# OutputResponse

OutputResponse[sys,u[t],{t,tmin,tmax}]

gives the numeric output response of systems model sys to the input u[t] for tminttmax.

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 t0. »

# Examples

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## Basic 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:

Wolfram Research (2010), OutputResponse, Wolfram Language function, https://reference.wolfram.com/language/ref/OutputResponse.html (updated 2014).

#### 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_2024_outputresponse, author="Wolfram Research", title="{OutputResponse}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/OutputResponse.html}", note=[Accessed: 17-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_outputresponse, organization={Wolfram Research}, title={OutputResponse}, year={2014}, url={https://reference.wolfram.com/language/ref/OutputResponse.html}, note=[Accessed: 17-June-2024 ]}