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

Chapter	Output Feedback
Page	8-28
Line	-4
Version	3.1.5
Date	6 Oct 2021

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

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

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

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

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

The final controller then becomes

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

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.