Example 1: Transfer function of an interacting system
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Example 1: Transfer function of an interacting system
An example of a control system with multiple signal paths is a multi-legged robot.
... A similar analysis can be accomplished using block diagram reduction ...
Example 1: Transfer function of an interacting system A two-path signal-flow graph is shown in Figure (a) and the corresponding block diagram is shown in Figure (b). An example of a control system with multiple signal paths is a multi-legged robot. The paths connecting the input R(s) and output Y(s) are P1 = GXG2G2GA (path 1) and P2 = G5G6G7G8 (path 2).
There are four self-loops: L1= G2H2, L2 = H3G3, L3 = G6H6, and L4 = G7H7. Loops L1, and L2 do not touch L3 and L4. Therefore, the determinant is ∆ = 1 - (L1 + L2 + L3 + L4) + (L1L3 + L1L4 + L2L3 + L2L4). The cofactor of the determinant along path 1 is evaluated by removing the loops that touch path 1 from A. Hence, we have L1 = L2 = 0 and ∆1 = 1 - (L3 + L4). Similarly, the cofactor for path 2 is ∆2 = 1 - (L1 + L2)
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A similar analysis can be accomplished using block diagram reduction techniques. The block diagram shown in Figure (b) has four inner feedback loops within the overall block diagram. The block diagram reduction is simplified by first reducing the four inner feedback loops and then placing the resulting systems in series. Along the top path, the transfer function is
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Example 2: Transfer function of a multiple-loop system A multiple-loop feedback system is shown in Figure in block diagram form. There is no need to redraw the diagram in signal-flow graph form, and so we shall proceed as usual by using Mason's signal-flow gain formula. There is one forward path Px = G1G2G3G4. The feedback loops are
EXAMPLE 3: Transfer function of a complex system Consider a reasonably complex system that would be difficult to reduce by block diagram techniques. A system with several feedback loops and feed forward paths is shown in Figure below. The forward paths are
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Signal-flow graphs and Mason's signal-flow gain formula may be used profitably for the analysis of feedback control systems, electronic amplifier circuits, statistical systems, and mechanical systems, among many other examples.
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EXAMPLE 4: The position control system for a spacecraft platform is governed by the following equations:
Sketch a signal-flow diagram or a block diagram of the system, identifying the component parts and their transmittances; then determine the system transfer function P(s)/R(s). Solution:
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Dr.Laith Abdullah Mohammed
Question 1: A four-wheel antilock automobile braking system uses electronic feedback to control automatically the brake force on each wheel. A block diagram model of a brake control system is shown in Figure below, where Ff(s) and FR(s) are the braking force of the front and rear wheels, respectively, and R(s) is the desired automobile response on an icy road. Find Ff(s)/ R(s).
Question 2: Off-road vehicles experience many disturbance inputs as they traverse over rough roads. An active suspension system can be controlled by a sensor that looks "ahead" at the road conditions. An example of a simple suspension system that can accommodate the bumps is shown in Figure below. Find the appropriate gain K1 so that the vehicle does not bounce when the desired deflection is R(s) = 0 and the disturbance is Td(s).
