some new results in the theory of negative imaginary systems with symmetric transfer matrix function abstract this note represents a first attempt to provide a definition and characterisation of negative imaginary systems for not necessarily rational transfer functions via a sign condition expressed in the entire domain of analyticity along the same lines of the classic definition of positive real systems under the standing assumption of symmetric transfer functions we then derive a necessary and sufficient condition that characterises negative imaginary transfer functions in terms of a matrix sign condition restricted to the imaginary axis once again following the same line of argument of the standard positive real case using this definition even transfer functions with a pole at the origin with double multiplicity as well as with a possibly negative relative degree can be negative imaginary

on the ergodic convergence rates of a first order primal dual algorithm we revisit the proofs of convergence for a first order primal dual algorithm for convex optimization which we have studied a few years ago in particular we prove rates of convergence for a more general version with simpler proofs and more complete results the new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

feedback stability of negative imaginary systems this paper extends the robust feedback stability theorem of negative imaginary systems by removing restrictive assumptions on the instantaneous gains of the systems that were imposed in the earlier literature and it further generalizes this robust analysis result into the case that allows negative imaginary systems to have poles at the origin in doing so we extend the class of negative imaginary systems for which this robust stability theorem is applicable we also show that this new generalized necessary and sufficient result specializes to the earlier theorems under the same assumptions we additionally prove that the previously known dc gain condition is not only necessary and sufficient for robust feedback stability under the earlier specified instantaneous gain assumptions but is also necessary and sufficient for robust feedback stability under new different and equally simple assumptions the general robust feedback stability theorem for negative imaginary systems with free body dynamics i e poles at the origin derived in this paper also specializes to the case that is only applicable for the negative imaginary system without poles at the origin since the results for negative imaginary systems with free body dynamics developed in this paper depend on the existence of a matrix psi with certain properties we also propose a systematic construction of this matrix psi and show that construction of one such psi is sufficient for determining the feedback stability of the closed loop system finally examples are used to demonstrate the applicability of the results

design of strictly positive real systems using constant output feedback the authors present a linear matrix inequality lmi approach to the strictly positive real spr synthesis problem find an output feedback k such that the closed loop system t s is spr the authors establish that if no such constant output feedback k exists then no dynamic output feedback with a proper transfer matrix exists to make the closed loop system spr the existence of k to guarantee the spr property of the closed loop system is used to develop an adaptive control scheme that can stabilize any system of arbitrary unknown order and unknown parameters

output feedback negative imaginary synthesis under structural constraints the negative imaginary property is a property that many practical systems exhibit this paper is concerned with the negative imaginary synthesis problem for linear time invariant systems by output feedback control sufficient conditions are developed for the design of static output feedback controllers dynamic output feedback controllers and observer based feedback controllers based on the design conditions a numerical algorithm is suggested to find the desired controllers structural constraints can be imposed on the controllers to reflect the practical system constraints also the separation principle is shown to be valid for the observer based design finally three numerical examples are presented to illustrate the efficiency of the developed theory

on α and d negative imaginary systems this paper is concerned with and d negative imaginary systems the definition of negative imaginary transfer functions is firstly introduced then the relationship between negative imaginary and negative imaginary transfer functions is studied by means of the generalised inverse an negative imaginary lemma is proposed to test the negative imaginary property of transfer functions also a necessary and sufficient condition is provided for the stability of interconnection of negative imaginary systems a state feedback controller design condition is established such that the resulting system is negative imaginary moreover the concept of negative imaginary transfer functions is extended to that of d negative imaginary transfer functions finally the developed results are illustrated by numerical examples

lmi based algorithm for strictly positive real systems with static output feedback abstract an algorithm based on linear matrix inequalities lmis is proposed to find a constant output feedback matrix k o and a constant output tandem matrix f such that the controlled system a b k o c b f c is strictly positive real spr the number of output variables of the plant a b c is greater than or equal to the number of its input variables considering that spr systems with static output feedback are related to spr systems with static state feedback as shown in this manuscript the first step of the algorithm is to find a matrix f such that all transmission zeros of the system a b f c have negative real parts after finding this matrix f an output feedback matrix k o such that the system a b k o c b f c is spr is found another algorithm is proposed to specify a decay rate the results are applied to the simulation of electrical stimulation for paraplegic patients to vary knee joint angle from 0 to 60

comments on dissipative analysis and control of state space symmetric systems automatica 45 2009 1574 1579 we show that some results presented by meisami azad mohammadpour and grigoriadis meisami azad m mohammadpour j grigoriadis k m 2009 dissipative analysis and control of state space symmetric systems automatica 45 1574 1579 are erroneous first we present a counterexample contradicting the necessary and sufficient condition of lemma 5 then we point out that lemma 4 which is a key instrumental lemma in the proof of theorem 6 is also erroneous and we propose a new lemma that corrects and completes the proof of theorem 6 finally we show that h norm formulations claimed by theorems 6 7 and 8 are inaccurate which is also sustained by previously published results as well as numerical examples taken from the commented paper

stability robustness of a feedback interconnection of systems with negative imaginary frequency response a necessary and sufficient condition expressed simply as the dc loop gain i e the loop gain at zero frequency being less than unity is given in this note to guarantee the internal stability of a feedback interconnection of linear time invariant lti multiple input multiple output systems with negative imaginary frequency response systems with negative imaginary frequency response arise for example when considering transfer functions from force actuators to colocated position sensors and are commonly important in for example lightly damped structures the key result presented here has similar application to the small gain theorem which refers to the stability of feedback interconnections of contractive gain systems and the passivity theorem which refers to the stability of feedback interconnections of positive real or passive systems a complete state space characterization of systems with negative imaginary frequency response is also given in this note and also an example that demonstrates the application of the key result is provided

a proximal projection method for finding zeros of set valued operators in this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set valued operators in a reflexive banach space this algorithm which we call the proximal projection method is essentially a fixed point procedure and our convergence results are based on new generalizations of the browder s demiclosedness principle we show how the proximal projection method can be applied for solving ill posed variational inequalities and convex optimization problems with data given or computable by approximations only the convergence properties of the proximal projection method we establish also allow us to prove that the proximal point method with bregman distances whose convergence was known to occur for maximal monotone operators still converges when the operator involved in it is monotone with sequentially weakly closed graph

stability analysis of interconnected systems with mixed negative imaginary and small gain properties in this letter an analytical framework is proposed to examine stability of two stable linear time invariant systems interconnected in positive feedback where the systems have mixed properties of negative imaginary and small gain using the notion of dissipativity the interconnection of systems is guaranteed to be finite gain stable under the condition that the dc loop gain is contractive this work builds on griggs and exploits a new set of frequency dependent triplets that was introduced in above reference to mix two unconditional stability statements i e small gain and passivity unlike the above reference the present work explores the important question of how a conditional stability statement as needed when two negative imaginary systems are connected in a feedback loop can be mixed with an unconditional stability statement as needed when two contractive systems are connected in a feedback loop the usefulness of the proposed analytical framework is demonstrated via a numerical example

technical communique dissipative analysis and control of state space symmetric systems the paper addresses the problem of analysis and static output feedback control synthesis for strict quadratic dissipativity of linear time invariant systems with state space symmetry as a particular case of dissipative systems we consider the mixed h and positive real performance criterion and we develop an explicit expression for calculating the h norm of these systems subsequently an explicit parametrization of the static output feedback control gains that solve the mixed h and positive real performance problem is obtained numerical examples demonstrate the use and computational advantages of the proposed explicit solutions

an observer based control scheme using negative imaginary theory this paper presents a full order state observer based scheme that transforms a causal lti minimally realized system into a strongly strict negative imaginary system by defining an auxiliary output based on the observed states the auxiliary output is used for closed loop control with positive feedback invoking the internal stability condition of negative imaginary theory a set of lmi conditions is derived that determines the value of a design parameter required for the proposed scheme to transform a given system with a minimal state space realization into a strongly strict negative imaginary system the proposed scheme is applicable for both stable unstable and square non square systems in this paper the framework is further explored for robustness analysis of uncertain systems and numerical examples are given to elucidate effectiveness of the proposed results

multiplicative iterative algorithms for convex programming abstract we study multiplicative iterative algorithms for the minimization of a differentiable convex function defined on the positive orthant of r n if the objective function has compact level sets and has a locally lipschitz continuous gradient then these algorithms converge to a solution of the minimization problem moreover the convergence is nearly monotone in the sense of kullback leibler divergence special cases of these algorithms have been applied in position emission tomography and are formally related to the em algorithm for position emission tomography

partial controllability concepts the main results in theory of controllability are formulated for deterministic or stochastic control systems given in a standard form i e given as a first order differential equation driven by an infinitesimal generator of strongly continuous semigroup in an abstract hilbert or banach space and disturbed by a deterministic function or by a white noise process at the same time some deterministic or stochastic linear systems can be written in a standard form if the state space is enlarged respectively the ordinary controllability conditions for them are too strong since they assume extended state space it is reasonable to introduce partial controllability concepts which assume original state space in this paper we study necessary and sufficient conditions of partial controllability for deterministic and stochastic linear control systems given in a standard form and their implications to particular cases

a generalized negative imaginary lemma and riccati based static state feedback negative imaginary synthesis abstract in this paper we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition then an algebraic riccati equation method is given to determine if a system is negative imaginary also a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property the controller synthesis procedure is based on the proposed negative imaginary lemma using this procedure the closed loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty such as in the case of unmodeled spill over dynamics in a lightly damped flexible structure a numerical example is presented to illustrate the usefulness of the results

a negative imaginary lemma and the stability of interconnections of linear negative imaginary systems the note is concerned with linear negative imaginary systems first a previously established negative imaginary lemma is shown to remain true even if the system transfer function matrix has poles on the imaginary axis this result is achieved by suitably extending the definition of negative imaginary transfer function matrices secondly a necessary and sufficient condition is established for the internal stability of the positive feedback interconnections of negative imaginary systems meanwhile some properties of linear negative imaginary systems are developed finally an undamped flexible structure example is presented to illustrate the theory

properties and stability analysis of discrete time negative imaginary systems abstract this paper is concerned with discrete time negative imaginary dt ni functions first a new definition of dt ni functions is introduced then by means of the relations between discrete time positive real and dt ni functions two different versions of dt ni lemmas are established to characterize the dt ni properties based on state space realizations also a necessary and sufficient condition is presented to guarantee the internal stability of positive feedback interconnected dt ni systems meanwhile some other properties of dt ni functions are studied several numerical examples are presented to illustrate the main results of this paper compared to the previous results our results remove the symmetric assumption in rational case

a class of stable square root nonlinear information filters information filters can process nonlinear systems with uncertain prior knowledge and the particular square root form of adaptive filters can improve numerical stability based on a square root decomposition of information matrix and an extra positive definite matrix the unscented transform and the cubature rule are applied to the information filtering architecture for nonlinear estimation a class of stable square root nonlinear information filters is then proposed in this technical note in addition the boundedness of their estimation errors is also proven results from simulations of filtering a chaotic map demonstrate that the proposed square root nonlinear filters can improve numerical stability and has better filtering performance than other information filters

on non proper negative imaginary systems abstract this paper studies non proper negative imaginary systems first the concept of negative imaginary transfer functions that may have poles at the origin and infinity is introduced then a generalized lemma is presented to provide a sufficient condition to characterize the non proper negative imaginary properties of systems the generalized lemma is given in terms of complex variable s in the quarter domain of analyticity compared to previous results our result removes the symmetric restriction also a new relationship is established between lossless negative imaginary and lossless positive real transfer function matrices by using a minor decomposition the results in this paper give us the possibility to address non proper and non symmetric descriptor systems with negative imaginary frequency response several examples are presented to illustrate the results

a generalization of the proximal point algorithm the problem considered in this paper is to find a solution to the generalized equation 0 in t x y where t is a maximal monotone operator on the product h 1 times h 2 of two hilbert spaces h 1 and h 2 we give a generalization of the proximal map and the proximal point algorithm in which the proposed iterative procedure is based on just one variable applying to convex programming problems instead of adding a quadratic term for all variables as in the proximal point algorithm a quadratic term for a subset of variables is added this paper proves that under a mild assumption our algorithm has the same convergence properties as the regular proximal point algorithm

on positive realness negative imaginariness and h control of state space symmetric systems this paper studies positive realness and negative imaginariness of state space symmetric systems first a necessary and sufficient condition is provided for a state space symmetric system to be respectively positive real and negative imaginary then an explicit formulation for the infimum of h norm under a mixed h and negative imaginary performance and an explicit solution for the associated optimal static output feedback control gain are developed based on symmetric negative imaginary theorem finally three examples are provided to illustrate the main results of the paper

operator splittings bregman methods and frame shrinkage in image processing we examine the underlying structure of popular algorithms for variational methods used in image processing we focus here on operator splittings and bregman methods based on a unified approach via fixed point iterations and averaged operators in particular the recently proposed alternating split bregman method can be interpreted from different points of view as a bregman as an augmented lagrangian and as a douglas rachford splitting algorithm which is a classical operator splitting method we also study similarities between this method and the forward backward splitting method when applied to two frequently used models for image denoising which employ a besov norm and a total variation regularization term respectively in the first setting we show that for a discretization based on parseval frames the gradient descent reprojection and the alternating split bregman algorithm are equivalent and turn out to be a frame shrinkage method for the total variation regularizer we also present a numerical comparison with multistep methods

a negative imaginary lemma without minimality assumptions and robust state feedback synthesis for uncertain negative imaginary systems abstract negative imaginary systems have important engineering applications for example in lightly damped flexible structures with collocated position sensors and force actuators this paper proposes a new negative imaginary lemma for generalized negative imaginary systems where the minimality assumption is relaxed the proposed lemma facilitates both analysis and synthesis for negative imaginary systems invoking this result a systematic robust static state feedback synthesis method is proposed in an lmi framework for systems with strictly negative imaginary uncertainty e g unmodeled spill over dynamics in a lightly damped flexible structure assuming all states are available for state feedback two numerical examples are given to show the usefulness of the proposed results

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

necessary and sufficient conditions on negative imaginariness for interval siso transfer functions and their interconnection this article is concerned with the study of the negative imaginariness of interval transfer functions necessary and sufficient conditions for strictly negative imaginary interval transfer functions subject to parameter uncertainty are first derived based on kharitonov s theorem then certain properties of the interconnection system with negative imaginary interval transfer functions are analyzed finally an example is provided to illustrate the main results of the article

decentralized control of symmetric systems abstract this paper is concerned with the stabilizability problem of symmetric systems by means of decentralized controllers it is shown that the set of decentralized fixed modes of a symmetric system is equal to the set of uncontrollable and unobservable modes of the system and implies that the stabilizability of a state space symmetric system by means of a decentralized controller is equivalent to the stabilizability for the system by means of a centralized controller

a note on the existence of zeroes of convexly regularized sums of maximal monotone operators abstract many algorithms for solving the problem of finding zeroes of a sum of two maximal monotone operators t1 and t2 have regularized subproblems of the kind 0 t1 x t2 x d x where d is a convex function we develop an unified analysis for existence of solutions of these subproblems through the introduction of the concept of convex regularization which includes several well known cases in the literature finally we establish conditions either on d or on the operators which assure solvability of the subproblems

comments on design of strictly positive real systems using constant output feedback a recent publication states and proves the conditions under which a linear time invariant system with state space realization a b c can be made strictly positive real via constant output feedback this note is intended to briefly present the development of the proof and to give due credit to the first proofs of this statement

stabilization and h control of symmetric systems an explicit solution abstract in this paper we address the h control analysis the output feedback stabilization and the output feedback h control synthesis problems for state space symmetric systems using a particular solution of the bounded real lemma for an open loop symmetric system we obtain an explicit expression to compute the h norm of the system for the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state space symmetric systems for the h control synthesis problem we derive an explicit expression for the optimally achievable closed loop h norm and the optimal control gains extension to robust and positive real control of such systems are also examined these results are obtained from the linear matrix inequality formulations of the stabilization and the h control synthesis problems using simple matrix algebraic tools

on the convergence of the exponential multiplier method for convex programming in this paper we analyze the exponential method of multipliers for convex constrained minimization problems which operates like the usual augmented lagrangian method except that it uses an exponential penalty function in place of the usual quadratic we also analyze a dual counterpart the entropy minimization algorithm which operates like the proximal minimization algorithm except that it uses a logarithmic entropy proximal term in place of a quadratic we strengthen substantially the available convergence results for these methods and we derive the convergence rate of these methods when applied to linear programs

entropic proximal mappings with applications to nonlinear programming we introduce a family of new transforms based on imitating the proximal mapping of moreau and the associated moreau yosida proximal approximation of a function the transforms are constructed in terms of the a divergence functional a generalization of the relative entropy and of bregman s measure of distance an analogue of moreau s theorem associated with these entropy like distances is proved we show that the resulting entropic proximal maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems applications of the results to the construction of generalized augmented lagrangians for nonlinear programs and the minimax problem are presented