Relative gains

If an intensifier, such as the 85 mm presented here, is now replacing the screen, a relative gain of the order of x50 is obtained which results in a conversion factor of 1 to 7.5 (1 incident X photon --> 7.5 electrons). This conversion efficiency not only resolves the quantum sink problem but also increases the light level significantly to compensate for the low gamma fluxes obtained from radioactive sources. 

The most popular and widely used technique for determining the best controller pairing is the relative gain array (RGA) method (Bristol, On a New Measure of Process Interaction, IEEE Trans. Auto. Control, AC-11, 133, 1966). The RGA method provides two important items of information ... 

By definition, the relative gain between the ith manipulated variable and thejth controlled variable is defined as ... 

A SISO eontroller was designed (for mean erystal size), even though relative gain array analysis showed possible interaetions between all of the three eontrol... 

Fig. 1. EPR spectrum of the dithionite-reduced Fepr protein fromD. vulgaris [from (7)]. The protein was 272 ftmol dm" in 25 mmol dm Hepes buffer, pH 7.5, and was reduced under argon with 10 mmol dm sodium dithionite for 3 min at ambient temperature. EPR conditions microwave frequency, 9331 3 MHz modulation frequency, 100 kHz modulation amplitude, 0.63 mT microwave power, 200 mW temperature (relative gain) 16 K (6.3X).

Fig. 3. EPR of as-isolated Z). vulgaris Fepr protein [from (7)]. EPR conditions were the same as for Fig. 1, except for microwave power (0.32 mW), temperature (9 K), Emd relative gain (3.6X).

Thorium. Multiple-collector measurement protocols by TIMS for thorium isotopic analysis typically involve the simultaneous measurement of Th and °Th (for silicate rocks), or Th and °Th, then Th and Th (for low- Th samples), using an axial ion counter and off-axis Faraday collector (Table 1). Various methods are used to correct for the relative gain between the low-level and Faraday detectors and 2a-uncertainties of l-5%o are typically obtained (Palacz et al. 1992 Cohen et al. 1992 McDermott et al. 1993 Rubin 2001). Charge-collection TIMS protocols enable Th, °Th and Th to be monitored simultaneously on a multiple-Faraday array and can achieve measurement uncertainties at the sub-permil level (Esat et al. 1995 Stirling et al. 1995). 

Analyze a MIMO system with relative gain array, and assess the pairing of manipulated and controlled variables. 

You may not find observing the process gain matrix satisfactory. That takes us to the relative gain array (RGA), which can provide for a more quantitative assessment of the effect of changing a manipulated variable on different controlled variables. We start with the blending problem before coming back to the general definition. 

How do we interpret the relative gain The idea is that if m2 does not interfere with m,. the derivative in the denominator should not be affected by the closed-loop involving m2, and its value... 

Example 10.4. Evaluate the relative gain array matrix for the blending problem. The complete relative gain array matrix for the 2 x 2 blending problem is defined as... 

There are several notable and general points regarding this problem, /.< ., without proving them formally here. The sum of all the entries in each row and each column of the relative gain array A is 1. Thus in the case of a 2 x 2 problem, all we need is to evaluate one element. Furthermore, the calculation is based on only open-loop information. In Example 10.4, the derivation is based on (10-25) and (10-26). 

We can now state the general definition of the relative gain array, A. For the element relating the i-th controlled variable to they-th manipulated variable,... 

The relative gain array can be derived in terms of the process steady state gains. Making use of the gain matrix equation (10-32), we can find (not that hard see Review Problems)... 

For your information, relative gain array can be computed as the so-called Hadamard product, Ay = KjjKrH, which is the element-by-element product of the gain matrix K and the transpose of its inverse. You can confirm this by repeating the examples with MATLAB calculations. 

To sum up, the key is to pair the manipulated and controlled variables such that the relative gain parameter is positive and as close to one as possible. 

Find the gain matrix and show that the relative gain parameter is 1. Show how this partially decoupling scheme can be implemented as analogous to Fig. 10.13. 

Interaction among control loops in a multivariable system has been the subject of much research over the last 20 years. Various types of decouplers were explored to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the relative gain array (RGA) as an index of loop interaction... 

It depends on the solution properties, or the relative balance between the hydrophobic and hydrophilic properties of the surfactants at this temperature. This term reflects the relative gain in the free energy when the hydrocarbon part of the surfactants is transferred from the aqueous solution to a hydrocarbon environment (either the micelle interior or the surface layer). The reason for this free energy gain is that the surfactant molecules are oriented at the... 

Depending on the relative gains and losses in internal rotation, the intramolecular reaction is favored entropically by up to 190 J/deg/mol (45 cal/deg/mol) or 55 to 59 kJ/mol (13 to 14 kcal/mol) at 25°C. Substituting 190 J/deg/mol (45 cal/deg/mol) into the exp (ASVR) term of equation 2.7 gives a factor of 6 X 109. Taking into account the difference in molecularity between the second-order and first-order reactions, this may be considered as the maximum effective concentration of a neighboring group, i.e., 6 X 109 M. In other words, for B in equation 2.22 to react with the same first-order rate constant as A B in equation 2.23, the concentration of A would have to be 6 X 109 M. 

Fig. 8.4 Relative ground state photoionization cross section as a function of laser wavelength in Rb in the presence of a 4335 V/cm field. Note the relative gain and offset settings. For the light polarization parallel to the electric field (lower trace), field dependent resonance structure extends beyond the zero field limit. No structure is observed for the case of light polarized perpendicular to the field (upper trace) (from...

The degrees of interaction between different sets of control loops controlling a MIMO plant can usually be determined by the use of the relative gain array technique due to BRISTOL 36. This approach can be illustrated by considering the process shown in Fig. 7.74 which has two inputs and two outputs. The procedure is as follows ... 

Paragraphs (i) to (iv) above apply equally to numerical values of these relative gains. 

The two possible control configurations for a system with two inputs and two outputs are shown in Fig. 7.77. One example of this is illustrated in Fig. 7.73 where the overhead and bottoms product compositions of a distillation process are controlled using the reflux and steam-to-reboiler flowrates respectively as the manipulated variables. Theoretically, we could employ the reflux flowrate to control the bottoms product composition and the steam-to-reboiler flowrate to control the overhead product composition. It is possible to determine which configuration produces the least interaction by forming the system relative gain array A, where ... 

It can be shown that the sum of the relative gains in any row or column of the RGA is equal to one<36). Hence, it is only necessary to compute one of the relative gains in the array and the other three can then be calculated. There are five forms of the RGA which are worthy of note and which can be expressed in terms of different values or ranges of values of A, viz. ... 

Configurations which produce relative gains outside the range 0 to 1 indicate situations where control is difficult. 

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