lecture5 - 5. Fundamentals of Control System 5.1...

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  1 Laplace Transform Uses of Laplace Transform: Helps solve differential equations Defines transfer functions Definition 5. Fundamentals of Control System 5.1 Preliminary Mathematics - = 0 st dt ) t ( f e ) s ( F
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  2 Laplace Transform (cont.) Formulae F(t) F(s) F(t) F(s) δ 1 e -at f(t) F(s+a) 1 1/s sin ϖ t ϖ /(s 2 + ϖ 2 ) t 1/s 2 cos ϖ t s/(s 2 + ϖ 2 ) t 2 2/s 3 e -at cos ϖ t ( s+a 29 /[(s+a) + ϖ ] e -at 1/(s+a) e -at sin ϖ t ϖ /[(s+a) + ϖ ] L{e -at f(t)} = F(s+a); where L{f(t)} = F(s);
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  3 Laplace Transform (cont.) Denote L{.} as the Laplace operator. L{       } = sF(s) –f o L{        } = s 2  F(s) – sf o  – Where   f o  = f(t)     @ t = 0                  =           @ t = 0 ) ( t f dt d ) t ( f dt d 2 2 ' o f ) t ( f dt d ' o f
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  4 Examples of Laplace Transform   Find Laplace Transform of e -4t . Solution: Since   L{e -at } = therefore   L{e -4t } = a s 1 + 4 s 1 +
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  5 Another Example Find Laplace Transform of e -2t cos3t. Solution: Since   L{e -at f(t)} = F(s+a) and      L{cos ϖ t} =  therefore L{cos 3 t} =   so L{e -2t cos3t} = 2 2 s s ϖ + ( 29 13 s 4 s 2 s 9 2 s ) 2 s ( 2 2 + + + = + + + 9 s s 2 +
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lecture5 - 5. Fundamentals of Control System 5.1...

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