A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. [1] Its practicality was demonstrated in 2008 by Ross et al. [2] in a laboratory implementation of the concept. The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory. [3]
A Carathéodory-π solution addresses the fundamental problem of defining a solution to a differential equation,
when g(x,t) is not differentiable with respect to x. Such problems arise quite naturally [4] in defining the meaning of a solution to a controlled differential equation,
when the control, u, is given by a feedback law,
where the function k(x,t) may be non-smooth with respect to x. Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s. [5]
An ordinary differential equation,
is equivalent to a controlled differential equation,
with feedback control, . Then, given an initial value problem, Ross partitions the time interval to a grid, with . From to , generate a control trajectory,
to the controlled differential equation,
A Carathéodory solution exists for the above equation because has discontinuities at most in t, the independent variable. At , set and restart the system with ,
Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-π solution.
A Carathéodory-π solution can be applied towards the practical stabilization of a control system. [6] [7] It has been used to stabilize an inverted pendulum, [6] control and optimize the motion of robots, [7] [8] slew and control the NPSAT1 spacecraft [3] and produce guidance commands for low-thrust space missions. [2]
A Carathéodory-π solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. [1] Its practicality was demonstrated in 2008 by Ross et al. [2] in a laboratory implementation of the concept. The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory. [3]
A Carathéodory-π solution addresses the fundamental problem of defining a solution to a differential equation,
when g(x,t) is not differentiable with respect to x. Such problems arise quite naturally [4] in defining the meaning of a solution to a controlled differential equation,
when the control, u, is given by a feedback law,
where the function k(x,t) may be non-smooth with respect to x. Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s. [5]
An ordinary differential equation,
is equivalent to a controlled differential equation,
with feedback control, . Then, given an initial value problem, Ross partitions the time interval to a grid, with . From to , generate a control trajectory,
to the controlled differential equation,
A Carathéodory solution exists for the above equation because has discontinuities at most in t, the independent variable. At , set and restart the system with ,
Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-π solution.
A Carathéodory-π solution can be applied towards the practical stabilization of a control system. [6] [7] It has been used to stabilize an inverted pendulum, [6] control and optimize the motion of robots, [7] [8] slew and control the NPSAT1 spacecraft [3] and produce guidance commands for low-thrust space missions. [2]