Optimal control of time‐delay systems by dynamic programming

Dadebo, S. A.; Luus, Rein

doi:10.1002/oca.4660130103

Cited by 57 publications

(51 citation statements)

References 12 publications

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“…From Table it is observed that for the time step size h = 0.004 and constant time‐delay τ ( t ) = 1, a minimum value of the cost functional is obtained as J = 4.796786. As it is shown in Table , this value of J computed by the proposed scheme is compared very well with those achieved by the single‐term Walsh series , iterative dynamic programming , Legendre multi‐wavelets , modified line‐up competition algorithm and recursive shooting method . Comparison results reveal the validity of the proposed FD method for solving the TDOCP ()–().…”

Section: The Second‐order Two‐step Fd Methodssupporting

confidence: 59%

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An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Jajarmi

Hajipour

2016

Asian Journal of Control

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In this paper, an efficient finite difference method is presented for the solution of time‐delay optimal control problems with time‐varying delay in the state. By using the Pontryagin's maximum principle, the original time‐delay optimal control problem is first transformed into a system of coupled two‐point boundary value problems involving both delay and advance terms. Then the derived system is converted into a system of linear algebraic equations by using a second‐order finite difference formula and a Hermite interpolation polynomial for the first‐order derivatives and delay terms, respectively. The convergence analysis of the proposed approach is provided. The new scheme is also successful for the optimal control of time‐delay systems affected by external persistent disturbances. Numerical examples are included to demonstrate the validity and applicability of the new technique. Some comparative results are included to illustrate the effectiveness of the proposed method.

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Section: The Second‐order Two‐step Fd Methodssupporting

confidence: 59%

“…Then the transformed problem has been solved by a well‐developed optimization algorithm. An iterative dynamic programming (IDP) is applied for the solution of TDOCPs in . Then Chen et al.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Jajarmi

Hajipour

2016

Asian Journal of Control

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“…The convergence is speeded up considerably by introducing the penalty term 0.01u 1 (t) 2 in the cost functional and deleting the terminal conditions. Dadebo and Luus [13] treat a similar CSTR with n = 4 state variables, but no delay in the control variable and no terminal conditions. Using a fine grid with N = 20 000, the CPU time for computing the optimal solution of this CSTR problem is in the range of a minute.…”

Section: Remarkmentioning

confidence: 99%

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Göllmann

Kern

Maurer

2008

Optim Control Appl Methods

228

190

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Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. problems with a constant state delay. In [3], he gave similar results for control problems with pure control delays. Halany [4] proves a maximum principle for optimal control problems with multiple constant delays in state and control variables that, however, are chosen to be equal for state and control. Similar results were obtained by Ray and Soliman [5]. Guinn [6] sketches a simple method for obtaining necessary conditions for control problems with a constant delay in the state variable. He suggests to augment the delayed control problem that yields a higher-dimensional undelayed control problem to which the standard maximum principle is applicable. Banks [7] derives a maximum principle for control systems with a time-dependent delay in the state variable. Delays in the control are admitted for systems linear in the control variable. Colonius and Hinrichsen [8] provide a unified approach to control problems with delays in the state variable by applying the theory of necessary conditions for optimization problems in function spaces. All articles mentioned so far do not consider general control or state inequality constraints.Angell and Kirsch [9] treat functional differential equations with function-space state inequality constraints. However, they do not discuss the regularity of the multiplier associated with the state constraint and do not provide a numerical example with a pure state space constraint. To our knowledge, optimal control problems with constant delays in state and control variables and mixed control-state inequality constraints have not yet been considered in the literature. The first goal in this paper is to derive a Pontryagin-type minimum (maximum) principle for this class of delayed control problems. Concerning the development of numerical methods and the numerical treatment of practical examples, our impression is that this topic has not yet been adequately addressed in the literature. Bader [10] uses collocation methods to solve the boundary value problem for the retarded state variable and advanced adjoint variable. He successfully solves several academic examples, but his method does not give accurate results for the more difficult CSTR...

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“…The iterative dynamic programming (IDP) is a variant of the standard approach allowing to come across these difficulties ( [15], [3]). By IDP, the problem is solved in a series of iterations rather than in a single pass of the DP algorithm.…”

Section: Iterative Dynamic Programmingmentioning

confidence: 99%

Irrigation Optimization by Modeling of Plant-Soil Interaction

Li¹,

Cournède²,

Mailhol³

2011

Applied Simulation and Modelling

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Irrigation scheduling is an important issue for crop management, in a general context of limited water resources and increasing concern about agricultural productivity. Methods to optimize crop irrigation should take into account the impact of water stress on plant growth and the water balance in the plant-soil-atmosphere system. In this article, we propose a methodology to solve the irrigation scheduling problem. For this purpose, a plant-soil interaction model is used to simulate the structural-functional plant growth conditioned by water status. The system dynamics is driven by a delay differential system. By considering a price for the crop yield and for the water resource, an optimal control problem can be formulated in order to find the optimal irrigation strategy. The solution is obtained by dynamic programming. In order to handle the delay term of the system due to the continuous mechanism of plant senescence and the curse of dimensionality, the iterative approach of dynamic programming is used.

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Optimal control of time‐delay systems by dynamic programming

Cited by 57 publications

References 12 publications

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Irrigation Optimization by Modeling of Plant-Soil Interaction

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