Video 8.1 Vijay Kumar 1 Property of University of Pennsylvania, Vijay Kumar
Definitions State State equations Equilibrium 2 Property of University of Pennsylvania, Vijay Kumar
Stability • Stable • Unstable • Neutrally (Critically) Stable 3 Property of University of Pennsylvania, Vijay Kumar
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 x ( t ) =0 is asymptotically stable if and only if it is stable and there exists a d > 0 x 1 d such that e 4 Property of University of Pennsylvania, Vijay Kumar
Asymptotic Stability x ( t ) =0 is asymptotically x 2 stable if and only if it is stable and there exists a d > 0 d such that x 1 x ( t ) =0 is globally asymptotically stable if and only if it is asymptotically stable and it is independent of x ( t 0 ) 5 Property of University of Pennsylvania, Vijay Kumar
Example x 2 Viscous friction, c x 1 q m � � Suppose you want x 1 6 E ( t ) cannot increase 6 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
Linear Autonomous Systems eigenvalues eigenvectors Solution for non defective A but similar story for Exponential of a matrix, X defective A Eigenvalues and eigenvectors for non defective X 8 Property of University of Pennsylvania, Vijay Kumar
Video 8.2 Vijay Kumar 9 Property of University of Pennsylvania, Vijay Kumar
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 Not Significant if and only if the real parts of all eigenvalues are non Significant dynamics positive, and zero eigenvalue is not repeated • Unstable if and only if there is one eigenvalue of A whose real part is positive 10 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
Example • Equation of motion Viscous friction, c • State space representation • Equilibrium points • Change of variables 12 Property of University of Pennsylvania, Vijay Kumar
Example • Equilibrium point number 1 Viscous friction, c q • Equilibrium point number 2 m 13 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
Example (c=0) • Equilibrium point number 1 No friction q m • Linearization Real parts of both eigenvalues are non negative No conclusive results 16 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
Lyapunov ’ s Direct Method • Avoids linearization (hence direct) 18 Property of University of Pennsylvania, Vijay Kumar
Example x 2 Viscous friction, c x 1 q m E ( t ) cannot increase 19 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
Example 1 • Equation of motion Viscous friction, c • State space representation q • Equilibrium point m What is a candidate Lyapunov function? 23 Property of University of Pennsylvania, Vijay Kumar
Example 1 Viscous friction, c q m 24 Property of University of Pennsylvania, Vijay Kumar
Example 2 • One-dimensional spring-mass-dashpot with a nonlinear spring x k M O Linearized system does not have significant dynamics What is a candidate Lyapunov function? 25 Property of University of Pennsylvania, Vijay Kumar
Video 8.3 Vijay Kumar 26 Property of University of Pennsylvania, Vijay Kumar
Fully-actuated robot arm ( n joints, n actuators) Equations of Motion symmetr n - n - n - ic, positive dimensi definite dimensiona dimensi inertia onal l vector of onal matrix vector Coriolis vector of of and gravitati actuator centripetal onal forces forces forces and torques 27 Property of University of Pennsylvania, Vijay Kumar
Fully-actuated robot arm (continued) 28 Property of University of Pennsylvania, Vijay Kumar
PD Control of Robot Arms Reference trajectory Error assume Proportional + Derivative Control 29 Property of University of Pennsylvania, Vijay Kumar
Assume no gravitational forces PD Control achieves Global Asymptotic Stability Lyapunov function candidate 0 Proof Identity 30 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
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 Property of University of Pennsylvania, Vijay Kumar
With gravitational forces PD Control achieves Global Asymptotic Stability but with a new equilibrium point g q 33 Property of University of Pennsylvania, Vijay Kumar
PD control with gravity compensation Global Asymptotic Stability with the correct equilibrium configuration Use the same Lyapunov function candidate: 34 Property of University of Pennsylvania, Vijay Kumar
Computed Torque Control Reference trajectory Compensate for gravity and inertial forces Global Asymptotic Stability 35 Property of University of Pennsylvania, Vijay Kumar
Computed Torque Control and Feedback Linearization v u y original nonline system ar feedbac new k system Nonlinear feedback transforms the original nonlinear system to a new linear system Linearization is exact (distinct from linear approximations to nonlinear systems) 36 Property of University of Pennsylvania, Vijay Kumar
Joint Space versus Task Space Control Task coordinates Reference trajectory Task space control Kinematics 37 Property of University of Pennsylvania, Vijay Kumar
Task Space Control Task coordinates Task space control Kinematics Commanded joint accelerations Computed torque control 38 Property of University of Pennsylvania, Vijay Kumar
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