers the addition of a popov multiplier to the anticausal zames falb multiplier is implemented by analogy with the causal search as a result a search over a noncausal subset of zames falb multipliers is obtained a comparison between all the search methods proposed in the literature is given

on the parametrization of all decentralized stabilizing controllers abstract a novel parametrization of all decentralized stabilizing controllers is provided in terms of a single parameter q that is constrained to satisfy a finite number of quadratic equations the results are presented in a general algebraic framework and apply to linear time invariant systems implications of this parametrization for decentralized control system performance are also discussed

lmi based stability criteria for discrete time lur e systems with monotonic sector and slope restricted nonlinearities this note presents new lmi based stability criteria for the discrete time multivariable lur e system with nonlinearities that are monotonic sector and slope restricted corresponding lur e lyapunov functions are constructed for such a system the new criteria are expressed in a reasonably general form that can be applied to both non diagonal and diagonal nonlinearities we explicitly compare the new approach to the existing lmi based popov like criteria in the literature and express the results in terms of an integral quadratic constraint iqc the applications of the new criteria to some control problems and strategies are briefly discussed numerical examples are included to show their performance and they are shown to be less conservative than the previous results

the aizerman and kalman conjectures using symmetry abstract using a decomposition of a lurie system in terms of symmetric and skew symmetric matrices this paper presents reformulations of the classical conjectures of aizerman and kalman which give valid conditions for absolute stability under this decomposition it is shown that a restatement of the aizerman conjecture implies stability while the re stated kalman conjecture implies contraction

comment on absolute stability analysis for negative imaginary systems automatica 67 2016 107 113 abstract this note provides the connection between the paper absolute stability analysis for negative imaginary systems and classical results in absolute stability strictly negative imaginary systems satisfy the aizerman conjecture

best achievable decentralized performance in this paper a novel parameterization of all decentralized stabilizing controllers is employed in mathematically formulating the best achievable decentralized performance problem as an infinite dimensional optimization problem finite dimensional optimization problems are then constructed that have values arbitrarily close to this infinite dimensional problem an algorithm which identifies the best achievable performance over all linear time invariant decentralized controllers is then presented it employs a global optimization approach to the solution of these finite dimensional approximating problems

decentralized stabilization characterization of all solutions and genericity aspects electrical and the decentralized stabilization problem of multivariable finite dimensional systems is considered in a fractional set up a new synthesis procedure for decentralized stabilizing compensators is proposed the class of all admissible local compensators that can be applied to a specified channel as an element of a decentralized compensator is identified the conditions under which the class of admissible local compensators is generic are investigated the problem of making a multi channel system stabilizable and detectable from a single channel applying decentralized feedback around the other channels has been shown to be generically solvable for a given set of dynamic compensators if and only if the plant is strongly connected

construction of inverses with prescribed zero minors and applications to decentralized stabilization we examine the following question suppose r is a principal ideal domain and that f is an n m matrix with elements in r with n m when does there exist an m n matrix g such that gf im and such that certain prescribed minors of g equal zero we show that there is a simple necessary condition for the existence of such a g but that this condition is not sufficient in general however if the set of minors of g that are required to be zero has a certain pattern then the condition is necessary as well as sufficient we then show that the pattern mentioned above arises naturally in connection with the question of the existence of decentralized stabilizing controllers for a given plant hence our result allows us to derive an extremely simple proof of the fact that a necessary and sufficient condition for the existence of decentralized stabilizing controllers is the absence of unstable decentralized fixed modes as well as to derive a very clean expression for these fixed modes in addition to the application to decentralized stabilization we believe that the result is of independent interest

zames falb multipliers for convergence rate motivating example and convex searches the bound of convergence rates for discrete time lurye sometimes written lur e systems has recently attracted much attention the contributions of this technical note are twofold firstly we sho

stability of higher order discrete time lur e systems abstract we consider discrete time lur e systems obtained by applying nonlinear feedback to a system of higher order difference equations arma models the arma model relates the inputs and outputs of the linear system and does not involve any internal or state variables a stability theory subsuming results of circle criterion type is developed including criteria for input to output stability a concept which is very much reminiscent of input to state stability

the divergence theorem for unbounded vector fields in the context of lebesgue integration we derive the divergence theorem for unbounded vector elds that can have singularities at every point of a compact set whose minkowski content of codimension greater than two is nite the resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions

on counter examples to aizerman and kalman conjectures counter examples to aizerman s and kalman s conjectures are considered investigation of the behaviour in the vicinity of the origin and at a distance from the origin is done the simultaneous exis

a multivariable extension of the tsypkin criterion using a lyapunov function approach for analyzing the stability of discrete time systems containing a feedback nonlinearity the tsypkin criterion is the closest analog to the popov criterion which is used for analyzing such systems in continuous time traditionally the proof of this criterion is based upon input output properties and function analytic methods in this paper we extend the tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector bounded memoryless time invariant nonlinearities along with providing a lyapunov function proof for this classical result

analysis and design for discrete time linear systems subject to actuator saturation we present a method to estimate the domain of attraction for a discrete time linear system under a saturated linear feedback a simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant moreover the condition can be expressed as linear matrix inequalities lmis in terms of all the varying parameters and hence can easily be used for controller synthesis the following surprising result is revealed for systems with single input suppose that an ellipsoid is made invariant with a linear feedback then it is invariant under the saturated linear feedback if and only if there exists a saturated nonlinear feedback which makes the ellipsoid invariant finally the set invariance condition is extended to determine invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements

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

input to state stability of discrete time lur e systems an input to state stability theory which subsumes results of circle criterion type is developed in the context of discrete time lur e systems the approach developed is inspired by the complexified aizerman conjecture

second order counterexamples to the discrete time kalman conjecture the kalman conjecture is known to be true for third order continuous time systems we show that it is false in general for second order discrete time systems by construction of counterexamples with stable periodic solutions we discuss a class of second order discrete time systems for which it is true provided the nonlinearity is odd but false in general this has strong implications for the analysis of saturated systems

topologies for the space of bv integrable functions in r n we introduce two topologies on the space of bv integrable functions in r n among the studied properties are barrelledness and integral representation of linear continuous functionals in one case we charctcterize the dual space we show the relations with the study of multipliers finally we also introduce and study a topology on the space of bounded bv functions in r n c 1997 academic press

computer aided design via optimization many design problems including control design problems involve infinite dimensional constraints of the form f z a 0 for all a e a where a denotes time or frequency or a parameter vector in other design problems tuning or trimming of certain parameters after manufacture of the system is permitted the corresponding constraint is that for each a in a there exists a value t of the tuning parameter in a permissible set t such that f z a t 0 recent algorithms for solving design problems having such constraints are summarized

brief dominant pole placement for multi loop control systems this paper proposes a method for multi loop pi controller design which can achieve dominant pole placement for two input two output processes it is an extension of the original dominant pole design pid controllers theory design and tuning instrument society of america research triangle park nc 1995 for siso systems unlike its siso counterpart where the controller parameters can be obtained analytically the multi loop version amounts to solving some coupled nonlinear equation with complex coefficients for which closed form formulae are not possible a novel approach is developed to solve the equation using a root trajectory method in which the solution to our pole placement problem is found from intersection points between the root trajectories and the positive real axis the design procedure is given and simulation examples are provided to show the effectiveness of the proposed method and comparisons are made with the blt method

the gauss green theorem abstract in the m dimensional euclidean space we establish the gauss green theorem for any bounded set of bounded variation and any bounded vector field continuous outside a set of m 1 dimensional hausdorff measure zero and almost differentiable outside a set of finite m 1 dimensional hausdorff measure

algebraic characterization of decentralized fixed modes an input output frequency domain characterization of decentralized fixed modes is given in this paper using only standard block diagram algebra well known determinantal expansions and the binet cauchy formula

computer aided stability analysis renders popov criterion obsolete a class of multipliers that can be used for producing alternative stability tests for systems having a single incrementally conic nonlinearity f x has been nearly parameterized by zames and falb 1 the present note demonstrates how linear programming can be used to compute optimal multipliers leading to a frequency domain stability criterion that is less conservative than the standard graphical criteria

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

comments on on the existence of stable causal multipliers for systems with slope restricted nonlinearities the above technical note presents a novel convex search within the zames falb multiplier class the aim of this correspondence is to correct the misuse of a relaxation on one condition in the search this relaxation leads to nontrivial numerical errors in all six examples discussed in the technical note correct application of the conditions still gives an improvement over absolute stability criteria in the literature for at least one example but some of the claims for lack of conservativeness in the above technical note should be moderated

discrete time convolution control systems in this paper we study a general class of linear time invariant discrete time distributed systems we consider both single input single output siso and multi input multi output mimo systems and study design procedures we develop a commutative algebra of transfer function b p0 for a general class of siso discrete time convolution systems which covers sampled distributed systems and of course lumped systems as a special case each element of b p0 is formulated as a ratio of two elements in an algebra l1 p0 of causal p0 stable transfer functions we demonstrate that l1 p0 indeed a euclidean ring give necessary and sufficient conditions for coprimeness between elements in l1 p0 and characterize poles and zeros for elements in b p0 in contrast to the algebra l1 the algebra b p0 includes both stable and unstable systems furthermore since p0 1 this formulation allows us to study the dominant poles inside the unit disc of the complex plane we study next mimo systems whose transfer functi

on a new method for h_ 2 based decomposition this note proposes a simple linear matrix inequality lmi system for the design of static precompensators to reduce the interactions of a multivariable system the approach is based on minimizing the h2 norm of a modified system and the lmis are derived from the well known technique of pseudodiagonalisation the approach is applied to two complex real life benchmark problems with high levels of interaction it is shown that its performance is significantly better than previously proposed lmi optimization techniques for designing static precompensators

zames falb multipliers for quadratic programming in constrained linear model predictive control a quadratic program must be solved on line at each control step and this constitutes a nonlinearity if zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded we show that if the nonlinearity is static then it is also monotone and slope restricted hence we show the existence of zames falb multipliers for such a nonlinearity the multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control

a less conservative lmi condition for stability of discrete time systems with slope restricted nonlinearities many conditions have been found for the absolute stability of discrete time lur e systems in the literature it is advantageous to find convex searches via lmis where possible in this technical note we construct two less conservative lmi conditions for discrete time systems with slope restricted nonlinearities the first condition is derived via lyapunov theory while the second is derived via the theory of integral quadratic constraints iqcs and noncausal zames falb multipliers both conditions are related to the jury lee criterion most appropriate for systems with such nonlinearities and the second generalizes it numerical examples demonstrate a significant reduction in conservatism over competing approaches

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