Cone Discrete-Time and Continuous-Time Linear Systems

Basic Definitions

Definition 7.1. Let $P=[p^T_1,\ldots,p^T_n]^T\in\mathbb R^{n\times n}$ be nonsingular and p _k be the k-th (k = 1,…,n) its row. The set

$$ \mathcal{P}=\left\{x_i\in\mathbb R^n : \bigcap\limits_{k=1}^np_kx_i\geq0\right\} \quad\quad (7.1)$$

is called a linear cone of the state variable x _i generated by the matrix P.