2 lec Eng

Mathematical Models of Systems

Differential equations of linear systems

u(t)

Input signal

System

y(t)

Output signal

n – the order of a system

– constant coefficients

State space description

variable substitution

State space description

In case of multi-inputs and multi-outputs (MIMO systems) we have:

	Inputs			Outputs
u1(t)							y1(t)
u2(t)			System				y2(t)
	...			...

um(t) yr(t)

m – number of inputs, r – number of outputs

State space description

In matrix representation we have:

– State equation

– State vector

– Output equation

–Vector of input signals

–Vector of output signals

Block diagram description

Typical elements of block diagrams

link

node (junction)

gain (amplification factor)

u

y

an arrow indicates the direction of signal transmission

one signal is transmitted to the different elements

y t Ku t ,

where K – a constant coefficient

Block diagram description

Typical elements of block diagrams

comparison (sum) element

u y

z

u y

z

product element

u y z

y t u t z t

y t u t z t

y t u t z t

Block diagram description

Typical elements of block diagrams

integrator

u(t)	y(t)	dy	u t
		dt
	or
u(s)	y(s)	y s		1	u s
				s

Graphical representation of State space

D

U				Y
	X	1	X
				C
		s
		A
A - System matrix			C - The observation matrix
B - Input transmission			D - Input to Output transmission
matrix			matrix

The relationship between the Transfer function and the Block diagram description