2-norm & infinity norms of a system are dependent or independent of delay?












2












$begingroup$


We are given some transfer function $G(s)$



And the 2-norm and $infty$ norms are given by



$$ {||G||}_2 = (frac{1}{2pi}int_{-infty}^{infty} |G(jomega)|^{2} text{d}omega)^{1/2} $$



$${||G||}_infty = text{sup}|G(jomega)| $$



I want to see if either or both of these norms are sensitive to some delay.



We can define the new system
$$H(s) = G(s) text{e}^{-stau}$$



And compare
${||G||}$ to ${||H||}$



I am trying to do this using the definition of norms that I provided, but I am not getting anywhere.



Is there an easier way to approach this? Perhaps in the time domain?






definite-integrals dynamical-systems norm







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    We are given some transfer function $G(s)$



    And the 2-norm and $infty$ norms are given by



    $$ {||G||}_2 = (frac{1}{2pi}int_{-infty}^{infty} |G(jomega)|^{2} text{d}omega)^{1/2} $$



    $${||G||}_infty = text{sup}|G(jomega)| $$



    I want to see if either or both of these norms are sensitive to some delay.



    We can define the new system
    $$H(s) = G(s) text{e}^{-stau}$$



    And compare
    ${||G||}$ to ${||H||}$



    I am trying to do this using the definition of norms that I provided, but I am not getting anywhere.



    Is there an easier way to approach this? Perhaps in the time domain?






    definite-integrals dynamical-systems norm







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      1



      $begingroup$


      We are given some transfer function $G(s)$



      And the 2-norm and $infty$ norms are given by



      $$ {||G||}_2 = (frac{1}{2pi}int_{-infty}^{infty} |G(jomega)|^{2} text{d}omega)^{1/2} $$



      $${||G||}_infty = text{sup}|G(jomega)| $$



      I want to see if either or both of these norms are sensitive to some delay.



      We can define the new system
      $$H(s) = G(s) text{e}^{-stau}$$



      And compare
      ${||G||}$ to ${||H||}$



      I am trying to do this using the definition of norms that I provided, but I am not getting anywhere.



      Is there an easier way to approach this? Perhaps in the time domain?






      definite-integrals dynamical-systems norm







      share|cite|improve this question











      $endgroup$




      We are given some transfer function $G(s)$



      And the 2-norm and $infty$ norms are given by



      $$ {||G||}_2 = (frac{1}{2pi}int_{-infty}^{infty} |G(jomega)|^{2} text{d}omega)^{1/2} $$



      $${||G||}_infty = text{sup}|G(jomega)| $$



      I want to see if either or both of these norms are sensitive to some delay.



      We can define the new system
      $$H(s) = G(s) text{e}^{-stau}$$



      And compare
      ${||G||}$ to ${||H||}$



      I am trying to do this using the definition of norms that I provided, but I am not getting anywhere.



      Is there an easier way to approach this? Perhaps in the time domain?






      definite-integrals dynamical-systems norm




      definite-integrals dynamical-systems norm






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 1 at 19:49







      Chemical Engineer

















      asked Jan 28 at 2:20









      Chemical EngineerChemical Engineer

      597




      597






















          1 Answer
          1






          active

          oldest

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          0





          +50







          $begingroup$

          Note that deriving the $H_2$ or $H_infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.



          On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.



          For the $H_infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.



          Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.






          share|cite|improve this answer









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            active

            oldest

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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0





            +50







            $begingroup$

            Note that deriving the $H_2$ or $H_infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.



            On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.



            For the $H_infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.



            Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.






            share|cite|improve this answer









            $endgroup$


















              0





              +50







              $begingroup$

              Note that deriving the $H_2$ or $H_infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.



              On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.



              For the $H_infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.



              Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.






              share|cite|improve this answer









              $endgroup$
















                0





                +50







                0





                +50



                0




                +50



                $begingroup$

                Note that deriving the $H_2$ or $H_infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.



                On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.



                For the $H_infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.



                Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.






                share|cite|improve this answer









                $endgroup$



                Note that deriving the $H_2$ or $H_infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.



                On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.



                For the $H_infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.



                Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 4 at 16:26









                polfosolpolfosol

                5,93931945




                5,93931945






























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