Parameter Identification

For on-line parameter identification, these simulations use a sequential least squares algorithm [3] (sometimes called recursive least squares, although it's not recursive) with a forgetting factor. This particular algorithm is known to have conditioning problems when the inputs vary slowly [4].

Equations 7 and 8 give the parameter vector and input vector, respectively, for the least squares algorithm.

$${\mathbf\theta} = \left[\begin{array}{ccccccccc} m_0 & m_1 & m_2 & m_3 & m_4 & m_5 & m_6 & m_7 & \ldots \end{array}\right]^T$$ (7)

$${\mathbf w} = \left[\begin{array}{ccccccccc} 1 & \alpha & \d... ... & \delta_e^3 & \alpha\delta_e^2 & \ldots \end{array}\right]^T$$ (8)

The output is $C_{M{\mathrm est}}={\mathbf w}^T{\mathbf \theta}$, and the output error is $C_{M{\mathrm est}} - C_{M_0}$.

The update formula used is given by Equations 9 and 10.

$${\mathbf P}_{k+1} = \lambda^{-1}{\mathbf P}_k - \frac{\lamb... ...}_k} {1+\lambda^{-1}{\mathbf w}_k^T{\mathbf P}_k{\mathbf w}_k}$$ (9)

$${\mathbf \theta}_{k+1} = {\mathbf \theta}_k + {\mathbf P}_k{\mathbf w}_k(C_{M{\mathrm est}} - C_{M_0})$$ (10)

where ${\mathbf P}_k$ is the covariance matrix at step $k$, and $\lambda$ is the forgetting factor. ${\mathbf P}_0$ is initialized to $\epsilon I$, where $\epsilon$ is a small positive.