Estimate unmeasured vector

The pole placement design predicates on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasureable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state X is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skip these measurable states. 

If it happens that 7 = 0 (all variables measured) then z = x, fW = and X e iM means simply g(x) = 0 (with S g U). In practice, if 7 > 0 then the frequent assumption is that the unmeasured vector y is uniquely determined by the condition g(z) = 0 given x = x g the decomposition (10.1.29) then the unique y = y is the estimate of the unmeasured vector. More generally, we have admitted the case that y is not uniquely determined see then the observability (and redundancy) classification in Section 8.5. 

We require that the estimate obeys the solvability condition (a) There exists some y (unmeasured vector) such that... 

When a gas chromatograph is used for composition measurements, estimates of the partial pressures are given directly. Or rather, estimates are made directly of a constituent vector c, that if properly normalized would give the partial pressures. If, on the other hand, some components of the outlet stream are unmeasured, then it is not immediately clear how to normalize c. In fact we may use a simplified version of the procedure described above to solve this problem for a mass spectrometer. Again we ignore the calibration matrix Mj and the measured peaks vector v (i.e. set Mi equal to the identity matrix in the formulas above), and obtain the following procedure. Partition ... 

Here, m j (subvector of m ) represents the measured (integrated) mass flowrates, y = m , the unmeasured ones, and the states of the inventories at time. The vector s. i is formally constant and the measured value of x thus x is composed of the actually measured value of (say, = (nifk) X and of x = s - k., where sj is the actually measured value of the constant represents the preceding estimate of 8,., . The covariance matrix of measurement errors in x used in the reconciliation is F (11.2.1). 

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