SecondOrder Integrator
Integrate input signal twice
 Library:
Simulink /
Continuous
Description
The SecondOrder Integrator block and the SecondOrder Integrator Limited block solve the secondorder initial value problem:
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. 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 several 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 Solver Selection Criteria.) 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
Defining Initial Conditions
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
Simulink does not allow initial condition values of inf
or NaN
.
If you limit state x or state dx/dt by specifying saturation limits (see Limiting the States) and one or more initial conditions are outside the corresponding limits, then the respective states are initialized to the closest valid value and a set of consistent initial conditions is calculated.
Limiting the States
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. With both states limited, the block appears as follows.
For each state, you can use the block parameter dialog box to set appropriate saturation limits.
Limiting x Only
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.
Limiting dx/dt Only
As with state x, state dx/dt is set as limited by default on the dx/dt pane of the SecondOrder Integrator Limited block 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:
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.
Limiting Both States
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.
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
Resetting the State
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 Loop Concepts).
Enabling ZeroCrossing Detection
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 box, is set to Use local settings
. For more information, see ZeroCrossing Detection.
Reinitializing dx/dt When x Reaches Saturation
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 Simulation of a Bouncing Ball.
Disregarding State Limits and External Reset for Linearization
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.
Specifying the Absolute Tolerance for the Block Outputs
By default Simulink software uses the absolute tolerance value specified in the Configuration Parameters dialog box (see 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.
Specifying the Display of the Output Ports
You can control whether 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.
Specifying the State Names
You can specify the name of x states and dx/dt states using the StateNameX
and StateNameDXDT
parameters. However, you must specify names for 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.
Selecting All Options
When you select all options, the block icon looks like this.
Ports
Input
expand all
u
— Input signal u
scalar  vector  matrix
Input signal u
to the integrator system, specified as a scalar, vector, or matrix.
Data Types: double
x0
— Initial condition x0
scalar  vector  matrix
External signal specifying the initial condition x0
to the integrator system. You can specify the initial condition as a scalar, vector, or matrix.
Dependencies
To enable this input port, set the Initial condition source x parameter to external
.
Data Types: double
dx0
— Initial condition dx0
scalar  vector  matrix
External signal specifying the initial condition dx0
to the integrator system. You can specify the initial condition dx0
as a scalar, vector, or matrix.
Dependencies
To enable this input port, set the Initial condition source dx/dt parameter to external
.
Data Types: double
Output
expand all
x
— Output signal x
scalar  vector  matrix
x
state output signal, provided as a scalar, vector, or matrix.
Data Types: double
dx
— Output signal dx
scalar  vector  matrix
dx
state output signal, specified as a scalar, vector, or matrix.
Data Types: double
Parameters
expand all
x
Initial condition source x
— Source of initial condition for state x
internal
(default)  external
Specify the source of the initial conditions for state x.
Limitations
Simulink software does not allow the initial condition of this block to be inf
or NaN
.
Dependencies
Selecting internal
enables the Initial condition x parameter and removes the X_{0} input port.
Selecting external
disables the Initial condition x parameter and enables the X_{0} input port.
Programmatic Use
Block Parameter: ICSourceX 
Type: character vector, string 
Values: 'internal'  'external' 
Default: 'internal' 
Initial condition x
— Initial condition of state x
0.0
(default)  scalar  vector  matrix
Specify the initial condition of state x.
Limitations
Simulink software does not allow the initial condition of this block to be inf
or NaN
.
Dependencies
To enable this parameter, set Initial condition source x to internal
.
Setting Initial condition source x to external
disables this parameter and enables the X_{0} input port.
Programmatic Use
Block Parameter: ICX 
Type: character vector, string 
Values: scalar  vector  matrix 
Default: '0.0' 
Limit x
— Limit values of state x
off
 on
Limit state x of the block to a value between the Lower limit x and Upper limit x parameters. The default value of the SecondOrder Integrator block is off
. The default value of the SecondOrder Integrator Limited is on
.
To limit state x to a value between the Lower limit x and Upper limit x parameters, select this check box.
To remove range limitations on state x, clear this check box.
Dependencies
Selecting this check box enables the Upper limit x and Lower limit x parameters.
Programmatic Use
Block Parameter: LimitX 
Type: character vector, string 
Values: 'off'  'on' 
Default: 'off' (SecondOrder Integrator)  'on' (SecondOrder Integrator Limited) 
Upper limit x
— Upper limit of state x
1
 inf
 scalar  vector  matrix
Specify the upper limit of state x. The default value for the SecondOrder Integrator block is inf
. The default value for the SecondOrder Integrator Limited block is 1
.
Tips
The upper saturation limit for state x must be strictly greater than the lower saturation limit.
Dependencies
To enable this parameter, select the Limit x check box.
Programmatic Use
Block Parameter: UpperLimitX 
Type: character vector, string 
Values: '1'  'inf'  scalar  vector  matrix 
Default: '1' (SecondOrder Integrator Limited)  'inf' (SecondOrder Integrator) 
Lower limit x
— Lower limit of state x
0
(default)  inf
 scalar  vector  matrix
Specify the lower limit of state x. The default value for the SecondOrder Integrator block is inf
. The default value for the SecondOrder Integrator Limited block is 0
.
Tip
The lower saturation limit for state x must be strictly less than the upper saturation limit.
Dependencies
To enable this parameter, select the Limit x check box.
Programmatic Use
Block Parameter: LowerLimitX 
Type: character vector, string 
Values: '0'  'inf'  scalar  vector  matrix 
Default: '0' (SecondOrder Integrator Limited)  'inf' (SecondOrder Integrator) 
Wrap x
— Enable wrapping of x
off
(default)  on
Enable wrapping of x between the Wrapped upper value x and Wrapped lower value x parameters. Enabling wrapping of x eliminates the need for zerocrossing detection, reduces solver resets, improves solver performance and accuracy, and increases simulation time span when modeling rotary and cyclic state trajectories.
If you specify Wrapped upper value x as inf
and Wrapped lower value x as inf
, wrapping will never occur.
Dependencies
Selecting this check box enables Wrapped upper value x and Wrapped lower value x.
Programmatic Use
Block Parameter: WrapX 
Type: character vector, string 
Values: 'off'  'on' 
Default: 'off' 
Wrapped upper value x
— Upper value for wrapping x
pi
(default)  scalar  vector  matrix
Specify the upper value for wrapping x.
Dependencies
To enable this parameter, select the Wrap x check box.
Programmatic Use
Block Parameter: WrappedUpperValueX 
Type: character vector, string 
Values: scalar  vector  matrix 
Default: 'pi' 
Wrapped lower value x
— Lower value for wrapping x
pi
(default)  scalar  vector  matrix
Specify the lower value for wrapping x.
Dependencies
To enable this parameter, select the Wrap x check box.
Programmatic Use
Block Parameter: WrappedLowerValueX 
Type: character vector, string 
Values: scalar  vector  matrix 
Default: 'pi' 
Absolute tolerance x
— Absolute tolerance for computing state x
auto
(default)  1
 scalar  vector
Specify the absolute tolerance for computing state x.
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.
Programmatic Use
Block Parameter: AbsoluteToleranceX 
Type: character vector, string 
Values: 'auto'  '1'  any positive real scalar or vector 
Default: 'auto' 
State name x
— Name for state x
''
(default)  character vector  string
Assign a unique name to state x.
Tips
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 character vector, string or a cell array.
Programmatic Use
Block Parameter: StateNameX 
Type: character vector, string 
Values: ' '  userdefined 
Default: ' ' 
dx/dt
Initial condition source dx/dt
— Source of initial condition for state dx/dt
internal
(default)  external
Specify the source of initial conditions for state dx/dt as internal
or external
.
Limitations
Simulink software does not allow the initial condition of this block to be inf
or NaN
.
Dependencies
Programmatic Use
Block Parameter: ICSourceDXDT 
Type: character vector 
Values: 'internal'  'external' 
Default: 'internal' 
Initial condition dx/dt
— Initial condition of state dx/dt
0.0
(default)  scalar  vector  matrix
Specify the initial condition of state dx/dt.
Limitations
Simulink software does not allow the initial condition of this block to be inf
or NaN
.
Dependencies
Programmatic Use
Block Parameter: ICDXDT 
Type: character vector 
Values: scalar  vector  matrix 
Default: '0.0' 
Limit dx/dt
— Limit values of state dx/dt
off
 on
Limit the dx/dt state of the block to a value between the Lower limit dx/dt and Upper limit dx/dt parameters. The default value of the SecondOrder Integrator block is off
. The default value of the SecondOrder Integrator Limited is on
.
Tip
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.
Dependencies
Selecting this check box enables the Upper limit dx/dt and Lower limit dx/dt parameters.
Programmatic Use
Parameter: LimitDXDT 
Type: character vector 
Values: 'off'  'on' 
Default: 'off' (SecondOrder Integrator)  'on' (SecondOrder Integrator Limited) 
Upper limit dx/dt
— Upper limit of state dx/dt
inf
(default)  scalar  vector  matrix
Specify the upper limit for state dx/dt.
Dependencies
If you limit x, then this parameter must have a strictly positive value.
To enable this parameter, select the Limit dx/dt check box.
Programmatic Use
Block Parameter: UpperLimitDXDT 
Type: character vector 
Values: scalar  vector  matrix 
Default: 'inf' 
Lower limit dx/dt
— Lower limit of state dx/dt
inf
(default)  scalar  vector  matrix
Specify the lower limit for state dx/dt.
Dependencies
If you limit x, then this parameter must have a strictly negative value.
To enable this parameter, select the Limit dx/dt check box.
Programmatic Use
Block Parameter: LowerLimitDXDT 
Type: character vector 
Values: scalar  vector  matrix 
Default: 'inf' 
Absolute tolerance dx/dt
— Absolute tolerance for computing state dx/dt
auto
(default)  1
 scalar  vector
Specify the absolute tolerance for computing state dx/dt.
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.
Programmatic Use
Block Parameter: AbsoluteToleranceDXDT 
Type: character vector, string, scalar,
or vector 
Values: 'auto'  '1'  any positive real scalar or vector 
Default: 'auto' 
State name dx/dt
— Name for state dx/dt
'
'
(default)  character vector  string
Assign a unique name to state dx/dt.
Tips
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 character vector,
string, or a cell array.
Programmatic Use
Block Parameter: StateNameDXDT 
Type: character
vector, string 
Values: ' '  userdefined 
Default: ' ' 
Attributes
External reset
— Reset states to their initial conditions
none
(default)  rising
 falling
 either
Reset the states to their initial conditions when a trigger event occurs in the reset signal.
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.
Programmatic Use
Block Parameter: ExternalReset 
Type: character vector, string 
Values: 'none'  'rising'  'falling'  'either' 
Default: 'none' 
Enable zerocrossing detection
— Enable zerocrossing detection
on
(default)  off
Select to enable zerocrossing detection. For more information, see ZeroCrossing Detection.
Programmatic Use
Parameter:
ZeroCross 
Type: character vector, string 
Values:
'on'  'off' 
Default:
'on' 
Reinitialize dx/dt when x reaches saturation
— Reset dx/dt when x reaches saturation
off
(default)  on
At the instant when state x reaches saturation, reset dx/dt to its current initial conditions.
Tip
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.
Programmatic Use
Block Parameter: ReinitDXDTwhenXreachesSaturation 
Type: character vector, string 
Values: 'off'  'on' 
Default: 'off' 
Ignore state limits and the reset for linearization
— Ignore state limits and external reset for linearization
off
(default)  on
For linearization purposes, have Simulink ignore the specified state limits and the external reset.
Programmatic Use
Block Parameter: IgnoreStateLimitsAndResetForLinearization 
Type: character vector, string 
Values: 'off'  'on' 
Default: 'off' 
Show output
— Output ports to display
both
(default)  x
 dxdt
Specify the output ports on the block.
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.
Programmatic Use
Block Parameter: ShowOutput 
Type: character vector, string 
Values: 'both'  'x'  'dxdt' 
Default: 'both' 
Block Characteristics
Data Types  double

Direct Feedthrough  yes

Multidimensional Signals  no

VariableSize Signals  no

ZeroCrossing Detection  yes

Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Not recommended for productionquality code. Relates to resource limits and restrictions on speed and memory often found in embedded systems. The code generated can contain dynamic allocation and freeing of memory, recursion, additional memory overhead, and widelyvarying execution times. While the code is functionally valid and generally acceptable in resourcerich environments, smaller embedded targets often cannot support such code.
In general, consider using the Simulink Model Discretizer to map continuous blocks into discrete equivalents that support production code generation. To start the Model Discretizer, select Analysis > Control Design > Model Discretizer
. One exception is the SecondOrder Integrator block because, for this block, the Model Discretizer produces an approximate discretization.
Introduced in R2010a