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Approximation of low rank solutions for linear quadratic control of partial differential equations

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Abstract

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modeled by partial differential equations. We use a modified Newton method to solve the ARE that takes advantage of several special features of these problems. The modified Newton method leads to a right-hand side of rank equal to the number of inputs regardless of the weights. Thus, the resulting Lyapunov equation can be more efficiently solved. The Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples. An example on controlling the deflection of the Euler-Bernoulli beam indicates that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

    Kirsten Morris

  2. Department of Mathematics, Clarkson University, Potsdam, NY, 13699, USA

    Carmeliza Navasca

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  1. Kirsten Morris
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  2. Carmeliza Navasca
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Correspondence to Kirsten Morris.

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Morris, K., Navasca, C. Approximation of low rank solutions for linear quadratic control of partial differential equations. Comput Optim Appl 46, 93–111 (2010). https://doi.org/10.1007/s10589-008-9192-7

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  • DOI: https://doi.org/10.1007/s10589-008-9192-7

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