Structural observability

Suppose that the dynamic behaviour of a real system can be described by means of a linear time-invariant (LTI) system given by the state space model 

Let n be the number of states. Then the system is numerically completely state observable according to Kalman s criterion if and only if the rank of the observability matrix 

Borutzky, Bond Graph Model-based Fault Diagnosis of Hybrid Systems, 

The disadvantage of this criterion is that the rank of the observability matrix depends on the numerical values of the system parameters, and that it does not give an indication for which eigenvalue of A the associated transient behaviour is not observable in the case rank(0) n. In other words, the criterion does not identify the unobservable modes. The Hautus test gives an answer to the latter question but still relies on numerical values of the system parameters. According to the Hautus test, an nth order LTI system (A, C) is completely observable if and only if 

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Out-of-Plane Vibrations, yCH and yCD. In accordance with all the proposed assignments (201-203), the bands at 797 and 716 cm correspond to yCH vibrators, which is confirmed by the C-type structure observed for these frequencies in the vapor-phase spectrum of thiazoie (Fig. 1-9). On the contrary, the assignments proposed for the third yCH mode are contradictory. According to Chouteau et al. (201), this vibration is located at 723 cm whereas Sbrana et al. (202) prefer the band at S49cm and Davidovics et al. (203) the peak at 877 cm This last assignment is the most compatible with the whole set of spectra for the thiazole derivatives (203) and is confirmed by the normal vibration mode calculations (205) (Table 1-25). The order of decreasing yCH frequencies, established by the study of isotopic and substituted thiazole derivatives, is (203) yC(4)H > 70(2)13 > yC(5)H. Both the 2- and 4-positions, which seem equivalent for the vCH modes, are quite different for the yCH out-of-plane vibrations, a fact related to the influence observed for the... 

A" Symmetry Vibrations. The first ring vibration of the A" type has been located at 650 cm as a result of the C-type structure observed for that band (Fig. 1-9) (202,203). 

For C70, molecular orbital calculations [60] reveal a large number of closely-spaced orbitals both above and below the HOMO-LUMO gap [60]. The large number of orbitals makes it difficult to assign particular groups of transitions to structure observed in the solution spectra of C70. UV-visible solution spectra for higher fullerenes (C n = 76,78,82,84,90,96) have also been reported [37, 39, 72]. 

The question might be addressed as to whether the hemispheroidal structures observed here may provide an indication that complete toroids may be formed from graphite. The structures appear to form as a result of optimal graphitisation of two adjacent concentric graphene tubes, when further extension growth is no longer feasible (i.e., as in the case shown in Fig. 2 at b-b where a four-walled tube section must reduce to a two-waUed one.) This suggests that perfect toroids... 

The structures observed in the mass spectra of fullerene molecules covered with alkaline earth metals, as described in the previous section, all seem to have a geometric origin, resulting in particularly stable cluster configurations every time a highly symmetrical layer of metal atoms around a central fullerene molecule was completed. When replacing the alkaline earth metals by an alkali metal (i.e., Li, Na, K, Rb, or Cs), a quite different situation arises. 

The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

Since the fine structure observed is only associated with the particular absorption edge being studied, and the energy of the absorption edge is dependent on the element and its oxidation state, EXAFS examines the local structure around one particular element, and in some cases, an element in a given oxidation state. A fuller picture can therefore be obtained by studying more than one absorbing element in the sample. 

Figure 6. Dislocation structures observed in (a) T1-S4 at.%Al and (b) T1-S6 at.%Al crystals with the [021] orientation deformed at room temperature.

The crystal structures observed during the oxidation of molybdenum consist of stable molybdenum dioxide in contact with the metal throughout the range 300-700°C. As the film thickens in the low-temperature range, the trioxide predominates on the surface. At 400°C, molybdenum trioxide is no longer observed and molybdenum dioxide is the only oxide observed. 

One of the simplest examples of line interference is impact broadening of H atom La Stark structure, observed in plasmas [176] (Fig. 4.1.(a)). For a degenerate ground state the impact operator is linear in the S-matrix ... 

The isotropic form has little graphitic characteristic and essentially no optical activity. It is composed of very fine grains without observable orientation and for this reason, it is known as isotropic carbon rather than isotropic graphite. It is often obtained in fluidized-bed deposition, possibly due to continuous surface regeneration by the mechanical rubbing action of the bed. An isotropic structure, observed by transmission electron microscopy, is shown in Fig. 7.4.111]... 

The experimental results of Mlura, ( ) are in agreement with these redispersion experiments. The cherry model structure observed for Pt-Ru/Al203 was more poorly defined than that observed for the Pt-Ru/Si02 clusters. 

Figure 8.8 Series of iniiared spectra during (a) CO2 production and (b) progressive oxidation of COaj[ on Pt3Sn(l 11) in 0.5 M H2SO4 saturated with CO each spectrum was accumulated ftom 50 interferometers at the potential indicated, (c, d) LEED pattern and schematic representation of the p(4 X 4) structure observed on PtsSnflll) after exposing the surface to O2 and electrolyte. The gray dicles are Pt surface atoms, the black circles are Sn atoms covered with OH, and the dotted circles are Sn atoms that are chemically different from Sn atoms modified with OH. (Reprinted with permission from Stamenkovic et al. [2003]. Copyright 1999. The American Chemical Society.)...

Possible stereochemistries for eight-coordinate [Ln(H20)8]3+ are the dodecahedron, the square antiprism, and the cube, although the last seems less likely (295). The proposed nine-coordinate [Ln(H20)9]3+ tri-capped trigonal prism transition state is similar to the [Ln(H20)9]3+ structures observed in the solid state. Thus, the possible water exchange... 

A sequence of STM images were obtained (Figure 7.2) of the Cu(l 10) surface before hydrogen exposure (A), at an ambient H2 pressure of 1 bar (B) and finally after evacuation under UHV conditions (C). It is clear that in the presence of H2 the surface reconstructs into the well-known (1x2) missing row structure and that an evacuation the surface reconstruction is lifted with the (2 x 1) structure observed. AES established that no impurities were present at the Cu(110) surface. 

At Rh(lll) under the same conditions as for CO adsorption at Pt(lll), the surface structures observed vary with the pressure. At the lowest pressure (10 8 Torr) a (2 x 1) overlayer forms but with increasing pressure the (v/7 x /7) 7 19° appears in the 10 7-l Torr region, above which (up to 700 Torr) only a (2 x 2) structure is present6. 

Figure 10.17 STM images of the changes in surface structure observed when meth-anethiol is adsorbed at a Cu(110) surface at room temperature, (a) Clean surface with terraces approximately lOnm wide separated by multiple steps, (b) After exposure to 2 L of methanethiol there has been considerable step-edge movement. On the terraces a local c(2 x 2) structure is evident, (c) After a further 7 L exposure, a view of a different area of the crystal shows rounded short terraces these still retain the c(2 x 2) local structure, (d) After 60 L gross changes to the surface are evident and the STM is unable to image at high resolution.

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