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Catégorie :Category: nCreator TI-Nspire
Auteur Author: octa69
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 7.65 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 17:06:14
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538704
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 7.65 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 17:06:14
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538704
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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-------------------------------------------------- Problem 1(a): Compute G_ZOH(z) for G(s) = 1 / [(s+1)(s-1)] -------------------------------------------------- Step 1: Define the ZOH Equivalent Transfer Function G_ZOH(z) = (1 - z^(-1)) Z{ G(s) / s } We need to compute: G(s)/s = 1 / [s(s+1)(s-1)] -------------------------------------------------- Step 2: Perform Partial Fraction Decomposition -------------------------------------------------- Express: 1 / [s(s+1)(s-1)] = A/s + B/(s+1) + C/(s-1) Multiply both sides by s(s+1)(s-1): 1 = A(s+1)(s-1) + B(s)(s-1) + C(s)(s+1) Expanding: A(s^2 - 1) + B(s^2 - s) + C(s^2 + s) = 1 Grouping terms: (A + B + C)s^2 + (-B + C)s + (-A) = 1 Setting up equations: 1. A + B + C = 0 2. -B + C = 0 3. -A = 1 Solving: A = -1 B + C = 1 B = C Since B = C: 2B = 1 B = 1/2 C = 1/2 Thus: 1 / [s(s+1)(s-1)] = -1/s + (1/2)/(s+1) + (1/2)/(s-1) -------------------------------------------------- Step 3: Compute the Inverse Laplace Transform -------------------------------------------------- Using: L^(-1){ 1/s } = 1 L^(-1){ 1/(s+a) } = e^(-at) Applying: g(t) = -1 + (1/2)e^(-t) + (1/2)e^(t) -------------------------------------------------- Step 4: Compute the Z-Transform -------------------------------------------------- Using: Z{ 1 } = 1 / (1 - z^(-1)) Z{ e^(-t) } = 1 / (1 - e^(-T)z^(-1)) Z{ e^(t) } = 1 / (1 - e^(T)z^(-1)) Substituting: Z{ g(t) } = -Z{ 1 } + (1/2)Z{ e^(-t) } + (1/2)Z{ e^t } Z{ g(t) } = -[1 / (1 - z^(-1))] + (1/2) [1 / (1 - e^(-T)z^(-1))] + (1/2) [1 / (1 - e^(T)z^(-1))] Applying: G_ZOH(z) = (1 - z^(-1)) Z{ g(t) } Expanding: G_ZOH(z) = (1 - z^(-1)) * [-1 / (1 - z^(-1)) + (1/2) / (1 - e^(-T)z^(-1)) + (1/2) / (1 - e^(T)z^(-1))] Final simplification provides the discrete-time transfer function. -------------------------------------------------- Problem 1(b): Compute x(k+1) = ¦ x(k) + u(k) -------------------------------------------------- Given: = F x + G u where: F = [ 1 1 ] [ 0 1 ] G = [ 1 ] [ 0 ] Sampling period: T = À/2 -------------------------------------------------- Step 1: Compute ¦ = e^(FT) -------------------------------------------------- The discrete-time state transition matrix is: ¦ = e^(F T) Since F is not diagonal, use the matrix exponential: e^(FT) = I + F T + (F T)^2 / 2! + (F T)^3 / 3! + ... Computing: ¦ = [ e^T T e^T ] [ 0 e^T ] Substituting T = À/2: ¦ = [ e^(À/2) (À/2)e^(À/2) ] [ 0 e^(À/2) ] -------------------------------------------------- Step 2: Compute = +(0 to T) e^(FÄ) G dÄ -------------------------------------------------- Using integral of matrix exponentials: = [ e^T - 1 ] [ 0 ] Substituting T = À/2: = [ e^(À/2) - 1 ] [ 0 ] -------------------------------------------------- Final Answer -------------------------------------------------- The discrete-time equivalent system is: x(k+1) = ¦ x(k) + u(k) where: ¦ = [ e^(À/2) (À/2)e^(À/2) ] [ 0 e^(À/2) ] = [ e^(À/2) - 1 ] [ 0 ] ---------------------------------------------------- Digital PI Controller Design - Fully Expanded and Step-by-Step Solution -------------------------------------------------- Given Information -------------------------------------------------- We need to design a **digital PI (Proportional-Integral) controller** of the form: D(z) = k_p + (k_i T z) / (z - 1) for the open-loop system: G(z) = z / (z - 1) where the **sampling time is T = 1**, such that the **closed-loop system transfer function** matches: H(z) = (2z) / (z^2 + 1) The goal is to determine the **controller gains ( k_p ) and ( k_i )** to achieve this desired system response. -------------------------------------------------- Step 1: Derive the Closed-Loop Transfer Function ------------------------------------------------ The **closed-loop transfer function** for a **unity feedback** system is given by: H(z) = (D(z) G(z)) / (1 + D(z) G(z)) ### **Substituting ( G(z) ):** Since we are given: G(z) = z / (z - 1) We substitute this into the closed-loop equation: H(z) = (D(z) * (z / (z - 1))) / (1 + D(z) * (z / (z - 1))) ### **Expanding ( D(z) ):** The **digital PI controller** is defined as: D(z) = k_p + (k_i T z) / (z - 1) Substituting this into the equation: H(z) = ((k_p + (k_i T z) / (z - 1)) * (z / (z - 1))) / (1 + (k_p + (k_i T z) / (z - 1)) * (z / (z - 1))) ### **Multiply numerator and denominator by ( (z - 1) ):** To clear the fractions, we multiply both the numerator and denominator by ( (z - 1) ): H(z) = ((k_p (z - 1) + k_i T z) * z) / ((z - 1) + k_p z (z - 1) + k_i T z) ### **Expanding the terms:** Expanding the numerator: k_p (z - 1) + k_i T z = k_p z - k_p + k_i T z Multiplying by ( z ): (k_p z - k_p + k_i T z) * z = k_p z^2 - k_p z + k_i T z^2 Expanding the denominator: (z -
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Compatible OS 3.0 et ultérieurs.
<<
-------------------------------------------------- Problem 1(a): Compute G_ZOH(z) for G(s) = 1 / [(s+1)(s-1)] -------------------------------------------------- Step 1: Define the ZOH Equivalent Transfer Function G_ZOH(z) = (1 - z^(-1)) Z{ G(s) / s } We need to compute: G(s)/s = 1 / [s(s+1)(s-1)] -------------------------------------------------- Step 2: Perform Partial Fraction Decomposition -------------------------------------------------- Express: 1 / [s(s+1)(s-1)] = A/s + B/(s+1) + C/(s-1) Multiply both sides by s(s+1)(s-1): 1 = A(s+1)(s-1) + B(s)(s-1) + C(s)(s+1) Expanding: A(s^2 - 1) + B(s^2 - s) + C(s^2 + s) = 1 Grouping terms: (A + B + C)s^2 + (-B + C)s + (-A) = 1 Setting up equations: 1. A + B + C = 0 2. -B + C = 0 3. -A = 1 Solving: A = -1 B + C = 1 B = C Since B = C: 2B = 1 B = 1/2 C = 1/2 Thus: 1 / [s(s+1)(s-1)] = -1/s + (1/2)/(s+1) + (1/2)/(s-1) -------------------------------------------------- Step 3: Compute the Inverse Laplace Transform -------------------------------------------------- Using: L^(-1){ 1/s } = 1 L^(-1){ 1/(s+a) } = e^(-at) Applying: g(t) = -1 + (1/2)e^(-t) + (1/2)e^(t) -------------------------------------------------- Step 4: Compute the Z-Transform -------------------------------------------------- Using: Z{ 1 } = 1 / (1 - z^(-1)) Z{ e^(-t) } = 1 / (1 - e^(-T)z^(-1)) Z{ e^(t) } = 1 / (1 - e^(T)z^(-1)) Substituting: Z{ g(t) } = -Z{ 1 } + (1/2)Z{ e^(-t) } + (1/2)Z{ e^t } Z{ g(t) } = -[1 / (1 - z^(-1))] + (1/2) [1 / (1 - e^(-T)z^(-1))] + (1/2) [1 / (1 - e^(T)z^(-1))] Applying: G_ZOH(z) = (1 - z^(-1)) Z{ g(t) } Expanding: G_ZOH(z) = (1 - z^(-1)) * [-1 / (1 - z^(-1)) + (1/2) / (1 - e^(-T)z^(-1)) + (1/2) / (1 - e^(T)z^(-1))] Final simplification provides the discrete-time transfer function. -------------------------------------------------- Problem 1(b): Compute x(k+1) = ¦ x(k) + u(k) -------------------------------------------------- Given: = F x + G u where: F = [ 1 1 ] [ 0 1 ] G = [ 1 ] [ 0 ] Sampling period: T = À/2 -------------------------------------------------- Step 1: Compute ¦ = e^(FT) -------------------------------------------------- The discrete-time state transition matrix is: ¦ = e^(F T) Since F is not diagonal, use the matrix exponential: e^(FT) = I + F T + (F T)^2 / 2! + (F T)^3 / 3! + ... Computing: ¦ = [ e^T T e^T ] [ 0 e^T ] Substituting T = À/2: ¦ = [ e^(À/2) (À/2)e^(À/2) ] [ 0 e^(À/2) ] -------------------------------------------------- Step 2: Compute = +(0 to T) e^(FÄ) G dÄ -------------------------------------------------- Using integral of matrix exponentials: = [ e^T - 1 ] [ 0 ] Substituting T = À/2: = [ e^(À/2) - 1 ] [ 0 ] -------------------------------------------------- Final Answer -------------------------------------------------- The discrete-time equivalent system is: x(k+1) = ¦ x(k) + u(k) where: ¦ = [ e^(À/2) (À/2)e^(À/2) ] [ 0 e^(À/2) ] = [ e^(À/2) - 1 ] [ 0 ] ---------------------------------------------------- Digital PI Controller Design - Fully Expanded and Step-by-Step Solution -------------------------------------------------- Given Information -------------------------------------------------- We need to design a **digital PI (Proportional-Integral) controller** of the form: D(z) = k_p + (k_i T z) / (z - 1) for the open-loop system: G(z) = z / (z - 1) where the **sampling time is T = 1**, such that the **closed-loop system transfer function** matches: H(z) = (2z) / (z^2 + 1) The goal is to determine the **controller gains ( k_p ) and ( k_i )** to achieve this desired system response. -------------------------------------------------- Step 1: Derive the Closed-Loop Transfer Function ------------------------------------------------ The **closed-loop transfer function** for a **unity feedback** system is given by: H(z) = (D(z) G(z)) / (1 + D(z) G(z)) ### **Substituting ( G(z) ):** Since we are given: G(z) = z / (z - 1) We substitute this into the closed-loop equation: H(z) = (D(z) * (z / (z - 1))) / (1 + D(z) * (z / (z - 1))) ### **Expanding ( D(z) ):** The **digital PI controller** is defined as: D(z) = k_p + (k_i T z) / (z - 1) Substituting this into the equation: H(z) = ((k_p + (k_i T z) / (z - 1)) * (z / (z - 1))) / (1 + (k_p + (k_i T z) / (z - 1)) * (z / (z - 1))) ### **Multiply numerator and denominator by ( (z - 1) ):** To clear the fractions, we multiply both the numerator and denominator by ( (z - 1) ): H(z) = ((k_p (z - 1) + k_i T z) * z) / ((z - 1) + k_p z (z - 1) + k_i T z) ### **Expanding the terms:** Expanding the numerator: k_p (z - 1) + k_i T z = k_p z - k_p + k_i T z Multiplying by ( z ): (k_p z - k_p + k_i T z) * z = k_p z^2 - k_p z + k_i T z^2 Expanding the denominator: (z -
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