Nonlinear optimal control: an enhanced quasi-LPV approach

• Main Author: Huang, Yun
• Format: Others
• Published: 1999
• Online Access: https://thesis.library.caltech.edu/1917/1/Huang_y_1999.pdf; Huang, Yun (1999) Nonlinear optimal control: an enhanced quasi-LPV approach. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5VNR-GF60. https://resolver.caltech.edu/CaltechETD:etd-05212007-082553 <https://resolver.caltech.edu/CaltechETD:etd-05212007-082553>

Realistic models of physical systems are often nonlinear. Our objective is to synthesize controllers for nonlinear systems that not only provide stability, but also deliver good closed-loop performance. The frozen Riccati equation approach is thoroughly examined. Although it suffers fundamental deficiencies due to its pointwise nature, it is proven that optimality is always possible under a certain assumption on the optimal value of the performance index. This is a consequence of the non-uniqueness of the pointwise linear model of the nonlinear dynamics. However, one cannot assess a priori the guaranteed global performance for a particular model choice. An alternative to the pointwise design is to treat nonlinear plants as linear parameter varying systems with the underlying parameters being functions of the state variables. By exploiting the variation rate bounds of the parameters, a controller that smoothly schedules on the parameters can be synthesized by solving a convex optimization problem. Depending upon the choice of the variation rate bounds, the resulting controller can range from replicating the pointwise design result, which comes with no guarantee on performance, to providing quadratic stability, in which case it can withstand arbitrarily fast parameter variation. Under the above quasi-LPV framework, we present a new scheme that incorporates the freedom of choosing the state-dependent linear representation into the control design process. It is shown that the L2-gain analysis can be reformulated as an infinite dimensional convex optimization problem, and an approximate solution can be obtained by solving a collection of linear matrix inequalities. The synthesis problem is cast as a minimization over an infinite dimensional bilinear matrix inequality constraint. An iterative algorithm, similar to the "D - K iteration" for µ synthesis, is proposed to compute the best achievable performance. It is demonstrated through several examples that this approach can effectively reduce conservatism of the overall design.