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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. NOLCOS 2013, Toulouse, France.

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

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  1. NORM-CONTROLLABILITY, or HowaNonlinearSystemRespondstoLargeInputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Joint work with Matthias Müller and Frank Allgöwer, Univ. of Stuttgart, Germany NOLCOS 2013, Toulouse, France TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

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

  3. CONTROLLABILITY: LINEAR vs. NONLINEAR Point-to-point controllability 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

  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 • process control: does more • reagent yield more product? • economics: does increasing • advertising lead to higher sales? This means that can be steered far away at least in some directions (output map will define directions) Possible application contexts:

  5. NORM-CONTROLLABILITY vs. ISS Input-to-state stability (ISS) [Sontag ’89]: Norm-controllability (NC): large large small/bounded small/bounded NC gain: lower bound on for worst-case ISS gain: upper bound on for all Lyapunov sufficient condition: Lyapunov characterization [Sontag–Wang ’95]: (to be made precise later)

  6. 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 : For dual notion of observability, [Hespanha–L–Angeli–Sontag ’05] followed a conceptually similar path where Precise duality relation – if any – between norm-controllability and norm-observability remains to be understood

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

  8. LYAPUNOV-LIKE SUFFICIENT CONDITION Theorem: is NC if satisfying (e.g., ) where continuous, • : s.t. , that gives where See paper for extension using higher-order derivatives

  9. IDEA of CONTROL CONSTRUCTION For simplicity take and . Fix . when , with s.t. ( is “good” for ) • Given , pick a “good” value For time s.t. , where is arb. small Repeat for until time s.t.

  10. IDEA of CONTROL CONSTRUCTION until we reach For simplicity take and . Fix . when , with s.t. ( is “good” for ) • Pick a “good” for For time s.t. Repeat for

  11. IDEA of CONTROL CONSTRUCTION Can prove: ’s and ’s don’t accumulate piecewise constant

  12. EXAMPLES Take For , where is arbitrary For each , “good” are and : when For , so any with is “good”: non-smooth at 0 can be increased from 0 at desired speed System is NC with

  13. EXAMPLES Assume Take For , “Good” inputs: when , For , – same as above System is NC with

  14. EXAMPLES • Relative degree = 2 need to work with Isothermal continuous stirred tank reactor with irreversible 2nd-order reaction from reagentA to productB 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 (globally) NC, but showing this is not straightforward: • Lyapunov sufficient condition only applies when initial conditions • satisfy – meaning that enough of • reagent A is present to increase the amount of product B • Several regions in state space need to be analyzed separately

  15. 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

  16. 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 • Study alternative (weaker) norm-controllability notions • Develop necessary conditions for norm-controllability • Treat other application-related examples

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