The Laplace Transform

The ODE $dy/dt + 2y = \sin(t)$ looks hard. In the Laplace domain: $sY - y_0 + 2Y = 1/(s^2+1)$ - that is first-degree algebra. The Laplace transform literally converts differential equations into high-school arithmetic. PID controllers in every industrial robot, Tesla Autopilot, and quadrotor drone are designed in the $s$-domain. The transfer function $H(s)$ shows how the system responds to each input frequency. Control theory is Laplace theory.

• **Control systems:** airplane autopilots, PID temperature controllers, anti-lock brakes - all designed via transfer functions H(s) in the s-domain
• **Circuit analysis:** RLC filters, amplifiers, power supplies - the frequency response H(iω) is obtained by substituting s = iω into the transfer function
• **DSP and digital filters:** Butterworth, Chebyshev, and Bessel filters are designed in the s-domain; the z-transform is the discrete-time analog for scipy.signal
• **Neural ODE (Chen 2018):** stability analysis via Laplace poles applied to continuous dynamical systems as an alternative to discrete-layer networks

Предварительные знания

• Power Series Methods

Definition and Transform Table

The ODE $y'' + 2y' + y = \sin(t)$ looks intimidating. In the Laplace domain: $(s^2 + 2s + 1)Y(s) = \frac{1}{s^2+1} + \text{ICs}$. That is polynomial algebra - high-school arithmetic. This single observation made the Laplace transform the default language of control engineering from the 1950s through today.

**Definition:** $\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st}\,dt$. A time-domain function $f(t)$ maps to a function of complex variable $F(s)$. The factor $e^{-st}$ damps any polynomial growth in $f$, making the integral converge when $\text{Re}(s)$ is large enough.

**Region of convergence:** the integral converges when $\text{Re}(s) > \sigma_0$, where $\sigma_0$ is the abscissa of absolute convergence. For $f(t) = e^{at}$ one needs $\text{Re}(s) > a$. For bounded functions $\sigma_0 \le 0$.

Derivative Rule and Solving IVPs

The property that makes Laplace indispensable for ODEs: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$. Second derivative: $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$. Initial conditions fold automatically into the equation - no separate substitution step is needed.

IVP algorithm: 1. apply $\mathcal{L}$ to both sides of the ODE 2. substitute initial conditions - they migrate into the numerator of $F(s)$ 3. solve the algebraic equation for $F(s)$ 4. recover $f(t) = \mathcal{L}^{-1}\{F(s)\}$ via table lookup or partial fractions.

**Transfer function** $H(s) = Y(s)/U(s)$ is the central object in control theory. For $y'' + ay' + by = u(t)$ one gets $H(s) = 1/(s^2 + as + b)$. The poles of $H(s)$ are the eigenvalues of the system. Stability: all poles in the left half-plane $\text{Re}(s) < 0$. PID controllers in Tesla Autopilot, drones, and industrial robots are all designed by placing poles of $H(s)$ in the left half-plane.

The Laplace transform assumes $f(t) = 0$ for $t < 0$. If $f(t)$ is nonzero before $t = 0$ or has a jump at $t = 0$, use the one-sided transform carefully and account for jump contributions.

Partial Fractions and Inverse Transform

After finding $F(s)$ algebraically, the task is to recover $f(t) = \mathcal{L}^{-1}\{F(s)\}$. The standard route: decompose $F(s)$ into **partial fractions**, then read each term from the table. The residue method turns the inversion into arithmetic.

**`sympy.apart(F, s)`** does partial fraction decomposition. **`sympy.inverse_laplace_transform(F, s, t)`** inverts directly. For numerical simulation use `scipy.signal.lsim` with `scipy.signal.TransferFunction`.

Applications: Circuits and Control Systems

1910. Oliver Heaviside - a self-taught engineer with no university degree - develops operational calculus for electrical circuits. Mathematicians mock it. Engineers adopt it immediately. Today it is the standard in every control course worldwide. Kirchhoff's law for an RLC circuit: $Li'' + Ri' + i/C = u'(t)$. After Laplace: $(Ls^2 + Rs + 1/C)I(s) = sU(s) - \ldots$ Transfer function: $H(s) = I(s)/U(s) = s/(Ls^2 + Rs + 1/C)$.

**Convolution theorem:** $\mathcal{L}\{(f*g)(t)\} = F(s)\cdot G(s)$. Convolution in the time domain equals multiplication in the $s$-domain. This is why the transfer function acts as a filter: output = input multiplied by $H(s)$, which in the time domain is convolution with the impulse response $h(t) = \mathcal{L}^{-1}\{H(s)\}$. Butterworth and Chebyshev digital filters are designed in the $s$-domain, then converted to discrete form via the $z$-transform.

**Final value theorem:** $\lim_{t\to\infty} f(t) = \lim_{s\to 0} s\cdot F(s)$. This gives the steady-state output without inverting the transform. For $H(s) = 1/(s+2)$ with a unit step: final value $= \lim_{s\to 0} s \cdot \frac{1}{s(s+2)} = \frac{1}{2}$.

Key Ideas

• **L{f}(s) = ∫₀^∞ f(t)e^{-st}dt** - integral transform mapping the time domain to the s-domain
• **L{f'} = sF(s) - f(0)** - derivatives become multiplication by s; ODEs become algebraic equations
• **Inverse transform** via partial fraction decomposition plus table lookup (or sympy.inverse_laplace_transform)
• **Transfer function H(s) = Y(s)/U(s)** encodes the complete dynamic behavior; poles determine stability

Related Topics

The Laplace transform bridges time-domain ODEs and frequency-domain analysis:

• Systems of ODEs — Transfer function of x' = Ax + Bu, y = Cx is H(s) = C(sI-A)⁻¹B
• Fourier Methods for ODEs — Fourier is Laplace restricted to s = iω: H(iω) is the frequency response

Вопросы для размышления

• The convolution theorem says multiplication in the s-domain equals convolution in time. How does this connect to the notion of a filter in signal processing?
• The poles of the transfer function equal the eigenvalues of the system matrix A. Can one verify this for y'' + ay' + by = u by writing it as a first-order system?
• Neural ODE uses Laplace stability analysis for continuous-depth networks. What does Re(s) < 0 for all poles mean in the context of training dynamics?

Связанные уроки

• de-03 — Systems of ODEs - special case; transfer function H(s) = C(sI-A)⁻¹B
• de-06 — Fourier transform is Laplace restricted to s = iω
• dyn-01 — Phase portraits and H(s) poles describe the same dynamics in two languages
• nm-06 — Finding poles of H(s) for large n requires numerical linear algebra
• calc-06-derivative-intro — The rule L{f'} = sF - f(0) builds directly on the derivative
• calc-13-techniques