reachability standard form

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An evaluation of estimation techniques for probabilistic reachability

Tell us more Hide this section if you want to rate later. Questions Courses. Reachability matrix for reachable canonical form Consider a system in reachable canonical form Oct 18 AM. Do you need an answer to a question different from the above?

Help us make our solutions better. We want to correct this solution. Tell us more. Was the final answer of the question wrong? Were the solution steps not detailed enough? Was the language and grammar an issue? We appreciate your Feedback Stay Solved :. Didn't find yours? Ask a new question Get plagiarism-free solution within 48 hours. Review Please. Next Previous.

Related Questions. Given the duality of the canonical forms for systems that are both reachable and observable, what Given the duality of the canonical forms for systems that are both reachable and observable, what does it mean for a system to be one and not the other?

Transformation to controllable canonical form. Show that a square matrix has an inverse if and only if its rows form a linearly independent set of vectors.In graph theoryreachability refers to the ability to get from one vertex to another within a graph. In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph.

The connected components of an undirected graph can be identified in linear time.

reachability standard form

The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph which, incidentally, need not be symmetric. Algorithms for determining reachability fall into two classes: those that require preprocessing and those that do not.

reachability standard form

If you have only one or a few queries to make, it may be more efficient to forgo the use of more complex data structures and compute the reachability of the desired pair directly. This can be accomplished in linear time using algorithms such as breadth first search or iterative deepening depth-first search. If you will be making many queries, then a more sophisticated method may be used; the exact choice of method depends on the nature of the graph being analysed.

Three different algorithms and data structures for three different, increasingly specialized situations are outlined below. The Floyd—Warshall algorithm [5] can be used to compute the transitive closure of any directed graph, which gives rise to the reachability relation as in the definition, above.

This algorithm is not solely interested in reachability as it also computes the shortest path distance between all pairs of vertices. For graphs containing negative cycles, shortest paths may be undefined, but reachability between pairs can still be noted. For planar digraphsa much faster method is available, as described by Mikkel Thorup in This algorithm can also supply approximate shortest path distances, as well as route information.

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An outline of the reachability related sections follows. The proof that such separators can always be found is related to the Planar Separator Theorem of Lipton and Tarjan, and these separators can be located in linear time. At each level of the recursion, only linear work is needed to identify the separators as well as the connections possible between vertices.

An even faster method for pre-processing, due to T. Kameda in[7] can be used if the graph is planaracyclicand also exhibits the following additional properties: all 0- indegree and all 0- outdegree vertices appear on the same face often assumed to be the outer faceand it is possible to partition the boundary of that face into two parts such that all 0-indegree vertices appear on one part, and all 0-outdegree vertices appear on the other i.

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Preprocessing performs the following steps.Computing reachability probability amounts to computing multidimensional integrals. In particular, we pay attention to QMC methods due to their theoretical benefits in convergence speed with respect to the MC method.

The Koksma-Hlawka inequality is a standard result that bounds the approximation of an integral by QMC techniques. However, it is not useful in practice because it depends on the variation of the integrand function, which is in general difficult to compute. The question arises whether it is possible to apply statistical or empirical methods for estimating the approximation error.

In this paper we compare a number of interval estimation techniques based on the Central Limit Theorem CLTand we also introduce a new approach based on the CLT for computing confidence intervals for probability near the borders of the [0,1] interval. Based on our analysis, we provide justification for the use of the developed approach and suggest usage guidelines for probability estimation techniques. Mariia Vasileva. Paolo Zuliani. Monte Carlo MC techniques are often used to estimate integrals of a mu Quasi-Monte Carlo qMC methods are a powerful alternative to classical We consider the problem of estimating the density of a random variable X The rough Bergomi rBergomi model, introduced recently in Bayer, Friz, Owenet al.

In this article we consider a number of models for the statistical data Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

Standard Form

Reachability is one of the fundamental problems in verification and model checking. Given a system model and a set of goal states indicating un wanted behaviourdoes the system eventually reach these states? The generalisation of this problem for stochastic systems is called probabilistic reachability, and it amounts to compute the probability that the system reaches a goal state.

Checking reachability in hybrid systems is an undecidable problem for all but the simplest systems timed automata [ 2 ].

Formal verification of hybrid systems can include verifying satisfiability of formulas involving real variables, which is known to be an undecidable problem when, e. This approach has been extended to a bounded probabilistic reachability method with statistically valid enclosures for the probability that a hybrid system can reach a goal state within a given time bound and number of steps [ 20 ].

In particular, we consider the k -step reachability probability for parametric hybrid systems with random parameters.In this paper we introduce the class of two-point boundary-value descriptor systems TPBVDSdiscrete-time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system 'state ' at the two endpoints of the system's interval of definition.

By introducing a standard form for regular pencils we obtain a new and simple generalized Cayley-Hamilton theorem that simplifies our investigation of well-posedness, Green's function solutions, and reachability and observability for TPBVDS. There are two distinct notions of reachability and observability that one can define for TPBVDS, associated with processes that propagate inward from and outward toward the boundaries.

We investigate each of these in detail, obtaining, among other things, far simpler forms for the reachability and observability results than found previously in the literature. In addition, we describe several methods for the efficient solution of TPBVDS, one involving recursions from each end of the interval toward the other and two others.

Documents: Advanced Search Include Citations. WillskytBernard C. Abstract In this paper we introduce the class of two-point boundary-value descriptor systems TPBVDSdiscrete-time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system 'state ' at the two endpoints of the system's interval of definition. Powered by:.We believe the best way to create an equal playing field is through prevention, not after-the-fact litigation.

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Origin Story. Are you ready to go somewhere? Let us help. Explore our programs.In linear algebraa Jordan normal formalso known as a Jordan canonical form [1] or JCF[2] is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in Kor equivalently if the characteristic polynomial of the operator splits into linear factors over K.

This condition is always satisfied if K is algebraically closed for instance, if it is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operatorand the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.

If the operator is originally given by a square matrix Mthen its Jordan normal form is also called the Jordan normal form of M.

Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocksthe order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.

The Jordan—Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matricesis a special case of the Jordan normal form.

The Jordan normal form is named after Camille Jordanwho first stated the Jordan decomposition theorem in Some textbooks have the ones on the subdiagonali.

The eigenvalues are still on the main diagonal. Or, equivalently, if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices.

Consider the following matrix:. The dimension of the eigenspace corresponding to the eigenvalue 4 is 1 and not 2so A is not diagonalizable. The matrix J is almost diagonal. This is the Jordan normal form of A. The section Example below fills in the details of the computation. In general, a square complex matrix A is similar to a block diagonal matrix. J is called the Jordan normal form of A.

reachability standard form

Each J i is called a Jordan block of A. In a given Jordan block, every entry on the superdiagonal is 1. This example shows how to calculate the Jordan normal form of a given matrix.

As the next section explains, it is important to do the computation exactly instead of rounding the results.

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