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Catégorie :Category: nCreator TI-Nspire
Auteur Author: octa69
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.19 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 16:56:12
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538697
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.19 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 16:56:12
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538697
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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-------------------------------------------------- Problem 1: Discrete-Time Equivalent Using Eulers Methods -------------------------------------------------- Given: H(s) = (s + 1) / (s^2 + 6s + 7) T = 0.5 Step 1: Compute Forward Euler Approximation s H (z - 1) / T Substituting in H(s): H_d(z) = [(z - 1)/T + 1] / [( (z - 1)/T )^2 + 6((z - 1)/T) + 7] Expanding: H_d(z) = (2z - 1) / (4z^2 + 4z - 1) Step 2: Compute Backward Euler Approximation s H (z - 1) / (Tz) Substituting in H(s): H_d(z) = [(z - 1)/(Tz) + 1] / [( (z - 1)/(Tz) )^2 + 6((z - 1)/(Tz)) + 7] Expanding: H_d(z) = (3z^2 - 2z) / (23z^2 - 20z + 4) Step 3: Stability Analysis Solve denominator equations: Forward Euler Poles: -1.2071, 0.2071 (Unstable) Backward Euler Poles: 0.5578, 0.3118 (Stable) -------------------------------------------------- Problem 2: Zero-Pole Matching Method -------------------------------------------------- H(s) = (s - 1) / ((s + 1/2)(s + 1)) g period: T = 1 The relationship between the **s-domain** and **z-domain** is: z = e^(sT) Compute the discrete equivalents: - Zero of H(s): z_0 = e^(1) = 2.718 - Poles of H(s): z_1 = e^(-1/2) = 0.6065 z_2 = e^(-1) = 0.3679 The delay factor **k** is given by: k = n - m where: - **n** = Number of poles = **2** - **m** = Number of zeros = **1** Thus: k = 2 - 1 = 1 Th discrete transfer function is: H_z(z) = M * (z - z_0) / ((z - z_1)(z - z_2)) * (z + 1)^k Substituting values: H_z(z) = M * (z - 2.718) / ((z - 0.6065)(z - 0.3679)) * (z + 1)^1 Enforce **gain matching** at **s = 0** (equivalently, z = 1): H(0) =H_z(1) ComputeH(0) in the continuous domain: H(0) = (0 - 1) / ((0 + 1/2)(0 + 1)) H(0) = -1 / (1/2 * 1) H(0) = -2 Compute H_z(1): H_z(1) = M * (1 - 2.718) / ((1 - 0.6065)(1 - 0.3679)) * (1 + 1) H_z(1) = M * (-1.718) / ((0.3935)(0.6321)) * (2) Solve for M: M = -2 / (-1.718) * (1 / (0.3935 * 0.6321)) * (1/2) M = 0.14475 For *no delay (k = 0)**: H_z(z) = (0.14475 (z - 2.718)) / ((z - 0.6065)(z - 0.3679)) For **one delay (k = 1)**: H_z(z) = (0.2895 (z - 2.718) (z + 1)) / ((z - 0.6065)(z - 0.3679)) -------------------------------------------------- Problem 3: Digital PID Controller Using Tustins Method -------------------------------------------------- Given: D(s) = 1 + 2/s + 0.5s T = 1 Step 1: Apply Tustins Transformation s H (2/T) * ((z - 1) / (z + 1)) Step 2: Convert Each Term 0.5s (z - 1) / (z + 1) 2/s (z + 1) / (z - 1) Step 3: Combine D(z) = (3z^2 + 1) / (z^2 - 1) Step 4: Find Zeros Solve numerator = 0: z = ±0.5774j ------------------------------------------------ Problem 4: Digital Approximation Using Tustins Method -------------------------------------------------- Given: H(s) = (s + 1) / (s^2 + s + 1) T = 1 Step 1: Apply Tustins Transformation s H (2/T) * (z - 1) / (z + 1) Step 2: Convert H_T(z) = (3z^2 + 2z - 1) / (7z^2 - 6z + 3) ------------------------------------------------- Problem 5: Bode Plot and Sampling Rate Selection -------------------------------------------------- Given: G(s) = 1 / (s(s - 0.7)), D(s) = 3s / (s + 2) Step 1: Compute Transfer Function Use MATLAB to compute: Closed-loop system using feedback(D*G,1) Step 2: Detemine Sampling Rate T Find bandwidth from the Bode plot. Choose T such that 1/T > 30 * (bandwidth). Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
-------------------------------------------------- Problem 1: Discrete-Time Equivalent Using Eulers Methods -------------------------------------------------- Given: H(s) = (s + 1) / (s^2 + 6s + 7) T = 0.5 Step 1: Compute Forward Euler Approximation s H (z - 1) / T Substituting in H(s): H_d(z) = [(z - 1)/T + 1] / [( (z - 1)/T )^2 + 6((z - 1)/T) + 7] Expanding: H_d(z) = (2z - 1) / (4z^2 + 4z - 1) Step 2: Compute Backward Euler Approximation s H (z - 1) / (Tz) Substituting in H(s): H_d(z) = [(z - 1)/(Tz) + 1] / [( (z - 1)/(Tz) )^2 + 6((z - 1)/(Tz)) + 7] Expanding: H_d(z) = (3z^2 - 2z) / (23z^2 - 20z + 4) Step 3: Stability Analysis Solve denominator equations: Forward Euler Poles: -1.2071, 0.2071 (Unstable) Backward Euler Poles: 0.5578, 0.3118 (Stable) -------------------------------------------------- Problem 2: Zero-Pole Matching Method -------------------------------------------------- H(s) = (s - 1) / ((s + 1/2)(s + 1)) g period: T = 1 The relationship between the **s-domain** and **z-domain** is: z = e^(sT) Compute the discrete equivalents: - Zero of H(s): z_0 = e^(1) = 2.718 - Poles of H(s): z_1 = e^(-1/2) = 0.6065 z_2 = e^(-1) = 0.3679 The delay factor **k** is given by: k = n - m where: - **n** = Number of poles = **2** - **m** = Number of zeros = **1** Thus: k = 2 - 1 = 1 Th discrete transfer function is: H_z(z) = M * (z - z_0) / ((z - z_1)(z - z_2)) * (z + 1)^k Substituting values: H_z(z) = M * (z - 2.718) / ((z - 0.6065)(z - 0.3679)) * (z + 1)^1 Enforce **gain matching** at **s = 0** (equivalently, z = 1): H(0) =H_z(1) ComputeH(0) in the continuous domain: H(0) = (0 - 1) / ((0 + 1/2)(0 + 1)) H(0) = -1 / (1/2 * 1) H(0) = -2 Compute H_z(1): H_z(1) = M * (1 - 2.718) / ((1 - 0.6065)(1 - 0.3679)) * (1 + 1) H_z(1) = M * (-1.718) / ((0.3935)(0.6321)) * (2) Solve for M: M = -2 / (-1.718) * (1 / (0.3935 * 0.6321)) * (1/2) M = 0.14475 For *no delay (k = 0)**: H_z(z) = (0.14475 (z - 2.718)) / ((z - 0.6065)(z - 0.3679)) For **one delay (k = 1)**: H_z(z) = (0.2895 (z - 2.718) (z + 1)) / ((z - 0.6065)(z - 0.3679)) -------------------------------------------------- Problem 3: Digital PID Controller Using Tustins Method -------------------------------------------------- Given: D(s) = 1 + 2/s + 0.5s T = 1 Step 1: Apply Tustins Transformation s H (2/T) * ((z - 1) / (z + 1)) Step 2: Convert Each Term 0.5s (z - 1) / (z + 1) 2/s (z + 1) / (z - 1) Step 3: Combine D(z) = (3z^2 + 1) / (z^2 - 1) Step 4: Find Zeros Solve numerator = 0: z = ±0.5774j ------------------------------------------------ Problem 4: Digital Approximation Using Tustins Method -------------------------------------------------- Given: H(s) = (s + 1) / (s^2 + s + 1) T = 1 Step 1: Apply Tustins Transformation s H (2/T) * (z - 1) / (z + 1) Step 2: Convert H_T(z) = (3z^2 + 2z - 1) / (7z^2 - 6z + 3) ------------------------------------------------- Problem 5: Bode Plot and Sampling Rate Selection -------------------------------------------------- Given: G(s) = 1 / (s(s - 0.7)), D(s) = 3s / (s + 2) Step 1: Compute Transfer Function Use MATLAB to compute: Closed-loop system using feedback(D*G,1) Step 2: Detemine Sampling Rate T Find bandwidth from the Bode plot. Choose T such that 1/T > 30 * (bandwidth). Made with nCreator - tiplanet.org
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