Array gain

On Array Gain Some readers may by now have noticed that very little attention has been given to the gain of the array. This should not be interpreted as lack of interest in that subject. Rather it reflects the fact that it can be obtained in a forward and simple manner. 

For those who still prefer to marvel over the element pattern and gain, just take the total array gain as calculated above and divide it by the number of elements. At some point down the road, they will most likely ask for the total array gain. Well, just multiply the element gain by the number of elements ... 

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 ... 

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

Thomas, R. E. Gaines, B. G. "Methodology for Designing Accelerated Aging Tests for Predicting Life of Photovoltaic Arrays" Department of Energy/National Bureau of Standards Workshop on Stability of Thin Film Solar Cells and Materials, Washington, D.C., May 1-3, 1978. 

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). 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

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. 

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. 

---