The alpha method for solving differential algebraic inequality (DAI) systems
Joey Peter 1, Nigam Chandra Parida 2, Soumyendu Raha 21 GE India Infrastructures (Energy), GE John F. Welch Technology Centre, Bangalore 560066, India.
2 Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore 560012, India.
Received by the editors November 15, 2008 and, in revised form, July 9, 2009.
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 alpha 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.
AMS subject classifications: 65L80, 49J15, 65K99, 65F99
Key words: Differential-algebraic equations, trajectory planning, numerical optimization, inequality path constraints.
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