AUTOMATIC CONTROL THEORY II

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

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 systemand some map

7. Hierarchically Consistent Control Systems • we would like to define a control systemwhich can produce as trajectories all functions of the formwhere 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 systemcharacterize linear quotient maps

14. Hierarchically Consistent Control Systems • the abstracted linear systemis 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 andthe Popov–Belevitch–Hautus (PBH) testsfor large-scale systems