Admissible region

Capdevila X, Dadure C. Perioperative management for one day hospital admission regional anesthesia is better than general anesthesia. Acta Anaesthesiol Belg. 2004 (suppl 55) 33—36. 

Since with the fixed value of vector Pbr and the lack of constraint (62) the admissible region of solutions is a polyhedron, F reaches its minimum at one of its vertices. With the rank of matrix A equal to m-1 and n unknowns the reference solution contains no less than n- m-1) zero components, which equals the number of chords of the system of independent loops of the network graph. In this case the graph tree is a polyhedron vertex and the optimal variant should be among the set of trees of a redundant scheme. 

The admissible region for closed-loop eigenvalues is denoted as F, a system is called F-stable if all its eigenvalues are located in this region and an imcertain system is called robustly F-stable if all eigenvalues for all operating conditions are contained in F. The definition of F-stability permits arbitrary regions in the complex 5 -plane and does not imderlie any restrictions. It also includes the special cases of the left half-plane for Hurwitz stability and the unit circle for Schur stability. 

Let first Qv O. Then the condition of solvability is (8.1.13) and adding the third column in (8.5.3) to the first and second, one finds immediately that this is also a necessary and sufficient condition for rankB to be full rank. Then, whatever be otherwise the measured values, Eq.(8.1.1) is uniquely solvable. Thus if the admissible region of measured values is restricted to a subset of variables obeying the inequalities (8.1.13) and (say) Q > 0, the system can be called observable in the mj, and at the same time we have rankB = 3. 

Let further the heat exchanger be absent. Then the variable Q is also absent, and we have the condition (8.1.14). Let in addition the condition (8.1.12) be satisfied in the admissible region of measured variables. Then, on by (8.1.1) where Q = 0, according to (8.5.4) we have... 

It is not difficult to show that if the condition (8.1.12) is fulfilled then y and rankB = 2 in the whole admissible region, in particular on iM. In the linearized system, we thus obtain the degree of redundancy equal to 1. On the other hand, so long as > 0, we obtain again the condition (8.1.14) and this condition satisfied, we have m >0 and m > 0. Further, because the two columns of B are linearly independent and because on as shown above, with the third column of (8.5.3) the rank of the matrix equals also 2, the m3-column of the Jacobi matrix is a linear combination of the columns of the new matrix B (8.5.6). According to (7.1.27), at points of Mthe. measured variable m3 can be qualified as nonredundant, in accord with the tentative qualification before formula (8.1.16). But observe that at points zi M, the matrix (8.5.3) is of rank 3, thus in the linearized system, again by (7.1.27), m3 will be qualified as redundant. If... 

We suppose that the state vector z can take its values in some N-dimensional interval Vet/ where ll is the admissible region (8.5.8). The interval can be assessed as some neighbourhood of a vector Zq e fSf. A first information can be obtained in the same manner as above, in the linear case. Taking different Zq e we can examine the behaviour of the Jacobi matrix Dg(Zo) on fW (restricted to t thus on r U). [We can also, in the case of balance models, start from different values of the independent parameters representing the degrees of freedom and determining Zq e fW see Sections 8.2 and 8.3. But such procedure may be rather tedious.] In the reconciliation, however, also the behaviour of Dg(z) in a neighbourhood of the solution manifold is relevant. 

Sunnnarizing the above arguments, we have shown that if the minimum of Q cannot be attained, i.e. does not exist in the admissible region (of positive mass flowrates) it exists only as a limit lying outside of the region. The (rigorously formulated) reconciliation problem is not solvable. 

The well-posedness of a problem can thus depend on the admissible region of variables. In the present example, it can be illustrated by the figure... 

If the admissible region of concentrations is (by the technology) limited by y < a and > b (a < b) then the problem is well-posed in our special example, all the measured variables are nonredundant and the minimized adjustments equal zero. 

While the initial value of x is the measured vector in contrast to the linear case we need also an initial guess (say) y of the unmeasured y. Clearly, this y" will be chosen somewhere in the admissible region of variables. The better is the guess, the better is our expectation that the procedure will converge. Given such y% we can compute the residual... 

Then u will be subjected to a condition requiring that y = y + u is not too distant from the initial guess y, to avoid y escaping from the admissible region. Formally, let us minimize the square norm... 

In each step of the whole procedure, we can also check if the computed z e U, thus if it remains in the admissible region. If z tends to escape from the admissible region then the solution gets lost and the reconciliation procedure has failed. If it is a single case in a series of measurements, most likely the initial guess (in particular the measured value x ) was too bad (perhaps due to a gross measurement error). If the case occurs frequently then, most likely, the reconciliation problem is not well-posed. 

It can again happen that the vector z escapes from the admissible region and the solution gets lost then the reconciliation procedure has failed. Both methods (short-cut as well as the previous one) were tested by the authors. They were found to work equally well, with the exception of single failures. 

The theoretical concepts of redundancy/nonredundancy are more tricky see Chapter 8, Section 8.5, finally Subsection 8.5.4. In practice, assuming the covariance matrix F diagonal, the concepts can be replaced by those of redundancy (adjustability)/nonadjustability. So if, in a series of measurements, a variable Xh is found nonadjustable (10.3.14) then it is, almost certainly, nonadjustable in the whole admissible region. Otherwise, at some points a variable can be also almost nonadjustable see Remark to Section 10.5 below. 

If thus the problem is not well-posed in the admissible region it can happen that, even if we have found some z where B(z) is of full column rank and y uniquely determined, at least one of the (psudo)standard deviations assessed by linear approximation is great and the estimated yj is uncertain the measurement is inappropriate for estimating this yj. which is an intuitive interpretation of the not-well-posedness . It can then also happen that in the course of successive approximations, some approximation yf escapes from the amissible region and the solution gets lost. 

The above characteristics make sense only if the minimum is found in an open admissible region Zl, not perhaps somewhere on its boundary. Indeed, in the linearized model we assume that the condition (10.7.6) is fulfilled the condition represents just the minimum in an open region. In addition, computing the confidence intervals such as in (9.3.56) we admit that the true value z of z... 

Figure 4.8 Examples of limit grading curves for fine and coarse aggregates, according to old ASTM recommendation (continuous line) and to past Polish Standard for Ordinary Concrete (broken line). Admissible regions are situated between the respective curves. 

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