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EN
Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper.
EN
Hybrid automata are a well-established modelling approach. The formalism is used in many real-time and control systems engineering projects, which makes model composition an increasingly relevant topic. A well-defined composition support allows concurrent engineering activities and the validation of larger systems. However, many existing publications seldom consider it or make unrealistic assumptions on the model design. The article discusses the common problems with hybrid automata composition and presents a new formalism, called linear time-invariant hybrid automata (LTI-HA), which targets specifically these issues. Our approach considers the superposition principle for flow functions, which makes it specifically useful for practical modelling purposes in the spacecraft and control domain. We compare the approach to well-known related ideas, such as hybrid I/O automata. Several properties of composition, such as commutativity, are proven.
EN
The article presents a class of perfect observers for standard continuous-time linear systems. Observers are resistant to significant changes in the dynamics of the system. The value of poles of the system can be unlimited as long as their number does not change and the conditions presented in this article are satisfied. There are presented numerical examples and simulations results.
4
Content available remote Oustaloup parallel approximation
EN
This article presents Oustaloup approximation method for fractional differential systems, which has been modified by the authors. Application of this method by using numerical experiments has been shown and behavior of discrete system has been analyzed.
PL
W artykule przedstawiono zmodyfikowaną przez autorów metodę aproksymacji Oustaloupa dla układów różniczkowych niecałkowitego rzędu. Pokazano jej wykorzystanie poprzez eksperyment numeryczny oraz przeanalizowano zachowanie systemu dyskretnego.
EN
In this paper, an operational method for solving linear and nonlinear systems described by ordinary differential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithm is done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials.
EN
The paper provides the minimal necessary modifications of linear matrix inequality conditions for the mixed H2/H∞ control design as well as for the augmented observer-based fault estimation to be mutually compatible in joint design of integrated fault estimation and fault tolerant control. To be possible, within this integration, to design the controller which guarantees a pre-specified H∞ norm disturbance attenuation level, the design conditions has to be regularized using the H2 performance index and, moreover, augmented fault observer must be of enforced dynamics. Analyzing the ambit of performances given on the mixed H2/H∞ design, the joint design conditions are formulated as a minimization problem subject to convex constraints expressed by a system of LMIs. The feasibility of the conditions is demonstrated by a numerical example.
7
Content available remote The poles method for second-order linear time-varying systems
EN
In dynamic linear systems described by differential equations with constant parameters, the poles of the rational function (transfer function of the system) play an important role. This article attempts to expand the poles concept in a situation where the system is described by the second-order linear system with timevarying parameters. It then introduces the concept of characteristic equations and time-dependent poles.
PL
W opisie liniowych systemów dynamicznych opisanych przez równania różniczkowe o stałych parametrach ważną rolę odgrywają bieguny funkcji wymiernej (transmitancji systemu). Ten artykuł rozszerza koncepcję biegunów na przypadek, gdy system jest opisany równaniem liniowym drugiego rzędu o zmiennych w czasie parametrach. Pojawiają się tu pojęcia zależnego od czasu równania charakterystycznego i zależnych od czasu biegunów transmitancji.
8
Content available remote The poles method for higher-order linear time-varying systems
EN
In dynamic linear systems described by differential equations with constant parameters, the poles of the rational function (transfer function of the system) play an important role. This article attempts to expand the poles concept in a situation where the system is described by the N-th order linear system with time-varying parameters. It then introduces the concept of characteristic equations and time-dependent poles.
PL
W opisie liniowych systemów dynamicznych opisanych przez równania różniczkowe o stałych parametrach ważną rolę odgrywają bieguny funkcji wymiernej (transmitancji systemu). Ten artykuł rozszerza koncepcję biegunów na przypadek, gdy system jest opisany równaniem liniowym N-tego rzędu o zmiennych w czasie parametrach. Pojawia się tu pojęcie zależnego od czasu równania charakterystycznego i zależnych od czasu biegunów transmitancji.
EN
We reconsider work by Bellin and Scott in the 1990s on R. Milner and S. Abramsky’s encoding of linear logic in the π-calculus and give an account of efforts to establish a tight connection between the structure of proofs and of the cut elimination process in multiplicative linear logic, on one hand, and the input-output behaviour of the processes that represent them, on the other, resulting in a proof-theoretic account of (a variant of) Chu’s construction. But Milner’s encoding of the linear lambda calculus suggests consideration of multiplicative co-intuitionistic linear logic: we provide a term assignment for it, a calculus of coroutines which presents features of concurrent and distributed computing. Finally, as a test case of its adequacy as a logic for distributed computation, we represent our term assignment as a λP system. We argue that translations of typed functional languages in concurrent and distributed systems (such as π-calculi or λP systems) are best typed with co-intuitionistic logic, where some features of computations match the logical properties in a natural way.
EN
In this paper, we examine P systems with a linear membrane structure, i.e., P systems in which only one membrane is elementary and the output of which is read out as the sequence of membrane labels in the halting configuration or vectors/numbers represented by this sequence. We investigate the computational power of such systems, depending on the number of membrane labels, kinds of rules used, and some other possible restrictions. We prove that two labels, elementary membrane creation and dissolution, together with the usual send-in and send-out rules, suffice to achieve computational completeness, even with the restriction that only one object is allowed to be present in any configuration of the system. On the other hand, limiting the number of labels to one reduces the computational power to the regular sets of non-negative integers. We also consider other possible variants of such P systems, e.g., P systems in which all membranes but one have the same label, P systems with membrane duplication rules, or systems in which multiple objects are allowed to be present in a configuration, and we describe the computational power of all these models.
EN
In the paper the problems of controllability, reachability and minimum energy control of a fractional discrete-time linear system with delays in state are addressed. A general form of solution of the state equation of the system is given and necessary and sufficient conditions for controllability, reachability and minimum energy control are established. The problems are considered for systems with unbounded and bounded inputs. The considerations are illustrated by numerical examples. Influence of a value of the fractional order on an optimal value of the performance index of the minimum energy control is examined on an example.
12
EN
An accurate numerical method is established for matrix inversion. It is shown theoretically that the scheme possesses the high order of convergence of seven. Subsequently, the method is taken into account for solving linear systems of equations. The accuracy of the contributed iterative method is clarified on solving numerical examples when the coefficient matrices are ill-conditioned. All of the computations are performed on a PC using several programs written in MATHEMATICA 7.
EN
Parsing Expression Grammar (PEG) encodes a recursive-descent parser with limited backtracking. The parser has many useful properties, and with the use of memoization, it works in a linear time. In its appearance, PEG is almost identical to a grammar in Extended Backus-Naur Form (EBNF), but usually defines a different language. However, in some cases only minor typographical changes are sufficient to convert an EBNF grammar into its PEG parser. As recently shown by Medeiros, this is, in particular, true for LL(1) grammars. But this is also true for many non-LL(1) grammars, which is interesting because the backtracking of PEG is often a convenient way to circumvent just the LL(1) restriction. We formulate a number of conditions for EBNF grammar to become its own PEG parser, and arrive at a condition that we call LL(1p), meaning that a top-down parser can choose its next action by looking at the input within the reach of one parsing procedure (rather than by looking at the next letter). An extension to LL(kp) for k > 1 seems possible.
14
Content available Optimal decoupling controllers revisited
EN
The problem of decoupling a linear system by dynamic compensation into multi-input multi-output subsystems is studied by applying proper and stable fractional representations of transfer matrices. A necessary and sufficient condition is given for a decoupling and a stabilizing controller to exist. The set of all controllers that decouple and simultaneously stabilize the system is determined in parametric form. Optimal decoupling controllers are then obtained by an appropriate selection of the parameter.
EN
The positivity of descriptor continuous-time and discrete-time linear systems with regular pencils are addressed. Such systems can be reduced to standard linear systems and can be decomposed into dynamical and static parts. Two definitions of the positive systems are proposed. It is shown that the definitions are not equivalent. Conditions for the positivity of the systems and the relationship between two classes of positive systems are established. The considerations are illustrated by examples of electrical circuits and numerical examples.
EN
The paper is devoted to the problem of observability and controllability analysis in nonlinear dynamic systems. Both continuous- and discrete-time systems described by nonlinear differential or difference equations, respectively, are considered. A new approach is developed to solve this problem whose features include (i) consideration of systems with non-differentiable nonlinearities and (ii) the use of relatively simple linear methods which may be supported by existing programming systems, e.g.,Matlab. Sufficient conditions are given for nonlinear unobservability/uncontrollability analysis. To apply these conditions, one isolates the linear part of the system which is checked to be unobservable/uncontrollable and, if the answer is positive, it is examined whether or not existing nonlinear terms violate the unobservability/uncontrollability property.
17
Content available Response of linear system to α-stable Levy input
EN
This article suggests the method of analysis stochastic processes in deterministic linear SISO system of first order. Using the definition of α-stable random variable, the definition of α-stable Levy process has been introduced and presented their properties. Transformation of α-stable white noise process in the system has been investigated as well. Obtained results has been illustrated by an example.
EN
A method for computation of positive asymptotically stable realizations of fractional descriptor continuous-time linear systems with regular pencil is proposed. The method is based on the decomposition of the improper transfer matrix into strictly proper matrix and a polynomial matrix. A procedure for decomposition of a positive asymptotically stable realization is given and illustrated by a numerical example.
EN
A method for checking of the positivity of descriptor continuous-time and discrete-time linear systems with singular pencil is proposed. The method is based on elementary row and column operations on the matrices of descriptor systems. Necessary and sufficient conditions for the positivity of the descriptor systems are established.
EN
Conditions for the existence of positive realizations for descriptor discrete-time linear systems are established. A procedure for computation of positive realizations for improper transfer matrices is proposed. The effectiveness of the method is demonstrated on numerical example.
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