An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

Li, Bin; Yu, Changqiu; Teo, Kok Lay; Duan, Guang Ren

doi:10.1007/s10957-011-9904-5

Cited by 71 publications

(56 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Results in Table show that nine penalty iterations are required to complete the solution process, and the final objective function value is 45.5716 with ρ = 1 × 10 9 and ε = 4.4444 × 10 −23 , which is better than the results in and . In addition, results in Table indicate that after nine penalty iterations,

\max_{1 \leq i \leq N} ({}, x_{1} ((), t) true/ 6 + u ((), t)) = 0

, which reveals that the inequality path constraint is well satisfied during the whole time horizon.…”

Section: Cases Testingmentioning

confidence: 90%

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

Liu

et al. 2016

Optim. Control Appl. Meth.

View full text Add to dashboard Cite

Summary An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l1 exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non‐differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

\max_{1 \leq i \leq N} ({}, x_{1} ((), t) true/ 6 + u ((), t)) = 0

, which reveals that the inequality path constraint is well satisfied during the whole time horizon.…”

Section: Cases Testingmentioning

confidence: 90%

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

Liu

et al. 2016

Optim. Control Appl. Meth.

View full text Add to dashboard Cite

show abstract

“…Example We consider the constrained fractional optimal control problem as follows:

\begin{matrix} \min_{x, u} & J false(x, u false) : = \int_{0}^{4.5} false(u^{2} false(t false) + x_{1}^{2} false(t false) false) d t, \\ s . t . & _{0}^{C} D_{t}^{α} x_{1} false(t false) = x_{2} false(t false),_{0}^{C} D_{t}^{α} x_{2} false(t false) = - x_{1} false(t false) + x_{2} false(t false) false(1.4 - 0.14 x_{2}^{2} false(t false) false) + 4 u false(t false), \\ - u false(t false) - \frac{1}{6} x_{1} false(t false) \geq 0, t \in false[0, 4.5 false], x_{1} false(0 false) = - 5, x_{2} false(0 false) = - 5 . \end{matrix}

This example for α = 1 is the Rayleigh problem with mixed state‐control constraint (both state and control are included in the path constraint) appeared in previous studies . Parameters TolFun, TolCon, and TolX of the fmincon function are set as 10 −14 .…”

Section: Numerical Examplesmentioning

confidence: 99%

B‐spline spectral method for constrained fractional optimal control problems

Ejlali

Hosseini

Yousefi

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.

show abstract

“…This shows the equality of the minima and of their set of minimizers. In particular, any minimizer is a solution of (26), and hence, the ordinary calculus of variations shows that it satisfies @H @u D 0. This result proves that the proposed penalties can be used to generate a sequence of interior solutions.…”

Section: Proofmentioning

confidence: 99%

“…There exist also recent works in the field of exact penalty methods for various types of optimal control problem [24][25][26][27][28][29]. These methods are of particular interest because each solution of the sequence of optimal control problem is easily computed using classical stationarity conditions of the solution.…”

Section: Introductionmentioning

confidence: 99%

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

Malisani

Chaplais

Petit

2014

Optim Control Appl Methods

View full text Add to dashboard Cite

SUMMARYThis paper exposes a methodology to solve state and input constrained optimal control problems for nonlinear systems. In the presented 'interior penalty' approach, constraints are penalized in a way that guarantees the strict interiority of the approaching solutions. This property allows one to invoke simple (without constraints) stationarity conditions to characterize the unknowns. A constructive choice for the penalty functions is exhibited. The property of interiority is established, and practical guidelines for implementation are given. A numerical benchmark example is given for illustration.

show abstract

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

Cited by 71 publications

References 13 publications

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

B‐spline spectral method for constrained fractional optimal control problems

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

Contact Info

Product

Resources

About