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  2. Settling time - Wikipedia

    en.wikipedia.org/wiki/Settling_time

    The settling time for a second order, underdamped system responding to a step response can be approximated if the damping ratio by = ⁡ () A general form is T s = − ln ⁡ ( tolerance fraction × 1 − ζ 2 ) damping ratio × natural freq {\displaystyle T_{s}=-{\frac {\ln({\text{tolerance fraction}}\times {\sqrt {1-\zeta ^{2}}})}{{\text ...

  3. Time constant - Wikipedia

    en.wikipedia.org/wiki/Time_constant

    First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.

  4. Step response - Wikipedia

    en.wikipedia.org/wiki/Step_response

    A typical step response for a second order system, illustrating overshoot, followed by ringing, all subsiding within a settling time. The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step ...

  5. Exponential response formula - Wikipedia

    en.wikipedia.org/wiki/Exponential_response_formula

    Physically, time invariance means system’s response does not depend on what time the input begins. For example, if a spring-mass system is at equilibrium, it will respond to a given force in the same way, no matter when the force was applied. When the time-invariant system is also linear, it is called a linear time-invariant system (LTI system).

  6. Damping - Wikipedia

    en.wikipedia.org/wiki/Damping

    The effect of varying damping ratio on a second-order system. The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), [7] that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator ...

  7. Crank–Nicolson method - Wikipedia

    en.wikipedia.org/wiki/Crank–Nicolson_method

    The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.

  8. Lax–Wendroff method - Wikipedia

    en.wikipedia.org/wiki/Lax–Wendroff_method

    What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(x, t)) at half time steps, t n + 1/2 and half grid points, x i + 1/2. In the second step values at t n + 1 are calculated using the data for t n and t n + 1/2.

  9. Double integrator - Wikipedia

    en.wikipedia.org/wiki/Double_integrator

    Feedback system with a PD controller and a double integrator plant In systems and control theory , the double integrator is a canonical example of a second-order control system. [ 1 ] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input u {\displaystyle {\textbf {u}}} .

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