1. AUTOMATIC CONTROL THEORY II Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. Hierarchically Consistent Control Systems large-scale systems are systems of very high complexity complexity is reduced by imposing a hierarchical structure on the system architecture systems of higher functionality reside at higher levels of the hierarchy

3. Hierarchically Consistent Control Systems two-layer control hierarchy

4. Hierarchically Consistent Control Systems two-layer control hierarchy  is frequently used in the quite common planning and control hierarchical systems each layer has different objectives the higher level uses a coarser system model than the lower level

5. Hierarchically Consistent Control Systems the main challenge in hierarchical systems is  the extraction of a hierarchy of models at various levels of abstraction which are compatible with the functionality and objectives of each layer abstraction or aggregation  grouping the system states into equivalence classes

6. Hierarchically Consistent Control Systems depending on the cardinality of the resulting quotient space, we may have  discrete  or continuous abstractions given a control system and some map

7. Hierarchically Consistent Control Systems we would like to define a control system which can produce as trajectories all functions of the form where x(t) is a trajectory of system the function h is the “quotient map” which performs the state aggregation

8. Hierarchically Consistent Control Systems the control input v of the coarser model is not the same input u of system v should be thought of as a macroinput u v can be velocity inputs of a kinematic model u may be force and torque inputs of a dynamic model

9. Hierarchically Consistent Control Systems difference between model reduction and abstraction

10. Hierarchically Consistent Control Systems generalizing the geometric notion of Φ-related vector fields to control systems notion of Φ-related control systems  allow us to push forward control systems through quotient maps  and obtain well-defined control systems describing the aggregate dynamics

11. Hierarchically Consistent Control Systems notion of Φ-related control systems mathematically formalizes the concept of virtual inputs used in backstepping designs aggregation is not independent of the functionality of the layer at which the abstracted system will be used

12. Hierarchically Consistent Control Systems when an abstracted model is extracted from a more detailed model  one would also like to ensure that certain properties propagate from the macromodel to the micromodel properties of interest at each layer include  optimality  controllability  stabilizability  trajectory tracking

13. Hierarchically Consistent Control Systems the macromodel is a consistent abstraction of the micromodel  controllability requests from the macromodel are implementable by the micromodel Given the linear control system characterize linear quotient maps

14. Hierarchically Consistent Control Systems the abstracted linear system is controllable if and only if (iff) given system is controllable checking the desired property on the abstracted system should be equivalent or sufficient to checking the property on the original system

15. Hierarchically Consistent Control Systems having characterized consistent linear abstractions we obtain a hierarchical controllability criterion  which has computational and conceptual advantages over the Kalman rank condition and the Popov–Belevitch–Hautus (PBH) tests for large-scale systems