The transfer function of a real system, H ( s) , is given as :

H(s) = As + B/ s² +Cs +D

where A, B ,C and D are positive constants. This system cannot operate as

The correct options are: an integrator and high pass filter.

Let's analyze the given transfer function of a real system:
$H\left(s\right)=\frac{As+B}{s^2+Cs+D}$
where ^A, ^B, ^C, and ^D are positive real constants.

To determine what kind of system or filter this transfer function can represent, we evaluate its frequency response by substituting $s=j\omega$:
$H\left(j\omega \right)=\frac{jA\omega +B}{-{\omega }^2+jC\omega +D}$

Now, let's examine the behavior at extreme frequencies:

1. Low Frequency Limit ($\omega \to 0$):
$\underset{\omega \to 0}{\lim }H\left(j\omega \right)=\frac{B}{D}$
Since ^B and ^D are positive constants, the response at zero frequency is a non-zero finite value ($\frac{B}{D}>0$).

2. High Frequency Limit ($\omega \to \infty$):
$\underset{\omega \to \infty }{\lim }H\left(j\omega \right)=\underset{\omega \to \infty }{\lim }\frac{jA\omega }{-{\omega }^2}=0$
The response at extremely high frequencies goes to zero.

Now we evaluate each of the filter and system types based on these boundary conditions:

• Low Pass Filter: Requires a non-zero response at $\omega \to 0$ and zero response at $\omega \to \infty$. The transfer function satisfies this, so it can act as a low pass filter (for example, if ^A is very small or zero, though here ^A is positive).
• Band Pass Filter: Requires zero response at $\omega \to 0$ and zero response at $\omega \to \infty$. If $B=0$, it would be a bandpass filter. However, the problem statement defines ^B as a positive constant ($B>0$). Thus, strictly speaking, $H\left(0\right)\ne 0$. However, depending on the relative values of the coefficients, it can approximate a band-pass characteristic if there is a peak at some middle frequency. But let's look at the remaining options which are absolutely impossible.
• High Pass Filter: A high pass filter requires a non-zero response at $\omega \to \infty$ and a zero response at $\omega \to 0$. Since $H(j\infty )=0$, this system cannot operate as a high pass filter under any conditions.
• Integrator: An ideal integrator has a transfer function of the form $H_{\mathrm{int}}\left(s\right)=\frac{K}{s}$, which goes to infinity as $s\to 0$ ($\omega \to 0$). Since the given transfer function has a finite value $\frac{B}{D}$ at $s=0$ (because $D>0$), this system cannot operate as an integrator.

Therefore, the system cannot operate as an integrator or a high pass filter.