Bridging the Gap

Abstract

Many systems that are required to be controlled—for example, the boiler-turbine unit discussed in earlier chapters—consist of several inputs and several outputs, and are known as multivariable systems, whereas the systems that have been analysed so far throughout this book are exclusively single-input single-output (SISO) systems. Thus it should be asked at this stage whether the SISO techniques are applicable to multivariable systems. Unfortunately, the techniques are not directly applicable, because of the presence of interaction between the variables. Figure 9.1 represents a

multivariable system with two inputs u₁ (s) and u₂ (s) and two outputs x₁ (s) and x₂(s), where

$$\left. {\begin{array}{*{20}{c}} {{x_1}\left( s \right) = {G_{11}}\left( s \right){u_1}\left( s \right) + {G_{12}}\left( s \right){u_2}\left( s \right)} \\ {{x_2}\left( s \right) = {G_{21}}\left( s \right){u_1}\left( s \right) + {G_{22}}\left( s \right){u_2}\left( s \right)} \end{array}} \right\}$$

(9.1)

The above-mentioned interaction is contributed by the transfer functions G₁₂(s) and G₂₁(s), which connect output 2 to input 1 and output 1 to input 2, respectively. If the transfer functions G₁₂(s) and G₂₁(s) were noth zero, the multivariable system would reduce to tow independent SISO systems, that is

$$\left. {\begin{array}{*{20}{c}} {{x_1}\left( s \right) = {G_{11}}\left( s \right){u_1}\left( s \right)} \\ {{x_2}\left( s \right) = {G_{22}}\left( s \right){u_2}\left( s \right)} \end{array}} \right\}$$

(9.2)

which can be analysed using the classical techniques of the previous chapters.