Volume 7, Issue 2
The $\alpha$ Method for Solving Differential Algebraic Inequality (DAI) Systems

Int. J. Numer. Anal. Mod., 7 (2010), pp. 240-260.

Published online: 2010-07

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• Abstract

This paper describes an algorithm for "direct numerical integration" of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation’s (DAE) Jacobian and this difficulty is addressed by a regularization property of the $α$ method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

• Keywords

Differential-algebraic equations, trajectory planning, numerical optimization, inequality path constraints.

65L80, 49J15, 65K99, 65F99

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@Article{IJNAM-7-240, author = {J. and Peter and and 20945 and and J. Peter and N. C. and Parida and and 20946 and and N. C. Parida and S. and Raha and and 20947 and and S. Raha}, title = {The $\alpha$ Method for Solving Differential Algebraic Inequality (DAI) Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2010}, volume = {7}, number = {2}, pages = {240--260}, abstract = {

This paper describes an algorithm for "direct numerical integration" of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation’s (DAE) Jacobian and this difficulty is addressed by a regularization property of the $α$ method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/718.html} }
TY - JOUR T1 - The $\alpha$ Method for Solving Differential Algebraic Inequality (DAI) Systems AU - Peter , J. AU - Parida , N. C. AU - Raha , S. JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 240 EP - 260 PY - 2010 DA - 2010/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/718.html KW - Differential-algebraic equations, trajectory planning, numerical optimization, inequality path constraints. AB -

This paper describes an algorithm for "direct numerical integration" of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation’s (DAE) Jacobian and this difficulty is addressed by a regularization property of the $α$ method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

J. Peter, N. C. Parida & S. Raha. (1970). The $\alpha$ Method for Solving Differential Algebraic Inequality (DAI) Systems. International Journal of Numerical Analysis and Modeling. 7 (2). 240-260. doi:
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