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Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t)...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta-...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) =...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) = 0 -1 ] y(t) = (1 out) u(t) , 0, \t <1 (1, t > 1 a) Find the state transition matrix and the impulse response matrix of the system. 2D) Determine whether the system is (i) completely state controllable, (ii) differentially control lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. (c) Repeat part (b) for to = 1. gd) Determine...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0 -1 4 1 a. Show that the system is not (internally) asymptotically stable b. Show that the system is both controllable and observable. c. Find matrices F e R1x2 and GE R2x1 such that o(A+ BF) C C_ and o (A GC) C C_ d. Find matrices (K, L, M, N) such that the feedback controller w(t) Kw(t) Ly(t) u(t) Mw(t)Ny(t) is internally stabilizing...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a)...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a) Verify that the system is observable and controllable. Then, design an output-feedback controller (based on a full-order observer) by placing the poles of the closed loop system at -1 j, -3, 12 ±j2. and-30 (mention which desired poles you select for your observer design and why).
(1 point) Consider the linear system 3 2 ' = y. -5 -3 a. Find the...
(1 point) Consider the linear system 3 2 ' = y. -5 -3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 = 15 and 2 V2 b. Find the real-valued solution to the initial value problem Syi ly 3y1 + 2y2, -541 – 3y2, yı(0) = 0, y2(0) = -5. Use t as the independent variable in your answers. yı(t) y2(t)
The transfer function of a linear system is G(s) = Y(s) S-1 U(s) 5? + 4s...
The transfer function of a linear system is G(s) = Y(s) S-1 U(s) 5? + 4s +3 a. Express this system in the modal form. b. Express this system in the standard controllable form (SCF). (Parts d, e, f, and g use this system) c. In the standard controllable form, suppose the output is replaced by y=[-1 a] | [x2] Give a value for a which makes the system unobservable. d. What is y(t) if y(0-)=-3, ay = 6 and...
(1 point) Consider the initial value problem -51เซี. -4 มี(0) 0 -5 a Find the eigenvalue λ, an eigenvector ul and a gen...
(1 point) Consider the initial value problem -51เซี. -4 มี(0) 0 -5 a Find the eigenvalue λ, an eigenvector ul and a generalized eigenvector u2 for the coefficient matrix of this linear system -5 u2 = b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers c2 c. Solve the original initial value problem m(t) = 2(t)-
(1 point) Consider the initial value problem -51เซี. -4 มี(0)...
(1 point) Consider the linear system 3 y y 5 3 a. Find the eigenvalues and...
(1 point) Consider the linear system 3 y y 5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 0 and A2 = -1 02 -3- -3+1 b. Find the real-valued solution to the initial value problem Svi C = -3y - 2y2, 591 +372 y.(0) = 6, 32(0) = -15. Use t as the independent variable in your answers. yı() y2(t) = 0
1 4 bea linear/matrix transformation such that Let T: 3 Fi 1 4 1 T 1...
1 4 bea linear/matrix transformation such that Let T: 3 Fi 1 4 1 T 1 1 C 6 h 3 Use the fact that T is linear to find the standard matrix [T of T and find T 1 Find a match for each of the following questions or choose NO MATCH if you can't find a match. What is the domain of T? What is the codomain of T? 4 4 How many rows does [T] have? How...
(1 point) Consider the initial value problem -2 j' = [ y, y(0) +3] 0 -2...
(1 point) Consider the initial value problem -2 j' = [ y, y(0) +3] 0 -2 a. Find the eigenvalue 1, an eigenvector 1, and a generalized eigenvector ū2 for the coefficient matrix of this linear system. = --1 V2 = b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. g(t) = C1 + C2 c. Solve the original initial value problem. yı(t) = y2(t) ==
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta-...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) = 0 -1 ] y(t) = (1 out) u(t) , 0, \t <1 (1, t > 1 a) Find the state transition matrix and the impulse response matrix of the system. 2D) Determine whether the system is (i) completely state controllable, (ii) differentially control lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. (c) Repeat part (b) for to = 1. gd) Determine...
Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0 -1 4 1 a. Show that the system is not (internally) asymptotically stable b. Show that the system is both controllable and observable. c. Find matrices F e R1x2 and GE R2x1 such that o(A+ BF) C C_ and o (A GC) C C_ d. Find matrices (K, L, M, N) such that the feedback controller w(t) Kw(t) Ly(t) u(t) Mw(t)Ny(t) is internally stabilizing...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a) Verify that the system is observable and controllable. Then, design an output-feedback controller (based on a full-order observer) by placing the poles of the closed loop system at -1 j, -3, 12 ±j2. and-30 (mention which desired poles you select for your observer design and why).
(1 point) Consider the linear system 3 2 ' = y. -5 -3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 = 15 and 2 V2 b. Find the real-valued solution to the initial value problem Syi ly 3y1 + 2y2, -541 – 3y2, yı(0) = 0, y2(0) = -5. Use t as the independent variable in your answers. yı(t) y2(t)
The transfer function of a linear system is G(s) = Y(s) S-1 U(s) 5? + 4s +3 a. Express this system in the modal form. b. Express this system in the standard controllable form (SCF). (Parts d, e, f, and g use this system) c. In the standard controllable form, suppose the output is replaced by y=[-1 a] | [x2] Give a value for a which makes the system unobservable. d. What is y(t) if y(0-)=-3, ay = 6 and...
(1 point) Consider the initial value problem -51เซี. -4 มี(0) 0 -5 a Find the eigenvalue λ, an eigenvector ul and a generalized eigenvector u2 for the coefficient matrix of this linear system -5 u2 = b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers c2 c. Solve the original initial value problem m(t) = 2(t)-
(1 point) Consider the initial value problem -51เซี. -4 มี(0)...
(1 point) Consider the linear system 3 y y 5 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. 2 0 and A2 = -1 02 -3- -3+1 b. Find the real-valued solution to the initial value problem Svi C = -3y - 2y2, 591 +372 y.(0) = 6, 32(0) = -15. Use t as the independent variable in your answers. yı() y2(t) = 0
1 4 bea linear/matrix transformation such that Let T: 3 Fi 1 4 1 T 1 1 C 6 h 3 Use the fact that T is linear to find the standard matrix [T of T and find T 1 Find a match for each of the following questions or choose NO MATCH if you can't find a match. What is the domain of T? What is the codomain of T? 4 4 How many rows does [T] have? How...
(1 point) Consider the initial value problem -2 j' = [ y, y(0) +3] 0 -2 a. Find the eigenvalue 1, an eigenvector 1, and a generalized eigenvector ū2 for the coefficient matrix of this linear system. = --1 V2 = b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. g(t) = C1 + C2 c. Solve the original initial value problem. yı(t) = y2(t) ==
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