Ramp Function
The ramp function r(t) = t·u(t) is a signal that starts at zero and increases linearly with time for t ≥ 0. Its Laplace transform is L{t·u(t)} = 1/s². The ramp is the integral of the unit step function: r(t) = ∫₀ᵗ u(τ)dτ. Conversely, the derivative of the ramp is the step function. In control systems, the ramp input tests a system's ability to track a constantly increasing reference — the steady-state error to a ramp input equals 1/Kᵥ, where Kᵥ is the velocity error constant.
Definition and Properties of the Ramp Function
The ramp function is defined as r(t) = t for t ≥ 0 and r(t) = 0 for t < 0, or equivalently r(t) = t·u(t). It belongs to the family of test signals used in control theory: impulse (δ), step (u), ramp (tu), and parabola (t²u/2). Each is the integral of the previous one. The ramp represents a constant-velocity input — like a motor that must track a steadily rotating reference, a satellite dish tracking a moving target, or an elevator moving at constant speed. Its Laplace transform L{tu(t)} = 1/s² follows directly from the property L{tf(t)} = −F'(s) applied to the unit step.
Key Formulas
Ramp Response of First and Second-Order Systems
For a first-order system H(s) = 1/(τs+1), the ramp response is Y(s) = 1/[s²(τs+1)]. Inverse transforming gives y(t) = t − τ + τe^{−t/τ}. After transients die out, y(t) ≈ t − τ, meaning the output tracks the ramp but lags behind by a constant τ seconds (or equivalently, τ units of amplitude). This steady-state error is the velocity lag. For a second-order system, the ramp response is more complex and the steady-state error depends on the system type number. Type-0 systems have infinite ramp error (they can't track ramps at all). Type-1 systems have finite error. Type-2 or higher systems track ramps with zero steady-state error.
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Open CalculatorVelocity Error Constant and System Type
The velocity error constant Kᵥ = lim(s→0) s·G(s) determines the steady-state error to a ramp input: e_ss = 1/Kᵥ. A system with one free integrator (Type 1) has finite Kᵥ and finite ramp error. A system with two free integrators (Type 2) has Kᵥ = ∞ and zero ramp error. This is why integral controllers are added to feedback systems — each integrator increases the system type by one, reducing steady-state error to polynomial inputs. However, adding integrators also reduces stability margins, creating the fundamental design tradeoff between steady-state accuracy and dynamic stability.
Ramp Function in Signal Processing
Beyond control theory, the ramp function appears in signal processing as a linear amplitude modulation. A ramp multiplied by a sinusoid creates a signal whose amplitude grows linearly — like a gradually increasing alarm tone. The ramp is also the integral of the rectangular window function, making it relevant to filter design. In digital signal processing, discrete ramp sequences r[n] = n·u[n] are used in testing digital filters and analyzing system behavior. The Z-transform of the discrete ramp is Z{n·u[n]} = z/(z−1)², the discrete counterpart to the Laplace transform 1/s².
Scaled and Delayed Ramps in Piecewise Functions
Many practical signals are constructed from scaled and shifted ramp functions. A triangular pulse can be written as r(t) − 2r(t−T) + r(t−2T): a ramp starting at 0, reversed at T by subtracting twice the ramp, and cancelled at 2T. Trapezoidal signals, sawtooth waves, and piecewise linear approximations all decompose into sums of shifted ramps and steps. This decomposition is useful because each piece has a known Laplace transform, and linearity allows you to transform the entire signal by transforming each piece. The LAPLACE Calculator handles these piecewise functions through the time-shifting property.
Related Topics in step response analysis
Understanding ramp function connects to several related concepts: ramp signal. Each builds on the mathematical foundations covered in this guide.
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