Data-driven optimal controller synthesis : a frequency domain approach

Abstract

Mechatronic systems play an important role in many industrial production facilities and consumer products. To attain desired dynamic responses of these systems, proper design of the embedded feedback control systems is essential. Ever increasing performance demands, however, put a severe challenge to the available methodologies to design these controllers. To meet design specifications of next generation motion systems, control design methodologies are required that explicitly take the multivariable nature of multi-degree-of-freedom motion systems into account while guaranteeing robust performance and stability in the presence of plant variations or uncertainties. Norm-based controller design offers a methodology to deal with these design specifications via the framework of generalized plants. Existing norm-based controller synthesis methodologies, however, have to rely on a parametric model of the system at hand. The plant identification process, required to obtain such a model, can be seen as a bottleneck in the practical applicability of norm-based controller design procedures. On the other hand, the dynamic responses of high performance motion systems can be well predicted via non-parametric descriptions such as frequency response data samples or impulse responses. This research presents a methodology to perform norm-based controller design based on frequency response data samples of the plant directly. To come up with such a methodology, the underlying phenomena that are inherent to limited availability of samples are studied and alternative criteria for stability and performance are posed that can be evaluated based on data samples only. An overview of the main results described in the thesis is given.A novel data-based stability test is proposed that enables validation of stability of a system based on frequency response samples only. This test is well suited for numerical evaluation and does not rely on knowledge about the number of unstable open-loop poles. Research efforts have been focussed to bridge the gap between a finite number of samples, and the set of all possible underlying frequency responses. It appears that sampling theory, well known in the context of harmonic decompositions of time domain signals, can be generalized towards the analysis of sampling of transfer functions. By combining analytical properties of transfer functions with this sampling framework, an explicit expression for the set of all frequency responses is given in terms of the available prior knowledge about the system. Two controller design approaches are proposed. As a first case, the H2 controller synthesis problem is considered. By composing the Youla parameter from a set of stable basis functions, the H2 controller synthesis problem can be rendered into a least squares optimization problem that can be solved via available tools. The performed analysis, however, shows that controller synthesis based on a frequency sampled performance criterion, unavoidably induces inter-frequency-grid performance degradation. Practical means to reduce this effect are given. Furthermore, a robust stability criterion is posed in terms of the Youla parameter that assures robustness with respect to plant uncertainty and therefore eliminates the usual assumption that infinitely many data samples are available. Alternative to the data-based H2 problem, fixed structure controller parameter optimization is considered. A two-step optimization algorithm is posed that sequentially focusses on stability and performance optimization via a steepest descend algorithm. A novel cost function is introduced that enables convergence from a destabilizing controller parameter set to a stable parameter set. For performance optimization, the gradient of the maximum singular values with respect to the controller parameters is found via a linearization approach. This approach enables to optimize the coefficients of a given fixed structure controller with respect to the H1 norm of the closed-loop system. The proposed data-based controller design methodology is evaluated on frequency response data samples that are obtained from an experimental setup. This validates the proposed stability test and illustrates the effectiveness of the proposed approach for fixed structure controller optimization.