# Why does the negative reward function in LQR encourage convergence to the origin?

I was reading Stanford's CS 229 materials on Linear Quadratic Regulation (LQR) (Lecture note 13, YouTube Lecture 18, around minute 36), and it mentions that:

[...] the quadratic formulation of the reward is equivalent to saying that we want our state to be close to the origin. For example, if $$U_t = I_n$$(the identity matrix) and $$W_t = I_d$$, then $$R_t = -\Vert(s_t)\Vert^2-\Vert(a_t)\Vert^2$$, meaning that we want to take smooth actions (small norm of $$a_t$$) to go back to the origin (small norm of $$s_t$$).

Why is that? Why the norm of the origin is smaller? $$s_t$$ for any time point are just a vector of n dimension which can take any value, right?

• The norm of $s_t$ is smaller the closer it is to the origin. Feb 18 '18 at 7:35
• Please provide links. Nov 29 '21 at 22:39

## 1 Answer

Minimizing $$s^TUs=s^Ts=\|s\|^2$$ minimizes the distance from the origin, because the norm of a vector is its Euclidean distance from the origin:

$$\|s\|= \sqrt{s_1^2+s_2^2+\dots+s_n^2} = \sqrt{(s_1-0)^2+(s_2-0)^2+\dots+(s_n-0)^2}$$