arXiv subjects

Kai-Feng He

Publications and source records attributed to Kai-Feng He.

5 recordsLinked to original sources

arXiv2021

Transform the Non-linear Programming Problem to the Initial-value Problem to Solve

A dynamic method to solve the Non-linear Programming (NLP) problem with Equality Constraints (ECs) and Inequality Constraints (IECs) is proposed. Inspired by the Lyapunov continuous-time dynamics stability theory in the control field, the optimal solution is analogized to the stable equilibrium point of a finite-dimensional dynamic system and it is solved in an asymptotic manner. Under the premise that the Karush-Kuhn-Tucker (KKT) optimality condition exists, the Dynamic Optimization Equation (DOE), which has the same dimension to that of the optimization parameter vector, is established and its solution will converge to the optimal solution of the NLP globally with a theoretical guarantee. Using the matrix pseudo-inverse, the DOE is valid even without the linearly independent regularity requirement on the nonlinear constraints. In addition, the analytic expressions of the Lagrange multipliers and KKT multipliers, which adjoin the ECs and the IECs respectively during the entire optimization process, are also derived. Via the proposed method, the NLP may be transformed to the Initial-value Problem (IVP) to be solved, with mature Ordinary Differential Equation (ODE) integration methods. Illustrative examples are solved and it is shown that the dynamic method developed may produce the right numerical solutions with high efficiency.

math.OC↗

arXiv2020

Variation Evolving for Optimal Control Computation, a Compact Way

A compact version of the variation evolving method (VEM) is developed in the primal variable space for optimal control computation. Following the idea that originates from the Lyapunov continuous-time dynamics stability theory in the control field, the optimal solution is analogized to the stable equilibrium point of a dynamic system and obtained asymptotically through the variation motion. With the introduction of a virtual dimension, namely the variation time, the evolution partial differential equation (EPDE), which seeks the optimal solution with a theoretical guarantee, is developed for the optimal control problem (OCP) with free terminal states, and the equivalent optimality conditions with no employment of costates are established in the primal space. These conditions show that the optimal feedback control law is generally not analytically available because the optimal control is related to the future states. Since the derived EPDE is suitable to be computed with the semi-discrete method in the field of PDE numerical calculation, the optimal solution may be obtained by solving the resulting finite-dimensional initial-value problem (IVP).

eess.SY↗

arXiv2018

The Third Evolution Equation for Optimal Control Computation

The Variation Evolving Method (VEM) that originates from the continuous-time dynamics stability theory seeks the optimal solutions with variation evolution principle. After establishing the first and the second evolution equations within its frame, the third evolution equation is developed. This equation only solves the control variables along the variation time to get the optimal solution, and its definite conditions may be arbitrary since the equation can eliminate possible infeasibilities. With this equation, the dimension of the resulting Initial-value Problem (IVP), transformed via the semi-discrete method, is greatly reduced. Therefore it might relieve the computation burden in seeking solutions. Illustrative examples are solved and it is shown that the proposed equation may produce more precise numerical solutions than the second evolution equation, and its computation time may be shorter for the dense discretization.

eess.SY↗

arXiv2018

Computation of Optimal Control Problems with Terminal Constraint via Modified Evolution Partial Differential Equation

The Variation Evolving Method (VEM), which seeks the optimal solutions with the variation evolution principle, is further developed to be more flexible in solving the Optimal Control Problems (OCPs) with terminal constraint. With the first-order stable dynamics to eliminate the infeasibilities, the Modified Evolution Partial Differential Equation (MEPDE) that is valid in the infeasible solution domain is proposed, and a Lyapunov functional is constructed to theoretically ensure its validity. In particular, it is proved that even with the infinite-time convergence dynamics, the violated terminal inequality constraints, which are inactive for the optimal solution, will enter the feasible domain in finite time. Through transforming the MEPDE to the finite-dimensional Initial-value Problem (IVP) with the semi-discrete method, the OCPs may be solved with common Ordinary Differential Equation (ODE) numerical integration methods. Illustrative examples are presented to show the effectiveness of the proposed method.

eess.SY↗

arXiv2017

A Variation Evolving Method for Optimal Control

A new method for the optimal solutions is proposed. Originating from the continuous-time dynamics stability theory in the control field, the optimal solution is anticipated to be obtained in an asymptotically evolving way. By introducing a virtual dimension, the variation time, a dynamic system that describes the variation motion is deduced from the Optimal Control Problem (OCP), and the optimal solution is its equilibrium point. Through this method, the intractable OCP is transformed to the Initial-value Problem (IVP) and it may be solved with mature Ordinary Differential Equation (ODE) numerical integration methods. Especially, the deduced dynamic system is globally stable, so any initial value will evolve to the extremal solution ultimately.

eess.SY↗