Consistency of Approximation of Bernstein Polynomial-Based Direct Methods for Optimal Control

Round 1

Reviewer 1 Report

Report on paper submitted to journal of "Machines" with title

"Consistency of approximation of Bernstein polynomial-based direct methods for optimal control"

 This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation.

 All theorems and corollaries are investigated and appear to be correct.

The paper is well written and can be appealing to the readers. But there are some comments that need to be made.

 Minor Comment:

 ·     Please recheck the whole of the paper about using $=$ or $\approx$, such as Equations 3, 8, 13, 19, and so on.

 ·       Please check punctuation in the whole of paper.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

This paper provides a direct method based on the Bernstein polynomials to solve a class of optimal control problems with a mixed input and state constraints. The proposed method transforms solving the problem under study into solving a system of algebraic equations.

This paper can be considered for publication after the authors address the following comments and suggestions:

1.     Throughout the manuscript typos should be thoroughly checked. Meanwhile the language needs to be polished.

2.      Please ended the relation with “,” or “.”.

3.      What are the main advantageous of the Bernstein polynomials compared other basis polynomials, such as Legendre and Chebyshev polynomials?  

4.      How to use Bernstein to solve problems with piecewise solutions?

5.      Please add more details for the title of figures.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

1) I would recommend the authors to explicit the proof of Theorem 3 as it is not immediate.

2) Figure 5 shown the control variables for Example 2. From the plot, it is clear that u3 doesn't satisfy the constraint <=1. Moreover, even thought a closed form solution of the optimal control problem is not available, I would still propose a numerical solution for comparison (state and controls).

3) The authors should clarify if, in Example 3, t_f is assigned (I am assuming). If t_f is not assigned, i.e. it is a free parameter, then the authors should clarify why the Hamiltonian convergence to zero (Figure 8) indicates a close approximation to optimality. Figure 7 appears before it is cited. Figure 7,8 are poorly organized (missing labels and or legends,...). The authors should also provide, for Example 3, some plots that shows that the control variables satisfy their respective constraints.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

The authors of the paper “Consistency of approximation of Bernstein polynomial-based direct method for optimal control” proposed the use of Bernstein polynomials for costate estimation of nonlinear constrained optimal control problems. The paper presents numerous lemmas related to the convergence properties and bounds of the approximation of the states and rates of the states.

This reviewer after a quick review of the literature finds that a large body of literature related to the use of Bernstein polynomials in solving optimal control problems is not cited precluding the reader from clearly assessing the contributions of the work.

Some of these papers include:

·         Split-Bernstein Approach to Chance-Constrained Optimal Control, Journal of Guidance, Control and Dynamics, Vol 40, No 11, Nov. 2017

·         Direct Transcription of Optimal Control Problems with Finite Elements on Bernstein Basis, Journal of Guidance, Control and Dynamics, Vol 42, No 2, Feb. 2019

·         Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints, Journal of Dynamic Systems, Measurement and Control, Vol. 140, No. 7, 2018.

·         AN EFFICIENT NUMERICAL SOLUTION OF FRACTIONAL OPTIMALCONTROL PROBLEMS BY USING THE RITZ METHOD ANDBERNSTEIN OPERATIONAL MATRIX, Asian Journal of Control, Vol 18, No 6., Nov. 2016.

 Next, the authors present three example to illustrate the use of Bernstein polynomial in solving Optimal control problems. It is unclear how their approach compares to other approaches to estimate costates such as:

·         A Gauss pseudospectral transcription for optimal control, David Benson, Ph.D. thesis, MIT , 2005.

·         Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method, Journal of Guidance, Control and Dynamics, Vol 29, No 6, Nov. 2006

In conclusion, this reviewer cannot recommend that this paper be accepted in its current form. There might be contributions in the paper, but due to a lack of comprehensive review of the literature and the lack of comparison with existing methods for estimating co-states, it is difficult to gauge the contributions of the paper and this reviewer has to regretfully recommend that the paper not be accepted for publication.

 

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

No further edits are requested.