14.5 Block-Diagram Analysis

Consider the multivariable block diagram shown in Figure 14-8. Each of the variables shown on this diagram is a vector and the blocks represent matrix transfer functions. That is, for two process outputs, two manipulated inputs, and one disturbance, the corresponding vectors are given below.

Setpoints, outputs, manipulated inputs, and the disturbance input are

The controller, process, and disturbance matrices are

Notice that if SISO controllers are used, the controller transfer function matrix is

where g_c1(s) and g_c2(s) will usually be PID-type controllers. The relationship between inputs (both manipulated and disturbance) and outputs is

Since the error signal is e(s) = r(s) - y(s), we can derive (where I is the identity matrix)

and using matrix inversion to solve for y(s), we find

Equation 14.20

This last equation is the closed-loop transfer function relationship for multivariable systems. Notice that the order of multiplication is very important. In general, G_P(s)G_c(s) G_c(s)G_p(s). This is clear if, for example, G_p is a 4 x 3 matrix and G_c is a 3 x 4 matrix.

Recall that the SISO closed-loop transfer function is

For SISO systems, the order of multiplication does not matter; for MIMO systems, it is crucial.