Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

Gong, Qi; Ross, I. Michael; Fahroo, Fariba

doi:10.1007/s10957-016-0909-y

Cited by 25 publications

(33 citation statements)

References 60 publications

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“…As shown in the authors' paper [12] and from the present study, the spline interpolation is a natural choice for the G-SPIN formula. For completeness, we repeat the main results of the earlier paper in this Section with detailed procedures for the derivation of the G-SPIN formula for spline interpolation.…”

Section: Extension To Spline Interpolationmentioning

confidence: 78%

“…Efforts to utilize arbitrary nodes include recent studies by Ross et al [4] and Gong et al [12] showing that convergence for the solution of nonlinear optimal control problems with pseudospectral methods can be guaranteed only when all quadrature weights are positive. They pointed out that analyses with more than ten uniform nodes might fail, because at least one of the weights becomes negative when Lagrange polynomials are used on uniform nodes.…”

Section: Extension To Spline Interpolationmentioning

confidence: 99%

“…One is generated using the coordinate transformation devised by the authors [11] for more uniformly distributed nodes than those obtained using the traditional spectral method. The other is a set of uniform nodes known as the worst possible choice for polynomial interpolation, with which the derived formulas are extremely inaccurate as mentioned in [4] and [12]. The Lagrange and tension spline interpolations over these sets of nodes are investigated by applying the present method to three carefully selected test functions and by comparing the resultant absolute errors in predicting integration and differentiation for those functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

Kim¹,

Park²,

Sung³

2015

Journal of Electrical Engineering and Technology

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-A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

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Section: Extension To Spline Interpolationmentioning

confidence: 78%

Section: Extension To Spline Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

Kim¹,

Park²,

Sung³

2015

Journal of Electrical Engineering and Technology

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“…Similarly, if P m t are chosen to be Chebyshev polynomials, we generate a Chebyshev PS method. Typical choices in PS optimal control theory are these "big two" methods [12] because other polynomial basis functions do not have desirable properties for optimal control applications [19,20]. The coefficients a m in Eq.…”

Section: A Standard Pseudospectral Optimal Control Theorymentioning

confidence: 99%

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

Ross

Proulx

Karpenko

et al. 2015

Journal of Guidance, Control, and Dynamics

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Motivated by uncertain parameters in nonlinear dynamic systems, we define a nonclassical optimal control problem where the cost functional is given by a Riemann-Stieltjes "functional of a functional." Using the properties of Riemann-Stieltjes sums, a minimum principle is generated from the limit of a semidiscretization. The optimal control minimizes a Riemann-Stieltjes integral of the Pontryagin Hamiltonian. The challenges associated with addressing the noncommutative operations of integration and minimization are addressed via cubature techniques leading to the concept of hyper-pseudospectral points. These ideas are then applied to address the practical uncertainties in control moment gyroscopes that drive an agile spacecraft. Ground test results conducted at Honeywell demonstrate the new principles. The Riemann-Stieltjes optimal control problem is a generalization of the unscented optimal control problem. It can be connected to many independently developed ideas across several disciplines: search theory, viability theory, quantum control and many other applications involving tychastic differential equations.

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“…Since 1990s, the application of the pseudospectral methods for solving optimal control problems has been popular due to their computational efficiency (Li, 2017;Limebeer, Perantoni, & Rao, 2014;Ross & Karpenko, 2012;Shamsi, 2011). For recent advances in the pseudospectral methods, see, for example, Gong, Ross, and Fahroo (2016) ;Tang, Liu, and Hu (2016). Pseudospectral methods approximate the state and control variables using interpolating polynomials with specific collocation points such as Legendre-Gauss-Lobatto(LGL), Legendre-Gauss(LG) (Mehrpouya, Shamsi, & Azhmyakov, 2014) and Legendre-Radau(LR) points (Garg et al, 2010).…”

Section: Introductionmentioning

confidence: 99%

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

Foroozandeh

Shamsi

Pinho

2017

International Journal of Control

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This paper presents a new approach for the efficient and accurate solution of Singular Optimal Control Problems (SOCP). A novel feature of the proposed method is that it does not require a priori knowledge of the structure of the solution. As the first step of this method, the SOCP is converted into a binary optimal control problem. Then, by utilizing the pseudospectral method, the resulting problem is transcribed to a mixed-binary non-linear programming problem. This mixed-binary non-linear programming problem, which can be solved by well-known solvers, allows us to detect the structure of the original optimal control and to compute the approximating solution of it (getting both the optimal state and control). The main advantages of the present method are that: (i) without a priori information, the structure of optimal control is detected; (ii) it produces good results even using a small number of collocation points; and (iii) the switching times can be captured accurately. These advantages are illustrated through a numerical implementation of the method on four examples.

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Spectral and Pseudospectral Optimal Control Over Arbitrary Grids

Cited by 25 publications

References 60 publications

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems

A mixed-binary non-linear programming approach for the numerical solution of a family of singular optimal control problems

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