AUTOMATIC CONTROL THEORY II

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

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

Hierarchically Consistent Control Systems two-layer control hierarchy

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

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

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

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

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

Hierarchically Consistent Control Systems difference between model reduction and

abstraction

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

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

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

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

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

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