Sliding mode control techniques for steerable systems

A system comprising:

• a directional drilling tool disposed in the wellbore and having a plurality of computing devices;
• one or more processors, communicatively coupled with the computing devices, and having a memory having stored therein instructions which, when executed, cause the one or more processors to:
• define, by a sliding mode controller, a sliding hypersurface for reducing a trajectory error in one or more error dimensions, the one or more error dimensions includes at least a first dimension that corresponds to a position based error and a second dimension that corresponds to an attitude based error;
• determine, by the slide mode controller, a current trajectory error between a current trajectory of the directional drilling tool and a reference trajectory for a curved path, the current trajectory error corresponds to a current error position in the one or more error dimensions;
• calculate, by the sliding mode controller, a sliding mode vector that originates from the current error position and substantially conforms to the sliding hypersurface in the one or more error dimensions;
• determine, by the sliding mode controller, a feedback control input for the directional drilling tool based on the sliding mode vector;
• instruct, by the sliding mode controller, the directional drilling tool to generate a wellbore path according to the feedback control input;
• and
• update the current trajectory error based on at least one of a change in position or a change in attitude for the directional drilling tool.

A non-transitory computer-readable storage medium having instructions stored thereon which, when executed by one or more processors, cause the one or more processors to:

• define, by a sliding mode controller, a sliding hypersurface for reducing a trajectory error in one or more error dimensions, the one or more error dimensions includes at least a first dimension that corresponds to a position based error and a second dimension that corresponds to an attitude based error;
• determine, by the slide mode controller, a current trajectory error between a current trajectory of a directional drilling tool and a reference trajectory for a curved path, the current trajectory error corresponds to a current error position in the one or more error dimensions;
• calculate, by the sliding mode controller, a sliding mode vector that originates from the current error position and substantially conforms to the sliding hypersurface in the one or more error dimensions;
• determine, by the sliding mode controller, a feedback control input for the directional drilling tool based on the sliding mode vector;
• instruct, by the sliding mode controller, the directional drilling tool to generate a wellbore path according to the feedback control input;
• and
• update the current trajectory error based on at least one of a change in position or a change in attitude for the directional drilling tool.