convergence and stability of distributed stochastic iterative processes a mathematical framework is proposed for convergence analysis of stochastic iterative processes arising in applications of pseudogradient optimization algorithms the framework is based upon the concepts of stochastic difference inequalities and vector lyapunov functions which are ideally suited for reducing the dimensionality problem arising in testing convergence of distributed parallel schemes by applying the m matrix conditions to a test matrix having a dimension equal to the number of the processor in the scheme one can use the framework to select suitable scaling factors for each individual processor producing a satisfactory convergence rate of the overall iterative process the results provide new convergence tests for distributed iterative processes arising in decentralized extremal regulation adaptation and parameter estimation schemes

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

decentralized techniques for the analysis and control of takagi sugeno fuzzy systems this paper discusses decentralized parallel distributed compensator design for takagi sugeno fuzzy systems the fuzzy system is viewed as an interconnection of subsystems some of which are strongly connected while others being weakly connected the necessary theory is developed so that one can associate this fuzzy system with another one in a higher dimensional space the so called expanded space design decentralized parallel distributed compensators in the expanded space then contract the solution for implementation on the original fuzzy system in this respect connective stability of the open loop and closed loop of the interconnected system is analyzed via the concepts of vector lyapunov functions and m matrices different lyapunov functions generate different results for the discrete time fuzzy system quadratic lyapunov generating the superior of the two following a similar approach stabilization of the closed loop fuzzy system using local parallel distributed compensators is investigated

controllability observability of expanded composite systems abstract the relation between original and expanded systems within the inclusion principle from the point of view of controllability observability of both subsystems and composite systems is studied it is proved that complementary matrices always exist ensuring that the subsystems and the overall expanded system are simultaneously controllable observable two practically important large classes of complementary matrices are identified to offer results computationally attractive first the existence of complementary matrices ensuring controllability observability of decoupled subsystems is proved then using this result the same property is proved for the composite expanded system

disturbance decoupling for linear time invariant systems a matrix pencil approach we give a new systematic analysis of disturbance decoupling problems for standard linear time invariant systems based on the theory of matrix pencils this approach is based on the computation of condensed forms under orthogonal equivalence transformations from these forms which can be computed in a numerically stable way we obtain new necessary and sufficient conditions that are numerically verifiable and furthermore we immediately obtain numerically stable algorithms to compute the desired compensators we present a numerical example that demonstrates the properties of the new approach

inclusion principle for linear time varying systems the main objective of this paper is to consider time varying linear systems within the framework of the inclusion principle starting from general definitions conditions are derived for a dynamic system to include a dynamic system of smaller dimension particular attention is paid to restriction and aggregation the most important special cases of inclusion it is proved using geometric arguments within the general time varying context that any inclusion relationship on a given time interval can be decomposed into a sequence of restriction aggregation pairs connections between inclusion and zero state equivalence are discussed the paper also presents a formulation of the contractibility expandability conditions for time varying state feedback controllers

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

contractibility of dynamic lti controllers using complementary matrices a generalized structure of complementary matrices involved in the input state output inclusion principle for linear time invariant systems including contractibility conditions for static state feedback controllers is well known in this note it is shown how to further extend this structure when considering contractibility of dynamic controllers necessary and sufficient conditions for contractibility are proved in terms of both unstructured and block structured complementary matrices for general expansion contraction transformation matrices explicit sufficient conditions for blocks of complementary matrices ensuring contractibility are proved for general expansion contraction transformation matrices moreover these conditions are further specialized for a particular class of transformation matrices

generalized selection of complementary matrices in the inclusion principle this paper presents a strategy for choosing complementary matrices in the framework of the inclusion principle with state lq optimal control of lti systems it is based on translating the basic restrictions given by the inclusion principle into explicit block structures for these matrices the degree of freedom given by these structures is illustrated by means of an example of overlapping decentralized control design

decentralized estimation and control with overlapping input state and output decomposition abstract the extension principle which was introduced recently is generalized to the case of input state and output expansion estimator and controller design problems are considered within the framework of the extension principle decentralized estimator and controller design with overlapping decompositions is also discussed within the same framework it is shown that if the extension principle is used then any estimator or controller designed in the expanded spaces is contractible to the original spaces for implementation furthermore it is shown that if an estimator designed for the expanded system achieves good estimation and or asymptotic estimation then the contracted estimator also achieves good estimation and or asymptotic estimation for the original system similarly it is also shown that if a controller designed for the expanded system achieves stability and or good performance then the contracted controller achieves stability and or good performance for the original system

reduction and non linear controllability of symmetric distributed systems this paper considers the reduction problem for distributed control systems in particular we consider controllability of systems containing multiple instances of diffeomorphic components where the overall system dynamics is invariant with respect to a discrete group action a subclass of such systems are systems with a set of identical components where the overall system dynamics are invariant with respect to physically interchanging these components the main result is a proposition which shows that for an equivalence class of symmetric systems of this type controllability of the entire class of systems can be determined by analysing the member of the equivalence class with the smallest state space the reduction methods developed are illustrated by considering the controllability of a team of mobile robots and a platoon of underwater vehicles

brief paper controllability and observability of expanded systems with overlapping decompositions the expanded system concept of siljak and his coworkers enables the design of robust overlapping controllers and observers for large scale systems to be carried out in a purely decentralised way in this paper we study the stability and detectability properties of expanded systems and show that it is only in certain special cases that these properties are preserved when the system is expanded these special cases define the classes of systems where the expanded system concept can be used for controller or observer design

on two sided modules which are left projective abstract the purpose of this paper is to study an analogue for non commutative rings r of projective modules we consider r bimodules of the form m v p where v rn is the module of row vectors and p is any two sided submodule of v these are finitely generated central r bimodules our main results 1 11 1 12 are these let ik denote the ideal generated by the k k minors of matrices with rows in p then p is projective and of rank d as left module if in d r and in d 1 0 the converse holds if every primitive ideal of r is maximal thus the familiar criterion of commutative algebra carries over directly to this class of rings and left projectivity is equivalent with right projectivity for central bimodules 1 13 some related questions are discussed in sections 2 and 3

overlapping block diagonal dominance and existence of liapunov functions the main objective of this paper is to formulate a generalization of block diagonal dominance which can be used to establish nonsingularity of matrices via overlapping diagonal blocks a number of stability results are derived in the new setting by exploiting the well known m matrix properties as well as extensions of the normalization scaling and alternative norm utilization a link between generalized block diagonal dominance and vector liapunov functions is established which can be applied in the stability analysis of interconnected dynamic systems

brief paper on the v stability of complex dynamical networks this paper introduces the concept of lyapunov v stability for complex dynamical networks under the new framework each dynamical node is associated with a passivity degree which indicates to what extent an effort is required for stabilizing the node from this approach the network stability problem is converted to measuring the negative definiteness of one simple matrix that characterizes the topology of the network pinning control is then suggested and investigated based on the new v stability formulation as an illustrative example a network with different node dynamics and non uniform coupling strengths is simulated to verify the analytic results moreover a comparison study for three different kinds of networks is provided to further illustrate the novelty and efficacy of the proposed v stability criterion and stabilization scheme

an inclusion principle for dynamic systems the purpose of this paper is to present a detailed study of the inclusion concept in dynamic systems which is a suitable mathematical framework for comparing systems with different dimensions the framework offers immediate results in reduced order modeling and the overlapping decentralized control of complex systems the presentation which is limited to linear constant systems relies on both the matrix algebra computations and the geometric elements structure to provide a balanced view of the issues involved in the concept of inclusion the framework is quite broad and has been used to consider nonlinear and time varying systems as well as systems with hereditary and stochastic effects

interconnection based performance analysis for a class of decentralized controllers this paper is concerned with decentralized controller design for large scale interconnected systems of pseudo hierarchical structure given such a system one can use existing techniques to design a decentralized controller for the reference hierarchical model obtained by eliminating certain weak interconnections of the original system although this indirect controller design is appealing as far as the computational complexity is concerned it does not necessarily result in satisfactory performance for the original pseudo hierarchical system an lq cost function is defined in order to evaluate the performance discrepancy between the pseudo hierarchical system and its reference hierarchical model under the designed decentralized controller a discrete lyapunov equation is then solved to compute this performance index however due to the large scale nature of the system this equation cannot be handled efficiently in many real world systems thus attaining an upper bound on this cost function can be more desirable than finding its exact value in practice for this purpose a novel technique is proposed which only requires solving a simple lmi optimization problem with three variables the problem is then reduced to a scalar optimization problem for which an explicit solution is provided it is also shown that when the original model is exactly hierarchical then the upper bounds obtained from the lmi and scalar optimization problems will both be equal to zero

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

contractibility of overlapping decentralized control abstract the purpose of this paper is to generalize the concept of contractibility of decentralized control laws in the inclusion principle after a system with overlapping subsystems is expanded into a larger space and decentralized control laws are formulated for the disjoint subsystems the laws need to be contracted for implementation in the original space we propose broader definitions of restriction and aggregation in the framework of inclusion which provide more flexibility in the contraction phase of the expansion contraction process in particular we discuss contractibility conditions for dynamic output controllers including state observers which have been of special interest in applications

generalized decompositions of dynamic systems and vector lyapunov functions the notion of decomposition is generalized to provide more freedom in constructing vector lyapunov functions for stability analysis of nonlinear dynamic systems a generalized decomposition is defined as a disjoint decomposition of a system which is obtained by expanding the state space of a given system an inclusion principle is formulated for the solutions of the expansion to include the solutions of the original system so that stabliity of the expansion implies stability of the original system stability of the expansion can then be established by standard disjoint decompositions and vector lyapunov functions the applicability of the new approach is demonstrated using the lotka voiterra equations

partial exponential stability of semi state systems the general class of non linear non autonomous semi state systems is considered from the viewpoint of stability of system motions a theorem is proved giving necessary and sufficient conditions for partial exponential stability of arbitrary motion of these systems over finite or infinite time intervals the results are illustrated by the analysis of a non linear electrical circuit

model abstraction and inclusion principle a comparison this paper discusses connections between the inclusion principle and model abstraction in large scale hierarchical control systems it also provides conditions for implementation of controllability and observability in linear continuous time dynamic systems satisfying either aggregation or restriction conditions as well as for observability implementation in linear continuous time dynamic systems satisfying conditions for abstraction

hierarchically consistent control systems large scale control systems typically possess a hierarchical architecture in order to manage complexity higher levels of the hierarchy utilize coarser models of the system resulting from aggregating the detailed lower level models in this layered control paradigm the notion of hierarchical consistency is important as it ensures the implementation of high level objectives by the lower level system in this paper we define a notion of modeling hierarchy for continuous control systems and obtain characterizations for hierarchically consistent linear systems with respect to controllability objectives as an interesting byproduct we obtain a hierarchical controllability criterion for linear systems from which we recover the best of the known controllability algorithms from numerical linear algebra

integral ring homomorphisms abstract the purpose of this paper is to extend the classical notion of integral extensions of commutative rings to homomorphisms of affine pi rings it generalizes previous work r pare and w schelter j algebra 53 1978 477 479 w schelter j algebra 40 1976 245 254 errata 44 1979 576 which assumed the existence of centralizing elements without such elements the close relation between integrality and module finiteness breaks down cf examples 1 3 1 4 but the geometric implications of integrality remain our main result theorem 6 3 is an analogue of chevalley s theorem that a homomorphism r s is integral if and only if the induced map on spectra is proper it is proved in sections 6 8 by combining a geometric analysis with some explicit estimates of degrees a corollary is that the composition of integral homomorphisms is integral proposition 6 10 the main new tools we use are central homomorphisms of a ring r to orders over dedekind domains which we call curves we prove theorem 5 13 that the boundary of any constructible set in x spec r contains a dense set of points accessible along such a curve this fact contains a description corollary 5 14 of the zariski topology on x as well as the theorem of bergman and small on degrees of representations corollary 5 15 some of the results of this paper were announced in m artin in proceedings international symposium on algebraic geometry kyoto 1977 pp 237 247

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

contractible controller design and optimal control with state and input inclusion abstract a new inclusion concept which involves both state and input expansion is introduced it is shown that with the approach undertaken here any control law designed in the expanded spaces is always contractible to the original spaces the inclusion of the cost functions is also discussed and necessary and sufficient conditions are derived it is shown that unlike the case when the two input spaces are identical the inclusion conditions must in general depend on the control laws however certain sufficient conditions which do not involve final control laws can be found if the inclusion is taken with respect to the optimal control finally controller design with overlapping decompositions is discussed within the framework of the new inclusion concept and optimal control

overlapping vs partitioning in block iteration methods application in large scale system theory abstract the concept of overlapping decomposition in splitting theory is introduced matrix splitting hamiltonian operator splitting and riccati operator splitting are successively considered for the case of block overlapping and are compared with classical corresponding splitting methods based on block partitioning numerical simulations are given overlapping decomposition is shown to succeed where classical partitioning decomposition fails

quantum and quasi plucker coordinates abstract we demonstrate a passage from the quasi plucker coordinates of gelfand and retakh to the quantum plucker coordinates built from q generic matrices in the process we rediscover the defining relations of the quantum grassmannian of taft and towber and provide that algebra with more concrete geometric origins

a decomposition technique for nonlinear dynamical system analysis a method for analyzing large scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented it is shown that the decomposition approach can be used to scale the sum of squares programming framework for nonlinear systems analysis the method constructs subsystem lyapunov functions which are used to form a composite lyapunov function for the whole system further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem lyapunov functions

robust overlapping guaranteed cost control of uncertain state delay discrete time systems this note presents a new extension of the inclusion principle to cope with the problem of designing robust overlapping controllers for state delayed discrete time systems with norm bounded uncertainties using the concept of guaranteed cost control expansion contraction relations for systems and contractibility conditions for output guaranteed cost memoryless controllers are proved including conditions on the equality of guaranteed performance bounds the controllers are designed in the expanded space using a linear matrix inequality lmi delay independent procedure specifically adapted to this class of problems the designed controllers are then contracted and implemented into the original system the results are specialized for the overlapping decentralized control design the method enables an effective construction of block tridiagonal controllers a numerical illustrative example is supplied

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

a canonical form for the inclusion principle of dynamic systems the inclusion principle provides a mathematical framework for comparing behavior of dynamic systems having different dimensions our main objective is to derive a canonical form for larger systems expansions that are obtained by expanding smaller systems contractions the form offers full freedom in selecting appropriate matrices for the expansion contraction process we will broaden the form to include feedback and propose an explicit characterization of contractible control laws subject to overlapping information structure constraints

an inclusion principle for hereditary systems abstract given two hereditary dynamic systems having different dimensions the conditions are provided under which a part of the motion of the larger system is reproduced by the smaller system that is the larger system includes the smaller one the conditions for inclusion are useful in applying the concept of vector liapunov functions to stability analysis of systems composed of overlapping subsystems by expanding the systems into a larger space the overlapping subsystems appear as disjoint and standard methods can be used to conclude stability of the expanded system under the inclusion conditions stability of the expansion implies stability of the original system an example is provided to show stability where the standard disjoint decompositions fail

nested e decompositions and clustering of complex systems abstract a graph theoretic method of extreme simplicity is proposed for decomposition and clustering of complex systems into interconnected subsystems with prescribed interaction strength the method is suitable for stability analysis via vector lyapunov functions and decentralized control and estimation of large scale systems the new decomposition scheme can be used in clustering aggregation of models in economics queuing networks and computer systems as well as decompositions in block iterative methods for solutions of linear algebraic equations

sequential design of multi overlapping controllers for longitudinal multi overlapping systems in this paper a strategy to design multi overlapping controllers for systems with longitudinal multi overlapping structure is presented the design process uses the inclusion principle to carry out a multi step expansion that allows the decentralized design of a sequence of expanded local controllers these expanded local controllers are subsequently contracted to produce a multi overlapping controller that can be implemented in the original system to illustrate the proposed methodology the response of a four story building under a seismic excitation is considered

preservation of controllability observability in expanded systems the result contributed by the article is that controllability observability of an original continuous time lti dynamic system can always be simultaneously preserved in expanded systems within the inclusion principle when using block structured complementary matrices this new structure offers more degrees of freedom for the selection of specific complementary matrices than well known used cases such as aggregations and restrictions which enable such preservation only in certain special cases a complete unrestricted transmission of these qualitative properties from the original controllable observable system to its expansion is a basic requirement on the expansion contraction process mainly when controllers observers are designed in expanded systems to be consequently contracted for implementation in initially given systems an original system composed of two overlapped subsystems is adopted as a general prototype ease a numerical example is supplied

decentralized control and overlapping decomposition of mechanical systems part 1 system decomposition this paper improves a new inclusion concept called txtension to mechanical systems described by matrix second order equations recent work on specialization of the inclusion principle has been generalized to the expansion and contraction of input state and output spaces but without the possibility of constructing completely general contractible controllers in this paper necessary and sufficient conditions for extensions and contractible feedback controllers are proved they enable the construction of general contractible controllers without any constraint on the structure of controllers for mechanical systems

hierarchical liapunov functions abstract a complex system is considered as a collection of dynamic components which may be combined in different ways to form stable subassemblies subsystems which may serve again as units to build still more complex stable configurations a hierarchical liapunov function is proposed which can be used to determine stability of the overall system obtained by the multilevel interconnection process stability when established in this way is hierarchically connective that is if at any instant in time the hierarchical evolution of the system is interrupted the system can fall apart in exactly the same way it was constructed and by re starting the process put together again without loss of stability this fact may be used to explain the evolution of stable forms in natural as well as man made systems

quantum deformation of the grassmannian manifold abstract in this paper we work out a deformation of g r n the grassmannian of r subspaces in a vector space of dimension n over a field k of characteristic 0 g r n is deformed as an homogeneous space for sl n k the special linear group of k n this means that k g r n the coordinate ring of g r n gets deformed together with with the coaction of k sl n the coordinate ring of sl n k on it our deformation comes together with a coaction of the corresponding deformation of sl n k at the end we give an explicit presentation of the deformed grassmannian in terms of generators and relations

stabilization of fixed modes in expansions of lti systems the objective of this note is to propose a method for stabilization of structurally fixed modes in expansions of lti dynamic systems in the scope of overlapping decentralized control design based on the expansion contraction framework enabling successful design for a broader class of problems than considered so far the method is based on a judicious choice of complementary matrices in the expanded space numerical examples are provided to illustrate simplicity and efficiency of the proposed approach

decentralized control using quasi block diagonal dominance of transfer function matrices the purpose of this paper is to present a generalization of the nyquist array method to blockwise decompositions which is based upon a new version of block diagonal dominance an additional flexibility of the proposed method is in partitioning of the system matrices into disjoint as well as overlapping submatrices which increases considerably the class of control systems which can be designed via block diagonal dominance within this framework the controllers for individual subsystems can be designed independently of each other so that their union represents a decentralized controller for the overall system