Structural Observability Matrix

In order to check whether an nth order LTI system (A, C) is completely observable independent of any parameter values, i.e. whether it is structurally completely observable, one may replace its matrices A, C in the observability matrix O by the structurally equivalent interconnection matrices A, C and determine the rank of the resulting stmctural observability matrix O. The structural equivalent matrices are obtained by replacing each non-zero entry in A, C by the value 1. Every zero entry in A, C remains a zero entry in A, C. An nth order LTI system (A, C) is then said to be structurally observable if the stmctural observability matrix O has full rank, i.e. 

Note that from rank(0 ) n it cannot be concluded that O does not have full rank, i.e. that the system is not observable. 

1 Bond Graph-based Construction of Interconnection Matrices 

2 Matrix-based Analysis of Structural Observability of Switched LTI Systems 

3 Example Electrical Network with Independent Switches 

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With these two interconnection matrices the structural observability matrix reads... 

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 ... 

In order to analyze estimability utilizing such ideas, we first include some notions related to structure. Then we define the concepts of structural observability and the generic rank of a matrix. 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

FIGURE 5.7 Generation of the 2T2 and 22A, states of CH4 +, by VB mixing of the four localized structures. The matrix elements between the structures, shown graphically, lead to the three-below-one splitting of the states, and to the observations of two ionization potential peaks in the PES spectrum. 

Two sets of notation are commonly used to describe overlayer structures observed in diffraction experiments, the Wood notation [92] and a matrix notation. Although the latter is more flexible, the former is more widely used and we shall restrict ourselves to it in this review. The nomenclature is based on a comparison between the unit mesh of the topmost layer, the overlayer, and that of the second, unreconstructed, substrate layer. If a and b are the unit mesh vectors of the substrate layer and a, and bg the unit mesh vectors of the overlayer, then Wood s notation for an overlayer of adsorbed species A on the hkl plane of a crystal M is... 

In Eq. (14), K is the matrix of transfer coefficients. The components of K are basic or structural parameters of the model. If the observations are linear combinations of the compartments, the observation function is given by Eq. (15), in which C is the observation matrix. 

Crystal stracture, twinning, dislocation stmetures and orientation relationships of MgCu2 type precipitates in Zr rich alloys have been studied by [1986Menl]. Polytypic structures assumed by Zr(Fe,Cr)2 Laves phases have been studied by [1986Men2, 1991Bur] both in bulk materials and in precipitates included in Zn rich alloys. Samples were analyzed by [1991 Bur] in as cast conditions as well as after heat treatment at 850 or 900°C. The variety of polytypic structures observed in precipitates was attributed to the lattice misfit and associated stresses between precipitates and the aZr matrix. 

It both conditions are met, then the model is structurally completely observable with the given set of sensors and the rank of the observability matrix O equals the number of states n. If rank(A) = n then one single sensor is sufficient to assure complete state observability. Its type and position is to be chosen so that both conditions are fulfilled. 

Figure 3.9 Craze-like deformation structures observed in SBM diblock copolymers having hexagonally packed PBMA cylinders in the PS matrix the deformation direction is shown by arrows [21] ...

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