Parity space

And model-based methods which are composed of quantitative model-based methods (such as analytical redundancy (Chow and Willsky, 1984), parity space (Gertler and Singer, 1990), state estimation (Willsky, 1976), or fault detection filter (Franck, 1990)) and qualitative model-based methods (such as causal methods digraphs (Shih and Lee, 1995), or fault tree (Venkatasubramanian, et ah, 2003)). 

F Kosebalaban and A Cinar. Integration of multivariate SPM and FDD by parity space technique for a food pasteurization process. Comput. Chem. Engg., 25 473-391, 2001. 

The parity space approach to FDI is based on a comparison of the behaviour of a real process with that of a model that describes the non-faulty process. Any discrepancies between the two are described by residuals. The development of parity relations for residuals using a state space model goes back to Chow and WiUsky [35] and has been presented in various publications, e.g. [8, 34, 36]. In the following, only the basic idea is outlined. 

Suppose that the dynamic behaviour of a system can be described by a linear LTI model (1,4a, 1.4b). Let n be the order of the system,the number of inputs, the dimension of the output vector y and q [Pg.13]

The direction and the magnitude of the parity vector depend on the faults that have occurred. All parity vectors build a tir dimensional so-called parity space. Any linear combination of rows in (1.10) is called a parity relation [8]. 

As (1.18) indicates, residual ri (r) is independent of the states a (r) and the unknown disturbances rf (f). It depends only on the faults / t) and if there are none, the residual vanishes. Besides w, further vectors can be chosen that satisfy (1.17) Hr such vectors constitute the rir y. q + l) o matrix W in (1.10). Residuals generated in this way are called parity space residuals [28, 37]. 

Equation (1.18) further shows that the computation of parity space residuals requires the derivatives of measured inputs and outputs up to the order q. As these measured signals carry noise, they need to be filtered which can be done by so-called... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

It has been discovered recently that the spectrum of solutions for growth in a channel is much richer than had previously been supposed. Parity-broken solutions were found [110] and studied numerically in detail [94,111]. A similar solution exists also in an unrestricted space which was called doublon for obvious reasons [94]. It consists of two fingers with a liquid channel along the axis of symmetry between them. It has a parabolic envelope with radius pt and in the center a liquid channel of thickness h. The Peclet number, P = vp /2D, depends on A according to the Ivantsov relation (82). The analytical solution of the selection problem for doublons [112] shows that this solution exists for isotropic systems (e = 0) even at arbitrary small undercooling A and obeys the following selection conditions ... 

The last bit, encodes the space-tinie parity of each site and is used to prevent the simultaneous CA-like updating of all lattice sites that we know must somehow be circumvented if the updating is to remain strictly synchronous. At any given time, exactly half of all sites are updated this is in fact nothing more than Vichniac s [vich84] checkboard updating. 

In this section, Hv means the adiabatic rate of change of the Hamiltonian.) By construction, the odd work has odd phase space parity, Vk (r, t) = -14 (1, t). With it the nonequilibrium probability distribution for mechanical work is [5]... 

The space inversion transformation is x —> —x and the corresponding operator on state vector space is called the parity operator (P). The parity operator reverses... 

Diagnostic observers consist in the definition of a set of observers from which it is possible to define residuals specific of only one failure [8]. Parity relations are relations derived from an input-output model or a state-space model [11] checking the consistency of process outputs and known process inputs. 

Let us make clear now the correspondence between our treatment here and Erdahl s 1978 treatment [4, Sec. 8]. Erdahl works in general Fock space and his operators conserve only the parity of the number of nuclei. He exhibits two families of operators that are polynomials in the annihilation and creation operators containing a three-body and a one-body term. Generic instances of these operators are denoted y and w. The coefficients are real, and Erdahl stresses that this is essential for his treatment. The one-body term is otherwise unrestricted, but the three-body term must satisfy conditions to guarantee that y+y or H +w does not contain a six-body term. For the first family the conditions amount to the three-body term being even under taking the adjoint, and for... 

In summary, Erdahl s treatment is more general and allows a more concise formulation because he works in Fock space, conserving only the parity of the number of particles however, he finds it necessary to restrict the coefficients to be real. We work at fixed particle number and have no reason for the restriction to real coefficients. If the Hamiltonian should be general Hermitian, in which case the RDM must likewise be assumed to be general Hermitian, then our approach leads to Hermitian semidefinite conditions. 

This section of the appendix is based on Appendix B of Ref 80. It outlines the transformation of the space-fixed form of the continuum wavefunction, Eq. (4.3), to a body-fixed form. It differs from the previous development in that the angular functions used in the final equations are all parity-adapted. 

The total angular momentum eigenfunctions, 7(R,f), have parity (—1)- therefore, the summation in Eqs. (A.4) and (A.5) extends over both positive and negative parities. The function is the space-fixed radial scattering... 

We now use Ae fact that the space-fixed functions (R, f) have parity (—1) = (—1). We can therefore multiply the second Wigner D matrix element in the curly bracket in Eq. (A.ll) by 1 = (—1) - This yields... 

The identity operator within the space of functions with a hxed value of J and the parity (denoted by p), and that are associated assymptotically with a quantum number K of the body-fixed z component of the total angular momentum, is... 

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