Canonical format

Fig. 7-3. Canonical format of matrix of linear system (7.2.3) (The verbal classification is explained below)...

Let further 7 > 0 thus L > 0 (else B would be zero matrix). Then Gauss-Jordan elimination on matrix B has been carried out, and we have the canonical format according to Fig. 7-3. 

Consequently, our transformation is just one of those admitted in (7.1.16). Although we have not needed computing the matrix L explicitly, formally the transformation to canonical format is a matrix projection, according to the terminology introduced after formula (7.1.16). Only the variables have been re-numbered to obtain the arrangments (7.2.3 and 7). 

We have thus completed the classification. The group I of columns represents the observable variables, the group II the unobservable ones. In this manner, the canonical format of the extended matrix enables one to identify uniquely the following invariants of the given linear system (independent of the admissible equivalent transformations). 

Notice that ranging the equations in the order c, b, a, multiplying Eq.c by -1, and taking the yj-variable as first we have the canonical format of the equivalent matrix of the system. Thus, in accord with direct inspection, the variable is observable, y, and yz unobservable. Taking for instance Jc, = 1 thus = 0, and Jcj = 0 the equations read... 

The arbitrariness of the above matrix L makes possible to replace the theoretical construction (using a basis of KerB ) by Gauss-Jordan elimination-, see Section 7.2. By a procedure described in the section and leading to the final arrangement according to Fig. 7-3, we obtain the canonical format of the matrix of the system (7.4.2). The extended matrix (B, A, c) can be transformed into equivalent form... 

As summarized at the end of Section 7.2, before Remarks, the canonical format determines certain invariants of the linear system (7.4.2). For the solvability in z (of Cz + c = 0) and for the regularity (7.4.1), see above (absence of linearly dependent equations). Then the invariants are the number H (degree... 

The method of matrix projection was introduced, in the context of balance equations, by Crowe et al. (1983). The observability/ redundancy classification criteria were clearly formulated also by Crowe (1989). The transformation of the set of linear equations to canonical format was presented in Madron (1992), Section 4.2, and in special form with more details in Madron and Veverka (1992). 

Explicitly, the solution can be found when using the matrix projection (elimination) according to Chapter 7. We suppose again that the matrix (B,A) is of full row rank M, and we designate L - rankB. Using in particular the canonical format (7.4.4) with (7.4.5 and 6), we can rearrange and partition the unmeasured variables... 

In the frequent case where the whole y is observable we have yg = y and Ag = A" with the special transformation (canonical format) according to Section 7.2 (Gauss-Jordan elimination). 

As the reader can notice, the latter values do not differ considerably from the initially guessed ones. The Jacobi matrix evaluated at this point and transformed to canonical format is again (10.6.9), with no zero column in submatrix A . In a series of such measurements, the canonical format (with different numerical values) will remain the same. We thus conclude in addition that... 

As the second example, let in addition the variable be measured. The Jacobi matrix evaluated again at the standard operation conditions and transformed to canonical format can be found immediately. In (10.6.9), we only shift the y -column and place it after the column y Then for instance in (10.6.9a) we have... 

The canonical format of the Jacobi matrix evaluated at this point is the same as with the standard values (only numerically different) and will remain such also in a series of measuren nts. We conclude that... 

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