Optimal and Robust Control, 2nd Edition

Appendix A. Norms

In this Appendix the main norms used in the text or, in any case, significant for the control theory, are summarized.

Norm of a Vector

The norm of a vector $x\in {ℂ}^n$ is a function $f:{ℂ}^n\to ℝ$ satisfying the following properties:

• $f(x)\ge 0$ ;

• $f(x)=0\leftrightarrow x=0$ ;

• $f(x+y)\le f(x)+f(y)$ with x and $y\in {ℂ}^n$ ;

• $f(\alpha x)=\left|\alpha \right|f(x)$ with $\alpha \in ℝ$.

Let $x=\left[\begin{array}{l}{x}_1\hfill \\ x_2\hfill \\ ⋮\hfill \\ x_n\hfill \end{array}\right]$, the 1-norm is:

$$\begin{array}{l}\mathrm{\Vert }x{\mathrm{\Vert }}_1={\displaystyle \sum _{i=1}^n\left|x_i\right|}\hfill \end{array}$$(A.1)

The 2-norm is instead defined by:

$$\begin{array}{l}\mathrm{\Vert }x{\mathrm{\Vert }}_2={\left({\displaystyle \sum _{i=1}^n|x_i{|}^2}\right)}^{\frac{1}{2}}\hfill \end{array}$$(A.2)

The 1-norm and the 2-norm are special cases of a more general family of norms, the p-norm of a vector:

$$\begin{array}{l}\mathrm{\Vert }x{\mathrm{\Vert }}_p={\left({\displaystyle \sum _{i=1}^n|x_i{|}^p}\right)}^{\frac{1}{p}}\hfill \end{array}$$(A.3)

Another commonly used norm of the p-norm family is the ∞-norm defined as:

$$\begin{array}{l}\mathrm{\Vert }x{\mathrm{\Vert }}_{\infty }={\max }_i\left|x_i\right|\hfill \end{array}$$(A.4)

Norm of a Matrix

By considering the role of matrices as linear operators, the norm of a matrix ...