Detailed Algorithm for Implementing Circle Criterion in QFT Framework for Saturation Nonlinearity

The control of a spacecraft’s orbit and attitude is one of the most complicated problems in the field of control engineering. In practice, engineers encounter uncertainty and nonlinearity due to large flexible appendages and saturated actuators. Control design methods for dealing with such problems may involve a lot of calculations. Matlab software automates many design methods in its control system toolbox that provides algorithms for systematically analyzing, designing and tuning linear control systems. To achieve fast and accurate controller design for more complex control systems many other advanced design methods need to be automated, too. As a sample, plant uncertainty, which is prevalent in spacecraft systems, requires robust consideration in the design process. Quantitative feedback theory (QFT), as a powerful method for addressing such complex issues, requires plenty of calculations that makes it necessary the method to be automated. A QFT design toolbox is developed by Tersoft company. But this toolbox cannot consider the treatment of some practical issues such as actuators saturation in its design process. In the QFT framework, saturation can be deal by Horowitz architecture or by non-interfering loop architecture, containing an inner loop around the saturation element in control loop. The circle criterion is a critical constraint on the inner loop transfer function, ensuring stability. This paper presents a comprehensive algorithm for automating the process of obtaining inner loop compensator design constraints, with detailed flowcharts to facilitate software development. An example is solved using the proposed algorithm implemented in the Matlab environment. Intermediate and final results are presented to follow the calculations step by step up. Finally, the algorithm's validity is confirmed by directly checking the satisfaction of the circle criterion for a complex plant with a high degree of uncertainty.