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I am searching for a Formal Definition of bandwidth given the transfer function of a system, or the bode plot.
Is it the frequency $\omega_b$ in which $$20log|H(j\omega_b)| = -3\text{dB}$$ or $$20log|H(j\omega_b)| = 20log|H(j0)| - 3\text{dB}$$
And if there is a definition without dB, I would like to know about it. Thanks in advance!
EDIT-homework: Application of definition: In which interval is $\omega_b$ contained? 
Observation: Bandwidth is defined as the frequency range $[\omega_1 \ \omega_2]$ over which the control of the system is "effective". Usually, $\omega_1 = 0$ and then, by definition, $\omega_2 = \omega_B$ is the bandwidth.
Definition: The (closed-loop) bandwidth, $\omega_b$, is the frequency where the norm of the sensitivity function, $|S(j\omega)|$, first crosses the $-3\text{dB}$ line from below. This means that: $$ \big|S(j\omega_B)\big| = \frac{1}{\sqrt{2}} = 0.707 \approx -3\text{dB} $$
The bandwidth in terms of the closed loop transfer function $T(s)$, $\omega_{BT}$, is the highest frequency at which the norm of the closed loop transfer function, $|T(j\omega)|$, crosses the $-3\text{dB}$ line from above. However, this is usually a poor indicator of performance and is mathematically stated as: $$ \big|T(j\omega_{BT})\big| = \frac{1}{\sqrt{2}} = 0.707 \approx -3\text{dB} $$
Τhe gain crossover frequency, $\omega_c$, is the frequency where the norm of the loop transfer function, $|L(j\omega)|$, first crosses the $0\text{dB}$ line from above. And mathematically: $$ \big|L(j\omega_c)\big| = 1 = 0\text{dB} $$ These three frequencies, and for systems which have a phase margin of the order $PM < 90^o$, are related with one another by the following inequality: $$ \omega_B < \omega_c < \omega_{BT} $$ All of the above definitions and frequencies of the system can be used as specifications for controller design and especially when using the loop shaping design method, where the designer tries to impose certain behaviour to the loop transfer function $L(s)$. As a last comment, I would like to write down the relations between the functions mentioned: $L(s), S(s), T(s)$. Suppose we have a plant described by $G(s)$ and which is controlled using the controller $K(s)$. Then:
$$ L(s) = G(s)\cdot K(s) \rightarrow \text{Loop Transfer Function} $$ $$ S(s) = \frac{1}{1+L(s)} \rightarrow \text{Sensitivity Function} $$ $$ T(s) = \frac{L(s)}{1+L(s)} \rightarrow \text{Closed Loop Transfer Function} $$ $$ P(s) = 1 + L(s) \rightarrow \text{System's Characteristic Polynomial} $$ Note that for the loop transfer function the multiplication should be $G(s)\cdot K(s)$ because when dealing with multivariable systems (MIMO) this is how you get the proper loop transfer matrix. Only if both matrices $G(s), K(s)$ are diagonal then the multiplications $G(s)\cdot K(s) = K(s)\cdot G(s)$ produce the same results. However, for single-input single-output systems (SISO), it doesn't matter.
