Section 3.1 Norms and System Gains Theorem The 2-norm of T(s)is finite if and only if T(s)is strictly proper and has no poles on the imaginary axis.The oo-norm of T(s)is finite if and only if T(s)is proper and has no poles on the imaginary axis. Proof. Assume that T(s)is strictly proper and has no poles on the imaginary axis.Then the Bode magnitude plot rolls off at high frequencies.It is not hard to see that the plot of c/(rs+1)is higher than that of T(s)for sufficiently large positive c and sufficiently small positive r,but the 2-norm of c/(rs +1)equals c/V2T.Hence T(s)has finite 2-norm. The rest of the proof follows similar lines 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 6/69Section 3.1 Norms and System Gains Theorem The 2-norm of T(s) is finite if and only if T(s) is strictly proper and has no poles on the imaginary axis. The ∞-norm of T(s) is finite if and only if T(s) is proper and has no poles on the imaginary axis. Proof. Assume that T(s) is strictly proper and has no poles on the imaginary axis. Then the Bode magnitude plot rolls off at high frequencies. It is not hard to see that the plot of c/(τ s + 1) is higher than that of T(s) for sufficiently large positive c and sufficiently small positive τ , but the 2-norm of c/(τ s + 1) equals c/ √ 2τ . Hence T(s) has finite 2-norm. The rest of the proof follows similar lines. Zhang, W.D., CRC Press, 2011 Version 1.0 6/69