Tutorial on Lyapunov's Stability

Tools for Analysis of Nonlinear Systems: Lyapunov’s MethodsStanis ław H. ŻakSchool of Electrical andComputer EngineeringECE 675Spring 20081‡‡‡‡‡‡OutlineNotation using simple examples of dynamical system modelsObjective of analysis of a nonlinear systemEquilibrium pointsStabilityLyapunov functionsBarbalat’s lemma2A Spring-Mass Mechanical Systemx---displacement of the mass from the rest position3‡‡‡‡Modeling the Mass-Spring SystemAssume a linear mass, where k is the linear spring constantApply Newton’s law to obtainDefine state variables: x =x and x =dx/dt1 2The model in state-space format:4‡‡Analysis of the Spring-Mass System ModelThe spring-mass system model is linear time-invariant (LTI)Representing the LTI system in standard state-space format5Modeling of the Simple PendulumThe simple pendulum6‡‡The Simple Pendulum ModelApply Newton’s second lawJ θ = −mgl sin θwhere J is the moment of inertia,2J = mlCombining givesgθ = − sin θl7‡‡‡State-Space Model of the Simple PendulumRepresent the second-order differential equation as an equivalent system of two first-order differential equationsFirst define state variables,x =θ and x =d θ/dt1 2Use the above to obtain state–space model (nonlinear, time invariant)8"‡‡Objectives of Analysis of Nonlinear SystemsSimilar to the objectives pursued when investigating complex linear systemsNot interested in detailed solutions, rather one seeks to ...