Gradients and Directional Derivatives: A Closed Curve Perspective

A gentle introduction to the fundamentals of nonlinear dynamics and controls

Let us define the state vector $x = \begin{bmatrix}q \\ \dot{q}\end{bmatrix}$

Gradient Vectors at multiple points on the level curve $$V(x)=<x,x>=r^2$$ which is a closed curve.

The gradient is defined as the vector of partial derivatives of a function with respect to each of its independent variables. The gradient of a scalar function $V(x)$ is denoted as $\nabla V(x)$ and is defined as:

\[\nabla V(x) = \begin{bmatrix}\frac{\partial V(x)}{\partial q} \\ \frac{\partial V(x)}{\partial \dot{q}}\end{bmatrix}\]

note that upon plotting the gradient vectors at multiple points on the curve $V(x)=<x,x>=r^2$, we get a vector field that points in the direction of the steepest ascent of the closed curve $V(x)=r^2$.

Notice the gradient vector is orthogonal to the red level curve of the function $V(x)=r^2$ with $r=3$.

This is because the gradient vector is orthogonal to the level curve of the function $V(x)$ at a point $x$.

Gradient Vectors at multiple points on the level curve $$V(x)=<kx,kx>=1$$

where $k = \begin{bmatrix}1/a \\ 1/b \end{bmatrix}$

let us now conder the dynamics of a nonlinear system given by:

\[\dot{x} = f(x)\]

where $x \in \mathbb{R}^n$ and $f(x) \in \mathbb{R}^n$

The directional derivative of $V(x)$ along the trajectory of the system is given by:

\[\frac{dV(x)}{dt} = \frac{\partial V(x)}{\partial x} \frac{dx}{dt} = \nabla V(x)^T f(x)\]

The directional derivative of $V(x)$ along the trajectory of the system is the dot product of the gradient of $V(x)$ and the vector field $f(x)$

\[(or)\]

The directional derivative of $V(x)$ along the trajectory of the system is the rate of change of $V(x)$ along the trajectory of the system.

A key idea arises from the fact that the gradient at each point on the curve $V(x)$ is orthogonal to the level curve but when taken an inner product with the vector field $f(x)$ resolves into a scalar that indicates how aligned the gradient vector is with the vector field $f(x)$ at that point.

if the scalar is positive, $\frac{dV(x)}{dt} > 0$ then the gradient vector is aligned with the vector field $f(x)$ and hence the curve $V(x)$ is expanding in the direction of the vector field $f(x)$.

if the scalar is negative, $\frac{dV(x)}{dt} < 0$ then the gradient vector is anti-aligned with the vector field $f(x)$ and hence the curve $V(x)$ is squishing in the direction of the vector field $f(x)$.

if the scalar is zero, $\frac{dV(x)}{dt} = 0$ then the gradient vector is orthogonal to the vector field $f(x)$ and hence the curve $V(x)$ is neither expanding nor squishing in the direction of the vector field $f(x)$.

KEY-IDEA: Think of it as a measure that indicates the rate of squish/expansion of the curve $V(x)$ in the directions of the vector field $f(x)$.