Transfer Functions
A transfer function is the Laplace-domain ratio of output to input for a linear time-invariant system, capturing its dynamics as a rational function.
From Differential Equation to Ratio
For a linear time-invariant (LTI) system with zero initial conditions, the Laplace transform turns a differential equation into an algebraic one. The transfer function G(s) = Y(s)/U(s) is the ratio of the transformed output to the transformed input. It is a complete description of the system's input-output behavior.
Rational form
Transfer functions of lumped LTI systems are ratios of polynomials in the complex variable s: G(s) = N(s)/D(s). The roots of N(s) are the zeros and the roots of D(s) are the poles. The order of D(s) is the system order.
Worked example
A mass-spring-damper with mass m, damping c, and stiffness k, driven by force u and measured by position y, obeys m*y'' + c*y' + k*y = u. Transforming gives (m*s^2 + c*s + k)*Y = U, so G(s) = 1/(m*s^2 + c*s + k). Its two poles set the natural frequency and damping.
What the transfer function tells you
- Poles govern stability and the character of the transient (decay rate, oscillation).
- Zeros shape how strongly each mode appears in the output and can cause undershoot.
- The DC gain G(0) gives the steady-state output for a unit step input.
- The frequency response G(jw) is read directly by setting s = jw.
Assumptions and limits
Transfer functions require linearity and time invariance, and they discard initial conditions. Nonlinear or time-varying plants must be linearized about an operating point first, after which the transfer function is a local approximation. Multi-input multi-output systems generalize to a matrix of transfer functions, though state-space is usually cleaner there.
Despite these limits the transfer function is the workhorse of classical control: Bode plots, root locus, and Nyquist analysis all operate directly on G(s), making it the natural language for single-loop design.