Stabilization and Optimization of Design Parameters for Control of Inverted Pendulum

In this work, we study the dynamic response of the most popular unstable control problem, the inverted pendulum in terms of classical control theory. The theoretical and experimental results presented here explore the relationship between changes in the indirect tuning parameters from the linear quadratic regulator (LQR) design, and the final system performance effected using the feedback gains specified as the LQR weight constraints are changed. First, we review the development of the modern control approach using full state-feedback for stabilization and regulation, and present simulation and experimental comparisons as we change the optimization targets for the overall system and as we change one important system parameter, the length of the pendulum. Second, we explore the trends in the response by developing the generalized root locus for the system using incremental changes in the LQR weights. Next, we present a family of curves showing the local root locus and develop relationships between the weight changes and the system performance. We describe how these locus trends provide insight that is useful to the control designer during the effort to optimize the system performance. Finally, we use our general results to design an effective feedback controller for a new system with a longer pendulum and present experiment results that demonstrate the effectiveness of our analysis.