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SYDE 252 (2)
Lecture

Zero Input Response

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Department
Systems Design Engineering
Course
SYDE 252
Professor
John Zelek
Semester
Fall

Description
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
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