Robust control of nonlinear systems from data

A general problem in the control field is to derive a model of the system to be controlled. The classical approach consists in building a mathematical model on the basis of the laws governing the system (e.g. mechanical, physical, economical) and then exploit it for de- signing a model-based control law that fulfills the desired specifications. However, such approach is not always possible for two main reasons: the uncomplete knowledge of the system laws and its nonlinearity which, respectively, do not enable to derive an accurate and tractable model. On the other hand, considering that the most of the existing sys- tems are nonlinear, the model to be derived should be a trade-off between accuracy and tractability. Indeed, the accuracy of the model, employed to design the control system, plays a crucial role since the performance achievable by the controlled system strongly depends on the size of the modeling error. In the presence of a poor accurate model, not only a performance degradation may occurr, but the closed loop stability may also be missed. The literature of nonlinear control usually assumes that the system to be controlled and its model are well known, altough this is not always true as pointed out above. In par- ticular, the nonlinear models usually employed are neural networks or parametric models whose parameters are identified from input/output data of the process. Due to the nature of these models, does not exist a systematic procedure to obtain a suitable description of the uncertainty associated with them, which in turn hampers a systematic dealing of the robust stability. Thus, in order to investigate the control of nonlinear and unknown systems, guaranteeing the stability of the closed loop system in the presence of model uncertainty, objective of this thesis is study the robust control of nonlinear dynamic systems from experimental data. At this aim, a Nonlinear Set Membership (NSM) identification methodology is used to derive a data-based model. Such technique identifies a model from input/output data col- lected in previous experiments and provides a finite measure of the uncertainty associated to the model (see [1] for more details). The obtained model results to be accurate both in linear and nonlinear conditions and its accuracy can be further improved using a greater quantity of data related to several experiments performed in different conditions for its identification. Moreover, when a NSM model is employed within a control scheme, the knowledge of its uncertainty bound is fundamental to study the robust stability of the closed loop control scheme. Among the model-based control techniques, this thesis focuses on Nonlinear Internal Model Control (NIMC) and Nonlinear Model Predictive Control (NMPC), which both require a tractable as well as accurate model also in the presence of highly nonlinear dynamics and parameter uncertainties. For this purpose, two methodologies which em- ploy a NSM model are proposed in this thesis: a Set Membership Internal Model Control (SIMC) and a Set Membership Model Predictive Control (SMPC) . The novelty of SIMC consists in deriving the controller by cascading a filter describing the desired input/output system behavior and the inverse of the system model (see [2]). Such a novel procedure exploits the recent results on the right–inversion (see [3]) and does not require the knowledge and the invertibility of the system. This is a not negligible advantage because the inversion of nonlinear systems is not trivial and sometimes impossible. Moreover, a robust stability study shows that the obtained SIMC control structure is input/output stable with finite gain by imposing a small gain condition in the control design phase (see [4], [5]). This is the second main result proposed because usually, in literature, the stability of NIMC control loops is empirically verified a posteriori. The SMPC methodology, instead, uses a NSM model to predict the system behavior and the its uncertainty bound to assess the robust stability of the proposed scheme (see [6]). In fact, at first, exploiting the uncertainty measure, it is shown that the SMPC control structure is robustly stable through an a posteriori stability analysis (see [7]). Then, a procedure to design a robust SMPC control law is proposed (see [8], [9]) and applied to control a nonlinear oscillator. In the case of SIMC and SMPC robust analysis, a vehicle yaw control system is designed to show the effectiveness of the proposed methodologies. A minor research theme, dealt in this thesis, is the design of robust control law using the technology of Direct Virtual Sensors (DVSs). DVSs are software algorithms derived directly from input/output data by means the NSM identification technique. They are able to estimate a variable of interest of a system exploiting some measures already avail- able [10]. It is shown that the direct identification from data implies a greater accuracy of the estimate w.r.t. the classical two steps approach (e.g. Kalman filter) [11]. Further, it is shown that using data collected in closed loop fashion allows to obtain a much accurate estimate than using open loop data [10, 11]. In this thesis, DVSs are used to develop a fault tolerant vehicle yaw control system: the DVS gets on duty and replaces the physical yaw rate sensor when a fault of the last one occurs. The DVS provides the estimate of the yaw rate which is the feedback variable guaranteeing the right working of the stability control system and hence the vehicle safety (see [12]). The novelty consists in the use of an estimated variable from experimental data for control purposes and, in particular, to replace the feedback variable. Moreover, the system in closed loop which employs the DVS results to be robustly stable through an a posteriori analysis.