A controllable system is a control system where we can change the state at any infinite time. In a controllable system, we have full freedom to modify, control, manipulate, or change the state or any variable parameter of the system. And this is called Controllability. Generally, in a controllable system, there is a control law or control signal named as u(t) exists that can drive the state from any initial value to the final value with respect to a finite interval of time. If anyone variable cannot be changed then it cannot be called a completely controllable system. A control system will be called as a completely controllable system when all the parameters or variables can be changed. Controllability gives us permission to obtain more features and functions.

Controllable systems are used in various applications from simple household appliances to complex industrial processes. Here are a few examples of Controllable systems,
1. Thermostat Control
2. Robotic Arm Control
3. Industrial Process Control

A controllable system typically consists of input devices, a processing unit, and output devices. The input devices provide signals or data to the processing unit, which analyzes and processes the input information. Based on this analysis, the processing unit generates control signals that are sent to the output devices, resulting in specific actions or changes in the system.

Actually, controllability is most suitable for a linear system. It is seen that for a nonlinear system, it is not perfect or unsuitable. For nonlinear systems, another tool such as Accessibility or Differential flatness can be used.

Also, note that an uncontrollable system does not mean that you can control anything. An uncontrollable system may have two parts - 1. Controllable Part and 2. Uncontrollable Part

In the uncontrollable part where all the variables or parameters are stable or constant, you cannot change anything. On the other hand, in the controllable part, where you can change the parameters, stable variables. Also, note that if the uncontrollable part is stable then it is easy to stabilize otherwise it is not stabilizable. When you can control only a few parameters of a control system then it is called partially controllable. Remember that controllability is mostly used with the word Observability. The controllability is related to the input of the system whereas Observability is related to the output of the system. There are huge equations and formulas to measure and find out the controllability and observability. In this article, I have given the basic theoretical formula.

The main difference between a controllable system and an uncontrollable system is that, A controllable system is one that can be controlled, manipulated, or influenced by external factors or inputs. In other words, the behavior or output of a controllable system can be altered or adjusted through the application of control signals. On the other hand, an uncontrollable system refers to a system that cannot be directly controlled or manipulated by external factors. These systems operate independently or are governed by internal mechanisms or natural processes that cannot be altered easily through external inputs. The behavior of uncontrollable systems is typically influenced by inherent properties, environmental factors, or physical laws.