System and Method for Feasibly Positioning Servomotors with Unmodeled Dynamics

A computing system for generating optimal tracking control (TC) policies for controlling a machine having machine uncertainties to track a given time-varying reference (GTVR) trajectory, comprising:

• a memory for storing data and program instructions, the data includes the GTVR trajectory, a constraint-admissible invariant set (CAIS) of machine states satisfying machine state constraints and a corresponding TC policy mapping a machine state within the CAIS to a control input satisfying control input rate constraints;
• a transceiver to receive data including an updated augmented state of the machine;
• and
• a processor of the computing system is configured to jointly control the computing system to control the machine operation to drive an augmented state of the machine to zero, and update the CAIS and the TC policy, wherein, for executing the joint control, the processor is configured to
• control the computing system using a sequence of control inputs generated using the TC policy and a sequence of augmented machine states within the CAIS corresponding to the sequence of tracking control inputs;
• and
• execute a constrained tracking approximate dynamic program (TADP) using the received data to update the value function, update the CAIS and the corresponding TC policy;
• and
• control the computing system using the updated TC policy, the updated CAIS and the corresponding updated value function to control the machine to track the GTVR trajectory.

A method having a computing system for generating optimal tracking control (TC) policies for controlling a machine having machine uncertainties to track a given time-varying reference (GTVR) trajectory, comprising:

• storing data and program instructions in a memory, the data includes the GTVR trajectory, a constraint-admissible invariant set (CAIS) of machine states satisfying machine state constraints and a corresponding TC policy mapping a machine state within the CAIS to a control input satisfying control input rate constraints;
• receiving data via a transceiver, the data including an updated augmented state of the machine;
• and
• using a processor of the computing system to jointly control the computing system to control the machine operation to drive an augmented state of the machine to zero, and update the CAIS and the TC policy, wherein, for executing the joint control, the processor is configured for
• controlling the computing system using a sequence of control inputs generated using the TC policy and a sequence of augmented machine states within the CAIS corresponding to the sequence of tracking control inputs;
• and
• executing a constrained tracking approximate dynamic program (TADP) using the received data to update the value function, update the CAIS and the corresponding TC policy;
• and
• control the computing system using the updated TC policy, the updated CAIS and the corresponding updated value function to control the machine to track the GTVR trajectory.

An apparatus having a computing system for generating optimal tracking control (TC) policies for controlling a servomotor having servomotor uncertainties to track a given time-varying reference (GTVR) trajectory, comprising:

• a memory for storing data and program instructions, the data includes the GTVR trajectory, a constraint-admissible invariant set (CAIS) of servomotor states and tracking errors, the CAIS satisfies servomotor state constraints and a corresponding control policy mapping a servomotor state within the CAIS to a control input satisfying control input constraints, wherein a control of the computing system to control an augmented servomotor state within the CAIS according to the corresponding TC policy maintains the augmented servomotor within the CAIS, while tracking the GTVR trajectory;
• a transceiver to receive data including an updated augmented state of the servomotor that includes a difference between updated states of the servomotor and states of the servomotor and the tracking error from the memory;
• and
• a processor of the computing system is configured to jointly control the computing system to control the servomotor operation to drive an augmented state of the servomotor to zero, update the CAIS and the TC policy, wherein, for executing the joint control, the processor is configured to
• control the computing system using the including a sequence of control inputs generated using the TC policy and a sequence of augmented servomotor states within the CAIS corresponding to the sequence of tracking control inputs;
• and
• execute a constrained tracking approximate dynamic program (TADP) using the received data to update the value function, update the CAIS and the corresponding TC policy, wherein, to execute the constrained TADP, the processor is configured to:
• execute a constrained policy evaluation to produce an updated CAIS and a corresponding updated value function of an augmented servomotor state, wherein the updated value function improves a tracking cost function of the operation of the servomotor and the updated CAIS enforces satisfaction of the augmented servomotor state and incremental control input constraints, for the GTVR trajectory;
• execute a constrained policy improvement to generate an updated TC policy that improves the tracking cost function of the servomotor operation according to the updated CAIS and the corresponding updated value function, for the GTVR trajectory;
• execute a reference scaling operation that employs the stored data to generate a modified reference trajectory, wherein the modified reference trajectory is a reformulation of the given reference trajectory to avoid constraint violation at time instants where the given reference trajectory exhibits jump discontinuities;
• and
• control the computing system using the updated TC policy and the updated CAIS and the corresponding updated value function to control the servomotor to track the GTVR trajectory.