Unfortunately the answer is not as straightforward as you might like.
Let's define to Open Loop Bandwidth as the frequency at which the loop gain is 1 (standard definition). Above this frequency the loop can not respond effectively to commands or feedback.
In an ideal system Sampling Rate compared to Open Loop bandwidth does not really matter so long as the Nyquist criterion is satisfied, but in any real system delay is usually associated with the sample rate clock (e.g. digital controller calculation time), then the relationship between Sampling Rate and Open Loop bandwidth really does matter. As a guideline the crossover frequency must be less than the reciprocal of the time delay in the loop : Wc < 1/Td .
There are several factors involved in loop delay: Time to measure the sensed value, Time to calculate the response, time to Activate the response.
The time to activate the response may be misleading. It's not just the response of the actuator. It's also the delay involved in converting from Discrete time to Continuous time (e.g. a DAC). The delay here is due to the Zero-Order-Hold effect of the conversion. All Linear Discrete Time control systems have this delay. The delay = T/2 where T is the sample period. So, assuming (as you have stated) all delays in the loop are negligible, then we only have to account for the ZOH delay (T/2) and therefore the max achievable bandwidth is Wc < 2fs (where fs is the sample rate) which gives the max achievable bandwidth in Hertz as fc < fs/pi .
Keeping the open loop bandwidth less that 1/10 sample rate is a good idea for a discrete time controller, controlling a continuous time plant. That's because you can reduce the effect of frequency warping in the model conversions from continuous to discrete.
However, despite all that, the most important aspects are not covered by any of the above discussion. You really do need to know what you are controlling because several factors may influence the bandwidth of the controller as well as the type and structure.
Is it a stable open-loop system? If not then this open loop bandwidth must be significantly greater than the frequency of the unstable mode.
Is it linear? If not then a linear controller may be unsuitable. You may need something like Receding Horizon Control (which does not have a bandwidth as such).
Is it minimum phase? If it is not the open loop bandwidth is usually required to be less than the RHP zero frequencies.
Without looking at the paper, and responding only to your question- the heating system described is called proportional control in the industry (assuming the setpoint is temperature, the measured variable is temperature and the output is power-- as is typical). In fact in many heating systems the gain can be high enough that the droop due to demand changes is negligible (and the system can be correct either by changing the setpoint or by applying a correction factor called 'manual reset' that the error is nulled a given setpoint with nominal demand.
So, you are correct that if there was a large demand change from nominal there would be a persistent error with proportional control, but it's not necessarily of any practical importance.
The thermal capacity of the object being heated does indeed introduce a pole in the response, but the heat loss increases with temperature difference (at least proportionally, but often much faster with convection or radiation losses) so the result is not an integral control response.
If you had a block of material that was sufficiently isolated from the environment (almost no conductive, convective or radiation losses) then you could consider it to be an integral controller, but that does not represent even a rough approximation to reality in any of the thousands of systems I've worked with.
Best Answer
As you know, poles and zeroes are the respective roots of the denominator and the numerator of a complex transfer function. Once they are known, they can be placed on a map - the \$s\$-plane - depending on their real and imaginary components. The below drawing shows an example for a transfer function featuring one zero and three poles:
The left-side of the map is called the left-half-plane abbreviated LHP while the right-side is the right-half-plane or RHP. Depending on the real value of the roots, poles and zeroes can be in either side. Without entering into the details (the literature abounds on the subject), you can infer the position of the zero or the pole if the transfer function is written in the following way for a pole:
In the above drawing, the "+" indicates a pole located in the left-half plane, leading to a damped response. The phase starts from 0° and asymptotically hits -90° as frequency approaches infinity. This is a classical pole also named stable pole. Take the same transfer function and replace the sign by a minus, and you propel the pole in the right-half plane to make it a RHPP. The magnitude response is unchanged but the phase response is now the opposite as before. It still starts from 0° but increases to 90° as frequency approaches infinity. Having a RHPP in the open-loop gain implies a robust compensation strategy bringing the pole back in the left-half plane once the loop is closed. A control system featuring a closed-loop transfer function with a RHPP cannot be operated.
Similar observations apply to the zero whose phase response also changes depending where it stays in the \$s\$-plane:
With a LHPZ, the phase starts from 0° and increases up to 90°. We say the zero boosts the phase when it appears in a transfer function. Now take the same zero and push it in the right-half plane then the phase response changes: the RHPZ no longer boosts the phase but lags it down to 90° as frequency approaches infinity.
Let's have a look at a transfer function having a LHP zero and two poles. The response is shown below. The phase response of the process to be compensated nicely lands to -90° as the LHP zero response compensates the lag of the high-frequency LHP pole.
Assume you want to crossover at 5 kHz or so. You think of a compensation strategy featuring some response to obtain a good phase margin at 5 kHz. I have arranged a pole and a zero to meet this goal as shown in the low side of the above drawing.
Now, think of the same transfer function but having an extra RHP zero on top of the existing LHP zero. The plant response is no longer the same with the RHPZ arbitrarily placed at 8 kHz:
The phase now hits -180° at high frequency, consequence of the RHP zero. Should you try to keep the same compensator, the phase margin would no longer be as high as before:
To maintain a good phase margin despite the RHP zero, you will have to crossover at a lower frequency, where the effects of the phase lag brought by the RHP zero are less observable. In other words, stay away from the RHPZ and its phase stress. By doing this, you slow down your converter but enjoy an acceptable phase margin:
Now let's see what it implies in terms of control system. I will take the example of a switching converter as it is my field of expertise. Think of a converter transferring the energy in a two-step approach, like a boost or a buck-boost converter: first you store the energy in the inductor during the on-time then transfer it to the load during the off-time. Should a sudden power demand occur, the converter can't immediately react and must go first through another storing-energy phase before answering the demand. This intermediate phase naturally introduces a delay in the response to a change: the current in the inductor has to grow cycle by cycle (it cannot instantaneously jump to the next current setpoint) but this current increase is hampered by the inductor value and the available volt-seconds. The mathematical model of this delay in the response is the familiar RHP zero which appears in the control-to-output transfer functions of the said converters. The only way to stabilize them is to select a crossover far before the RHPZ phase lag brings troubles. Usually, people adopt a crossover placed 20-30% below the worst-case RHPZ position.
A pure delay also introduces a phase lag while its magnitude is constant to 1 or 0 dB. It can be inserted in the transfer function as shown below. This is excerpted from an APEC seminar I taught in 2012 The Dark Side of Control Theory:
This delay \$\tau\$ which in Laplace can be expressed as \$e^{-s\tau}\$ will lag the phase and affects the plant response. The phase margin can suffer and you should account for its presence (and variability) especially if you shoot for a high bandwidth. Delays are caused by propagation times, conversion times etc. Here, you see that a 250-ns delay incurs a phase lag of 9° at 100 kHz. Not a big deal if you plan on closing the loop for a 1-kHz crossover but if you shoot for a much higher figure, you need to account for its contribution.
Finally, keep in mind that the Bode stability argument is meant for minimum-phase transfer functions (no delay or RHP poles and zeroes in the expression). When delays are present, the Bode plot can mislead you in determining the system stability.