Description
A dynamical system with continuous time and deterministic behaviour can be summarized by its update equation and output equation. The update rule handles possible continuous inputs, while observations can be gathered by retrieving the outputs. While systems with no input are common, systems with no outputs have limited practicality.
\[\begin{cases}
\dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000118}{t}))
\\
\htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000056}{O}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}))
\end{cases}\]
Derivation
(See: Continuous-Time Update Operator (ODE), and Continuous-Time System with Input)
- A continuous-time dynamical system has an update rule:
\[\dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \frac{d}{dt}\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}))\] - The update equation can be extended with a continuous input term:
\[\dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000118}{t}))\] - 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 \( \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000118}{t}) \) should be a (continuous) function of the current system state \( \htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}) \), we get:
\[ \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000056}{O}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t})) \] - Grouping the two system equations, we get the system:
\[ \begin{cases} \dot{\htmlClass{sdt-0000000046}{\mathbf{x}}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000027}{T}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t}), \htmlClass{sdt-0000000078}{\mathbf{u}}(\htmlClass{sdt-0000000118}{t})) \\ \htmlClass{sdt-0000000086}{\mathbf{y}}(\htmlClass{sdt-0000000118}{t}) = \htmlClass{sdt-0000000056}{O}(\htmlClass{sdt-0000000046}{\mathbf{x}}(\htmlClass{sdt-0000000118}{t})) \end{cases} \]
as required.
