Lecture #14 Computational methods to construct multiple Lyapunov functions & Applications

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

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

3. Current-state dependent switching c › {cq2 Rn: q2 Q} ´ (not necessarily disjoint) covering of Rn, i.e., [q2Qcq = Rn Current-state dependent switching S[c] ´ set of all pairs (s, x) with s piecewise constant and x piecewise continuous such that 8t, s(t) = q is allowed only if x(t) 2 cq c2 c1 s = 1 or 2 s = 1 s = 2 Thus (s, x) 2S[c] if and only if x(t) 2cs(t)8t

4. Multiple Lyapunov functions Theorem: Suppose there exists a12 K1, a22 K and a family of continuously differentiable, functions Vq: Rn!R, q 2 Q such that and at any z 2 Rn where a switching signal in S can jump from p to q Then the origin is (glob) uniformly asymptotically stable.

5. Multiple Lyapunov functions (linear) no resets Theorem: Suppose there exist constants d, e ¸ 0 and symmetric matrices Pq 2Rn£n, q 2 Q such that and at any z 2 Rn where a switching signal in S can jump from p to q check! Then the origin is uniformly asymptotically stable (actually exponentially). Given the Aq, q 2 Q can we find Pq,q 2 Q and c› { cq: q 2 Q } so that we have stability? stabilization through choice of switching regions

6. Multiple Lyapunov functions (linear) no resets Theorem: Suppose there exist constants d, e ¸ 0 and symmetric matrices Pq 2Rn£n, q 2 Q such that and at any z 2 Rn where a switching signal in S can jump from p to q check! Then the origin is uniformly asymptotically stable (actually exponentially). Given Pq,q 2 Q, suppose we choose: sigma must be equal to (one of) the largest Vq(z) ›z’ Pq z for switching to be possible at z 2 Rn from p to q, we must have Vq(z) = Vp(z) jump conditionis always satisfied ) )

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

8. Finding multiple Lyapunov functions We need to find Pq,q 2 Q such that 8q 2 Q Suppose we can find Pq,q 2 Q and constants gp,q, mp,q¸ 0, p,q 2 Q such that Then ¸ 0 when z’ Pq z¸z’ Pp z8p2Q

9. Constructing multiple Lyapunov functions no resets Theorem: Suppose there exist constants d, e, gp,q, mp,q¸ 0, p,q 2 Q and symmetric matrices Pq 2Rn£n, q 2 Q such that Then, for the origin is uniformly asymptotically stable (actually exponentially).

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

11. Back to constructing multiple Lyapunov functions no resets Pq – Pq’ = 0 Theorem: Suppose there exist constants d, e, gp,q, mp,q¸ 0, p,q 2 Q and symmetric matrices Pq 2Rn£n, q 2 Q such that Then, for the origin is uniformly asymptotically stable (actually exponentially). MI on the unknowns d, e > 0, gp,q, mp,q 2R, Pq 2Rn£n, p,q 2 Q, (only linear for fixed mp,q) Only useful if we have the freedom to choose the switching region cq

12. Convex polyhedral regions cq, q2 Q´ convex polyhedral with disjoint interiors c2 c1 c3 c5 c4 cell q2 Q: component-wise boundary between regions p, q 2 Q:

13. Constructing multiple Lyapunov functions • Theorem: • Suppose there exist • symmetric matrices Pq2R(n+1)£(n+1), q 2 Q • vectors kpq2Rn+1, p,q 2 Q • constants e, d > 0 • symmetric matrices with nonnegative entries Uq,Wq, q 2 Q (not nec. pos. def) • such that for every p,q 2 Q for which cq and cp have a common boundary could be generalized to affine systems, i.e., (homework!) and 8q 2 Q the origin is uniformly asymptotically stable (actually exponentially). LMI on the unknowns Pq, kpq, d, e, Uq, Wq

14. Constructing multiple Lyapunov functions * Why? Consider multiple Lyapunov functions: 1st On the boundary between cq and cp:

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

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

17. Example: Vision-based control of a flexible manipulator kflex mtip mtip bflex  qtip qtip bbase bbase qbase qbase u u Ibase Ibase flexible manipulator very lightly damped 4th dimensional small bending approximation Control objective: drive qtip to zero, using feedback from qbase! encoder at the base qtip ! machine vision (essential to increase the damping of the flexible modes in the presence of noise)

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

19. Switched process manipulator y u

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

21. Switched system feedback connection with controller 2 (only qbase available) feedback connection with controller 1 (qbase and qtip available) -qmax qmax qtip

22. Closed-loop response 5 0 -5 0 10 20 30 40 50 60 5 0 -5 0 10 20 30 40 50 60 no switching(feedback only from qbase) qtip(t) with switching (close to 0 feedback also from qtip ) qtip(t) qmax = 1

23. Visual servoing Goal: Drive the position x of the end-effector to a reference point r x x r r   u u controllaw controllaw rc encoder-based control mixed vision/encoder-based control • reference r in pre-specified position • end-effector position x determined from the joint angles • reference r determined from the camera measurement rc • 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 mixedvision/encoder-based control x r x r  x r controllaw u  controllaw u rc, xc encoder-based control controllaw u rc vision-based control • reference r determined from the camera measurement rc • end-effector position x determined from the camera measurement xc Typically guarantees zero error, even in the presence of manipulator and camera modeling errors

25. Prototype problem Goal: Drive the position (x1, x2) of the end-effector to the reference point (r1, r2), using visual feedback end-effector reference (x1, x2) (r1, r2) (x3, 0) • For simplicity: • Normalized pin-hole camera moves along a straight line perpendicular to its optical axis camera measurements (robot & reference) • Feedback linearized kinematic model for the motions of robots and camera

26. Stabilizability by output-feedback Can we design an output-feedback controller that asymptotically stabilizes the equilibrium point x1 = r1x2 = r2x3 =  for some   R ? Local linearization around equilibrium yields: not detectable Manifestation of a basic limitation on the class of controllers that asymptotically stabilize the system…

27. Stabilizability by output-feedback (1) Can we design an output-feedback controller that asymptotically stabilizes the equilibrium point x1 = r1x2 = r2x3 =  for some   R ? NO Lemma: There exist no locally Lipschitz functionsF: Rn R2 [0,)  RnG: Rn R2 [0,)  R3for which the closed-loop system defined by (1) and has an asymptotically stable equilibrium point with x1 = r1x2 = r2

28. Stabilizability by output-feedback (1) This limitation is not caused by lack of controllability/observability Lemma 2: For every r1, r2  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 ( t0, t0+T ). Indeed, for u1 = u2 = 0  robot stopped u3 = 1  camera moving at constant speed there exists a state reconstruction function hT for which measured outputs final state relative position with respect to reference

29. Limitation on asymptotic stabilization Goal: Drive the position (x1, x2) of the end-effector to the reference point (r1, r2), using visual feedback (x1, x2) system is at equilibrium (I) reference (r1, r2) end-effector (x1, x2) system is not at equilibrium (II) (, 0) 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

30. Limitation on asymptotic stabilization Goal: Drive the position (x1, x2) of the end-effector to the reference point (r1, r2), using visual feedback (x1, x2) system is at equilibrium (I) reference (r1, r2) end-effector (x1, x2) system is not at equilibrium (II) (, 0) any position for camera 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 := { [r1r2  z´]´  Rn+3 :   R , z  Rn }asymptotically stable any state for controller The set  is asymptotically stable if  > 0,  > 0 such that t0  0d(x0,  ) <   d( x(t),  ) <  and d(x(t),  )  0 as t   solution to closed-loop (required to exist globally)

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

32. Hybrid controller look mode:  robot stopped, camera moving  collect information for reconstruction  timer • > Tlook •  := 0 • > Tmove •  := 0 move mode:  move towards reference at constant speed( (0,1] ) Theorem: For any Tlook,Tmove > 0,  (0,1], the hybrid controller renders the set := { [r1r2  z´]´  Rn+3 :   R , z  Rn }asymptotically stable for the closed-loop system. For  = 1, the the set  is reached in finite time.

33. Stateflow simulation 1 0 -1 0 5 10 15 end-effector reference (  = 1 ) (x1, x2) (r1, r2) = ( 0, 1) (x3, 0) x2 x3 x1

34. Next lecture… Applications: ???