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Two Degree of Freedom Optimal Control for Nonlinear Systems with Parameter Uncertainty

  • Author(s): Shaffer, Richard
  • Advisor(s): Gong, QI
  • et al.
Abstract

This dissertation explores some of the ways that optimal control can be used to mitigate the effects of parametric uncertainty on the control's ability to achieve a desired endpoint condition on the state variables. Standard optimal control solutions rely on precise knowledge of parameter values which are difficult to measure in many practical applications. The result is that, when implementing the designed controls, there is likely to be some error between the desired state trajectories and the actual system outputs. Usually these errors are managed using feedback control using sensor measurements to correct the state deviations on the fly. However, recently there has been promising work done for generating controls which operate in the open-loop for generating state trajectories which are inherently robust to uncertainty in the parameters.

One avenue of applying optimal control for managing parameter uncertainty is unscented optimal control which borrows the low sample discretization of the Unscented Transform for approximating a Riemann-Stieltjes integral of an objective function. This method has proven effective for generating open-loop controls. In this dissertation we explore a family of sensitivity function based optimal control problems, which utilizes an approximation of the response to parameter deviations through a Taylor series, which can be used to likewise generate open-loop controls. We then use these sensitivity based problems as a lens for exploring the unscented control problem. Furthermore, we identify conditions for equivalence between the two optimal control problems. These inherently robust open-loop trajectories represent a single degree of freedom control.

In addition to generating the open-loop controls for a more robust state trajectory, this dissertation provides a problem formulation for generating a set of time-varying feedback gains. These gains use sensor measurement for both the sensitivity function based problem as well as the unscented problem frameworks. The unscented problem avoids linearizing the error dynamics which is shown to be advantageous for designing the feedback gains. The use of feedback gains represents a second degree of freedom when combined with a more robust open-loop control significantly improving the performance.

These concepts are demonstrated on a simulated nonlinear double gimbal mechanism with flexible effects where the unscented feedback control gives a nearly $100\%$ decrease in variance of the error of the trajectory endpoints and $82\%$ in the case of the purely open-loop. The mathematical model of the double gimbal as a second order set of differential equations represents a prevalent mechanical system in engineering. The controls generated by the neighboring unscented problem demonstrate a significant improvement in the robustness to the parameter uncertainty over the standard controls. The utility of these concepts is further illustrated experimentally using a nonlinear two link robotic arm with flexible joints.

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