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Lecture Signals, systems & inference – Lecture 9: Observers for state estimation

The following will be discussed in this chapter: Hidden modes of composite systems: series (cascade) connections, hidden modes of composite systems: feedback and parallel connections, performance of real-time simulation, observer configuration, observer performance (with no measurement noise),. | Lecture Signals, systems & inference – Lecture 9: Observers for state estimation Observers for state estimation 6.011, Spring 2018 Lec 9 1 Hidden modes of composite systems: series (cascade) connections x(t) = x1(t) y1(t) = x2(t) y2(t) = y(t) H1(s) H2(s) 2 Hidden modes of composite systems: feedback and parallel connections x1(t) y1(t) = y(t) x(t) + H1(s) - H2(s) y2(t) x2(t) x1(t) y1(t) H1(s) x(t) y(t) + x2(t) y2(t) H2(s) 3 Observers 4 System (“plant”) w[n] x[n] q[n] y[n] + A, b, cT, d 1[n] 5 A good model w[n] x[n] b[n q[n] yb[n] y[n] + A, b, cT, d 1[n] 6 Performance of real-time simulation Actual Damping parameter b = 0.2, Estimate input torque x(t) = unit-amplitude pulse Error of duration 5 seconds 0.15 0.1 0.05 Pendulum angle (q1) 0 ˜q1 -0.05 -0.1 -0.15 q1 -0.2 -0.25 q1 -0.3 0 2 4 6 8 10 12 14 16 18 20 Time 7 (sec) Observer configuration w[n] x[n] q[n] y[n] + A, b, cT Plant Z[n] y[n] q[n] q[n] y[n] - + T A, b, c Observer B 8 Observer performance (with no measurement noise) Undamped suspended pendulum, input torque x(t) = unit-amplitude pulse Actual of duration 5 seconds Estimate Error Observer gains /1 = -7 and /2 = -2 1.5 1 q1 Pendulum angle (q1) 0.5 ˜q1 0 -0.5 q1 -1 -1.5 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 9 Observer performance (with measurement noise) 3 Actual 2 q1 Estimate q1 1 0 Pendulum angle (q1) -1 -2 -3 -4 -5 -6 0.0 0.5 1.0 1.5 2.0 Time 10 (s) Observer for ship heading error Desired heading Actual q1[n] heading x[n] Rudder angle q1 [n + 1] 1 q1 [n] ⇢ q[n + 1] = = + x[n] q2 [n + 1] 0 ↵ q2 [n] = Aq[n] + bx[n] . 11 MIT OpenCourseWare https://ocw.mit.edu 6.011 Signals, Systems and Inference Spring 2018 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. 12