March30Lecture_Till19

March30Lecture_Till19 - CIS 540 Principles of Embedded...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
CIS 540 Principles of Embedded Computation Spring 2017 http://www.seas.upenn.edu/~cis540/ Instructor: Rajeev Alur [email protected]
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ?
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern