1. NOISE SENSITIVITY OF BALANCE WITH DELAYED ON-OFF CONTROL
Jiaxing Wang1, Rachel Kuske,2, David Simpson3
1School of Mathematical Sciences, Peking University , Beijing, 100871, China
2Department of Mathematics, University of British Columbia, Vancouver BC V6K1T5, CA
3 Mathematics, Massey University, Palmerston North, 4442, NZ
POSTER ID: FS09-004
1. Introduction and Models:
We consider motion of inverted pendulum with delayed feedback as a model for
human and mechanical balancing tasks. We study here the interaction of noise
and delay, with discontinuous control and feedback control, which influences
transient dynamics and the stability of the balanced state.
5. Conclusion
Noise sensitivity for on-off control
 Act and Wait Control
 Noise sensitivities due to variation of the eigenvalues of the underlying systems
 Sensitive to external fluctuations in the “waiting” intervals when control is off
Additional fluctuations when error in choosing deadbeat controls
 Feedback Control
 Noise sensitivities when nonlinearities, delays, and on-off control support
transition to bi-stable states
 Need to understand underlying dynamical structure
XXIII ICTAM, 19-24 August 2012, Beijing, China
•Jiaxing Wang, School of Mathematical Sciences, Beijing, P.R.China
Email: yolandwjx@gmail.com Phone: 8613811408190
6. Reference
[1] Stépán, G., Insperger T.:. Robust Time-periodic Control of Time Delayed Systems, in IUTAM Symposium on Dynamics and Control
of Nonlinear Systems with Uncertainty, H. Y. Hu and E. Kreuzer (Eds.), p. 343–352.
[2] Simpson D.J.W., Kuske R., Li Y.-X.: Dynamics of Simple Balancing Models with State Dependent Switching Control. To appear, J.
Nonlinear Science.
[3] Asai Y., Tasaka Y., Nomura K., T. Nomura, M. Casadio, and P. Morasso. A model of postural control in quiet standing: Robust
compensation of delay-induced instability using intermittent activation of feedback control. PLoS ONE, 4(7):e6169, 2009.
[4] Landry M., Campbell S.A., Morris K., Aguilar C.O.: Dynamics of an inverted pendulum with delayed feedback control. SIAM J.
Applied Dynamical Systems 2005;4(2):333–51
[5] J.G. Milton, T. Ohira, J.L. Cabrera, R.M. Frasier, J.B. Gyorffy, F.K. Ruiz, M.A. Strauss, E.C. Balch, P.J. Marin, and J.L. Alexander.
Balancing with vibration: A prelude for “Drift and Act” balance control. PLoS ONE, 4(10):e7427, 2009
SM16 Vibrations and control of structures
Figure 1: Phase-plane for 𝜃,𝜙(𝑖. 𝑒. 𝜃) for
feedback control, ON-OFF regions
determined by (3)
Canonical Model:
• Inverted pendulum with pivot or moveable base control
• Non-dimensionalized and streamlined to capture key effects
𝜽−𝐬𝐢𝐧𝛉 = 𝐅𝐜𝐨𝐬𝛉, 𝐅(𝐭) = −𝐃𝜽(𝐭 − 𝛕) − 𝐏𝛉(𝐭 − 𝝉 (1)
• 𝜃 : the angular displacement from vertical
• F : control of the balancing actions
• P,D: parameters chosen by the operator appropriately
• 𝝉 : the time of delayed feedback in the system
Model with act-and-wait control:
• Characterized by cycling between sampling period with a feedback gains (act),
followed by no feedback for a certain number of periods (wait).
• Discrete model with time intervals Δt (based on the approach in [1]).
𝑛 = 𝜏/Δ𝑡, 𝑝 = 𝑃Δ𝑡2
, 𝑑 = 𝐷Δ𝑡, k: the period steps of control
𝑿 = (𝜃, 𝜃) 𝑇
𝑫 = (−𝑝, −𝑑) 𝑇
then
𝐗 j + 1 = 𝐀𝐗 j + 𝐁F j − n ，F j = g(j)𝐃X(j) (2)
where:
𝐀 = 𝑒 𝑨, 𝐁 = 𝑒 𝑨(1−s) 𝑑𝑠
1
0
𝑩, 𝑨 =
0 1
∆𝑡2
0
, 𝑩 =
0
1
,
g j =
1 𝑖𝑓 𝑗 = hk, h ∈ 𝐙
0 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒
Considering the augmented system for
𝒁𝒋 = (𝑿 𝑇 j , F j − 1 , … , F(j − n)) 𝑇
Over a full act-and-wait period
𝒁 𝒌 = 𝑮 𝒌−𝟏 𝒁 𝒌−𝟏 = 𝑮 𝒌−𝟏 … 𝑮 𝟎 𝒁 𝟎 = 𝚿𝒁 𝟎
• Stability properties :given by the largest modulus of the eigenvalues of the
matrix Ψ, denoted as ρ here.
Figure 2: Left: Stability region for vertical position 𝜃 = 0 with 𝜏 = 0.2 in the p-d plane
for k=1, n=5. Right: For k>n, the stability region is given by a triangular region, with
optimal values of p*,d* corresponding to deadbeat control.
2.RESULTS FOR ACT-AND-WAIT CONTROL
• Larger parametric region for stability with the potential for dead-beat control.
• For k≤n the stability region is D-shaped in a relatively small range of values.
• For k>n, the stability boundaries consists of three straight lines in terms of 𝜏, k
and n and the stability region is larger
Fig.3: Left: Solid line: Simulations of
system (1) with p=p*, d=d*. Jagged line:
The case n=5, k=6 with act-and-wait
control. Smooth line: The case where
control is always on. Dash-dotted line:
Realization with p,d uniformly distributed
about p*,d*, for n=10, k=n+1,.
Right: Realization with additive noise for
n=10, k=n+1,with p=p*, d=d*.
 Two possible sources of noise
• P and D as random variables
• From additive noise in (1), that is:
𝜽 − 𝒔𝒊𝒏𝜽 = 𝑭𝒄𝒐𝒔𝜽 + 𝜼𝝃(𝒕) (5)𝛈 : a constant 𝛏 :white noise
 Comparison of two sources of noise
• Minimal difference between perfect deadbeat control and control with random
parameters due to the abrupt dampening of the deadbeat control
• Additive noise results in heavier fluctuations
 Explanation for the difference
• For random p and d, the probability distribution of ρ is determined by the
following formula combining equation (4):
Pr ρ ∈ A = I ρ∈A ∙ dPr(p, d)
Once p, d takes values relatively close to deadbeat control p*, d* there is strong
damping, and subsequent values of p, d would result in very small growth rates.
For the system with external noises, it can be approximated a convolution of O-U
processes. Perturbations from the Brownian motion grow in the "wait" part of the
cycle.
Fig. 5:Top:the noise-driven behavior in the
phase-plane (dotted blue line) and the
deterministic behaviour (red line),
Bottom: the time series for the noisy dynamics
(solid blue line) and deterministic dynamics
(red dashed line).
4. RESULTS FOR FEEDBACK CONTROL
 In this section we consider (5) with the on-off feedback control given in (3).
• For smaller values of 𝜏 and s, with other parameters not too large, the system
exhibits zig-zag type solutions (see Fig.1).
• For larger values of the delay, the spiral behavior is observed
 The bifurcation structure for zig-zag periodic solutions (see Fig.4 based on [2])
• There is a range of P for which the vertical position θ = 0 and the zig-zag
periodic solution are bistable.
• For these values, noise can then drive the system to the attracting oscillating
dynamics away from the vertical solution.
• The closer the value of P is to the bifurcation value, the more sensitive the
system is to additive noise.
Figure 4:Schematic of bifurcation curves
for fixed points in 𝜃 = 0 and values of on
the zig-zag periodic solutions.
This is because when k>n, the system has n poles at zero and the remaining 2
poles are the eigenvalues of the 2x2 matrix M with
𝑴 = 𝑨 𝑘
+ 𝑨 𝑘−n−1
𝑩𝑫
And the decay ratio 𝜌 thus be given by:
𝜌 =
1
2
𝑀𝑎𝑥
|𝑓𝜏,𝑘,𝑛(𝑝, 𝑑) + 𝑓𝜏,𝑘,𝑛
2
(𝑝, 𝑑) − 𝑔𝜏,𝑘,𝑛(𝑝, 𝑑)|,
|𝑓𝜏,𝑘,𝑛(𝑝, 𝑑) − 𝑓𝜏,𝑘,𝑛
2
(𝑝, 𝑑) − 𝑔𝜏,𝑘,𝑛(𝑝, 𝑑)|
(4)
where fτ,k,n(p, d) and gτ,k,n(p. d) are linear functions of p and d for given τ, k and n.
Model with feedback
control:
As given in [3],using (1), P and D are not
zero when:
𝛉 𝐭 − 𝛕 𝛟 𝐭 − 𝛕 − 𝐬𝛉 𝐭 − 𝛕 > 𝟎
(3)
• Switching manifold: defined by
parameter s < 0(usually 𝑠 < 1 )
• Divides ON and OFF regions in phase
plane(see Fig.1)
• Control ON in regions where
dynamics drift away from origin