27595005-Chapter-5-Reduction-of-Multiple-Subsystems-Block

Chapter 5 Reduction of Multiple Subsystems Block Diagrams  –  Transformation and Reduction Techniques • Cascade form • Parallel form • …. • feedback

Signal-Flow graphs Signal-Flow Graphs of State Equations  Alternative Representations in State Space? Similarity Transformations?

Components of a block diagram for a linear, time-invariant system

Parallel Subsystems

equivalent transfer function

Transformation Rules

Loading in cascaded systems

Feedback Control System simplified model;

 E ( s ) = R ( s ) ∓ C ( s ) H ( s )

equivalent transfer function

Open Loop Transfer Function = Loop Gain = G(s)H(s)

 E ( s ) =

C ( s ) G(s)

Example

More examples ?

Block diagram Reduction Example (1) **

Block diagram Reduction Example (2)

Example:

Chapter 5 Reduction of Multiple Subsystems Block Diagrams Signal-Flow graphs  –  components  –  Mason’s Rule

Analysis and Design of Feedback Systems

Signal-flow graph components

system

signal

Definitions 1) Branch: unidirectional segment 2) Node 3) Path 4) Loop

interconnection of systems and signals

Example of Building signal-flow graphs 1) cascaded system nodes

2) parallel system nodes

3) feedback system nodes

Example

Signal-flow graph development: a. signal nodes; b. signal-flow graph; c. simplified signalflow graph

Chapter 5 Reduction of Multiple Subsystems Block Diagrams Analysis and Design of Feedback Systems Signal-Flow graphs – components  –  Mason’s Rule

Motivation Example

A 2-input and 2-output system

Mason’s Rule Defintions Loop gain: The product of branch gains found by traversing a path that starts at a node and ends at the same node, following the direction of the signal flow, without passing through any other  node more than once. Forwards-path gain: The product of gains found by traversing a path from the input node to the output node of the signals-flow graph in the direction of signal flow. Nontouching loops: loops that do not have any nodes in common. Nontouching-loop gain: the product of loop gains from nontouching loops taken two, three, four, or more at a time. Mason’s Rule: The transfer function, C(s) / R(s), of a system represented by a signal-flow graph is: C ( s ) ∑k  k  k  where G ( s) = =  R s ( ) ∆ k = number of forward paths Tk  = the kth forward-path gain ∆ = 1- Σ loop gains + Σ nontouching-loop gains taken two at a time –  Σ nontouching-loop gains taken three at a time, + Σ nontouching-loop gains taken four at a time …. ∆k  = ∆- Σ loop gain terms in ∆ that touch the kth forward path. In other words, ∆k is formed by eliminating from ∆ those loop gains that touch the kth forward path T  ∆

Mason’s Rule: The transfer  function, C(s) / R(s), of a system represented by a signalflow graph is:

G2 ( s) H 1 ( s)

Loop gain:

G4 ( s) H 2 ( s) G4 ( s)G5 ( s) H 3 ( s) G4 ( s)G6 ( s) H 3 ( s)

G1 ( s )G2 ( s )G3 ( s )G4 ( s )G5 ( s )G7 ( s ) Forwards-path gain:

G1 ( s )G2 ( s )G3 ( s )G4 ( s )G6 ( s)G7 ( s )

Nontouching loops: e.g. loop G2(s)H1(s) dose not touch loops G4(s)H2(s), G4(s)G5(s)H3(s), and G4(s)G6(s)H3(s).

Nontouching-loop gain: the product of loop gains from nontouching loops taken two, three, four, or more at a time.

[G2 ( s ) H 1 ( s )][G4 ( s ) H 2 ( s )] [G2 ( s ) H 1 ( s )][G4 ( s )G5 ( s ) H 3 ( s )] [G ( ) H  ( )][G ( )G ( ) H  ( )]

Mason’s Rule illustration

More examples