Admissible region restriction

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. 

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. 

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