real-time optimization for large scale nonlinear processes -diehl

8/6/2019 Real-Time Optimization for Large Scale Nonlinear Processes -DIEHL

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G

G

Gs

H Hact

H

Hs.act

H

Hw.act

H

N

p

Poc(x0) x0 P(x0) x0

P(t)

t

P(t)

P(t)

Pk(xk) k xk Pk

Pk(xk) q

q0, . . . , q N1 qi i s s0, . . . , sN si si = (s

xi , s

zi )

sxi szi


16/201

T T(x0) Poc(x0) t

t

P(t)

= t/T

u

uS uf(x)

uf(x) ufT(x) T

u(t; x0) Poc(x0) w

w = (q, s)

x

xcl x0 xS x(t; x0) Poc(x0) y

y = (w,,)

y

= y0 y

yk

Pk

z

zcl z(t; x0) Poc(x0)


17/201

t0

[t0, t0 + T]

t0 + [t0 + , t0 + T + ]

+


18/201


19/201

+


20/201

+

+

k


21/201

k

k

k

k

k

k


22/201

x0

x0

+

+

+


23/201


24/201

+

+

+


25/201

B(x(t), z(t), u(t), p , t) x(t) = f(x(t), z(t), u(t), p , t)0 = g(x(t), z(t), u(t), p , t).

x Rnx z Rnz

u Rnu p Rnp

gz

() B()

[t0, tf]

t0 tftft0

L(x(t), z(t), u(t), p , t) dt + E(x(tf), z(tf), p , tf),

L

E


26/201

l(x(t), z(t), u(t), p , t)

Rnl

lr(p,t) Rnl t [t0, tf]

Q

tft0

Q 12 (l(x(t), z(t), u(t), p , t) lr(t, p))22 dt

l(x(t), z(t), u(t), p , t)

Q = I lr(p,t) = 0, t [t0, tf]

e(x(tf), z(tf), p , tf) Rne tf

t0

l(x(t), z(t), u(t), p , t)22 dt + e(x(tf), z(tf), p , tf)22.

h(x(t), z(t), u(t), p , t) 0, t [t0, tf].

re(x(tf), z(tf), p , tf) = 0

ri

(x(tf), z(tf), p , tf) 0

x(t0) = x0.

l22 :=

nl

i=1 l2i


27/201

p

t0

p = p

t0 = t0

t0 p

p

t

x :=

xp

t

x0 :=

x0p

t0

,

B() x = f()

B() :=B() 0 00 Inp 0

0 0 1

,

f() :=f()0

1

,

t(t)

t

p

t0 = 0 T

tf

re(x(T)) := t(T) tf = 0.

T

x0

tf = t(T) = t(0) + T = t0 + T T = tf t0


28/201

Poc(x0) : minu(), x(),

z(), (T)

T0

L(x(t), z(t), u(t)) dt + E(x(T), z(T))

B(x(t), z(t), u(t)) x(t) f(x(t), z(t), u(t) ) = 0, t [0, T],

g(x(t), z(t), u(t) ) = 0, t [0, T],

x(0)

x0 = 0,

re(x(T), z(T) ) = 0,

ri(x(T), z(T)) 0, h(x(t), z(t), u(t)) 0, t [0, T].

T

x0

x(t; x0) z(t; x0) u

(t; x0) t [0, T(x0)]

x0


29/201

0

cA0 V

cA cB cC cD

Ak1 B k2 C

2Ak3 D.

ki

ki() = ki0 exp

Ei/oC + 273.15

, i = 1, 2, 3.

K

QK C D


30/201

cA =V

VR(cA0 cA) k1()cA k3()c2A

cB = VVR

cB +k1()cA k2()cB

=V

VR(0 ) + kwAR

CpVR(K )

1

Cpk1()cAH1 + k2()cBH2 + k3()c2AH3

K =1

mKCPK

QK + kwAR( K)

.

CPK Cp

H1 H2 H3

ki0 Ei i = 1, 2, 3

VR AR kW mK x

u

x =

cA

cB

K

u =

VVR

QK

x = f(x, u).

=cB |ScA0

0

xS =

2.1402 moll

1.0903 moll114.19 C112.91 C

uS =

14.19 h1

1113.5 kJh

.

xS, uS

f(xS, uS) = 0


31/201

k10 1012h1 kglk20

1012h1 Cp

kJ

kgKk30 109h1 kw kJhm2KE1 AR m

2

E2 VR lE3 mK kgH1

kJmol

CPK kJ

kgK

H2 kJmol

cA0 mol

l

H3 kJmol

0 C

x0 Poc(x0)

xS

L(x, u) := (x xS)TQ(x xS) + (u uS)TR(u uS)

Q :=

0.2 mol2 l2 0 0 00 1.0 mol2 l2 0 00 0 0.5 C2 00 0 0 0.2 C2

R :=

0.5h2 0

0 5.0 107 kJ2 h2

.

uLB u(t) uUB

uLB :=

3.0 h1

9000 kJh

uUB :=

35.0 h1

0 kJh

,

h(x(t), u(t)) :=

u(t) uLBuUB u(t)

0.

Poc(x0)


32/201

0 200 400 600 800 1000 1200 1400 1600 1800 2000

1

2

3

x1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5

1

1.5

x2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x4

0 200 400 600 800 1000 1200 1400 1600 1800 200010

15

20

25

u1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1000

0

u2

time [s]

x(t; x0) u(t; x0)


33/201

minu(),x()

T0

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

x(t) = f(x(t), u(t)) , t [0, T],x(t0) = x0,

h(x(t), u(t)) 0, t [0, T].

T = 2000

T =

x(t; x0) u(t; x0)

x0 :=

1.0 moll

0.5 moll

100 C100 C

x0 Rnx

Poc(x0) u(t; x0)

x(t; x0) z(t; x0)

t [0, T(x0)] T

T

tf t

x

x0 x(; x0)

z(; x0) u(; x0) Poc(x0) t1 [0, T(x0)]

x1 := x

(t1; x0)

x

(; x0)

Poc(x1) x(; x1) z(; x1) u(; x1) [0, T(x1)] Poc(x0)

T(x1) = T(x0) t1

Poc(x1)

Poc(x0) t1

x(t; x1) = x(t1 + t; x0)

z(t; x1) = z(t1 + t; x0)

u(t; x1) = u(t1 + t; x0)

t [0, T(x1)].


34/201

t = 0

t1 [0, T(x0)]

u(; x0)

u(t1; x0) = u(0; x(t1; x0)), t1 [0, T(x0)].

uf

uf(x0) := u(0; x0).

B() xcl(t) = f(xcl(t), zcl(t), uf(xcl(t)))0 = g(xcl(t), zcl(t), u

f(xcl(t))).

nx

uf(x0)

uf(xcl(t))

x(; x0) z(; x0) u(; x0) Poc(x0)

x(t) x(t; x0) x(t)

uf(xcl(t)) = u(0; xcl(t)), t [0, T(x0)],

ulnfc(xcl(t)) := u(t; x0) + K(t)(xcl(t) x(t; x0)),

K(t) :=uf

x(x(t; x0))

t [0, T(x0)] u(t; x0) uf(x(t; x0)) = u

(0; x(t; x0))

K


35/201

x(; x0) z(; x0) u(; x0)

K

uf(x)

ulnfc(x)

x := xS + (x0 xS), xS x0

uf(x) := u(0; x)

T =

x {0.20, 0.19, . . . , 1.19, 1.20} ulnfc(x) := uS+K(0)(xxS)

uf() xS

T

T

T =

x(; x0) z(; x0)

u(; x0)

uf(x0) := u(0; x0).


36/201

0 0.2 0.4 0.6 0.8 1

15

20

25

u1

0 0.2 0.4 0.6 0.8 14000

2000

0

2000

4000

u2

x := xS+ (x0 xS)

xS = 0) x0 = 1 t = 0

0

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

L(x, z, u) > 0

(x,z,u) = (xS, zS, uS) L(xS, zS, uS) = 0

V(x0) :=

0

L(x(t; x0), z(t; x0), u

(t; x0)) dt,

V() C1

u(; x0) C0 x0 V() Rnx

V(xS) = 0 V(x0) > 0, x0 = xS

d

dtV(xcl(t)) < 0, xcl(t) = xS,

xcl(t) t [0, ) xS


37/201

xcl() x0 x(; x0)

d

dtV(x

(t; x0)) < 0, x(t; x0) = xS.

V(x(t; x0)) = V(x0)

t0

L(x(; x0), z(; x0), u

(; x0)) d,

t

d

dt

V(x(t; x0)) =

L(x(t; x0), z

(t; x0), u

(t; x0)) < 0,

x(t; x0)

= xS.

T

T

xT(; x0)

zT(; x0) uT(; x0)

x(; x0) z(; x0) u(; x0) [0, T]

ufT(x0) := uT(0; x0)

uf(x)

ufT(x) +

ufT(x0)

E(x(T), z(T))

VT(x0) :=

T0

L(xT(t; x0), zT(t; x0), u

T(t; x0)) dt + E(x

T(T; x0))


38/201

T xS

VT(x) VT(x), 0, T, x T.

VT(x0) =

0

L(xT(t; x0), zT(t; x0), u

T(t; x0)) dt + VT(x

T(; x0))

x0 = xT(0; x0)

VT(xT(; x0)) VT(xT(0; x0)) VT(xT(; x0)) VT(xT(0; x0))=

0L(xT(t; x0), z

T(t; x0), u

T(t; x0)) dt.

ddt

VT(xT(0; x0)) =

VTx

(xT(0; x0)) xT(0; x0)

L(xT(0; x0), zT(0; x0), uT(0; x0)) < 0, x0 = xS.

x0 := xcl(t) t

t

xcl(t) = x0 = xT(0; x0)

ufT(xcl(t)) := uT(0; x0)

zcl(t) = zT(0; x0)

xcl(t) = xT(0; x0).

d

dtVT(xcl(t)) =

VTx

(xcl(t)) xcl(t) < 0, xcl(t) = xS.

re(x(T)) := x(T) xS,

xS E(x(T))

xT(t; x0) zT(t; x0) u

T(t; x0) Poc,T(x0)

[0, T]

Poc,T(x0)

[0, T]

t [T, T] xT(t; x0) := xS zT(t; x0) := zS

uT(t; x0) := uS


39/201

Poc,T(x0)

Poc,T(x0)

x0 xT(t)

uT(t)

xS, uS T

T

Poc,T(x0)

Poc,T(x0)

P Rnxnx

E(x(T)) := x(T) xS2P := P12 (x(T) xS)22

ri(x(T)) := x(T) xS2P 0,

> 0

x(T)

:= {x Rnx|xxS2P }

P

u(x) = uS + K (x xS) P

h(x, z, uS + K (x xS)) 0 {(x, z) Rnx Rnz |x , g(x, z, uS + K (x xS)) = 0}

E(x(T)) :=

T

L(x(t; x(T)), z(t; x(T)), u(t; x(T))) dt


40/201

x(t; x(T))

z(t; x(T))

[T, )

B() x(t) = f(x(t), z(t), K(x(t))),0 = g(x(t), z(t), K(x(t))),

x(T) = x(T).

K : Rnx Rnu

ri(x(T)) 0 x(T) x(t; x(T))

z(t; x(T))

u(t; x(T)) := K(x(t; x(T)))

[0, T]

[0, T]

u(t)

t [T, T]

u(t) = K(x(t))

ri(x(T)) 0

E(x(T))

E(x(T))

K() := uS


41/201

urc(x)


42/201


43/201

Poc(x0)

P(x0)

T

t : [0, 1] [0, T], t(, T) := T

[0, 1]

x

z

u

[0, 1]

[0, 1]

T

T() = 0

T


44/201

q0q1

q2

q3

q4

0 1 2 3 4 1

u()

s0

s1

s2s3

s4

s5

0 1 2 3 4 1

x()

N = 5

u()

[0, 1]

0 = 0 < 1 < . . . < N= 1,

u()

u() := qi [i, i+1), i = 0, 1, . . . N 1,

N

qi Rnu

u(1) := qN := qN1,

qN qN1 u(1)

z(1)

g(x(1), z(1), u(1)) = 0

[i, i+1 ]

2(N+ 1)

sx0 , sx1 , . . . , s

xN s

z0, s

z1, . . . , s

zN

nx nz


45/201

si := (sxi , s

zi )

N

[i, i+1]

B() xi() = T f(xi(), zi(), qi) 0 = g(xi(), zi(), qi) exp

i

i+1 i

g(sxi , s

zi , qi)

xi(i) = sxi .

> 0

sxi , szi , qi

(sxi , szi , qi)

N

xi() zi()

[i, i+1] si qi

xi(; si, qi) zi(; si, qi)

xi() zi() L

Li(si, qi)

Li(si, qi) := i+1

i

T L(xi(), zi(), qi)) d.

sxi szi

g(sxi , szi , qi) = 0 i = 0, 1, . . . , N.

sxi+1

xi

sxi+1

= xi(

i+1; s

i, q

i), i = 0, . . . , N

1.

sx0 x0

s0 = x0.

si i = 0, . . . , N

si


46/201

0 . . . N N + 1

h(sxi , szi , qi) 0, i = 0, 1, . . . , N.

P(x0) : minq0, . . . , q N1,

s0, . . . , sN

N1i=0

Li(sxi , s

zi , qi) + E(s

xN, s

zN)

sxi+1 xi(i+1; sxi , szi , qi) = 0, i = 0, . . . , N 1, g(sxi , s

zi , qi) = 0, i = 0, . . . , N,

sx

0 x0 = 0, re(sxN, s

zN) = 0,

ri(sxN, szN) 0,

h(sxi , szi , qi) 0, i = 0, . . . , N.

q :=

q0

qN1

Rnq , s :=

s0

sN

Rns, w := qs

Rnw

nq := Nnu ns := (N + 1)(nx + nz) nw = nq + ns F(w) :=

N1i=0 Li(si, qi) + E(sN) G : R

nw


47/201

q0

q1q2

qN1

s0 s1s2

s3

sN

RnG

H : Rnw RnH

minw Rnw

F(w)

G(w) = 0,H(w) 0.

w Rnw

P(x0) x0

x0 P(x0)

P(x0)

P(x0) : minsx0

nx ,w (nwnx)F(sx0 , w)

sx0

x0 = 0,

G(sx0, w) = 0,H(sx0, w) 0.

T

T() = 0, [0, 1]


48/201

T

x0 T

sx0 x0 = 0 (Inx| 0)sx0 x0 = 0 x0 Rnx

sxi Rnx+1

q = (q0, q1, . . . , q N1) s = (s0, s1, . . . , sN)

s

q

sx0nx+1 q

minq Rnq , s Rns

F(q, s)

G(q, s) = 0,

G

Gs

s

Gs

Inxgsx0

gsz0

x0(1)

sx0 x0(1)

sz0Inx

xN1(N)sxN1

xN1(N)szN1

Inx

gsxN

gszN

Inx gszi

gszi

q

q

s

G

s

N =

nw = nq + ns = Nnu + (N + 1)nx = 100 2 + 101 4 = 604


49/201

0 10 20 30 40 50 60 70 80 90 1001

2

3

sx1

0 10 20 30 40 50 60 70 80 90 100

0.5

1

1.5

sx2

0 10 20 30 40 50 60 70 80 90 100

100

110

120

sx3

0 10 20 30 40 50 60 70 80 90 100

100

110

120

sx4

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

q1

0 10 20 30 40 50 60 70 80 90 100

2000

1000

0

q2

node index i

si qi

Nnx = 400

nx = 4 ns nw ns = 604 404 = 200 = nq

P(x0) x0


50/201


51/201

P(t) : minw Rnw

F(t, w)

G(t, w) = 0H(t, w) 0

C2

F : R Rnw R G : R Rnw RnG H : R Rnw R

nH

t

t

x0

P(x0)

w

q0, . . . , q N1, s0, . . . , sN

w(t)

P(t)

t

w(t)

t

w(t)

w(t)

P(t)

t

P(t + )

P(t)

w(t+)

O(2)

t


52/201

t

minw Rnw

F(w)

G(w) = 0H(w) 0

F : Rnw R G : Rnw RnG H : Rnw RnH

F F wG G

wG(w) :=Gw

(w)T

.

w Rnw G(w) = 0 H(w) 0

w

F(w) F(w)

w

w

F(w) < F(w)

w = w

w

Hj(w) H(w)

Hj(w) = 0

w

G(w) :=

G(w)Hact(w)

.

w

w

G(w)T G(w)T

RnG RnH

G

H

L

L(w,,) := F(w) TG(w) TH(w).

w


53/201

w Rnw

RnG

RnH

(w, , )

wL(w, , ) = 0 G(w) = 0,

H(w) 0, 0,

j Hj(w) = 0, j = 1, 2, . . . , nH.

(w, , )

j = 0 Hj(w

) > 0 Hj(w) = 0

Hact(w)

(w, , )

Hact(w) =:

Hs.act

Hw.act

(w).

minw Rnw

1

2wTAw + aTw

b + Bw = 0c + Cw 0

a Rnw b RnG c RnH A Rnw Rnw B RnG Rnw

C RnH Rnw


54/201

(w, , )

Aw + a

BT

CT = 0

b + Bw = 0,

c + Cw 0, 0,

j (cj + Cj,w) = 0, j = 1, 2, . . . , nH.

A BT

B 0

w

=

ab

.

B

nG A B

A Rn Rn B Rm Rn m n A B

A BT

B 0

(BT, CT)

nG+nH

w Rnw

w

RnG RnH (w, , )


55/201

2wL(w, , ) Ns

Gs(w) :=

GHs.act

(w),

w Ns Ns := {w Rnw |wGs(w)Tw = 0},

wT 2wL(w, , ) w > 0.

2wL(w, , ) Nw := {w Ns|wHw.act(w)Tw 0}.

(w, , )

F

G

H

P(t)

minw Rnw

F(t, w)

G(t, w) = 0H(t, w) 0

F : R Rnw R G : R Rnw RnG H : R Rnw RnH

C2

(w(t), (t), (t))

t

loc := {(t, w) R Rnw|w P(t))}.

(t, w(t))

loc

2w

L := 2Lw2


56/201

(w(t), (t), (t))

loc

P(t)

minw R

1

2w2

t + sinh(w) 0

w(t), (t)

w(t) = max(0, arcsinh(t)),

(t) =w(t)

cosh(w(t)).

w

t(t) =

0,

t < 0,cosh(arcsinh(t))1,

t > 0,

t(t) = cosh(w(t))1(1 tanh(w(t))) w

t(t).

w(t)

w(0)

P(t)

P(0)

(w(0), (0), (0))

Hs.act

Hw.act

(w, , s.act , w.act )

(w(0), (0), (0)) t = 0

minw nw

1

2wT 2wL w +

twL

Tw

Gt

+ wGTw = 0Hs.act

t+ (wHs.act)T w = 0.

Hw.act

t+ (wHw.act)T w 0.


57/201

5 0 5

3

2

1

0

1

2

3

w*(t)

w(t)

w.act

> 0

v : [0, ) Rnw RnG RnH

t w(t)(t)

(t)

P(t)

t [0, )

t = 0

limt0,t>0

1

tw(t)

w(0)

(t) (0)(t) (0)

=

w

:= w

s.actw.act

0

.

(w, , s.act ,

w.act )

2wL

wGsT w = 0

(wG, wHs.act, wHw.act) w(0)


58/201

> 0

P(t)

t (0, )

(w(t), (t), (t))

t [0, )

(w , ,

s.act

, w.act)

minw nw

1

2wT 2wL w +

twL

Tw

Gt + wGTw = 0Hs.actt

+ (wHs.act)T w = 0.

Hw.actt

+ (wHw.act)T w 0,

t (, 0] (w(t), (t), (t)) > 0

limt0,t


59/201

5 0 5

3

2

1

0

1

2

3

w*(t)

t

w* t

w* t

w(t)

P(t)

t = 0

w(0) = (0) = 0

minw R

1

2w2

1 + w 0,

w = 1 w.act = 1,

w(t), (t)

t 0, t > 0

minw R

1

2w2

1 + w 0,

w = 0 w.act

= 0.

w(t), (t)

t 0, t < 0


60/201

(w, , )

minw Rnw

F(w)

G(w) = 0H(w) 0

y0 = (w0, 0, 0)

yk+1 = yk + kyk

k (0, 1]

yk =

wkkk

:= wkk k

k k

(wk, k, k)

minw k

1

2wT Ak w + wF(wk)Tw

G(wk) + wG(wk)Tw = 0

H(wk) + wH(wk)Tw 0

k

Ak k Rnw yk

y = (w, , )

k := 1 k := Rnw

Ak := 2wL(wk, k, k).

(wk, k)

2wL(wk, k)wk + wF(wk) wG(wk)k = 0,G(wk) + wG(wk)Twk = 0.


61/201

k = k + k

w

L(wk, k) +

2

wL(wk, k) wk

wG(wk) k = 0,

G(wk) +wG(wk)T wk = 0

wL(wk, k)G(wk)

+

(w, )

wL(wk, k)G(wk)

wkk

= 0,

wL(w, )G(w)

= wF(w) wG(w)G(w)

= 0.

(w, )

w

0

w0

(w, )

w0 w

0 2wL(w0, 0) wG(w0)

wG(w0)T 0

(wk, k) 2wL(wk, k) wG(wk)wG(wk)T 0

wk+1 wk

k+1

=

wF(wk)G(wk)

(w, )

wk+1 wk+1

C

wk wk

2

C 0


62/201

(w, , )

y = (w, , )

F(w)

wG(w)

H(w) = 0

G(w) = 0

H(w) 0 0

j H(w)j = 0, j = 1, 2, . . . , nH.

y0

y0 = y

A0

(w0, 0, 0)

A0 w0 + F(w

) wG(w

)0 H(w

)0 = 0G(w) + wG(w)Tw0 = 0

H(w) + wH(w)Tw0 00 0

0,j (H(w) + wH(w)Tw0)j = 0, j = 1, 2, . . . , nH.

(w0, 0, 0) = (0, , )

A0 wG(w)T

w

w

A0

(w, , )


63/201

P(t) : mint R, w Rnw

F(t, w)

t t = 0G(t, w) = 0H(t, w) 0

F : R Rnw R G : R Rnw RnG H : R Rnw RnH

C2

P(t)

t

t t = 0 t

t

w :=

tw

, :=

t

,

G(w) :=

t t

G(t, w)

,

L

P(t)

L( w, , ) := F( w) TG( w) TH(w)= F(t, w) t(t t) TG(t, w) TH(t, w)= L(t,w,,) t(t t),

L(t,w,,) = F(t, w) TG(t, w) TH(t, w)

P(t)

P()

(w(0), (0), (0))

P(0)

(w(), (), ()) P(t)

( w

(0),

(0),

(0))

P(0)

P()

> 0

(w, , )

(w(), (), ()) P(t)

1

w

=

w

= lim

t0,t>0

1

t

w(t) w(0)(t) (0)

(t) (0)


64/201

2w L = 2wL t = 0 t

( w(0), (0), (0))

P(0)

(w(0), (0), (0))

P(0)

t (0) =t

L(0, w(0), (0), (0))

mint,w

1

2wT2wLw + t

twLTw + wFTw + F

tt +

1

2

2Lt2

t2

t = 0G + G

tt + wGTw = 0

H+ Ht

t + wHTw 0,

t = 0

w(0)

(0)

(0)

t =

G(0, w(0)) = 0

Hact(0, w(0)) = 0

wF(0, w(0)) = wG(0, w(0))(0) + wH(0, w(0))(0)

Ft

+ 122Lt2

2

minw

1

2wT2wLw +

twLTw + (wG(0) + wH (0))T w

Gt

+ wGTw = 0Hact

t + wHactTw 0

Hinact + Hinact

t + wHinactTw 0,


65/201

(w,,)

= (0) +

= (0) +

2wLw +

twL wG wH = 0,

G

t + wGTw = 0,

Hs.act

t + wHs.actTw 0,

Hw.act

t + wHw.actTw 0,

Hinact + H

inact

t + wHinactTw 0, ,s.act(0) + s.act 0,

w.act 0, inact 0,

((0) + )s.actj

Hs.act

t + wHs.actTw

j

= 0,

w.actj

Hw.act

t + wHw.actTw

j

= 0,

inactj

Hinact + Hinact

t + wHinactTw

j

= 0.

w

Hinact + Hinact

t + wHinactTw > 0

inactj = 0 ((0) + )s.actj > 0

Hs.act

t + wHs.actTw = 0.

w :=w

, :=

,

:=

,


66/201

2wLw + t wL wG wH = 0,G

t+ wGTw = 0,

Hs.act

t+ wHs.actTw = 0,

Hw.act

t+ wHw.actTw 0,

w.act 0,

w.actj

Hw.actt + wH

w.actT

wj

= 0,

inact = 0,

w

w(0)

y(t)

y(t) + y()

y(t + )

y(t + ) (y(t) + y()) = O(2)

P(t2)

P(t1)

w(t)

t1

t2

P(t)


67/201

5 0 5

3

2

1

0

1

2

3

w* +

w

t*

+

t*=0.5

w*=0.48

w

t = 0.5, w = 0.48 = arcsin(0.5)

mint

R, w

R

1

2w2

t t = 0

t + sinh(w)

0

t = t

w = max(0, arcsin(t)

= w/ cosh(w)

P(t)

P(t + )

mint R, w R

1

2wT (1 tanh(w)w w + w w

t = 0

t + sinh(w) t + cosh(w) w 0,

t = ,

w = max( w1tanh(w)w

, tsinh(w)+

cosh(w)),

= w+(1tanh(w)w)w

cosh(w)

t, w

t = 0, w = 0


68/201


69/201

t

x0(t)

t

x0(t) u(0; x0(t)) t

P(x0(t))

. . . , ti, ti+1, . . .


70/201

i = ti+1 ti

P(x0(ti))

x0(ti) ti ti+1

i

x0(ti+1) ti

P(x0(ti+1)) [ti, ti+1] ti+1

P(x0(ti+1))

[ti, ti+1]

+

i

t0 = 0

x0

N = 100

= 100


71/201

0 200 400 600 800 1000 1200 1400 1600 1800 20001

2

3

x1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5

1

1.5

x2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x4

0 200 400 600 800 1000 1200 1400 1600 1800 200010

15

20

25

u1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1000

0

u2

time [s]


72/201

i

x0

i

i i

P(x0) x0 s

x0 s

x0 x0 = 0

min

sx

0

nx

,w

(nwnx)F(sx0 , w)

sx0 x0 = 0,G(sx0 , w) = 0,

H(sx0 , w) 0,

w = (sx0 , w) sx0

t

x0 t

y(x0) = (w(x0),

(x0), (x0))

P(x0) P(x0 + )

y(x0 + )


73/201

0 0.2 0.4 0.6 0.8 1

15

20

25

q1 0

0 0.2 0.4 0.6 0.8 14000

2000

0

2000

4000

q2 0

initializationfirst iterate

0 0.2 0.4 0.6 0.8 1

15

20

25

q1 0

0 0.2 0.4 0.6 0.8 14000

2000

0

2000

4000

q2 0

first iterate=initialization

second iterate

q10 q20

= 1

x0

P(x)

x := xS + (x0 xS),

xS x0

P(xS) = 0 nw

q0

q0

= 1


74/201

x(t)

z(t)

u(t)

t [ti, tf] x0(ti) [ti+1, tf] =[ti + i, tf] x0(ti+1) = x

(ti+1)

T(i+1

i) i = ti+1

ti

N

P(x0(t0)) [t0, t0 +T] w = (q0, . . . , q N1, s0, . . . , sN) tk = t0 +

ki=1 i = t0 + T k

Nk

xk := x0(tk) Pk(xk)

wk k wk = (qk, . . . , q N1, sk, . . . , sN)

Pk(xk) k = 0, . . . , N 1

Pk(xk) : minqk, . . . , q N1,

sk, . . . , sN

N1i=k

Li(sxi , s

zi , qi) + E(s

xN, s

zN)

sxi+1 xi(i+1; sxi , szi , qi) = 0, i = k, . . . , N 1, g(sxi , s

zi , qi) = 0, i = k, . . . , N ,

sxk xk = 0, re(sxN, s

zN) = 0,

ri(sxN, szN) 0,

h(sxi , szi , qi) 0, i = k, . . . , N .

P0(x0) P(x0)

yk = (wk,

k,

k) Pk(xk )

xk+1 xk+1 = (sxk+1)

k

yk+1

Pk+1(xk+1) xk+1 +


75/201

Sk

qk sk yk = (wk, k, k) Pk() Pk+1() Sk

yk+1 = Skyk.

y00 = (w00,

00,

00) P0()

x0 k = 0, . . . , N 1

tk xk

ykk Pk(xk) y

k+1k

(qk)k+1k

yk+1k

yk+1k Pk+1()

yk+1k+1 := Skyk+1k .

Pk+1() xk+1 y

k+1k+1

k

N = 100

k = 20 k = 0, . . . , N

1

y00 P(xS)

u1

u1


76/201

0 200 400 600 800 1000 1200 1400 1600 1800 20001

2

3

x1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5

1

1.5

x2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x4

0 200 400 600 800 1000 1200 1400 1600 1800 200010

15

20

25

u1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1000

0

u2

time [s]

y00

P0(x

0)

x1, . . . , xN

k < N

y00 Pk() y0k Pk(xk)

y00 y0k

Pk()


77/201

uk [tk, tk+1] uk = uk Kk(xk xk ) uk Kk

xkxk

y00

y00 = (w00,

00,

00) y

0k =

SkSk1 . . . S 1y00 Pk(xk)

Pk() xk

k = 0, . . . , N 1

tk xk

Pk(xk) y1k

(qk)1k

y1k

k

y1k

yk Pk(xk ) y1k yk = O(xk xk2)

xk xk


78/201

0 200 400 600 800 1000 1200 1400 1600 1800 20001

2

3

x1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.5

1

1.5

x2

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x3

0 200 400 600 800 1000 1200 1400 1600 1800 2000

100

110

120

x4

0 200 400 600 800 1000 1200 1400 1600 1800 200010

15

20

25

u1

0 200 400 600 800 1000 1200 1400 1600 1800 2000

2000

1000

0

u2

time [s]

sxi


79/201

Pk(xk)

(Inx|0)sxk xk = 0.

Pk(xk)

Pk+1(xk+1) xk+1 (Inx|0)sxk+1

P0(x0), P1(x1), . . . , P N1(xN1)

P() = P0() P(x0), P(x1), . . . , P (xN1)

xk

P()

w

q0, . . . , q N1 s0, . . . , sN


80/201

P(xk+1) yk+1k = (wk+1k , k+1k , k+1k )

P(xk)

yk+1k+1

wk+1k

wk+1k = (q0, q1, . . . , q N2, qN1; s0, s1, . . . , sN2, sN1, sN)

wk+1

k := (q1, q2, . . . , q N1, qnew

N1; s1, s2, . . . , sN1, snew

N1, snew

N ),

qnewN1 snewN1 s

newN

qnewN1 := qN1, s

newN1 := sN1, s

newN := sN,

N1

sN

qN1

snew

N (sN, qN1)

qnewN1 := qN1, snewN1 := sN, s

newN := s

newN (s

newN1, q

newN1).

N1

N0 = g(sN, qN) = g(sN, qN1) qN1 qN

snewN


81/201

sN

qN1 N1

xk

yk+1k+1 P(xk+1) yk+1k P(xk)

xk+1 = sx1

yk+1k

yk+1k+1 := yk+1k

yk+1k P(xk) P(xk+1)

xk+1 = xk

yk+2k+1

yk+2k+1

yk+1

=

O(xk+1 xk2)

P(xk+1)

P(xk + (xk+1 xk)) = 0

= 1

xk, xk+1

k, k + 1, . . .


82/201

x0

x0

tk x0(tk) P(x0(tk))

tk+1 tk+1 tk

u(x0(t))


83/201

tk

x0(tk)

x0(tk)

u(x0(t))

x0(t)

tk

2000


84/201

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

time [s]

CPUtim

e[s]

overall CPU time per iteration

feedback response time

N = 100


85/201

N


86/201

w

q Rnq s Rns

P(x0) : minq,s

F(q, s)

G(q, s) = 0

F : D Rnq Rns R G : D Rnq Rns Rns

G

Gs

(q, s) D

q

s

L : D Rns RL(q, s , ) := F(q, s) TG(q, s)

(q,s,)L(q,s,) = qF(q, s) qG(q, s)sF(q, s) sG(q, s)

G(q, s)

= 0.

y Rn n := nq + ns + ns

R : D D Rns Rn

y :=

qs

R(y) :=

qF(q, s) qG(q, s)sF(q, s) sG(q, s)

G(q, s)

,


87/201

R(y) = 0

y0

y1, y2, . . .

yk+1 = yk + yk,

yk

R(yk) +R

y(yk)yk = 0,

qLsLG(q, s)

+ 2L

q2

2L

qs

T G

q

T

2Lqs

2Ls2

Gs

T

Gq

Gs

0

qk

sk

k = 0.

2L(q,s)2

A(y) = Aqq ATqs

Aqs Ass

,

R

J(y) :=

Aqq A

Tqs

Gq

T

Aqs AssGs

T

Gq

Gs

0

.

R(yk

) + J(yk

)yk

= 0.

F

F(q, s) =1

2l(q, s)22


88/201

l : D Rnl nl 1

A

A(q, s) :=

l(q, s)

T

l(q, s)

.

2L(q,s)2

A

2L(q,s)2

2L(q, s)2

l

(q, s)

Tl

(q, s)

=

nli=1

li2li

(q, s)2+

nsi=1

i2Gi

(q, s)2.

y = (q, s, )

sF sG = ls

Tl(q, s) sG = 0,

= sG1 ls

T

l(q, s) = O (l(q, s)) ,

2L(q, s)2

A(q, s))

= O (l(q, s)) .

l(q, s)

A(q, s)

l(q, s)

Rnl

H

, H H

F(q, s) =1

2l(q, s), l(q, s)H,

A(q, s)ij := Re

l

(q, s)i,

l

(q, s)j

H

,

i, j

(q, s)

A(q, s)


89/201

(yk) k = 0, 1, . . . Rn

yk+1 = yk + yk = yk J(yk)1R(yk),

R(y) = 0.

R : D Rn Rn

J : D Rn R(nn)

D J(y1)1 J(y2) Ry (y2)

< 1, y1, y2 D J(y1)1 (J(y2) J(y3)) y2 y3, y1, y2, y3 D.

y0 := J(y0)1R(y0)

y0

0 := +

2y0 < 1

D0 :=

y Rn| y y0 y

01 0

D.

(yk)

D0 y D0 R(y) = 0

yk

(yk)

yk+1

=

J(yk+1)1

R(yk+1)

= J(yk+1)1 (R(yk+1) R(yk) J(yk) yk)= J(yk+1)1 1

0(Ry

(yk + tyk) J(yk)) yk dt= J(yk+1)1 1

0(Ry

(yk + tyk) J(yk + tyk))yk dt+J(yk+1)1 1

0(J(yk + tyk) J(yk))yk dt

10

J(yk+1)1 (Ry

(yk + tyk) J(yk + tyk))yk dt+1

0J(yk+1)1 (J(yk + tyk) J(yk))yk dt

yk + 10

tyk2 dt=

+

2yk

yk =: kyk.


90/201

k 1 yk+1 yk k+1 k 1

yk+1 0yk, k 0

yk+m yk 11 0y

k k0

1 0 y0, k, m 0.

ym y0 11 0y

0, m 0.

(yk)

D0 y R J1

0 = limk

yk = limk

J(yk)1R(yk) = J(y)1R(y)

R(y) = 0

J(y)

J(y)

J(y) =

Aqq A

Tqs

Gq

T

Aqs AssGs

T

Gq

Gs

0

Gs

Ar(y) := I GqT

Gs T

Aqq A

Tqs

Aqs Ass I

G

s1 G

q

J(y)

J(y)1 = C1(y)Ar(y)1C1(y)

T + C2(y)

C1(y) :=

I

Gs

1 Gq

Gs

TAqs Ass

Gs

1 Gq


91/201

C2(y) := 0 0 00 0 Gs 1

0Gs

T Gs

TAss

Gs

1 .

B

B = (Gq

|Gs

)

Ar

J(y)(C1(y)Ar(y)1C1(y)

T + C2(y)) = I

J(y)C1(y) =

Ar(y)0

0

J(y)C2(y) = I C1(y)T0

0

.

R : D Rn

Ar(y)

D

Ar(y)1 A < , y D.

C1 C2 D C1(y) C1 < , y D,

C2(y) C2 < , y D.

:= C1AC1 + C2

J(y1) J(y2) y1 y2, < , y1, y2 D,

J(y)

A(y) 2L

(q, s)2(y)

, < 1, y D.


92/201

0 1 2 3 4 5 6 76

5

4

3

2

1

0

log10

||w

k||

index k

Newton

GaussNewton

y0 := J(y0)1R(y0)

y0

0 := +

2y0 < 1

D0 :=

y Rn| y y0 y

01 0

D.

(yk)

y = (q, s, ) D

(q, s)

w

0.1


93/201

q

N

T

k

k N

k=1 k = T

T = N

tk := T k tk tk1 = k

t0 = 0, tk =k

i=1

i, k = 1, . . . , N.

x0, x1, . . . , xN t0, t1, . . . , tN Pk(xk)

t0 x0

u0

t1

t1

P1(x1) x1

x(t1)

B()x(t) = f(x(t), z(t), u0), t [t0, t1],0 = g(x(t), z(t), u0), t [t0, t1],

x0(t0) = x0,

P1(x1)

sx1 = x1.


94/201

P0(x0)

q0 u0 = q

0

0+ q0

0= q1

0

sx1 x0(t1; sx0, sz0, q0) = 0,g(sx0, s

z0, q0) = 0,

sx0 x0 = 0,q0 u0 = 0,

sx0 sz0 q0 s

x1 s

x1 = x1

B() x0(t) = f(x0(t), z0(t), q0), t [t0, t1],0 = g(x0(t), z0(t), q0) e

tt0t1t0 g(sx0, s

z0, q0), t [t0, t1],

x0(t0) = sx0 ,

sx0 = x0 q0 = u0 g(sx0, s

z0, q0) = 0

x0(t1; sx0 , s

z0, q0) = x(t1) = x1 s0, q0

sx1 = x1

sx0 , q0

u0

q0

sz0 g(sx0, s

z0, q0) = 0 x0(t1; sx0, sz0, q0)

x(t1)

q0, . . . , q N1

P(x0) P0(x0)

P0 := P0(x0) = P(x0)

Pk

Pk(xk)

Pk

k = 0, . . . , N

Pk : minq,s

F(q, s)

G(q, s) = 0,qi ui = 0, i = 0, . . . , k 1.


95/201

q1

0

q11q12

q1N1

x1

s1

0s11

s12s13

s1N

t0 t1 t2 t3 tN

P1

q0

x1

q2

0

q2

1

s2

1

x2

s22 s23

s2

0

t0 t1 t2

P2

q0 q1 x2


96/201

q = (q0, q1, . . . , q N1)

Pk : minq,s

F(q, s)

G(q, s) = 0,

Qk1T

(q uk) = 0.

Qk1 Rnqmk mk = knu

Qk1 :=

Imk

0

,

uk Rnq

uk :=

u0

uk1

, Qk1T(quk) =

q0 u0

qk1 uk1

,

uk

PN

q

Pk

Pk

k

minq,s

F(q, s)

G(q, s) = 0

QT1 (q u) = 0

Q1 Rnqm m nq m

Q1

Q2 :=

0

I(nqm)

Rnq(nqm).

L L(q,s,,) := L(q,s,) QTQT1 (q u),

L

(q,s,,Q)L(q,s,,Q) =

qL Q1QsL

G(q, s)QT1 (q u)

= 0.


97/201

(Q1|Q2)T Rnqnq

Q + QT1 qLQT2 qL

sL

G(q, s)QT1 (q u)

= 0

Q := QT1 qL.

Q

y Rn n = nq+ ns + ns

R(y) :=

QT1 (q u)QT2 qLsL

G(q, s)

=

QT1 (q u)000

+

0QT2

I

I

R(y)

R

y=

QT1000

+

0QT2

I

I

Ry

J(y)

J(y) :=

QT10

00

+ 0

QT2I

I

J(y)

=

QT1 0 0

QT2 Aqq QT2 A

Tqs Q

T2Gq

T

Aqs AssGs

T

Gq

Gs

0

.


98/201

J : D Rn

J(y1)1

J(y2) Ry

(y2) < 1, y1, y2 D, J(y1)1 J(y2) J(y3) y2 y3, y1, y2, y3 D,

yk+1

+

2 yk

yk

J(y)1

J(y)

J(y)1 = Q

I

C1A1r CT1 + C2

I

QT2 AqqQ1 IAqsQ1 IG

qQ1 I

,

Q :=

Q1 Q2 I

I

,

Ar(y) := QT2 Ar(y)Q2,

C1(y) :=

QT2 II

C1(y)Q2,

C2(y) :=

QT2 I

I

C2(y)

Q2 I

I

,

Ar(y) C1(y) C2(y)


99/201

J(y)J(y)1 = I

J(y) Q =

I

QT2 AqqQ1 QT2 AqqQ2 Q

T2 A

Tqs Q

T2Gq

T

AqsQ1 AqsQ2 AssGs

T

Gq

Q1Gq

Q2Gs

0

.

QT2 Q2 = I

QT2 AqqQ2 Q

T2 A

Tqs Q

T2Gq

T

AqsQ2 AssGs

T

G

qQ2

G

s0

1

= C1(y)A1r (y)C

T1 (y) + C2.

J(y) Q

I

C1A1r C

T1 + C2

=

I

QT2 AqqQ1 IAqsQ1 IGq

Q1 I

,

C1(y) C2(y)

C1(y) C2(y)

C1(y) C1(y) C1, y D,

C2(y) C2(y) C2, y D.

Ar(y)1

Ar(y) Ar(y) = QT2 Ar(y)Q2

Ar(y) Ar(y) Ar(y)

Ar(y) =

QT1QT2

Ar(y) (Q1|Q2) =

Ar(y)

1(y), . . . , (nqm)(y) Ar(y) Ar(y)

k(y) [1(y), nq (y)] 1(y) 1(y)


100/201

Ar(y)1 = 11(y)

11(y)

= A1r (y) A, y D.

J(y)1 C1(y)Ar(y)1C1(y)T + C2(y) C1AC1 + C2 = , y D.

J(y2)

R

y (y2) =

0QT2

I

I

J(y2) R

y (y2)

J(y2) J(y3) =

0QT2

I

I

(J(y2) J(y3))

J1(y1)

0QT2

I

I

=

Q2 I

I

C1(y1)Ar(y1)1C1(y1)T + C2(y1)

QT2 I

I

,

J1(y1)

J(y2) R

y(y2)

=

J1(y1)

0QT2

I

I

J(y2) Ry

(y2)

=

J(y2) Ry

(y2)

< 1, y1, y2 D,


101/201

J1(y1) J(y2) J(y3)

= J(y2) J(y3) y2 y3, y1, y2, y3 D.

y

yk

Pk

k = 0, 1, . . .

PN+k := PN

k > 0

N

k

Pk

Pk : minq,s

F(q, s)

G(q, s) = 0

Qk1T

(q uk) = 0

Qk1 Rnqmk Qk2 Rnq(nqmk)

Qk1 := Imk0 , Qk2 := 0

I(nqmk)

mk 0 = m0 mk nq

Qk+12 = Qk2k k :=

0

I(nqmk+1)

R(nqmk)(nqmk+1).

uk

yk = (qk, sk, k)

uk := qk, k = 0, . . .

P0

m0 = 0

P0

P1, P2, . . .

y0

Pk

Rk

Rk(y) :=

Qk1T

(q uk)Qk2

TqLsL

G(q, s)

= 0.

N


102/201

R0(y) = 0

R(y) = 0

Jk(y)

Jk(y) =

Qk1T

000

+

0

Qk2T

I

I

J(y),

J(y)

R(y)

yk = yk+1 yk

Pk

yk

yk

:= Jk

(yk

)1

Rk

(yk

).

(yk)

y D0 =

y Rn| y y0 y0

1 0

D, 0 = +

2y0.

yk

k k = N

PN

y1, y2, . . . , yN

D0

y0

y = (q, s, )

yk+1

+

2 yk

yk

(q, s)

< 1

Jk

yk

yk+1

Rk

Rk+1


103/201

yk

yk+1

Pk+1

Rk+1

Qk+11T

(q uk+1) = Qk+11 T(q qk+1) = 0

yk+1 = (qk+1, sk+1, k+1)

qk+1

yk

yk := Jk+1(yk)1 Rk+1(yk)

yk := Jk(yk)1 Rk(yk)

yk

Rk(yk)+Jk(yk)yk = 0

Rk+1(yk) + Jk+1(yk)yk = 0

Rk+1(yk) + Jk+1(yk) yk

=

Qk+11T(qkqk+1)

Qk+12T

qLsL

G(q, s)

+ Qk+11

T

Qk+12

T

Aqq Qk+12

T

ATqs Qk+12

TGq

T

Aqs AssGs

T

Gq

Gs

0

qk+1

qk

sk+1skkk+1

=

0

Qk+12T

I

I

qLsL

G(q, s)

+

Aqq A

Tqs

Gq

T

Aqs AssGs

T

Gq

Gs

0

yk

=

0

Tk Qk2T

I

I

qLsL

G(q, s)

+

Aqq ATqs

Gq

T

Aqs AssGs

T

Gq

Gs 0

y

k

=

0Tk

I

I

Qk2TqLsL

G(q, s)

+

Q

k2TAqq Q

k2TATqs Q

k2TGq

T

Aqs AssGs

T

Gq

Gs

0

yk

=

0Tk

I

I

00

0

= 0.


104/201

0 10 20 30 40 50 60 70 80 90 1001

2

3

sx1

0 10 20 30 40 50 60 70 80 90 100

0.5

1

1.5

sx2

0 10 20 30 40 50 60 70 80 90 100

100

110

120

sx3

0 10 20 30 40 50 60 70 80 90 100

100

110

120

sx4

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

q1

0 10 20 30 40 50 60 70 80 90 100

2000

1000

0

q2

node index k

w

y D0

Pk0

k0 = N Rk = Rk0, k k0

0 = limk

Jk(yk)1Rk(yk) = limk

Jk0(yk)1Rk0(yk) = Jk0(y)1Rk0(y),

Rk0(y) = 0

G(q, s) = 0

w = (q, s)


105/201

0 1 2 3 4 5 6 76

5

4

3

2

1

0

log10

||w

k||

index k

Newton

GaussNewton

P0

y0

y

y

yk Pk

k

1

y y0

yk Pk

y yk

y = (q,s,)

y0

y0

y0

y

y0 P0

y y0 20y0

1 0 , 0 = +

2y0.

(yk)

y0

y1


106/201

< 1

0

yk+m yk 11 0y

k k0

1 0 y0, k, m 0,

y y1 = limm

y1+m y1 01 0 y

0.

y0 P0

y y0 y y1 + y0 y1 20

1 0 y0.

y

yk k Pk

y yk 2kyk

1 k 2k+10 y0

1 0 .

y

Pk

k < N

k N yk = y

yk Pk

k < N

y

y0

y0

P0 = P(x0)

P0 : minq,s

F(q, s)

sx0 x0 = 0,G(q, s) = 0,

Rx0(y) :=

qLsL

sx0 x0G(q, s)

.


107/201

Ry

J

x0 y0

y

0

y0 P0(x0) x0 = x0+

y0

P0 = P0(x0)

y0 := y0

y0 ,

Rx0(y0) = Rx0(y

0) = 0 y

0

y0 = J(y0)1Rx0(y0)

= J(y0)1

Rx0(y0) +

000

= J(y0)1

000

.

J(y0

)1

y0 P0(x0)

y

y0

P0(x0)

= x0 x0

y y0 2 +

2

1

( +

2

),

y y0

1 2

22.

y

F(q, s)


108/201

R

y D0 Ry (y)

BR, y D0. y = (q, s, )

y0 = (q0, s

0,

0)

F(q, s) F(q0, s0) 1

2BRy y02.

(q0

, s0

)

(q, s)

G(q0, s0) = G(q

, s) = 0

L(q,s,) = F(q, s) TG(q, s)

L(y)L(y0) =1

0Ly

(y0 + t1(y y0)) (y y0) dt1

real-time optimization for large scale nonlinear processes -diehl

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