GATE Control Systems formula sheet

Here the feedback signal is subtracted at the comparator. The characteristic equation is $1+G(s)H(s)=0$. After clearing fractions, its roots are the closed-loop poles.

Here the feedback signal is added at the comparator. The characteristic equation is $1-G(s)H(s)=0$.

For a unit step into $T(s)=K/(\tau s+1)$. At $t=\tau$ the output has made 63.2% of its final change, and its initial slope is $K/\tau$.

The exact settling values are $-\ln 0.02=3.91$ and $-\ln 0.05=3.00$ time constants. A first-order system without a zero has no peak time and no overshoot.

For a causal LTI system with rational, proper $T(s)$ (numerator degree $\le$ denominator degree), with poles counted after cancelling common factors. If $T(s)$ is proper but not strictly proper, $h(t)$ also contains an impulse $D\delta (t)$, which does not affect BIBO stability. An improper $T(s)$, such as the pure differentiator $T(s)=s$, is never BIBO stable. A single pole on the $j\omega$-axis, even a simple one, is enough to make the system not BIBO stable.

Here $G=N_G/D_G$ and $H=N_H/D_H$, written without cancelling any factor. The roots are the closed-loop poles. For positive feedback, $T(s)=\frac{G(s)}{1-G(s)H(s)}$ and the characteristic equation is $1-G(s)H(s)=0$, that is $D_G{D}_H-N_G{N}_H=0$.

Negative feedback, with $G(s)H(s)=N(s)/D(s)$ in monic form. The ordinary root locus is for $K\ge 0$.

A point $s$ that is not itself an open-loop pole or zero is on the root locus ($K\ge 0$) if and only if the net angle is an odd multiple of $180^{\circ }$. $K$ does not appear, so this condition alone decides the shape of the locus.

Valid only for a stable $G(s)$, after the transients decay. The frequency does not change, only the amplitude and phase.

For $K>0$ and real poles and zeros in the left half plane ($T_{zi},T_{pk}>0$). A quadratic factor adds its own magnitude and phase terms (see the Bode table below). $N$ is the system type, the number of poles at the origin. This additive structure is why Bode plots are built factor by factor.

The contour is traversed clockwise, encloses $Z$ zeros and $P$ poles of $F(s)$, and passes through none of them. $N_0$ = net number of clockwise encirclements of the ORIGIN by the image of $F(s)$. A counterclockwise encirclement counts as $-1$.

For a type-$r$ loop with low-frequency gain constant $K_r$. The image angle runs from $+r\cdot 90^{\circ }$ (at $\omega =0^{-}$) to $-r\cdot 90^{\circ }$ (at $\omega =0^{+}$), with radius $\to \infty$ and always clockwise. If $K_r<0$ (for example, when an open-loop RHP pole makes the low-frequency gain negative), the arc is rotated by $180^{\circ }$ but still runs clockwise.

Applies to unity feedback with a stable closed loop, for inputs $Au(t)$, $Atu(t)$ and $\frac{A}{2}{t}^{2}u(t)$. A type-$N$ loop ($N$ poles at the origin) has zero error for inputs of lower order than $t^N$, a finite error for $t^N$, and an infinite error for higher orders.

Exact for the unity-feedback loop $\frac{{\omega }_n^2}{s(s+2\zeta {\omega }_n)}$, whose closed loop is a pure second-order system. The straight-line rule $\text{PM}\approx 100\zeta$ degrees stays within about $3.5^{\circ }$ of the exact value for $0<\zeta \le 0.6$ and drifts off above that ($\zeta =0.7$ gives $65.2^{\circ }$, not $70^{\circ }$). For a higher-order loop it is only an estimate, valid when one complex pole pair dominates. Lead and lag design from the Bode plot start by turning a $\zeta$ or overshoot specification into a required PM this way.

$n$ states, $m$ inputs, $p$ outputs. $A,B,C,D$ are constant matrices because the system is time-invariant.

The denominator must be monic. The last row of $A$ holds the negated denominator coefficients in ascending powers of $s$, and $C$ holds the numerator coefficients in ascending powers. This form is always controllable. Some texts number the states in reverse, which puts the coefficients in the first row; $G(s)$ is the same either way.

Holds at every $\omega$ ($\omega$ in rad/s, $T$ in s). The magnitude plot is untouched. The phase lag grows linearly with $\omega$, so on a logarithmic frequency axis it falls away more and more steeply.

$P{M}_0$ is the margin of the delay-free loop and ${\omega }_{gc}$ is its gain-crossover frequency (where $|GH|=1$). This is valid because the delay leaves ${\omega }_{gc}$ where it was. Read stability from $P{M}_T>0$ only when the magnitude crosses 0 dB once and the open loop has no right-half-plane poles.