Method and apparatus for state space trajectory control of uncertain dynamical systems

A computer-implemented method for generating and executing a guidance control policy operatively associated with transitioning a dynamically controlled system controlling an object through a state space from an initial state to a final state over a time interval and an object trajectory using a processor, where the processor does not comprise the object, the method comprising:

• communicating the initial state of the dynamically controlled system in the state space from the object to the processor using a communications interface providing communication between the object and the processor, where the initial state comprises a vector of initial system states in the state space and where the initial state further comprises an initial location in a three-dimensional physical space;
• providing additional data to the processor, the additional data comprising:
• the final state of the dynamically controlled system in the state space, where the final state is a vector of final system states in the state space;
• a final location in the three-dimensional physical space;
• a set of parameters of interest;
• and
• a nominal value for every parameter of interest in the parameters of interest;
• receiving the additional data at the processor and determining the guidance command policy using processor-performed steps comprising:
• providing a plurality of Hyper-Pseudospectral (HS) points, where each HS point comprises a set of dimension values, and where each dimension value in the set of dimension values corresponds to a specific parameter of interest in the set of parameters of interest, and where the each dimension value represents an uncertainty associated with the specific parameter of interest;
• storing the plurality of Hyper-Pseudospectral (HS) points in a memory;
• formulating a nominal differential flow {right arrow over (F)}0(t) by substituting the nominal value for each parameter of interest into a dynamical model {right arrow over (F)}(t), where the dynamical model {right arrow over (F)}(t), comprises:
• a vector of system states {right arrow over (x)}K(t) where the vector of system states {right arrow over (x)}K(t) comprises the vector of initial system states provided by the object and the vector of final system states provided by the additional data;
• a vector of system controls {right arrow over (u)}(t) where the vector of system controls {right arrow over (u)}(t) comprises control variables within the dynamically controlled system over the time interval from the initial state to the final state;
• and
• a set of system parameters {right arrow over (p)}K where each system parameter in the set of system parameters {right arrow over (p)}K describes an uncertain constant within the dynamical model {right arrow over (F)}(t), and where each parameter of interest provided by the additional data is a member of the set of system parameters {right arrow over (p)}K;
• formulating an off nominal flow {right arrow over (F)}k(t) for each HS point in the plurality of HS points by:
• retrieving each HS point from the memory;
• and
• substituting the each HS point into the dynamical model {right arrow over (F)}(t), thereby generating one or more off nominal flows {right arrow over (P)}(t);
• receiving a desired system state, where the desired system state comprises one or more vectors of system states {right arrow over (x)}K(t) within the time interval from the initial state to the final state;
• solving for optimized system controls {right arrow over (u)}(t) where the optimized system controls {right arrow over (u)}(t) is the vector of system controls {right arrow over (u)}(t) within the nominal differential flow {right arrow over (F)}0(t) and the one or more off nominal flows {right arrow over (P)}(t) which optimizes a performance metric, where the performance metric compares the desired system state with a resulting system state, where the resulting system state comprises one or more a vectors of system states {right arrow over (x)}K(t) generated by the optimized system controls {right arrow over (u)}(t);
• and
• providing the guidance command policy where the guidance command policy comprises the optimized system controls {right arrow over (u)}(t);
• communicating the guidance command policy from the processor to a guidance control module comprising the object through the communications interface providing communication between the object and the processor;
• and
• utilizing the guidance control module to control the dynamically controlled system in accordance with the guidance control policy and transitioning the object through the state space from the initial state to the final state over the time interval and the object trajectory, where the object trajectory is a path of the object in the three-dimensional physical space from the initial location to the final location.