Minimal path

Method Path minimization Path maximization Hyperplane minimization Hypersphere minimization Global minimization Points moved... 

Figure 14 Minimization paths by different methods for the two-dimensional Rosen-brock function f(xx,x2) = (1 — x,)2 + 100(x2 - x,2)2 (a) truncated Newton, (b) conjugate gradient, (c) BFGS quasi-Newton, (d) limited-memory BFGS, (e) steepest descent, first 150 iterations. See program output in Figure 15.

Table 2 shows results for minimization in the two-dimensional case. We use this example to display the various minimization paths on the function s... 

The function 0(x) is called the structure function of the system. The structure function is represented by using the concept of minimal path and minimal cut. A minimal path is a minimal set of components whose functioning ensures the functioning of the system. A minimal cut is a minimal set of components whose failures cause the system to fail. Let aj(x) be the jth minimal path series structure for path Aj, j = 1, p and x) be the h minimal parallel cut structure for cut k =1,. .. s. Then we have... 

For the bridge structure [Figure 5(c)] we have four minimal paths and four minimal cuts, and their structure functions are given as... 

The detector chosen must be insensitive. A refractive index detector (with a preparative cell if required) or a UV detector with an optical path length of 0.1-0.5 mm is the most suitable. Minimal path length is obtained with a film detector whereas the eluate flows over an inclined glass plate. ... 

The final result allows the easy calculation of the probability of A1 and shows, moreover, the minimal paths ( minimum cut sets , that is the minimum number of components involved) which may lead to the final event (the top event) Al. They are A, B, F, G and (C D). The calculation of the final probability... 

Nahman, J. (1992). Minimal paths cuts of networks exposed to common-cause failiues. IEEE Trans. Reliability 41(1), 76-80,84. 

Let Lxy, the set of minimal paths of one point with coordinates (x,y) such as... 

Kobayashi, K. and Yamamoto, H. (1999). A new algorithm in enumerating all minimal paths in a sparse network. Reliability Engineering System Safety, vol. 65, issue 1, p 11-15. 

Samaxl, M.A. (1987). An efficient algorithm for simultaneously deducing minimal paths as well as cuts of a communication network. Microelectronic Reliability, vol.27, no.3, p 437-441. 

It is 93 % for the maximum path length d (0.67 mm) in the crystal, and 98 % for the minimal path length (0.2 mm). Thus, no absorption correction is required with MoKa radiation the lower intensity of the reflections compared with Cu K radiation must, however, be taken into consideration (Eq. 14). 

We see for instance that the chemical distance between graphs and G2 is equal to twice their graph distance where by the term "graph distance" [1] we understand the length (number of edges) of a minimal path between the vertices. 

Theorem 2.4. For a triplet of graphs Gjthe graph G2 lies on a minimal path connecting the graphs Gj and Gj in iff the... 

We have, for instance, d(G 2 8 minimal path from G12 to Gg (or from Gg to Gj g) composed of six edges. This result can be simply verified by (3.13). The maximal common subgraph is empty. 

A safety analysis via minimal path sets detection for object-oriented models... 

A new method is developed that integrates safety analysis with multi-domain object-oriented modelling. This is the contribution aimed by this paper. In essence, the proposed method automatically detects the minimal path sets of a technical system. 

Object-oriented modelling based on differential algebraic equations offers large expressive power for establishing multi-disciplinary engineering tools. In addition, the structure of an Object-Oriented Model (OOM) resembles the functional paths that cause a technical system to operate. This property is exploited by the minimal path sets detection method. 

This paper is organised as follows Section 2 recaps the basics of OOM, describes the selected modelling approach and establishes an example model of a safety relevant aircraft system. Section 3 describes the minimal path sets detection and exemplifies the method by means of the established system model. Section 4 concludes the paper. 

Elmqvist et al. (2014) propose and Bouissou et al. (2014) use a structure variant approach that is even more flexible for the modelling of failures. Future work will combine this approach with the minimal path sets detection method described in this paper. 

In general, the concept of minimal path sets distinguishes only between operation and failure. Thus, the proposed detection method addresses two states (intact, failed) at component and at system level. 

Insertion of component failure rates Failure rates X are stored in each component model that is enhanced with failures. Constant failure rates (exponentially distributed lifetimes) are assumed per default. Since the stress level of a component is known in the simulation, its failure rate can be adapted accordingly. Failure rates are used to compute probability of system operation Rsyff) or failure from the detected minimal path sets. 

This section describes the multi-domain object-oriented model of a stabiliser trim control and actuation system. This model of a safety relevant aircraft system is based on generally available information (Airbus Training, Boeing B737 NG, Persson et al. 2001). It is established for exemplification of the automated minimal path sets detection method. 

The minimal path sets detection method reveals those conditions that cause a system to operate. The example system is considered operational if it moves the stabiliser surface according to the actual command and holds it in place if no command is present. 

Normal operation (mode = 0) or disconnection (mode = 1) of the spur gear are considered for the detection of minimal path sets. 

As explained in section 2.2.2, a system status output (not shown in the images) is computed for use by the minimal path sets detection (section 3). This output indicates if the system operates normally or if it fails (no motion or uncommanded motion). To this end, the actual motion rate of the stabiliser surface is compared with the command (trim switch inputs). 

This section describes the proposed method for automated detection of the minimal path sets of a technical system. It is based on the idea that the related object-oriented model structure resembles the functional paths that cause the system to operate. 

After detection of the minimal path sets, probabilities of system operation R,yJ(i) or failure are computed from failure rates (section 2.2.3) using the method described by Heidtmann (1989). 

Properties of minimal path sets in an object-oriented model... 

In such a graph, minimal path sets are not just simple s-t paths. An s-t path connects two nodes s and t of a graph through an unbranched, open walk ... 

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