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Continuous-Time System with Input

Prerequisites

Continuous-Time Update Operator (ODE) | \(\dot{\mathbf{x}}(t) = \frac{d}{dt}\mathbf{x}(t) = T(\mathbf{x}(t))\)

Description

Dynamical systems with continuous time formulations can also be given continuous inputs. These will impact the evolution of the system, and are often necessary to ensure the system can function.

\[\dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000118}{t}))\]

Symbols Used:

This is the symbol for a dynamical system's update operator.
\( \mathbf{x} \)

This symbol represents a state of the dynamical system at some time point.
\( \mathbf{u} \)

This symbol represents the input of a dynamical system.
\( t \)

This symbol represents time. It is often measured by its SI unit seconds (\( \htmlClass{udt-0000000002}{s} \)).

Derivation

  1. The update equation for a continuous-time, deterministic system is given by:
    \[\dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \frac{d}{dt}\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}))\]
  2. Adding input to the system can be encoded by a modification to the update operator, adding it as a parameter:
    \[ \dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000118}{t})) \]
    as required.

References

  1. Jaeger, H. (n.d.). Neural Networks (AI) (WBAI028-05) Lecture Notes BSc program in Artificial Intelligence. Retrieved May 17, 2024, from https://www.ai.rug.nl/minds/uploads/LN_NN_RUG.pdf