1. Lecture #14 Computational methods to construct multiple Lyapunov functions & Applications João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems

2. Summary Computational methods to construct multiple Lyapunov functions—Linear Matrix Inequalities (LMIs) Applications (vision-based control)

3. Current-state dependent switching  [  ] ´ set of all pairs ( , x) with  piecewise constant and x piecewise continuous such that 8 t,  (t) = q is allowed only if x(t) 2  q Current-state dependent switching  ›  q 2 R n : q 2  } ´ (not necessarily disjoint) covering of R n, i.e., [ q 2   q = R n 11 22  = 1  = 2  = 1 or 2 Thus ( , x) 2  [  ] if and only if x(t) 2   (t) 8 t

4. Multiple Lyapunov functions Theorem: Suppose there exists    2  1,    2  and a family of continuously differentiable, functions V q : R n ! R, q 2  such that Then the origin is (glob) uniformly asymptotically stable. and at any z 2 R n where a switching signal in  can jump from p to q

5. Multiple Lyapunov functions (linear) Theorem: Suppose there exist constants ,  ¸ 0 and symmetric matrices P q 2 R n £ n, q 2  such that Then the origin is uniformly asymptotically stable (actually exponentially). and at any z 2 R n where a switching signal in  can jump from p to q check! no resets Given the A q, q 2  can we find P q, q 2  and  › {  q : q 2  } so that we have stability? stabilization through choice of switching regions

6. Multiple Lyapunov functions (linear) Theorem: Suppose there exist constants ,  ¸ 0 and symmetric matrices P q 2 R n £ n, q 2  such that Then the origin is uniformly asymptotically stable (actually exponentially). and at any z 2 R n where a switching signal in  can jump from p to q check! no resets Given P q, q 2 , suppose we choose: sigma must be equal to (one of) the largest V q (z) › z’ P q z ) for switching to be possible at z 2 R n from p to q, we must have V q (z) = V p (z) ) jump condition is always satisfied

7. Multiple Lyapunov functions (linear) Theorem: Suppose there exist constants ,  ¸ 0 and symmetric matrices P q 2 R n £ n, q 2  such that Then the origin is uniformly asymptotically stable (actually exponentially). and at any z 2 R n where a switching signal in  can jump from p to q check! no resets It is sufficient to find P q, q 2  such that or equivalently 8 q 2  :

8. Finding multiple Lyapunov functions We need to find P q, q 2  such that 8 q 2  Suppose we can find P q, q 2  and constants  p,q,  p,q ¸ 0, p,q 2  such that Then ¸ 0 when z’ P q z ¸ z’ P p z 8 p 2 

9. Constructing multiple Lyapunov functions Theorem: Suppose there exist constants , ,  p, q,  p, q ¸ 0, p,q 2  and symmetric matrices P q 2 R n £ n, q 2  such that the origin is uniformly asymptotically stable (actually exponentially). Then, for no resets

10. Linear Matrix Inequality (LMI) where F : R m ! R n £ n ´ affine function (i.e., F(x)–F(0) is linear) F(x) ¸ 0 System of linear matrix inequalities:, single LMI System of linear matrix equalities and inequalities,, There are very efficient methods to solve LMIs (type ‘help lmilab’ in MATLAB) single LMI (however most numerical solvers handle equalities separately more efficiently)

11. Back to constructing multiple Lyapunov functions Theorem: Suppose there exist constants , ,  p,q,  p,q ¸ 0, p,q 2  and symmetric matrices P q 2 R n £ n, q 2  such that the origin is uniformly asymptotically stable (actually exponentially). Then, for no resets MI on the unknowns ,  > 0,  p,q,  p,q 2 R, P q 2 R n £ n, p,q 2 , (only linear for fixed  p,q ) P q – P q ’ = 0 Only useful if we have the freedom to choose the switching region  q

12. Convex polyhedral regions  q, q 2  ´ convex polyhedral with disjoint interiors 22 11 33 44 55 component-wise boundary between regions p, q 2  : cell q 2  :

13. Constructing multiple Lyapunov functions Theorem: Suppose there exist 1.symmetric matrices P q 2 R (n+1) £ (n+1), q 2  2.vectors k pq 2 R n+1, p,q 2  3.constants e,  > 0 4.symmetric matrices with nonnegative entries U q,W q, q 2  (not nec. pos. def) such that for every p,q 2  for which  q and  p have a common boundary the origin is uniformly asymptotically stable (actually exponentially). and 8 q 2  LMI on the unknowns P q, k pq, , , U q, W q could be generalized to affine systems, i.e., (homework!)

14. Constructing multiple Lyapunov functions Why? Consider multiple Lyapunov functions: 1 st On the boundary between  q and  p : *

15. Constructing multiple Lyapunov functions Why? Consider multiple Lyapunov functions 2 nd On  q : * ¸ 0

16. Constructing multiple Lyapunov functions Why? Consider multiple Lyapunov functions 3 rd On  q : *

17. Example: Vision-based control of a flexible manipulator flexible manipulator 4 th dimensional small bending approximation Control objective: drive  tip to zero, using feedback from  base ! encoder at the base  tip ! machine vision (essential to increase the damping of the flexible modes in the presence of noise) very lightly damped b flex k flex u m tip b base  base  tip I base u m tip b base I base  base  tip

18. Example: Vision-based control of a flexible manipulator u m tip b base I base  base  tip To achieve high accuracy in the measurement of  tip the camera must have a small field of view feedback output: Control objective: drive  tip to zero, using feedback from  base ! encoder at the base  tip ! machine vision (essential to increase the damping of the flexible modes in the presence of noise)

19. Switched process u manipulator y

20. Switched process controller 1 controller 2 u manipulator y controller 1 optimized for feedback from  base and  tip and controller 2 optimized for feedback only from  base E.g., LQG controllers that minimize

21. Switched system feedback connection with controller 1 (  base and  tip available) feedback connection with controller 2 (only  base available)  tip  max  max

22. Closed-loop response 0102030405060 -5 0 5  tip (t) no switching (feedback only from  base ) 0102030405060 -5 0 5  tip (t) with switching (close to 0 feedback also from  tip )  max = 1

23. Visual servoing Goal:Drive the position x of the end-effector to a reference point r encoder-based control control law u x r  reference r in pre-specified position end-effector position x determined from the joint angles control law u x r  mixed vision/encoder-based control rcrc reference r determined from the camera measurement r c end-effector position x determined from the joint angles

24. Visual servoing Goal:Drive the position x of the end-effector to a reference point r encoder-based control control law u x r  mixed vision/encoder -based control control law u x r  rcrc u x r vision-based control r c, x c reference r determined from the camera measurement r c end-effector position x determined from the camera measurement x c Typically guarantees zero error, even in the presence of manipulator and camera modeling errors

25. Prototype problem end-effector reference (x 1, x 2 ) (r 1, r 2 ) (x 3, 0) For simplicity: 1.Normalized pin-hole camera moves along a straight line perpendicular to its optical axis camera measurements (robot & reference) 2.Feedback linearized kinematic model for the motions of robots and camera Goal:Drive the position (x 1, x 2 ) of the end-effector to the reference point (r 1, r 2 ), using visual feedback

26. Stabilizability by output-feedback Local linearization around equilibrium yields: not detectable Manifestation of a basic limitation on the class of controllers that asymptotically stabilize the system… Can we design an output-feedback controller that asymptotically stabilizes the equilibrium point x 1 = r 1 x 2 = r 2 x 3 =  for some   R ?

27. Stabilizability by output-feedback Can we design an output-feedback controller that asymptotically stabilizes the equilibrium point x 1 = r 1 x 2 = r 2 x 3 =  for some   R ? (1) NO Lemma: There exist no locally Lipschitz functions F: R n  R 2  [0,  )  R n G: R n  R 2  [0,  )  R 3 for which the closed-loop system defined by (1) and has an asymptotically stable equilibrium point with x 1 = r 1 x 2 = r 2

28. Lemma 2: For every r 1, r 2  R, the system (1) is observable. Moreover, there exists a constant input which allows one to uniquely determine the initial state by observing the output over any finite interval ( t 0, t 0 +T ). Indeed, for u 1 = u 2 = 0  robot stopped u 3 = 1  camera moving at constant speed there exists a state reconstruction function h T for which Stabilizability by output-feedback (1) This limitation is not caused by lack of controllability/observability measured outputs final state relative position with respect to reference

29. Limitation on asymptotic stabilization To disambiguate between cases I and II, the camera must move away from ( , 0) This would violate the stability of the equilibrium point in case I Goal:Drive the position (x 1, x 2 ) of the end-effector to the reference point (r 1, r 2 ), using visual feedback end-effector reference (x 1, x 2 ) (r 1, r 2 ) ( , 0) (x 1, x 2 ) system is not at equilibrium (II) system is at equilibrium (I)

30. Limitation on asymptotic stabilization end-effector reference (x 1, x 2 ) (r 1, r 2 ) ( , 0) (x 1, x 2 ) system is not at equilibrium (II) system is at equilibrium (I) There is no need to stabilize the camera and the state of the controller to a fixed equilibrium. It is sufficient to make the set  := { [r 1 r 2  z´]´  R n+3 :   R, z  R n } asymptotically stable Goal:Drive the position (x 1, x 2 ) of the end-effector to the reference point (r 1, r 2 ), using visual feedback any position for camera any state for controller The set  is asymptotically stable if  > 0,  > 0 such that  t 0  0 d(x 0,  ) <   d( x(t),  ) <  and d(x(t),  )  0 as t   solution to closed-loop (required to exist globally)

31. Hybrid controller  robot stopped, camera moving  collect information for reconstruction  move towards reference at constant speed (   (0,1] ) Relies on the camera motion to estimate the robot’s position w.r.t. the target. look mode: move mode:  > T look  := 0  > T move  := 0  timer

32. Hybrid controller look mode: move mode:  > T look  := 0  > T move  := 0  robot stopped, camera moving  collect information for reconstruction  move towards reference at constant speed (   (0,1] )  timer Theorem: For any T look,T move > 0,   (0,1], the hybrid controller renders the set  := { [r 1 r 2  z´]´  R n+3 :   R, z  R n } asymptotically stable for the closed-loop system. For  = 1, the the set  is reached in finite time.

33. 051015 0 1 end-effector reference (x 1, x 2 ) (r 1, r 2 ) = ( 0, 1) (x 3, 0) Stateflow simulation x1x1 x2x2 x3x3 (  = 1 )

34. Next lecture… Applications: ???