riccati differential equation in optimal filtering of periodic non stabilizable systems this paper discusses the periodic solutions of the matrix riccati differential equation in the optimal filtering of periodic systems special emphasis is given to non stabilizable systems and the question addressed is the existence and uniqueness of a steady state periodic non negative definite solution of the periodic riccati differential equation which gives rise to an asymptotically stable steady state filter the results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution the convergence of the general solution of the periodic riccati differential equation to a periodic equilibrium solution is also investigated the results are extensions of existing time invariant systems results to the case of periodic systems

on the assignment of invariant factors by time varying feedback strategies abstract in this paper the realization of arbitrary invariant factors or cyclic components of the closed loop system matrix is discussed it is shown that the use of time varying feedback strategies enables one to realize arbitrary invariant factors in contrast with time invariant feedback control

brief paper the extended periodic lyapunov lemma in this paper the so called lyapunov lemma for linear systems with periodically time varying coefficients is dealt with by this lemma the asymptotic stability of a given periodic system is linked with the existence of a periodic positive definite solution of the associated lyapunov equation under the assumption on the system being observable by relaxing the observability assumption to detectability an extended version of the lemma is obtained where the system stability is linked with the existence of a periodic positive semidefinite solution of the lyapunov equation

controllability stabilizability and matrix riccati equations for periodic systems in this note controllability stabilizahility and related concepts for periodically time varying systems are discussed especially it is proved that the definition of controllability employed in connection with the existence of periodic solutions of periodic matrix riccati equations is equivalent to kalman s original definition

exact model matching of linear systems using generalized sampled data hold functions a new technique is presented for the solution of the exact model matching problem of linear time invariant systems using generalized sampled data hold functions this technique reduces the problem to that of solving a nonhomogeneous algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions are established and the expressions of the controller matrices are derived

a fast high quality and reproducible parallel lagged fibonacci pseudorandom number generator we study the suitability of the additive lagged fibonacci pseudo random number generator for parallel computation this generator has relatively short period with respect to the size of its seed however the short period is more than made up for with the huge number of full period cycles it contains these different full period cycles are called equivalence classes we show how to enumerate the equivalence classes and how to compute seeds to select a given equivalence class in addition we present some theoretical measures of quality for this generator when used in parallel next we conjecture on the size of these measures of quality for this generator extensive empirical evidence supports this conjecture in addition a probabilistic interpretation of these measures leads to another conjecture similarly supported by empirical evidence finally we give an explicit parallelization suitable for a fully reproducible asynchronous mimd implementation

the control of linear time periodic systems using floquet lyapunov theory in this paper we use floquet lyapunov theory to derive the floquet factors of the state transition matrix of a given linear time periodic system we show how the periodicity of one of the factors can be determined a priori using a constant matrix which we call the yakubovich matrix based upon the signs of the eigenvalues of the monodromy matrix we then describe a method for the numerical computation of the floquet factors relying upon a boundary value problem formulation and the yakubovich matrix further we show how the invertibility of the controllability gramian and a specific form for the feedback gain matrix can be used to derive a control law for the closed loop system the controller can be full state or observer based it also allows the engineer to assign all the invariants of the system i e the full monodromy matrix deriving the feedback matrix requires solving a matrix integral equation for the periodic floquet factor of the new state transition matrix of the closed loop system this is

the dynamics of a business game a 2d piecewise smooth nonlinear map abstract in this paper we give a proper investigation of the problem of remanufacturing duopoly game 1 a smooth piecewise map is introduced to describe such game a rich analysis either analytical or numerical is presented in this paper in order to investigate the stability of the fixed points of the game s model our obtained results show that those fixed points lose its stability due to flip and neimark sacker border bifurcations furthermore we study the stability of those points when there are periodic cycles with points jump from region to other region

thermodynamics of random number generation we analyze the thermodynamic costs of the three main approaches to generating random numbers via the recently introduced information processing second law given access to a specified source of randomness a random number generator rng produces samples from a desired target probability distribution this differs from pseudorandom number generators prng that use wholly deterministic algorithms and from true random number generators trng in which the randomness source is a physical system for each class we analyze the thermodynamics of generators based on algorithms implemented as finite state machines as these allow for direct bounds on the required physical resources this establishes bounds on heat dissipation and work consumption during the operation of three main classes of rng algorithms including those of von neumann knuth and yao and roche and hoshi and for prng methods we introduce a general trng and determine its thermodynamic costs exactly for arbitrary target distributions the results highlight the significant differences between the three main approaches to random number generation one is work producing one is work consuming and the other is potentially dissipation neutral notably trngs can both generate random numbers and convert thermal energy to stored work these thermodynamic costs on information creation complement landauer s limit on the irreducible costs of information destruction

dead beat control of linear periodic discrete time systems in this paper the dead beat control of linear periodic discrete time systems subject to periodic disturbances is studied necessary and sufficient conditions are given for the existence of a linear periodic controller such that both output dead beat control under periodic disturbances and state dead beat control under no disturbance are performed moreover synthesis procedures of the controller as well as of state dead beat observers are given which can be applied also to a class of bilinear systems

nonlinear h control around periodic orbits abstract in this paper we study the nonlinear h control of systems with periodic orbits we develop the notion of an induced l 2 gain so called nonlinear h norm for systems where the no disturbance behavior of the system is a periodic orbit and provide conditions under which the induced l 2 gain of the system around the orbit can be made less than a specified value by state feedback this work is a natural extension of results on nonlinear h control of nonlinear systems in a neighborhood of a stable equilibrium point to the periodic orbit case synthesis of a nonlinear h state feedback controller is facilitated by the use of transverse coordinates and in particular the transverse linearization of the system

pole placement for linear discrete time systems by periodic output feedbacks abstract necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single input single output linear discrete time systems are established an algorithm for finding a sequence of output feedback gains is presented and illustrated by a simple numerical example

on stabilization and spectrum assignment in periodically time varying continuous time systems this note discusses the stabilization and spectrum assignment problems in linear periodically time varying lptv continuous time systems with sampled state or output feedback the hybrid nature of the overall feedback system in this case imposes some carefulness in handling classical concepts related to purely lptv continuous time systems in particular this note points out the fact that the stabilization of such systems by periodic feedback gains with sampled state or output does not imply the relocation of the original characteristic exponents of the lptv systems it is also shown that the concept of monodromy matrix as extended to lptv hybrid systems has not all the features of a true monodromy matrix

interval algorithm for random number generation the problem of generating a random number with an arbitrary probability distribution by using a general biased m coin is studied an efficient and very simple algorithm based on the successive refinement of partitions of the unit interval 0 1 which we call the interval algorithm is proposed a fairly tight evaluation on the efficiency is given generalizations of the interval algorithm to the following cases are investigated 1 output sequence is independent and identically distributed i i d 2 output sequence is markov 3 input sequence is markov 4 input sequence and output sequence are both subject to arbitrary stochastic processes

a simple unpredictable pseudo random number generator two closely related pseudo random sequence generators are presented the 1 p generator with input p a prime outputs the quotient digits obtained on dividing 1 by p the x 2 bmod n generato

feedback stabilization of linear diffusion systems with periodically scanning sensors the feedback stabilization of linear diffusion systems with periodically scanning sensors is considered the control inputs to the system are given by the feedback of the sakawa type observer output in this case since the output operator becomes time periodic the characteristic multiplier assignment is used to design the observer and the bounded perturbation theorem of the evolution operators is used to prove the closed loop stability

subspace model identification part 2 analysis of the elementary output error state space model identification algorithm the elementary moesp algorithm presented in the first part of this series of papers is analysed in this paper this is done in three different ways first we study the asymptotic properties of the estimated state space model when only considering zero mean white noise perturbations on the output sequence it is shown that in this case the moespl implementation yields asymptotically unbiased estimates an important constraint to this result is that the underlying system must have a finite impulse response and subsequently the size of the hankel matrices constructed from the input and output data at the beginning of the computations depends on the number of non zero markov parameters this analysis however leads to a second implementation of the elementary moesp scheme namely moesp2 the latter implementation has the same asymptotic properties without the finite impulse response constraint secondly we compare the moesp2 algorithm with a classical state space model identification scheme the latter

multirate robust servomechanism controllers of linear delay systems using a hybrid structure a multirate robust servomechanism controller design for a linear time delay system is presented the proposed controller is constructed by using a preparatory continuous internal model followed by a multirate error feedback compensator an optimal algorithm for choosing the controller parameters is also derived the algorithm is derived by minimizing a squared deviation error performance index and the practical optimal solution can simply be solved from a discrete time algebraic riccati equation a distinctive feature of the algorithm is that an optimal pulse compensation is also employed to improve the transient characteristics such a pulse compensation tends to eliminate the undesired influence arising from the initial deviation from the ultimate steady state in particular if the initial deviation from the ultimate steady state is known then the optimal pulse compensation can drive the system into the steady state in a time no larger than the input delay time plus the sampling period

periodically varying compensation of time invariant systems linear time invariant discrete time scalar systems under periodically varying output feedback and the associated problems of stabilization and arbitrary eigenvalue assignment are investigated the general form of the characteristic polynomial of the closed loop periodically varying system is obtained and some detailed results are provided for the second order case interesting research questions are pointed out

on stability and stabilization of periodic discrete time systems with an application to satellite attitude control an alternative stability analysis theorem for nonlinear periodic discrete time systems is presented the developed theorem offers a trade off between conservatism and complexity of the corresponding stability test in addition it yields a tractable stabilizing controller synthesis method for linear periodic discrete time systems subject to polytopic state and input constraints it is proven that in this setting the proposed synthesis method is strictly less conservative than available tractable synthesis methods the application of the derived method to the satellite attitude control problem results in a large region of attraction

output stabilization via pole placement of discrete time linear periodic systems the output stabilization problem for discrete time linear periodic systems is solved both the state feedback control law and the state predictor are based on a suitable time invariant state sampled reformulation associated with a periodic system preliminary concepts of periodic system theory are briefly recalled in particular the structural properties of a linear discrete time periodic system are properly related to those of a time invariant system associated with it by resorting to such a time invariant reformulation the output stabilization problem via pole placement is solved the stabilizing controller is constituted by a control law and an asymptotic state predictor both of which are shown to be periodic

cost translation and a lifting approach to the multirate lqg problem it is shown how to translate an instance of a multirate sampled data lqg problem into an equivalent modified single rate shift invariant problem via a lifting isomorphism approach using this approach one can solve the multirate lqg problem without using periodic system theory or solving periodic riccati equations and without suffering any increases in state dimension this translation procedure shows the correct way to translate rms noise specification to the lifted domain for a multirate q design computer aided design package

monodromy eigenvalue assignment in linear periodic systems this note addresses the problem of assigning the eigenvalues of the monodromy matrix of a linear continuous time periodic system by periodic state feedback a controllability criterion is given which slightly strengthens known results an explicit expression for a periodic feedback gain which performs the pole assignment is given when the system is controllable over one period the whole monodromy matrix is assignable by periodic feedback an example is given

characteristic multiplier assignment in continuous time linear periodic systems a characteristic multiplier assignment problem for continuous time periodic systems with analytic coefficients is considered first the necessary and sufficient condition for controllability is shown under this condition an algorithm for assigning a set of characteristic multipliers is proposed the algorithm is constructed by a simple repetitive procedure and is suited to computer programming the given feedback gain matrix is composed of a sequence of dirac delta functions such a feedback system can easily be implemented using proper approximation to the delta function lastly the design method is applied to the attitude control problem of a spin stabilized satellite and some simulation results are given

inertia theorems for the periodic liapunov equation and periodic riccati equation the inertia theorems for the time invariant liapunov equation and time invariant riccati equation are generalized to the case were the coefficient matrices are periodically time varying

a gramian based controller for linear periodic systems this note proposes a new design method for the control of linear time periodic systems the method is based on the reachability gramian and a specific form for the feedback gain matrix to build a novel control law for the closed loop system the new controller allows assignment of all the invariants of the system calculating the feedback requires solving a matrix integral equation for the periodic floquet factor of the state transition matrix of the closed loop system

chaotic dynamics in nonlinear duopoly game with heterogeneous players in this study we investigate the dynamics of a nonlinear discrete time duopoly game where the players have heterogeneous expectations two players with different expectations are considered one is boundedly rational and the other thinks with adaptive expectations the stability conditions of the equilibria are discussed we show how the dynamics of the game depend on the model parameters we demonstrate that as some parameters of the game are varied the stability of nash equilibrium is lost through period doubling bifurcation the chaotic features are justified numerically via computing lyapunov exponents sensitive dependence on initial conditions and the fractal dimension

multirate control a new approach abstract a new multirate digital control of linear periodic and time invariant systems is proposed an explicit multirate law is given which transforms the overall control design into that of a relatively small dimensional discrete time invariant system for which standard design techniques can be used it is shown that the proposed multirate scheme is in general more effective as a design tool than current multirate designs the controllability properties of the multirate system are studied consequently necessary and sufficient conditions for pole placement and stabilizability of the multirate system are given in terms of the continuous plant matrices moreover the multirate lqr problem is solved in this case the solution is obtained by solving an algebraic discrete riccati equation here necessary and sufficient conditions for the existence of a unique nonnegative definite solution of the riccati equation and the asymptotic stability of the corresponding closed loop multirate system are obtained these conditions are given in terms of the continuous time parameters

inertia theorems for the periodic lyapunov difference equation and periodic riccati difference equation abstract the periodic lyapunov difference equation plde and periodic riccati difference equation prde are dealt with the inertia i e the number of positive null and negative eigenvalues of any symmetric periodic solution of such equations is linked with the pattern of eigenvalues of the monodromy matrix associated with the open loop for plde or closed loop for prde underlying systems different results are obtained by imposing requirements with decreasing strength to the original system precisely assumptions of observability reconstructibility and detectability are successively introduced some results are also given for the particular case of positive semidefinite periodic solutions

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

dynamic cournot duopoly games with nonlinear demand function in this paper dynamic duopolistic cournot models are investigated with discrete time scales under the assumption of unknown inverse demand function and linear cost with this motivation we consider different types of models bounded rational duopoly puu s duopoly bounded rational duopoly with delay and bounded rational multi team model in these models the firms use two important adjustment mechanism the bounded rationality and puu s approach to update their quantity in each period the locally asymptotic stability of the fixed point of each model is investigated and complex dynamic characteristics including period doubling bifurcation strang attractors and chaotic phenomena are also discussed numerical simulations are carried out to show such complex behavior of the four models and to point out the impact of the models parameters on the stability of the fixed points

conditions for minimizing the norm sensitivity of characteristic roots the sensitivity of a characteristic root of an n by n real matrix is measured by the euclidean norm of the root s n2derivatives with respect to the elements of the matrix let denote a real root and sigma j omega a complex root conditions for minimizing the sensitivity norms based on lambda sigma omega and sigma j omega are obtained since the conditions apply for all n and involve simple algebraic properties of the matrix they may have useful applications

simultaneous pole assignment in linear periodic systems by constrained structure feedback the authors present methods for pole assignment by feedback with constrained structure in linear periodic systems using generalized sampled data hold functions gshf constrained structure is taken to mean that the input of each channel is restricted to depend only on the measurements of some specific output channels the basic idea of gshf control is to use sample and hold but to consider the hold function as a design parameter four strategies are proposed for closing a control loop using gshf the key to the results obtained is that when gshf control with constrained structure is applied to a periodic system a discrete time time invariant decentralized system is obtained for which control design methods are available

periodic lyapunov equation based approaches to the stabilization of continuous time periodic linear systems this note is concerned with stabilization of continuous time periodic linear cpl systems with state feedback the design is based on solutions to a class of parametric periodic lyapunov differential equations pldes resulting from the problem of minimal energy control with guaranteed convergence rate by carefully studying the properties of the pldes and their solutions a continuous periodic state feedback is designed the plde based approach is effective in designing stabilizing controller for cpl systems as the designers need only to solve a linear differential equation whose solution can be obtained analytically if the system is relative simple and can be computed numerically by integration in general cases necessary and sufficient conditions on the free parameter in the plde are proposed to guarantee the stability of the closed loop system whose characteristic multiplier set can even be exactly computed accordingly a numerical example is worked out to show the effectiveness of the proposed approach

a correspondence between lti controller and generalized hold function abstract generalized sampled data hold function gshf has been studied in a number of works and it is known to be a nonstandard sampled data control scheme in this paper we show that some linear time invariant controllers become gshf when a standard zero order sample and hold is applied thus establishing a correspondence between gshf and a standard sampled data control scheme

on a general optimal algorithm for multirate output feedback controllers for linear stochastic periodic systems a modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed by combining the discrete time linear quadratic regulation lqr control problem and the discrete time stochastic linear quadratic regulation slqr control problem to obtain an extended linear quadratic regulation elqr control problem one derives a general optimal algorithm to balance the advantages of the optimal transient response of the lqr control problem and the optimal steady state regulation of the slqr control problem in general the solution of this algorithm is obtained by solving a set of coupled matrix equations special cases for which the coupled matrix equations can be reduced to a discrete time algebraic riccati equation are discussed a reducable case is the optimal algorithm derived by h m al rahmani and g f franklin 1990 where the system has complete state information and the discrete time quadratic performance index is transformed from a continuous time one

linear periodic systems eigenvalue assignment using discrete periodic feedback the eigenvalue assignment problem of a t periodic linear system using discrete periodic state feedback gains is discussed for controllable systems an explicit formula for the feedback law is given that can be used for the arbitrary assignment of the eigenvalues of phi sub c1 t 0 the closed loop state transition matrix from 0 to t for the special case of periodic systems controllable over one period this control law can be used to obtain any desired phi sub c1 t 0

decoupling of continuous time linear time invariant systems using generalized sampled data hold functions abstract we investigate the possibility of input output decoupling without stability a square continuous time lti plant c a b by designing discrete devices the generalized sampled data hold functions by adopting sequential design procedures for the resulting diagonalizing problem two cases are found to be solvable and yield nontrivial solutions 1 cb nonsingular 2 m n 2m rk c b n m where n m are the number of states and inputs outputs respectively c i n c t cc t 1 c

asymptotic stabilization of a class of continuous time linear periodic systems abstract in this paper the problem of the asymptotic stabilization of a class of continuous time linear periodic systems is considered first under a simplifying assumption necessary conditions involving the concept of left eigenvector are derived for the asymptotic stabilization of continuous time linear periodic systems belonging to such a class then under the same simplifying assumption conditions are given under which a static state feedback control law can be designed in an iterative manner so that the closed loop system is stable

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

subspace model identification part 1 the output error state space model identification class of algorithms in this paper we present two novel algorithms to realize a finite dimensional linear time invariant state space model from input output data the algorithms have a number of common features they are classified as one of the subspace model identification schemes in that a major part of the identification problem consists of calculating specially structured subspaces of spaces defined by the input output data this structure is then exploited in the calculation of a realization another common feature is their algorithmic organization an rq factorization followed by a singular value decomposition and the solution of an overdetermined set or sets of equations the schemes assume that the underlying system has an output error structure and that a measurable input sequence is available the latter characteristic indicates that both schemes are versions of the mimo output error state space model identification moesp approach the first algorithm is denoted in particular as the elementary moesp scheme

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

a class of subspace model identification algorithms to identify periodically and arbitrarily time varying systems abstract subspace model identification algorithms that allow the identification of a linear time varying ltv state space model from an ensemble set of input output measurements are presented in this paper each pair of input and output sequences in this ensemble is recorded when the underlying system to be identified undergoes the same time varying behavior the algorithms operate directly on the available ensemble of input output data and are a generalization of the recently proposed multivariable output error state space moesp class of algorithms to this ensemble type of identification problems a special case is considered in this paper where the repetition of this time varying behavior is intrinsic namely in periodically time varying systems an example of identifying a multirate sampled data system from a recorded input and output sequence demonstrates some of the capabilities of the presented subspace model identification algorithms

optimal hold functions for sampled data regulation abstract this paper investigates the use of generalized sampled data hold function control gshf to optimize quadratic measures of performance in sampled data control loops the idea of gshf control is to use sampled data feedback but consider the hold function as a design parameter explicit solutions are given for the gshf versions of the optimal lq and lqg regulators the questions of existence uniqueness and computation of optimal hold functions are treated an example is presented

reference governors and maximal output admissible sets for linear periodic systems abstractin this paper we consider the problem of constraint management in linear periodic lp systems using reference governors rg first we introduce the periodic invariant maximal output adm

static output feedback control for periodically time varying systems most control designs for periodically time varying systems use either full state feedback or observer based state feedback in this paper it is shown that static output feedback is sufficient for the exponential stabilization of a periodical system under both the controllability and observability assumptions in fact by incorporating a new generalized hold function in the control design one is able to arbitrarily shift all the poincare exponents of the periodical system most importantly the control signal is guaranteed to be continuous in time while the control signal from previous designs may be discontinuous

h controllability and observability of linear periodic systems with reference to linear periodic systems the classical notion of controllability introduced by kalman k controllability is shown to be equivalent to the characterization of controllability h controllability first proposed by hewer as a natural extension to the periodic case of an eigenvalue eigenvector controllability condition independently introduced by several authors for time invariant systems only the proof of such an equivalence also leads to the conclusion that if a t periodic system s is k controllable and k is the degree of the minimal polynomial of the monodromy matrix associated with s then at any time t s is controllable over t t kt since k is lower than or equal to the system order a well known result due to brunovsky is slightly strengthened by duality the corresponding observability results follow

lyapunov and riccati equations periodic inertia theorems the lyapunov and riccati differential equations with periodically time varying coefficients are considered under the assumption of detectability of the underlying periodic system two inertia theorems are provided linking the inertia of the solution to the one of the so called monodromy matrix

h sub infinity state estimation for linear periodic systems this note deals with the h sub infinity state estimation problem for linear periodic systems the question addressed is the design of an unbiased linear periodic and asymptotically stable estimator that achieves a prescribed h sub infinity performance on an infinite horizon necessary and sufficient conditions for the existence of a periodic estimator have been derived asymptotic properties of the finite horizon estimation problem when the time horizon tends to infinity are also investigated

a canonical decomposition of linear periodic discrete time systems in this paper a linear periodic discrete time system is studied on the basis of simple properties of the subspaces of controllable states and of unreconstructible ones a canonical structure theorem is derived this generalizes 1o such a system the classical kalman decomposition while preserving the constant dimensionality of the four subsystems which arise when a periodic continuous time system is decomposed the dynamic matrices of the non controllable and or non reconstructible subsystems are shown to be non singular at each time instant as those for a time invariant discrete time system are

converse lyapunov functions for exponentially stable periodic orbits abstract in this paper we present a technique for constructing a class of quadratic lyapunov functions for exponentially stable periodic orbits this construction is facilitated by the use of a special set of local coordinates that serve to highlight the tangential and transverse dynamics of the system