Section 3.1 Norms and System Gains Proof. 川Ti(s)+T2(s) 去/iTU)+TG-)Pdo Ih(sl呢+asl6+2Re[2∫rGT.G-)d. Now,it suffices to show the last integral equals zero.Convert it into a contour integral by closing the imaginary axis with an infinite radius semicircle in the LHP: 去∫0网Uojt=2动n(-sres 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 10/69
Section 3.1 Norms and System Gains Proof. kT1(s) + T2(s)k 2 2 = 1 2π Z |T1(jω) + T2(jω)| 2 dω = kT1(s)k 2 2 + kT2(s)k 2 2 + 2Re 1 2π Z T1(jω)T2(jω)dω . Now, it suffices to show the last integral equals zero. Convert it into a contour integral by closing the imaginary axis with an infinite radius semicircle in the LHP: 1 2π Z T1(jω)T2(jω)dω = 1 2πj I T1(−s)T2(s)ds. Zhang, W.D., CRC Press, 2011 Version 1.0 10/69
Section 3.1 Norms and System Gains Proof (ctd.1). Recall Cauchy's theorem,which concludes that if a function does not have poles in a simply connected region,then its integral on any closed contour contained in the region equals zero.Therefore, the right-hand side of the above equation equals zero. The oo-norm of T(s)equals the distance from the origin to the farthest point on the Nyquist plot of (s).It is also appears as the peak on the Bode magnitude plot of T(s) Property of o-norm: It is sub-multiplicative Ti(s)2(s)≤五(s)T2(s) 4口，+@，4定4定90C Zhang.W.D..CRC Press.2011 Version 1.0 11/69
Section 3.1 Norms and System Gains Proof (ctd.1). Recall Cauchy’s theorem, which concludes that if a function does not have poles in a simply connected region, then its integral on any closed contour contained in the region equals zero. Therefore, the right-hand side of the above equation equals zero. The ∞-norm of T(s) equals the distance from the origin to the farthest point on the Nyquist plot of T(s). It is also appears as the peak on the Bode magnitude plot of T(s) Property of ∞-norm: It is sub-multiplicative: kT1(s)T2(s)k∞ ≤ kT1(s)k∞kT2(s)k∞ Zhang, W.D., CRC Press, 2011 Version 1.0 11/69
Section 3.1 Norms and System Gains Proof (ctd.1). Recall Cauchy's theorem,which concludes that if a function does not have poles in a simply connected region,then its integral on any closed contour contained in the region equals zero.Therefore, the right-hand side of the above equation equals zero. The oo-norm of T(s)equals the distance from the origin to the farthest point on the Nyquist plot of T(s).It is also appears as the peak on the Bode magnitude plot of T(s) Property of oo-norm: It is sub-multiplicative: ‖T1(s)T2(s)川o≤‖T1(s)川∞l‖T2(s)川o 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 11/69
Section 3.1 Norms and System Gains Proof (ctd.1). Recall Cauchy’s theorem, which concludes that if a function does not have poles in a simply connected region, then its integral on any closed contour contained in the region equals zero. Therefore, the right-hand side of the above equation equals zero. The ∞-norm of T(s) equals the distance from the origin to the farthest point on the Nyquist plot of T(s). It is also appears as the peak on the Bode magnitude plot of T(s) Property of ∞-norm: It is sub-multiplicative: kT1(s)T2(s)k∞ ≤ kT1(s)k∞kT2(s)k∞ Zhang, W.D., CRC Press, 2011 Version 1.0 11/69
Section 3.1 Norms and System Gains System Gains An important question in design:If it is known how large the input is,how large is the output going to be?For example,the input is a signal with its 2-norm less than or equal to 1.What is the least upper bound on the 2-norm of the output? Answer:The answer to this question correlates with an important concept called system gain Table:System gains for SISO systems r(t)=(t) r(a ly(t) ly( ( 7(s 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 12/69
Section 3.1 Norms and System Gains System Gains An important question in design: If it is known how large the input is, how large is the output going to be? For example, the input is a signal with its 2-norm less than or equal to 1. What is the least upper bound on the 2-norm of the output? Answer: The answer to this question correlates with an important concept called system gain Table: System gains for SISO systems r(t) = δ(t) kr(t)k2 ky(t)k2 kT(s)k2 kT(s)k∞ ky(t)k∞ kT(t)k∞ kT(s)k2 Zhang, W.D., CRC Press, 2011 Version 1.0 12/69
Section 3.1 Norms and System Gains System Gains An important question in design:If it is known how large the input is,how large is the output going to be?For example,the input is a signal with its 2-norm less than or equal to 1.What is the least upper bound on the 2-norm of the output? Answer:The answer to this question correlates with an important concept called system gain Table:System gains for SISO systems r(t)=6(t) (t)l2 ly(t)1l2 T(s)2 IIT(s)ll o lly(t)llo T(t)川x I‖T(s)Il2 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 12/69
Section 3.1 Norms and System Gains System Gains An important question in design: If it is known how large the input is, how large is the output going to be? For example, the input is a signal with its 2-norm less than or equal to 1. What is the least upper bound on the 2-norm of the output? Answer: The answer to this question correlates with an important concept called system gain Table: System gains for SISO systems r(t) = δ(t) kr(t)k2 ky(t)k2 kT(s)k2 kT(s)k∞ ky(t)k∞ kT(t)k∞ kT(s)k2 Zhang, W.D., CRC Press, 2011 Version 1.0 12/69