1 of 17 NORM - CONTROLLABILITY, or How a Nonlinear System Responds to Large Inputs Daniel Liberzon Univ. of Illinois at Urbana-Champaign, U.S.A. Workshop in honor of Andrei Agrachev’s 60 th birthday, Cortona, Italy Joint work with Matthias Müller and Frank Allgöwer (Univ. of Stuttgart, Germany)

2 of 17 TALK OUTLINE Motivating remarks Definitions Main technical result Examples Conclusions

3 of 17 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 of 17 NORM - CONTROLLABILITY: BASIC IDEA Possible application contexts: 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)

5 of 17 NORM - CONTROLLABILITY vs. NORM - OBSERVABILITY where For dual notion of observability, [ Hespanha–L–Angeli–Sontag ’05 ] followed a conceptually similar path Instead of reconstructing precisely from measurements of, norm-observability asks to obtain an upper bound on from knowledge of norm of : Precise duality relation – if any – between norm-controllability and norm-observability remains to be understood

6 of 17 NORM - CONTROLLABILITY vs. ISS Lyapunov characterization [ Sontag–Wang ’95 ]: Norm-controllability (NC): large Lyapunov sufficient condition: (to be made precise later) Input-to-state stability (ISS) [ Sontag ’89 ]: small / bounded

7 of 17 FORMAL DEFINITION (e.g., ) Definition: System is norm-controllable (NC) if where is nondecreasing in and class in fixed scaling function id reachable set at from using radius of smallest ball around containing

8 of 17 LYAPUNOV - LIKE SUFFICIENT CONDITION Here we use where continuous, where and set s.t. is norm-controllable if s.t. : Theorem: [ CDC’11, extensions in CDC’12 ]

9 of 17 SKETCH OF PROOF For simplicity take and 1)Pick For time s.t., where is arb. small Repeat for time s.t. until

10 of 17 SKETCH OF PROOF 2)Pick For simplicity take and For time s.t. Repeat for until we reach

11 of 17 SKETCH OF PROOF piecewise constant ’s and ’s don’t accumulate Can prove:

12 of 17 EXAMPLES 1) For, 2) and So can take if we choose as in Example 1 For, Can take

13 of 17 EXAMPLES 3) Since we can’t have with as for fixed (although we do have as if ) x Not norm-controllable with but norm-controllable with 4)

14 of 17 EXAMPLES 5) For, Can take Can take s.t.

15 of 17 EXAMPLES 6) Isothermal continuous stirred tank reactor with irreversible 2nd-order reaction from reagent A to product B 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 but need to extend the theory to prove this: Several regions in state space need to be analyzed separately Relative degree = 2 need to work with

16 of 17 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.

17 of 17 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 (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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