Laplace transforms turn differential equations in control systems into algebraic equations in the s-domain. This cheat sheet covers the transform pairs and properties most often used for modeling, transfer functions, and system response. Students need these formulas to analyze stability, transient response, steady-state behavior, and input-output relationships efficiently.
Key Facts
- The unilateral Laplace transform is F(s) = integral from 0 to infinity of f(t)e^(-st) dt, which is the standard form used for causal control systems.
- The transform of a unit step is L{u(t)} = 1/s, and the transform of a ramp is L{t u(t)} = 1/s^2.
- The transform of an exponential is L{e^(at)} = 1/(s - a), valid when the real part of s is greater than a.
- The sinusoidal pairs are L{sin(bt)} = b/(s^2 + b^2) and L{cos(bt)} = s/(s^2 + b^2).
- Time differentiation gives L{df/dt} = sF(s) - f(0+), which is essential for including initial conditions.
- Time integration gives L{integral from 0 to t of f(tau) d tau} = F(s)/s.
- A transfer function is G(s) = Y(s)/U(s) with all initial conditions set to zero.
- The final value theorem states that f(infinity) = limit as s approaches 0 of sF(s), if all poles of sF(s) are in the open left half-plane.
Vocabulary
- Laplace transform
- A mathematical operation that converts a time-domain function f(t) into an s-domain function F(s).
- s-domain
- The complex-frequency domain where s = sigma + j omega is used to analyze system dynamics algebraically.
- Transfer function
- The ratio G(s) = Y(s)/U(s) that relates output to input for a linear time-invariant system with zero initial conditions.
- Pole
- A value of s that makes the denominator of a transfer function equal to zero and strongly affects stability and response speed.
- Zero
- A value of s that makes the numerator of a transfer function equal to zero and shapes the system response.
- Final value theorem
- A theorem used to find the steady-state value of a signal from limit as s approaches 0 of sF(s), when its stability conditions are satisfied.
Common Mistakes to Avoid
- Forgetting initial conditions in derivative transforms is wrong because L{df/dt} equals sF(s) - f(0+), not just sF(s), unless the initial value is zero.
- Using transfer functions with nonzero initial conditions is wrong because G(s) = Y(s)/U(s) is defined only when all initial conditions are set to zero.
- Applying the final value theorem without checking poles is wrong because the theorem fails if sF(s) has poles in the right half-plane or repeated poles on the imaginary axis.
- Confusing 1/s and 1/s^2 is wrong because 1/s represents a unit step, while 1/s^2 represents a unit ramp.
- Ignoring time shifts is wrong because L{f(t - a)u(t - a)} = e^(-as)F(s), so the delay factor must be included.
Practice Questions
- 1 Find the Laplace transform of f(t) = 4e^(-3t)u(t).
- 2 Find the inverse Laplace transform of F(s) = 10/(s^2 + 25).
- 3 For G(s) = 12/(s^2 + 5s + 6), identify the poles and determine whether the system is stable.
- 4 Explain why transfer functions are usually derived with zero initial conditions in control-system analysis.