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Discrete-Time Dynamical System

Prerequisites

Discrete-Time Update Operator | \(\mathbf{x}(n+1) = T( \mathbf{x}(n) )\)
Discrete-Time System with Input | \(\mathbf{x}(n+1) = T\left( \mathbf{x}(n), \mathbf{u}(n) \right)\)
Output Function | \( O \)

Description

For deterministic discrete time systems, sequential inputs can be added by extending the update equation. If the dynamical system generates outputs, an output function is added as a system equation. While systems with no input are common, systems with no outputs have limited practicality.

\[\begin{cases} \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}+1) = \htmlClass{sdt-0000000027}{T}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000117}{n}) ) \\ \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000117}{n}) = \htmlClass{sdt-0000000056}{O}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}) ) \end{cases}\]

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.
\( O \)

This symbol represents the output function of a dynamical system.
\( \mathbf{u} \)

This symbol represents the input of a dynamical system.
\( \mathbf{y} \)

This symbol represents the output of a dynamical system.
\( n \)

This symbol represents any given whole number, \( n \in \htmlClass{sdt-0000000014}{\mathbb{W}}\).

Derivation

  1. The update operator \( \htmlClass{sdt-0000000027}{T} \) for a discrete-time deterministic system is given:
    \[\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}+1) = \htmlClass{sdt-0000000027}{T}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}) )\]
  2. This is extended to allow for sequential inputs along with the states \( \htmlClass{sdt-0000000046}{\mathbf{x}} \):
    \[\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}+1) = \htmlClass{sdt-0000000027}{T}\left( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000117}{n}) \right)\]
  3. The output is generated through an appropriately formed output function:

    The symbol \( O \) represents a function that generates outputs of a dynamical system upon observing a particular state. This corresponds to a "measurement" of the system, often an appropriate analogy given the partial observability of real-life dynamical systems.


    Since \( \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000117}{n}) \) should be a function of the current system state \( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}) \), we get:
    \[ \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000117}{n}) = \htmlClass{sdt-0000000056}{O}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}) ) \]
  4. The output equation is added to the update equation, forming the given system:
    \[ \begin{cases} \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}+1) = \htmlClass{sdt-0000000027}{T}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000117}{n}) ) \\ \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000117}{n}) = \htmlClass{sdt-0000000056}{O}( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000117}{n}) ) \end{cases} \]
    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