# Errata: Example 8.10 missing factor of v, a1 and a2 flipped

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Chapter Output Feedback 8-28 -4 3.1.5 6 Oct 2021

Example 8.10 (Steering control with velocity scheduling) is missing a factor of ${\displaystyle v}$ in the definition of ${\displaystyle A_{\text{d}}}$ that propagates through the example. In addition, the coefficients ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ are swapped in the definition of ${\displaystyle w_{2}}$.

The equation for ${\displaystyle A_{\text{d}}}$ should read

${\displaystyle A_{\text{d}}=\left.{\frac {\partial f}{\partial x}}\right|_{(x_{\text{d}},u_{\text{d}})}=\left.{\begin{bmatrix}0&0&-{\color {blue}v}\sin \theta \\0&0&{\color {blue}v}\cos \theta \\0&0&0\end{bmatrix}}\right|_{(x_{\text{d}},u_{\text{d}})}={\begin{bmatrix}0&0&0\\0&0&{\color {blue}v_{\text{r}}}\\0&0&0\end{bmatrix}},}$

This leads to an error in the definition of ${\displaystyle w_{2}}$ later in the example. In addition, the coefficients ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$ are swapped and there was an error in the sign, so the definition of ${\displaystyle w_{2}}$ should be

${\displaystyle w_{2}={\color {blue}-}{\frac {l}{v_{\text{r}}}}({\frac {\color {blue}a_{2}}{\color {blue}v_{\text{r}}}}e_{y}+{\color {blue}a_{1}}e_{\theta }).}$

The final controller then becomes

${\displaystyle {\begin{bmatrix}v\\\phi \end{bmatrix}}=-{\begin{bmatrix}\lambda _{1}&0&0\\0&{\dfrac {{\color {blue}a_{2}}l}{v_{\text{r}}^{\color {blue}2}}}&{\dfrac {{\color {blue}a_{1}}l}{v_{\text{r}}}}\end{bmatrix}}{\begin{bmatrix}x-v_{\text{r}}t\\y-y_{\text{r}}\\\theta \end{bmatrix}}+{\begin{bmatrix}v_{\text{r}}\\0\end{bmatrix}}.}$

These corrections also change the controller response in Figure 8.13. The corrected response is

Acknowledgements: Thanks to Kaivalya Bakshi and Theunis Botha for pointing out the errors in this example.