Section 3.1 Norms and System Gains Frequently Used Signal Norms 1-norm.The 1-norm of r(t)is the integral of its absolute value: r()=Ir(e)l de 2-norm.The 2-norm of r(t)is 11/2 Ir(2()d Suppose that r(t)is the current through a 10 resistor.The instantaneous power equals r(t)2 and the energy equals (t)2 x-norm.The oo-norm of r(t)is the least upper bound of its absolute value: r(t)oo=sup r(t) 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 4/69
Section 3.1 Norms and System Gains Frequently Used Signal Norms 1-norm. The 1-norm of r(t) is the integral of its absolute value: kr(t)k1 := Z ∞ −∞ |r(t)| dt 2-norm. The 2-norm of r(t) is kr(t)k2 := Z ∞ −∞ r 2 (t) dt1/2 Suppose that r(t) is the current through a 1Ω resistor. The instantaneous power equals r(t) 2 and the energy equals kr(t)k 2 2 ∞-norm. The ∞-norm of r(t) is the least upper bound of its absolute value: kr(t)k∞ := sup t |r(t)| Zhang, W.D., CRC Press, 2011 Version 1.0 4/69
Section 3.1 Norms and System Gains Frequently Used Signal Norms 1-norm.The 1-norm of r(t)is the integral of its absolute value: r()=Ir(e)l de 2-norm.The 2-norm of r(t)is 11/2 Ir(()d Suppose that r(t)is the current through a 10 resistor.The instantaneous power equals r(t)2 and the energy equals lr(t)2 oo-norm.The oo-norm of r(t)is the least upper bound of its absolute value: lr(t)川o:=supr(t)l 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 4/69
Section 3.1 Norms and System Gains Frequently Used Signal Norms 1-norm. The 1-norm of r(t) is the integral of its absolute value: kr(t)k1 := Z ∞ −∞ |r(t)| dt 2-norm. The 2-norm of r(t) is kr(t)k2 := Z ∞ −∞ r 2 (t) dt1/2 Suppose that r(t) is the current through a 1Ω resistor. The instantaneous power equals r(t) 2 and the energy equals kr(t)k 2 2 ∞-norm. The ∞-norm of r(t) is the least upper bound of its absolute value: kr(t)k∞ := sup t |r(t)| Zhang, W.D., CRC Press, 2011 Version 1.0 4/69
Section 3.1 Norms and System Gains Frequently Used System Norms Consider a linear time-invariant and causal system T(t),of which the input is r(t)and the output is y(t): y(t)=T(t)*r(t) T(t-T)r(T)dr Let T(s)denote the transfer function of T(t).Norms can also be defined for the system T(s): 2-norm. 11/2 Irse=层ITVP d oo-norm. I‖T(s)Ilo:=sup|TUw)川 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 5/69
Section 3.1 Norms and System Gains Frequently Used System Norms Consider a linear time-invariant and causal system T(t), of which the input is r(t) and the output is y(t): y(t) = T(t) ∗ r(t) = Z ∞ −∞ T(t − τ )r(τ ) dτ Let T(s) denote the transfer function of T(t). Norms can also be defined for the system T(s): 2-norm. kT(s)k2 := 1 2π Z ∞ −∞ |T(jω)| 2 dω 1/2 ∞-norm. kT(s)k∞ := sup ω |T(jω)| Zhang, W.D., CRC Press, 2011 Version 1.0 5/69
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 6/69
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 ∞-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
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/(Ts+1)is higher than that of T(s)for sufficiently large positive c and sufficiently small positive T,but the 2-norm of c/(Ts+1)equals c/v27.Hence T(s)has finite 2-norm. The rest of the proof follows similar lines. 4口，44定4生，定QC Zhang.W.D..CRC Press.2011 Version 1.0 6/69
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 ∞-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