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NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs

NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs. Daniel Liberzon. Univ. of Illinois at Urbana-Champaign, U.S.A. Joint work with Matthias Müller and Frank Allg öwer (Univ. of Stuttgart, Germany).

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NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs

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  1. NORM-CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Joint work with Matthias Müller and Frank Allgöwer(Univ. of Stuttgart, Germany) Workshop in honor of Andrei Agrachev’s 60th birthday, Cortona, Italy

  2. TALK OUTLINE • Motivating remarks • Definitions • Main technical result • Examples • Conclusions

  3. CONTROLLABILITY: LINEAR vs. NONLINEAR Linear systems: • can check controllability via matrix rank conditions • Kalman contr. decomp. controllable modes Nonlinear control-affine systems: similar results exist but more difficult to apply General nonlinear systems: controllability is not well understood Point-to-point controllability

  4. NORM-CONTROLLABILITY: BASIC IDEA • Instead of point-to-point controllability: • look at the norm • ask if can get large by applying large for long time This means that can be steered far away at least in some directions (output map will define directions) • Possible application contexts: • process control: does more • reagent yield more product ? • economics: does increasing • advertising lead to higher sales ?

  5. NORM-CONTROLLABILITY vs. NORM-OBSERVABILITY Instead of reconstructing precisely from measurements of , norm-observability asks to obtain an upper bound on from knowledge of norm of : where For dual notion of observability, [Hespanha–L–Angeli–Sontag ’05] followed a conceptually similar path Precise duality relation – if any – between norm-controllability and norm-observability remains to be understood

  6. NORM-CONTROLLABILITY vs. ISS Input-to-state stability (ISS) [Sontag ’89]: Norm-controllability (NC): small/bounded small/bounded large large Lyapunov characterization [Sontag–Wang ’95]: Lyapunov sufficient condition: (to be made precise later)

  7. FORMAL DEFINITION (e.g., ) fixed scaling function id reachable set at from using radius of smallest ball around containing Definition: System is norm-controllable (NC) if where is nondecreasing in and class in

  8. LYAPUNOV-LIKE SUFFICIENT CONDITION Theorem: is norm-controllable if s.t. : where continuous, • set s.t. and where Here we use [CDC’11, extensions in CDC’12]

  9. SKETCH OF PROOF For simplicity take and • Pick For time s.t. , where is arb. small Repeat for until time s.t.

  10. SKETCH OF PROOF For simplicity take and • Pick For time s.t. Repeat for until we reach

  11. SKETCH OF PROOF piecewise constant Can prove: ’s and ’s don’t accumulate

  12. EXAMPLES For , For , So can take Can take For , if we choose as in Example 1 and

  13. EXAMPLES Since we can’t have with as for fixed (although we do have as if ) x Not norm-controllable with but norm-controllable with

  14. EXAMPLES For , For , Can take s.t. Can take

  15. EXAMPLES concentrations of species A and B, concentration of reagent A in inlet stream, amount of product B per time unit, flow rate, reaction rate, , reactor volume This system is NC from initial conditions satisfying • Relative degree = 2 need to work with Isothermal continuous stirred tank reactor with irreversible 2nd-order reaction from reagent A to product B but need to extend the theory to prove this: • Several regions in state space need to be analyzed separately

  16. LINEAR SYSTEMS If is a real left eigenvector of not orthogonal to then system is NC w.r.t. ( ) Corollary: If is controllable and has real eigenvalues, then is NC Let be real. Then is controllable if and only if system is NC w.r.t. left eigenvectors of More generally, consider Kalman controllability decomposition If is a real left eigenvector of , then system is NC w.r.t. from initial conditions

  17. CONCLUSIONS Contributions: • Introduced a new notion of norm-controllability for general • nonlinear systems • Developed a Lyapunov-like sufficient condition for it • (“anti-ISS Lyapunov function”) • Established relations with usual controllability for linear systems Future work: • Identify classes of systems that are norm-controllable • (in particular, with identity scaling function) • Study alternative (weaker) norm-controllability notions • Develop necessary conditions for norm-controllability • Treat other application-related examples

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