MATH319 Exercises 2 Workshop Exercise 1 4 Workshop Exercise 2

3 Assessed Exercise 1

A1.1 Express the following differential and integral equations as block diagrams

y=4{{d^{2}u}\over{dt^{2}}}+3\int u+6u;

(i)

y+4{{d^{2}y}\over{dt^{2}}}=2{{du}\over{dt}}+7\int u.

(i i)

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A1.2 (i) Express the following coupled differential equations as a block diagram, where $u$ is the input, $y$ is the output, $x$ is a state variable, and $a, b, c$ and $d$ are constants:

{{dx}\over{dt}}=ax+bu,

and

{{dy}\over{dt}}=cx+du.

(ii) Express the following coupled differential and integral equations as a block diagram, where $u_{1}$ and $u_{2}$ are the inputs, $y$ is the output, $x$ is a state variable, and $a,c,b_{1},b_{2},d_{1}$ and $d_{2}$ are constants:

{{dx}\over{dt}}=ax+b_{1}u_{1}+b_{2}u_{2},

{{dy}\over{dt}}=cx+d_{1}u_{1}+d_{2}u_{2}.

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A1.3 A simple harmonic oscillator satisfies

m{{d^{2}x}\over{dt^{2}}}+kx=u,

where $t$ is time, $x$ is displacement, $u$ is the input, and $k$ and $m$ are positive constants. By introducing an extra state variable $v=dx/dt$ , write this as a first order system of differential equations in matrix form $(A,B,C,D)$ .

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A1.4 Let

A=\left[\begin{pmatrix}1&4&10\cr 0&2&0\cr 0&0&3\cr\end{pmatrix}\right].

Find $(sI-A)^{-1}$ , where $s$ is an algebraic variable.

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