Block Diagram Algebra and Transfer Functions of Systems - Chap.7 - Unity Feedback Systems

Oct 26, 2026

Block diagram algebra rules and transfer-function transformations are presented for linear feedback control systems. The chapter explains how to combine cascade and parallel blocks, eliminate forward and feedback loops, remove blocks from signal paths, and rearrange summing points to obtain equivalent block diagrams. It derives the unity feedback form, identifies the resulting characteristic equation from 1 ± G = 0, and develops superposition for multiple inputs under zero initial conditions. Finally, it outlines systematic reduction steps to convert complicated multi-loop diagrams into canonical form and isolates blocks using selected transformations.

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7.6 Unity Feedback Systems

7.7 Superposition of Multiple Inputs

7.8 Reduction of Complicated Block Diagrams

CHAP.7]BLOCK DIAGRAM ALGEBRA AND TRANSFER FUNCTIONS OF SYSTEMS | Transformation | | Equation | Block Diagram | Equivalent Block
Diagram | | --- | --- | --- | --- | --- | | 1 | Combining Blocks
in Cascade | Y=(P₁P₂)X | | | | 2 | Combining Blocks
in Parallel;or
Eliminating a
Forward Loop | Y=P₁X±P₂X | | | | 3 | Removing a Block
from a Forward
Path | Y =P₁X±P₂X | | | | 4 | Eliminating a
Feedback Loop | Y=P₁(X=P₂Y) | | | | 5 | Removing a Block
from a Feedback
Loop | Y=P₁(X=P₂Y) | | | | 6a | Rearranging
Summing Points | Z =W±X±Y | | | | 6b | Rearranging
Summing Points | Z=W±X±Y | | | | 7 | Moving a Summing
Point Ahead of a
Block | Z=PX±Y | | | | 8 | Moving a Summing
Point Beyond a
Block | Z =P[X±Y] | | | [CHAP.7BLOCK DIAGRAM ALGEBRA AND TRANSFER FUNCTIONS OF SYSTEMS | Transformation | | Equation | Block Diagram | Equivalent Block
Diagram | | --- | --- | --- | --- | --- | | 9 | Moving a Takeoff
Point Ahead of a
Block | Y=PX | | | | 10 | Moving a Takeoff
Point Beyond a
Block | Y=PX | | | | 11 | Moving a Takeoff
Point Ahead of a
Summing Point | Z =X±Y | | | | 12 | Moving a Takeoff
Point Beyond a
Summing Point | Z =X±Y | | | Fig.7-6 Continued # 7.6 UNITY FEEDBACK SYSTEMS Definition 7.7:A unity feedback system is one in which the primary feedback b is identically equalto the controlled output c. EXAMPLE 7.6.H=1 for a linear,unity feedback system(Fig.7-7). Fig.7-7 Any feedback system with only linear time-invariant elements can be put into the form of a unityfeedback system by using Transformation 5. EXAMPLE 7.7. 三 Fig.7-8 CHAP.7]BLOCK DIAGRAM ALGEBRA AND TRANSFER FUNCTIONS OF SYSTEMS The characteristic equation for the unity feedback system,determined from 1±G=0,is (7.7) DG±NG=0 where DG is the denominator and Ng the numerator of G. # 7.7 SUPERPOSITION OF MULTIPLE INPUTS Sometimes it is necessary to evaluate system performance when several inputs are simultaneouslyapplied at different points of the system. When multiple inputs are present in a linear system,each is treated independently of the others.The output due to all stimuli acting together is found in the following manner.We assume zero initialconditions,as we seek the system response only to inputs. Step 1:Set all inputs except one equal to zero. Step 2:Transform the block diagram to canonical form,using the transformations of Section 7.5. Step 3:Calculate the response due to the chosen input acting alone. Step 4:Repeat Steps 1 to 3 for each of the remaining inputs. Step 5:Algebraically add all of the responses(outputs)determined in Steps 1 to 4.This sum is thetotal output of the system with all inputs acting simultaneously. We reemphasize here that the above superposition process is dependent on the system being linear. EXAMPLE 7.8.We determine the output C due to inputs U and R for Fig.7-9. Fig.7-9 Step 1: Put U=0. Step 2: The system reduces to Step 3: By Equation(7.3),the output CR due to input R is CR=[G₁G₂/(1+G₁G₂)]R. Step 4a: Put R=0. Step 4b: Put -1 into a block,representing the negative feedback effect: Rearrange the block diagram: Let the -1 block be absorbed into the summing point: Step 4c: By Equation(7.3),the output Cu due to input U is Cu=[G₂/(1+G₁G₂)]U. # 7.8 REDUCTION OF COMPLICATED BLOCK DIAGRAMS The block diagram of a practical feedback control system is often quite complicated.It may includeseveral feedback or feedforward loops,and multiple inputs.By means of systematic block diagramreduction,every multiple loop linear feedback system may be reduced to canonical form.Thetechniques developed in the preceding paragraphs provide the necessary tools. The following general steps may be used as a basic approach in the reduction of complicated

FAQs

What is a unity feedback system, and how is it defined?

A unity feedback system is one in which the primary feedback signal equals the controlled output identically. The chapter states b is identically equal to c in this form.

How can the characteristic equation for a unity feedback system be determined?

For the unity feedback system, the characteristic equation is determined from 1 ± G = 0. It is written as DG ± NG = 0, where DG is the denominator and NG the numerator of G.

How is the output computed when multiple inputs act simultaneously on a linear system?

Use the superposition process: set all inputs except one to zero, reduce the diagram to canonical form, compute the response for each input alone, then algebraically add all partial outputs. The method requires the system to be linear and typically assumes zero initial conditions.

Oct 26, 2026

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