Control Systems/List of Equations

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.

Fundamental Equations

 * $$e^{j\omega} = \cos(\omega) + j\sin(\omega)$$


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


 * $$\mathcal{L}[f(t) * g(t)] = F(s)G(s)$$
 * $$\mathcal{L}[f(t)g(t)] = F(s) * G(s)$$


 * $$|A - \lambda I| = 0$$
 * $$Av = \lambda v$$
 * $$wA = \lambda w$$


 * $$dB = 20 \log(C)$$

Basic Inputs

 * $$u(t) = \left\{

\begin{matrix} 0, & t < 0 \\  1, & t \ge 0 \end{matrix}\right. $$


 * $$r(t) = t u(t)$$


 * $$p(t) = \frac{1}{2}t^2 u(t)$$

Error Constants

 * $$K_p = \lim_{s \to 0} G(s)$$
 * $$K_p = \lim_{z \to 1} G(z)$$


 * $$K_v = \lim_{s \to 0} s G(s)$$
 * $$K_v = \lim_{z \to 1} (z - 1) G(z)$$


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

System Descriptions

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


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


 * $$Y(s) = H(s)X(s)$$
 * $$Y(z) = H(z)X(z)$$


 * $$ x'(t) = A x(t) + B u(t)$$
 * $$ y(t) = C x(t) + D u(t)$$


 * $$C[sI - A]^{-1}B + D = \mathbf{H}(s)$$
 * $$C[zI - A]^{-1}B + D = \mathbf{H}(z)$$


 * $$\mathbf{Y}(s) = \mathbf{H}(s)\mathbf{U}(s)$$
 * $$\mathbf{Y}(z) = \mathbf{H}(z)\mathbf{U}(z)$$


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

Feedback Loops

 * $$ H_{cl}(s) = \frac{KGp(s)}{1 + KGp(s)Gb(s)}$$


 * $$H_{ol}(s) = KGp(s)Gb(s)$$


 * $$F(s) = 1 + H_{ol}$$

Transforms

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


 * $$f(t)

= \mathcal{L}^{-1} \left\{F(s)\right\} = {1 \over {2\pi}}\int_{c-i\infty}^{c+i\infty} e^{st} F(s)\,ds$$


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


 * $$f(t)

= \mathcal{F}^{-1}\left\{F(j\omega)\right\} = \frac{1}{2\pi}\int_{-\infty}^\infty F(j\omega) e^{-j\omega t} d\omega$$


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


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


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


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

Transform Theorems

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


 * $$x(0) = \lim_{s \to \infty} s X(s)$$

State-Space Methods

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


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


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


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

Root Locus

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


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


 * $$N_a = P - Z$$


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


 * $$\sigma_0 = \frac{\sum_P - \sum_Z}{P - Z}$$


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

Lyapunov Stability

 * $$MA + A^TM = -N$$

Controllers and Compensators

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