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Mathematical description of the system properties functions

Mathematical methods and tools are summarized for the characterization of the linear systems having constant coefficients in the followings. Let’s use the previously applied system description way being modeled by $\textbf{\underline{L}} operator. Consequently, the system response of a linear invariant system by a $f=f(t) input function is as follow:

$\textbf{\underline{L}}y(t) = f(t), where $f(t) = \int_{-\infty}^{\infty}\left|f(t) \right|}dt < \infty, as well as $\textbf{\underline{L}} = \sum_{k=1}^{n}a_{k}\frac{d^{k}}{dt^{k}},

and the initial condition are: $\frac{d^{m}y(t)}{dt^{m}}\bigg{|}_{t=0} = y_{m}, where $m = 0,1,...,n-1

Thus, the differential equation of system response:

$\sum_{k=0}^{n}a_{k}\frac{d^{k}y(t)}{dt^{k}} = f(t); \quad \frac{d^{m}y(t)}{dt^{m}}\bigg{|}_{t=0} = y_{m}; \quad m = 0,1,2,...,n-1

Let’s execute the Laplace transformation on both sides of the differential equation:

$\mathcal{L}\left\{\sum_{k=0}^{n}a_{k} \frac{d^{k}y(t)}{dt^{k}}\right\} = \mathcal{L}\left\{f(t)\right\}

$\sum_{k=0}^{n}a_{k}\mathcal{L}\left\{\frac{d^{k}y(t)}{dt^{k}}\right\} = \mathcal{L}\left\{f(t)\right\} = F(s)

Then apply the rules of Laplace transformation

$\mathcal{L}\left\{\frac{d^{k}y(t)}{dt^{k}}\right\} = s^{k}Y(s) - \sum_{k=1}^{n}s^{(n-k)} \frac{d^{(k-1)}y(t)}{dt^{(k-1)}}\bigg{|}_{t=0}

Így $\sum_{k=0}^{n}a_{k}\left[s^{k}Y(s)-\sum_{m=1}^{k}s^{(m-k)}\frac{d^{(m-1)}y(t)}{dt^{m-1}}\bigg{|}_{t=0}\right] = F(s)

$\sum_{k=0}^{n}a_{k}\left[s^{k}Y(s) - \sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right] = F(s)

$\sum_{k=0}^{n}a_{k}Y(s)s^{k}-\sum_{k=0}^{n}a_{k}\left[\sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right] = F(s)

$\sum_{k=0}^{n}a_{k}Y(s)s^{k} = F(s) + \sum_{k=0}^{n}a_{k}\left[\sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right]

$Y(s) = \frac{F(s) + \sum_{k=0}^{n}a_{k}\left[ \sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right]}{\sum_{k=0}^{n}a_{k}s^{k}}

The consequence of obtained result by the driving is, that the linear differential equation (or integral-differential equation by more broad interpretation) having constant coefficients has been transformed into polynomial, while the response function $-\;y=y(t) \;- into rational function of complex polynomials in $, i.e. in the extended complex frequency domain (supposing, $F(s) is polynomial or rational function in $ also). Furthermore, it is possible to see, “s” has operator properties in the extended complex frequency domain, because any operations by $ will appear in the real parameter space like a mapping.

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