Modeling of Dynamical System

Transfer Functions, Block Diagram

and Signal Flow Graph (Week 2 & 3)Transfer FunctionA general n-th order LTIV differential equation (DE),

(2.50)

where c(t) is the output, r(t) is the input and as , bs are the coefficients of the DE that represent the system. Taking Laplace,

(2.51)

If we assume all initial condition are zero,

(2.52)

The transfer function of the system is

Notice that the system output could be obtained using

(2.54)

The transfer function can be represented as a following block diagram.

The roots of numerator are called zeros and roots of denominator are called poles.

Block DiagramsBasic components of a block diagram for a LTIV system

Cascade or series subsystems,

Parallel Subsystems,

Feedback Form

a. Feedback control system;b. simplified model;c. equivalent transfer functionMoving blocks to create familiar forms,

Example 1

Reduce the following block diagram to form a single transfer function.

Solution,

Example 2

Reduce the following block diagram to form a single transfer function.

Solution,

Signal Flow GraphsSFG may be viewed as a simplified form of block diagram. SFG consists of arrows (represent systems) and nodes (represent signals).

Signal-flow graph components:a. system (arrows);b. signal (nodes);c. interconnection of systems and signals

Converting common block diagrams to SFG

Converting a block diagram to SFG

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

Mason Gain Formula

The transfer function of a given system represented by a SFG is:

where

k = no. of paths

= the kth forward-path gain

= 1 - loop gains + non-touching loop gains 2 at a time -

non-touching loop gains 3 at a time + non-touching

loop gains 4 at a time -

=- loop gains terms in that touch the kth forward path. In other words, is formed by eliminating from those loop gains that touched the kth forward path. Example 1

Example 2Use Masons Gain formula to obtain the transfer function of the system represented by the following SFG.

Moving block G1(s) to the right of summing junction, then apply feedback formula

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