Linear quadratic regulator (LQR)

The settings of a controller governing either a machine or process (like a self-driving car or chemical reactor) can found by using a mathematical algorithm that minimizes a cost function with weighting factors supplied by a human (engineer).

The cost function is often defined as a sum of the deviations of key measurements (like altitude) from their desired values. The algorithm thus finds those controller settings that minimize undesired deviations. The magnitude of the control action itself may also be included in the cost function.

The LQR algorithm reduces the amount of work done by an engineer to optimize the controller. However, the engineer still needs to specify the cost function parameters, and compare the results with the specified design goals. Often this means that controller construction will be an iterative process in which the engineer judges the “optimal” controllers produced through simulation and then adjusts the parameters to produce a controller more consistent with design goals.

The LQR algorithm is essentially an automated way of finding an appropriate state-feedback controller. As such, it is not uncommon for control engineers to prefer alternative methods, like full state feedback, also known as pole placement, in which there is a clearer relationship between controller parameters and controller behavior. Difficulty in finding the right weighting factors limits the application of the LQR based controller synthesis.

Extra information:

https://en.wikipedia.org/wiki/Linear–quadratic_regulator

Papers:

Autonomous Driving Motion Planning With Constrained Iterative LQR

Constrained iterative LQR for on-road autonomous driving motion planning

 

Video:

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