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Concepts

In this section, we will cover the main concepts that allow to understand the functionalities of vip-ivp.

Initial Value Problem (IVP)​

vip-ivp is a tool designed for solving Initial Value Problems (IVPs). You may already be familiar with other tools that deal with IVPs, such as Simulink, Amesim, Modelica, or Dymola.

An IVP is a type of differential equation where the solution involves variables that evolve over time, starting from a set of known initial conditions. These problems are common in various fields of physics and engineering, such as mechanics, electronics, and chemical reactions..

Temporal Variables​

In vip-ivp, Temporal Variables represent quantities that evolve over time. They are instances of the TemporalVar class. To build a system, you create Temporal Variables. It is that simple.

Here’s the behavior of Temporal Variables in vip-ivp:

  1. Before the system is solved, Temporal Variables have no value. Attempting to access their value will result in an error.
  2. After the system is solved, their values over time can be computed and accessed.

Temporal Variables can be created in many ways, including from constants, temporal functions, arithmetic expressions, or integration.

Solver​

The solver in vip-ivp is responsible for computing the evolution of variables over time using numerical integration schemes. It ensures that the computed solution meets the required precision.

One of the strengths of vip-ivp is the flexibility in how systems are defined. Unlike other solvers such as SciPy's solve_ivp(), which impose a strict structure on how systems must be defined, vip-ivp allows users to define systems in a more modular and flexible way.

warning

An inherent limitation of IVP solvers is that differentiation is not supported in a "clean" way. Since vip-ivp uses an integration scheme to solve the system, differentiation is not possible without relying on past values.

This implies the following considerations for system design:

  • vip.integrate() should always be preferred over vip.differentiate(), as integration guarantees the precision of the solution, while differentiation does not.
  • vip.differentiate() may be used for specific cases (e.g., a PID controller), but in those cases, a small time step is recommended to maintain accuracy.