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 Switching orchestrated by controller, Can be optimized for best performance  This work considers the second type, the switched systems  A switched system 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  [0,1] such that [Wicks, Peletics & Decarlo, 1998] A switching law can be constructed for quadratic stability  Find P 1, P 2 >0,     ≥  or     ≤  such that [Wicks & Decarlo, 1997] A switching law can be constructed based on V(x) = max{x T P 1 x, x T P 2 x}, or V(x) = min{x T P 1 x, x T P 2 x}.  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  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. Based on V min Based on V max

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

8 Switching law based on directional derivatives Consider a function V ( x ). The one-sided directional derivative at x along the direction  is, Then, For the family of linear systems  Let the switching law be constructed as V(x) can be V max (x), V min (x ), or V c (x) 

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

10 Directional derivatives along sliding surfaces A1xA1x A2xA2x  A 1 x+(1-  )A 2 x Different situations w.r.t V max and V min A2xA2x A1xA1x  A 1 x+(1-  )A 2 x Not necessary to require Not sufficient to require  How can we ensure along sliding direction?

11 Some key points in this work A1xA1x A2xA2x  A 1 x+(1-  )A 2 x For V max, If there exists a  s.t. at the non-differentiable point, then along the switching surface. For V min, no sliding motion exist in the set of non-differentiable points Note: A 1 x and A 2 x points away from switching surface Only need to consider the set of differentiable points A2xA2x A1xA1x  A 1 x+(1-  )A 2 x

12 Stabilization via min function The switching law: Proposition 1: There exist no sliding motion in the set of x where V min ( x ) is not differentiable. Matrix condition: Consider V min = min{x T P j x: j=1,2,…,J}. If there exist  ij ≥0,  ij ≥0,  i=1 N  ij =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 A i ’ s: The matrix inequalities [Wicks & Decarlo, 1997] ensures stabilization regardless of sliding motion. The number of matrices P j ’s (J) does not need to be equal to the number of A i ’s (N) : J≥N, or J<N. As J increases, the convergence rate  increases. Example: System cannot be stabilized via quadratic Lyapunov functions: No P>0 and  satisfy both neutrally stable With J=2, maximal  ; With J=3, maximal  ; With J=4, maximal   ;

14 A dual result for V max and V c Recall V max and V c are conjugate functions Consider the dual switched systems: Proposition: Suppose N=2. Sys 1 is stable iff Sys 2 is stable. Sys 1: Sys 2: Remarks: ₋ Results also obtained for the case N >2. ₋ It is easier to obtain matrix conditions via Sys. 2 since V max 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, V max decreases along the sliding surface iff it decreases along a certain convex combination of A 1 x and A 2 x.  A dual result obtained via max function and convex hull function  Condition for stabilization characterized by BMIs.