«NEW PARADIGMS FOR CONTROL THEORY»

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

2. Content • Background • Proposal • Examples

3. Facts • 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? • 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. Proposal • 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. Models • Control objectives • Controller design r s P : d z q D u y r s Known structure, q Q P d z q Uncertainty y u C z D D RH d d D L Prevailing paradigm Signal procesing viewpoint Models

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

8. i i i e c Unmodelled environment S S v S v v c e C I x Proposed alternative (Energy-based) Control by interconnection

9. controller, Hc(z) energy • power preserving S C S I 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 Controller • Focus on energy and dissipation • Shape and exchange pattern

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 qÎRn, pÎRn are the generalized position and momenta, respectively, M(q)=MT(q)>0 is the inertia matrix, and V(q) is the potential energy MODEL Control uÎRm, and assume rank(G)=m < n Convenient to decompose u=ues(q,p)+udi(q,p)

11. TARGET DYNAMICS Desired (closed loop) energy function where Md=MdT>0 and Vd(q) with port controlled Hamiltonian dynamics where

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

13. Examples BALL AND BEAM

14. Ball and Beam

15. Ball and Beam

16. Vertical take-off and landing aircraft

17. Cart with inverted pendulum

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

19. (Passive) walking Control objetive: Shape energy

20. (Passive) walking

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

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

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!!