Video 8.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar

Transcription

1 Video 8.1 Vijay Kumar 1

2 Definitions State State equations Equilibrium 2

3 Stability Stable Unstable Neutrally (Critically) Stable 3

4 Stability Translate the origin to x e x(t) =0 is stable (Lyapunov stable) if and only if for any e > 0, there exists a d(e) > 0 such that x 2 d e x 1 x(t) =0 is asymptotically stable if and only if it is stable and there exists a d > 0 such that 4

5 Asymptotic Stability x 2 d x 1 x(t) =0 is asymptotically stable if and only if it is stable and there exists a d > 0 such that x(t) =0 is globally asymptotically stable if and only if it is asymptotically stable and it is independent of x(t 0 ) 5

6 Example Viscous friction, c x 2 q x 1 m E(t) cannot increase Suppose you want x 1 6 6

7 Global Asymptotic Stability of Linear Systems Global Asymptotic Stability if and only if the real parts of all eigenvalues of A are negative Lyapunov Stability, not Global Asymptotic Stability if and only if the real parts of all eigenvalues are non positive, and zero eigenvalue is not repeated Unstable if and only if there is one eigenvalue of A whose real part is positive 7

8 Linear Autonomous Systems Solution eigenvectors eigenvalues Exponential of a matrix, X for non defective A but similar story for defective A Eigenvalues and eigenvectors for non defective X 8

9 Video 8.2 Vijay Kumar 9

10 Stability of Almost Linear Systems Global Asymptotic Stability if and only if the real parts of all eigenvalues of A are negative Lyapunov Stability, not Global Asymptotic Stability if and only if the real parts of all eigenvalues are non positive, and zero eigenvalue is not repeated Not Significant Significant dynamics Unstable if and only if there is one eigenvalue of A whose real part is positive 10

11 Lyapunov s theorem Nonlinear, autonomous systems Near equilibrium points If the linearized system exhibits significant behavior, then the stability characteristics of the nonlinear system near the equilibrium point are the same as that of the linear system. 11

12 Example Equation of motion Viscous friction, c State space representation Equilibrium points Change of variables 12

13 Example Equilibrium point number 1 Viscous friction, c Equilibrium point number 2 q m 13

14 Example Equilibrium point number 1 Viscous friction, c Linearization q m If c>0 and g>0, real parts of both eigenvalues are always negative The system is locally asymptotically stable 14

15 Example Equilibrium point number 2 Viscous friction, c Linearization q m If c>0 and g>0, both eigenvalues are real, one is positive. The system is unstable 15

16 Example (c=0) Equilibrium point number 1 No friction q Linearization m Real parts of both eigenvalues are non negative No conclusive results 16

17 Summary for Nonlinear Autonomous Systems Write equations of motion in state space notation Solve f(x)=0 Identify equilibrium point(s), x e Linearize equations of motion to get the coefficient matrix A Compute eigenvalues of A. Use Lyapunov s theorem. If the linearized system have significant dynamics, we can make an inference about stability. 17

18 Lyapunov s Direct Method Avoids linearization (hence direct) 18

19 Example Viscous friction, c x 2 q x 1 m E(t) cannot increase 19

20 Lyapunov s Direct Method V(x) is a continuous function with continuous first partial derivatives V(x) is positive definite Such a function V is called a Lyapunov Function Candidate V acts like a norm What if you can show that V never increases? 20

21 Theorem 1. The (above) system is stable if there exists a Lyapunov function candidate such that the time derivative of V is negative semi-definite along all solution trajectories of the system. 21

22 Theorem 2. The (above) system is asymptotically stable if there exists a Lyapunov function candidate such that the time derivative of V is negative definite along all solution trajectories of the system. 22

23 Example 1 Equation of motion Viscous friction, c State space representation q Equilibrium point What is a candidate Lyapunov function? m 23

24 Example 1 Viscous friction, c q m 24

25 Example 2 One-dimensional spring-mass-dashpot with a nonlinear spring x k O M What is a candidate Lyapunov function? Linearized system does not have significant dynamics 25

26 Video 8.3 Vijay Kumar 26

27 Fully-actuated robot arm (n joints, n actuators) Equations of Motion symmetr ic, positive definite inertia matrix n- dimensiona l vector of Coriolis and centripetal forces n- dimensi onal vector of gravitati onal forces n- dimensi onal vector of actuator forces and torques 27

28 Fully-actuated robot arm (continued) 28

29 PD Control of Robot Arms Reference trajectory Error assume Proportional + Derivative Control 29

30 Assume no gravitational forces PD Control achieves Global Asymptotic Stability Lyapunov function candidate Proof 0 Identity 30

31 Assume no gravitational forces PD Control achieves Global Asymptotic Stability Lyapunov function candidate Proof - decreasing as long as velocity is non zero can it reach a state where? 31

32 Assume no gravitational forces PD Control achieves Global Asymptotic Stability Proof decreasing as long as velocity is non zero can it reach a state where La Salle s theorem guarantees Global Asymptotic Stability 32

33 With gravitational forces PD Control achieves Global Asymptotic Stability but with a new equilibrium point g q 33

34 PD control with gravity compensation Global Asymptotic Stability with the correct equilibrium configuration Use the same Lyapunov function candidate: 34

35 Computed Torque Control Reference trajectory Compensate for gravity and inertial forces Global Asymptotic Stability 35

36 Computed Torque Control and Feedback Linearization v u y nonline ar feedbac k new system original system Nonlinear feedback transforms the original nonlinear system to a new linear system Linearization is exact (distinct from linear approximations to nonlinear systems) 36

37 Joint Space versus Task Space Control Task coordinates Reference trajectory Task space control Kinematics 37

38 Task Space Control Task coordinates Task space control Kinematics Commanded joint accelerations Computed torque control 38