List of Equations

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List of Equations
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The following is a list of the important equations from the text, arranged by subject. For more information about these equations, including the meaning of each variable and symbol, the uses of these functions, or the derivations of these equations, see the relevant pages in the main text.

1 Fundamental Equations

LaTeX: e^{j\omega} = \cos(\omega) + j\sin(\omega) Euler's Formula



LaTeX: (a*b)(t) = \int_{-\infty}^\infty a(\tau)b(t - \tau)d\tau Convolution



LaTeX: \mathcal{L}[f(t) * g(t)] = F(s)G(s)
LaTeX: \mathcal{L}[f(t)g(t)] = F(s) * G(s)
Convolution Theorem



LaTeX: |A - \lambda I| = 0
LaTeX: Av = \lambda v
LaTeX: wA = \lambda w
Characteristic Equation



LaTeX: dB = 20 \log(C) Decibels



2 Basic Inputs

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LaTeX: u(t) = \left\{</dd></dl>
<p>\begin{matrix} 
</p>
<pre> 0, & t < 0
</pre>
<p>\\ 
</p>
<pre> 1, & t \ge 0
</pre>
<p>\end{matrix}\right.


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LaTeX: r(t) = t u(t)


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LaTeX: p(t) = \frac{1}{2}t^2 u(t)

3 Error Constants

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LaTeX: K_p = \lim_{s \to 0} G(s)
LaTeX: K_p = \lim_{z \to 1} G(z)


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LaTeX: K_v = \lim_{s \to 0} s G(s)
LaTeX: K_v = \lim_{z \to 1} (z - 1) G(z)


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LaTeX: K_a = \lim_{s \to 0} s^2 G(s)
LaTeX: K_a = \lim_{z \to 1} (z - 1)^2 G(z)

4 System Descriptions

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LaTeX: y(t) = \int_{-\infty}^\infty g(t, r)x(r)dr


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LaTeX: y(t) = x(t) * h(t) = \int_{-\infty}^\infty x(\tau)h(t - \tau)d\tau


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LaTeX: Y(s) = H(s)X(s)
LaTeX: Y(z) = H(z)X(z)


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LaTeX: x' = A(t)x(t) + B(t)u(t)
LaTeX: y(t) = C(t)x(t) + D(t)u(t)


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LaTeX: C[sI - A]^{-1}B + D = \bold{H}(s)
LaTeX: C[zI - A]^{-1}B + D = \bold{H}(z)


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LaTeX: \bold{Y}(s) = \bold{H}(s)\bold{U}(s)
LaTeX: \bold{Y}(z) = \bold{H}(z)\bold{U}(z)


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LaTeX: M = \frac{y_{out}}{y_{in}} = \sum_{k=1}^N \frac{M_k \Delta\ _k}{ \Delta\ }

5 Feedback Loops

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LaTeX:  H_{cl}(s) =  \frac{KGp(s)}{1 + KGp(s)Gb(s)}


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LaTeX: H_{ol}(s) = KGp(s)Gb(s)


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LaTeX: F(s) = 1 + H_{ol}

Template:-

6 Transforms

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LaTeX: F(s) = \mathcal{L}[f(t)] = \int_0^\infty f(t) e^{-st}dt


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LaTeX: f(t) </dd></dl>
<pre>       = \mathcal{L}^{-1} \left\{F(s)\right\}
       = {1 \over {2\pi}}\int_{c-i\infty}^{c+i\infty} e^{st} F(s)\,ds


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LaTeX: F(j\omega) = \mathcal{F}[f(t)] = \int_0^\infty f(t) e^{-j\omega t} dt


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LaTeX: f(t) </dd></dl>
<pre>       = \mathcal{F}^{-1}\left\{F(j\omega)\right\}     
       = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{-j\omega t} d\omega


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LaTeX: F^*(s) = \mathcal{L}^*[f(t)] = \sum_{i = 0}^\infty f(iT)e^{-siT}


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LaTeX: X(z) = \mathcal{Z}\left\{x[n]\right\} = \sum_{i = -\infty}^\infty x[n] z^{-n}


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LaTeX:  x[n] = Z^{-1} \{X(z) \}= \frac{1}{2 \pi j} \oint_{C} X(z) z^{n-1} dz \


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LaTeX: X(z, m) = \mathcal{Z}(x[n], m) = \sum_{n = -\infty}^{\infty} x[n + m - 1]z^{-n}

7 Transform Theorems

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LaTeX: x(\infty) = \lim_{s \to 0} s X(s)
LaTeX: x[\infty] = \lim_{z \to 1} (z - 1) X(z)


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LaTeX: x(0) = \lim_{s \to \infty} s X(s)

8 State-Space Methods

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LaTeX: x(t) = e^{At-t_0}x(t_0) + \int_{t_0}^{t}e^{A(t - \tau)}Bu(\tau)d\tau
LaTeX: x[n] = A^nx[0] + \sum_{m=0}^{n-1}A^{n-1-m}Bu[n]


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LaTeX: y(t) = Ce^{At-t_0}x(t_0) + C\int_{t_0}^{t}e^{A(t - \tau)}Bu(\tau)d\tau + Du(t)
LaTeX: y[n] = CA^nx[0] + \sum_{m=0}^{n-1}CA^{n-1-m}Bu[n] + Du[n]


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LaTeX: x(t) = \phi(t, t_0)x(t_0) + \int_{t_0}^{t} \phi(\tau, t_0)B(\tau)u(\tau)d\tau
LaTeX: x[n] = \phi[n, n_0]x[t_0] + \sum_{m = n_0}^{n} \phi[n, m+1]B[m]u[m]


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LaTeX:  G(t, \tau) = \left\{\begin{matrix}C(\tau)\phi(t, \tau)B(\tau) & \mbox{ if } t \ge \tau \\0 & \mbox{ if } t < \tau\end{matrix}\right.
LaTeX: G[n] = \left\{\begin{matrix}CA^{k-1}N & \mbox{ if } k > 0 \\ 0 & \mbox{ if } k \le 0\end{matrix}\right.

9 Root Locus

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LaTeX: 1 + KG(s)H(s) = 0
LaTeX: 1 + K\overline{GH}(z) = 0


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LaTeX: \angle KG(s)H(s) = 180^\circ
LaTeX: \angle K\overline{GH}(z) = 180^\circ


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LaTeX: N_a = P - Z


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LaTeX: \phi_k = (2k + 1)\frac{\pi}{P - Z}


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LaTeX: \sigma_0 = \frac{\sum_P - \sum_Z}{P - Z}


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LaTeX: \frac{G(s)H(s)}{ds} = 0 or LaTeX: \frac{\overline{GH}(z)}{dz} = 0

10 Lyapunov Stability

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LaTeX: MA + A^TM = -N

11 Controllers and Compensators

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LaTeX: D(s) = K_p + {K_i \over s} + K_d s
LaTeX: D(z) = K_p + K_i \frac{T}{2} \left[ \frac{z + 1}{z - 1} \right] + K_d \left[ \frac{z - 1}{Tz} \right]