An utterly impartial history of Germany: An anniversary (7th October 1989) Michail Sergejewitsch Gorbatschow and...

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1 An utterly impartial history of Germany: An anniversary (7th October 1989) Michail Sergejewitsch Gorbatschow and... 1

2 An utterly impartial history of Germany: An anniversary (7th October 1989)... the chair of the Staatsrat and the first secretary of the central committee of the SED the comrade Erich Honecker ( der Vorsitzende des Staatsrates und der Generalsekretär des Zentralkomitees der SED der Genosse Erich Honecker ) 2

3 An utterly impartial history of Germany: An anniversary (7th October 1989) Those who are late will be punished by life itself ( wer zu spät kommt den bestraft das Leben ) 3

4 An utterly impartial history of Germany: An anniversary (7th October 1989) Those who are late will be punished by life itself ( wer zu spät kommt den bestraft das Leben ) Some time later (9th November 1989) Time delay is bad (or good) 4

5 Some time earlier (don t mention the...) On the theory of balancing Fritz Schürer, Math. Nachr., 1948 (Phew!) 5

6 Some physics, the pendulum: x(t) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0) cos (Ωt) (+ ) 6

7 Some physics, the pendulum: Remark: linear systems x(t) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0) cos (Ωt) (+ ) x(t) = e λt λ 2 e λt = Ω 2 e λt λ 2 = Ω 2 ( characteristic equation ) λ = ±iω x(t) = e iωt (+e iωt ) 7

8 Some physics, the pendulum: x(t) x(t) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0) cos (Ωt) (+ ) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0)e Ωt (+ ) 8

9 Some physics, the pendulum: x(t) x(t) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0) cos (Ωt) (+ ) ẍ(t) = Ω 2 x(t), Ω 2 = g l x(t) = x(0)e Ωt (+ ) x(0) = 0 stable ( Ω 2 < 0) x(0) = 0 unstable (Ω 2 > 0) 9

10 How to balance (I): x(t) K x(t) ẍ(t) = Ω 2 x(t) Kx(t) = `Ω 2 K x(t) x(0) = 0 stable if K > Ω 2 (!) 10

11 How to balance (I): x(t) 2 Ω =g/l K x(t) unstable ẍ(t) = Ω 2 x(t) Kx(t) = `Ω 2 K x(t) stable x(0) = 0 stable if K > Ω 2 (!) x(0)=0 always stable for some gain K (!??) K 11

12 How to balance (II): latency x(t) K x(t τ) ẍ(t) = Ω 2 x(t) Kx(t τ) Differential-difference equation (!?!) 12

13 How to balance (II): latency x(t) K x(t τ) x(t) = e λt λ 2 e λt = Ω 2 e λt Ke λ(t τ) λ 2 = Ω 2 Ke λτ ẍ(t) = Ω 2 x(t) Kx(t τ) Differential-difference equation (!?!) transcendental equation yields infinitely many (!) solutions. always at least one solution with Re(λ) > 0, i.e., unstable state x(0) = 0. Time delay is devastating (!). Kx(t τ) Kx(t)+Kτẋ(t)

14 How to balance (II): latency x(t) K x(t τ) x(t) = e λt λ 2 e λt = Ω 2 e λt Ke λ(t τ) λ 2 = Ω 2 Ke λτ ẍ(t) = Ω 2 x(t) Kx(t τ) Differential-difference equation (!?!) transcendental equation yields infinitely many (!) solutions. always at least one solution with Re(λ) > 0, i.e., unstable state x(0) = 0. Time delay is devastating (!). Kx(t τ) Kx(t)+Kτẋ(t) +... So how do we then balance a stick: Our brain uses the velocity as well for feedback (!), i.e., K 1 x(t τ) K 2 ẋ(t τ) (and engineer would call it a PD controller). 14

15 Some (neuro) biology: Stabilisation possible if τ < τ cr where 2 τ cr = i.e. l cr = 3 Ω 4 gτ cr 2 15

16 Some (neuro) biology: Stabilisation possible if τ < τ cr where 2 τ cr = i.e. l cr = 3 Ω 4 gτ cr 2 Remark: theoretical analysis P (λ) = λ 2 Ω 2 + K 1 e λτ + K 2 λe λτ P (0) = 0, P (0) = 0, P (0) = 0 16

17 Some more or less successful examples how to balance a stick: 17

18 Some more or less successful examples how to balance a stick: l cr = 3 4 gτ 2 cr (!!) 18

19 How to cope with latency: 19

20 How to cope with latency: Slow is not bad! 20

21 How to balance (epilogue): 21

22 How to balance (epilogue): 22