Advanced Modern Engineering Mathematics 4th. Edition by James, Glyn - Burley, David - Clements, Dick - Dyke, Phil - Searl, John | 9780273719236 | 0273719238 | Holooly

Advanced Modern Engineering Mathematics

171 SOLVED PROBLEMS

A 3 × 3 symmetric matrix A has eigenvalues 6, 3 and 1. The eigenvectors corresponding to the eigenvalues 6 and 1 are [1 2 0]^T and [-2 1 0]^T respectively. Find the eigenvector corresponding to the eigenvalue 3, and hence determine the matrix A.

Verified Answer:

Since the matrix A is symmetric the eigenvectors [...

A feedback control system modelled by the differential equation x+ax+kx=0 is known to be asymptotically stable, for k> 0 , a> 0 Set up the state-space form of the equation and show that V(x1,x2)=kx1^2+(x2+ax1)^2 , x1=x, x2=x is a suitable Lyapunov function for verifying this.

Verified Answer:

State-space form is

x^{\cdot }=\begin{bmatr...

A second-order system is governed by the state equation Using a suitable transformation x(t) = Mz(t), reduce this to the canonical form

Verified Answer:

Taking A =

\begin{bmatrix}3 & 4\\ 2 &...

A system is characterized by the state Given that the input is the unit step function and initially x1(0)=x2(0)=1 deduce

Verified Answer:

the solution is given by

x(t)=e^{AT}x(0)+\i...

A third-order system is characterized by the state-space model x^{\cdot }=\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -5 & -6\end{bmatrix}x+\begin{bmatrix}1\\ -3\\ 18\end{bmatrix}u , y=\begin{bmatrix}1 & 0 & 0\end{bmatrix}x where x=\begin{bmatrix}x_{1} &x_{2}& x_{3}\end{bmatrix}^{^{T}} Obtain the equivalent canonical representation of the model and then obtain the response of the system to a unit step u(t ) = H(t ) given that initially x(0)=\begin{bmatrix}1 & 1 & 0\end{bmatrix}^{T}.

Verified Answer:

The eigenvalues of the matrix A =

\begin{bm...

By considering the Gerschgorin circles, show that the eigenvalues of the matrix A= [ 2 -1 0 -1 2 -1 0 -1 2 ] satisfy the inequality 0≤λ≤4 Hence determine the smallest modulus eigenvalue of A correct to two decimal places.

Verified Answer:

A is a symmetric matrix its eigenvalues are real. ...

By representing the data in the matrix form A_{\zeta }=y where \zeta =\begin{bmatrix}m & c\end{bmatrix}^{T} use the pseudo inverse to find the values of m and c which provide the least squares fit to the linear model y = mx + c for the following data. k 1 2 3 4 5 x_{k} 0 1 2 3 4 y_{k} 1 1 2 2 3

Verified Answer:

Data may be represented in the matrix form

...

calculate A^{K} A = \begin{bmatrix}0 & 1\\ -2 & -3\end{bmatrix}

Verified Answer:

A has eigenvalues

\lambda _{1}=-1

a...

Calculate e^{At} and sin At when A=\begin{bmatrix}1 & -1\\ 0 & 1\end{bmatrix}

Verified Answer:

Again A has repeated eigenvalues, with

\lam...

Classify the following quadratic form: -3x_{1}^{2}-5x_{2}^{2}-3x_{3}^{2}+2x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3}

Verified Answer:

The matrix corresponding to the quadratic form is ...

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