On several composite quadratic Lyapunov functions for switched systems

1. On several composite quadratic Lyapunov functions for switched systems Tingshu Hu, Umass Lowell Liqiang Ma, Umass Lowell Zongli Lin, Univ. of Virginia

2. Outline Background on switched systems and sliding motion Approach of this work, main issues Use three Lyapunov functions to construct switching laws How to ensure stability in the presence of sliding motion? Main Results Switching law based on directional derivatives Directional derivatives along sliding surface Stabilization via min function Dual result for max function and convex hull function Conclusions

3. Switched systems and LDIs Given a family of linear systems • A linear differential inclusion (LDI): Switching controlled by unknown force Expect the worst case • A switched system Switching orchestrated by controller, Can be optimized for best performance • This work considers the second type, the switched systems Two approaches: • Analytical methods for lower-order sys [Antsaklis, Michel, Hu,Xu, Zhai] • Using Lyapunov functions for switching laws construction and • stability analysis

4. Earlier constructive approaches • Find P > 0 and a[0,1] such that[Wicks, Peletics & Decarlo, 1998] A switching law can be constructed for quadratic stability • Find P1, P2>0,b1, b2 ≥ 0 (or b1,b2 ≤ 0) such that [Wicks & Decarlo, 1997] • A switching law can be constructed based on • V(x) = max{xTP1 x, xTP2x}, or V(x) = min{xTP1 x, xTP2x}. • Stability ensured only if no sliding motion occurs. • Both methods involving BMIs, harder than LMIs but numerically possible • More recent development based on BMIs:[Decarlo, Branicky, Pettesson, • Lennartson, Zhai, etc].

5. Effect of sliding motion Assume the matrix condition is satisfied [Wicks & Decarlo, 1997] When sliding motion occurs, the system can be stable or unstable Based onVmin Based onVmax • Sliding motion not unusual in switched systems; • It may be a result of optimization • Not realizable but can be approximated via • hysteresis, delay, fast sampling, etc.

6. Approach of this work, main issues Approach: • Use three types of Lyapunov functions to construct switching laws • Functions composed from a family of quadratic functions; • Lead to semi-definite matrix conditions, numerically possible; • Two types of functions are not everywhere differentiable • Main issues: • How to dealing with non-differentiable Lyapunov functions? • Use directional derivatives • How to address stability in the presence of possible sliding motion? • Exclude the existence of sliding motion • Characterize directional derivatives along sliding direction

7. Level set = Level set = Level set = Three Lyapunov functions Given matrices: . Let 1) The min function: 2) The max function: 3) The convex hull function: • Only the convex hull function is everywhere differentiable • Vc and Vmax are convex conjugate pairs, they have been successfully applied to LDIs and saturated systems [Goebel, Hu, Lin, Teel, Zaccarian]

8. Switching law based on directional derivatives Consider a function V(x). The one-sided directional derivative at x along the direction xis, Then, x For the family of linear systems • Let the switching law be constructed as V(x) can be Vmax(x), Vmin(x), or Vc(x)

9. How sliding motion complicates the analysis? Along sliding direction: What really matters is • Sliding along the set of differentiable points is easy to deal with • Sliding along the set of non-differentiable points is more complicated

10. A2x A1x aA1x+(1-a)A2x • How can we ensure along sliding direction? Directional derivatives along sliding surfaces Different situations w.r.t Vmax and Vmin aA1x+(1-a)A2x A2x A1x Not sufficient to require Not necessary to require

11. aA1x+(1-a)A2x A2x A2x A1x A1x aA1x+(1-a)A2x Some key points in this work For Vmax, If there exists a b[0,1] s.t. For Vmin, no sliding motion exist in the set of non-differentiable points at the non-differentiable point, then A1x and A2x points away from switching surface along the switching surface. Only need to consider the set of differentiable points Note:

12. Stabilization via min function The switching law: Proposition 1: There exist no sliding motion in the set of x where Vmin(x) is not differentiable. Matrix condition: Consider Vmin= min{xTPj x: j=1,2,…,J}. If there exist h, bij≥0, aij≥0, Si=1Naij=1, such that Then for every solution, Stability ensured by matrix condition even if sliding motion occurs.

13. Stabilization via min function Special case with two Ai’s: The matrix inequalities [Wicks & Decarlo, 1997] ensures stabilization regardless of sliding motion. The number of matrices Pj’s (J) does not need to be equal to the number of Ai’s (N) : J≥N, or J<N. As J increases, the convergence rate h increases. Example: both neutrally stable System cannot be stabilized via quadratic Lyapunov functions: No P>0 and a[0,1] satisfy With J=2, maximal h = 0.3375; With J=3, maximal h > 0.3836; With J=4, maximal h> 0.4656;

14. A dual result for Vmax and Vc Recall Vmax and Vc are conjugate functions Consider the dual switched systems: Sys 1: Sys 2: Proposition: Suppose N=2. Sys 1 is stable iff Sys 2 is stable. Remarks: • Results also obtained for the case N >2. • It is easier to obtain matrix conditions via Sys. 2 since Vmax is • piecewise quadratic.

15. Example: A pair of dual systems Sys 1: Sys 2: Sliding motion occurs for both systems. They are stable with the same convergence rate w.r.t correspoding Lyapunov function.

16. Conclusions • Switching laws constructed via three Lyapunov functions • The min function • The max function • The convex hull function • Sliding motion carefully considered by using the • directional derivatives • When min function is used, sliding motion does not exist in the set of non-differentiable points • When max function is used, Vmax decreases along the sliding surface iff it decreases along a certain convex combination of A1xand A2x. • A dual result obtained via max function and convex hull function • Condition for stabilization characterized by BMIs.