Usually, when data is collected in an experiment, scientists have a control. Experiments are seen as flawed without controls. I am just curious as to where else scientists have 'built' controls using models. Obviously, we cannot find a real 'control Earth', so we must make climate models, but then why can we even proceed with the experiment (how are we so sure the models act within a small error of the 'control Earth')?
It's true that, usually, when data is collected in an experiment scientists have a control, and very often not having a control results in a flawed experiment - that is, an experiment from which less conclusions can be drawn.
Experiments are a very efficient way to gather high quality data, but in some fields they might even be impossible. However, experiments aren't the only way to get knowledge and they aren't mandatory to make science.
In some fields experiments are quite easy and therefore most research is based in experiments. For example, it's usually easy for chemist to test the way substances react by making experiments.
In other fields some questions can't be resolved by direct experiments due to practical or ethical reasons. In a known textbook example, sociologists are interested in what circumstances cause suicide rates to rise or drop, but ethics prevents them from performing an experiment by submitting a sample of people to circumstances supposed to induce suicide and compare suicide rate of this sample with suicide rate in a control sample. Therefore, sociologist need to draw conclusions from observational data - that is, observing real population under natural circumstances they can't adjust - or from experiments of related phenomena that can be performed without ethical concerns.
We in Earth sciences face a similar problem: some experiments can't be done because of scale or just because we just have an only Earth. For example, how do we know that rivers cause deltas? Ideally, we'd like to do an experiment by selecting several coast tracts, putting rivers in a randomly chosen sample of them, waiting a few millions of years and comparing evolution of the coast tracts with and without rivers. We can't do this experiment, but it doesn't prevent geologists to have learned a lot about deltas, because we have a lot of data from observation of naturally occurring deltas, from small scale experiments, and from study of auxiliary sciences (physics, hydraulics, etc.).
Climate is similar to deltas. Just as we can experiment with setting and removing rivers, we can't change CO2 levels of the whole atmosphere to measure what happens, but we can observe what actually happens, search for geological register of what happened in the past, do a lot of small scale experiments, and use our knowledge of basic sciences and draw conclusions about climate evolution from them.
Clearly, interpretation of that data is not as straightforward as data from a big experiment would be, but it is still possible and useful. In fact, it's the way most of current knowledge in many fields has been built.
All that without needing a random sample of Earths nor a control Earth.
What do you mean by a control Earth? There is a control Earth: all of history until now. We don't know everything that happened on this control Earth, but we have some reasonable estimates of how temperature, carbon dioxide levels, and other factors changed in the past, and so we us those as best we can to develop how the Earth did act.
However, we have another data point to use, other than our limited information about the past Earth. We have physics to describe heat transfer, radiation emission and absorption and more. We could use this knowledge to create a model wholly independent of any knowledge of past Earth conditions. Since the various physics we use is all experimentally confirmed, we can be much more confident in it than we can in past Earth conditions.
When you build a climate model, you make sure that the things that did happen can be modeled correctly by that model. The climate models that I am familiar with meet that criteria, for some subsection of the past that was used to create the model. Admittedly, the model is only as good as the past data we are using to calibrate it; but in the physics of the model we can be certain.
So we have created models that can correctly predict the temperature changes in some subset of the past, and we are using those to extrapolate into the future.