1. 1 A Lyapunov Approach to Frequency Analysis Tingshu Hu, Andy Teel UC Santa Barbara Zongli Lin University of Virginia

2. 2 Outline  Introduction – Background – Motivation and problem formulation  Frequency response: main results – Linear systems – Homogeneous systems – Nonlinear systems  Frequency analysis for linear differential inclusions – Numerical analysis with quadratic Lyapunov functions – An observation on frequency response vs L 2 gain  Conclusions

3. 3 Background  Frequency analysis for linear systems – A complete theory with fully developed numerical tools  Closely related to L 2 gain, robust stability and performance  Frequency analysis for nonlinear systems – Attempted since 1950s – Approximate analysis tool: describing function (no longer persuaded) – Complex phenomena observed: jump, subharmonic oscillations,   No systematic approach, as far as we know  Input-output property for nonlinear systems – Lyapunov approach (Sontag and Wang) – LMI for linear systems (Boyd et al)  May be conservative when u belongs to a specific class, e.g., periodical  Our objective: Develop systematic approach to frequency analysis for nonlinear systems

4. 4 Frequency Response for Linear Systems For a SISO linear system: Frequency response: For a nonlinear system -- no transfer function, the output not necessarily sinusoidal, not even periodic -- how to define frequency response?

5. 5 A Bridge: An Equivalent Definition for Linear Systems The input u= k in sin (  t    can be written as: Consider the combined system Claim: k out / k in is the least positive number  such that there exists K > 0 and  satisfying

6. 6 Extension to Nonlinear Systems A nonlinear system: where u is the output of an oscillator : Consider the combined system Let  K  be a locally Lipschitz function. We call  an upper bound for the frequency response if there exist K > 0 and  satisfying The infimum of such functions  ’s  is called the frequency response

7. 7 Characterization of Frequency Response Through A Lyapunov Approach Our Objective: For  Linear systems  Homogeneous systems  Nonlinear systems

8. 8 FR: Linear Systems Main result: With Lyapunov approach, FR can be exactly characterized with quadratic functions. The system: Assumptions: Theorem: The FR is the least  such that there exist P =P T  ,  satisfying

9. 9 FR: Homogeneous Systems Main result: FR can be exactly characterized with homogeneous Lyapunov functions. The system: Assumptions:  w, g  for all g  G( w ). ( |w( t )|  w 0 ) F and h are homogeneous of degree one and globally Lipschitz.  Denote

10. 10 FR: Homogeneous Systems Consider a C 1 function W : R n+l  R  0 and numbers  p  satisfying Theorem: The FR is the infimum of  such that there exist p  and a C 1 function W, homogeneous of degree p, satisfying (1). (1)

11. 11 FR: Nonlinear Systems Main result: Any upper bound for FR can be arbitrarily approximated with Locally Lipschitz Lyapunov functions. The system: Assumptions:  w, g  for all g  G( w ). ( |w( t )|  w 0 ) F is locally Lipschitz with nonempty, convex and compact values. Denote

12. 12 FR: Nonlinear Systems Given locally Lipchitz. Consider a locally Lipschitz function W : R n+l  R  0 and numbers  p  satisfying Theorem: is an upper bound of FR if and only if there exist p  and a locally Lipschitz function W satisfying (2). (2)

13. 13 Nonlinear Systems: An Example The system: If d is arbitrary, the steady state gain from d to y is unbounded, e.g., a constant d > 1 will drive y unbounded. Now assume d ( t ) = k sin (  t  Then By using Lyapunov approach with p = 2, The steady state x is bounded by 2|w 0 |/  2k/   The gain is less than 

14. 14 LDIs: Numerical Analysis with Quadratic Lyapunov Functions A linear differential inclusion (LDIs): For polytopic LDIs, Given  Consider W (  T P    is an upper bound for the FR if there exist P = P T > 0,  such that  LMI - based algorithm has been developed to minimize the bound   Other numerically tractable homogeneous Lyapunov functions are under consideration.

15. 15 LDIs: An Observation of FR vs L 2 Gain Linear system: Peak of FR  L 2 Gain What do we expect for LDIs, Peak of FR  L 2 Gain ? If this is the case, we can suppress FR indirectly by minimizing L 2 gain. A counter example: A second order LDI,  L 2 gain  0.9906. -- By LMI algorithm in Boyd et al.  Peak of FR  1.0917. -- By simulation, with  Peak of FR  1.2169. -- With LMI-based algorithm, Two asymptotic resp.

16. 16 Conclusions  FR definition extended to nonlinear systems – Based on a bridge established for linear systems  Lyapunov approach developed to evaluate FR – Linear systems  with quadratic functions – Homogeneous systems  with homogeneous functions – Nonlinear systems  with locally Lipchitz functions  LMI- based algorithm for FR analysis of LDIs  An obeservation: peak of FR  L 2 gain  Future works –  Numerical analysis for LDIs with homogeneous Lyapunov functions –  Extension to discrete-time systems –  Applications