1. «NEW PARADIGMS FOR CONTROL THEORY» Romeo Ortega LSS-CNRS-SUPELEC Gif-sur-Yvette, France

2. ContentContent Background Proposal Examples

3. FactsFacts Modern (model-based) control theory is not providing solutions to new practical control problems Prevailing trend in applications: data-based « solutions » Neural networks, fuzzy controllers, etc They might work but we will not understand why/when New applications are truly multidomain There is some structure hidden in «complex systems » Revealed through physical laws Pattern of interconnection is more important than detail

4. Why?Why? Signal processing viewpoint is not adequate: = Input-Output-Reference-Disturbance. Classical assumptions not valid: linear + «small » nonlinearities interconnections with large impedances time-scale separations lumped effects Methods focus on stability (of a set of given ODEs) no consideration of the physical nature of the model.

5. ProposalProposal Reconcile modelling with, and incorporate energy information into, control design.How? Propose models that capture main physical ingredients: energy, dissipation, interconnection Attain classical control objectives (stability, performance) as by-products of: Energy-shaping, interconnection and damping assignment. Confront, via experimentation, the proposal with current practice.

6. Prevailing paradigm Models Control objectives Controller design Signal procesing viewpoint Models  C P  d u z y z d rs P  r d u s z y : Uncertainty  Known structure,   RH d DL

7. Control objectives  z-z d « small »  effect of d on z « small » Controller y z u :C d Class of admissible systems TOO LARGE !!  Conservativeness (min max designs)  High gain (sliding modes, backstepping…)  Complexity Practically useless Intrinsic to signal-processing viewpoint Drawbacks!!!

8.   CI Unmodelled environment ii i x c ec e vvv (Energy-based) Control by interconnection Proposed alternative

9. Models PLANT: H(x) energy function, x state, (v,i) conjugated port variables, Geometric (Dirac) structure capturing energy exchange Dissipation ENVIRONMENT: Passive port Flexibility and dissipation effects Parasitic dynamics Control objectives Focus on energy and dissipation Shape and exchange pattern Controller   C I controller, Hc(z) energy power preserving

10. IDA-PBC of mechanical systems To stabilize some underactuated mechanical devices it is necessary to modify the total energy function. In open loop Where q  R n, p  R n are the generalized position and momenta, respectively, M(q)=M T (q)>0 is the inertia matrix, and V(q) is the potential energy MODEL Control u  R m, and assume rank(G)=m < n Convenient to decompose u=u es (q,p)+u di (q,p)

11. TARGET DYNAMICS Desired (closed loop) energy function where M d =M d T >0 and V d (q) with port controlled Hamiltonian dynamics where

12. All assignable energy functions are characterized by a PDE!! All assignable energy functions are characterized by a PDE!! The PDE is parameterized by two free matrices (related to physics)

13. ExamplesExamples BALL AND BEAM

14. Ball and Beam

16. Vertical take-off and landing aircraft

17. Cart with inverted pendulum

18. ExamplesExamples (PASSIVE) WALKING Plant: double pendulum Environement: elastic (stiff) Model

19. (Passive) walking Control objetive: Shape energy

20. (Passive) walking

21. other mechatronic systems: teleoperators, robots in interaction (with environement)

22. Plant: (controlled) wave eq. Environment: passive mech. contact model control objective: shape energy Piezoelectric actuators

23. Control through long cables E.g., overvoltage in drives model control objective: change interconnection to suppress waves

27. Dual to teleoperators Many examples in power electronics and power systems

28. Thank you!!