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docs(mpc_lateral_controller): modify mathjax visualization error (#6486)
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* docs: modified mathjacs error

* test

* fix

* docs(mpc_lateral_controller): modify mathjax visualization error

Signed-off-by: root <root@DESKTOP-EP0T4HP>

---------

Signed-off-by: root <root@DESKTOP-EP0T4HP>
Co-authored-by: Daiki Sakanoue <[email protected]>
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sak-0822 and Daiki Sakanoue authored Mar 31, 2024
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88 changes: 44 additions & 44 deletions control/mpc_lateral_controller/model_predictive_control_algorithm.md
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Expand Up @@ -22,7 +22,7 @@ In the linear MPC formulation, all motion and constraint expressions are linear.

$$
\begin{gather}
x_{k+1}=Ax_{k}+Bu_{k}+w_{k}, y_{k}=Cx_{k} \tag{1} \\
x_{k+1}=Ax_{k}+Bu_{k}+w_{k}, y_{k}=Cx_{k} \tag{1} \\\
x_{k}\in R^{n},u_{k}\in R^{m},w_{k}\in R^{n}, y_{k}\in R^{l}, A\in R^{n\times n}, B\in R^{n\times m}, C\in R^{l \times n}
\end{gather}
$$
Expand All @@ -47,29 +47,29 @@ Then, when $k=2$, using also equation (2), we get

$$
\begin{align}
x_{2} & = Ax_{1} + Bu_{1} + w_{1} \\
& = A(Ax_{0} + Bu_{0} + w_{0}) + Bu_{1} + w_{1} \\
& = A^{2}x_{0} + ABu_{0} + Aw_{0} + Bu_{1} + w_{1} \\
& = A^{2}x_{0} + \begin{bmatrix}AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\ u_{1} \end{bmatrix} + \begin{bmatrix}A & I \end{bmatrix}\begin{bmatrix}w_{0}\\ w_{1} \end{bmatrix} \tag{3}
x_{2} & = Ax_{1} + Bu_{1} + w_{1} \\\
& = A(Ax_{0} + Bu_{0} + w_{0}) + Bu_{1} + w_{1} \\\
& = A^{2}x_{0} + ABu_{0} + Aw_{0} + Bu_{1} + w_{1} \\\
& = A^{2}x_{0} + \begin{bmatrix}AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\\ u_{1} \end{bmatrix} + \begin{bmatrix}A & I \end{bmatrix}\begin{bmatrix}w_{0}\\\ w_{1} \end{bmatrix} \tag{3}
\end{align}
$$

When $k=3$ , from equation (3)

$$
\begin{align}
x_{3} & = Ax_{2} + Bu_{2} + w_{2} \\
& = A(A^{2}x_{0} + ABu_{0} + Bu_{1} + Aw_{0} + w_{1} ) + Bu_{2} + w_{2} \\
& = A^{3}x_{0} + A^{2}Bu_{0} + ABu_{1} + A^{2}w_{0} + Aw_{1} + Bu_{2} + w_{2} \\
& = A^{3}x_{0} + \begin{bmatrix}A^{2}B & AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\ u_{1} \\ u_{2} \end{bmatrix} + \begin{bmatrix} A^{2} & A & I \end{bmatrix}\begin{bmatrix}w_{0}\\ w_{1} \\ w_{2} \end{bmatrix} \tag{4}
x_{3} & = Ax_{2} + Bu_{2} + w_{2} \\\
& = A(A^{2}x_{0} + ABu_{0} + Bu_{1} + Aw_{0} + w_{1} ) + Bu_{2} + w_{2} \\\
& = A^{3}x_{0} + A^{2}Bu_{0} + ABu_{1} + A^{2}w_{0} + Aw_{1} + Bu_{2} + w_{2} \\\
& = A^{3}x_{0} + \begin{bmatrix}A^{2}B & AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\\ u_{1} \\\ u_{2} \end{bmatrix} + \begin{bmatrix} A^{2} & A & I \end{bmatrix}\begin{bmatrix}w_{0}\\\ w_{1} \\\ w_{2} \end{bmatrix} \tag{4}
\end{align}
$$

If $k=n$ , then

$$
\begin{align}
x_{n} = A^{n}x_{0} + \begin{bmatrix}A^{n-1}B & A^{n-2}B & \dots & B \end{bmatrix}\begin{bmatrix}u_{0}\\ u_{1} \\ \vdots \\ u_{n-1} \end{bmatrix} + \begin{bmatrix} A^{n-1} & A^{n-2} & \dots & I \end{bmatrix}\begin{bmatrix}w_{0}\\ w_{1} \\ \vdots \\ w_{n-1} \end{bmatrix}
x_{n} = A^{n}x_{0} + \begin{bmatrix}A^{n-1}B & A^{n-2}B & \dots & B \end{bmatrix}\begin{bmatrix}u_{0}\\\ u_{1} \\\ \vdots \\\ u_{n-1} \end{bmatrix} + \begin{bmatrix} A^{n-1} & A^{n-2} & \dots & I \end{bmatrix}\begin{bmatrix}w_{0}\\\ w_{1} \\\ \vdots \\\ w_{n-1} \end{bmatrix}
\tag{5}
\end{align}
$$
Expand All @@ -78,8 +78,8 @@ Putting all of them together with (2) to (5) yields the following matrix equatio

$$
\begin{align}
\begin{bmatrix}x_{1}\\ x_{2} \\ x_{3} \\ \vdots \\ x_{n} \end{bmatrix} = \begin{bmatrix}A^{1}\\ A^{2} \\ A^{3} \\ \vdots \\ A^{n} \end{bmatrix}x_{0} + \begin{bmatrix}B & 0 & \dots & & 0 \\ AB & B & 0 & \dots & 0 \\ A^{2}B & AB & B & \dots & 0 \\ \vdots & \vdots & & & 0 \\ A^{n-1}B & A^{n-2}B & \dots & AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\ u_{1} \\ u_{2} \\ \vdots \\ u_{n-1} \end{bmatrix} \\ +
\begin{bmatrix}I & 0 & \dots & & 0 \\ A & I & 0 & \dots & 0 \\ A^{2} & A & I & \dots & 0 \\ \vdots & \vdots & & & 0 \\ A^{n-1} & A^{n-2} & \dots & A & I \end{bmatrix}\begin{bmatrix}w_{0}\\ w_{1} \\ w_{2} \\ \vdots \\ w_{n-1} \end{bmatrix}
\begin{bmatrix}x_{1}\\\ x_{2} \\\ x_{3} \\\ \vdots \\\ x_{n} \end{bmatrix} = \begin{bmatrix}A^{1}\\\ A^{2} \\\ A^{3} \\\ \vdots \\\ A^{n} \end{bmatrix}x_{0} + \begin{bmatrix}B & 0 & \dots & & 0 \\\ AB & B & 0 & \dots & 0 \\\ A^{2}B & AB & B & \dots & 0 \\\ \vdots & \vdots & & & 0 \\\ A^{n-1}B & A^{n-2}B & \dots & AB & B \end{bmatrix}\begin{bmatrix}u_{0}\\\ u_{1} \\\ u_{2} \\\ \vdots \\\ u_{n-1} \end{bmatrix} \\\ +
\begin{bmatrix}I & 0 & \dots & & 0 \\\ A & I & 0 & \dots & 0 \\\ A^{2} & A & I & \dots & 0 \\\ \vdots & \vdots & & & 0 \\\ A^{n-1} & A^{n-2} & \dots & A & I \end{bmatrix}\begin{bmatrix}w_{0}\\\ w_{1} \\\ w_{2} \\\ \vdots \\\ w_{n-1} \end{bmatrix}
\tag{6}
\end{align}
$$
Expand All @@ -88,7 +88,7 @@ In this case, the measurements (outputs) become; $y_{k}=Cx_{k}$, so

$$
\begin{align}
\begin{bmatrix}y_{1}\\ y_{2} \\ y_{3} \\ \vdots \\ y_{n} \end{bmatrix} = \begin{bmatrix}C & 0 & \dots & & 0 \\ 0 & C & 0 & \dots & 0 \\ 0 & 0 & C & \dots & 0 \\ \vdots & & & \ddots & 0 \\ 0 & \dots & 0 & 0 & C \end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2} \\ x_{3} \\ \vdots \\ x_{n} \end{bmatrix} \tag{7}
\begin{bmatrix}y_{1}\\\ y_{2} \\\ y_{3} \\\ \vdots \\\ y_{n} \end{bmatrix} = \begin{bmatrix}C & 0 & \dots & & 0 \\\ 0 & C & 0 & \dots & 0 \\\ 0 & 0 & C & \dots & 0 \\\ \vdots & & & \ddots & 0 \\\ 0 & \dots & 0 & 0 & C \end{bmatrix}\begin{bmatrix}x_{1}\\\ x_{2} \\\ x_{3} \\\ \vdots \\\ x_{n} \end{bmatrix} \tag{7}
\end{align}
$$

Expand Down Expand Up @@ -124,8 +124,8 @@ Substituting equation (8) into equation (9) and tidying up the equation for $U$.

$$
\begin{align}
J(U) &= (H(Fx_{0}+GU+SW)-Y_{ref})^{T}Q(H(Fx_{0}+GU+SW)-Y_{ref})+(U-U_{ref})^{T}R(U-U_{ref}) \\
& =U^{T}(G^{T}H^{T}QHG+R)U+2\left\{(H(Fx_{0}+SW)-Y_{ref})^{T}QHG-U_{ref}^{T}R\right\}U +(\rm{constant}) \tag{10}
J(U) &= (H(Fx_{0}+GU+SW)-Y_{ref})^{T}Q(H(Fx_{0}+GU+SW)-Y_{ref})+(U-U_{ref})^{T}R(U-U_{ref}) \\\
& =U^{T}(G^{T}H^{T}QHG+R)U+2\lbrace\{(H(Fx_{0}+SW)-Y_{ref})^{T}QHG-U_{ref}^{T}R\rbrace\}U +(\rm{constant}) \tag{10}
\end{align}
$$

Expand All @@ -141,9 +141,9 @@ For a nonlinear kinematic vehicle model, the discrete-time update equations are

$$
\begin{align}
x_{k+1} &= x_{k} + v\cos\theta_{k} \text{d}t \\
y_{k+1} &= y_{k} + v\sin\theta_{k} \text{d}t \\
\theta_{k+1} &= \theta_{k} + \frac{v\tan\delta_{k}}{L} \text{d}t \tag{11} \\
x_{k+1} &= x_{k} + v\cos\theta_{k} \text{d}t \\\
y_{k+1} &= y_{k} + v\sin\theta_{k} \text{d}t \\\
\theta_{k+1} &= \theta_{k} + \frac{v\tan\delta_{k}}{L} \text{d}t \tag{11} \\\
\delta_{k+1} &= \delta_{k} - \tau^{-1}\left(\delta_{k}-\delta_{des}\right)\text{d}t
\end{align}
$$
Expand Down Expand Up @@ -171,9 +171,9 @@ We make small angle assumptions for the following derivations of linear equation

$$
\begin{align}
y_{k+1} &= y_{k} + v\sin\theta_{k} \text{d}t \\
y_{k+1} &= y_{k} + v\sin\theta_{k} \text{d}t \\\
\theta_{k+1} &= \theta_{k} + \frac{v\tan\delta_{k}}{L} \text{d}t - \kappa_{r}v\cos\theta_{k}\text{d}t
\tag{12} \\
\tag{12} \\\
\delta_{k+1} &= \delta_{k} - \tau^{-1}\left(\delta_{k}-\delta_{des}\right)\text{d}t
\end{align}
$$
Expand Down Expand Up @@ -202,9 +202,9 @@ Substituting this equation into equation (12), and approximate $\Delta\delta$ to

$$
\begin{align}
\tan\delta &\simeq \tan\delta_{r} + \frac{\text{d}\tan\delta}{\text{d}\delta} \Biggm|_{\delta=\delta_{r}}\Delta\delta \\
&= \tan \delta_{r} + \frac{1}{\cos^{2}\delta_{r}}\Delta\delta \\
&= \tan \delta_{r} + \frac{1}{\cos^{2}\delta_{r}}\left(\delta-\delta_{r}\right) \\
\tan\delta &\simeq \tan\delta_{r} + \frac{\text{d}\tan\delta}{\text{d}\delta} \Biggm|_{\delta=\delta_{r}}\Delta\delta \\\
&= \tan \delta_{r} + \frac{1}{\cos^{2}\delta_{r}}\Delta\delta \\\
&= \tan \delta_{r} + \frac{1}{\cos^{2}\delta_{r}}\left(\delta-\delta_{r}\right) \\\
&= \tan \delta_{r} - \frac{\delta_{r}}{\cos^{2}\delta_{r}} + \frac{1}{\cos^{2}\delta_{r}}\delta
\end{align}
$$
Expand All @@ -213,9 +213,9 @@ Using this, $\theta_{k+1}$ can be expressed

$$
\begin{align}
\theta_{k+1} &= \theta_{k} + \frac{v\tan\delta_{k}}{L}\text{d}t - \kappa_{r}v\cos\delta_{k}\text{d}t \\
&\simeq \theta_{k} + \frac{v}{L}\text{d}t\left(\tan\delta_{r} - \frac{\delta_{r}}{\cos^{2}\delta_{r}} + \frac{1}{\cos^{2}\delta_{r}}\delta_{k} \right) - \kappa_{r}v\text{d}t \\
&= \theta_{k} + \frac{v}{L}\text{d}t\left(L\kappa_{r} - \frac{\delta_{r}}{\cos^{2}\delta_{r}} + \frac{1}{\cos^{2}\delta_{r}}\delta_{k} \right) - \kappa_{r}v\text{d}t \\
\theta_{k+1} &= \theta_{k} + \frac{v\tan\delta_{k}}{L}\text{d}t - \kappa_{r}v\cos\delta_{k}\text{d}t \\\
&\simeq \theta_{k} + \frac{v}{L}\text{d}t\left(\tan\delta_{r} - \frac{\delta_{r}}{\cos^{2}\delta_{r}} + \frac{1}{\cos^{2}\delta_{r}}\delta_{k} \right) - \kappa_{r}v\text{d}t \\\
&= \theta_{k} + \frac{v}{L}\text{d}t\left(L\kappa_{r} - \frac{\delta_{r}}{\cos^{2}\delta_{r}} + \frac{1}{\cos^{2}\delta_{r}}\delta_{k} \right) - \kappa_{r}v\text{d}t \\\
&= \theta_{k} + \frac{v}{L}\frac{\text{d}t}{\cos^{2}\delta_{r}}\delta_{k} - \frac{v}{L}\frac{\delta_{r}\text{d}t}{\cos^{2}\delta_{r}}
\end{align}
$$
Expand All @@ -224,7 +224,7 @@ Finally, the linearized time-varying model equation becomes;

$$
\begin{align}
\begin{bmatrix} y_{k+1} \\ \theta_{k+1} \\ \delta_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & v\text{d}t & 0 \\ 0 & 1 & \frac{v}{L}\frac{\text{d}t}{\cos^{2}\delta_{r}} \\ 0 & 0 & 1 - \tau^{-1}\text{d}t \end{bmatrix} \begin{bmatrix} y_{k} \\ \theta_{k} \\ \delta_{k} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ \tau^{-1}\text{d}t \end{bmatrix}\delta_{des} + \begin{bmatrix} 0 \\ -\frac{v}{L}\frac{\delta_{r}\text{d}t}{\cos^{2}\delta_{r}} \\ 0 \end{bmatrix}
\begin{bmatrix} y_{k+1} \\\ \theta_{k+1} \\\ \delta_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & v\text{d}t & 0 \\\ 0 & 1 & \frac{v}{L}\frac{\text{d}t}{\cos^{2}\delta_{r}} \\\ 0 & 0 & 1 - \tau^{-1}\text{d}t \end{bmatrix} \begin{bmatrix} y_{k} \\\ \theta_{k} \\\ \delta_{k} \end{bmatrix} + \begin{bmatrix} 0 \\\ 0 \\\ \tau^{-1}\text{d}t \end{bmatrix}\delta_{des} + \begin{bmatrix} 0 \\\ -\frac{v}{L}\frac{\delta_{r}\text{d}t}{\cos^{2}\delta_{r}} \\\ 0 \end{bmatrix}
\end{align}
$$

Expand All @@ -243,23 +243,23 @@ Using this equation, write down the update equation likewise (2) ~ (6)
$$
\begin{align}
\begin{bmatrix}
x_{1} \\ x_{2} \\ x_{3} \\ \vdots \\ x_{n}
x_{1} \\\ x_{2} \\\ x_{3} \\\ \vdots \\\ x_{n}
\end{bmatrix}
= \begin{bmatrix}
A_{1} \\ A_{1}A_{0} \\ A_{2}A_{1}A_{0} \\ \vdots \\ \prod_{i=0}^{n-1} A_{k}
A_{1} \\\ A_{1}A_{0} \\\ A_{2}A_{1}A_{0} \\\ \vdots \\\ \prod_{i=0}^{n-1} A_{k}
\end{bmatrix}
x_{0} +
\begin{bmatrix}
B_{0} & 0 & \dots & & 0 \\ A_{1}B_{0} & B_{1} & 0 & \dots & 0 \\ A_{2}A_{1}B_{0} & A_{2}B_{1} & B_{2} & \dots & 0 \\ \vdots & \vdots & &\ddots & 0 \\ \prod_{i=1}^{n-1} A_{k}B_{0} & \prod_{i=2}^{n-1} A_{k}B_{1} & \dots & A_{n-1}B_{n-1} & B_{n-1}
B_{0} & 0 & \dots & & 0 \\\ A_{1}B_{0} & B_{1} & 0 & \dots & 0 \\\ A_{2}A_{1}B_{0} & A_{2}B_{1} & B_{2} & \dots & 0 \\\ \vdots & \vdots & &\ddots & 0 \\\ \prod_{i=1}^{n-1} A_{k}B_{0} & \prod_{i=2}^{n-1} A_{k}B_{1} & \dots & A_{n-1}B_{n-1} & B_{n-1}
\end{bmatrix}
\begin{bmatrix}
u_{0} \\ u_{1} \\ u_{2} \\ \vdots \\ u_{n-1}
u_{0} \\\ u_{1} \\\ u_{2} \\\ \vdots \\\ u_{n-1}
\end{bmatrix} +
\begin{bmatrix}
I & 0 & \dots & & 0 \\ A_{1} & I & 0 & \dots & 0 \\ A_{2}A_{1} & A_{2} & I & \dots & 0 \\ \vdots & \vdots & &\ddots & 0 \\ \prod_{i=1}^{n-1} A_{k} & \prod_{i=2}^{n-1} A_{k} & \dots & A_{n-1} & I
I & 0 & \dots & & 0 \\\ A_{1} & I & 0 & \dots & 0 \\\ A_{2}A_{1} & A_{2} & I & \dots & 0 \\\ \vdots & \vdots & &\ddots & 0 \\\ \prod_{i=1}^{n-1} A_{k} & \prod_{i=2}^{n-1} A_{k} & \dots & A_{n-1} & I
\end{bmatrix}
\begin{bmatrix}
w_{0} \\ w_{1} \\ w_{2} \\ \vdots \\ w_{n-1}
w_{0} \\\ w_{1} \\\ w_{2} \\\ \vdots \\\ w_{n-1}
\end{bmatrix}
\end{align}
$$
Expand All @@ -280,7 +280,7 @@ As an example, let's determine the weight matrix $Q_{1}$ of the evaluation funct

$$
\begin{align}
Q_{1} = \begin{bmatrix} q_{e} & 0 & 0 & 0 & 0& 0 \\ 0 & q_{\theta} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & q_{e} & 0 & 0 \\ 0 & 0 & 0 & 0 & q_{\theta} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}
Q_{1} = \begin{bmatrix} q_{e} & 0 & 0 & 0 & 0& 0 \\\ 0 & q_{\theta} & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & q_{e} & 0 & 0 \\\ 0 & 0 & 0 & 0 & q_{\theta} & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}
\end{align}
$$

Expand All @@ -302,7 +302,7 @@ For instance, write $Q_{2}$ as follows for the $n=2$ system.

$$
\begin{align}
Q_{2} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & q_{d} & 0 & 0 & -q_{d} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -q_{d} & 0 & 0 & q_{d} \end{bmatrix}
Q_{2} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & q_{d} & 0 & 0 & -q_{d} \\\ 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & -q_{d} & 0 & 0 & q_{d} \end{bmatrix}
\end{align}
$$

Expand Down Expand Up @@ -340,35 +340,35 @@ We can also put constraints on the input deviations. As the derivative of steeri

$$
\begin{align}
\dot{u}_{min} < \dot{u} < \dot{u}_{max}
\dot u_{min} < \dot u < \dot u_{max}
\end{align}
$$

We discretize $\dot{u}$ as $\left(u_{k} - u_{k-1}\right)/\text{d}t$ and multiply both sides by dt, and the resulting constraint become linear and convex

$$
\begin{align}
\dot{u}_{min}\text{d}t < u_{k} - u_{k-1} < \dot{u}_{max}\text{d}t
\dot u_{min}\text{d}t < u_{k} - u_{k-1} < \dot u_{max}\text{d}t
\end{align}
$$

Along the prediction or control horizon, i.e for setting $n=3$

$$
\begin{align}
\dot{u}_{min}\text{d}t < u_{1} - u_{0} < \dot{u}_{max}\text{d}t \\
\dot{u}_{min}\text{d}t < u_{2} - u_{1} < \dot{u}_{max}\text{d}t
\dot u_{min}\text{d}t < u_{1} - u_{0} < \dot u_{max}\text{d}t \\\
\dot u_{min}\text{d}t < u_{2} - u_{1} < \dot u_{max}\text{d}t
\end{align}
$$

and aligning the inequality signs

$$
\begin{align}
u_{1} - u_{0} &< \dot{u}_{max}\text{d}t \\ +
u_{1} + u_{0} &< -\dot{u}_{min}\text{d}t \\
u_{2} - u_{1} &< \dot{u}_{max}\text{d}t \\ +
u_{2} + u_{1} &< - \dot{u}_{min}\text{d}t
u_{1} - u_{0} &< \dot u_{max}\text{d}t \\\ +
u_{1} + u_{0} &< -\dot u_{min}\text{d}t \\\
u_{2} - u_{1} &< \dot u_{max}\text{d}t \\\ +
u_{2} + u_{1} &< - \dot u_{min}\text{d}t
\end{align}
$$

Expand All @@ -384,6 +384,6 @@ Thus, putting this inequality to fit the form above, the constraints against $\d

$$
\begin{align}
\begin{bmatrix} -1 & 1 & 0 \\ 1 & -1 & 0 \\ 0 & -1 & 1 \\ 0 & 1 & -1 \end{bmatrix}\begin{bmatrix} u_{0} \\ u_{1} \\ u_{2} \end{bmatrix} \leq \begin{bmatrix} \dot{u}_{max}\text{d}t \\ -\dot{u}_{min}\text{d}t \\ \dot{u}_{max}\text{d}t \\ -\dot{u}_{min}\text{d}t \end{bmatrix}
\begin{bmatrix} -1 & 1 & 0 \\\ 1 & -1 & 0 \\\ 0 & -1 & 1 \\\ 0 & 1 & -1 \end{bmatrix}\begin{bmatrix} u_{0} \\\ u_{1} \\\ u_{2} \end{bmatrix} \leq \begin{bmatrix} \dot u_{max}\text{d}t \\\ -\dot u_{min}\text{d}t \\\ \dot u_{max}\text{d}t \\\ -\dot u_{min}\text{d}t \end{bmatrix}
\end{align}
$$

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