## The Laplace Transform

Gerald M. Lemer

Laplace transformation is a technique used to relate time- and frequency-dependent linear systems. A linear system is a collection of electronic components (e.g., resistors, capacitors, inductors) or physical components (e.g., masses, springs, oscillators) arranged so that the system output is a linear function of system input The input and output of an electronic system are commonly voltages, whereas the input to an attitude control system is a sensed angular error and the output is a restoring torque. Most systems are linear only for a restricted range of input

Laplace transformation is widely used to solve problems in electrical engineering or control theory (e.g., attitude control) that may be reduced to linear differential equations with constant coefficients. The Laplace transform of a real function, /(/), defined for real t > 0 is e(/('))= *■(')- jyO)cxp(-sOdt (F-l)

where 0+ indicates that the lower limit of the integral is evaluated by taking the limit as /—>0 from above. The argument of the Laplace transform, F(s), is complex, s=o + iu where ;'=V- 1 . For most physical applications, t and u denote time and frequency, respectively, and o is related to the decay time. The inverse Laplace transform is

where the real constant C is chosen such that F(s) exists for all Re(i) > C, that is, to the right of any singularity.

Properties of the Laplace Transform and the Inverse Laplace Transform*. The

Laplace transform and its inverse are linear operators, thus

==aF(s) + bG(s) (F-3) £ "\aF(s)+bG (*))=a£~ '(/"(j))+bt~ l(G(s))

= af(t) + bg(t) (F-4) where a and b are complex constants.

*For further details, see DiStefano, et al., [1967]

The initial value theorem relates the initial value of f(t),f(0*), to the Laplace transform, f(0+)= YnnsF(s) (F-5)

and the final value theorem, which is widely used to determine the steady-state response of a system, relates the final value of /(/), /(oo), to the Laplace transform,*

The Laplace and inverse Laplace transformations may be scaled in either the time domain (time scaling) by

or the frequency domain (frequency scaling) by

The Laplace transform of the time-delayed function, /(/ — /„), is

where/(/ - /q)=0 for / < /q. The inverse Laplace transform of the frequency shifted function, f(i-io), is e-'(f(i-io)) = exp(io/)/(/) (F-10)

Laplace transforms of exponentially damped, modulated, and scaled functions are e(exp (~at)f(t))=F(s + a) (F-lla)

6 (sinutf(t)) = [ F(s- /to)- F(s + iu)]/2i (F-1 lb)

e (cos«//(/)) = [ F(s - /to) + F(i + /to)]/2 (F-1 lc)

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