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

• US 9,849,785 B1
• Filed: 06/21/2016
• Issued: 12/26/2017
• Est. Priority Date: 12/10/2012
• Status: Active Grant

• Current Assignee: The Government Of The United States Of America, As Represented By The Secretary Of The Navy
• Original Assignee: the united states of america as represented by the secretary of the navy
• Inventors: Ross, Isaac Michael, Karpenko, Mark, Proulx, Ronald Joseph
• Primary Examiner(s): Cass, Jean-Paul
• Application Number:
  US15/188,505
• Time in Patent Office: 553 Days
• Field of Search: 701 25
• CPC Class Codes: B60K 35/10 Input arrangements, i.e. from user to vehicle, associated with vehicle functions or specially adapted therefor
• CPC Class Codes: B64G 1/245 Attitude control algorithms for spacecraft attitude control
• CPC Class Codes: B64G 1/286 using control momentum gyroscopes (CMGs)
• CPC Class Codes: G05B 17/02 electric

Abstract

Methods, systems and computer readable media are presented for computing a guidance control policy to transition an uncertain dynamical system from an initial state to a final state, in which a set of points are computed to provide discreet and accurate representation of uncertainty, and in which a guidance control policy is computed based on a set of equations involving the initial state, the final state, state variables, control variables, and parameters, as well as designated parameters of interest, a set of constraints corresponding to state and control variables, a performance metric, statistical distribution types corresponding to the parameters of interest, statistical moments individually corresponding to the parameters of interest, and weighting values corresponding to the parameters of interest. A guidance control policy which defines control variables for transitioning from the initial state to the final state which is robust to the considered system uncertainty is computed according to the computed set of points and the performance metric.

Claims

1. 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, the method comprising:

providing input data to a processor, the input data comprising;

the initial state of the dynamically controlled system in the state space, where the initial state comprises a vector of initial system states in the state space;

the final state of the dynamically controlled system in the state space, where the final state comprises a vector of final system states in the state space;

a set of parameters of interest; and

a nominal value for every parameter of interest in the parameters of interest;

receiving the input 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 and the vector of final system states provided by the input 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 input 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 system comprising the object; and

controlling 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.

2. The method of claim 1 further comprising:

communicating one or more constraints to the processor, where the one or more constraints comprise a constraint on the vector of system states {right arrow over (x)}_K(t), a constraint on the vector of system controls {right arrow over (u)}(t), a constraints on the set of system parameters {right arrow over (p)}_K, or combinations thereof; and

optimizing the performance metric based on the one or more constraints.

3. The method of claim 1 where the performance metric minimizes uncertainty in the vector of system states {right arrow over (x)}_K(t) over the time interval, at a specific instant within the time interval, or a combination thereof.

4. The method of claim 3 wherein g({right arrow over (F)}(t_f) is any function whose argument is related to the set of the parameters of interest as described by the plurality of HS points and where the final state is defined at a final time t_f, and where the performance metric comprises J=g({right arrow over (F)}(t_f)) where g({right arrow over (F)}(t_f)) is evaluated at the final time t_f.

5. The method of claim 3 wherein g({right arrow over (F)}(t_f)) is any function whose argument is related to the set of the parameters of interest as described by the plurality of HS points and where the performance metric comprises an integral of g({right arrow over (F)}(t)) with respect to time.

6. The method of claim 3 wherein g({right arrow over (F)}(t)) is any function whose argument is related to the set of the parameters of interest as described by the plurality of HS points and where the final state is defined at a final time t_f, and where the performance metric comprises a linear combination of an integral of g({right arrow over (F)}(t)) with respect to time plus a function h({right arrow over (F)}(t_f)), wherein h({right arrow over (F)}(t_f)) is any function whose argument is related to the set of the parameters of interest as described by the plurality of HS points and where h({right arrow over (F)}(t_f)) is evaluated at the final time t_f.

7. The method of claim 1 further comprising providing the plurality of HS points using processor-performed steps by:

generating a space of uncertain parameters, where the space of uncertain parameters has a quantity of dimensions equal to a quantity of parameters of interest comprising the set of parameters of interest, and assigning an axis to each dimension of the space of uncertain parameters;

retrieving one or more statistical distribution types from the memory;

assigning one statistical distribution type to each specific parameter of interest where the statistical distribution type corresponds to an expected variation of the each specific parameter of interest;

mapping the statistical distribution type assigned to the each specific parameter of interest to a single axis of the space of uncertain parameters, such that each axis assigned to the each dimension of the space of uncertain parameters reflects the expected variation of one specific parameter of interest; and

providing the each HS point in the plurality of HS points by selecting a point within the space of uncertain parameters, and providing the set of dimension values for the each HS point by describing a location of the each HS point using the expected variation on every single axis of the space of uncertain parameters, such that every dimension value in the set of dimension values for the each HS point represents an uncertainty associated with a particular specific parameter of interest.

8. The method of claim 7 where the statistical distribution type comprises a uniform distribution, a Gaussian distribution, a Beta distribution, a Poisson distribution, a frequency distribution, a binomial distribution, or combinations thereof.

9. The method of claim 7 further comprising assigning a weight to each HS point in the plurality of HS points where the weight is a positive real number and where a summation of the weights assigned is equal to one, and where the plurality of HS points are provided by optimization of a cost junction J comprising the each HS point and the weight assigned to the each HS point.

10. The method of claim 1 where the input data further comprises an initial location in a three-dimensional physical space and a final location in the three-dimensional physical space, and where the object trajectory is a path of the object in the three-dimensional physical space from the initial location to the final location.

11. The method of claim 10 where the object is a spacecraft and where the spacecraft comprises a plurality of control moment gyroscopes, and further comprising:

providing the guidance command policy as a vector of gimbal rate commands; and

operating the plurality of control moment gyroscopes in accordance with the vector of gimbal rate commands.

12. The method of claim 1 where the processor does not comprise the object and further 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; and

communicating the guidance command policy from the processor to the guidance system comprising the object using the communications interface.

13. 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.

14. The method of claim 13 further comprising providing the plurality of HS points using processor-performed steps by:

generating a space of uncertain parameters, where the space of uncertain parameters has a quantity of dimensions equal to a quantity of parameters of interest comprising the set of parameters of interest, and assigning an axis to each dimension of the space of uncertain parameters;

retrieving one or more statistical distribution types from the memory;

assigning one statistical distribution type to each specific parameter of interest where the statistical distribution type corresponds to an expected variation of the each specific parameter of interest;

mapping the statistical distribution type assigned to the each specific parameter of interest to a single axis of the space of uncertain parameters, such that each axis assigned to the each dimension of the space of uncertain parameters reflects the expected variation of one specific parameter of interest; and

providing the each HS point in the plurality of HS points by selecting a point within the space of uncertain parameters, and providing the set of dimension values for the each HS point by describing a location of the each HS point using the expected variation on every single axis of the space of uncertain parameters, such that every dimension value in the set of dimension values for the each HS point represents an uncertainty associated with a particular specific parameter of interest.