An iterative method for suboptimal control of linear time-delayed systems

Mirhosseini-Alizamini, Seyed Mehdi; Effati, Sohrab; Heydari, Aghileh

doi:10.1016/j.sysconle.2015.04.013

Cited by 18 publications

(12 citation statements)

References 37 publications

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“…Table also indicates that for h = 0.001 the cost functional value is computed as J = 3.101717 within a 1.442331 s of CPU time. This value of J is very close to its optimal value J ∗ ≈3.1017 and compares favourably with those achieved by the approximation schemes in . Comparison results are listed in Table .…”

Section: The Second‐order Two‐step Fd Methodssupporting

confidence: 74%

“…This example was first introduced by Banks and Burns , and has been subsequently considered by Pananisamy and Rao and Mirhosseini‐Alizamini et al. to illustrate their methods. Find the optimal control u ∗ ( t ) which minimizes the quadratic cost functional

J = \frac{3}{2} x^{2} (2) + \frac{1}{2} {falsefalse}_{\int}^{0} u^{2} (t) dt,

subject to the time‐delay system

({, \begin{matrix} arrayleft \\ \dot{x} (t) = x (t) + x (t - 1) + u (t), 0 \leq t \leq 2, \\ x (t) = 1, - 1 \leq t \leq 0 . \end{matrix})

…”

Section: The Second‐order Two‐step Fd Methodsmentioning

confidence: 99%

“…Example III.1. This example was first introduced by Banks and Burns [25], and has been subsequently considered by Pananisamy and Rao [26] and Mirhosseini-Alizamini et al [27] to illustrate their methods. Find the optimal control u * (t) which minimizes the quadratic cost functional…”

Section: Illustrative Examplesmentioning

confidence: 99%

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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: 74%

J = \frac{3}{2} x^{2} (2) + \frac{1}{2} {falsefalse}_{\int}^{0} u^{2} (t) dt,

subject to the time‐delay system

({, \begin{matrix} arrayleft \\ \dot{x} (t) = x (t) + x (t - 1) + u (t), 0 \leq t \leq 2, \\ x (t) = 1, - 1 \leq t \leq 0 . \end{matrix})

…”

Section: The Second‐order Two‐step Fd Methodsmentioning

confidence: 99%

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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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“…Example Consider the following example:

J = \frac{3}{2} X^{2} false(2 false) + \frac{1}{2} true \int_{0}^{2} U^{2} false(t false) d t,

subject to the time‐delay system

\begin{matrix} alignleft \\ align-1 X^{'} (t) & align-2 = X (t) + X (t - 1) + U (t), 0 \leq t \leq 2, \\ align-1 X (t) & align-2 = 1, - 1 \leq t \leq 0, \end{matrix}

in which the analytic solution for U ( t ) is

U (t) = \{\begin{matrix} δ false(e^{2 - t} + false(1 - t false) e^{1 - t} false), & 0 \leq t \leq 1, \\ δ e^{2 - t}, & 1 \leq t \leq 2, \end{matrix}

and with δ =−0.3932, J ≃3.1017. This example solved by several numerical techniques such as averaging approximations method, single‐term Walsh series, variational iteration method, and finite difference method with h = 0.01. In Table , these numerical results are compared to the results obtained using the present method for different values of k , M .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…and with = −0.3932, J ≃ 3.1017. This example solved by several numerical techniques such as averaging approximations method, 50 single-term Walsh series, 51 variational iteration method, 56 and finite difference method 57 with h = 0.01. In Table 6, these numerical results are compared to the results obtained using the present method for different values of k, M. Numerical methods J Averaging approximations method, 50 3.0833 Single-term Walsh series 51 3.0879 Variational iteration method 56 3.1091 Finite difference method 57 3 Example 7.…”

Section: Examplementioning

confidence: 99%

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

Sabermahani

Ordokhani

Yousefi

2019

Optim Control Appl Methods

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Summary In this paper, a numerical method for solving time‐varying delay equations and optimal control problems with time‐varying delay systems is discussed. This method is based upon Fibonacci wavelets and Petrov‐Galerkin method. To solve these problems, first, the Fibonacci wavelets are presented. With the aid of operational matrices of integration and delay for Fibonacci wavelets and using Petrov‐Galerkin method and Newton's iterative method, we solve two classes of time‐varying delay problems, numerically. The approximate solutions achieved by this method satisfy all the initial conditions. In addition, an estimation of the error is given. Numerical results are included to demonstrate the accuracy and applicability of the present technique.

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Approximate optimal control design for quadrotors: A computationally fast solution

Yao,

Rafee Nekoo,

Xin

2023

Optim Control Appl Methods

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SummaryAn approximate closed‐form optimal control design is proposed for the flight control of quadrotor unmanned aerial vehicles. The nonlinear dynamic equation is rewritten as a pseudo‐linear form without approximations. The quadratic cost function is modified by adding perturbation terms to the state weighting matrix. A co‐state, which is associated with the solution to the partial differential Hamilton–Jacobi–Bellman (HJB) equation, is approximated by a power series of an instrumental variable with symmetric matrices as the coefficients. The solution to the intractable HJB equation can be reduced to solving these coefficient matrices, which are in forms of a differential Riccati equation and a series of linear Lyapunov equations, These equations can be solved recursively and analytically. Specifically, the differential Riccati equation can be solved offline and only once, and the linear Lyapunov equations can be solved analytically. These approximations lead to a closed‐form suboptimal state feedback control law, which is computationally more efficient than the similar finite‐time state‐dependent Riccati equation (SDRE) technique that requires the solution of the state‐dependent differential Riccati equation at each time step and demands a high computational cost. The proposed control law is applied to the flight control design of quadrotors. Numerical simulations validate the effectiveness of the proposed optimal control technique with superior performance of control accuracy and robustness. It is compared favorably with the finite‐time SDRE technique in terms of computation efficiency and control effort, especially when onboard implementations and experiments are needed.

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An iterative method for suboptimal control of linear time-delayed systems

Cited by 18 publications

References 37 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

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

Approximate optimal control design for quadrotors: A computationally fast solution

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