Control systems block diagrams show how signals move through an engineered system and how each part changes the input. This cheat sheet helps students read, simplify, and analyze diagrams used in robotics, electronics, vehicles, and automation. It is useful because complex systems become easier to understand when each block, summing junction, and feedback path is clearly identified.
Key Facts
- A block represents a transfer function, and its output is found by multiplying the input by the block value: Y(s) = G(s)X(s).
- Blocks in series multiply together, so two blocks G1(s) and G2(s) combine as Geq(s) = G1(s)G2(s).
- Blocks in parallel add together when their outputs are summed, so Geq(s) = G1(s) + G2(s) for a positive summing junction.
- For negative feedback, the closed-loop transfer function is T(s) = G(s) / (1 + G(s)H(s)).
- For positive feedback, the closed-loop transfer function is T(s) = G(s) / (1 - G(s)H(s)).
- The error signal in a basic negative feedback loop is E(s) = R(s) - H(s)Y(s).
- The characteristic equation of a closed-loop negative feedback system is 1 + G(s)H(s) = 0.
- A continuous-time system is stable when all closed-loop poles have negative real parts.
Vocabulary
- Block diagram
- A visual model that uses blocks, arrows, and junctions to show how signals pass through a control system.
- Transfer function
- A ratio that describes the output of a system divided by its input in the s-domain, usually written as G(s) = Y(s) / X(s).
- Summing junction
- A point in a block diagram where two or more signals are added or subtracted.
- Feedback
- A signal path that sends part of the output back to compare with or modify the input.
- Closed-loop system
- A system in which the output is measured and fed back to affect the system input.
- Pole
- A value of s that makes the denominator of a transfer function equal to zero and strongly affects system stability and response.
Common Mistakes to Avoid
- Forgetting the feedback sign is wrong because negative feedback uses 1 + G(s)H(s), while positive feedback uses 1 - G(s)H(s).
- Adding blocks in series is wrong because series blocks multiply, so the equivalent transfer function is G1(s)G2(s), not G1(s) + G2(s).
- Ignoring the feedback block H(s) is wrong because the returned signal is H(s)Y(s), not always just Y(s).
- Moving a summing junction across a block without adjusting the signal is wrong because moving junctions can require multiplying or dividing by the block transfer function.
- Judging stability from the open-loop transfer function alone is wrong because closed-loop stability depends on the poles of the closed-loop transfer function.
Practice Questions
- 1 Two blocks are in series with G1(s) = 4 and G2(s) = 3 / (s + 2). What is the equivalent transfer function?
- 2 A negative feedback system has G(s) = 10 / (s + 5) and H(s) = 1. Find the closed-loop transfer function T(s).
- 3 Two parallel paths have transfer functions G1(s) = 2s and G2(s) = 5, and their outputs are added. What is the equivalent transfer function?
- 4 Why does negative feedback often improve control system accuracy and reduce sensitivity to disturbances?