Data-based control design for linear and recurrent neural network models with stability guarantees
In this doctoral dissertation a framework is defined for the application of data-based methods to control linear and recurrent neural network (RNN) systems.
The main goal of this work is to define sound and reliable methods for data-driven control design. The proposed methods are of hybrid nature, mixing direct and indirect data-driven algorithms. On the one hand, direct data-driven techniques are used to enforce the desired performances. On the other hand, an indirect approach is resorted to in order to define closed-loop stability constraints applied to uncertainty sets or identified system models, the latter being linear or RNN-based ones. The proposed approach leads to computationally lightweight unifying optimization problems based on linear matrix inequalities (LMIs).
As a starting point, a method for the application of virtual reference feedback tuning (VRFT) to linear time-invariant single-input single-output discrete-time systems affected by measurement noise is firstly defined, guaranteeing robust closed-loop stability properties by design. In order to guarantee robust stability, we resort to the definition of a polytopic uncertainty set constructed via a scenario-based set membership identification procedure. Extensions to cope with disturbance rejection requirements and input saturations, and to alternative ellipsoidal uncertainty sets are explored as well.
Secondly, the attention is shifted towards RNN-based models, in view of the remarkable modelling capabilities of this class of models, where a method for an accurate selection of the RNN model class in presence of noisy input/output data is proposed. We first focus on the stability properties of some classes of recurrent neural networks, as they have been only marginally investigated so far in the literature. In this regard, an incremental input-to-state stability sufficient condition is proposed for a general class of RNN models, proving to be less conservative than other conditions available in the literature. As an alternative, two regional (or local) stability conditions for the origin of RNN systems are also presented, jointly to procedures to maximize the dimension of the estimated subset of the basin of attraction.
Since the previously derived conditions can be enforced in the form of LMIs to guarantee incremental input-to-state stability or local stability also to RNN-based closed-loop systems, the possible inclusion of these conditions in unifying LMI-based optimization problems is investigated, where the performances can be enforced via cost functions based on VRFT or H2 control methods.
The previous methodologies are successfully tested on linear and nonlinear simulation examples, among which also some realistic case studies, i.e., the pH neutralization process and a water-heating benchmark system.
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