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1 of 16 NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. NOLCOS 2013, Toulouse, France Joint work with Matthias Müller and Frank Allgöwer, Univ. of Stuttgart, Germany TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A

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2 of 16 TALK OUTLINE Motivating remarks Definitions Main technical result Examples Conclusions

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3 of 16 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

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4 of 16 NORM - CONTROLLABILITY: BASIC IDEA process control: does more reagent yield more product ? economics: does increasing advertising lead to higher sales ? 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:

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5 of 16 NORM - CONTROLLABILITY vs. ISS Lyapunov characterization [ Sontag–Wang ’95 ] : Lyapunov sufficient condition: (to be made precise later) Input-to-state stability (ISS) [ Sontag ’89 ] : small / bounded ISS gain: upper bound on for all Norm-controllability (NC): large NC gain: lower bound on for worst-case

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6 of 16 NORM - CONTROLLABILITY vs. NORM - OBSERVABILITY 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 Instead of reconstructing precisely from measurements of, norm-observability asks to obtain an upper bound on from knowledge of norm of :

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7 of 16 FORMAL DEFINITION (e.g., ) Definition: System is norm-controllable (NC) if where is nondecreasing in and class in reachable set at from using radius of smallest ball around containing

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8 of 16 LYAPUNOV - LIKE SUFFICIENT CONDITION where where continuous, is NC if satisfying Theorem: : s.t., that gives See paper for extension using higher-order derivatives ( e.g., )

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9 of 16 IDEA of CONTROL CONSTRUCTION For time s.t., where is arb. small Repeat for time s.t. until when, with s.t. For simplicity take and. Fix. 1)Given, pick a “good” value ( is “good” for )

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10 of 16 2)Pick a “good” for until we reach For time s.t. Repeat for IDEA of CONTROL CONSTRUCTION For simplicity take and. Fix. when, with s.t. ( is “good” for )

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11 of 16 piecewise constant ’s and ’s don’t accumulate Can prove: IDEA of CONTROL CONSTRUCTION

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12 of 16 EXAMPLES 1) For, Take For, System is NC with where is arbitrary when For each, “good” are and : non-smooth at 0 can be increased from 0 at desired speed so any with is “good”:

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13 of 16 EXAMPLES 2) For, For, – same as above Assume Take “Good” inputs: when, System is NC with

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14 of 16 EXAMPLES 3) Isothermal continuous stirred tank reactor with irreversible 2nd-order reaction from reagent A to product B, reactor volume concentrations of species A and B concentration of reagent A in inlet stream amount of product B per time unit flow rate, reaction rate This system is (globally) NC, but showing this is not straightforward: Several regions in state space need to be analyzed separately Relative degree = 2 need to work with Lyapunov sufficient condition only applies when initial conditions satisfy – meaning that enough of reagent A is present to increase the amount of product B

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15 of 16 LINEAR SYSTEMS More generally, consider Kalman controllability decomposition If is a real left eigenvector of, then system is NC w.r.t. from initial conditions ( ) 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 If is a real left eigenvector of not orthogonal to then system is NC w.r.t.

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16 of 16 CONCLUSIONS 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 Contributions: 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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