Problem 13.7 of Additional Exercises by Stephen Boyd: Energy consumption optimization problem

Question

Consider a unit mass electric vehicle with position $x(t)∈ℝ$, moving with velocity $v(t)$ and acceleration $a(t)$. The vehicle needs to be maneuvred from initial position of rest to final position $x(T)$ at time $t=T$ satisfying $x(T)≥x_f$. Requirements are $\dot v≤A$, $0≤v(t)≤V$, $v(T)≤v_f$.

Let $p(t),p_b(t),p_d(t)$ be power delivered by battery, power lost in braking, power lost due to drag respectively and $k(t)$ be kinetic energy of vehicle which is $v(t)^2/2$. Then, $$p(t)=\dot k(t)+p_b(t)+p_d(t)$$ where where, $p_d(t)=cv(t)^3$ and the power used for braking $p_b(t)$ can be controlled.

Take $T=nh$ and divide interval [$0,T]$ into $n$ subintervals of length $h$ to discretize the vehicle equations of motion, $\dot x=v,\dot v=a$. In discrete form we get following equations

$$x_{r+1}=x_r+h(v_r+v_{r+1})/2$$ $$v_{r+1}=v_r+ha_r$$ $$p_r=(k_{r+1}−kr)/h+p_{b,r}+p_{d,r}$$ $$k_r=v^2_r/2$$

Total energy consumed is $E=h \sum_{r=0}^{n}p_r$.

1. Formulate an optimization problem for minimum energy consumption. Write all variables and constraints on them. Indicate decision variables. Comment on the nature of the problem.

2. Solve the problem by taking $x_f=12 m, v_f=2 m/s, V=10 m/s, A=2 m/s^2, c=2, h=0.1, T=5 s$.

I am fairly new to convex optimization and solving these kinds of problems. I am not sure how to go around with this one. I think formulating the optimization problem is the tough part. I think I can approach solving the problem, once I know what exactly needs to be done. Thoughts and hints are appreciated.

Thanks

Edit: I recently found out that this problem is the same as problem 13.7 of the additional exercises in Stephen Boyd's book- 2020 edition.

Answer 1

$E = h \sum_{r=0}^{n}p_r$

This seems to be an error because it sums $n+1$ intervals of width $h$.

The optimization problem is almost written out in full already: \begin{align} \min \quad & h \sum_{r=0}^{n-1}p_r \\ \text{s.t.} \quad & x_{r+1}=x_r+h(v_r+v_{r+1})/2 \quad \forall r \in \{0,1,\ldots,n-1\} \\ &v_{r+1}=v_r+ha_r \quad \forall r \in \{0,1,\ldots,n-1\} \\ &p_r=(k_{r+1}−k_r)/h+p_{b,r}+p_{d,r} \quad \forall r \in \{0,1,\ldots,n-1\} \\ &p_{d,r}=cv_r^3 \quad \forall r \in \{0,1,\ldots,n-1\} \\ &k_r=v^2_r/2 \quad \forall r \in \{0,1,\ldots,n\} \\ & x_0 = 0 \\ & x_n \geq x_f \\ & a_r \leq A \quad \forall r \in \{0,1,\ldots,n-1\} \\ & 0 \leq v_r \leq V \quad \forall r \in \{0,1,\ldots,n\} \\ & v_n \leq v_f \\ & a,p,p_b,p_d \in \mathbb{R}^{n-1}, \; x,v,k \in \mathbb{R}^n \end{align}

The optimization variables and their dimensions are listed on the last line. The constraints $p_{d,r}=cv_r^3$ and $k_r=v^2_r/2$ are not convex, but they can be rephrased as inequalities: $p_{d,r} \geq cv_r^3$ and $k_r \geq v^2_r/2$, which are convex inequalities on the domain $v_r \geq 0$. At optimality, the inequalities will be tight.