Linear System Theory and Design_chi-tsong chen--Solution Manual答案外文.pdf

Linear System Theory and Design S LING QING 2.1 Consider the memoryless system with characteristics shown in Fig 2.19, in which u denotes the input and y the output. Which of them is a linear system? Is it possible to introduce a new output so that the system in Fig 2.19(b) is linear? Figure 2.19 Translation: 考虑具有图2.19 中表示的特性的无记忆系统。其中u 表示输入，y 表示输出。 下面哪一个是线性系统？可以找到一个新的输出，使得图 2.19(b) 中的系统是线性 的吗？ Answer: The input-output relation in Fig 2.1(a) can be described as: y a *u Here a is a constant. It is a memoryless system. Easy to testify that it is a linear system. The input-output relation in Fig 2.1(b) can be described as: y a *u +b Here a and b are all constants. Testify whether it has the property of additivity. Let: y 1 a *u1 +b y 2 a *u2 +b then: (y y ) a * (u u ) 2 *b + + + 1 2 1 2 So it does not has the property of additivity, therefore, is not a linear system. But we can introduce a new output so that it is linear. Let: z y −b z a *u z is the new output introduced. Easy to testify that it is a linear system. The input-output relation in Fig 2.1(c) can be described as: y a(u) *u a(u) is a function of input u. Choose two different input, get the outputs: y 1 a1 *u1 1 Linear System Theory and Design S LING QING