Determination of the Allowable Parametric Uncertainty in the Closed Loop Via a Skew-μ Based Technique with an Application to the Yaw-Roll Vehicle Model

Abstract

In this dissertation, a methodical approach is presented that quantifies how well model parameters need to be known or by how much they can vary in order that the performance of a controller in closed loop is maintained. Unique in the formulation of this parametric approach is the ability to include parameters that are coupled together. This methodical skew μ approach with real parametric uncertainty is applied to the coupled yaw-roll vehicle model. An analysis of the μ-sensitivity of the yaw-roll vehicle model with full parametric uncertainty is also analyzed to ascertain which parameters are the most sensitive and which parameters can be most logically grouped together in the varying subset of the skew μ uncertainty block. A comparison between the vehicle model and high-fidelity simulation model is provided to show the potential limitations of the approach. Typically, robust control design techniques are used to ensure a system has guaranteed performance within certain margins of operation. However, it is often required to know which parameters and by how much these parameters affect the closed loop performance. In order to achieve guaranteed robustness, robust design techniques tend to broaden the output response often at the cost of a reduction in the performance of closed loop response characteristics. In this dissertation, the method presented seeks to determine how accurate the parameters of system model need to be in order to maintain the performance of an existing controller without broadening the original model response. In effect, a quantification of parameter variance allowed is calculated for a given system model with an existing controller and initial parametric variance assumptions and measurement specifications. To accomplish this quantification of allowable uncertainty, the infinite norm concept (H∞) and the extension to the structured singular value or μ value is utilized. Typically, μ analysis and structured uncertainty are used to account for various types of uncertainty for the synthesis of a controller. The μ analysis, however, is shown to also be useful in analyzing the allowable uncertainty of a closed loop system for a previously designed controller. Addi- tionally, the linear fractional transformation (LFT) is leveraged to formulate a system model that incorporates uncertainty on the parameters while maintaining the effects of parameter coupling. With a closed loop around the yaw rate of the coupled yaw-roll vehicle it is shown that the parametric eigenvalue analysis may not reveal accurate sensitivity information about the closed loop system. The μ sensitivity analysis indicates that the center of gravity height and weight split parameters are the most sensitive parameters for the coupled vehicle model. The vehicle system is also understandably sensitive to velocity. Initial uncertainties representative of typical variances and measurement capabilities are applied to the vehicle model for several cases of fixed and varying subsets within the uncertainty block. The skew μ for these cases is then calculated. It is shown that the specific parameters in the fixed subset affect the additional variance allowed on the parameters in the varying subset. Finally, spectral analysis using a CarSim high fidelity vehicle model is performed to confirm the validity of the additional allowable variance on the center of gravity height and weight split parameters that was found via the skew μ based analysis. In summary, the skew μ analysis is used to calculate the remaining “space” that each parameter can vary given an initial uncertainty for each parameter. If the initial uncertainty is increased for some or all of the parameters then there is less space remaining for further variation of the parameters in the varying subset. Additionally, μ sensitivity and skew μ analysis are shown to be a helpful tool to determine which parameters to focus on for estimation as well as how much the parametric terms of the model can vary for a given controller. With the assumption that a given model is accurate representation of the system a designer determine how much uncertainty is allowable for each parameter in the closed loop under specific initial uncertainty conditions.