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


1 Answer 1


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. $\endgroup$ Commented Jul 27, 2021 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
    Commented Jul 27, 2021 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. $\endgroup$ Commented Jul 27, 2021 at 20:37
  • $\begingroup$ @FedericoPoloni: I'm not sure if you got my note, hoping you would find time to write an Answer based on your upvoted Comment above. $\endgroup$
    – hardmath
    Commented Aug 23, 2022 at 15:40
  • $\begingroup$ @hardmath I can see your answer if I go to my account page and click on "helpful flags", but it is without signature (I don't know who wrote it), and I didn't get any notifications about it. $\endgroup$ Commented Aug 23, 2022 at 16:14

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