Date: Sat, 24 Feb. 2024 06:23:13 +00:00
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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 {EQUATION(size="80")}$\textbf{\underline{L}}{EQUATION} operator. Consequently, the system response of a linear invariant system by a {EQUATION(size="80")}$f=f(t){EQUATION} input function is as follow:
{EQUATION(size="80")}$\textbf{\underline{L}}y(t) = f(t){EQUATION}, where {EQUATION(size="80")}$f(t) = \int_{-\infty}^{\infty}\left|f(t) \right|}dt < \infty{EQUATION}, as well as {EQUATION(size="80")}$\textbf{\underline{L}} = \sum_{k=1}^{n}a_{k}\frac{d^{k}}{dt^{k}}{EQUATION},
and the initial condition are: {EQUATION(size="80")}$\frac{d^{m}y(t)}{dt^{m}}\bigg{|}_{t=0} = y_{m},{EQUATION} where {EQUATION(size="80")}$m = 0,1,...,n-1{EQUATION}
Thus, the differential equation of system response:
{EQUATION(size="80")}$\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{EQUATION}
Let’s execute the Laplace transformation on both sides of the differential equation:
{EQUATION(size="80")}$\mathcal{L}\left\{\sum_{k=0}^{n}a_{k} \frac{d^{k}y(t)}{dt^{k}}\right\} = \mathcal{L}\left\{f(t)\right\}{EQUATION}
{EQUATION(size="80")}$\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){EQUATION}
Then apply the rules of Laplace transformation
{EQUATION(size="80")}$\mathcal{L}\left\{\frac{d^{k}y(t)}{dt^{k}}\right\} = s^{k}Y(s) - \sum_{k=1}^{n}s^{(n-k)} \frac{d^{(k-1)}y(t)}{dt^{(k-1)}}\bigg{|}_{t=0}{EQUATION}
Így {EQUATION(size="80")}$\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){EQUATION}
{EQUATION(size="80")}$\sum_{k=0}^{n}a_{k}\left[s^{k}Y(s) - \sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right] = F(s){EQUATION}
{EQUATION(size="80")}$\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){EQUATION}
{EQUATION(size="80")}$\sum_{k=0}^{n}a_{k}Y(s)s^{k} = F(s) + \sum_{k=0}^{n}a_{k}\left[\sum_{m=1}^{k}s^{(m-k)}y_{m-1}\right]{EQUATION}
{EQUATION(size="80")}$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}}{EQUATION}
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 {EQUATION(size="80")}$-\;y=y(t) \;-{EQUATION} into rational function of complex polynomials in {EQUATION(size="80")}$"s"{EQUATION}, i.e. in the extended complex frequency domain (supposing, {EQUATION(size="80")}$F(s){EQUATION} is polynomial or rational function in {EQUATION(size="80")}$"s"{EQUATION} also). Furthermore, it is possible to see, “s” has operator properties in the extended complex frequency domain, because any operations by {EQUATION(size="80")}$"s"{EQUATION} will appear in the real parameter space like a mapping.