hw1
DownloadTélécharger
Actions
Vote :
ScreenshotAperçu
Informations
Catégorie :Category: nCreator TI-Nspire
Auteur Author: octa69
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 4.21 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 16:49:11
Mis à jour Updated: 18/03/2025 - 16:49:28
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538688
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 4.21 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 16:49:11
Mis à jour Updated: 18/03/2025 - 16:49:28
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538688
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
<<
-------------------------------------------------- Problem 1: State-Space Representation & Transfer Function -------------------------------------------------- Given: y(k) = -y(k-2) - u(k-1) - u(k-2) Step 1: Define State Variables x1(k) = y(k), x2(k) = y(k-1) Step 2: Express State Equations x1(k+1) = x2(k) x2(k+1) = -x1(k) - u(k) - u(k-1) Matrix form: [ x1(k+1) ] = [ 0 1 ] [ x1(k) ] + [ -1 ] u(k) + [ -1 ] u(k-1) [ x2(k+1) ] [ -1 0 ] [ x2(k) ] [ -1 ] Output equation: y(k) = [ 1 0 ] [ x1(k) ] -------------------------------------------------- Step 3: Compute Transfer Function from State-Space -------------------------------------------------- Formula: T(z) = C (zI - ¦)^(-1) Compute: (zI - ¦) = [ z -1 ] [ -1 z ] Determinant: det(zI - ¦) = z^2 + 1 Inverse: (zI - ¦)^(-1) = (1 / (z^2 + 1)) * [ z 1 ] [ -1 z ] Multiply by : (zI - ¦)^(-1) = (1 / (z^2 + 1)) * [ -z - 1 ] [ 1 - z ] Multiply by C: T(z) = (-z - 1) / (z^2 + 1) -------------------------------------------------- Step 4: Compute Transfer Function Using Z-Transform -------------------------------------------------- a) : Understanding the Given Difference Equation -------------------------------------------------- The original discrete-time system equation is: y(k) = -y(k-2) - u(k-1) - u(k-2) We want to find the **transfer function** T(z) = Y(z) / U(z) using the **Z-transform**. -------------------------------------------------- b) Apply the Z-Transform -------------------------------------------------- Using the **Z-transform properties**: 1. Basic Z-transform rule: Z{y(k-n)} = z^(-n) Y(z) 2. Apply the Z-transform to both sides: Z{y(k)} = -Z{y(k-2)} - Z{u(k-1)} - Z{u(k-2)} Since: Z{y(k)} = Y(z) Z{y(k-2)} = z^(-2) Y(z) Z{u(k-1)} = z^(-1) U(z) Z{u(k-2)} = z^(-2) U(z) We substitute these into the equation: Y(z) + z^(-2) Y(z) = - (z^(-1) U(z) + z^(-2) U(z)) -------------------------------------------------- c) Solve for Transfer Function T(z) -------------------------------------------------- Factor out Y(z) on the left-hand side: Y(z) (1 + z^(-2)) = - U(z) (z^(-1) + z^(-2)) Solve for T(z) = Y(z) / U(z): T(z) = (- (z^(-1) + z^(-2))) / (1 + z^(-2)) -------------------------------------------------- d): Convert to Standard Transfer Function Form -------------------------------------------------- Multiply both numerator and denominator by z^2 to remove negative exponents: T(z) = (- (z + 1)) / (z^2 + 1) -------------------------------------------------- e): Interpretation -------------------------------------------------- - The **numerator** (- (z + 1)) defines the **zeros** of the system. - The **denominator** (z^2 + 1) defines the **poles** of the system. - The **Z-transform method matches the previous state-space method**, confirming correctness. -------------------------------------------------- Final Verified Result -------------------------------------------------- T(z) = (-z - 1) / (z^2 + 1) This is the **transfer function**, which fully describes the system in the Z-domain --------------------------------------------------- --------------------------------------------------- -------------------------------------------------- Step 5: Stability Analysis -------------------------------------------------- Poles: Solve denominator equation: z^2 + 1 = 0 z = ±j Zeros: Solve numerator equation: -z - 1 = 0 z = -1 Since poles are on the unit circle, the system is marginally stable. -------------------------------------------------- Problem 2: Inverse of Transfer Function -------------------------------------------------- Perform long division: (-z - 1) / (z^2 + 1) Step 1: Divide leading term: -z / z^2 = -z^(-1) Step 2: Multiply: (-z^(-1)) * (z^2 + 1) = -z - z^(-1) Step 3: Subtract: (-z - 1) - (-z - z^(-1)) = -1 + z^(-1) Step 4: Repeat for remaining terms: -1 / z^2 = -z^(-3) (z^(-3)) * (z^2 + 1) = z^(-1) + z^(-3) (-1 + z^(-1)) - (z^(-1) + z^(-3)) = -1 - z^(-3) Repeat until: T(z) = -z^(-1) + z^(-3) - z^(-5) + ... -------------------------------------------------- Problem 3: Characteristic Equation & Stability -------------------------------------------------- Given: y(k) = 4y(k-1) - 4y(k-2) Step 1: Assume solution y(k) = z^k z^k = 4z^(k-1) - 4z^(k-2) Divide by z^(k-2): z^2 = 4z - 4 Characteristic equation: z^2 - 4z + 4 = 0 Solve: (z - 2)(z - 2) = 0 z = 2, 2 Since |z| > 1, the system is unstable. -------------------------------------------------- Step 4: Compute Solution with Initial Conditions -------------------------------------------------- Assume: y(k) = C1 * 2^k + C2 * k * 2^k Use given initial conditions: y(0) = 1 -> C1 = 1 y(1) = 0 -> 2C1 + 2C2 = 0 -> C2 = -1 Final solution: y(k) = 2^k (1 - k) -------------------------------------------------- Problem 4: Final V
[...]
>>
Compatible OS 3.0 et ultérieurs.
<<
-------------------------------------------------- Problem 1: State-Space Representation & Transfer Function -------------------------------------------------- Given: y(k) = -y(k-2) - u(k-1) - u(k-2) Step 1: Define State Variables x1(k) = y(k), x2(k) = y(k-1) Step 2: Express State Equations x1(k+1) = x2(k) x2(k+1) = -x1(k) - u(k) - u(k-1) Matrix form: [ x1(k+1) ] = [ 0 1 ] [ x1(k) ] + [ -1 ] u(k) + [ -1 ] u(k-1) [ x2(k+1) ] [ -1 0 ] [ x2(k) ] [ -1 ] Output equation: y(k) = [ 1 0 ] [ x1(k) ] -------------------------------------------------- Step 3: Compute Transfer Function from State-Space -------------------------------------------------- Formula: T(z) = C (zI - ¦)^(-1) Compute: (zI - ¦) = [ z -1 ] [ -1 z ] Determinant: det(zI - ¦) = z^2 + 1 Inverse: (zI - ¦)^(-1) = (1 / (z^2 + 1)) * [ z 1 ] [ -1 z ] Multiply by : (zI - ¦)^(-1) = (1 / (z^2 + 1)) * [ -z - 1 ] [ 1 - z ] Multiply by C: T(z) = (-z - 1) / (z^2 + 1) -------------------------------------------------- Step 4: Compute Transfer Function Using Z-Transform -------------------------------------------------- a) : Understanding the Given Difference Equation -------------------------------------------------- The original discrete-time system equation is: y(k) = -y(k-2) - u(k-1) - u(k-2) We want to find the **transfer function** T(z) = Y(z) / U(z) using the **Z-transform**. -------------------------------------------------- b) Apply the Z-Transform -------------------------------------------------- Using the **Z-transform properties**: 1. Basic Z-transform rule: Z{y(k-n)} = z^(-n) Y(z) 2. Apply the Z-transform to both sides: Z{y(k)} = -Z{y(k-2)} - Z{u(k-1)} - Z{u(k-2)} Since: Z{y(k)} = Y(z) Z{y(k-2)} = z^(-2) Y(z) Z{u(k-1)} = z^(-1) U(z) Z{u(k-2)} = z^(-2) U(z) We substitute these into the equation: Y(z) + z^(-2) Y(z) = - (z^(-1) U(z) + z^(-2) U(z)) -------------------------------------------------- c) Solve for Transfer Function T(z) -------------------------------------------------- Factor out Y(z) on the left-hand side: Y(z) (1 + z^(-2)) = - U(z) (z^(-1) + z^(-2)) Solve for T(z) = Y(z) / U(z): T(z) = (- (z^(-1) + z^(-2))) / (1 + z^(-2)) -------------------------------------------------- d): Convert to Standard Transfer Function Form -------------------------------------------------- Multiply both numerator and denominator by z^2 to remove negative exponents: T(z) = (- (z + 1)) / (z^2 + 1) -------------------------------------------------- e): Interpretation -------------------------------------------------- - The **numerator** (- (z + 1)) defines the **zeros** of the system. - The **denominator** (z^2 + 1) defines the **poles** of the system. - The **Z-transform method matches the previous state-space method**, confirming correctness. -------------------------------------------------- Final Verified Result -------------------------------------------------- T(z) = (-z - 1) / (z^2 + 1) This is the **transfer function**, which fully describes the system in the Z-domain --------------------------------------------------- --------------------------------------------------- -------------------------------------------------- Step 5: Stability Analysis -------------------------------------------------- Poles: Solve denominator equation: z^2 + 1 = 0 z = ±j Zeros: Solve numerator equation: -z - 1 = 0 z = -1 Since poles are on the unit circle, the system is marginally stable. -------------------------------------------------- Problem 2: Inverse of Transfer Function -------------------------------------------------- Perform long division: (-z - 1) / (z^2 + 1) Step 1: Divide leading term: -z / z^2 = -z^(-1) Step 2: Multiply: (-z^(-1)) * (z^2 + 1) = -z - z^(-1) Step 3: Subtract: (-z - 1) - (-z - z^(-1)) = -1 + z^(-1) Step 4: Repeat for remaining terms: -1 / z^2 = -z^(-3) (z^(-3)) * (z^2 + 1) = z^(-1) + z^(-3) (-1 + z^(-1)) - (z^(-1) + z^(-3)) = -1 - z^(-3) Repeat until: T(z) = -z^(-1) + z^(-3) - z^(-5) + ... -------------------------------------------------- Problem 3: Characteristic Equation & Stability -------------------------------------------------- Given: y(k) = 4y(k-1) - 4y(k-2) Step 1: Assume solution y(k) = z^k z^k = 4z^(k-1) - 4z^(k-2) Divide by z^(k-2): z^2 = 4z - 4 Characteristic equation: z^2 - 4z + 4 = 0 Solve: (z - 2)(z - 2) = 0 z = 2, 2 Since |z| > 1, the system is unstable. -------------------------------------------------- Step 4: Compute Solution with Initial Conditions -------------------------------------------------- Assume: y(k) = C1 * 2^k + C2 * k * 2^k Use given initial conditions: y(0) = 1 -> C1 = 1 y(1) = 0 -> 2C1 + 2C2 = 0 -> C2 = -1 Final solution: y(k) = 2^k (1 - k) -------------------------------------------------- Problem 4: Final V
[...]
>>
