Task-Oriented Identification

When the dynamic system is controlled by a continuous-valued control signal vector $\mathbf{u}(t)$, it can be taken into account by replacing the equation of dynamics (2) with one of these two options:

$$\mathbf{s}(t) = \mathbf{g}\left(\left[ \begin{array}{c} \mathbf{u}(t-1) \\ \mat... ...s}(t-1) \end{array} \right],\boldsymbol{\theta}_\mathbf{g}\right)+\mathbf{m}(t)$$ (7)

$$\left[ \begin{array}{c} \mathbf{u}(t) \\ \mathbf{s}(t) \end{array} \right] = \mathbf{g}\left(\left[ \begin{array}{c} \mathbf{u}(t-1) \\ \mat... ...}(t-1) \end{array} \right],\boldsymbol{\theta}_\mathbf{g}\right)+\mathbf{m}(t).$$ (8)

The first one (7) assumes that the control signal is coming outside the model. The latter one (8) is called task-oriented identification because it predicts the control signals $\mathbf{u}(t)$ within the model. Figure 1 illustrates these two options.

Figure: Traditional model (left) and task-oriented identification (right). Traditionally, the control signals $\mathbf{u}(t)$ are coming from outside the model, but in task-oriented identification they are within the model.

We choose to use task-oriented identification (Eq. 8) in this paper for the following reasons. Firstly, it allows for three different control schemes described in the next section. Secondly, it creates an opportunity to learn more. The learning algorithm finds such a state space that the prediction of observations and control signals is as accurate as possible. A well-learned state space should thus make control easier. Thirdly, it is biologically motivated. Different parts of the cerebellum can be used for motor control and cognitive processing depending on where their outputs are directed [5].