Class Notes (834,936)
Canada (508,830)
SYDE 252 (2)

Zero Input Response

18 Pages
Unlock Document

Systems Design Engineering
SYDE 252
John Zelek

Lecture 3 Time-domain analysis: Zero-input Response (Lathi 2.1-2.2) Zero-input response basics  Remember that for a Linear System Total response = zero-input response + zero-state response  In this lecture, we will focus on a linear system’s zero-input response, y(t), which is the solution of the system equation when input x(t) = 0. 0 ⇒ ⇒ ⇒ General Solution to the zero-input response equation(1)  From maths course on differential equations, we may solve the equation: ……… (3.1) by letting , where c and λ are constants  Then: Substitute into (3.1) L2.2 p153 General Solution to the zero-input response equation(2)  We get: ……… (3.1)  This is identical to the polynomial Q(D) with λ replacing D, i.e.  We can now express Q(λ) in factorized form: ………. (3.2)  Therefore λ has N solutions: λ , λ , …. , λ , assuming that all λ are 1 2 N i distinct. L2.2 p152 General Solution to the zero-input response equation(3)  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). L2.2 p154 Characteristic Polynomial of a system  Q(λ) is called the characteristic polynomial of the system  Q(λ) = 0 is the characteristic equation of the system  The roots to the characteristic equation Q(λ) = 01 i2e. λ ,Nλ , …. , λ , are extremely important.  They are called by different names: • Characteristic values • Eigenvalues • Natural frequencies  The exponentials are the characteristic modes (also known as natural modes) of the system Characteristics modes determine the system’s behaviour L2.2 p154 Example 1 (1) Find y0(t), the zero-input component of the response, for a LTI system described by the following differential equation: (D ++D 2)=()y t ()Dx t when the initial conditions are  For zero-input response, we want to find the solution to:  The characteristic equation for this system is therefore:  The characteristic roots are therefore λ1= -1 and 2 = -2.  The zero-input response is L2.2 p155 Example 1 (2)  To find the two unknowns c1 and c2, we use the initial conditions  This yields to two simultaneous equations:  Solving this gives:  Therefore, the zero-input response of y(t) is given by: Repeated Characteristic Roots  The discussions so far assume that all characteristic roots are distinct. If there are repeated roots, the form of the solution is modified.  The solution of the equation: is given by:  In general, the characteristic modes for the differential equation: are:  The solution fo0 y (t) is Example 2 Find y0(t), the zero-input component of the respo2se for a LTI system described by the following differential equation: +6+= 9+) D3 5t) ( ) when the initial conditions are  The characteristic polynomial for this system is:  The repeated roots are therefore λ = -3 and λ = -3. 1 2  The zero-input response is  Now, determine the constants using the initial conditions giv1s c = 3 and c = 2. 2  Therefore: L2.2 p156 Complex Characteristic Roots
More Less

Related notes for SYDE 252

Log In


Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.