How to compute 2-norm or infinity norm of following system?


i am confused whether to calculate using simple matrix theory "where it don't regard for s domain" or H2 and H-infinty norm. Thanks in advance. The problem is 4.4 from book Essentials of Robust Control


You probably know that matrix norms can be defined by the vector norms in the following way: \begin{equation} ||A||:= \max_{x\neq 0} \frac{||Ax||}{||x||} \end{equation} for a matrix $A$. So you just look up the definition for the infinity or 2-norm and plug it into the expression above. When doing this you will further realize that the following equalities hold: \begin{equation} ||A||_\infty = \max\limits_i \sum\limits_{j}|a_{ij}|\\ ||A||_2= \sqrt{\lambda_{max}(A^HA)} \end{equation} So for the infinity norm you sum up the absolute values in each row and take the the row with the largest sum. For the 2-norm you need to find the eigenvalue with the largest absolute value for the matrix $A^HA$. Note when you have the variable $s$ in your matrix the norms might also be functions of $s$.

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    $\begingroup$ I am afraid that this answer is missing some important context. In control theory there is a standard definition of norms for control systems that does not coincide with the definition of matrix norms; see e.g. Section 2.2 in users.abo.fi/htoivone/courses/robust/rob2.pdf . Clearly OP is confused about those. $\endgroup$ Jul 27 at 6:21
  • $\begingroup$ Yes you are right, I wasn't aware of that since I don't work in control theory. But your link should provide the definitions needed. $\endgroup$
    – Pepe
    Jul 27 at 11:20
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    $\begingroup$ thanks ..... i read the link provided by Federico Poloni, the above question would be solved using H2 and H-infinite norm definitions. $\endgroup$ Jul 27 at 20:37

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