State Space Modeling nonlinear terms of vehicle’s longitudinal motion

Question

My project is to build a software to help drivers achieve ecodriving thanks to optimal control based advices. Consequently, to feed a linear quadratic regulator, vehicle longitudinal equation of motion should be transformed into a state space form.

Considering a state having :

$$ \begin{align*} & \dot{s} = v\\ & \dot{v} = \mathit{u}_{t} - \mathit{u}_{b} - \frac{1}{m} 0.5 \mathit{\rho}_{air} A \mathit{C}_{d}v^2 - g(sin \theta(s) + \mathit{C}_{rr} cos \theta(s))\\ \\ \end{align*} $$

where

$\mathit{u}_{t}$ is tractive acceleration

$\mathit{u}_{B}$ is braking acceleration

$m$ is vehicle mass

$\mathit{\rho}_{air}$ is air density

$A$ is the vehicle frontal area

$\mathit{C}_{d}$ is the aerodynamic drag coefficient

$v$ is vehicle velocity

$g$ is gravity

$\theta(s)$ is road grade at position s

$\mathit{C}_{rr}$ is coefficient of rolling

How to linearize nonlinear terms $sin \theta(s)$, $cos \theta(s)$ and $v^2$ ?

Answer 1

I guess your project is a school project, therefore my answer will be generic.

Consider the nonlinear state space system $$ \dot{x} = f(x,u), $$ with state $x(t) \in \mathbb{R}^n$ and input $u(t) \in \mathbb{R}^k$. The linearized dynamics around the selected operating point $\left(x_0, u_0\right)$ are then given by $$ \dot{\tilde{x}} = A\tilde{x} + B\tilde{u}, $$ with $\tilde{x}(t) = x(t) - x_0$ and $\tilde{u}(t) = u(t) - u_0$, $A = \frac{\partial f(x_0,u_0)}{\partial x}$ and $B = \frac{\partial f(x_0,u_0)}{\partial u}$.

In your case, I would use the state vector $x = \begin{bmatrix}s \\ v\end{bmatrix}$ and input vector $ u = \begin{bmatrix} u_t \\ u_B \\ \theta \end{bmatrix}$.

Answer 2

In the real world cos(theta)=1 and sin(theta)=theta in radians, assuming you aren't rockhopping in a Jeep.