Objectives may vary depending upon the application but in simpler terms you are looking for the following in time-based systems:
1. Improved transient response of the system in terms of overshoots, settling time and rise time.
2. (1) can be performed by designing a controller based on frequency response or through root locus by carefully selecting the poles and zeros of your controller.
3. Some optimization based controllers, will try to minimize or optimize your objective function. The objective function can either be eigenvalue based function or time based function. The eigenvalue based objective function will try to push the dominant poles further to the left hand side of the root locus until the objective function is optimized. The time based function, as the name suggests, optimizes a parameter while the system is running e.g. minimizing the deviation of the system frequency from the nominal value in a electrical system.
First Part Analysis: First analyze the existing system and obtain the picture of it, like it is stable or not. If stable what is its relative stability (may be in terms of overshoot & settling times, phase and gain margins etc.). Also controllability and observability of the system.
Second part Controller Design: Design a suitable controller based on the above system analysis. You may use a traditional PID controller, state feedback controllers, optimal control or new generation controllers like ADRC.
I'll try to answer your question in a philosophical way, so that you can grasp the overall picture. Since you asked about the main objectives, perhaps you can find them in the answers of “What are the main objectives of Medical Diagnosis and Treatment?”
Medical Diagnosis is a process that the doctors take to determine which disease or illness explains a patient's symptoms and signs, before they can recommend the available treatments to deal with the disease or illness, with the goal of curing it (best outcome) or preventing it from getting worse and without undesired side effects.
Similarly, in Control System Analysis, engineers use tools to carry out examination to understand the characteristics and identify the properties of a plant, before designing a feedback controller to make a plant behave in a desired manner (meeting a set of performance specifications). In technical words, the control goal is to make a plant stable that operates in a predictable way, either by eliminating the error or by regulating the error bounded within the tolerance band, which ultimately leads to SAFETY (for users & environments), RELIABILITY (for operators), and PROFITABILITY (for stakeholders).
If you are in some industry, the main objectives of Control analysis and design and is first of all the guarantee of your system stability in various possible environments and then, not less important, guaranteeing some desired performance. While stable operation is always needed, performance usually must be continuously improved, which for many systems means faster and faster and also more and more precise.
Now, if you are a student towards some degree, the same objectives could remain relevant, yet the main objective is to satisfy your adviser’s requirements and get the degree. :-)