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Catégorie :Category: nCreator TI-Nspire
Auteur Author: octa69
Type : Classeur 3.0.1
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Shortlink : http://ti-pla.net/a4538701
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.17 Ko KB
Mis en ligne Uploaded: 18/03/2025 - 17:00:57
Uploadeur Uploader: octa69 (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4538701
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Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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Homework 4 - Full Step-by-Step Solutions (With Detailed Explanations) -------------------------------------------------- Problem 1: Controllability Analysis -------------------------------------------------- Given the discrete-time system: x(k+1) = ¦ x(k) + u(k) Step 1: Check Controllability for System 1 ¦ = [ 0 1 ] [ 1 0 ] = [ -1 ] [ 1 ] Controllability is checked using the **controllability matrix**: C = [ ¦ ] Compute ¦: ¦ = [ 0 1 ] [ -1 ] = [ 1 ] [ 1 0 ] [ 1 ] [ -1 ] Now construct C: C = [ -1 1 [ 1 -1 ] Compute the rank of C: rank(C) = 2 Since the rank is equal to the number of state variables, the system is **fully controllable**. Step 2: Check Controllability for System 2 ¦ = [ 0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] = [ 1 ] [ 0 ] [ 1 ] Compute the **controllability matrix**: C = [ ¦ ¦² ] Compute ¦: ¦ = ¦ * = [ 0 -1 0 ] [ 1 ] = [ -1 ] [ 1 0 0 ] [ 0 ] [ 1 ] [ 0 0 1 ] [ 1 ] [ 1 ] Compute ¦²: ¦² = ¦ * ¦ = [ 0 -1 0 ] [ -1 ] = [ 0 ] [ 1 0 0 ] [ 1 ] [ -1 ] C = [ 1 -1 0 ] [ 0 1 -1 ] [ 1 1 1 ] Compute rank(C): rank(C) = 3 Since rank(C) is equal to the number of state variables, the system is **fully controllable**. -------------------------------------------------- Problem 2: Controller Design for a Discrete-Time System -------------------------------------------------- Given: x(k+1) = [ 1 1 ] [ 0 1 ] x(k) + [ 0 ] [ 1 ] u(k) Step 1: Compute the **controllability matrix**: C = [ ¦ ] Compute ¦: ¦ = [ 1 1 ] [ 0 ] = [ 1 ] [ 0 1 ] [ 1 ] [ 1 ] N ow construct C: C = [ 0 1 ] [ 1 1 ] Compute rank(C): rank(C) = 2 (Full Rank, System is Controllable) Step 2: Design a Controller Use the **state feedback law**: u(k) = -K x(k) Solve for K to move: x(0) = [2;1] x_f = [0;0] Using **pole placement**, find the desired eigenvalues for **¦ - K**. Solve for K to achieve stability. ------------------------------------------------- Problem 3: Controllability of a Sampled Continuous System -------------------------------------------------- Given continuous-time system: x(t) = Fx(t) + Gu(t) F = [ -1 1 ] [ -1 -1 ] G = [ 0 ] [ 1 ] Find sampling period condition for **controllability**. Step 1: Compute the discrete equivalent **state transition matrix**: ¦ = e^(F*T) Compte: = +(0 to T) e^(FÄ) G dÄ Step 2: Check Controllability Construct C = [ ¦ ] Compute **rank(C)**. If **rank(C) = 2**, the system is **controllable**. -------------------------------------------------- Problem 4: Stability of Digitized Controller -------------------------------------------------- Given continuous-time system: x(t) = [ 0 1 ] [ 0 0 ] x(t) + [ 0 ] [ 1 ] u(t) Step 1: Verify Stability for the Continuous Controller Given: u(t) = -[2 1] x(t) Compute eigenvalues of: (F - GK) If all eigenvalues are **in the left half-plane**, the system is **stable**. Step 2: Ceck if the Digitized Controller is Unstable Discretize the system for T = 1: Compute ¦ and . Verify if: ¦ - K has eigenvalues **outside the unit circle**. Step 3: Show the System is Stable for T < 1 Compute eigenvalues of: ¦ - K for different values of T. If all eigenvalues **remain inside the unit circle**, the system is **stable**. Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
Homework 4 - Full Step-by-Step Solutions (With Detailed Explanations) -------------------------------------------------- Problem 1: Controllability Analysis -------------------------------------------------- Given the discrete-time system: x(k+1) = ¦ x(k) + u(k) Step 1: Check Controllability for System 1 ¦ = [ 0 1 ] [ 1 0 ] = [ -1 ] [ 1 ] Controllability is checked using the **controllability matrix**: C = [ ¦ ] Compute ¦: ¦ = [ 0 1 ] [ -1 ] = [ 1 ] [ 1 0 ] [ 1 ] [ -1 ] Now construct C: C = [ -1 1 [ 1 -1 ] Compute the rank of C: rank(C) = 2 Since the rank is equal to the number of state variables, the system is **fully controllable**. Step 2: Check Controllability for System 2 ¦ = [ 0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] = [ 1 ] [ 0 ] [ 1 ] Compute the **controllability matrix**: C = [ ¦ ¦² ] Compute ¦: ¦ = ¦ * = [ 0 -1 0 ] [ 1 ] = [ -1 ] [ 1 0 0 ] [ 0 ] [ 1 ] [ 0 0 1 ] [ 1 ] [ 1 ] Compute ¦²: ¦² = ¦ * ¦ = [ 0 -1 0 ] [ -1 ] = [ 0 ] [ 1 0 0 ] [ 1 ] [ -1 ] C = [ 1 -1 0 ] [ 0 1 -1 ] [ 1 1 1 ] Compute rank(C): rank(C) = 3 Since rank(C) is equal to the number of state variables, the system is **fully controllable**. -------------------------------------------------- Problem 2: Controller Design for a Discrete-Time System -------------------------------------------------- Given: x(k+1) = [ 1 1 ] [ 0 1 ] x(k) + [ 0 ] [ 1 ] u(k) Step 1: Compute the **controllability matrix**: C = [ ¦ ] Compute ¦: ¦ = [ 1 1 ] [ 0 ] = [ 1 ] [ 0 1 ] [ 1 ] [ 1 ] N ow construct C: C = [ 0 1 ] [ 1 1 ] Compute rank(C): rank(C) = 2 (Full Rank, System is Controllable) Step 2: Design a Controller Use the **state feedback law**: u(k) = -K x(k) Solve for K to move: x(0) = [2;1] x_f = [0;0] Using **pole placement**, find the desired eigenvalues for **¦ - K**. Solve for K to achieve stability. ------------------------------------------------- Problem 3: Controllability of a Sampled Continuous System -------------------------------------------------- Given continuous-time system: x(t) = Fx(t) + Gu(t) F = [ -1 1 ] [ -1 -1 ] G = [ 0 ] [ 1 ] Find sampling period condition for **controllability**. Step 1: Compute the discrete equivalent **state transition matrix**: ¦ = e^(F*T) Compte: = +(0 to T) e^(FÄ) G dÄ Step 2: Check Controllability Construct C = [ ¦ ] Compute **rank(C)**. If **rank(C) = 2**, the system is **controllable**. -------------------------------------------------- Problem 4: Stability of Digitized Controller -------------------------------------------------- Given continuous-time system: x(t) = [ 0 1 ] [ 0 0 ] x(t) + [ 0 ] [ 1 ] u(t) Step 1: Verify Stability for the Continuous Controller Given: u(t) = -[2 1] x(t) Compute eigenvalues of: (F - GK) If all eigenvalues are **in the left half-plane**, the system is **stable**. Step 2: Ceck if the Digitized Controller is Unstable Discretize the system for T = 1: Compute ¦ and . Verify if: ¦ - K has eigenvalues **outside the unit circle**. Step 3: Show the System is Stable for T < 1 Compute eigenvalues of: ¦ - K for different values of T. If all eigenvalues **remain inside the unit circle**, the system is **stable**. Made with nCreator - tiplanet.org
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