local factorization of trajectory lifting morphisms for single input affine control systems trajectory preserving and lifting maps have been implicitly used in many recursive or hierarchical control design techniques well known systems theoretic concepts such as differential flatness or more recent ones such as simulation and bisimulation can be also understood through the trajectory lifting maps they define in this paper we initiate a study of trajectory preserving and lifting maps between affine control systems our main result shows that any trajectory lifting map between two single input control affine systems can be locally factored as the composition of two special trajectory lifting maps a projection onto a quotient system followed by a differentially flat output with respect to another control system we use this decomposition result to show that under mild regularity conditions trajectory preserving maps between single input affine control systems also lift trajectories as an additional application of the main result we also show how the hierarchical stabilization method known as back stepping can be used based on the existence of a trajectory preserving and lifting map having a feedback stabilizable control system as codomain

bisimulation relations for dynamical control and hybrid systems the fundamental notion of bisimulation equivalence for concurrent processes has escaped the world of continuous and subsequently hybrid systems inspired by the categorical framework of joyal nielsen and winskel we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems we prove that these notions can be captured by the abstract notion of bisimulation as developed by joyal nielsen and winskel this is the first unified notion of system equivalence that transcends discrete and continuous systems furthermore this enables the development of a novel and natural notion of bisimulation for hybrid systems which is the final goal of this paper

a connection between simulation relations and feedback transformations in nonlinear control systems abstract the recent introduction of simulation and bisimulation relations in the study of nonlinear control systems provides a broad potential framework for the study and classification of such systems this paper reviews the definitions of simulation and bisimulation relations and develops connections between them and the well established notions of feedback transformations and feedback equivalences

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

approximate reduction of dynamic systems abstract the reduction of dynamic systems has a rich history with many important applications related to stability control and verification reduction of nonlinear systems is typically performed in an exact manner as is the case with mechanical systems with symmetry which unfortunately limits the type of systems to which it can be applied the goal of this paper is to consider a more general form of reduction termed approximate reduction in order to extend the class of systems that can be reduced using notions related to incremental stability we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors i e when it is behaviourally similar to a dynamic system on a lower dimensional space these concepts are illustrated on a series of examples

the structure of nonlinear control systems possessing symmetries a concept of symmetry is defined for general nonlinear control systems it is shown under various technical conditions that nonlinear control systems with symmetries admit local and or global decompositions in terms of lower dimensional subsystems and feedback loops the structure of the individual subsystems is dependent on the structure of the symmetry group for example if the symmetry group is abelian one of the subsystems is a quadrature an additional feature of the decomposition is that the state space dimensions of the subsystems sum to the state space dimension of the original system

ifra properties of some markov jump processes with general state space consider a stochastically monotone increasing regular markov jump process on a partially ordered measurable space x it is shown that the first passage time to any upper set u is ifra a conjecture of brown and chaganty brown m chaganty n r 1983 on the first passage time distribution for a class of markov chains ann probab 11 1000 1008 is verified and results of brown and chaganty brown m chaganty n r 1983 on the first passage time distribution for a class of markov chains ann probab 11 1000 1008 and of ross ross s m 1981 generalized poisson shock models ann probab 9 896 898 are extended

optimal control of systems possessing symmetries it is shown that a symmetry in an optimization problem induces a decomposition of the optimal feedback control law into factors one factor can be calculated algebraically and depends only on the symmetry the other factor corresponds to a lower dimensional optimization problem this gives a priori information about the structure of the optimal feedback control law and indicates a possibly more efficient method for optimizing such systems

multivariate classes of life distributions in reliability theory four classes of life distributions which have been useful in describing aging where systems are assumed to have independent univariate component lifetimes are 1 the increasing failure rate ifr class 2 the increasing failure rate average ifra class 3 the new better than used nbu class and 4 the new better than used in expectation nbue class these classes together with multivariate analogs of the ifr and ifra classes are reviewed and discussed an up to date survey of the literature on this topic is included new multivariate definitions of the nbu and nbue classes are introduced

on the projections of isotropic distributions the class of distributions on rm 1 c m oo which are the m dimensional marginal distributions of orthogonally invariant distributions on rm n is characterized this result is then used to provide a partial answer to the following question given a symmetric distribution on r and an integer n 2 under what conditions will there exist a random vector x e rn such that a x has the given distribution up to a positive scale factor for all a 0 a e rm

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

multidimensional ifra processes abstract two types of multidimensional processes are defined the first of these generalizes a univariate ifra process due to ross and relates to a multivariate concept of ifra due to esary and marshall the second of these relates to a multivariate concept of ifra due to the present authors decompositions for multistate monotone structure functions are given and behavior of nonincreasing stochastic processes such as those given above is analyzed various coherence assumptions for multistate systems are also analyzed

multivariate phase type distributions a univariate random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing markov chain univariate phase type random variables are useful because they arise from processes that are often encountered in applications they have densities that can be written in a closed form they possess some useful closure properties and they can approximate any nonnegative random variable this paper introduces and discusses several extensions to the multivariate case it shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest some examples are included

a stochastic process approach to multivariate reliability systems notions based on conditional stochastic order the purpose of this paper is to introduce extensions of the well known notions irf dfr nbu and nwu in reliability theory these extensions are applicable when considering complex multicomponent systems where the components are not necessarily behaving independently and where by continuous observation of the system its past at any time t is known to some degree the definitions are in terms of conditional stochastic order of the system s or its components residual lifetimes given the field corresponding to observed past

multivariate new better than used distributions a general theory of multivariate new better than used nbu distributions is developed using the general theory some multivariate nbu conditions are introduced and studied some of the results apply as well to multivariate increasing hazard rate average ihra and exponential distributions proofs of some related well known implications are simplified

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

the failure and hazard processes in multivariate reliability systems the component failures in a complex multi component system are treated in the paper as a multivariate point process and the system failure as a univariate point process derived from this our main concern is to extend the well known concept of a hazard function to this general case it is suggested that e g the system failure hazard be defined as a stochastic process viz the compensator of the system failure counting process relative to the fields describing the system s past the connections between such a hazard process and the lifetime distribution classes of arjas arjas e 1981 a stochastic process approach to multivariate reliability systems notions based on conditional stochastic order math oper res 6 263 276 are also discussed

equivalence of switching linear systems by bisimulation a general notion of hybrid bisimulation is proposed for the class of switching linear systems connections between the notions of bisimulation based equivalence state space equivalence algebraic and input output equivalence are investigated an algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived hybrid state space reduction is performed by hybrid bisimulation between the hybrid system and itself by specializing the results obtained on bisimulation also characterizations of simulation and abstraction are derived connections between observability bisimulation based reduction and simulation based abstraction are studied

consistent abstractions of affine control systems in this paper we consider the problem of constructing abstractions of affine control systems that preserve reachability properties and in particular local accessibility in this framework showing local accessibility of the higher level abstracted model is equivalent to showing local accessibility of the more detailed lower level model given an affine control system and a smooth surjective map we present a canonical construction for extracting an affine control system describing the trajectories of the abstracted variables we then obtain conditions on the abstraction maps that render the original and abstracted system equivalent from a local accessibility point of view such consistent hierarchies of accessibility preserving abstractions of nonlinear control systems are then considered for various classes of affine control systems including linear bilinear drift free and strict feedback systems

on decomposition of nonlinear control systems in this paper the problem is studied when the system x i k 1 u t g t x x 0 x 0 defined on an n dimensional analytic manifold can be decomposed into either independent subsystems or a cascade of subsystems necessary and sufficient conditions for the existence of such local decompositions are given in terms of lie algebras generated by the system

on the first failure time of dependent multicomponent reliability systems in this paper multicomponent reliability systems are considered where component failure and repair completion rates depend on the state ages and current repair durations of the other components this is a generalization of a model of ross ross s m 1984 a model in which component failure rates depend on working set naval res logist quart 31 297 300 sufficient conditions on the sets of rates which imply stochastic ordering between first failure times of two such systems are found sufficient conditions on the rates which imply that the first failure time of such a system is new better than used nbu are given some results of barlow and proschan barlow r e proschan f 1976 theory of maintained systems distribution of time to first system failure math oper res 1 32 42 chiang and niu chiang d t niu s c 1980 on the distribution of time to first system failure j appl probab 17 481 489 and ross ross s m 1976 on the time to first failure in multicomponent exponen

equivalence of dynamical systems by bisimulation a general notion of bisimulation is defined for linear input state output systems using analogies with the theory of concurrent processes a characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory bisimulation is shown to be a notion which unifies the concepts of state space equivalence and state space reduction and which allows to study equivalence of systems with nonminimal state space dimension the notion of bisimulation is especially powerful for nondeterministic dynamical systems and leads in this case to a notion of equivalence which is finer than equality of external behavior for abstractions of systems it is shown how the results specialize to previously obtained results by other authors extensions of the main results to the nonlinear case are provided

the achievable dynamics via control by interconnection we consider the problem of finding a controller such that when interconnected to the plant we obtain a system that is equivalent to a desired system here equivalence is formalized as bisimilarity we give necessary and sufficient conditions for the existence of such a controller the systems we consider are linear input state output systems a comparison is made to previously obtained results about achievable implementable behaviors in the behavioral approach to systems theory among the advantages of using the notion of bisimilarity is the fact that it directly applies to state space systems while the computations involved are operations on constant matrices

on controlled invariance for fully nonlinear systems we give a necessary and sufficient condition for local controlled invariance that applies to fully nonlinear control systems this generalizes an earlier result of nijmeijer and van der schaft by eliminating some of the constant rank assumptions for the distributions that occur in the statement of their result our work is related to the recent work of cheng and tarn who showed that such constant rank assumptions can be weakened when discussing the local controlled invariance of nonlinear systems that are affine in the control

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

controlled invariance for affine control systems the paper gives a solution of the disturbance decoupling problem for non linear systems the main difference between this and related works in the field is the construction of a now state dependent basis for the inputs

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

lifting of trajectories of control systems related by smooth mappings abstract given a pair of control systems x f x u and y h y v whose state spaces are differentiable manifolds m and n respectively we focus on situations where there exists a mapping m n that induces some type of correspondence between the trajectories of the systems an important instance of this situation occurs when f is a complex system for example one having many states h is a subsystem of f presumably with fewer states and a simpler structure than f and is a mapping from the full state space m to a reduced state space n some recent research has concentrated on the concept of related systems where it is required that send trajectories of f onto trajectories of h here we approach the problem from a different direction and ask under what conditions trajectories of the subsystem h can be lifted to trajectories of the full system f we provide computable sufficient conditions for the local and global lifting of trajectories from n to m connections between the lifting problem and the problem of global f g invariance are also discussed briefly

bisimulation equivalence of differential algebraic systems in this paper the notion of bisimulation relation for linear input state output systems is extended to general linear differential algebraic dae systems geometric control theory is used to deri

extensions of the freund distribution with applications in reliability theory the freund distribution models situations in which the exponential residual lifetime of one component depends on the working status of another component the literature has discussed generalizations and analogs of the freund distribution such generalizations can be obtained by replacing exponential random variables by weibull or gamma random variables or by compounding mixing freund distributions this paper shows how to unify various well known generalizations of the freund distribution by using one simple functional representation this functional representation has a physical meaning that sheds new light on the meaning of some of these generalizations furthermore using this functional representation we derive various properties of the freund distribution and its extensions in particular we show that some random variables which have the freund distribution or one of its extensions are positively associated and also belong to some classes of multivariate increasing failure rate average and new better than used distributions these properties provide bounds for various probabilistic quantities of interest in reliability theory

nbu processes with general state space waiting times for many kinds of events can be represented as first passage times for appropriate stochastic processes in some applications the most natural stochastic process is multidimensional in other applications it may be more natural to consider a stochastic process which takes on values in a more general space in this paper we consider new better than used nbu stochastic processes on partially ordered separable banach spaces that is processes which have nbu first passage times to closed upper sets various equivalent conditions for a process to be nbu are derived some of the results can be used also to characterize increasing hazard rate average ihra processes examples of such processes include processes with shocks and recovery and processes with random repair times some applications in reliability theory and in other areas of probability are given

approximate bisimulation relations for constrained linear systems in this paper we define the notion of approximate bisimulation relation between two continuous systems while exact bisimulation requires that the observations of two systems are and remain identical approximate bisimulation allows the observations to be different provided the distance between them remains bounded by some parameter called precision approximate bisimulation relations are conveniently defined as level sets of a so called bisimulation function which can be characterized using lyapunov like differential inequalities for a class of constrained linear systems we develop computationally effective characterizations of bisimulation functions that can be interpreted in terms of linear matrix inequalities and optimal values of static games we derive a method to evaluate the precision of the approximate bisimulation relation between a constrained linear system and its projection this method has been implemented in a matlab toolbox matisse an example of use of the toolbox in the context of safety verification is shown

symmetries and conservation laws for hamiltonian systems with inputs and outputs a generalization of noether s theorem the definition of symmetries and conservation laws for autonomous i e without external cases hamiltonian systems are generalized to hamiltonian systems with inputs and outputs it is shown that a symmetry implies the existence of a conservation law and vice versa thereby generalizing noether s theorem for autonomous hamiltonian systems

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

a class of multivariate new better than used distributions a class of multivariate distributions with new better than used nbu marginals is introduced a number of necessary and sufficient conditions for a distribution to be a member in the class are given closure results which are useful for the identification or construction of members in the class are derived in particular simple proofs of some well known preservation properties of the class of the univariate nbu distributions are obtained two examples of replacement models that give rise to multivariate nbu distributions in the class are discussed

nonlinear norm observability and simulation of control systems abstract the bi simulation relation has recently been attracting growing interest in the study of nonlinear control systems in the hope that through such a relation the behaviors and properties of a nonlinear system can be inferred from those of another system which is easier to handle in this paper we consider the propagation of the property of nonlinear norm observability through a simulation relation given two control systems that are related by a graph simulation relation we derive conditions under which the norm observability of the simulating system implies the norm observability of the simulated system the obtained results are given in terms of set valued functions several examples are included to illustrate various applications of our results

bisimilar linear systems the notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics and in particular reachability in this paper the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems given a discrete time or continuous time linear system and a finite observation map we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system interestingly the characterizations for discrete time systems are more restrictive than for continuous time systems due to the existence of an atomic time step we show that computing the coarsest bisimulation which results in maximum complexity reduction corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map these results establish strong connections between complexity reduction concepts in control theory and computer science

analysis of the neumann problem along a straight edge we show there exists an l p solution for p 2 to the open image in new window neumann problem on an edge domain in open image in new window 2 for 0 1 forms and we explicitly compute the singularities which are of complex logarithmic and arctangent type along the edge of the solution

hierarchical trajectory refinement for a class of nonlinear systems trajectory generation for nonlinear control systems is an important and difficult problem in this paper we provide a constructive method for hierarchical trajectory refinement the approach is based on the recent notion of f related control systems given a control affine system satisfying certain assumptions we construct a f related control system of smaller dimension trajectories designed for the smaller abstracted system are guaranteed by construction to be feasible for the original system constructive procedures are provided for refining trajectories from the coarser to the more detailed system