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OutputResponse
✖
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 all

Basic Examples  (4)Summary of the most common use cases

The step response of a second-order system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-40yp3u
Out[1]=1

The output response of a transfer-function model to a sinusoidal input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-hnozhm
Out[1]=1

Visualize the response:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-i6ykrd
Out[2]=2

The response of a state-space model from nonzero initial conditions:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-v66wvk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-hgbhcb
Out[2]=2

The response of a discrete-time system to a sampled sinusoid:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-9vvbo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-bjdg59
Out[2]=2

Scope  (47)Survey of the scope of standard use cases

Basic Uses  (19)

Find the initial value response for a scalar continuous-time state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d19nyw
Out[1]=1

Find the zero initial condition response for a symbolic input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d15h6b
Out[1]=1

Find the numeric response of a fourth-order system to a sinusoidal input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-dujgi2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-ru83r
Out[2]=2

Find the numeric step response for a continuous-time transfer-function model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-kwxetd
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-bigsap
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-g3s5rq
Out[3]=3

Get the symbolic solution:

In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-gx1dl4
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0g7kxbwzgfy-oa4do
Out[5]=5

Find the numeric step response for a discrete-time state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-gzy1l
Out[1]=1

Find the step response for a discrete-time transfer function with a numeric simulation:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-gsj1z
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-cdxda4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-bt9eij
Out[3]=3

Get the symbolic solution:

In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-b04qok
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0g7kxbwzgfy-dlk53w
Out[5]=5

Find the symbolic response for an affine state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-km3fgi
Out[1]=1

Find the symbolic response for an affine state-space model with a nonzero equilibrium:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-bncpdt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-icy3s
Out[2]=2

Find the numeric response for an affine state-space model with multiple equilibria:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-mw0lr1
Out[1]=1

Find the symbolic response for a nonlinear state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-buug13
Out[1]=1

Find the numeric response for a nonlinear state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-j7go6n
Out[1]=1

Find the numeric response of a two-output fourth-order system to a triangle wave:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-f8ukoj
Out[1]=1

Find the symbolic response of a three-output transfer-function model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-nfpzu0
Out[1]=1

Find the response of a state-space model with output delays:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d6f76g
Out[1]=1

A system with two inputs:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-lsdvw2
Out[1]=1

When a multiple-input system receives a single input, it is applied separately to each input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-egga0x
Out[1]=1

A numeric response for a multiple-input, multiple-output transfer-function model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d3u57
Out[1]=1

A second-order system settling from nonzero initial states:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d39h86
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-2uppi
Out[2]=2

A nonlinear state-space model with multiple equilibria:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-nd243

The steady-state position depends on the initial condition:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-iucwc
Out[2]=2

An alternating input signal can cause the system to switch between equilibria:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-b9tr3k
Out[3]=3

Continuous-Time Systems  (19)

The output response of a continuous-time system to a step input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-n99pas
Out[2]=2

The response for various damping ratios:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-i3zuiw
Out[3]=3

The response to a unit step input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-ulo3
Out[1]=1

The response of a descriptor StateSpaceModel:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-tjrx4p
Out[1]=1

The response when there is an algebraic equation:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-fvws2s
Out[1]=1

The response of a state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-7h9utw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-iiv2ry
Out[2]=2

The initial values of the states are assumed to be zero:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-jhk0wh
Out[3]=3

The response of a two-output system to a delayed step input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-drydme
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-vzm17u
Out[2]=2

The output response for nonzero initial conditions:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-6vvnrr
Out[1]=1

Plot the response:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-28p6ux
Out[2]=2

The output response for a system with two inputs:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-qt6t23
Out[1]=1

A second-order system step response goes from oscillations at to overdamped at :

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-zycipu
Out[1]=1

If there are fewer input signals than system inputs, the remaining signals are set to zero:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-s4q3ry
Out[1]=1

A multi-input system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-nrn3sk

When a scalar input signal is given, it is applied to each input in turn:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-gsgt0t
Out[2]=2

If the time interval is specified, the result is computed numerically:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-s0qohv
Out[1]=1

The symbolic result:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-6slnxm
Out[2]=2

The results are equivalent:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-p92t8i
Out[3]=3

The response of a generic continuous-time system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-h8aoux
Out[1]=1

The response to a sine wave:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-2iy76j
Out[1]=1

Step response of a time-delay transfer-function model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-ei2620
Out[1]=1

Step response of a time-delay state-space model:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-fz4mvh
Out[1]=1

A StateSpaceModel with a singular descriptor matrix:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-qfg82
Out[1]=1

Plot the response:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-indaxy
Out[2]=2

The output response of an AffineStateSpaceModel to a UnitStep input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-4ooqbv
Out[1]=1

Plot the response:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-vpv6nm
Out[2]=2

The response from nonzero initial conditions:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-qqzq1k
Out[3]=3

The output response of a NonlinearStateSpaceModel to a UnitStep input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-1i33z2
Out[1]=1

Plot the response:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-j7031r
Out[2]=2

Discrete-Time Systems  (9)

The output response of a single-input system to a sampled sinusoid:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-f2xe2i
Out[1]=1

Plot of the sampled output with a zero-order hold:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-y3xko
Out[2]=2

The response for a generic discrete-time system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-n23rsa
Out[1]=1

The response to a unit step sequence:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-i5i745
Out[1]=1

The response for a symbolic descriptor system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-chq17
Out[1]=1

The response of a two-input system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d3h0kd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-e384o4
Out[2]=2

The response of a first-order discrete-time system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-8z324x
Out[1]=1

The response to a unit step sequence:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-dxaum5
Out[1]=1

The output response of a discrete-time system to a time-dependent input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-bg8uuu
Out[1]=1

The response for τ=0.1:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-b6tvrr
Out[2]=2

Ramp response of a time-delay system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-envn49
Out[1]=1

Generalizations & Extensions  (3)Generalized and extended use cases

If the initial time is not specified, it is assumed to be zero:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-x8dm9
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-fzwokp
Out[2]=2

When a system has state delays, the initial states can include history:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-bxz4vh
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-hp35e4
Out[2]=2

For discrete-time systems with delays, the initial states can be given as a sequence:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-d55no9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-cmntpm
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Determine the steady-state output value of a stable first-order system in response to a unit step input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-tl8d37
Out[1]=1

The time constant:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-m4jekb
Out[2]=2

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-mactfh
Out[3]=3

Visualize the response of an unstable system and its response after feedback stabilization:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-imid4o
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-l3u35x
Out[2]=2

A compensator for the antenna:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-dlvtdw

The closed-loop system:

In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-nry2d
Out[4]=4

Plot the response:

In[5]:=5
✖
https://wolfram.com/xid/0g7kxbwzgfy-bp9e8e
Out[5]=5

The zero-input response of a system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-7jxtx0
Out[1]=1

Properties & Relations  (5)Properties of the function, and connections to other functions

The natural response is determined by the poles of the system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-pr0h4t
Out[1]=1

The poles:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-dbicwb
Out[2]=2

The results of StateResponse and OutputResponse match for state output:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-p6ayjb
Out[1]=1

A discrete-time system responding to a continuous-time input:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-kpat8y
Out[1]=1

For a smaller sampling period, more sample points are needed:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-isqhi0
Out[2]=2

The impulse response of a system:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-ck9v9s
Out[1]=1

OutputResponse assumes that the input is zero for :

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-zxj2z
Out[2]=2

Thus the solution obtained using InverseLaplaceTransform is different for :

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-pehr45
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-hhlccu
Out[4]=4

The initial states for a descriptor systems are chosen to be consistent for the inputs:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-huclh
In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-knj6o
Out[2]=2

The second output equals the derivative of the input:

In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-r2c57
Out[3]=3

When inconsistent conditions are given, they are replaced:

In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-e1l7za
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0g7kxbwzgfy-klh2ji
Out[5]=5

Consistent initial states depend on the slow subsystem from KroneckerModelDecomposition:

In[6]:=6
✖
https://wolfram.com/xid/0g7kxbwzgfy-624bj
Out[6]=6

For continuous-time systems, the initial conditions are given by :

In[7]:=7
✖
https://wolfram.com/xid/0g7kxbwzgfy-i46028
Out[7]=7

Possible Issues  (4)Common pitfalls and unexpected behavior

A continuous-time system cannot be simulated with sampled inputs:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-4ykcw
Out[1]=1

Discretize the system:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-8puu9y
Out[2]=2

Computations with machine numbers can be unstable:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-zdudbq
Out[1]=1

Rationalize the system:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-mrbbbj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0g7kxbwzgfy-6rsceh
Out[3]=3

Or compute the numeric response:

In[4]:=4
✖
https://wolfram.com/xid/0g7kxbwzgfy-b3ei5i
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0g7kxbwzgfy-20vnp
Out[5]=5

Symbolic output responses do not support time delays:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-erfy8
Out[1]=1

Try a numeric simulation:

In[2]:=2
✖
https://wolfram.com/xid/0g7kxbwzgfy-ce9uu3
Out[2]=2

For descriptor systems, solutions only exist when Det[λ e - a]≠0 for some λ:

In[1]:=1
✖
https://wolfram.com/xid/0g7kxbwzgfy-hfa3f1
Out[1]=1
Wolfram Research (2010), OutputResponse, Wolfram Language function, https://reference.wolfram.com/language/ref/OutputResponse.html (updated 2014).
✖
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).

✖
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.

✖
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

✖
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: 25-July-2025 ]}

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@misc{reference.wolfram_2025_outputresponse, author="Wolfram Research", title="{OutputResponse}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/OutputResponse.html}", note=[Accessed: 25-July-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: 25-July-2025 ]}

✖
@online{reference.wolfram_2025_outputresponse, organization={Wolfram Research}, title={OutputResponse}, year={2014}, url={https://reference.wolfram.com/language/ref/OutputResponse.html}, note=[Accessed: 25-July-2025 ]}