a combined phase i phase ii scaled potential algorithm for linear programming we develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program without the addition of any m terms the method and its lower bounding procedure are particularly simple compared with previous interior algorithms not requiring feasibility

double interpolation for digital to analog conversion interpolative digital to analog converters generate an output that has only a few analog levels they provide fine resolution by oscillating rapidly between these levels in such a manner that the average output represents the value of the applied code here we describe an improved method of interpolating that results in reduced noise in the signal band a theory of the interpolation confirmed by experiments demonstrates that switching between only two levels at 1 3 mhz could provide 16 bit resolution for telephone signals

lpv system analysis via quadratic separator for uncertain implicit systems this paper considers a class of linear systems containing time varying parameters whose behavior is not known exactly we assume that the parameters vary within known intervals and there are known bounds on their rates of variation our objective is to give a computationally verifiable condition that guarantees stability of the system for all possible parameter variations we first point out that the information on the rate bounds can be exploited by considering an augmented system described by an implicit model we then develop a tool called the quadratic separator for stability analysis of uncertain implicit systems using this tool a sufficient stability condition is obtained for the original linear parameter varying system moreover we show that the stability condition obtained is equivalent to the existence of a lyapunov function that depends on the parameters in a linear fractional manner finally the computational aspects of the proposed stability conditions are addressed

on anstreicher s combined phase i phase ii projective algorithm for linear programming anstreicher has proposed a variant of karmarkar s projective algorithm that handles standard form linear programming problems nicely we suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm much of the analysis depends on a two constraint linear programming problem that is a relaxation of the scaled original problem

signal reconstruction via h infty sampled data control theory beyond the shannon paradigm this paper presents a new method for signal reconstruction by leveraging sampled data control theory we formulate the signal reconstruction problem in terms of an analog performance optimization problem using a stable discrete time filter the proposed h performance criterion naturally takes inter sample behavior into account reflecting the energy distributions of the signal we present methods for computing optimal solutions which are guaranteed to be stable and causal detailed comparisons to alternative methods are provided we discuss some applications in sound and image reconstruction

brief an analysis and design method for linear systems subject to actuator saturation and disturbance we present a method for estimating the domain of attraction of the origin for a 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 this condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis 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 this condition is then extended to determine the invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements the effectiveness of the developed methods is illustrated with examples

refined error analysis in second order spl sigma spl delta modulation with constant inputs although the technique of sigma delta spl sigma spl delta modulation is well established in practice for performing high resolution analog to digital a d conversion theoretical analysis of the error between the input signal and the reconstructed signal has remained partial for modulators of order higher than 1 the only rigorous error analysis currently available that matches practical and numerical simulation results is only applicable to a very special configuration namely the standard and ideal k bit k loop spl sigma spl delta modulator moreover the error measure involves averaging over time as well as possibly over the input value at the second order it is known in practice that the mean squared error decays with the oversampling ratio spl lambda at the rate o spl lambda sup 5 in this paper we introduce two new fundamental results in the case of constant input signals we first establish a framework of analysis that is applicable to all second order modulators provided that the built in quantizer has uniformly spaced output levels and that the noise transfer function has its two zeros at the zero frequency in particular this includes the one bit case a rigorous and deterministic analysis of which is still not available this generalization has been possible thanks to the discovery of the mathematical tiling property of the state variables of such modulators the second aspect of our contribution is to perform an instantaneous error analysis that avoids infinite time averaging until now only an o spl lambda sup 4 type error bound was known to hold in this setting under our generalized framework we provide two types of squared error estimates one that is statistically averaged over the input and another that is valid for almost every input in the sense of lebesgue measure in both cases we improve the error bound to o spl lambda sup 4 5 up to a logarithmic factor for a general class of modulators including some specific ones that are covered in this paper in detail in the particular case of the standard and ideal two bit double loop configuration our methods provide a previously unavailable instantaneous error bound of o spl lambda sup 5 again up to a logarithmic factor

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

absolute stability criteria for multiple slope restricted monotonic nonlinearities absolute stability criteria such as the classical popov criterion guarantee stability for a class of sector bounded nonlinearities although the sector restriction bounds the admissible class of the nonlinearities the local slope of the nonlinearity may be arbitrarily large in this paper the authors derive absolute stability criteria for multiple slope restricted time invariant monotonic nonlinearities like the popov criterion in the single input single output case the authors results provide a simple graphical interpretation involving a straight line in a modified popov plane

a potential function reduction algorithm for solving a linear program directly from an infeasible warm start this paper develops an algorithm for solving a standard form linear program directly from an infeasible warm start i e directly from a given infeasible solution hat x that satisfies a hat x b but hat x ngeqslant 0 the algorithm is a potential function reduction algorithm but the potential function is somewhat different than other interior point method potential functions and is given by f x b q ln c t x b sum limits j 1 n ln x j h j c t x b where q n sqrt n is a given constant h is a given strictly positive shift vector used to shift the nonnegativity constaints andb is a lower bound on the optimal value of the linear program the duality gapc t x b is used both in the leading term as well as in the barrier term to help shift the nonnegativity constraints the algorithm is shown under suitable conditions to achieve a constant decrease in the potential function and so achieves a constant decrease in the duality gap and hence also in the infeasibility in o n iterations under more restrictive assumptions regarding the dual feasible region this algorithm is modified by the addition of a dual barrier term and will achieve a constant decrease in the duality gap and in the infeasibility in o sqrt n iterations

an extension of the potential reduction algorithm for linear complementarity problems with some priority goals abstract we extend the potential reduction algorithm to solve the restricted convex linear complementarity problem lcp x t s 0 x i 0 s j 0 s mx q and 0 x s r n for some index sets i and j in polynomial time the algorithm will either discover that no solution exists or generate a sequence of pairs x k s k which simultaneously drives x k t s k x k i and s k j to zero in particular we discuss how to apply the algorithm to solving the combined phase i phase ii convex lcp and to identifying a vertex linear programming solution

design of delta sigma modulators via generalized kalman yakubovich popov lemma this paper is concerned with the design of delta sigma modulators via the generalized kalman yakubovich popov lemma the shaped noise transfer function ntf is assumed to have infinite impulse response and the optimization objective is minimizing the maximum magnitude of the ntf over the signal frequency band by virtue of the gkyp lemma the optimization of an ntf is converted into a minimization problem subject to quadratic matrix inequalities and then an iterative algorithm is proposed to solve this alternative minimization problem each iteration of the algorithm contains linear matrix inequality constraints only and can be effectively solved by the existing numerical software packages moreover specifications on the ntf zeros are also integrated in the design method a design example demonstrates that the proposed design method has an advantage over the benchmark one in improving the signal to noise ratio

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

brief stability analysis of discrete time systems with actuator saturation by a saturation dependent lyapunov function in this paper stability of discrete time linear systems subject to actuator saturation is analyzed using a saturation dependent lyapunov function this saturation dependent lyapunov function captures the real time information on the severity of actuator saturation and leads to less conservative estimate of the domain of attraction which is based on the solution of an lmi optimization problem numerical examples are presented to show the effectiveness of the proposed method

multiple description coding by dithered delta sigma quantization we address the connection between the multiple description md problem and delta sigma quantization the inherent redundancy due to oversampling in delta sigma quantization and the simple linear additive noise model resulting from dithered lattice quantization allow us to construct a symmetric and time invariant md coding scheme we show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity the entropy rate of the dithered delta sigma quantization scheme approaches the symmetric two channel md rate distortion function for a memoryless gaussian source and mean square error mse fidelity criterion at any side to central distortion ratio and any resolution in the optimal scheme the infinite order noise shaping filter must be minimum phase and have a piecewise flat power spectrum with a single jump discontinuity an important advantage of the proposed design is that it is symmetric in rate and distortion by construction so the coding rates of the descriptions are identical and there is therefore no need for source splitting

technical communique lmi relaxations for robust stability of linear systems with saturating controls a method is proposed for studying robust stability of uncertain linear systems subject to actuator saturation it hinges upon successive lmi relaxations and therefore can readily be implemented using widely available numerical tools the technique is illustrated on two examples and is shown to significantly improve previously published results

augmented lyapunov function approach to l2 gain analysis for discrete time systems with saturation nonlinearities abstract this paper presents a method for l 2 gain analysis of discrete time systems with saturation nonlinearity by using an augmented lyapunov functional by using sector and slope properties of the saturation nonlinearity the l 2 gain performance criterion is obtained via lmi formulation the proposed lyapunov function have augmented quadratic form with states as well as the sector bounded nonlinear function numerical examples show that the proposed method is less conservative than the existing ones

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

anti windup based dynamic controller synthesis for nonlinear systems under input saturation this paper describes the design of dynamic controller and static anti windup compensator awc for lipschitz nonlinear systems under input saturation global and local awc based control schemes for stabilization of the nonlinear systems are proposed and necessary conditions for feasibility of the control approaches are investigated a one step approach for simultaneous design of h controller and awc by means of linear matrix inequalities lmis is presented herein which supports multi objective synthesis to attain stabilization or tracking robustness against disturbance and noise and penalization of large and high frequency control signals this multi objective synthesis can be accomplished by incorporating design weights as commonly used in the standard h control theory to design a performance oriented anti windup based control scheme lmis for the global control of the nonlinear systems subject to input saturation are derived by application of a quadratic lyapunov function the lipschitz condition the global sector condition l 2 gain reduction substantial matrix algebra and variable transformation in order to cope with unstable and oscillatory nonlinear systems lmi based local results are established using a local sector condition additional conditions are derived by incorporating properties of the saturation function to ensure well posedness of the controller two simulation examples are provided to show the effectiveness of the proposed control schemes for control of stable and chaotic nonlinear systems under input saturation

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

following a balanced trajectory from an infeasible point to an optimal linear programming solution with a polynomial time algorithm this paper is concerned with the problem of following a trajectory from an infeasible starting point directly to an optimal solution of the linear programming problem a class of trajectories for the problem is defined based on the notion of a balanced solution to the problem given a prespecified positive balancing constant an infeasible solution x is said to be balanced if the optimal value gap is less than or equal to times the infeasibility gap mathematically this can be written as ctx z tx where the linear form tx is the phase i objective function and z is the optimal objective value of the linear program the concept of a balanced solution is used to define a class of trajectories from an infeasible point to an optimal solution of a given linear program each trajectory has the property that all points on or near the trajectory in a suitable metric are balanced the main thrust of the paper is the development of an algorithm that traces a given balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial time more specifically the algorithm allows for fixed improvement in the bound on the phase i and the phase ii objectives in on newton steps

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

piecewise quadratic lyapunov functions for systems with deadzones or saturations a piecewise quadratic lyapunov function is developed for the analysis of the global and regional performances for systems with saturation deadzone in a general feedback configuration with an algebraic loop this piecewise quadratic lyapunov function effectively incorporates the structure of the saturation deadzone nonlinearity several sector like conditions are derived to describe the complex nonlinear algebraic loop these conditions transform several performance analysis problems into optimization problems with linear or bilinear matrix inequalities the effectiveness of the results is demonstrated with numerical examples

stability criteria of sector and slope restricted lur e systems presents stability criteria of sector and slope restricted lur e systems in terms of linear matrix inequalities by fully exploiting inherent properties of sector and slope restrictions in the time domain interpreting the time domain criteria in the frequency domain furthermore supplies simpler expressions several examples show excellent performances of these criteria

frequency domain min max optimization of noise shaping delta sigma modulators this paper proposes a min max design of noise shaping delta sigma modulators we first characterize the all stabilizing loop filters for a linearized modulator model by this characterization we formulate the design problem of lowpass bandpass and multi band modulators as minimization of the maximum magnitude of the noise transfer function ntf in fixed frequency band s we show that this optimization minimizes the worst case reconstruction error and hence improves the snr signal to noise ratio of the modulator the optimization is reduced to an optimization with a linear matrix inequality lmi via the generalized kyp kalman yakubovich popov lemma the obtained ntf is an fir finite impulse response filter which is favorable in view of implementation we also derive a stability condition for the nonlinear model of delta sigma modulators with general quantizers including uniform ones this condition is described as an norm condition which is reduced to an lmi via the kyp lemma design examples show advantages of our design

comparative analysis of affine scaling algorithms based on simplifying assumptions we analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions we show that under a strong probabilistic assumption regarding the distribution of the data in an iteration the decrease in the primal potential function will be omega rho sqrt log n with high probability compared to the guaranteed 1 2n is a parameter in the potential function andn is the number of variables under the same assumption we further show that the objective reduction rate of dikin s affine scaling algorithm is 1 1 sqrt log n with high probability compared to no guaranteed convergence rate

design of interpolative sigma delta modulators via semi infinite programming this correspondence considers the optimized design of interpolative sigma delta modulators sdms the first optimization problem is to determine the denominator coefficients the objective of the optimization problem is to minimize the passband energy of the denominator of the loop filter transfer function excluding the dc poles subject to the continuous constraint of this function defined in the frequency domain the second optimization problem is to determine the numerator coefficients in which the cost function is to minimize the stopband ripple energy of the loop filter subject to the stability condition of the noise transfer function ntf and signal transfer function stf these two optimization problems are actually quadratic semi infinite programming sip problems by employing the dual parameterization method global optimal solutions that satisfy the corresponding continuous constraints are guaranteed if the filter length is long enough the advantages of this formulation are the guarantee of the stability of the transfer functions applicability to design of rational infinite impulse response iir filters without imposing specific filter structures and the avoidance of iterative design of numerator and denominator coefficients our simulation results show that this design yields a significant improvement in the signal to noise ratio snr and have a larger stability range compared with the existing designs

sigma delta spl sigma spl delta quantization and finite frames the k level sigma delta spl sigma spl delta scheme with step size spl delta is introduced as a technique for quantizing finite frame expansions for spl ropf sup d error estimates for various quantized frame expansions are derived and in particular it is shown that spl sigma spl delta quantization of a unit norm finite frame expansion in spl ropf sup d achieves approximation error where n is the frame size and the frame variation spl sigma f p is a quantity which reflects the dependence of the spl sigma spl delta scheme on the frame here spl par spl middot spl par is the d dimensional euclidean 2 norm lower bounds and refined upper bounds are derived for certain specific cases as a direct consequence of these error bounds one is able to bound the mean squared error mse by an order of 1 n sup 2 when dealing with sufficiently redundant frame expansions this represents a significant improvement over classical pulse code modulation pcm quantization which only has mse of order 1 n under certain nonrigorous statistical assumptions spl sigma spl delta also achieves the optimal mse order for pcm with consistent reconstruction

on the approximate solution of a class of large discrete quadratic programming problems by delta sigma modulation the case of circulant quadratic forms we show that modulators can be interpreted as heuristic solvers for a particular class of optimization problems then we exploit this theoretical result to propose a novel technique to deal with very large unconstrained discrete quadratic programming udqp problems characterized by quadratic forms entailing a circulant matrix the result is a circuit based optimization approach involving a recast of the original problem into signal processing specifications then tackled by the systematic design of an electronic system this is reminiscent of analog computing where untreatable differential equations were solved by designing electronic circuits analog to them the approach can return high quality suboptimal solutions even when many hundreds of variables are considered and proved faster than conventional empirical optimization techniques detailed examples taken from two different domains illustrate that the range of manageable problems is large enough to cover practical applications

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

a combined phase i phase ii projective algorithm for linear programming we devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution inclusion of such a constraint allows the algorithm to be applied to a possibly infeasible standard form linear program without the addition of any bigm terms or conversion to a primal dual problem

approximating a bandlimited function using very coarsely quantized data a family of stable sigma delta modulators of arbitrary order digital signal processing has revolutionized the storage and transmission of audio and video signals as well as still images in consumer electronics and in more scientific settings such as medical imaging the main advantage of digital signal processing is its robustness although all the operations have to be implemented with of necessity not quite ideal hardware the a priori knowledge that all correct outcomes must lie in a very restricted set of well separated numbers makes it possible to recover them by rounding off appropriately bursty errors can compromise this scenario as is the case in many communication channels as well as in memory storage devices making the perfect data unrecoverable by rounding off in this case knowledge of the type of expected contamination can be used to protect the data prior to transmission or storage by encoding them with error correcting codes this is done entirely in the digital domain these advantages have contributed to the present widespread use of digital signal processing many signals however are not digital but analog in nature audio signals for instance correspond to functions f t modeling rapid pressure oscillations which depend on the continuous time t i e t ranges over or an interval in and not over a discrete set and the range of f typically also fills an interval in f or this reason the first step in any digital processing of such signals must consist in a conversion of the analog signal to the digital world usually abbreviated as a d conversion for different types of signals different a d schemes are used in this paper we restrict our attention to a particular class of a d conversion schemes adapted to audio signals note that at the end of the chain after the signal has been processed stored retrieved transmitted all in digital form it needs to be reconverted to an analog signal that can be understood by a human hearing system we thus need a d a conversion there

mean squared error analysis of quantizers with error feedback quantization is a fundamental process in digital signal processing delta sigma modulators are often utilized for quantization which can be easily implemented with static uniform quantizers and error feedback filters in this paper we analyze the mean squared quantization error of the quantizer with error feedback including the delta sigma modulators first we study the quantizer with an ideal optimal error feedback filter that minimizes the mean squared error mse of quantization we show that the amplitude response of the optimal error feedback filter can be parameterized by one parameter this parameterization enables us to find the optimal error feedback filter numerically second the relationship between the number of bits used for the quantizer and the achievable mse is clarified by using the optimal error feedback filter this makes it possible to investigate the efficiency of the quantizer with the optimal error feedback filter in terms of mse then ideal optimal error feedback filters are approximated by practically implementable filters using the yule walker method and the linear matrix inequality based method numerical examples are provided for demonstrating our analysis and synthesis

a potential reduction algorithm with user specified phase i phase ii balance for solving a linear program from an infeasible warm start this paper develops a potential reduction algorithm for solving a linear programming problem directly from a warm start initial point that is neither feasible nor optimal the algorithm is an interior point variety that seeks to reduce a single potential function which simultaneously coerces feasibility improvement phase i and objective value improvement phase ii the key feature of the algorithm is the ability to specify beforehand the desired balance between infeasibility and nonoptimality in the following sense given a prespecified balancing parameter beta 0 the algorithm maintains the following phase i phase ii beta balancing constraint throughout left c t x z right beta xi t x where c t x is the objective function z is the unknown optimal objective value of the linear program and xi t x measures the infeasibility of the current iterate x this balancing constraint can be used to either emphasize rapid attainment of feasibility set beta large

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

linear time algorithms for two and three variable linear programs o n time algorithms for linear programming problems with two or three variables and n constraints are described the approach uses convexity dominance of linear functions and linear time median finding algorithms the algorithms improve the previously known best bounds of o n log n time for both of these problems

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

a stability inequality for nonlinear feedback systems with slope restricted nonlinearity using lyapunov s direct method a stability inequality in frequency domain is derived for nonlinear feedback systems with slope restricted nonlinearity in the present approach a transformed system that involves the slope of the nonlinearity is considered thus leading to a stability inequality that incorporates the slope information but not the sector information of the nonlinearity the approach is essentially an extension of a recent idea of the author

theoretical efficiency of a shifted barrier function algorithm for linear programming abstract this paper examines the theoretical efficiency of solving a standard form linear program by solving a sequence of shifted barrier problems of the form minimize x c t x e j 1 n ln x j eh j s t a x b x eh 0 for a given and fixed shift vector h 0 and for a sequence of values of 0 that converges to zero the resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program the advantage of using the shifted barrier approach is that a starting feasible solution is unnecessary and there is no need for a phase i phase ii approach to solving the linear program either directly or through the addition of an artificial variable furthermore the algorithm can be initiated with a warm start i e an initial guess of a primal solution x that need not be feasible the number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible and optimal the number of iterations will also depend on the judicious choice of the shift vector h if an approximate center of the dual feasible region is known in advance then h can be chosen so that the guaranteed fractional decrease in at each iteration is 1 1 6 n which contributes afactor of 6 n to the number of iterations needed to solve the problem