OutputResponse
OutputResponse [sys,u[t],{t,tmin,tmax}]
gives the numeric output response of systems model sys to the input u[t] for tmin≤t≤tmax.
OutputResponse [sys,{u[0],u[1],…}]
gives the output response of the discrete-time system sys to the input sequence u[i].
OutputResponse [sys,u[t],t]
gives the symbolic output response of system sys to the input u[t] as a function of time t.
OutputResponse [sys,{u1[t],…,um[t]},…]
gives the output response for multiple inputs ui[t].
Details
- OutputResponse is also known as impulse response, step response, and ramp response.
- OutputResponse solves the underlying differential or difference equations for the given input.
- The systems model sys can be a TransferFunctionModel , a StateSpaceModel , a continuous-time AffineStateSpaceModel , or a continuous-time NonlinearStateSpaceModel .
- A linear TransferFunctionModel or StateSpaceModel sys can also be a descriptor and delay system.
- The initial values for the differential and difference equations are taken to be zero for a TransferFunctionModel . For the state-space models, they are taken to be the state operating values of sys unless specified.
- OutputResponse [{sys,{x10,x20,…,xn0}},…] can be used to specify the initial state for a state-space model sys.
- For descriptor state-space systems, the initial states need to be consistent.
- For delay state-space systems, the initial states include history and can be given as xi0[t] for t≤0. »
Examples
open allclose allBasic Examples (4)
The step response of a second-order system:
The output response of a transfer-function model to a sinusoidal input:
Visualize the response:
The response of a state-space model from nonzero initial conditions:
The response of a discrete-time system to a sampled sinusoid:
Scope (47)
Basic Uses (19)
Find the initial value response for a scalar continuous-time state-space model:
Find the zero initial condition response for a symbolic input:
Find the numeric response of a fourth-order system to a sinusoidal input:
Find the numeric step response for a continuous-time transfer-function model:
Get the symbolic solution:
Find the numeric step response for a discrete-time state-space model:
Find the step response for a discrete-time transfer function with a numeric simulation:
Get the symbolic solution:
Find the symbolic response for an affine state-space model:
Find the symbolic response for an affine state-space model with a nonzero equilibrium:
Find the numeric response for an affine state-space model with multiple equilibria:
Find the symbolic response for a nonlinear state-space model:
Find the numeric response for a nonlinear state-space model:
Find the numeric response of a two-output fourth-order system to a triangle wave:
Find the symbolic response of a three-output transfer-function model:
Find the response of a state-space model with output delays:
A system with two inputs:
When a multiple-input system receives a single input, it is applied separately to each input:
A numeric response for a multiple-input, multiple-output transfer-function model:
A second-order system settling from nonzero initial states:
A nonlinear state-space model with multiple equilibria:
The steady-state position depends on the initial condition:
An alternating input signal can cause the system to switch between equilibria:
Continuous-Time Systems (19)
The output response of a continuous-time system to a step input:
The response for various damping ratios:
The response to a unit step input:
The response of a descriptor StateSpaceModel :
The response when there is an algebraic equation:
The response of a state-space model:
The initial values of the states are assumed to be zero:
The response of a two-output system to a delayed step input:
The output response for nonzero initial conditions:
Plot the response:
The output response for a system with two inputs:
A second-order system step response goes from oscillations at to overdamped at :
If there are fewer input signals than system inputs, the remaining signals are set to zero:
A multi-input system:
When a scalar input signal is given, it is applied to each input in turn:
If the time interval is specified, the result is computed numerically:
The symbolic result:
The results are equivalent:
The response of a generic continuous-time system:
The response to a sine wave:
Step response of a time-delay transfer-function model:
Step response of a time-delay state-space model:
A StateSpaceModel with a singular descriptor matrix:
Plot the response:
The output response of an AffineStateSpaceModel to a UnitStep input:
Plot the response:
The response from nonzero initial conditions:
The output response of a NonlinearStateSpaceModel to a UnitStep input:
Plot the response:
Discrete-Time Systems (9)
The output response of a single-input system to a sampled sinusoid:
Plot of the sampled output with a zero-order hold:
The response for a generic discrete-time system:
The response to a unit step sequence:
The response for a symbolic descriptor system:
The response of a two-input system:
The response of a first-order discrete-time system:
The response to a unit step sequence:
The output response of a discrete-time system to a time-dependent input:
The response for τ=0.1:
Ramp response of a time-delay system:
Generalizations & Extensions (3)
If the initial time is not specified, it is assumed to be zero:
When a system has state delays, the initial states can include history:
For discrete-time systems with delays, the initial states can be given as a sequence:
Applications (3)
Determine the steady-state output value of a stable first-order system in response to a unit step input:
The time constant:
Visualize it:
Visualize the response of an unstable system and its response after feedback stabilization:
A compensator for the antenna:
The closed-loop system:
Plot the response:
The zero-input response of a system:
Properties & Relations (5)
The natural response is determined by the poles of the system:
The poles:
The results of StateResponse and OutputResponse match for state output:
A discrete-time system responding to a continuous-time input:
For a smaller sampling period, more sample points are needed:
The impulse response of a system:
OutputResponse assumes that the input is zero for :
Thus the solution obtained using InverseLaplaceTransform is different for :
The initial states for a descriptor systems are chosen to be consistent for the inputs:
The second output equals the derivative of the input:
When inconsistent conditions are given, they are replaced:
Consistent initial states depend on the slow subsystem from KroneckerModelDecomposition :
For continuous-time systems, the initial conditions are given by :
Possible Issues (4)
A continuous-time system cannot be simulated with sampled inputs:
Discretize the system:
Computations with machine numbers can be unstable:
Rationalize the system:
Or compute the numeric response:
Symbolic output responses do not support time delays:
Try a numeric simulation:
For descriptor systems, solutions only exist when Det [λ e - a]≠0 for some λ:
Related Guides
History
Introduced in 2010 (8.0) | Updated in 2012 (9.0) ▪ 2014 (10.0)
Text
Wolfram Research (2010), OutputResponse, Wolfram Language function, https://reference.wolfram.com/language/ref/OutputResponse.html (updated 2014).
CMS
Wolfram Language. 2010. "OutputResponse." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/OutputResponse.html.
APA
Wolfram Language. (2010). OutputResponse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OutputResponse.html
BibTeX
@misc{reference.wolfram_2025_outputresponse, author="Wolfram Research", title="{OutputResponse}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/OutputResponse.html}", note=[Accessed: 14-April-2025 ]}
BibLaTeX
@online{reference.wolfram_2025_outputresponse, organization={Wolfram Research}, title={OutputResponse}, year={2014}, url={https://reference.wolfram.com/language/ref/OutputResponse.html}, note=[Accessed: 14-April-2025 ]}