Section 2 - Block Diagrams & Signal Flow Graphs - ESE 430 - Feedback Control Systems

Oct 25, 2026

SECTION 2 explains how block diagrams represent feedback control systems. It defines blocks as subsystems modeled with transfer functions and signals as inputs/outputs with arrow directions. Summing junctions are described for algebraic summation, including subtraction using a negative sign. It then derives single-block equivalents for interconnections in cascade, parallel, and feedback, giving the corresponding relationships and closed-loop transfer function forms, including negative-feedback and notes on positive-feedback loop gain.

LanguageEnglish

CategoryTechnology

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Pages63

Reading Time2 h 39 min

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2 Block Diagrams

Blocks and signals in block diagrams

Input/output relationship for a single block

Interconnection forms: cascade, parallel, feedback

Cascade equivalent transfer function

Parallel equivalent transfer function

Feedback form and closed-loop transfer function

Negative vs positive feedback and loop gain

SECTION 2: BLOCK DIAGRAMS & SIGNAL FLOW GRAPHS ESE 430 – Feedback Control Systems 2 Block Diagrams K. Webb ESE 430 3 􀂅 In the introductory section we saw examples of block diagrams to represent systems, e.g.: 􀂅 Block diagrams consist of 􀂆 Blocks – these represent subsystems – typically modeled by, and labeled with, a transfer function 􀂆 Signals – inputs and outputs of blocks – signal direction indicated by arrows – could be voltage, velocity, force, etc . 􀂆 Summing junctions – points were signals are algebraically summed – subtraction indicated by a negative sign near where the signal joins the summing junction K. Webb ESE 430 4 􀂅 The basic input/output relationship for a single block is: 􀜻 􀝏 = 􀜷 􀝏 ⋅ 􀜩 􀝏 􀂅 Block diagram blocks can be connected in three basic forms: 􀂆 Cascade 􀂆 Parallel 􀂆 Feedback 􀂅 We’ll next look at each of these forms and derive a singleblock equivalent for each K. Webb ESE 430 5 􀂅 Blocks connected in cascade: 􀜺1 􀝏 = 􀜷 􀝏 ⋅ 􀜩 1 􀝏 , 􀜺2 􀝏 = 􀜺1 􀝏 ⋅ 􀜩 2 􀝏 􀜻 􀝏 = 􀜺2 􀝏 ⋅ 􀜩 3 􀝏 = 􀜺1 􀝏 ⋅ 􀜩 2 􀝏 ⋅ 􀜩 3 􀝏 􀜻 = 􀜷 ⋅ 􀜩 1 ⋅ 􀜩 2 ⋅ 􀜩 3 = 􀜷 ⋅ 􀜩􀯘􀯤 􀜩􀯘􀯤 = 􀜩 1 ⋅ 􀜩 2 ⋅ 􀜩 3 􀂅 The equivalent transfer function of cascaded blocks is the product of the individual transfer functions K. Webb ESE 430 6 􀂅 Blocks connected in parallel: 􀜺1 􀝏 = 􀜷 􀝏 ⋅ 􀜩 1 􀝏 􀜺2 􀝏 = 􀜷 􀝏 ⋅ 􀜩 2 􀝏 􀜺3 􀝏 = 􀜷 􀝏 ⋅ 􀜩 3 􀝏 􀜻 􀝏 = 􀜺1 􀝏 ± 􀜺2 􀝏 ± 􀜺3 􀝏 􀜻 􀝏 = 􀜷 􀝏 ⋅ 􀜩 1 􀝏 ± 􀜷 􀝏 ⋅ 􀜩 2 􀝏 ± 􀜷 􀝏 ⋅ 􀜩 3 􀝏 􀜻 = 􀜷 􀜩 1 ± 􀜩 2 ± 􀜩 3 = 􀜷 ⋅ 􀜩􀯘􀯤 􀜩􀯘􀯤 = 􀜩 1 ± 􀜩 2 ± 􀜩 3 􀂅 The equivalent transfer function is the sum of the individual transfer functions: K. Webb ESE 430 7 􀂅 Of obvious interest to us, is the feedback form: 􀜻 􀝏 = 􀜧 􀝏 􀜩 􀝏 􀜻 􀝏 = 􀜴 􀝏 − 􀜺 􀝏 􀜩 􀝏 􀜻 􀝏 = 􀜴 􀝏 − 􀜻 􀝏 􀜪 􀝏 􀜩 􀝏 􀜻 􀝏 1 + 􀜩 􀝏 􀜪 􀝏 = 􀜴 􀝏 􀜩 􀝏 􀜻 = 􀜴 ⋅ 􀜩 1 + 􀜩 􀝏 􀜪 􀝏 􀂅 The closed-loop transfer function, 􀜶 􀝏 , is 􀜻 􀝏 􀜩 􀝏 􀜶 􀝏 = ~~ ~~ = ~~ ~~ 􀜴 􀝏 1 + 􀜩 􀝏 􀜪 􀝏 K. Webb ESE 430 8 􀜶 􀝏 = 􀂅 Note that this is negative feedback, for positive feedback: 􀜶 = 􀜩 1 − 􀜩 􀝏 􀜪 􀝏 􀂅 The 􀜩 􀝏 􀜪 􀝏 factor in the denominator is the loop gain or open-loop transfer function 􀂅 The gain from input to output with the feedback path broken is the forward path gain – here, 􀜩 􀝏 􀂅 In general: 􀜶 = fo~~ ~~rw1rldopoapth~~ ~~gagainin K. Webb ESE 430 9 􀂅 Calculate the closed-loop transfer function 􀂅 􀜦 􀝏 and 􀜩 􀝏 are in cascade 􀂅 􀜪1 􀝏 is in cascade with the feedback system consisting of 􀜦 􀝏 ,􀜩 􀝏 , and 􀜪2 􀝏 􀜶 = 􀜪1 ⋅ 􀜦 􀜩 1 + 􀜦 􀝏 􀜩 􀝏 􀜪2 􀝏 􀜪1 􀝏 􀜦 􀝏 􀜩 􀝏 􀜶 􀝏 = ~~ ~~ 1 + 􀜦 􀝏 􀜩 􀝏 􀜪2 􀝏 K. Webb ESE 430 10 􀂅 We’re often interested in unity-feedback systems 􀂅 Feedback path gain is unity 􀂆 Can always reconfigure a system to unity-feedback form 􀂅 Closed-loop transfer function is: K. Webb ESE 430

FAQs

What do blocks, signals, and summing junctions represent in a block diagram?

Blocks represent subsystems and are typically labeled with transfer functions. Signals represent inputs and outputs, with direction indicated by arrows. Summing junctions algebraically combine signals, with subtraction shown using a negative sign.

How is the equivalent transfer function for cascade-connected blocks obtained?

For cascade connections, the equivalent transfer function equals the product of the individual transfer functions of the blocks.

What is the closed-loop transfer function for a feedback system and how do negative feedback and loop gain relate?

For the feedback form, the closed-loop transfer function is the forward-path gain divided by 1 plus the product of forward path gain and feedback-path gain. The denominator’s feedback factor corresponds to the loop gain (open-loop transfer function). Negative feedback uses the plus sign in the denominator, while positive feedback uses the corresponding opposite sign as shown in the notes.

Oct 25, 2026

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