Linear controllers performance criteria

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Schweickhardt and Allgower in Chapter A3 mainly concentrate on the nonlinearity assessment of processes. A comprehensive overview of general nonlinearity measures and a thorough investigation of the predictive and computational dimension of open loop measures are presented. As the main objective becomes the development of a tool to judge whether a nonlinear controller should be benefieial or needed for a particular process with specific nonlinear characteristics, the controller relevant nonlinearity is quantified. The selected measure is based on the relative differences between the output of nonlinear state feedback law and that of an equivalent linear state feedback law. The controller relevant nonlinearity measure depends not only on the plant dynamics and region of operation but also on the performance criterion used in the derivation of the controller law. 

Autosampler precision can be checked by replicate injections of a control sample, with wash injection intervals between every two sample injections. The repeatability of peak areas, mathematically expressed as RSD, is used as a criterion for autosampler precision. For example, when 10 consecutive injections of 10 pL of a solution are performed, the expected RSD for peak area precision ranges from 0.5% to 1.0%. Single injections of different volumes, such as 5, 10, 50, or 80 pL, can also be used, simultaneously checking the linearity of the injector, the detector, and the data system. Another approach to qualify the autosampler involves the gravimetric determination of the average volume of water per injection withdrawn from a tared vial after six 50-pL injections. The procedure... 

Example In the manufacture of aircraft engines and gas turbines superalloy is widely used due to its excellent performance, which to a great extent depends on the content of trace elements in the alloy. Critical control of the alloy s composition is therefore essential for the safe operation of such engines. The obvious method to avoid isobaric interferences is to select isotopes where no interferences exist. While H, Pb, and Bi meet this criterion, in case of Sn only the Sn ion is free of superimpositions, whereas Sn has serious interferences from Ars, Mo N2, and "Te". To determine the correct content of elements in superalloy on a quadrupole GD instrument also when no suitable isotope exists, both matching the sample matrix by use of external standards and multivariable linear regression have extensively been used (Fig. 15.10) [61]. 

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