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consider the system X(t) = ax(1) + bu(t) with a = 0.001,b= 1,x(0) = 5. (a)...
consider the system
X(t) = ax(1) + bu(t) with a = 0.001,b= 1,x(0) = 5. (a) Simulate this system using the Matlab command initial (b) Now use u(t) = -kx(t) where k is found as the optimal gain by minimizing the performance index J= ax (1) + ru (1) dt Use q=1, r=1 to simulate this system.
Problem 5. Given the system in state equation form, x=Ax + Bu where (a) A=10-3 01, B=10 0 0-2 (b)...
Problem 5. Given the system in state equation form, x=Ax + Bu where (a) A=10-3 01, B=10 0 0-2 (b) A=10-2 01,B=11 Can the system be stabilized by state feedback u-Kx, where K [k, k2 k3l?
Problem 5. Given the system in state equation form, x=Ax + Bu where (a) A=10-3 01, B=10 0 0-2 (b) A=10-2 01,B=11 Can the system be stabilized by state feedback u-Kx, where K [k, k2 k3l?
Consider a (continuous-time) linear system x=Ax + Bu. We introduce a time discretization tk-kAT, ...
Consider a (continuous-time) linear system x=Ax + Bu. We introduce a time discretization tk-kAT, where ΔT = assume that the input u(t) is piecewise constant on the equidistant intervals tk, tk+1), , and N > 0, and N 1 a(t) = uk for t E [tk, tk+1). (a) Verify that the specific choice of input signals leads to a discretization of the continuous-time system x = Ax + Bu in terms of a discrete-time system with states x,-2(tr) and inputs...
(a) Consider the system i = Ax + bu where 1-1::) and b = (b1 b2)....
(a) Consider the system i = Ax + bu where 1-1::) and b = (b1 b2). Derive a necessary and sufficient condition on the matrix b (i.e., bi and b2) for complete controllability of the system. (b) Same as part (a), except that 0 -1 A= 1 0
6. Consider a state-space system x = Ax+ Bu, y = Cx for which the control...
6. Consider a state-space system x = Ax+ Bu, y = Cx for which the control input is defined as u- -Kx + r, with r(t) a reference input. This results in a closed-loop system x (A-BK)x(t)+ Br(t) = with matrices 2 -2 K=[k1 K2 For this type of controller, ki, k2 ER do not need to be restricted to positive numbers - any real number is fine (a) What is the characteristic equation of the closed-loop system, in terms...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the c...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the controllability matrix, and ii) the Hautus-Rosenbrock test. b) Identify the controllable and uncontrollable subspaces, and convert the system to a Kalman con- 0 trollable canonical form c) Suppose that we start from the initial state z(0) (1,1, 1)T. Is there a control u(t) that drives the state to (1(3,-1,1)7 at some time t? Is there a control u(t) that...
Consider the linear system X' = AX where A is defined by , where a and...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
Q2. Consider the system described by the following differential equation x(t) Ax(t) where and x(0) -
Q2. Consider the system described by the following differential equation x(t) Ax(t) where and x(0) -
just 1,2,4 Problem 1 Consider the linear system of equations Ax = b, where x €...
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
please help !!!! 10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a) Show that v = | 1 | and w = 1-2) are eigenvectors of A. b) Identify the defective eigenvalu...
please help !!!!
10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a) Show that v = | 1 | and w = 1-2) are eigenvectors of A. b) Identify the defective eigenvalue of A, and find a corresponding generalized eigenvector Ax c) Write out the general solution of x
10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a)...
consider the system
X(t) = ax(1) + bu(t) with a = 0.001,b= 1,x(0) = 5. (a) Simulate this system using the Matlab command initial (b) Now use u(t) = -kx(t) where k is found as the optimal gain by minimizing the performance index J= ax (1) + ru (1) dt Use q=1, r=1 to simulate this system.
Problem 5. Given the system in state equation form, x=Ax + Bu where (a) A=10-3 01, B=10 0 0-2 (b) A=10-2 01,B=11 Can the system be stabilized by state feedback u-Kx, where K [k, k2 k3l?
Problem 5. Given the system in state equation form, x=Ax + Bu where (a) A=10-3 01, B=10 0 0-2 (b) A=10-2 01,B=11 Can the system be stabilized by state feedback u-Kx, where K [k, k2 k3l?
Consider a (continuous-time) linear system x=Ax + Bu. We introduce a time discretization tk-kAT, where ΔT = assume that the input u(t) is piecewise constant on the equidistant intervals tk, tk+1), , and N > 0, and N 1 a(t) = uk for t E [tk, tk+1). (a) Verify that the specific choice of input signals leads to a discretization of the continuous-time system x = Ax + Bu in terms of a discrete-time system with states x,-2(tr) and inputs...
(a) Consider the system i = Ax + bu where 1-1::) and b = (b1 b2). Derive a necessary and sufficient condition on the matrix b (i.e., bi and b2) for complete controllability of the system. (b) Same as part (a), except that 0 -1 A= 1 0
6. Consider a state-space system x = Ax+ Bu, y = Cx for which the control input is defined as u- -Kx + r, with r(t) a reference input. This results in a closed-loop system x (A-BK)x(t)+ Br(t) = with matrices 2 -2 K=[k1 K2 For this type of controller, ki, k2 ER do not need to be restricted to positive numbers - any real number is fine (a) What is the characteristic equation of the closed-loop system, in terms...
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the controllability matrix, and ii) the Hautus-Rosenbrock test. b) Identify the controllable and uncontrollable subspaces, and convert the system to a Kalman con- 0 trollable canonical form c) Suppose that we start from the initial state z(0) (1,1, 1)T. Is there a control u(t) that drives the state to (1(3,-1,1)7 at some time t? Is there a control u(t) that...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
Q2. Consider the system described by the following differential equation x(t) Ax(t) where and x(0) -
just 1,2,4
Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L,...
please help !!!!
10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a) Show that v = | 1 | and w = 1-2) are eigenvectors of A. b) Identify the defective eigenvalue of A, and find a corresponding generalized eigenvector Ax c) Write out the general solution of x
10. 20 points Consider the homogeneous system x' Ax, where 4 0 0 A 1 0 2 02 3 a)...
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