stable manifold approach for hamilton -jacobi equations in...
TRANSCRIPT
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Stable manifold approach for Hamilton-Jacobi equations in optimal control theory with applications
Noboru Sakamoto Nanazan University Nagoya University
9th CDPS 2015 at Beijing July 3rd 2015
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Outline of talk
Hamilton-Jacobi equation and stable manifold
Stable manifold algorithm
Applications
Non-uniqueness of solution for HJE
Summary
• DC motor over-current avoidance
• Swing-up and stabilization of inverted pendulum
• Swing-up and stabilization of flexible inverted pendulum
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HJE and optimal control
Optimal control problem
Solution
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Problem 0
HJE & stable manifold
Problem 1
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Stable manifold
Stable manifold
Invariance property
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FACT
FACT
HJE & stable manifold
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Problem 0
Problem 1
HJE & stable manifold
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Stable manifold algorithm
Stable manifold algorithm (Sakamoto & van der Schaft 2008)
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The idea of this iteration
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How to compute optimal control?
No computation for V(x)
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Computation : a numerical example
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Computation of solution on SM Numerically obtain SM Interpolation/functional fitting
Summary of stable manifold approach
High accuracy Fast convergence (exponential) Iterative, suitable for computer impl. Non-analytic function can be handled (saturation)
Advantages
Challenges Higher dimensional systems (6
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Applications (Experimental verifications)
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Model of DC motor
Stabilization & (overcurrent prevention)
Control objective
DC motor over-current avoidance
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DC motor over-current avoidance
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0.02 0.04 0.06 0.08 0.1 0 Time [s]
Stable manifold approximation
Optimal control
Experimental result
DC motor over-current avoidance
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How much can we enlarge the domain of attraction of the closed loop system
Compute large stable manifold Real implementation Robustness
Challenging issues
Swing up & stabilization of inverted pnedulum
Upright position (equilibrium)
Pending position (to be included in the domain of attraction)
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(Eq. for the body)
(Eq. for the pendulum)
Modeling
Swing up & stabilization of inverted pnedulum
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What if sat(u) = 7[V] ?
Swing up & stabilization of inverted pnedulum
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HJ equation (Hamiltonian system)
Stable manifold computation
Swing up with 2 swings !
Swing up & stabilization of inverted pnedulum
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saturation
HJ equation
2 swing control
= Original HJ equation ???
2 swing control ? 1 swing 2 swing
Pending pos.
dθ/d
t
θ
Non-uniqueness and curvature of SM
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θ
dθ/d
t 1 swing, 2swing, 3swing solutions coexist in one HJE
1 swing
2 swings
3 swings 4 swings
Non-uniqueness and curvature of SM
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Stable manifold including １swing, ２swing, 3 swing solution
Non-uniqueness and curvature of SM
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φencoder DC motor
θencoder
Spring steel flexible beam
©JAXA
Light weight
Agility
Nonlinear control of flexible structure
Swing up & stabilization of flexible IP
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LQ control causes spillover instability
Frequency-dependent LQ control can stabilize the system
Swing up control with filter?
No switching is allowed!
Swing up & stabilization of flexible IP
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φencoder DC motor
θencoder
Cut-off filter Delay compensation Spring steel flexible beam
Controller structure
Swing up & stabilization of flexible IP
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Optimal controller:
Swing up & stabilization of flexible IP
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Other applications
２．Optimal control for Burgers eq
３．Optimal control for input delay system
１．Swing-up of the Acrobot
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Summary
Numerical computation method for HJE
Computation of stable manifold
Non-uniqueness of solution for HJE
Experimental verifications
• DC motor over-current avoidance • Swing-up control for inverted pendulum • Swing-up control for flexible inverted pendulum
A. J. van der Schaft (Univ. Groningen), Y. Umemura (Aishin AW, Co, Ltd), students of Control System Lab of NUAE Matlab programme is available upon request: [email protected]
Acknowledgement
Thank you !
Slide Number 1Outline of talkHJE and optimal controlHJE & stable manifoldStable manifoldHJE & stable manifoldHJE & stable manifoldStable manifold algorithmThe idea of this iterationHow to compute optimal control?Computation : a numerical exampleSummary of stable manifold approach Applications� (Experimental verifications)DC motor over-current avoidanceDC motor over-current avoidanceDC motor over-current avoidanceSwing up & stabilization of inverted pnedulumSwing up & stabilization of inverted pnedulumSwing up & stabilization of inverted pnedulumSwing up & stabilization of inverted pnedulumNon-uniqueness and curvature of SMNon-uniqueness and curvature of SMNon-uniqueness and curvature of SMSwing up & stabilization of flexible IPSwing up & stabilization of flexible IPSwing up & stabilization of flexible IPSwing up & stabilization of flexible IPOther applicationsSummary