pade approximants and noise rational functions abstract we continue a study of pade approximants pa for a series perturbed by random noise this time we consider more general rational functions we begin with the simple case of a sum of two geometric series and then show how these considerations can be extended to a general rational function we do not study the most general case but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered we show that froissart doublets are a universal feature and we construct an analog of the froissart polynomial introduced in the earlier paper

lqg optimal control of multirate sampled data systems an lqg control synthesis technique for multirate sampled data systems where each input has its own frequency of updating and each output has its own frequency of measurement is presented the skew of the output sampling mechanism renders the prediction problem singular for the lack of measures at some time instants an ad hoc prediction algorithm is derived by enlarging the dimension of the state vector to be estimated and feeding the state filter with a combination of measured outputs over a past and finite interval of time the constraints on the input updating mechanism are dealt with by inserting a dynamical system corresponding to a zero order hold into the loop

multirate lqg control of continuous time stochastic systems abstract the lqg problem for stochastic continuous time systems subject to multirate sampling of both input and output variables is considered by restating the problem as a discrete time periodic lqg problem a sufficient condition for the existence of an optimal stabilizing regulator is given in terms of the structural properties of the original system and the cost function this improves on previous contributions where optimal control schemes were proposed without addressing existence and or stability issues the possibility of incorporating an integral action within the optimal lqg regulator is also briefly discussed

robust pade approximation via svd pade approximation is considered from the point of view of robust methods of numerical linear algebra in particular the singular value decomposition this leads to an algorithm for practical computation that bypasses most problems of solution of nearly singular systems and spurious pole zero pairs caused by rounding errors for which a matlab code is provided the success of this algorithm suggests that there might be variants of pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to infty unlike traditional pade approximants which converge only in measure or capacity

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

pursuing a maneuvering target which uses a random process for its control since a pursuer pursuing a maneuvering target does not know what maneuvers an evading target will make the maneuvers the target s control law appear as a random process to the pursuer however he has opinions about what the evader will do from these he can assign a prior probability distribution to the evader s maneuvers for a linear pursuit evasion problem in which the evader s control law is modeled as a random process in which the pursuer has partial noisy linear measurements of his own and the evader s relative position and a quadratic optimality criterion is used past results of the authors imply that the optimal control is a linear function of the predicted miss determining the predicted miss involves estimating the evader s terminal position from past system measurements nonlinear filtering techniques are used to give expressions for computing the conditional expectation of the evader s terminal position even in the presence of the random unknown maneuvers of the evader

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

froissart doublets in pade approximation in the case of polynomial noise first we study the relation between the zeros of random polynomials rn 1 and the zeros and poles of their pade approximants n n rn 1 next we consider the distribution of zeros and poles of pade approximants to the geometric series perturbed by a random polynomial noise we observe numerically interesting connections between two above problems some numerical observations on the froissart doublets have been also made

a periodic fixed structure approach to multirate control we develop an approach to designing reduced order multirate controllers a discrete time model that accounts for the multirate timing sequence of measurements is presented and is shown to have periodically time varying dynamics using discrete time stability theory the optimal projection approach to fixed order i e full and reduced order dynamic compensation is generalized to obtain reduced order periodic controllers that account for the multirate architecture it is shown that the optimal reduced order controller is characterized by means of a periodically time varying system of equations consisting of coupled riccati and lyapunov equations in addition the multirate static output feedback control problem is considered for both problems the design equations are presented in a concise unified manner to facilitate their accessibility for developing numerical algorithms for practical applications

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

state space solution to the periodic multirate h sub spl infin control problem a lifting approach a two riccati equation solution to the h sub spl infin control problem for periodic multirate systems is derived via the lifting method the solution is expressed in terms of two algebraic riccati equations the causality constraints are represented by a set of positive definiteness conditions and coupling criteria as a by product the study shows that there is a close connection between the solution of the causally constrained lifted problem and the solution obtained by solving the problem directly as a periodic h sub spl infin problem defined for the multirate system

an algebraic riccati equation for the discrete time periodic prediction problem abstract in this paper the periodic problem is reformulated as a prediction problem relative to a suitable time invariant system this enables one to establish a one to one correspondence between the solutions of the discrete time periodic riccati equation dpre and the solutions of a suitable discrete time and time invariant riccati equation in particular the constant solutions of this last equation are the periodic generators of the original periodic riccati equation this enables one to easily derive existence uniqueness and convergence results for the drpe from known results relative to the time invariant case

a new optimal multirate control of linear periodic and time invariant systems the optimal multirate design of linear continuous time periodic and time invariant systems is considered it is based on solving the continuous linear quadratic regulation lqr problem with the control being constrained to a certain piecewise constant feedback necessary and sufficient conditions for the asymptotic stability of the resulting closed loop system are given an explicit multirate feedback law that requires the solution of an algebraic discrete riccati equation is presented such control is simple and can be easily implemented by digital computers when applied to linear time invariant systems multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly compared to the classical single rate sampled data feedback in which the state is always sampled at the same rate the multirate system can provide a better response with a considerable reduction in the optimal cost in general the multirate scheme offers more flexibility in choosing the sampling rates

optimal control of discrete time singularly perturbed systems in this paper we present a new algorithm for solving the optimal control of discrete time singularly perturbed systems the main idea of this algorithm is based on two steps first the hamiltonian difference equation is reduced to the backward recursive form rather than the forward recursive form second the bilinear transformation is applied to transform the derived non symmetric discrete time riccati equations into continuous time equations in order to improve the efficiency of this scheme two matrix permutations are introduced into this algorithm by taking into account the previous work of gajic and shen 1991 therefore substantial numerical advantages are gained namely computation and memory requirements the f 8 aircraft model is used to illustrate the efficiency of the proposed method

stability of periodically time varying systems this paper proposes a novel approach to stability analysis of discrete time nonlinear periodically time varying systems the contributions are as follows firstly a relaxation of standard lyapunov conditions is derived this leads to a less conservative lyapunov function that is required to decrease at each period rather than at each time instant secondly for linear periodic systems with constraints it is shown that compared to standard lyapunov theory the novel concept of periodic lyapunov functions allows for the calculation of a larger estimate of the region of attraction an example illustrates the effectiveness of the developed theory

time invariant orthonormal wavelet representations a simple construction of an orthonormal basis starting with a so called mother wavelet together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications an orthonormal basis may not however always be a suitable representation of a signal particularly when time or space invariance is a required property the conventional way around this problem is to use a redundant decomposition we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions we show that it is possible to achieve time invariance and preserve the orthonormality we subsequently propose an efficient approach to obtain such a decomposition we demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

a tensor optimization algorithm for bezier shape deformation in this paper we propose a tensor based description of the bezier shape deformation bsd algorithm denoted as t bsd the bsd algorithm is a well known technique based on the deformation of a bezier curve through a field of vectors a critical point in the use of real time applications is the cost in computational time recently the use of tensors in numerical methods has been increasing because they drastically reduce computational costs our formulation based in tensors t bsd provides an efficient reformulation of the bsd algorithm more precisely the evolution of the execution time with respect to the number of curves of the bsd algorithm is an exponentially increasing curve as the numerical experiments show the t bsd algorithm transforms this evolution into a linear one this fact allows to compute the deformation of a bezier with a much lower computational cost

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

a stability criterion for asynchronous multirate linear systems a stability criterion is developed for linear asynchronous multirate sampled data systems such systems typically include a linear time variant continuous time plant and several unsynchronized digital controllers the analysis method uses a time domain approach based on the closed loop state transition matrix the result is sufficient stability criterion which gives an objective measure of long term and short term stability a simple example illustrates the method

a translation invariant wavelet representation algorithm with applications we address the time varying problem of wavelet transforms and a new translation invariant wavelet representation algorithm is proposed using the algorithm introduced by beylkin see siam j numer anal vol 29 p 1716 1740 1992 we compute the wavelet transform for all the circular time shifts of a length n signal in o n log n operations the wavelet coefficients of the time shift with minimal cost are selected as the best representation of the signal using a binary tree search algorithm with an appropriate cost function we apply the translation invariant representation algorithm to a geoacoustic data compression application the results show that the new algorithm can reduce the distortion the squared error in our case substantially if the input signals are transients that are sensitive to time shifts

frequency shift invariant orthonormal wavelet packet representations it is commonly known that the dyadic structure of wavelet expansions results in both time and frequency translation sensitivity of an input signal we develop the first efficient method to reduce frequency alignment sensitivity by introducing a wavelet packet decomposition that is invariant to frequency shifts of a signal a frequency shifted wavelet packet library is presented and an efficient best basis algorithm is developed to determine the best signal representation among all frequency shifts of a signal the algorithm computes all frequency shifted coefficients in o n sup 2 operations followed by an efficient depth first tree search of the same complexity

on reducing the order of kalman filters for discrete time stochastic systems having singular measurement noise a reduced order least squares state estimator is developed for estimating the states of a linear discrete time stochastic system having some outputs which are noise free

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

pade approximants and noise a case of geometric series abstract we study effects on pade approximants pa of a random noise added to coefficients of a power series numerical experiments performed during the last 30 years have shown that such noise results in the appearance of the so called froissart doublets fd pairs of zeros and poles separated by a distance of a scale of the noise for the first time we can prove that this effect actually takes place for geometric series we can also show what happens with noise induced poles or zeros that do not form fd we construct a polynomial that has as its roots mean positions of fd coefficients of this polynomial are expressed by random perturbations of coefficients of the series we study numerically this polynomial for low orders to show where in these orders fd coalesce numerical study of this polynomial to see a behaviour of fd for large orders is much easier than a study of the entire pa we also express a deviation of a pole of pas approximating the true pole of 1 1 z from 1 by perturbations of coefficients and show numerically that when order of pas increases this deviation drops down

brief paper mpc of constrained discrete time linear periodic systems a framework for asynchronous control strong feasibility stability and optimality via periodic invariance state feedback model predictive control mpc of discrete time linear periodic systems with time dependent state and input dimensions is considered the states and inputs are subject to periodically time dependent hard convex polyhedral constraints first periodic controlled and positively invariant sets are characterized and a method to determine the maximum periodic controlled and positively invariant sets is derived the proposed periodic controlled invariant sets are then employed in the design of least restrictive strongly feasible reference tracking mpc problems the proposed periodic positively invariant sets are employed in combination with well known results on optimal unconstrained periodic linear quadratic regulation lqr to yield constrained periodic lqr control laws that are stabilizing and optimal one motivation for systems with time dependent dimensions is efficient control law synthesis for discrete time systems with asynchronous inputs for which a novel modeling framework resulting in low dimensional models is proposed the presented methods are applied to a multirate nano positioning system

brief performance limitations in the robust servomechanism problem for discrete time periodic systems fundamental limitations for error tracking regulation are obtained for the robust servomechanism problem rsp for discrete time periodic systems in studying this problem the rsp for a multi input multi output discrete time system is considered application of these results is then made to the periodic system robust servomechanism problem and explicit expressions for the limiting costs for error tracking regulation are obtained these limitations can be characterized completely by the number and location of the non minimum phase transmission zeros of the system s associated lifted system together with a term which depends on the order of the system the number of inputs the number of transmission zeros and the system s period

from singular to nonsingular filtering of periodic systems filling the gap with the spectral interactor matrix with reference to periodic discrete time systems the concept of the periodic spectral interactor matrix operator psimo is introduced such an interactor enables one to transform the singular system into a nonsingular one with identical spectral properties this concept is useful to characterize the family of periodic solutions of the singular periodic riccati equation arising from the singular filtering problem the psimo is also a synthetic tool to represent the invariant zeros structure at infinity delays of periodic systems in an operational fashion

two matrix norm conditions for asymptotic stability in the presence of controller disturbances the stability analysis for digital control systems is extended to consider controller misbehavior this approach shows that it is possible for a system to survive repeated controller perturbations because of plant dynamics this idea is placed on a quantitative basis by deriving results about asymptotic stability in terms of matrix norms these results are intended to be both general in application and convenient for experiments

a sampled data approach to pyragas type delayed feedback stabilization of periodic orbits a sampled data pyragas type delayed feedback control framework is proposed for stabilizing both periodic solutions of linear periodic systems and unstable periodic orbits of nonlinear systems the proposed framework uses the difference between two consecutive samples of the state as a feedback input furthermore the controller is turned on and off periodically in an alternating fashion allowing one to obtain a monodromy matrix for the closed loop system asymptotic behavior of the closed loop system state trajectories is investigated through an analysis of the obtained monodromy matrix then an efficient feedback gain design method is developed by exploiting the properties of the monodromy matrix finally the efficacy of the proposed framework is demonstrated with a numerical example