estimation and control of systems with unknown covariance and multiplicative noise the problem of estimation and control for systems with multiplicative noise and unknown second order statistics is considered conditions are found for the existence of a solution based on game theoretic ideas the conditions for the existence of a saddle point for the time invariant filtering problem are necessary and sufficient whereas for all other cases only necessary the central idea of the solution is to convert the stochastic problem to a deterministic optimal control problem whose minimax point is sought with respect to the control filter and unknown statistics parameters the results that are derived show that the problem of estimation for systems with unknown covariances depends on the costate matrix which in turn is a function of the performance measure thus the filter loses one of its best known properties that of independence of the performance functional this property holds not only for the classical kalman filter but also for multiplicative systems

pareto optimal strategy for stochastic weakly coupled large scale systems with state dependent system noise this note is concerned with the decentralized infinite horizon stochastic pareto optimal static output feedback strategy for a class of weakly coupled systems with state dependent noise first pareto optimal control problems are formulated using a static output feedback strategy the necessary conditions are given by the cross coupled stochastic algebraic riccati type equations csares for proving the existence of the static output feedback strategy that minimizes the quadratic cost function after determining the asymptotic structure of the solutions of the csares a new sequential numerical algorithm and newton s method for solving the csares are described the resulting numerical solution is used to develop the pareto optimal strategy finally the efficiency of the proposed algorithm is demonstrated by solving a numerical example for a practical megawatt frequency control problem

control effort considerations in the stabilization of uncertain dynamical systems abstract this paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system we focus attention on the amplitude norm of the controller which is used to achieve this end the uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded a lyapunov function is used to establish the stability of the closed loop system the paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given lyapunov function furthermore a procedure is indicated for the construction of the desired feedback control law

split group codes we construct a class of codes of length n such that the minimum distance d outside of a certain subcode is up to a constant factor bounded below by the square root of n a well known property of quadratic residue codes the construction using the group algebra of an abelian group and a special partition or splitting of the group yields quadratic residue codes duadic codes and their generalizations as special cases we show that most of the special properties of these codes have analogues for split group codes and present examples of new classes of codes obtained by this construction

on the robustness of lq regulators it is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed loop modes

robust discrete controllers guaranteeing l sub 2 and l sub infinity performances an algorithm is given for designing discrete time controllers which are robust with respect to parameter variations and which also guarantee specified l sub 2 performance constraints for stochastic inputs and l sub infinity performance constraints for deterministic inputs this is not an analysis but a design method guaranteeing the specified robust performances also an upper bound on the covariance matrix is provided

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

robust linear quadratic designs with real parameter uncertainty this note derives a linear quadratic regulator which is robust to real parametric uncertainty by using the overbounding method of petersen and hollot 1986 the resulting controller is determined from the solution of a single modified riccati equation this controller has the same guaranteed robustness properties as standard linear quadratic designs for known systems it is proven that when applied to a structural system the controller achieves its robustness by minimizing the potential energy of uncertain stiffness elements and minimizing the rate of dissipation of energy by the uncertain damping elements

duadic codes a new family of binary cyclic n n 1 2 and n n 1 2 codes are introduced which include quadratic residue qr codes when n is prime these codes are defined in terms of their idempotent generators and they exist for all odd n p 1 a 1 p 2 a 2 cdots p r a r where each p i is a prime equiv pm 1 pmod 8 dual codes are identified the minimum odd weight of a duadic n n 1 2 code satisfies a square root bound when equality holds in the sharper form of this bound vectors of minimum weight hold a projective plane the unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent 73 37 9 duadic codes all duadic codes of length less than 127 are identified and the minimum weights of their extensions are given one of the duadic codes of length 113 has greater minimum weight than the qr code of that length

a double commutator guaranteed cost bound for robust stability and performance in this paper we present a new guaranteed cost bound motivated by the maximum entropy equation this bound is of interest due to its sensitivity to skew symmetric structured uncertainty which is difficult to address by means of conventional guaranteed cost bounds

state feedback control of systems with multiplicative noise via linear matrix inequalities we consider lti systems perturbed by parametric uncertainties modeled as white noise disturbances we show how to maximize via state feedback control the smallest norm of the noise intensity vector producing instability in the mean square sense using convex optimization over linear matrix inequalities we also show how to maximize performance robustness where performance is measured by expected output energy with either bounded initial conditions and zero inputs classical lqg cost or zero initial conditions and deterministic inputs of bounded energy a generalization of the h norm

polyadic constacyclic codes for any given positive integer m a necessary and sufficient condition for the existence of type i m adic constacyclic codes is given furthermore for any given integer s a necessary and sufficient condition for s to be a multiplier of a type i polyadic constacyclic code is given as an application some optimal codes from type i polyadic constacyclic codes including generalized reed solomon codes and alternant maximum distance separable codes are constructed

duadic codes over z sub 2k duadic codes constitute a well known family of binary cyclic codes they are generalized in this correspondence to the setting of abelian codes over the ring z sub 2k self duality isoduality and type ii properties are studied

lyapunov equations energy functionals and model order reduction of bilinear and stochastic systems we discuss the relation of a certain type of generalized lyapunov equations to gramians of stochastic and bilinear systems together with the corresponding energy functionals while gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems the relation of gramians and energy functionals for bilinear systems is less obvious we discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic gramians satisfying generalized lyapunov equations in any of the considered cases the definition of algebraic gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems we illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems

analysis of nonsmooth vector valued functions associated with infinite dimensional second order cones abstract given a hilbert space h the infinite dimensional lorentz second order cone k is introduced for any x h a spectral decomposition is introduced and for any function f r r we define a corresponding vector valued function f h x on hilbert space h by applying f to the spectral values of the spectral decomposition of x h with respect to k we show that this vector valued function inherits from f the properties of continuity lipschitz continuity differentiability smoothness as well as s semismoothness these results can be helpful for designing and analyzing solution methods for solving infinite dimensional second order cone programs and complementarity problems

compensatability and optimal compensation of systems with white parameters the optimal compensation problem is considered in the case of linear discrete time systems with stationary white parameters and quadratic criteria a generalization of the notion of mean square stabilizability namely mean square compensatability is introduced it is shown that suitable mean square compensatability and detectability conditions are sufficient and necessary in general for the existence of a unique optimal mean square stabilizing compensator tests are given to determine whether or not a system is mean square compensatable it is indicated how to calculate numerically the tests and the optimal mean square stabilizing compensator the results are illustrated with examples

on extremal self dual codes of length 96 let c be a binary extremal self dual code of length 96 we prove that an automorphism of c of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points moreover if all automorphisms of order 3 are fixed point free then aut c is solvable and its order divides 253 or 255 or aut c is the alternating group a5 which is the only possible group of order 60 furthermore aut c 20 or 40 cannot occur

maximum entropy type lyapunov functions for robust stability and performance analysis we present two lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency each lyapunov function involves the sum of two matrices the first being the solution to the so called maximum entropy equation and the second being a constant auxiliary portion the significant feature of these lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix while the performance bounds are relatively tight compared to alternative approaches thus these lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis kevwords maximum entropy function robust stability robust performance

duadic codes corresp the notion of duadic codes over gf 2 is generalized to arbitrary fields duadic codes of composite length are constructed an upper bound is given for the minimum distance of duadic codes of length a prime power

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

controller design of systems with multiplicative noise general linear continuous stochastic systems are considered with multiplicative noise in the control and state channels and stabilizing optimal inputs are synthesized for both time varying and time invariant situations in the nonstationary case a set of nonlinear matrix differential equations has to be solved in the stationary problem a set of nonlinear matrix algebraic equations provides the solution the separation principle is not valid and both the state filtering and the control synthesis problem should be treated simultaneously

the optimal projection equations with petersen hollot bounds robust stability and performance via fixed order dynamic compensation for systems with structured real valued parameter uncertainty a feedback control design problem involving structured real valued plant parameter uncertainties is considered a quadratic lyapunov bound suggested by recent work of i r petersen and c v hollot 1986 is utilized in conjunction with the guaranteed cost approach of s s l chang and t k c peng 1972 to guarantee robust stability with robust performance bound necessary conditions that generalize the optimal projection equations for fixed order dynamic compensation are used to characterize the controller that minimizes the performance bound the design equations thus effectively serve as sufficient conditions for synthesizing dynamic output feedback controllers that provide robust stability and performance

design of robust state feedback laws existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices the uncertainties are modelled as sector bounded non linearities and feedback laws ensuring global asymptotic closed loop stability are obtained using lyapunov e direct method the algorithms essentially amount to repeatedly solving a parameter dependent riccati equation until the maximal solution becomes positive definite

robust stabilization of uncertain systems in this paper we consider the systems described by dx axdt budt sum i sigma i f i xd beta i qquad text or qquad dot x ax bu sum i b i f i x t c i x and we will derive conditions under which there exists a feedback control law u kx such that the closed loop system is stable for all sigma i or smooth nonlinearities f i the nonlinearities f i and the noisy gains sigma i d beta i are unknown uncertainties in the system and the problem considered is to obtain a control law which is robust against these uncertainties as far as stability is concerned

stochastic h sub 2 h sub spl infin control with state dependent noise this paper discusses the stochastic h sub 2 h sub spl infin control problem with state dependent noise by means of the stabilization exact observability and stochastic detectability of stochastic systems the infinite horizon stochastic h sub 2 h sub spl infin control design is developed for the finite horizon h sub 2 h sub spl infin control problem our results generalize the corresponding deterministic ones to the stochastic models finally the observer based suboptimal stochastic h sub 2 h sub spl infin control is discussed in which the state variables cannot be measured directly and a feasible design algorithm is proposed

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

the constrained lyapunov problem and its application to robust output feedback stabilization given a dynamical system whose description includes time varying uncertain parameters it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point when designing such a controller one may consider static i e memoryless or dynamic compensation in this paper we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained lyapunov problem clp the clp can be stated in purely algebraic terms once the clp is described we provide necessary and sufficient conditions for its solution to exist subsequently we consider application of the clp to a number of robust stabilization problems involving static output feedback and observer based feedback

a riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems this paper presents a procedure for designing a full state observer and feedback control law which will stabilize a given uncertain linear system the uncertain linear systems under consideration are described by state equations which depend on uncertain parameters these uncertain parameters may be time varying their values however are constrained to lie within known compact bounding sets the design procedure involves solving two algebraic riccati equations a feature of the design procedure presented is the fact that it reduces to the standard lqg design procedure if the system contains no uncertain parameters

mean square stability analysis of stochastic continuous time linear networked systems in this technical note we study the mean square stability based analysis of stochastic continuous time linear networked systems the stochastic uncertainty is assumed to enter multiplicatively in system dynamics through input and output channels of the plant necessary and sufficient conditions for mean square exponential stability are expressed in terms of the input output property of deterministic or nominal system dynamics captured by the mean square system norm and variance of channel uncertainty the stability results can also be interpreted as a small gain theorem for continuous time stochastic systems linear matrix inequalities based optimization formulation is provided for the computation of mean square system norm for the stability analysis for a special case of single input channel uncertainty we also prove a fundamental limitation result that arises in the mean square exponential stabilization of the continuous time linear system overall the contributions in this technical note generalize the existing results on stability analysis from discrete time linear systems to continuous time linear systems with multiplicative uncertainty simulation results are presented for western system coordinating council 9 bus power system to demonstrate the application of the developed framework

parameter space methods for robust control design a guided tour the results obtained in studies of robust stability and stabilizability of control systems with parametric structured uncertainties are reviewed both the algebraic methods based upon characteristic equations and the methods using lyapunov functions and riccati equations are discussed and compared in the context of algebraic methods most promising are the kharitonov type approach and the optimization procedure of embedding a geometric figure of some kind inside the stability regions of the parameter space maximizing its size using minimax or some other mathematical programming technique in the framework of lyapunov s direct method the dominant approach has been a quadratic function estimation of stability regions in the parameter space in large sale systems the concept of vector lyapunov functions has been used with the possibility of choosing quadratic forms norm like functions and their combinations

on the weight enumerators of duadic and quadratic residue codes in this correspondence we compute the weight enumerators of various quadratic residue codes over f sub 2 and f sub 3 together with certain codes of related families like the duadic and the quadratic double circulant codes we use a parallel algorithm to find the number of codewords of a given not too high weight from which we deduce by usual classical methods for self dual and formally self dual codes over f sub 2 and f sub 3 their associated previously unknown weight enumerators we compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

parametrization of all linear compensators for discrete time stochastic parameter systems abstract for discrete time stochastic parameter systems this paper presents a characterization of all state covariances assignable by a linear controller and a parametrization of all controllers that achieves a desired covariance these results indirectly provide the parametrization of all linear fixed order compensators which are mean square stabilizing for this class of systems the paper also includes robustification of the derived compensators and an example to illustrate the results

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

a lyapunov robustness bound for linear systems with periodic uncertainties abstract for linear systems involving uncertain parameters with known constant nominal values and uncertain perturbations that vary sinusoidally with time lyapunov robustness anslysis is used to determine a stability bound or margin for the amplitudes of the parameter perturbations this bound is the size of a hypercube in parameter space for which asymptotic stability is guaranteed the bound which is based on a quadratic lyapunov function that depends linearly on parameter perturbations varies with the frequency of the uncertain parameter perturbations the bound is asymptotically proportional to the square root of this frequency as it becomes large

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

polyadic codes revisited we generalize the notions of duadic codes triadic codes polyadic codes and split group codes to include noncyclic abelian codes necessary and sufficient conditions for the existence of such codes and properties such as a duality property and a lower bound on the minimum weight of the subcode of odd like codewords are studied this construction and its modification lead to many good codes eight of which have minimum distance better than the lower bound given in brouwer s table

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

consta abelian polyadic codes in this correspondence the class of polyadic codes is generalized to the class of consta abelian polyadic codes which in particular includes the class of constacyclic polyadic codes properties such as the equivalence of polyadic codes and the mth root lower bound for the minimum weight of a subcode of certain types of polyadic codes are preserved in the consta abelian case sufficient conditions for the existence of this class of codes are established for the special case of constacyclic codes the characterization of negacyclic self dual codes of length coprime to the characteristic of the field in terms of negacyclic duadic codes is also given

generalized cyclotomic codes of length p sub 1 sup e 1 p sub t sup e t we first introduce a generalized cyclotomy of order 2 with respect to p sub 1 sup e 1 p sub t sup e t we then present two classes of new error correcting binary cyclic codes of length p sub 1 sup e 1 p sub t sup e t based on this generalized cyclotomy we prove either a square root bound or a similar bound on the minimum odd weight of those codes with length n p sub 1 sup e 1 p sub 2 sup e 2

cyclic concatenated codes with constacyclic outer codes the known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code the outer code is cyclic and the lengths of the inner and outer codes are relatively prime it is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code moreover it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes this description of cyclic codes of composite length leads to the introduction of the concept of a poor code length pcl all but low rate codes of this length have a poor minimum distance a pcl is directly related to the existence of irreducible binomials in the binary case the first few pcls are 9 25 27 45 49 75 81 and 99 arbitrarily long binary cyclic codes that are better than binary bch codes of primitive length are constructed

robust optimal parametric lq control with a guaranteed cost bound and applications for an optimal parametric linear quadratic lq control problem a design objective is to determine a controller of constrained structure such that the closed loop system is asymptotically stable and an associated performance measure is optimized in the presence of system uncertainty the system via a parametric lq design is further required to be robust in terms of maintaining the closed loop stability with a guaranteed cost bound this problem is referred to as robust optimal parametric lq control with a guaranteed cost bound and is addressed in this work a new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations the results presented generalize some previous work in this area a versatile numerical algorithm is also given for computing the robust optimal gains the usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a vtol helico

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

the automorphism groups of the generalized quadratic residue codes we give the full automorphism groups as groups of semiaffine transformations of the extended generalized quadratic residue codes we also present a proof of the gleason prange theorem for the extended generalized quadratic residue codes that relies only on their definition and elementary theory of linear characters

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

the majorant lyapunov equation a nonnegative matrix equation for robust stability and performance of large scale systems a new robust stability and performance analysis technique is developed the approach involves replacing the state covariance by its block norm matrix i e the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics a bound i e majorant for the block norm matrix is given by the majorant lyapunov equation a lyapunov type nonnegative matrix equation existence uniqueness and computational tractability of solutions to the majorant lyapunov equation are shown to be completely characterized in terms of m matrices two examples are considered for a damped simple harmonic oscillator with uncertain but constant natural frequency the majorant lyapunov equation predicts unconditional stability and for a pair of nominally uncoupled oscillators with uncertain coupling the majorant lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators a result not predictable from quadratic or vector lyapunov functions

robust controller synthesis using the maximum entropy design equations this note presents an application of the optimality conditions obtained in 1 for dynamic compensation in the presence of state control and measurement dependent noise by solving these equations which represent a fundamental generalization of standard steady state lqg theory a series of increasingly robust control designs is obtained for the example considered in 2

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

on the stability robustness of the continuous time lqg optimal control the robustness properties of the lqg optimal controller that is obtained by the asymptotic recovery method are discussed it is shown that in spite of the good recovered stability margins the resulting closed loop design is inherently unrobust in the sense that small parameter variations in the plant and or small realization errors in the controller may lead to an instantaneous instability of the design

brief paper robust multi objective feedback design with linear guaranteed cost bounds this paper considers the problem of designing a state feedback control law which minimizes multiple quadratic performance objectives in the presence of uncertain linear plant dynamics vector optimization techniques are used to minimize upper bounds on each performance objective guaranteed cost by exploiting the linear uncertainty bound introduced by bernstein 1987 the vector optimization problem is scalarized and pareto optimal solutions are computed the theory is applied to a numerical example with two quadratic performance measures

cyclotomic constructions of cyclic codes with length being the product of two primes cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms in this paper three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1 n2 and dimension n1 n2 1 2 are presented and analyzed where n1 and n2 are two distinct primes bounds on their minimum odd like weight are also proved some of the codes presented in this paper are among the best cyclic codes

linear time varying system control based on the inversion transformation this paper proposes a new state feedback control design for uniformly controllable linear time varying systems the new design is based on a nonlinear state transformation that converts the stabilization problem of the system state into a destabilization problem of the transformed state such a transformation results in a new control that has two advantages over previous controls 1 in some of the previous control designs forecast of future information of the time varying system matrices over some time interval t t is required to calculate the state feedback gain at any time instant t while in the new control design no such forecast is needed 2 in other previous control designs consecutive time differentiation on the time varying system matrices is required while in the new control design time integration instead of differentiation is applied to the system matrices the new control can therefore be applied to systems with time varying parameters that are piecewise continuous in time

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

robust decentralized optimal output feedback sufficient conditions are developed for characterizing robust decentralized static output feedback controllers the approach involves deriving necessary conditions for minimizing a bound on closed loop performance over a specified range of uncertain parameters the effect of plant parameter variations on the closed loop covariance is overbounded by means of a modified lyapunov equation whose solutions are guaranteed to provide robust stability and performance

robust stability and performance via fixed order dynamic compensation the discrete time case a discrete time feedback control design problem involving parametric uncertainty is considered a quadratic bound suggested by recent work on discrete time state space h sub infinity theory is utilized in conjunction with the guaranteed cost approach to guarantee robust stability with a robust performance bound the principal result involves sufficient conditions for characterizing robust full and reduced order controllers with a worst case h sub 1 performance bound

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen