SYDE252 Lecture Notes - Eigenvalues And Eigenvectors, Linear Time-Invariant Theory, The Roots

114 views18 pages

Document Summary

Total response = zero-input response + zero-state response. In this lecture, we will focus on a linear system"s zero-input response, y0(t), which is the solution of the system equation when input x(t) = 0. (3. 1) by letting , where c and are constants. From maths course on differential equations, we may solve the equation: This is identical to the polynomial q(d) with replacing d, i. e. we can now express q( ) in factorized form: Therefore has n solutions: 1, 2, . , n, assuming that all i are (3. 2) distinct. Therefore, equation (3. 1): has n possible solutions: where are arbitrary constants. It can be shown that the general solution is the sum of all these terms: In order to determine the n arbitrary constants, we need to have n constraints (i. e. initial or boundary or auxiliary conditions). Q( ) is called the characteristic polynomial of the system. Q( ) = 0 is the characteristic equation of the system.

Get access

Grade+20% off
$8 USD/m$10 USD/m
Billed $96 USD annually
Grade+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
40 Verified Answers
Class+
$8 USD/m
Billed $96 USD annually
Class+
Homework Help
Study Guides
Textbook Solutions
Class Notes
Textbook Notes
Booster Class
30 Verified Answers

Related Documents