Integrate input signal twice
Continuous
The SecondOrder Integrator block and the SecondOrder Integrator Limited block solve the secondorder initial value problem:
$\begin{array}{l}\frac{{d}^{2}x}{d{t}^{2}}=u,\\ {\frac{dx}{dt}}_{t=0}=d{x}_{o},\\ {x}_{t=0}={x}_{o},\end{array}$
where u is the input to the system. The block is therefore a dynamic system with two continuous states: x and dx/dt.
Note: These two states have a mathematical relationship, namely, that dx/dt is the derivative of x. In order to satisfy this relationship throughout the simulation, Simulink places various constraints on the block parameters and behavior. 
The SecondOrder Integrator Limited block is identical to the SecondOrder Integrator block with the exception that it defaults to limiting the states based on the specified upper and lower limits. For more information, see Limiting the States.
Simulink^{®} software can use a number of different numerical integration methods to compute the outputs of the block. Each has advantages for specific applications. Use the Solver pane of the Configuration Parameters dialog box to select the technique best suited to your application. (For more information, see Choose a Solver.) The selected solver computes the states of the SecondOrder Integrator block at the current time step using the current input value.
Use the block parameter dialog box to:
Specify whether the source of each state initial condition is internal or external
Specify a value for the state initial conditions
Define upper and lower limits on either or both states
Specify absolute tolerances for each state
Specify names for both states
Choose an external reset condition
Enable zerocrossing detection
Reinitialize dx/dt when x reaches saturation
Specify that Simulink disregard the state limits and external reset for linearization operations
You can define the initial conditions of each state individually as a parameter on the block dialog box or input one or both of them from an external signal.
To define the initial conditions of state x as
a block parameter, use the Initial condition source x dropdown
menu to select internal
and enter the value
in the Initial condition x field.
To provide the initial conditions from an external
source for state x, specify the Initial
condition source x parameter as external
.
An additional input port appears on the block.
To define the initial conditions of state dx/dt as
a block parameter, use the Initial condition source dx/dt dropdown
menu to select internal
and enter the value
in the Initial condition dx/dt field.
To provide the initial conditions from an external
source for state dx/dt, specify Initial
condition source dx/dt as external
.
An additional input port appears on the block.
If you choose to use an external source for both state initial conditions, your block appears as follows.
Note:

When modeling a secondorder system, you may need to limit the block states. For example, the motion of a piston within a cylinder is governed by Newton's Second Law and has constraints on the piston position (x). With the SecondOrder Integrator block, you can limit the states x and dx/dt independent of each other. You can even change the limits during simulation; however, you cannot change whether or not the states are limited. An important rule to follow is that an upper limit must be strictly greater than its corresponding lower limit.
The block appearance changes when you limit one or both states. Shown below is the appearance of the block with both states limited.
For each state, you can use the block parameter dialog box to set appropriate saturation limits.
If you use the SecondOrder Integrator Limited block, both states are limited by default. But you can also manually limit state x on the SecondOrder Integrator block by selecting Limit x and entering the limits in the appropriate parameter fields.
The block then determines the values of the states as follows:
When x is less than or equal to its lower limit, the value of x is held at its lower limit and dx/dt is set to zero.
When x is in between its lower and upper limits, both states follow the trajectory given by the secondorder ODE.
When x is greater than or equal to its upper limit, the value of x is held at its upper limit and dx/dt is set to zero.
You can choose to reinitialize dx/dt to a new value at the time when x reaches saturation. See Reinitializing dx/dt when x reaches saturation
As with state x, state dx/dt is set as limited by default on the dx/dt pane of the SecondOrder Integrator Limited parameter dialog box. You can manually set this parameter, Limit dx/dt, on the SecondOrder Integrator block. In either case, you must enter the appropriate limits for dx/dt.
If you limit only the state dx/dt, then the block determines the values of dx/dt as follows:
When dx/dt is less than or equal to its lower limit, the value of dx/dt is held at its lower limit.
When dx/dt is in between its lower and upper limits, both states follow the trajectory given by the secondorder ODE.
When dx/dt is greater than or equal to its upper limit, the value of dx/dt is held at its upper limit.
When state dx/dt is held at it upper or lower limit, the value of x is governed by the firstorder initial value problem:
$$\begin{array}{l}\frac{dx}{dt}=L,\\ x({t}_{L})={x}_{L,}\end{array}$$
where L is the dx/dt limit (upper or lower), t_{L} is the time when dx/dt reaches this limit, and x_{L} is the value of state x at that time.
When you limit both states, Simulink maintains mathematical consistency of the states by limiting the allowable values of the upper and lower limits for dx/dt. Such limitations are necessary to satisfy the following constraints:
When x is at its saturation limits, the value of dx/dt must be zero.
In order for x to leave the upper limit, the value of dx/dt must be strictly negative.
In order for x to leave its lower limit, the value of dx/dt must be strictly positive.
Consequently, for such cases, the upper limit of dx/dt must be strictly positive and the lower limit of dx/dt must be strictly negative.
When both states are limited, the block determines the states as follows:
Whenever x reaches its limits, the resulting behavior is the same as that described in "Limiting x only".
Whenever dx/dt reaches one of its limits, the resulting behavior is the same as that described in "Limiting dx/dt only" — including the computation of x using a firstorder ODE when dx/dt is held at one of its limits. In such cases, when x reaches one of its limits, it is held at that limit and dx/dt is set to zero.
Whenever both reach their respective limits simultaneously, the state x behavior overrides dx/dt behavior to maintain consistency of the states.
When you limit both states, you can choose to reinitialize dx/dt at the time when state x reaches saturation. If the reinitialized value is outside specified limits on dx/dt, then dx/dt is reinitialized to the closest valid value and a consistent set of initial conditions is calculated. See Reinitializing dx/dt when x reaches saturation
The block can reset its states to the specified initial conditions based on an external signal. To cause the block to reset its states, select one of the External reset choices on the Attributes pane. A trigger port appears on the block below its input port and indicates the trigger type.
Select rising
to reset
the states when the reset signal rises from zero to a positive value,
from a negative to a positive value, or a negative value to zero.
Select falling
to reset
the states when the reset signal falls from a positive value to zero,
from a positive to a negative value, or from zero to negative.
Select either
to reset
the states when the reset signal changes from zero to a nonzero value
or changes sign.
The reset port has direct feedthrough. If the block output feeds back into this port, either directly or through a series of blocks with direct feedthrough, an algebraic loop results (see Algebraic Loops).
This parameter controls whether zerocrossing detection is enabled
for this block. By default, the Enable zerocrossing detection parameter
is selected on the Attributes pane. However,
this parameter is only in affect if the Zerocrossing control,
on the Solver pane of the Configuration Parameters
dialog, is set to Use local settings
. For
more information, see ZeroCrossing Detection.
For certain modeling applications, dx/dt must be reinitialized when state x reaches its limits in order to pull x out of saturation immediately. You can achieve this by selecting Reinitialize dx/dt when x reaches saturation on the Attributes pane.
If this option is on, then at the instant when x reaches saturation, Simulink checks whether the current value of the dx/dt initial condition (parameter or signal) allows the state x to leave saturation immediately. If so, Simulink reinitializes state dx/dt with the value of the initial condition (parameter or signal) at that instant. If not, Simulink ignores this parameter at the current instant and sets dx/dt to zero to make the block states consistent.
This parameter only applies at the time when x actually reaches saturation limit. It does not apply at any future time when x is being held at saturation.
Refer to the sections on limiting the states for more information.
For an example, see the sldemo_bouncesldemo_bounce
example.
For cases where you simplify your model by linearizing it, you can have Simulink disregard the limits of the states and the external reset by selecting Ignore state limits and the reset for linearization.
By default Simulink software uses the absolute tolerance value specified in the Configuration Parameters dialog box (see Specifying Error Tolerances for VariableStep Solvers) to compute the output of the integrator blocks. If this value does not provide sufficient error control, specify a more appropriate value for state x in the Absolute tolerance x field and for state dx/dt in the Absolute tolerance dx/dt field of the parameter dialog box. Simulink uses the values that you specify to compute the state values of the block.
You can control whether or not to display the x or
the dx/dt output port using the ShowOutput
parameter.
You can display one output port or both; however, you must select
at least one.
You can specify the name of x states and dx/dt states
using the StateNameX
and StateNameDXDT
parameters.
However, you must specify names for either both or neither; you cannot
specify names for just x or just dx/dt.
Both state names must have identical type and length. Furthermore,
the number of names must evenly divide the number of states.
When you select all options, the block icon looks like this.
The Integrator block accepts and outputs signals of type double
on
its data ports. The external reset port accepts signals of type double
or Boolean
.
For more information, see Data Types Supported by Simulink in the Simulink documentation.
The x pane of the SecondOrder Integrator block dialog box appears as follows:
The dx/dt pane of the SecondOrder Integrator block dialog box appears as follows:
The Attributes pane of the SecondOrder Integrator block dialog box appears as follows:
Specify the initial condition source for state x.
Default: internal
internal
Get the initial conditions of state x from the Initial condition x parameter.
external
Get the initial conditions of state x from an external block.
Simulink software does not allow the initial condition of this
block to be inf
or NaN
.
Selecting internal
enables the Initial
condition x parameter.
Selecting external
disables the Initial
condition x parameter..
Parameter: ICSourceX 
Type: string 
Value: 'internal'  'external' 
Default: 'internal' 
Specify the initial condition of state x.
Default: 0.0
Simulink software does not allow the initial condition of this
block to be inf
or NaN
.
Setting Initial condition source x to internal
enables
this parameter.
Setting Initial condition source x to external
disables
this parameter.
Parameter: ICX 
Type: scalar or vector 
Value: '0' 
Default: '0' 
Limit state x of the block to a value between the Lower limit x and Upper limit x parameters.
Default: Off
for
SecondOrder Integrator, On
for SecondOrder Integrator
Limited
Limit state x to a value between the Lower limit x and Upper limit x parameters.
Do not limit the state x output to a value between the Lower limit x and Upper limit x parameters.
This parameter enables Upper limit x parameter.
This parameter enables Lower limit x parameter.
Parameter: LimitX 
Type: string 
Value: 'off'  'on' 
Default: 'off' 
Specify the upper limit of state x.
Default: inf
for
SecondOrder Integrator, 1
for SecondOrder Integrator
Limited
The upper saturation limit for state x must be strictly greater than the lower saturation limit.
Limit x enables this parameter.
Parameter: UpperLimitX 
Type: scalar or vector 
Value: 'inf' 
Default: 'inf' 
Specify the lower limit of state x.
Default: –inf
for
SecondOrder Integrator, 0
for SecondOrder Integrator
Limited
The lower saturation limit for state x must be strictly less than the upper saturation limit.
Limit x enables this parameter.
Parameter: LowerLimitX 
Type: scalar or vector 
Value: 'inf' 
Default: 'inf' 
Specify the absolute tolerance for computing state x.
Default: auto
You can enter auto
, –1,
a positive real scalar or vector.
If you enter auto
or –1,
Simulink uses the absolute tolerance value in the Configuration Parameters
dialog box (see Solver Pane) to compute state x.
If you enter a real scalar value, that value overrides the absolute tolerance in the Configuration Parameters dialog box and is used for computing all x states.
If you enter a real vector, the dimension of that vector must match the dimension of state x. These values override the absolute tolerance in the Configuration Parameters dialog box.
Parameter: AbsoluteToleranceX 
Type: string, scalar, or vector 
Value: 'auto'  '1' 
any positive real scalar or vector 
Default: 'auto' 
Assign a unique name to state x.
Default: ''
To assign a name to a single state, enter the name between quotes, for example, position'.
To assign names to multiple x states, enter a commadelimited list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.
The state names apply only to the selected block.
If you specify a state name for x, you must also specify a state name for dx/dt.
State names for x and dx/dt must have identical types and lengths.
The number of states must be evenly divided by the number of state names. You can specify fewer names than x states, but you cannot specify more names than x states. For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states. However, you must be consistent and apply the same scheme to the state names for dx/dt.
To assign state names with a variable in the MATLAB^{®} workspace, enter the variable without quotes. A variable can be a string or a cell array.
Parameter: StateNameX 
Type: string 
Value: ' ' 
userdefined 
Default: ' ' 
Specify the initial condition source for state dx/dt.
Default: internal
internal
Get the initial conditions of state dx/dt from the Initial condition dx/dt parameter.
external
Get the initial conditions of state dx/dt from an external block.
Simulink software does not allow the initial condition of this
block to be inf
or NaN
.
Selecting internal
enables the Initial
condition dx/dt parameter.
Selecting external
disables the Initial
condition dx/dt parameter.
Parameter: ICSourceDXDT 
Type: string 
Value: 'internal'  'external' 
Default: 'internal' 
Specify the initial condition of state dx/dt.
Default: 0.0
Simulink software does not allow the initial condition of this
block to be inf
or NaN
.
Setting Initial condition source dx/dt to internal
enables
this parameter.
Setting Initial condition source dx/dt to external
disables
this parameter.
Parameter: ICDXDT 
Type: scalar or vector 
Value: '0' 
Default: '0' 
Limit the dx/dt state of the block to a value between the Lower limit dx/dt and Upper limit dx/dt parameters.
Default: Off
for
SecondOrder Integrator, On
for SecondOrder Integrator
Limited
Limit state dx/dt of the block to a value between the Lower limit dx/dt and Upper limit dx/dt parameters.
Do not limit state dx/dt of the block to a value between the Lower limit dx/dt and Upper limit dx/dt parameters.
If you set saturation limits for x, then the interval defined by the Upper limit dx/dt and Lower limit dx/dt must contain zero.
This parameter enables Upper limit dx/dt.
This parameter enables Lower limit dx/dt.
Parameter: LimitDXDT 
Type: string 
Value: 'Off'  'On' 
Default: 'Off' 
Specify the upper limit for state dx/dt.
Default: 'inf'
If you limit x, then this parameter must have a strictly positive value.
Limit dx/dt enables this parameter.
Parameter: UpperLimitDXDT 
Type: scalar or vector 
Value: 'inf' 
Default: 'inf' 
Specify the lower limit for state dx/dt.
Default: 'inf'
If you limit x, then this parameter must have a strictly negative value.
Limit dx/dt enables this parameter.
Parameter: LowerLimitDXDT 
Type: scalar or vector 
Value: 'inf' 
Default: 'inf' 
Specify the absolute tolerance for computing state dx/dt.
Default: auto
You can enter auto
, –1,
a positive real scalar or vector.
If you enter auto
or –1,
then Simulink uses the absolute tolerance value in the Configuration
Parameters dialog box (see Solver Pane) to compute the dx/dt output
of the block.
If you enter a numeric value, that value overrides the absolute tolerance in the Configuration Parameters dialog box.
Parameter: AbsoluteToleranceDXDT 
Type: string, scalar, or vector 
Value: 'auto'  '1' 
any positive real scalar or vector 
Default: 'auto' 
Assign a unique name to state dx/dt.
Default: ''
To assign a name to a single state, enter the name between quotes, for example, 'velocity'.
To assign names to multiple dx/dt states, enter a commadelimited list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.
The state names apply only to the selected block.
If you specify a state name for dx/dt, you must also specify a state name for x.
State names for x and dx/dt must have identical types and lengths.
The number of states must be evenly divided by the number of state names. You can specify fewer names than dx/dt states, but you cannot specify more names than dx/dt states. For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states. However, you must be consistent and apply the same scheme to the state names for x.
To assign state names with a variable in the MATLAB workspace, enter the variable without quotes. A variable can be a string or a cell array.
Parameter: StateNameDXDT 
Type: string 
Value: ' ' 
userdefined 
Default: ' ' 
Reset the states to their initial conditions when a trigger event occurs in the reset signal.
Default: none
none
Do not reset the state to initial conditions.
rising
Reset the state when the reset signal rises from a zero to a positive value or from a negative to a positive value.
falling
Reset the state when the reset signal falls from a positive value to zero or from a positive to a negative value.
either
Reset the state when the reset signal changes from zero to a nonzero value or changes sign.
Parameter: ExternalReset 
Type: string 
Value: 'none'  'rising'  'falling'  'either' 
Default: 'none' 
Select to enable zerocrossing detection. For more information, see ZeroCrossing Detection.
Default: On
Enable zerocrossing detection.
Do not enable zerocrossing detection.
Parameter: ZeroCross 
Type: string 
Value: 'on'  'off' 
Default: 'on' 
At the instant when state x reaches saturation, reset dx/dt to its current initial conditions.
Default: Off
Reset dx/dt to its initial conditions when x becomes saturated.
Do not reset dx/dt to its initial conditions when x becomes saturated.
The dx/dt initial condition must have a value that enables x to leave saturation immediately. Otherwise, Simulink ignores the initial conditions for dx/dt to preserve mathematical consistency of block states.
Parameter: ReinitDXDTwhenXreachesSaturation 
Type: string 
Value: 'off'  'on' 
Default: 'off' 
For linearization purposes, have Simulink ignore the specified state limits and the external reset.
Default: Off
Ignore the specified state limits and the external reset.
Apply the specified state limits and the external reset setting.
Parameter: IgnoreStateLimitsAndResetForLinearization 
Type: string 
Value: 'off'  'on' 
Default: 'off' 
Specify the output ports on the block.
Default: both
both
Show both x and dx/dt output ports.
x
Show only the x output port.
dx/dt
Show only the dx/dt output port.
Parameter: ShowOutput 
Type: string 
Value: 'both'  'x'  'dxdt' 
Default: 'both' 