![Q. 2. (a) State Sylvesters interpolation formula for finding any function (particularly eAt) of an m x m matrix A both when all the eigenvalues of A are distinct and when some (say, the first k) are identical. (b) For the matrix A obtained in 1 (b) find the state transition matrix eAt using Sylvesters formula. (c) Find the state transition matrix ø(t) using the formula ø(t) = e At-L-1 [(sl-A)-1] where L-1 is (2 Points) (4 marks) inverse Laplace transform, and verify that it tallies with that found in 2 (b). (4 marks)](http://img.homeworklib.com/questions/44de95b0-b4b5-11ea-b9c9-ff89401272b6.png?x-oss-process=image/resize,w_560)
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Q. 2. (a) State Sylvester's interpolation formula for finding any function (particularly eAt) of an m...
2. For the simple pendulum shown in Figure 2, the nonlinear equations of motion are given by θ(t)...
do (b) and (c) only.
2. For the simple pendulum shown in Figure 2, the nonlinear equations of motion are given by θ(t) + 믈 sin θ(t) + m 0(t)-0 Pivot point L, length Massless rod , mass Figure 2. A simple pendulum 3. Consider again the pendulum of Figure 2 of problem 2 when g = 9.8 m/s, 1 = 4.9m, k =0.3, and (a) Determine whether the system is stable by finding the characteristic equation obtained from setting...
($ ?) 4 2. (a) Find the eigenvalues and eigenvectors of the matrix 3 Hence or...
($ ?) 4 2. (a) Find the eigenvalues and eigenvectors of the matrix 3 Hence or otherwise find the general solution of the system = 4x + 2y = 3x - y 195 marks 5. (a) Give a precise definition of Laplace transform of a function f(t). Use your definition to determine the Laplace transform of 3. Osts 2 6-t, 2 <t f(t) = [20 marks] (b) A logistic initial value problem is given by dP dt kP(M-P), P(0) -...
PLEASE SOLVE ii ans iv, thank you Theorem 2.35 (Putzer Algorithm for Finding eAt) Let 11,...
PLEASE SOLVE ii ans iv, thank
you
Theorem 2.35 (Putzer Algorithm for Finding eAt) Let 11, 12, ..., In be the (not necessarily distinct) eigenvalues of the matrix A. Then n-1 eAt = ) Pk+1(t)Mk, k=0 where Mo :=1, Mk := || (A - l;I), i=1 for 1 <k<n and the vector function p defined by pi(t) p(t) = | P2(t) [ Pn(t) for tER, is the solution of the IVP [ 0 0 ... 01 1 12 0 ......
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 +...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
Reasonable chance of getting good scores on the exams. References: Lectures 8.1 and 8.2. Sections...
reasonable chance of getting good scores on the exams. References: Lectures 8.1 and 8.2. Sections 9.1, 9.2, and 9.3 in the textbook. You can make free use of the table of Laplace Transforms on Page 500 but you must justify any other formulas for Laplace transforms. When we speak of "the" inverse Laplace transform of a function F(s) we mean the unique continuous function / : [0,00) → R (if there is one!) whose Laplace transforn is F(s) (for sufficiently...
Determine e At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula eAt = X(t)X(0)1. 0...
Determine e At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula eAt = X(t)X(0)1. 0 2 2 2 0 2 2 2 0 First, find X(t). Choose the correct answer below 4t -2t 4t e -2t (1+t) e e -2t OA. X(t) (1+t)e4t 0 e2t B. X(t)= e4t 0 -2t -2t 2t - e - 2t (1t)e 4t e e 4t e 4t - sint sin t 0 (1t)e -2t O C....
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 |...
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
Please answer this in specific way,thanks. 1. A Markov chain X = (X2) >0 with state...
Please answer this in specific way,thanks.
1. A Markov chain X = (X2) >0 with state space I = {A, B, C} has a one-step transition matrix P given by 70 2/3 1/3) P= 1/3 0 2/3 (1/6 1/3 1/2) (a) Find the eigenvalues 11, 12, 13 of P. (b) Deduce pn can be written as pn = 10 + XU, + Aug (n > 0) and determine the matrices U1, U2, U3 by using the equations n = 0,1,2....
ELE480/580: Control Systems II Homework #6 Due: 4/29/2019 1. Consider a state equation with 0 0-2...
please answer all
ELE480/580: Control Systems II Homework #6 Due: 4/29/2019 1. Consider a state equation with 0 0-2 B 1C [0 0 1] A1 0 1 0 1 -3 0 Find the observer gain matrix L that places all three observer eigenvalues at -5. Write the state equation that defines the observer 2. For the state equation defined by the following state matrices x(t) 1 01x,(t)[1 h(t) | = | 0 0 111X2(t) | + | | | u(t)...
The state space model of an interconnected three tank water storage system is given by the following equation -3 1 o 1[hi] dt The heights of water in the tanks are, respectively, hi, h2, hz. Each tan...
The state space model of an interconnected three tank water storage system is given by the following equation -3 1 o 1[hi] dt The heights of water in the tanks are, respectively, hi, h2, hz. Each tank has an independent input flow; the volume flow rates of input water into the three tanks are, respectively, qii,qi2, Oi3. Each tank also has a water discharge outlet and the volume flow rates of water coming out of the tanks are, respectively, qo1,...
do (b) and (c) only.
2. For the simple pendulum shown in Figure 2, the nonlinear equations of motion are given by θ(t) + 믈 sin θ(t) + m 0(t)-0 Pivot point L, length Massless rod , mass Figure 2. A simple pendulum 3. Consider again the pendulum of Figure 2 of problem 2 when g = 9.8 m/s, 1 = 4.9m, k =0.3, and (a) Determine whether the system is stable by finding the characteristic equation obtained from setting...
($ ?) 4 2. (a) Find the eigenvalues and eigenvectors of the matrix 3 Hence or otherwise find the general solution of the system = 4x + 2y = 3x - y 195 marks 5. (a) Give a precise definition of Laplace transform of a function f(t). Use your definition to determine the Laplace transform of 3. Osts 2 6-t, 2 <t f(t) = [20 marks] (b) A logistic initial value problem is given by dP dt kP(M-P), P(0) -...
PLEASE SOLVE ii ans iv, thank
you
Theorem 2.35 (Putzer Algorithm for Finding eAt) Let 11, 12, ..., In be the (not necessarily distinct) eigenvalues of the matrix A. Then n-1 eAt = ) Pk+1(t)Mk, k=0 where Mo :=1, Mk := || (A - l;I), i=1 for 1 <k<n and the vector function p defined by pi(t) p(t) = | P2(t) [ Pn(t) for tER, is the solution of the IVP [ 0 0 ... 01 1 12 0 ......
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
reasonable chance of getting good scores on the exams. References: Lectures 8.1 and 8.2. Sections 9.1, 9.2, and 9.3 in the textbook. You can make free use of the table of Laplace Transforms on Page 500 but you must justify any other formulas for Laplace transforms. When we speak of "the" inverse Laplace transform of a function F(s) we mean the unique continuous function / : [0,00) → R (if there is one!) whose Laplace transforn is F(s) (for sufficiently...
Determine e At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula eAt = X(t)X(0)1. 0 2 2 2 0 2 2 2 0 First, find X(t). Choose the correct answer below 4t -2t 4t e -2t (1+t) e e -2t OA. X(t) (1+t)e4t 0 e2t B. X(t)= e4t 0 -2t -2t 2t - e - 2t (1t)e 4t e e 4t e 4t - sint sin t 0 (1t)e -2t O C....
Q10 The Hamiltonian of a two-state system is given by H E ( i)- I02)(2 | -i | ¢1)(2 | +i | ¢2) (¢1 1) where , p2) form a complete and orthonormal basis; E is a real constant having the dimensions of energy (a) Is H Hermitian? Calculate the trace of H (b) Find the matrix representing H in the | øı), | 42) basis and calculate the eigenvalues and the eigenvectors of the matrix. Calculate the trace of...
Please answer this in specific way,thanks.
1. A Markov chain X = (X2) >0 with state space I = {A, B, C} has a one-step transition matrix P given by 70 2/3 1/3) P= 1/3 0 2/3 (1/6 1/3 1/2) (a) Find the eigenvalues 11, 12, 13 of P. (b) Deduce pn can be written as pn = 10 + XU, + Aug (n > 0) and determine the matrices U1, U2, U3 by using the equations n = 0,1,2....
please answer all
ELE480/580: Control Systems II Homework #6 Due: 4/29/2019 1. Consider a state equation with 0 0-2 B 1C [0 0 1] A1 0 1 0 1 -3 0 Find the observer gain matrix L that places all three observer eigenvalues at -5. Write the state equation that defines the observer 2. For the state equation defined by the following state matrices x(t) 1 01x,(t)[1 h(t) | = | 0 0 111X2(t) | + | | | u(t)...
The state space model of an interconnected three tank water storage system is given by the following equation -3 1 o 1[hi] dt The heights of water in the tanks are, respectively, hi, h2, hz. Each tank has an independent input flow; the volume flow rates of input water into the three tanks are, respectively, qii,qi2, Oi3. Each tank also has a water discharge outlet and the volume flow rates of water coming out of the tanks are, respectively, qo1,...
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