2 lec Eng
.pdfMathematical 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:
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Outputs |
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u1(t) |
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y1(t) |
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u2(t) |
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System |
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y2(t) |
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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 |
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dt |
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or |
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u(s) |
y(s) |
y s |
1 |
u s |
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Graphical representation of State space
D
U |
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Y |
X |
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X |
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C |
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s |
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A |
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A - System matrix |
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C - The observation matrix |
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B - Input transmission |
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D - Input to Output transmission |
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matrix |
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matrix |
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The relationship between the Transfer function and the Block diagram description