Second-Order Integrator
Second-order integrator.
blockType: SecondOrderIntegrator
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Path in the library: |
Description
The block Second-Order Integrator performs second order integration. The behaviour of the block is described by the following equations:
where is the input signal. Thus, the block represents a dynamic system with two continuous states: and .
| These two states have a mathematical relationship, is the derivative of . To fulfil this relationship during simulation, Engee imposes various constraints on the parameters and behaviour of the block. |
These equations describe the exact dependence of output on input in continuous time. Numerical approximation methods are used to estimate them with finite accuracy. In Engee there are several types of solvers that implement different numerical integration methods to compute the output of a block. Each solver has its own advantages depending on the specific problem.
The selected solver calculates the output of the block Second-Order Integrator at the current time step using the current input value and the state value at the previous time step.
The initial values and can be set in the block parameters dialogue box.
Restriction of states
When modelling a second-order system, you may need to constrain the states of the blocks. For example, the motion of a piston inside a cylinder is governed by Newton’s second law and has constraints on the piston position . In a block Second-Order Integrator you can constrain the states and independently of each other. An important rule to follow is that the upper limit must be strictly greater than the corresponding lower limit.
Limit only x
You can restrict the state of , by selecting the checkbox Limit x and entering the restrictions in the corresponding parameters.
The block defines the state values as follows:
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When is less than or equal to the lower limit, the value of is held at the lower limit and is set to zero.
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When is between the lower and upper limits, both states correspond to a given second-order ODU.
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When is greater than or equal to its upper limit, the value of is held at its upper limit and is set to zero.
You can reinitialise with a new value while reaches its limit.
Limit dx/dt only
You can restrict the state of , by selecting the check box Limit dx/dt and entering the restrictions in the corresponding parameters.
If you restrict only the state , the block defines the values of as follows:
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When is less than or equal to the lower limit, the value of is held at the lower limit.
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When is between the lower and upper limits, both states correspond to a given second-order ODU.
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When is greater than or equal to its upper limit, the value of is held at its upper limit.
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When the state is held at its upper or lower limit, the value of is determined by the first-order initial value task:
,
,
where
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- is the limit (upper or lower);
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- the time when reaches this limit;
- the value of state at the moment .
Limit of both states
When you constrain both states, Engee maintains mathematical consistency between states by constraining the allowable upper and lower limit values for . Such constraints are necessary to satisfy the following constraints:
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When is within the constraints, the value of must be zero.
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For to go beyond the upper limit, the value of must be strictly negative.
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For to be outside the lower limit, the value of must be strictly positive.
In such cases, the upper limit of must be strictly positive and the lower limit of must be strictly negative.
When both states are bounded, the block determines the states as follows:
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Whenever reaches its limits, the resulting behaviour is the same as described in section Limit only x.
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Whenever reaches one of its limits, the resulting behaviour includes computing by usage of a first-order ODE when is held at one of its limits. In such cases, whenever reaches one of its limits, it is held at that limit and is set to zero.
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Whenever both reach their respective limits at the same time, the state behaviour of overrides the behaviour of to maintain state consistency.
When you limit both states, you can choose to re-initialise at the time the state reaches saturation. If the re-initialised value falls outside the specified limits , then is re-initialised to the nearest valid value and a consistent set of initial conditions is calculated.
Ports
Input
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u
—
input signal
scalar | vector | matrix
Details
The input signal u to the integrator as a scalar, vector or matrix.
| Data types |
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| Complex numbers support |
No |
Output
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x
—
output signal
scalar | vector | matrix
Details
The output of the x state signal as a scalar, vector or matrix.
| Data types |
|
| Complex numbers support |
No |
#
dx
—
output signal
scalar | vector | matrix
Details
The output of the dx state signal as a scalar, vector or matrix.
| Data types |
|
| Complex numbers support |
No |
Parameters
Parameters
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Initial condition x —
initial state value x
Scalar / array of real numbers
Details
Initial state value of x block.
| Default value |
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| Program usage name |
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| Tunable |
No |
| Evaluatable |
Yes |
#
Limit x —
state limit values x
Logical
Details
Limit the block output to a value between the parameters Lower limit x and Upper limit x. By default, this checkbox is unchecked.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
No |
#
Upper limit x —
upper limit of state value x
Scalar / array of real numbers
Details
Specify the upper limit of the status value x.
The upper limit for the state x must be strictly greater than the lower limit.
Dependencies
To use this parameters, select the checkbox Limit x.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
#
Lower limit x —
lower limit of state value x
Scalar / array of real numbers
Details
Specify the lower limit of the status value x.
The lower limit for the state x must be strictly less than the upper limit.
Dependencies
To use this parameters, select the checkbox Limit x.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
#
Wrap x —
transfer switch on x
Logical
Details
Enable carry states x between the parameters Wrapped upper value x and Wrapped Lower value x. Enabling carry states eliminates the need for zero crossing detection, reduces the number of solver resets, improves solver performance and accuracy, and increases simulation time when modelling rotational and cyclic state trajectories.
If you specify Wrapped upper value x as inf and Wrapped Lower value x as -inf, no carryover will occur.
| Default value |
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| Program usage name |
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| Tunable |
No |
| Evaluatable |
No |
#
Wrapped upper value x —
upper transfer value x
Scalar / array of real numbers
Details
Specify the upper value to carry x. The upper value for the carry x must be greater than the lower value.
Dependencies
To use this parameters, select the check box Wrap x.
| Default value |
|
| Program usage name |
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| Tunable |
No |
| Evaluatable |
Yes |
#
Wrapped lower value x —
lower value for transfer x
Scalar / array of real numbers
Details
Specify the lower value to carry x. The lower value for the carry x must be less than the upper value.
Dependencies
To use this parameters, tick the checkbox Wrap x.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
Parameters
#
Initial condition dx/dt —
initial state value dx/dt
Scalar / array of real numbers
Details
Initial state value dx/dt.
| Default value |
|
| Program usage name |
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| Tunable |
No |
| Evaluatable |
Yes |
#
Limit dx/dt —
state limit values dx/dt
Logical
Details
Limit the block output to a value between the parameters Lower limit dx/dt and Upper limit dx/dt. By default, this checkbox is unchecked.
If the state limits are set x, the interval defined by the parameters Lower limit dx/dt and Upper limit dx/dt must contain zero.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
No |
#
Upper limit dx/dt —
upper limit of state value dx/dt
Scalar / array of real numbers
Details
Specify the upper limit of the status value dx/dt.
Dependencies
If you limit x, this parameter must have a strictly positive value.
To use this parameter, tick the checkbox Limit dx/dt.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |
#
Lower limit dx/dt —
lower limit of condition dx/dt
Scalar / array of real numbers
Details
Specify the lower status limit dx/dt.
Dependencies
If you limit x, this parameter must have a strictly negative value.
To use this parameter, tick the checkbox Limit dx/dt.
| Default value |
|
| Program usage name |
|
| Tunable |
No |
| Evaluatable |
Yes |