1

2015 12 2

ohtsuka@i.kyoto-u.ac.jp

2

C

3

�

�

�

�

C

4

5

T x(T )

x(t) = f(x(t), u(t))

x(0) = x0 (given)

J = ϕ(x(T )) +∫ T

0L(x(t), u(t))dt

C(x(t), u(t)) = 0

x(t) u(t)

C ≤ 0 C + v2 = 0 v:

J u

6

x = f(x, u), x(0) = x0,

λ = −

(∂H

∂x

)T

(x, u, λ, μ), λ(T ) =

(∂ϕ

∂x

)T

(x(T )),

∂H

∂u(x, u, λ, μ) = 0,

C(x, u) = 0.

H(x, u, λ, μ) = L(x, u) + λTf(x, u) + μTC(x, u) ,

λ μ

[Animation]

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C

8

Model Predictive Control (MPC)

9

Model Predictive Control (MPC)

10

x(t) = f(x(t), u(t))

J = ϕ(x(t+ T )) +∫ t+T

tL(x(τ), u(τ))dτ

C(x(t), u(t)) = 0

C ≤ 0 C + v2 = 0 v:

x(t)

uopt(τ ; t, x(t)) (t ≤ τ ≤ t+ T )

u(t) = uopt(t; t, x(t)) · · ·

11

t

t t+T

u

Receding Horizon

recede · · ·

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•

•

•

•

•

•

13

�

�

�

�

C

14

t T

U(t) = [u∗T0 (t), μ∗T0 (t), u∗T1 (t), μ∗T

1 (t), . . . , u∗TN−1(t), μ∗TN−1(t)]

T

u(t) = u∗0(t) C = 0 μ∗i

� Δτ �= Δt

15

J(U(t), (t), t) = ϕ(x∗N(t)) +∑N−1

i=0 L(x∗i (t), u∗i (t))Δτ

x∗0(t) = x(t)

x∗i+1(t) = x∗i (t) + f(x∗i (t), u∗i (t))Δτ

λ∗N (t) =(∂ϕ∂x

)T(x∗N(t))

λ∗i (t) = λ∗i+1(t) +(∂H∂x

)T(x∗i (t), u

∗i (t), λ

∗i+1(t), μ

∗i (t))Δτ

∂H∂u

(x∗i (t), u∗i (t), λ

∗i+1(t), μ

∗i (t)) = 0

C(x∗i (t), u∗i (t)) = 0

H(x, λ, u, μ) = L+ λTf + μTC

x∗i (t), λ∗i (t) U(t) x∗0(t) = x(t)

16

F (U(t), x(t), t) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂H∂u

(x∗0(t), u∗0(t), λ

∗1(t), μ

∗0(t))

C(x∗0(t), u∗0(t))

...

∂H∂u

(x∗N−1(t), u∗N−1(t), λ

∗N (t), μ

∗N−1(t))

C(x∗N−1(t), u∗N−1(t))

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦= 0

U(t) = [u∗T0 (t), μ∗T0 (t), u∗T1 (t), μ∗T

1 (t), . . . , u∗TN−1(t), μ∗TN−1(t)]

T

F (U, x, t) 0 U

17

F (U(t), x(t), t) = 0

18

t U(t) F (U(t), x(t), t) = 0

⇔d

dtF (U(t), x(t), t) = 0, F (U(0), x(0), 0) = 0

⇔d

dtF (U(t), x(t), t) = −ζF (U(t), x(t), t), F (U(0), x(0), 0) = 0

ζ > 0 F �= 0 F = 0

⇔U =

(∂F∂U

)−1 (−ζF − ∂F

∂xx− ∂F

∂t

),

F (U(0), x(0), 0) = 0

U(t)

U(t)

F(U,x(t),t)=0

t

U

U(t)

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Continuation/GMRES

x(t)

U 1 ∂F∂UU = −ζF − ∂F

∂xx− ∂F

∂t

U(t) U(t) U(t+Δt) = U(t) + U(t)Δt

1 1

1 GMRES

� T 0 U(0)

( ’02; O. ’04)

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1

GMRES

1

21

�

�

�

�

C

22

2

q2

q1

s1

s2

l1

l2

m1

m2

u1

u2

J1

J2

x = [q1 q2 q1 q2]T

u = [u1 u2]T

J =1

2(x(t+ T )− xf )

TSf (x(t+ T )− xf )

+1

2

∫ t+T

t((x− xf )

TQ(x− xf ) + uTRu)dτ

( ’02; O. ’04)

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μ

u1 2 |u1| ≤ u1max

[Animation 1: |u1| ≤ 5 ] [Animation 2: |u1| ≤ 2.5 ]

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0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

100

Number of Grids on the Horizon, N

Ave

rage

Com

puta

tiona

l Tim

e pe

r Upd

ate

[ms]

C/GMRES

L−BFGS

Truncated Newton

N [ms]

L-BFGS, Truncated Newton · · ·

CPU: PentiumII 300 MHz, Tf = 0.2 [s], GMRES k = 2

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356 [mm] 212 [mm], 142 [mm], 0.894 [kg]

(Seguchi, O. ’03)

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x = [x y θ x y θ]T ∈ R6

u = [u1 u2]T ∈ {umin, 0, umax}

2

J =1

2(x(t+ T )− xf )

TSf (x(t+ T )− xf)

+1

2

∫ t+T

t((x− xf )

TQ(x− xf ) + uTRu)dτ

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0 10 20 30

−0.4

−0.2

0

0.2

0.4

Time [s]

x [m

]

0 10 20 30

−0.4

−0.2

0

0.2

0.4

Time [s]

y [m

]

0 10 20 30−1

−0.5

0

0.5

1

Time [s]

θ [ra

d]

0 10 20 30−0.2

0

0.2

0.4

Time [s]

u 1 [N]

0 10 20 30−0.2

0

0.2

0.4

Time [s]

u 2 [N]

CPU: Athlon 900 MHz, Tf = 1 [s],

1/120 [s] CCD

[Movie 1: θ(0) = −π ] [Movie 2: θ(0) = π/2 ] [Movie 3: disturbance ]

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x

y

u

v

Ψ

N

X

Y

(x,y)

WPi

WPi+1

xi

yi

xi+1

yi+1

Guide Point

L

Ship

Target Point

x = [x, u, y, v, ψ, r,X, Y,N ]T

u = [Xr, Yr, Nr]T

J = 12(x(t+ T )− xref )

TSf (x(t+ T )− xref )

+ 12

∫ t+Tt ((x− xref )

TQ(x− xref ) + uTRu)dτ

( ’02, ’08)

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-4 -2 0 2 4 6 8 10 12

-4

-2

0

2

4

6

8

10

x [m]

y [m]

-4 -2 0 2 4 6 8 10 12

-4

-2

0

2

4

6

8

10

x [m]

y [m]

3.5 [m], 0.59 [m], 200 [kg] 0.1 [sec], 30 [sec]

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(90× 30m 3m) ( ’05, ’08) [More]

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(Independent System Operator; ISO)

⇒

( ’14)

32

n

• d s

• v(d) c(s)

• HEMS (Home Energy Management System)

•

ISO

• λ

•

•

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ISO

00

J = xT(t+ T )Q1x(t+ T )︸ ︷︷ ︸+∫ t+T

t

(xT(τ)Q2x(τ)︸ ︷︷ ︸+

n∑i=1

Φi(λi))dτ

Φi = −r1(λisi − ci(si)︸ ︷︷ ︸ + vi(di)− λidi︸ ︷︷ ︸

)+ r2

(si − di︸ ︷︷ ︸

)2

vi(di) = 2√di, ci(si) = s2i /2

34

10

θi

(i = 1, · · · , 10)

λi

(i = 1, · · · , 10)

0.33 ms ⇒

35

(Okazaki, O. ’06)

MuPAL-α

( ’05, ’06, ’09)

( ’10)

LHC (Noga et al. ’10, ’11)

(Hashimoto et al. ’10, ’12)

Moving Horizon Receding Horizon

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�

�

�

�

C

37

Maple C

Mathematica

38

39

C

∂H/∂u

40

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http://www.symlab.sys.i.kyoto-u.ac.jp/˜ohtsuka/index j.htm

42

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[2] J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, 2ndEd., SIAM (2010)

[3] , , (2011)

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[13] T. Ohtsuka, “Quasi-Newton-Type Continuation Method for Nonlinear Receding HorizonControl,” J. of Guidance, Control, and Dynamics, Vol. 25, No. 4, pp. 685–692 (2002)

[14] T. Ohtsuka, “A Continuation/GMRES Method for Fast Computation of NonlinearReceding Horizon Control”, Automatica, Vol. 40, No. 4, pp. 563–574 (2004)

[15] Y. Shimizu, T. Ohtsuka, and M. Diehl, “A Real-Time Algorithm for Nonlinear RecedingHorizon Control Using Multiple Shooting and Continuation/Krylov Method,” Int. J. ofRobust and Nonlinear Control, Vol. 19, No. 8, pp. 919–936 (2009)

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[22] M. A. S. Kamal, M. Mukai, J. Murata, and T. Kawabe, “Model Predictive Control ofVehicles on Urban Roads for Improved Fuel Economy,” IEEE Trans. on Control Systems

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Technology, Vol. 21, No. 3, pp. 831–841 (2013)

[23] J. Lee, and M. Yamakita, “Nonlinear Model Predictive Control for ConstrainedMechanical Systems with State Jump,” Proc. of the 2006 IEEE Int. Conf. on ControlApplications, pp. 585–590 (2006)

[24] T. Murao, H. Kawai, and M. Fujita, “Visual Motion Observer-based Stabilizing RecedingHorizon Control via Image Space Navigation Function,” Proc. of the 19th IEEE Int. Conf.on Control Applications, pp. 1648–1653 (2010)

[25] M. Okazaki, and T. Ohtsuka, “Switching Control for Guaranteeing the Safety of a TetheredSatellite,” J. of Guidance, Control, and Dynamics, Vol. 29, No. 4, pp. 822-830 (2006)

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[29] M. Takekawa, J. Aoki, M. Nakaya, T. Ohtani, and T. Ohtsuka, “An Application ofNonlinear Model Predictive Control Using C/GMRES Method to a pH NeutralizationProcess,” Proc. of SICE Annual Conf. 2010, pp. 1494–1496 (2010)

[30] R. Noga, T. Ohtsuka, C. de Prada, E. Blanco, and J. Casas, “Simulation Study onApplication of Nonlinear Model Predictive Control to the Superfluid Helium CryogenicCircuit,” Reprints of the 18th IFAC World Congress, pp. 3647–3652 (2011)

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[31] T. Hashimoto, Y. Yoshioka, and T. Ohtsuka, “Receding Horizon Control for Hot Strip MillCooling Systems,” IEEE/ASME Trans. on Mechatronics, Vol. 18, No. 3, pp. 998–1005(2013)

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[41] N. Fujii, and T. Ohtsuka, “Nonlinear Adaptive Model Predictive Control via Immersionand Invariance Stabilizability,” , Vol. 25, No. 10, pp. 281–288(2012)

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