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Iterative Learning in Ballistic Control: Formulation of Spatial Learning Processes for Endpoint Control

[+] Author and Article Information
Jian-Xin Xu

e-mail: elexujx @nus.edu.sg

Deqing Huang

e-mail: elehd@nus.edu.sg
Department of Electrical and Computer Engineering,
National University of Singapore,
Singapore 117576

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received November 23, 2010; final manuscript received May 8, 2012; published online November 7, 2012. Assoc. Editor: Warren E. Dixon.

J. Dyn. Sys., Meas., Control 135(2), 024501 (Nov 07, 2012) (11 pages) Paper No: DS-10-1344; doi: 10.1115/1.4007236 History: Received November 23, 2010; Revised May 08, 2012

In this paper, we formulate and explore the characteristics of iterative learning in ballistic control problems, where the projectile experiences a constant gravitational force and a fluid drag force that is quadratic in speed. Three scenarios are considered in the spatial learning process, where the shooting speed, shooting angle, or their combination, are, respectively, the manipulated variables. The viewed endpoint displacement is the controlled variable. Under the framework of iterative learning control, ballistic learning convergence is derived in the presence of process uncertainties. In the end, an illustrative example is provided to verify the validity of the proposed ballistic learning control schemes.

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References

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Figures

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Fig. 1

The sketch of ballistic control problem under the effect of air resistance

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Fig. 2

In ballistic control with shooting angle learning, there is a unique critical shooting angle ϕc ∈ (0,π/2-θ) such that the projectile can achieve the furthest flight distance. (a) The motion trajectories of projectile when the shooting angle φ is less than φc, (b) the motion trajectories of projectile when the shooting angle φ is greater than φc. Due to the effect of air resistance, φc is usually less than π/4 − θ. In practice, case (b) with a larger incident angle is preferred when an obstacle exists in between P and O.

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Fig. 3

Variation of the parameter b versus flight height y in ballistic control process. When the flight height of projectile is increased from 0 m to 10 km, the magnitude of b is reduced by 2/3.

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Fig. 4

The output error profile versus iteration number. Within 13 iterations, the output error in dramatically decreased from 2517 m to 0.54 m.

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Fig. 5

The profile of initial shooting speed signal versus iteration number. The desired shooting speed is v0,d = 807.47 m/s.

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Fig. 6

The motion trajectory of projectile with given shooting angle φ = π/6 and the learned initial speed v0 = 807.47 m/s. It can be seen that the target located at zd = 2 × 104/cos(θ) is attacked accurately.

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Fig. 7

The profile of learning gain versus iteration number. Within 12 iterations, it changes from 0.74 to 81.35.

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Fig. 8

Gradient variation profile versus parametric perturbation and input change, where 9057 ≤ dz/dφ ≤ 7.76 × 104 and the process parameters are m = 4.7 kg, θ = π/12 rad, v0 = 900 m/s

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Fig. 9

The profile of output error versus iteration number. Within five iterations, the output error in dramatically decreased from 2313 m to 0.11 m.

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Fig. 10

The profile of input signal versus iteration number. Due to the parametric perturbation, the desired projection angle is 0.3328 rad.

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Fig. 11

The motion trajectory of projectile with given initial speed v0 = 900 m/s and the learned projection angle φd = 0.3328 rad

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Fig. 12

The variation of learning gain versus iteration number. Within four iterations, it increases from 1.289 × 10−5 to 4.349 × 10−5 and then decreases to 2.899 × 10−5.

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Fig. 13

Variation of the performance index J. Within 34 iterations, J is decreased from 5.348 × 104 to 0.0078. Accordingly, the output error is dramatically decreased from 2313 m to 0.88 m.

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Fig. 14

The profile of input components versus iteration number. The desired shooting angle 0.2654 rad and shooting speed v0 = 978.0 m/s are achieved within 34 iterations.

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Fig. 15

The output error profile versus iteration number under wind effect. After learning 12 iterations, the output error is decreased by 99.9%, from the initial discrepancy 2517 m to the final 2.44 m in average.

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