1. 1 Absolute Stability with a Generalized Sector Condition Tingshu Hu

2. 2 Outline  Background, problems and tools Absolute stability with a conic sector, circle criterion, LMIs The generalized sector bounded by PL functions (PL sector) Composite quadratic Lyapunov functions  Main results: Estimation of DOA with invariant level sets Quadratics : Invariant ellipsoid  LMIs Composite quadratics : Invariant convex hull of ellipsoids  BMIs An example  Building up the main results ─ Foundation: Stability analysis of systems with saturation Main idea: Describing PL sector with saturation functions Absolute stability  stability for a family of saturated systems  Summary

3. 3 System with a conic sector condition The conic sector condition : A system with a nonlinear and/or uncertain component : Question: what is the condition of robust stability for all possible  u,t) satisfying the sector condition?

4. 4 Stability for a nonlinear system Consider a nonlinear system: Stability is about the convergence of the state to the origin or an equilibrium point. Also, if it is initially close to the origin, it will stay close. Stability region: the set of initial x 0 such that the state trajectory converges to the origin. Global stability: the stability region is the whole state space.

5. 5 Quadratic function and level sets  Given a n  n real symmetric matrix P, P=P T. If x T Px>0 for all x  R n \{0}, we call P a positive definite matrix, and denote P > 0. (Negative definite can be defined similarly)  With P > 0, define V(x)= x T Px. Then V is a positive definite function, i.e., V(x) > 0 for all x  R n \{0}.  Level sets of a quadratic function: Ellipsoids. Given 

6. 6 Quadratic stability  The system :  Stability condition: If for all x  (P,    )\{0}, Then  (P,   ) is a contractively invariant set and a region quadratic stability. (*)  Condition (*) means that along the boundary of  (P,  ) for any  0, the vector points inward of the boundary In Lyapunov stability theory, the quadratic Lyapunov function is replaced with a more general positive-definite function Px

7. 7 Quadratic stability for linear systems  For a linear system:  Stability condition: If for all x  (P,    )\{0}, Then  (P,   ) is a contractively invariant set and a region of quadratic stability. (*)  (*) is equivalent to  Lyapunov matrix inequality.  As long as there exists a P satisfying the matrix inequality, the linear system is stable

8. 8 Absolute stability with conic sector The conic sector condition : Consider again the system with a nonlinear component : Absolute stability: the origin is globally stable for any  satisfying the sector condition  u,t) F(sI-A) -1 B

9. 9 Conditions for absolute stability  Popov criterion  Circle criterion  LMI condition Quadratic stability Description with linear differential inclusion (LDI): Quadratic stability: exists P=P’ >0 such that Px

10. 10 Motivation for a generalized sector Limitations of the conic sector: not flexible could be too conservative Note: Subclass of the conic sector has been considered, e.g., slope restricted, Monotone ( Dewey & Jury, Haddad & Kapila, Pearson & Gibson, Willems, Safonov et al, Zames & Falb, etc.) Our new approach  extend the linear boundary functions to nonlinear functions  basic consideration: numerical tractability  Our Choice: Piecewise linear convex/concave boundary functions 

11. 11 A piecewise linear (PL) sector Let   and   be    odd symmetric,  piecewise linear  convex or concave for u > 0 The generalized sector condition: Main feature: More flexible and still tractable

12. 12 A tool: the composite quadratic functionthe composite quadratic function Given J positive definite matrices: Denote The composite quadratic function is defined as:  The level set of V C is the convex hull of ellipsoids  Convex, differentiable

13. 13 Applying composite quadratics to conic sectors Recall: A systems with conic sector condition can be described with a LDI: Theorem: Consider V c composed from Q j ’s. If there exist ijk ≥ 0, i = 1,2, j,k =1,2,…,J, such that Then Example: A linear difference inclusion: x(k+1)  co{A 1 x, A 2 (a)x} With quadratics, the maximal a ensuring stability is a 1 =4.676; With composite quadratics (N=2), the maximal a is a 2 =7.546

14. 14  Main results: Invariant level sets Quadratics : Invariant ellipsoid LMIs Composite quadratics : Invariant convex hull of ellipsoids BMIs An example

15. 15 Absolute stability analysis via absolutely invariant level sets Consider the system: L V ( 1 ) is absolutely contractively invariant (ACI) if it is contractively invariant for all  co {      For a Lyapunov candidate V ( x ), its 1-Level set is The set L V (1) is contractively invariant (CI) if  Quadratics : ACI ellipsoids,  Composite quadratics: ACI convex hull of ellipsoids

16. 16 Result 1: contractive invariance of ellipsoid Consider the system, Theorem: An ellipsoid  ( Q  ) is contractively invariant iff and there exist such that

17. 17 Result 2: Quadratics → ACI ellipsoids The system, Theorem: An ellipsoid  ( Q  ) is ACI if and only if and there exist such that

18. 18 Result 3: ACI of convex hull of ellipsoids and there existsuch that Consider V c composed from Q j ’s. L Vc ( 1) is the convex hull of  (Q j -1 ). Theorem: L Vc (1) is ACI if there exist iqjk ≥ 0, i  0,1,…,N, q=1,2, j,k =1,2,…,J, such that

19. 19 Optimizing ACI level sets Choose reference points x 1,x 2,…,x K. Determine ACI L Vc ( 1) such that  x p ’s are inside L Vc (1) with  maximized.

20. 20 Example A second order system:     Reference point: Maximal   L Vc (1):  (Q 1 -1 ):  (Q 2 -1 ):

21. 21 Composite quadratics + PL sector  max  0.8718 Quadratics + PL sector  max = 0.6401 Quadratics + conic sector  max  0.4724 A closed-trajectory under the “worst switching” w.r.t V c ACI convex hull A diverging trajectory

22. 22  Building up the main results Stability analysis for systems with saturation Describing PL sector with saturation functions Stability for an array of saturated systems Absolute stability

23. 23 Stability analysis for systems with saturation The system Problem : To characterize the (contractive) invariance of Traditional approach: find k , 0 < k   ≤ 1, such that then use the traditional absolute stability analysis tools Note: The condition takes form of bilinear matrix inequalities

24. 24 New approach of dealing with saturation The basic idea: If |v| ≤ 1, then  -1  For any row vector h, Recall the traditional approach Further more, the resulting condition for invariance of ellipsoid  includes only LMIs  is necessary and sufficient We have full degree of freedom in choosing h as compared with the one degree of freedom in choosing k  in k  f.

25. 25 Foundation: The necessary and sufficient condition for invariance of ellipsoid Theorem: the ellipsoid  ( Q  ) is contractively invariant for if and only if there exists such that

26. 26 Building-up tool: description of PL functions with saturation functions Consider a PL function with only one bend The necessary and sufficient condition for invariance of ellipsoid follows.

27. 27 Key step: description of PL functions with saturation functions A PL function, Define Properties:

28. 28 Putting things together: Absolute stability via saturated systems The original system and N systems with saturation, ACI of a level set for S CI of the level set for all S iq Stability analysis results contained in: T. Hu, Z. Lin, B. M. Chen, Automatica, pp.351-359, 2002 T. Hu and Z. Lin, IEEE Trans. AC-47, pp.164-169, 2002 T. Hu, Z. Lin, R. Goebel and A. R. Teel, CDC04, to be presented.

29. 29 Summary The systems: subject to PL sector condition Tool: composite quadratic Lyapunov function Problem: determine ACI sets (convex hull of ellipsoids) Key step: description of PL functions with saturations Main feature: more flexible as compared with conic sector, and still tractable Future topics: under PL sector condition,  characterize the nonlinear L 2 gain  apply non-quadratics to study input-state, input-output, state-output properties