forcedresponse

Forced response is a concept in linear systems theory describing the portion of a system's output that is produced by external excitation. It is contrasted with the natural response, which arises from the system's initial conditions and its internal dynamics. For a linear differential equation L[y] = f(t), the total response y(t) can be written as y(t) = y_n(t) + y_p(t), where y_n solves L[y_n] = 0 and y_p is a particular solution to L[y] = f(t). The term y_p, or the forced response, depends on the input f(t) but not on the initial state, while y_n depends on the initial values.

In the time domain, common inputs such as step, impulse, or sinusoidal forcing produce corresponding forced responses that describe the steady or transient behavior due to the input. In electrical circuits, for example, an RC or RLC circuit driven by a voltage source has a forced response corresponding to the charging or sinusoidal steady-state, while the natural response decays according to the circuit's dissipative elements.

In Laplace transform terms, the forced response is associated with the particular solution obtained from F(s) multiplied by the transfer function H(s), representing the system's reaction to the input with zero initial conditions. Separating forced and natural responses helps engineers analyze transient performance and steady-state behavior, design controllers, and predict system behavior under different inputs.

Related concepts include natural response, transient, steady-state, impulse response, transfer function, and system stability.