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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.

Figure 14-8. Multivariable block diagram.

graphics/14fig08.gif

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

graphics/14equ19f.gif


The controller, process, and disturbance matrices are

graphics/14equ19g.gif


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

graphics/14equ19h.gif


where gc1(s) and gc2(s) will usually be PID-type controllers. The relationship between inputs (both manipulated and disturbance) and outputs is

graphics/14equ19j.gif


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

graphics/14equ19k.gif


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

Equation 14.20

graphics/14equ20.gif


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

Recall that the SISO closed-loop transfer function is

graphics/14equ20a.gif


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

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