# Diz 47 en

# Active Adaptive Control[edit]

**Autor**: Rathouský Jan

This thesis is concerned with stochastically optimal adaptive control strategies and their so-called active adaptive modifications, which represent computationally feasible approximations of dual control. A control strategy is called stochastically optimal, if it optimally solves a given control problem defined for a stochastic system, i.e. a system, the behavior of which is described by the means of probability theory. The thesis is particularly concerned with analysis of the cautious control strategy. The term active adaptive then means, that the control strategy adapts to new information about the system and at the same time actively examines the system and aims to induce such response from the system that brings as much information as possible, while not violating the control performance more than allowable.

The first part of this thesis contains derivation and analysis of the cautious controller of a general ARMAX model with known MA part. A complete analysis of convergence of the associated cautious Riccati-like equation is presented, which is important when extending the control horizon to infinity to find a steady state controller. It is also shown that a finite steady state control law exists even in the case of divergence of the cautious Riccati-like equation. Because the results are formulated for an ARMAX model, they are applicable to a wide range of linear dynamical systems.

The second part of the thesis proposes novel active adaptive control algorithms. It starts with a single-step algorithm for an ARX system based on cautious control. Extension of this algorithm to multiple step is possible, but has not been studied because of the inconvenient properties of cautious control derived in the first part of the thesis. Multiple step adaptive active algorithms based on information matrix properties are presented next, including the so-called ellipsoid algorithm that is studied in more detail. These algorithms are based on a two-phase bicriterial approach, which means that an initial control is first found using a classical control design method (MPC is usually used throughout the thesis) and this control is afterwards altered to achieve active excitation. The thesis also presents a conservative convexification of the ellipsoid algorithm that makes it solvable for higher dimensional systems, where the original nonconvex algorithm becomes infeasible.

**Jan Rathouský**, mailto:rathouskyj@gmail.com