CIS
March30Lecture_Till19

March30Lecture_Till19 - CIS 540 Principles of Embedded...

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CIS 540 Principles of Embedded Computation Spring 2017 Instructor: Rajeev Alur [email protected]
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Stability of Dynamical Systems Key correctness requirement for dynamical systems: stability Small perturbations in the input values should not cause disproportionately large changes in the outputs For cruise controller, correctness requirements: Safety: Speed should always be within certain threshold values Liveness: Actual speed should eventually get close to desired speed Stability: If grade of the road changes, speed should change only slowly Classical mathematical formalization of stability: Lyapunov stability of equilibria Bounded-Input-Bounded-Output stability of response CIS 540 Spring 2017; Lecture March 30
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Equilibria of Dynamical Systems Consider a closed (i.e. without inputs) continuous-time component H If H has inputs, then we can analyze equilibria by setting inputs to a fixed value Assume state x is k-dimensional, and dynamics is Lipschitz- continuous given by dx/dt = f(x) A state x e is called an equilibrium of H if f(x e ) = 0 If initial state of H equals an equilibrium state x e , then the system stays in this state at all times CIS 540 Spring 2017; Lecture March 30
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Pendulum Equilibria Dynamics when external torque is 0: d = ; d = - g sin / l Length l Torque u Weight mg Displacement mg sin Equilibrium states:  =0; sin =0 Equilibrium state 1: =0; =0; Pendulum is vertically downwards Equilibrium state 2: =0; =- ; Pendulum is vertically upwards CIS 540 Spring 2017; Lecture March 30
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Lyapunov Stability Consider a closed continuous-time component H with Lipschitz- continuous dynamics dx/dt = f(x) Given an initial state s, let x [s] denote the unique state response signal for the initial value problem x(0)=s and dx/dt=f(x) Consider an equilibrium state s e : if initial state is s e then the response x [s e ] is a constant function of time, always equal to s e Stability of an equilibrium: when the system is in an equilibrium state, if we perturb its state slightly As time passes, will the state stay close to the equilibrium state ? As time passes, will the system eventually return to the equilibrium state? CIS 540 Spring 2017; Lecture March 30
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Lyapunov Stability Conditions Suppose the initial state s is close to an equilibrium state s e , does the state along the response signal x [s] stay close to s e ?
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  • Fall '09
  • ALUR
  • Stability theory, Lyapunov stability, Pendulum Equilibria

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