Skip to content

Typos and 'Mayer term' --> 'terminal cost function'.md #4

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
fczyzyk wants to merge 1 commit into
base: main
Choose a base branch
from
Open

Typos and 'Mayer term' --> 'terminal cost function'.md #4

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
6 changes: 3 additions & 3 deletions ocp.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -120,10 +120,10 @@ This transformation is often helpful to simplify mathematical derivations (as we

Let's assume that an optimal control problem has been formulated in one of the forms presented earlier and has been given to us to solve. The following section explores numerical solutions applicable to these problems, focusing on trajectory optimization. Our goal is to output an optimal control (and state trajectory) based on the given cost function and dynamics structure. It's important to note that the methods presented here are not learning methods just yet; they don't involve ingesting data or inferring unknown quantities from it. However, these methods represent a central component of any decision-learning system, and we will later explore how learning concepts can be incorporated.

Before delving into the solution methods, let's consider an electric vehicle energy management problem which we will use this as a test bed throughout this section. Consider an electric vehicle traversing a planned route, where we aim to optimize its energy consumption over a 20-minute journey. Our simplified model represents the vehicle's state using two variables: $x_1$, the battery state of charge as a percentage, and $x_2$, denoting the vehicle's speed in meters per second. The control input $u$, ranging from -1 to 1, represents the motor power, with negative values indicating regenerative braking and positive values representing acceleration. The problem can be formally expressed as a mathematical program in Bolza form:
Before delving into the solution methods, let's consider an electric vehicle energy management problem which we will use as a test bed throughout this section. Consider an electric vehicle traversing a planned route, where we aim to optimize its energy consumption over a 20-minute journey. Our simplified model represents the vehicle's state using two variables: $x_1$, the battery state of charge as a percentage, and $x_2$, denoting the vehicle's speed in meters per second. The control input $u$, ranging from -1 to 1, represents the motor power, with negative values indicating regenerative braking and positive values representing acceleration. The problem can be formally expressed as a mathematical program in Bolza form:

$$ \begin{align*}
\min_{x, u} \quad & J = \underbrace{x_{T,1}^2 + x_{T,2}^2}_{\text{Mayer term}} + \underbrace{\sum_{t=1}^{T-1} 0.1(x_{t,1}^2 + x_{t,2}^2 + u_t^2)}_{\text{Lagrange term}} \\[2ex]
\min_{x, u} \quad & J = \underbrace{x_{T,1}^2 + x_{T,2}^2}_{\text{Bolza terminal cost function}} + \underbrace{\sum_{t=1}^{T-1} 0.1(x_{t,1}^2 + x_{t,2}^2 + u_t^2)}_{\text{Lagrange term}} \\[2ex]
\text{subject to:} \quad & x_{t+1} = f_t(x_t, u_t), \quad t = 1, \ldots, T-1 \\[1ex]
& x_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\[1ex]
& -1 \leq u_t \leq 1, \quad t = 1, \ldots, T-1 \\[1ex]
Expand Down Expand Up @@ -341,4 +341,4 @@ inequality constraint is about {glue:text}`ineq_constraint[None]:.2f` while that


```{bibliography}
```
```