Continuous system differential equations - Signals and Systems study notes

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Mathematical model of continuous-time systems

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Column writing system of differential equations based on the physical characteristics of the actual system.
Given the initial state and excitation conditions, in response to demand.
By mathematical analysis and then back to the physical reality.
FIG structure, for example, a block diagram and a signal flow diagram.

Note: the order is unknown amount of general differential equation of the highest order irrespective of the order of free items, and the right of the equal sign, the complete judgment is $v_c(t)$ ofthe highest order minus minimum orders.

The actual order of the equation by a number of independent dynamic element determined

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A block diagram representation of a continuous-time system

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In the block diagram of the learning process in addition to the basic operation symbol, as will encounter forward difference concepts (forward difference) the differential.
Differential operation corresponding to the differentiation operation in the electric mode, we studied the differential amplifier circuit, that is, when the voltage of two input terminals of the differential circuit, only the output voltage fluctuation, so called differential.

Forward differential function is often referred to as the differential function. For functions $and (n)$ , equally spaced nodes, we call $\ Bm {y (n + 1) -y (n)}$为一阶前向差分,
同理 $\bm{y(n)-y(n-1)}$为一阶后向差分；此外还有中心差分 $\bm{\frac{1}{2}[y(n+1)-y(n-1)]}$

其实求导中就包含差分公式：如用前向差分公式就可以表示为 $f^{'}(x)=\frac{f(x_{k+1})-f(x_k)}{x_{k+1}-x{k}}$

差分的阶：
$y(n)=x(n-2)+ay(n-1)$ 方程的阶次仍然是一阶后向差分方程，判别为响应的最高阶次减去响应的最小阶次，与自由项（激励）无关。

标准的差分方程应为 $y(n)-ay(n-1)=x(n-2)$，自由项全在等号右边。

根据框图写微分方程：

详细的求解过程中有一些习惯性的技巧：

1. 从后向前假设
2. 先找加法器，有几个加法器列几个等式
3. 列出后消去中间变量，最后只保留 $e(t)和r(t)$

当我们略去推导过程，最后通过式1和式2相互代入其实就可得到我们想要的方程。

根据微分方程画框图：