Számítógéppel irányított rendszerek elmélete Gyakorlat - Mintavételezés, DT-LTI rendszermodellek Hangos Katalin Villamosmérnöki és Információs Rendszerek Tanszék e-mail: hangos.katalin@virt.uni-pannon.hu CCS-Gy6 Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 1 / 1ren
Exercise 1 Given a CT LTI system: ẋ(t) = 2x(t) + 4u(t), x() = y(t) = x(t) Discretize the above system with the sampling time h = 1s. Give the value of the state variable at t = 1 and t = 2 (with u(t) = 1(t)) Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 2 / 1ren
Solution to Exercise 1 ẋ(t) = 2x(t) + 4u(t), x() = y(t) = x(t) Discrete time model Φ = e 2 ; Γ = 2 2e 2 ; C = 1 The value of the state vector with u(k) = 1(k) x(1) = 2 2e 2 ; x(2) = 2 2e 4 Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 3 / 1ren
Exercise 2 Given the following CT-LTI SISO system 1 ẋ = x + 4 1 u y = 1 1 x 1 Discretize the system if the sampling time is 1 sec, i.e. h = 1 s! 2 Compute the solution with x() =, u(k) 1 for k = 1, 2! Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 4 / 1ren
Solution to Exercise 2 1 ẋ = y = 1 4 1 1 x x + 1 u 1. Discrete time model Φ = e Ah = L 1 ( (si Ah) 1) h=1s = = e 1 e 4 = Γ = A 1 (e Ah I)B =.3679.183.2479 Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 5 / 1ren
Solution to Exercise 2 2 x(k + 1) =.3679.183 x(k) +.2479 u x() = 2. Solution of the discrete time model x(1) =.2479 x(2) =.2933 Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 6 / 1ren
Exercise 3 A DT-LTI SISO system is given by following input-output model: 2y(k + 3) + 6y(k + 2) 2y(k) = u(k + 1) + 3u(k) 1 Compute its pulse transfer operator! 2 Give a possible state space model of DT-LTI system! 3 Is this state space model unique? Why? 4 Show that the pulse transfer operator H(q) is independent of the representation of the state space model, i.e. it is invariant with respect to coordinate transformation of the state space model! Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 7 / 1ren
Solution to Exercise 3 1 A DT-LTI SISO system is given by following input-output model: 2y(k + 3) + 6y(k + 2) 2y(k) = u(k + 1) + 3u(k) 1. Compute its pulse transfer operator! Solution:.5q + 1.5 H(q) = q 3 + 3q 2 1 2. Give a possible state space model of DT-LTI system! Solution: controller form 3 1 1 (Φ, Γ, C) = 1,,.5 1.5 1 Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 8 / 1ren
Solution to Exercise 3 2 A DT-LTI SISO system is given by following input-output model: 2y(k + 3) + 6y(k + 2) 2y(k) = u(k + 1) + 3u(k) 3. Is this state space model unique? Why? Solution: NO, see state transformation below 4. Show that the pulse transfer operator H(q) is independent of the representation of the state space model, i.e. it is invariant with respect to coordinate transformation of the state space model! Solution: use the form H(q) = C(qI Φ) 1 Γ + D with the SSR matrix transformation formulaes. Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 9 / 1ren
Homework exercise Given the following CT-LTI SISO system 1 1 ẋ = x + 2 4 y = 1 1 x 1 u 1 Discretize the system if the sampling time is 1 sec, i.e. h = 1 s! 2 Compute the solution with x() =, u(k) 1 for k = 1, 2! Gyakorlat - Mintavételezés, CCS-Gy6 DT-LTI 1 / 1ren