I am going to give you my five cents on this situation, however I would strongly recommend you to consult an accountant practicing in Illinois to see what your options really are, and help you plan the right strategy.
Unfortunately I think this plan would not work. Since you are a resident of Illinois that's where you will report your taxes in, regardless of what state your income bearing company is registered in.
The only scenario I can come up with where you can somehow circumvent the need to pay income taxes for your company income in Illinois is if you form a (let's say Wyoming) corporation or LLC, elect it to be taxed as C-Corporation, and then keep all the profits in the company without distributing dividends to yourself. This way the income generated by this company would be taxed on federal level at some corporate tax rate (for details you need to consult an accountant), there will be no tax paid to Wyoming since Wyoming has no corporate income tax, and if you do not distribute any dividends to yourself the rest of the profit will stay in the company untaxed.
The problem of course is what are you doing with this profit, if you do not distribute it to yourself. Of course you could probably reinvest it into the business or spend on business related activities, which logically should increase the profit further and recursively make your dilemma of what to do with the money even bigger. You could pay yourself a salary as an officer, which brings you to a question of paying income tax on that salary, or distribute just as much dividend as you need to keep going and pay tax on that. Again, all of those questions should be addressed to practicing Illinois accountant.
I hope I gave you enough food for thought, but if you want to get a solid answer I do suggest you to consult an accountant.