LQR (Linear Quadratic Regulator) and LQG (Linear Quadratic Gaussian) are two closely related control techniques that are used in control systems engineering to design optimal feedback controllers. While both LQR and LQG are based on the same principles of optimization and feedback control, there are some significant differences between the two methods.

LQR is a control design method that is used to design an optimal feedback controller for a system with a known dynamic model. The objective of LQR is to minimize a quadratic cost function that is a function of the system’s inputs and outputs. The LQR controller is designed based on the system’s state feedback, which means that the controller’s output is a function of the system’s state variables. LQR is used for open-loop systems, meaning that the system is not subject to any disturbances or noise.

On the other hand, LQG is a control design method that is used to design an optimal feedback controller for a system with unknown or uncertain dynamics. The objective of LQG is to minimize a quadratic cost function that is a function of the system’s inputs and outputs while accounting for measurement noise and disturbances. The LQG controller is designed based on both the system’s state feedback and output feedback, which means that the controller’s output is a function of both the system’s state variables and the system’s outputs. LQG is used for closed-loop systems, meaning that the system is subject to disturbances and noise.

One of the main advantages of LQR and LQG is that they both allow for the design of optimal feedback controllers that can improve the performance of a system. Additionally, both techniques are based on mathematical models of the system, which makes them particularly useful for controlling complex systems that are difficult to model accurately.

However, LQR and LQG do have some limitations. One limitation is that they require accurate mathematical models of the system, which can be difficult to obtain in some cases. Another limitation is that both techniques can be computationally intensive, which can make them difficult to implement in real-time control applications.

When deciding between LQR and LQG, the choice depends on the type of system being controlled. If the system has a known dynamic model and is not subject to disturbances or noise, LQR is a good choice. If the system has unknown or uncertain dynamics and is subject to disturbances or noise, LQG is a better choice. In summary, both LQR and LQG are powerful control techniques that can be used to design optimal feedback controllers, but the choice between the two depends on the specific requirements of the control application.