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Integration

To create an integrated variable, use the vip.integrate(value, x0) function:

  • value is the variable to be integrated
  • x0 is the value of the integrated variable at t=0st=0 s
# Create a sine by integrating a cosine
cosine = vip.temporal(lambda t: np.cos(t))
sine = vip.integrate(cosine, 0)

Integrate a standard Python value​

Standard Python values can be integrated. The only restriction is that it must be a numeric value or a collection containing only numeric values.

If value is a collection, x0 must follow the same structure: it must be a dict with matching keys, or a list/ndarray with the same shape.

integrated_constant = vip.integrate(5, x0=0)
integrated_dict = vip.integrate({"a": 3, "b": lambda t: -t}, x0={"a": 1, "b": 3})
integrated_list = vip.integrate([1, 2, 3], x0=[3, 4, 5])

Integrate a Temporal Variable containing a collection​

You can also integrate TemporalVar instances that contain collections of numeric values. As with standard collections, x0 must match the structure of value.

source_dict = vip.temporal({"a": 3, "b": lambda t: -t})
source_list = vip.temporal([1, 2, 3])

integrated_dict = vip.integrate(source_dict, x0={"a": 1, "b": 3})
integrated_list = vip.integrate(source_list, x0=[3, 4, 5])

Set output limits​

To prevent an integrated variable from exceeding certain values — for example, in a PID controller with anti-windup — use the minimum and maximum arguments to set bounds on the output.

import numpy as np

source = vip.temporal(lambda t: np.cos(t))
# Variable with output limits
bounded_variable = vip.integrate(source, x0=0.0, maximum=0.5, minimum=-0.5)
# Variable without output limits
unbounded_variable = vip.integrate(source, x0=0.0)

unbounded_variable.to_plot("Without integration bounds")
bounded_variable.to_plot("With integration bounds")

vip.solve(10, time_step=0.01)

The following result is obtained:

Integration bounds

warning

If x0 lies outside the [minimum, maximum] bounds, it will raise a ValueError.

Variable bounds​

The minimum and maximum arguments can also be dynamic TemporalVar instances:

import numpy as np

source = vip.temporal(lambda t: np.cos(t))
upper_limit = vip.temporal(lambda t: 1 - 0.1 * t)

bounded_variable = vip.integrate(source, x0=0.0, maximum=upper_limit)
unbounded_variable = vip.integrate(source, x0=0.0)

unbounded_variable.to_plot("Without integration bounds")
bounded_variable.to_plot("With integration bounds")

vip.solve(10, time_step=0.01)

Integration with variable upper bound

warning

If both minimum and maximum are set, they must not cross over at any point in time — the minimum must always remain less than or equal to the maximum. Violating this condition will result in an error.