Nonlinear toolface control system for a rotary steerable drilling tool

A non-transitory machine-readable medium comprising instructions stored therein, the instructions executable by one or more processors to facilitate performing a method of forming a wellbore, the method comprising:

• determining a desired toolface of a drilling tool;
• calculating a toolface error by determining a difference between a current toolface of the drilling tool and the desired toolface;
• generating a model to describe a behavior of the drilling tool;
• modifying the model to create a modified model based on a one-to-one linear relationship between a torque of a turbine of the drilling tool and the current toolface;
• mapping a one-to-one nonlinear relationship between the torque of the turbine and an angular displacement of a motor of the drilling tool;
• calculating a correction to reduce the toolface error, the correction based on the modified model by applying the mapping to the torque of the turbine;
• transmitting a signal to the motor of the drilling tool such that the signal changes the angular displacement of the motor to adjust the current toolface based on the correction;
• and
• drilling a wellbore with a drill bit oriented at the desired toolface.

A downhole drilling tool control system comprising:

• a processor;
• a memory communicatively coupled to the processor with computer program instructions stored therein, the instructions configured to, when executed by the processor, cause the processor to:
• determine a desired toolface of a drilling tool;
• calculate a toolface error by determining a difference between a current toolface of the drilling tool and the desired toolface;
• generate a model to describe a behavior of the drilling tool;
• modify the model to create a modified model based on a one-to-one linear relationship between a torque of a turbine of the drilling tool and the current toolface;
• map a one-to-one nonlinear relationship between the torque of the turbine and an angular displacement of a motor of the drilling tool;
• calculate a correction to reduce the toolface error, the correction based on the modified model by applying the mapping to the torque of the turbine;
• transmit a signal to the motor of the drilling tool such that the signal changes the angular displacement of the motor to adjust the current toolface based on the correction;
• and
• drill a wellbore with a drill bit oriented at the desired toolface.