dc programming and dca thirty years of developments the year 2015 marks the 30th birthday of dc difference of convex functions programming and dca dc algorithms which constitute the backbone of nonconvex programming and global optimization in this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools the survey is comprised of three parts in the first part we present a brief history of the field while in the second we summarize the state of the art results and recent advances we focus on main theoretical results and dca solvers for important classes of difficult nonconvex optimization problems and then give an overview of real world applications whose solution methods are based on dca the third part is devoted to new trends and important open issues as well as suggestions for future developments

corrections to bisection algorithm for computing the frequency response gain of sampled data systems infinite dimensional congruent transformation approach this paper derives a bisection algorithm for computing the frequency response gain of sampled data systems with their intersample behavior taken into account the properties of the infinite dimensional congruent transformation i e the schur complement arguments and the sylvester law of inertia play a key role in the derivation specifically it is highlighted that counting up the numbers of the negative eigenvalues of self adjoint operators is quite important for the computation of the frequency response gain this contrasts with the well known arguments on the related issue of the sampled data h sub spl infin problem where the key role is played by the positivity of operators and the loop shifting technique the effectiveness of the derived algorithm is demonstrated through a numerical example

large scale molecular optimization from distance matrices by a d c optimization approach a so called dca method based on a d c difference of convex functions optimization approach algorithm for solving large scale distance geometry problems is developed different formulations of equivalent d c programs in the l 1 approach are stated via the lagrangian duality without gap relative to d c programming and new nonstandard nonsmooth reformulations in the l infty approach resp the l 1 l infty approach are introduced substantial subdifferential calculations permit us to compute sequences of iterations in the dca quite simply the computations actually require matrix vector products and only one cholesky factorization resp with an additional solution of a convex program in the l 1 approach resp the l 1 l infty approach and allow the exploitation of sparsity in the large scale setting two techniques respectively using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure are investigated in order to compute a good starting point for the dca finally many numerical simulations of the molecular optimization problems with up to 12567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations the globality of found solutions and the robustness and efficiency of our algorithms

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm the i th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one we show that near a local maximum however the largest singular value has a lipschitz second derivative but need not have a third derivative using this regularity result we give a quadratically convergent algorithm for computing the l norm of a transfer matrix

minimum action method for the kardar parisi zhang equation we apply a numerical minimum action method derived from the wentzell freidlin theory of large deviations to the kardar parisi zhang equation for the height profile of a growing interface in one dimension we find that the transition pathway between different height configurations is determined by the nucleation and subsequent propagation of facets or steps corresponding to moving domain walls or growth modes in the underlying noise driven burgers equation this transition scenario is in accordance with recent analytical studies of the one dimensional kardar parisi zhang equation in the asymptotic weak noise limit we also briefly discuss transitions in two dimensions

new conic sectors for sampled data feedback systems abstract new conic sectors are presented which can be used to analyze sampled data feedback systems first a cone is constructed which contains just the sampler next by using conic sector results for interconnected systems the cone is propagated to the entire sampled data operator the new cone result is less conservative has a smaller radius and removes the restriction of an open loop stable sampled data compensator

multivariable feedback a personal reminiscence my involvement in the development of linear multivariable feedback theory and its applications is described from a highly personal point of view concentrating on the origin of key ideas and their development in association with many colleagues detailed references are given to all my papers and books and explanatory notes provided where further material may be found

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

a sensitivity tradeoff for plants with time delay in this paper we show that the sensitivity function of a scalar feedback system must satisfy an integral constraint when the open loop transfer function is strictly proper and contains a time delay the integral constraint is identical to the classical bode sensitivity integral which holds provided the open loop gain has greater than a one pole rolloff the design implications of the two constraints are somewhat different however and this difference is discussed the relation between open loop phase and a conflict between bandwidth and sensitivity properties is explored

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

comments on robust tracking control for rigid robotic manipulators in the original paper by m zhihong and m palaniswami ibid vol 39 p 154 9 1994 a robust tracking control using a nominal feedback controller and a variable structure compensator for a rigid robotic manipulator with uncertain dynamics was presented it is assumed that the model uncertainty is bounded the author claims that this assumption is not necessarily correct and that the results obtained are therefore untenable

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

robust stabilization of sampled data systems to structured lti perturbations sampled data systems with stable structured lti linear time invariant perturbations in the half plane algebra applied to the continuous time plant are considered necessary and sufficient conditions are derived for robust l sub 2 stability of such systems an example is provided to illustrate the results which shows that the small gain theorem can be an extremely conservative robustness test in this sampled data context

on the gain margin improvement using dynamic compensation based on generalized sampled data hold functions this paper shows that the use of dynamic compensation based on generalized sampled data hold functions gshf can arbitrarily improve the gain margin for continuous time nonminimum phase linear systems the gshf compensator is a particular periodic digital controller the effect of sampling period on the gain margin is analyzed furthermore it is proved that under a mild condition the gain margin improvement can be achieved without forcing the sampling period small an important advantage of periodic compensation over lti compensation is found to be the capability of reducing conflict between gain margin and sensitivity

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

logarithmic integrals interpolation bounds and performance limitations in mimo feedback systems we study performance limitation issues found in linear multivariable feedback systems our main contributions include bode and poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions these results characterize and quantify explicitly how open loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance the role of time delay is also studied in this context most notably we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros and in particular on how pole and zero directions are aligned the latter is characterized by angles measuring the mutual orientation between zero and pole directions and it is shown to play a crucial role in multivariable system design

instantons in the burgers equation the instanton solution for the forced burgers equation is found this solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent that is for large positive velocity differences the results agree with the one found recently by polyakov who used the operator product conjecture if this conjecture is true then our wkb asymptotics of the wyld functional integral should be exact to all orders of perturbation expansion around the instanton solution we also generalized our solution for the arbitrary dimension of the burgers kpz equation as a result we found the asymptotics of the angular dependence of the velocity difference probability distribution function textcopyright 1996 the american physical society

convergence of a subgradient method for computing the bound norm of matrices abstract let and be any norms on r m and r n respectively we study a subgradient method for computing the associated bound norm s a sup ax x 1 a nonconvex optimization problem it is proved that homodual method converges when one of the norms and is polyhedral

robust stability and performance analysis of sampled data systems the authors consider the robust stability and performance analysis of a general linear interconnection of a continuous time plant and a discrete time controller via sample and hold devices they obtain necessary and sufficient conditions for the robust stability of the feedback system consisting of an uncertain plant represented as a known linear time invariant nominal model with time varying norm bounded uncertainty and a shift invariant discrete time controller these results show that the condition arising from the small gain theorem is necessary and sufficient when the uncertainties belong to a class of linear systems also similar results are derived for a case of structured uncertainty both l sub 2 and l sub infinity signal norms are considered robust performance analysis problem is solved by converting it to a robust stability analysis problem

performance analysis of sampled data control systems abstract this brief paper considers multivariable sampled data systems with bandlimited exogenous inputs a procedure is given for the computation of the l 2 induced input to output norm of such a system in addition sufficient conditions are given for input output stability the digital implementation of an analog controller will generally degrade the performance of the closed loop system in light of this two examples are given to illustrate how this degradation is quantified as a function of the sampling period in general this allows the determination of the minimum sampling rate required to recover the performance designed for using the original continuous time setup

robustness of zero shifting via generalized sampled data hold functions in this paper we study robustness and sensitivity properties of a sampled data feedback system with a generalized sampled data hold function gshf we argue that shifting non minimum phase zeros using gshf control can lead to difficulties unless the zero is outside the closed loop bandwidth

robust stability of sampled data systems under possibly unstable additive multiplicative perturbations applies the fr operator technique to the robust stability problem of sampled data systems against additive multiplicative perturbations where a reasonable class of perturbations consists of unstable as well as stable ones assuming that the number of unstable modes of the plant does not change we show that a small gain condition in terms of the fr operator representation which is actually equivalent to a small gain condition in terms of the l sub 2 induced norm is still necessary and sufficient for the sampled data system to be robustly stable against h periodic perturbations in spite of their possible instability the result is derived by a nyquist type of arguments next a necessary and sufficient condition for robust stability against linear time invariant lti perturbations is also given furthermore we show that if the plant is either single input or single output the condition can be reduced to a readily testable form finally we clarify when the small gain condition becomes a particularly poor measure for robust stability

kolmogorov turbulence in a random force driven burgers equation anomalous scaling and probability density functions high resolution numerical experiments described in this work show that velocity fluctuations governed by the one dimensional burgers equation driven by a white in time random noise with the spectrum ensuremath vert f k mathrm ensuremath vert 2 ifmmode bar else textasciimacron fi ensuremath propto mathit k mathrm ensuremath 1 exhibit a biscaling behavior all moments of velocity differences mathit s mathit n mathrm ensuremath le 3 r ensuremath vert u x r u x mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath equiv ensuremath vert ensuremath delta u mathrm ensuremath vert mathit n ifmmode bar else textasciimacron fi ensuremath propto mathit r mathit n 3 while mathit s mathit n g3 r ensuremath propto mathit r mathit n ensuremath xi with ensuremath xi mathit n ensuremath approxeq 1 for real ng0 chekhlov and yakhot phys rev e 51 r2739 1995 the probability density function which is dominated by coherent shocks in the interval ensuremath delta u0 is scrp ensuremath delta u r ensuremath propto ensuremath delta u mathrm ensuremath mathit q with q ensuremath approxeq 4 a phenomenological theory describing the experimental findings is presented

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

instanton filtering for the stochastic burgers equation we address the question of whether one can identify instantons in direct numerical simulations of the stochastically driven burgers equation for this purpose we first solve the instanton equations using the chernykh stepanov method 2001 phys rev e 64 026306 these results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of chernykh and stepanov with remarkable agreement

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

existence conditions and properties of the frequency response operators of continuous time periodic systems the definition of the frequency response operator via the steady state analysis in finite dimensional linear continuous time periodic fdlcp systems is revisited it is shown that the frequency response operator is guaranteed to be well defined only densely on the linear space l2 which is different from the usual understanding fortunately however it turns out that this frequency response operator can have an extension onto l2 so that the equivalence between the time domain h2 norm respectively the l2 induced norm and the frequency domain h2 norm respectively the h infty norm of the frequency response operator is recovered under some stronger assumptions it is also shown that the frequency response operator can be viewed as a bounded operator from l1 to l1 which can also be established via the steady state analysis

paper sensitivity integrals for multivariable discrete time systems the purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discrete time systems we present extensions to both the classical poisson integral and bode s sensitivity integral formulae the main utility of these results is that they can be used to quantify design limitations that arise in multivariable discrete time systems due to such system characteristics as open loop unstable poles and nonminimum phase zeros and to such design requirements as stability and bandwidth constraints these formulae are similar to those for multivariable continuous time systems obtained elsewhere and they reveal that feedback design limitations depend on directionality properties of the sensitivity function and on those of unstable poles and nonminimum phase zeros in the open loop transfer function

estimating hadamard operator norms with application to triangular truncation abstract it is shown how the power method can be used to estimate hadamard operator norms its application to the problem of finding the norm of the triangular truncation operator is considered

large negative velocity gradients in burgers turbulence we consider one dimensional burgers equation driven by large scale white in time random force the tails of the velocity gradients probability distribution function pdf are analyzed by saddle point approximation in the path integral describing the velocity statistics the structure of the saddle point instanton that is the velocity field configuration realizing the maximum of probability is studied numerically in details the numerical results allow us to find analytical solution for the long time part of the instanton its careful analysis confirms the result of balkovsky et al phys rev lett 78 1452 1997 based on short time estimations that the left tail of pdf has the form mathrm ln mathcal p u x ensuremath propto ensuremath u x 3 2

generalization of integral constraints on sensitivity to time delay systems extends results on integral constraints to time delay systems key technical issues in this extension include properties of nonminimum phase zeros the convergence of blaschke products and the poisson integral the results have implications for scalar and multivariable control system design as well as in filtering problems

estimating the largest elements of a matrix we derive an algorithm for estimating the largest p a a a 1 values a ij or a ij for an m a a n matrix a along with their locations in the matrix the matrix is accessed using only matrixa a a vector or matrixa a a matrix products for p 1 the algorithm estimates the norm a m max i j a ij or max i j a ij the algorithm is based on a power method for mixed subordinate matrix norms and iterates on n a a t matrices where t a a a p is a parameter for p t 1 we show that the algorithm is essentially equivalent to rook pivoting in gaussian elimination we also obtain a bound for the expected number of matrixa a a vector products for random matrices and give a class of counter examples our numerical experiments show that for p 1 the algorithm usually converges in just two iterations requiring the equivalent of 4t matrixa a a vector products and for t 2 the algorithm already provides excellent estimates that are usually within a factor 2 of the largest element and frequently exact for p 1 we incorporate deflation to improve the performance of the algorithm experiments on real life datasets show that the algorithm is highly effective in practice

algorithmes de calcul du maximum des formes quadratiques sur la boule unite de la norme du maximum certaines methodes directes et indirectes pour le calcul de max x t ax x 1 sont etudiees les methodes directes sont basees sur les proprietes particulieres des normes 1 2 et ces methodes sont tres simples mais ne s appliquent qu a certaines familles de matrices la methode indirecte est la methode autoduale introduite dans 25 26 avec 1 dans ce cas le choix du vecteur initial pour qu il y ait convergence vers une solution optimale est largement discute some direct and indirect methods are studied for computing max x t ax x 1 wherea is symmetric definite positive direct methods are constructed using particular properties of 1 2 norms these methods are very simple but uniquely suitable to certains families of matrices the indirect method is the autodual method introduced in 25 26 29 with 1 in this case the problem of choosing an initial vector so that convergence of the iterative sequence occurs to an optimal solution is largely discussed

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

tradeoffs in linear time varying systems an analogue of bode s sensitivity integral bode s integral characterizes some of the tradeoffs imposed by the open loop dynamics of a linear time invariant system here we present an analogous time domain characterization applicable to time varying systems

crossover frequency limitations in mimo nonminimum phase feedback systems this paper investigates limitations and design tradeoffs of the closed loop sensitivity performance of linear time invariant nonminimum phase uncertain multiple input multiple output plants with i inputs and m outputs where m spl les l it is shown that if rows i sub 1 i sub k of the plant transfer function form a k spl times l nonminimum phase transfer matrix and if the design is such that the sensitivity gain of k 1 rows among the rows i sub 1 i sub k of the closed loop transfer function is low then by necessity the sensitivity gain of the remaining row is high this sensitivity constraint is quantified with the help of the crossover frequency restriction of a specially constructed single input single output transfer function that includes the right half plane zeros and poles of the k spl times l transfer matrix

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

sensitivity integrals and transformation techniques a new perspective we provide alternative proofs for the classical bode and poisson type sensitivity integrals and their extensions for both continuous time and discrete time systems our derivation uses the well known properties of laplace and z transformations this derivation helps establish a connection between bode and poisson type sensitivity integrals and laplace and z transforms hence providing an alternative perspective in interpreting these fundamental integral results

spectral analysis and singular value computations of the noncompact frequency response and compression operators in sampled data systems this paper is motivated by the problem of computing the frequency response gain of general sampled data systems with noncompact frequency response operators we first show that with the j unitary transformation the computation in the noncompact operator case can be reduced in principle to that in the compact operator case to which an existing efficient and reliable bisection method can be applied at the same time however we point out that there arise some critical problems in this reduction to the compact case which could be serious enough to invalidate the apparent success in the reduction through some spectral analysis of operators involving or related to the frequency response operators we eventually prove that these critical problems can be circumvented after all and we give an explicit result that shows how to compute the frequency response gain with a bisection method dealing only with finite dimensional matrices extending the arguments we also give a bisection method to compute the singular values of the frequency response operators and the associated compression operators

generalized sample hold functions frequency domain analysis of robustness sensitivity and intersample difficulties over the past 5 years there has been substantial interest in the use of generalized sample hold functions for control in this correspondence the authors use a tool which is novel in this context namely amplitude modulation theory the authors employ this tool to analyze the quantitative and qualitative features of the intersample behavior in a frequency domain setting this offers new theoretical and practical insights into the method the authors conclusion is that the perceived benefits come at substantial cost which makes its practical use questionable

inherent design limitations for linear sampled data feedback systems there is a well developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller this theory describes limitations on achievable performance present when the plant has non minimum phase zeros unstable poles and or time delays the parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants this paper develops a theory of design limitations for sampled data feedback systems wherein the response of the analogue system output is considered this is done using the fact that the steady state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental this fact allows fundamental sensitivity and complementary sensitivity functions that re

analysis and design for ill conditioned plants part 1 lower bounds on the structured singular value abstract a study is made of a special case of the robust performance problem given by freudenberg 1989 when the weightings used to describe the uncertainty and performance specifications vary only with frequency then it is possible to strengthen the results in the above mentioned work by deriving both upper and lower bounds upon the structured singular value both sets of bounds are stated in terms of the coupling coefficients introduced by freudenberg 1989 and essentially yield necessary and sufficient conditions for the structured singular value to be small this information is used to suggest a strategy for compensator design to achieve robust performance despite plant ill conditioning applying this strategy to an example it can be seen how the design trade offs quantified by the bode gain phase relation manifest themselves in the robust performance problem finally the design is compared with one obtained using the synthesis approach

multivariable gain phase and sensitivity integral relations and design trade offs we present several extensions of the classical bode integral relations to multivariable systems the main contributions are an extended gain phase integral formula and an extended sensitivity integral relation these results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems in particular they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single input single output systems both results show that design limitations in multivariable systems depend strongly upon certain directionality properties

ripple free deadbeat control for sampled data systems in this paper the ripple free deadbeat control problem for sampled data systems is considered this control objective is to settle the error to zero for all time after some finite settling time in other words to eliminate the ripples between the sampling instants in deadbeat control of sampled data systems the necessary and sufficient conditions to solve this problem are derived and related to the system type and a method of constructing the ripple free deadbeat control system is presented

brief upper and lower bounds of the frequency response gain of sampled data systems this paper gives three independent methods to compute the upper and lower bounds of the norm of the frequency response operator of sampled data systems they are useful first in its own right as an approximate method and more importantly as a method to provide an effective set of the initial upper and lower bounds for the recently developed bisection algorithm it is also shown that our arguments have very close relationships to the sampled data h 2 and h problems

integral constraints and performance limits on complementary sensitivity discrete time systems abstract we study performance limitation issues for multivariable discrete time feedback systems the complementary sensitivity function is employed as a performance measure and bode and poisson type integral inequalities and h type performance limits are derived the results exhibit frequency dependent constraints as well as best achievable limits on the complementary sensitivity function which are shown to be determined by nonminimum phase zeros unstable poles and time delays in particular the directions of such zeros and poles are seen to play a central role to this effect

logarithmic integrals and system dynamics an analogue of bode s sensitivity integral for continuous time time varying systems a new time domain interpretation of bode s integral is presented this allows for a generalization to the class of time varying systems which possess an exponential dichotomy it is shown that the sensitivity function is constrained on average by the spectral values in the dichotomy spectrum of the antistable component of the open loop dynamics

brief paper generalized sampled data hold functions with asymptotic zero order hold behavior and polynomic reconstruction we present a generalized sampled data hold function that combines arbitrary z domain zero placement ability with zero order hold behavior under constant input thus exhibiting minimal intersample ripple by design our hold can be regarded as a generalization of the fractional order hold with a polynomial instead of a simple linear pattern and therefore with as many tuning parameters as desired moreover the polynomial approach turns out to provide a simple mechanism of control energy minimization among other benefits all these features help to achieve qualitatively better perfect model referencing because problematic sampling z zeros or the intersample issues of the conventional generalized holds need no longer be endured

connection between nonlinear energy optimization and instantons how systems transit between different stable states under external perturbation is an important practical issue we discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems in the context of the one dimensional swift hohenberg equation which has multiple stable equilibria we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances minimal seeds for transition to specific attractors from the ground state then after generalizing the technique to consider multiple equally spaced in time perturbations it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations importantly we find that the key features of the instanton can be captured by a low number of discrete perturbations typically one perturbation per basin of attraction crossed this suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton

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

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

gain phase relationships for discrete time systems the classical gain phase relations obtained by bode for minimum phase continuous time systems can be used to establish bounds on the performance of feedback systems it is shown that completely analogous relations can be obtained for discrete time systems and can be used for the same purpose in particular the limitations imposed by plant zeros and poles which lie outside the unit circle are investigated for both genuine discrete time systems namely those defined by difference equations and for sampled data systems one of the results shows that sampled data systems may practically be unstabilizable if their unstable poles lie outside a particular circular region the intended application of these results is to allow preliminary screening of proposed specifications for feasibility in a cad system before embarking on a detailed design

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

analytic properties of the singular values of a rational matrix abstract it is shown that the singular values of a rational matrix g s where s x jy are locally a set of real analytic functions of x and y a brief discussion is given of their global properties

instantons and intermittency we describe the method for finding the non gaussian tails of the probability distribution function pdf for solutions of a stochastic differential equation such as the convection equation for a passive scalar the random driven navier stokes equation etc the existence of such tails is generally regarded as a manifestation of the intermittency phenomenon our formalism is based on the wkb approximation in the functional integral for the conditional probability of large fluctuation we argue that the main contribution to the functional integral is given by a coupled field force configuration char22 the instanton as an example we examine the correlation functions of the passive scalar u advected by a large scale velocity field ensuremath delta correlated in time we find the instanton determining the tails of the generating functional and show that it is different from the instanton that determines the probability distribution function of high powers of u we discuss the simplest instantons for the navier stokes equation textcopyright 1996 the american physical society

brief damping of harmonic disturbances in sampled data systems parameterization of all optimal controllers optimal damping of harmonic disturbances of known frequencies is studied for sampled data systems a sampled data output feedback controller is designed to minimize the intersample variations of the controlled variable the set of all stabilizing optimal controllers is obtained in terms of the youla parameterization and a set of interpolation conditions at the disturbance frequencies which ensure that the stationary cost is minimized

design of a stable state feedback controller based on the multirate sampling of the plant output a new type of controller is proposed which detects the ith plant output n sub i times during a period of t sub 0 and changes the plant inputs once during t sub 0 it is shown that an arbitrary state feedback can be realized by such controllers if the plant is observable this implies for instance that arbitrary symmetric pole assignment is possible if the plant is controllable it is also shown that if the plant has no zeros at the origin the state transition matrix of the controller itself can be set arbitrarily without changing the state feedback to be realized that is to say inversely expressed any state feedback can be equivalently realized by a controller with any prescribed degree of stability

frequency responses for sampled data systems their equivalence and relationships abstract there are two ways to introduce the notion of frequency response for sampled data systems one is based on the so called lifting and the other based on an interpretation of steady state response in terms of impulse modulation this paper proves the equivalence of these two notions in particular it establishes a more direct link of the second approach to the h norm and also provides the first approach with a natural interpretation of steady state response as an infinite sum of sinusoidal signals this study also leads to a comprehensive account of impulse modulation from the lifting viewpoint

constraints on sensitivity characteristics in linear multivariable discrete time control systems abstract this paper is concerned with the limitations on the sensitivity characteristics for linear multivariable discrete time control systems some integral type constraints and the lower bounds of the weighted h norm imposed by the unstable poles the unstable zeros and their directions of the open loop system or the given plant are developed by a factorization approach these constraints and bounds which are tighter than those in the previous work are also characterized by the state space representations the descriptions are closely related to a special type of the algebraic riccati equation and the relation between the sampling period and the sensitivity performance is discussed for digital control systems the constraints on the more general function mixed sensitivity function are investigated in a similar way

design of ripple free multivariable robust servomechanisms the application of standard robust servomechanism theory to sampled data systems guarantees asymptotic tracking at sampling instances and between sampling instants the output will normally contain ripple in this note the robust servomechanism theory is extended to sampled data systems and a technique is proposed for ripple free tracking of sinusoids and polynomials it is shown that a continuous internal model is necessary and sufficient to provide ripple free response

periodic discrete time systems stability analysis and robust control using zero placement a simple method based on floquet theory for obtaining the characteristic equation and hence stability of periodic discrete time systems is presented using this method it is shown that 2 periodic controllers can be used to relocate the zeros of siso plants some examples are considered to illustrate the use of 2 periodic controllers for robust control of finite gain margin problems

on a key sampling formula relating the laplace and l transforms this note provides a new rigorous derivation of a key sampling formula for discretizing an analogue system the required conditions are formulated in time domain and give a clear characterization of the classes of signals and systems to which the formula applies

input output stability of sampled data systems the authors study the l sub p input output stability of a continuous time controller using the usual arrangement of periodic sampling and zero order hold it is noted that even if the hybrid system is exponentially stable this arrangement does not yield l sub p 1

finite time behavior of inner systems in this paper we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties for the class of linear time invariant dynamical systems these characteristics are determined by transmission zeros of the inner factor of the system transfer function the relation between nonminimum phase zeros and hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance the results are used to solve various tracking problems both on finite as well as on infinite time horizons a typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller

l sub 2 induced norms and frequency gains of sampled data sensitivity operators this paper develops exact computable formulas for the frequency gain and l sub 2 induced norm of the sensitivity operator in a sampled data control system with sampled data we refer to a system that combines both continuous time and discrete time signals which is studied in continuous time the expressions are obtained using lifting techniques in the frequency domain and have application in performance and stability robustness analysis taking into account full intersample information