## 42 Block diagram reduction 421 Control systems with multiple loops

A control system may have several feedback control loops. For example, with a ship autopilot, the rudder-angle control loop is termed the minor loop, whereas the heading control loop is referred to as the major loop. When analysing multiple loop systems, the minor loops are considered first, until the system is reduced to a single overall closed-loop transfer function.

To reduce complexity, in the following examples the function of s notation (V) used for transfer functions is only included in the final solution.

Example 4.1

Find the closed-loop transfer function for the system shown in Figure 4.2. Solution

In Figure 4.2, the first minor loop to be considered is G3H3. Using equation (4.4), this may be replaced by

First Minor Loop

First Minor Loop

Now Gml is multiplied by, or in cascade with G2. Hence the combined transfer function is

G2Gm1

The reduced block diagram is shown in Figure 4.3.

Following a similar process, the second minor loop Gm2 may be written

G2G3 1+G3H3

Multiplying numerator and denominator by 1 + G3H3

1 + G3H3 + G2G3H2 But Gm2 is in cascade with Gi, hence

G1G2G3

Transfer function (4.7) now becomes the complete forward-path transfer function as shown in Figure 4.4.

Cascade

Second Minor Loop

Cascade

G2G3 |
C(s) | |||

1 + G3H3 |
• w |

H2 | |||

H1 | ||

4- |

Fig. 4.3 First stage of block diagram reduction.

Fig. 4.3 First stage of block diagram reduction.

G1G2G3 |
C(s) | ||

* |
1 + G3H3 + G2G3H2 |
' w |

H | ||

Fig. 4.4 Second stage of block diagram reduction.

Fig. 4.4 Second stage of block diagram reduction.

The complete, or overall closed-loop transfer function can now be evaluated

Multiplying numerator and denominator by 1 + G3 H3 + G2G3H2

R( ) 1 + G3(s)H3(s) + G2(s)G3(s)H2(s) + G1(s)G2(s)Gs(s)H1(s) ( " )

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