Section 3.1 Norms and System Gains 3.1 Norms and System Gains Why Do We Need Norms The performance of a control system is usually specified in terms of the "size"of certain signals. Solution:The "size"of a signal can be defined by introducing norms."The signal is small"means its norm is small Consider a signal r(t).A norm is a nonnegative real number. denoted by r(t).that satisfying the following properties: r(t)=0 if and only if r(t)=0.vt. 3or(t)=ar(t)o is any real number. n(e)+2(c)≤n()+2(e 4口，+@，4定4=定0C Zhang.W.D..CRC Press.2011 Version 1.0 3/69Section 3.1 Norms and System Gains 3.1 Norms and System Gains Why Do We Need Norms The performance of a control system is usually specified in terms of the ”size” of certain signals. Solution:The ”size” of a signal can be defined by introducing norms. “The signal is small” means its norm is small Consider a signal r(t). A norm is a nonnegative real number, denoted by kr(t)k, that satisfying the following properties: 1 kr(t)k = 0 if and only if r(t) = 0 , ∀t. 2 kαr(t)k = |α|kr(t)k, α is any real number. 3 kr1(t) + r2(t)k ≤ kr1(t)k + kr2(t)k. Zhang, W.D., CRC Press, 2011 Version 1.0 3/69