Infinite-dimensional nonlinear predictivecontroller design for open-channel hydraulic

systems

D. Georges,Control Systems Dept - Gipsa-lab, Grenoble INP

Workshop on ”Irrigation Channels and Related Problems”,Salerno, Italy, october 2-4, 2008

Outline of the talk

1. Control of a single canal pool

2. Modelling of open-channel hydraulic systems: Saint VenantPDE

3. Some backgrounds on predictive control

4. Formulation of the associated optimal control problem

5. Necessary conditions for optimality: the adjoint PDEs

6. Computation of the related two-point boundary value problem

7. Description of the here-proposed predictive control scheme

8. Some simulation results

9. The multi-pool case: a decomposition approach viaLagrangian relaxation

10. Some conclusions and perspectives

Control problemRegulation of a single pool of an irrigation canal (pool = canalsection delimited by 2 gates) using the upstream regulator gate:

B

hDh

S

P

hQ

I

L

Upstream Downstream

Transversal section Longitudinal section

Why to use nonlinear predictive control?

1. In order to take nonlinear transportation and diffusionphenomena into account;

2. Because (finite-dimensional) nonlinear optimal controlprovides, under some specific assumptions, stabilizingfeedbacks (infinite-horizon optimal control, predictive control):possible extension in the infinite-dimensional framework?

Open-channel modelling: Saint Venant PDE

Rectangular canal section case:

S :

{B∂z∂t + ∂Q

∂x = q∂Q∂t + ∂

∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J(Q, z)) = kq Q

Bz

(1)z = relative water level,(m); B=canal width,(m); Q = water flow

rate, (m3

s ); A = wet section, (m2); I = canal slope ; g = acc. ofgravity, ( m

s2 ); J = friction (mm ) ; q(x)= withdrawal / length unit

(m2

s ) ; k=0, if q > 0, 1, if q < 0.

B.C .

{Q(0, t) = u(t) (integral control : upstream)z(L, t) = v(t) (disturbance : downstream)

(2)

Models of the gates may be introduced

Some backgrounds on predictive controlConsider a nonlinear system of the form:

x = F (x , u), x(0) = x0 (3)

with F (0, 0) = 0 (the origin is an equilibrium point).

1. At time t, x(t) = xt is measured or computed from a stateobserver.

2. Computation of an (open-loop) optimal control problem overa receding control interval [t, t + T ] starting with state xt

(so-called ”associated optimal control problem” (AOCP)):

minu(.)

∫ t+T

tL(x(τ), u(τ))dτ (4)

s.t. x = F (x , u), x(t) = xt and x(t + T ) ∈ E (5)

where E denotes a domain of the state space in whichx(t + T ) is assigned.

3. Application of the first control u(t) obtained from thesequence {u(t), u(t + ∆t), u(t + 2∆t), ...., u(t + N∆t)}.

4. Go back to 1), t + ∆t → t.

Some backgrounds on predictive control

Some sufficient conditions to ensure local asymptoticstability around the origin (Mayne & Michalska, TAC 90):

I L(., .) is positive definite in both x and u (in the sense ofHessian matrices w.r.t. x and u)

I under assumptions (mainly, F is 2 times cont. differentiabble,F is uniformly Lipschitz in x w.r.t. u, local uniformcontrollability) ensuring existence and uniqueness of theAOCP

I E = {0}

V (x , t) =

∫ t+T

tL(x(τ), u(τ))dτ is a control Lyapunov function.

Extension to the infinite-dimensional case?

Associated optimal control problem

minu

∫ T

0m(u(t))dt +

∫ T

0

∫ L

0l(z(x , t),Q(x , t))dxdt (6)

s.t.

S :

B ∂z∂t + ∂Q

∂x = q∂Q∂t + ∂

∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J) = kq Q

Bzµ = u

(7)

with

I .C .

Q(x , 0) = φ1(x), x ∈ [0, L]z(x , 0) = φ2(x)µ(0) = φ1(0)

(8)

B.C .

{Q(0, t) = µ(t) (control : upstream − end)z(L, t) = v(t) (disturbance : downstream − end)

(9)

and constraint on state at final time (terminal constraint):

T .C .

{Q(x ,T ) = Q0(x), x ∈ [0, L]z(x ,T ) = z0(x)

(10)

Regulation around an equilibrium state

I Regularization term:

m(u(t)) =r

2u(t)2, r > 0 (11)

I State cost function:

l(z(x , t),Q(x , t)) =1

2[q1(x)(z(x , t)− z0(x))2 (12)

+q2(x)(Q(x , t)− Q0(x))2], q1(.), q2(.) > 0

where z0(x), Q0(x), x ∈ [0, L] = equilibrium state, solution of:

Se :

{∂Q∂x = q0∂∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J) = kq0

QBz

(13)

with Boundary Conditions:

B.C .

{Q0(0) = Qo

z0(L) = zL(14)

Some remarks

I Some other possible control objectives by choosing other mand l functionals:

I Regulation of the downstream-end level z(x = L, t), withupstream-end boundary control and a downstream-endboundary condition Q(x = L, t) = v(t), where v is thedownstream-end water flow acting as a disturbance.

I Regulation of the downstream-end level z(x = L, t), withdownstream-end boundary control and an upstream-endboundary condition Q(x = 0, t) = v(t), where v is theupstream-end water flow acting as a disturbance.

I Control problems with withdrawals may be also consideredwithout restriction.

Lagrangian formulation

L(Q, z , λ1, λ2, λ3, u) =

∫ T

0{m(u) + λ3S3}dt

(15)

+

∫ T

0

∫ L

0{l(z ,Q) + λ1S1 + λ2S2}dxdt

with

S :

S1 = B∂z

∂t + ∂Q∂x − q

S2 = ∂Q∂t + ∂

∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J)− kq Q

Bz

= ∂Q∂t + ∂

∂x (Q2

Bz + 12gBz2)− gBz(I − J)− kq Q

BzS3 = µ− u

(16)

Necessary conditions for optimality: some adjointPDEs (D. Georges & M.L. Chen, ECC 01)

derived from the Lagrangian formulation + variational calculus:In order that both u(t) ∈ [0,T ] and the trajectory of system:

S :

B ∂z∂t + ∂Q

∂x = q∂Q∂t + ∂

∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J) = kq Q

Bzµ = u

(17)

with

I .C .

Q(x , 0) = φ1(x), x ∈ [0, L]z(x , 0) = φ2(x)µ(0) = φ1(0)

(18)

B.C .

{Q(0, t) = µ(t) (control)z(L, t) = v(t) (disturbance)

(19)

and

T .C .

Q(x ,T ) = Q0(x), x ∈ [0, L]z(x ,T ) = z0(x)µ(T ) = Q0(x = 0)

(20)

are optimal,

it is necessary that there exists λ = (λ1(x , t), λ2(x , t), λ3(t)),solution of the adjoint system of S:

Sadj :

∂l∂z − B ∂λ1

∂t + ∂λ2∂x ( Q2

Bz2 − gBz)− λ2(gB(I − J)

−gBz∂J

∂z− kq

Q

Bz2) = 0

∂l

∂Q− ∂λ1

∂x− ∂λ2

∂t− 2

∂λ2

∂x

Q

Bz− λ2(kq

1

Bz− gBz

∂J

∂Q) = 0

λ1(0, t) + 2λ2(0, t)Q(0, t)

Bz(0, t)+ λ3 = 0

with

I .T .C .

λ1(x , 0), λ1(x ,T ), free, x ∈ [0, L]λ2(x , 0), λ2(x ,T ), freeλ3(0), λ3(T ) free

(21)

and

B.C .

{λ2(0, t) = 0, t ∈ [0,T ]

λ1(L, t) + 2λ2(L, t) Q(L,t)Bz(L,t) = 0

(22)

For all t ∈ [0,T ], the optimal control u = u∗, is necessary solutionof:

m′(u(t))− λ3(t) = 0

⇒ A two-point boundary value problem

Computation of the two-point boundary valueproblem

Computation in two stages:

1. Spatial and temporal discretization of the canonical equations:Preissman numerical scheme (1962): applied to both S andSadj ;

2. Solution of the nonlinear algebraic equations derived from thetwo-dimensional grid via a Newton-Raphson method.

Preissman scheme: a finite-difference scheme

semi-implicit in timeApproximation of fonctions f and their derivatives:

f (x , t) =1− θ

2[fi+1 + fi ] +

θ

2[f +

i+1 + f +i ]

∂f

∂x(x , t) =

1− θ∆x

[fi+1 − fi ] +θ

∆x[f +

i+1 − f +i ]

∂f

∂t(x , t) =

1

2∆t[f +

i − fi + f +i+1 − fi+1] (23)

where i = spatial index, + = t + ∆t and 0 ≤ θ ≤ 1 = relaxationcoefficient.If θ ≥ 0, 5, we get an unconditionally stable integration scheme.

An algebraic set of nonlinear equations

For the case when m(u) = 12u2, if N is the number of spatial

discretization points and M, the number of temporal discretizationpoints:

I S + B.C. ⇒ (2N + 1)×M unknown variables(z(x , t),Q(x , t), µ(t)) with (2N + 1)× (M − 1) equations +4N + 2 constraints (the states (z ,Q, µ) are imposed at t = 0and t = T ).

I Sadj + B.C. ⇒ (2N + 1)×M unknown variables(λ1(x , t), λ2(x , t), λ3(t)) with (2N + 1)× (M − 1) equations.

⇒ 4N ×M + 2M variables(z(x , t),Q(x , t), µ(t), λ1(x , t), λ2(x , t), λ3(t)) for 4N ×M + 2Mequations: an implicit problem

Computation of both state and adjoint state via aNewton-Raphson scheme.

The predictive control scheme1. At time t : computation of the open-loop optimal control

problem, with both initial and final states fixed:

minu

∫ t+T

tm(u(t))dt +

∫ t+T

t

∫ L

0l(z(x , t),Q(x , t))dxdt (24)

s.t.

S :

B ∂z∂t + ∂Q

∂x = q∂Q∂t + ∂

∂x (Q2

Bz ) + gBz ∂z∂x − gBz(I − J) = kq Q

Bzµ = u

(25)with

I .C .

Q(x , 0) = Qt(x), x ∈ [0, L]z(x , 0) = zt(x)µ(0) = Qt(0)

(26)

Qt(x) et zt(x) = states at time t on [0, L],

B.C .

{Q(0, τ) = µ(τ), τ ∈ [t, t + T ] (control)z(L, τ) = v(τ) (prediction of downstream − end level)

(27)and

T .C .

{Q(x , τ = t + T ) = Q0(x), x ∈ [0, L]z(x , τ = t + T ) = z0(x)

(28)

The predictive control scheme

2 Apply u(t) defined as the first optimal control input sequencecomputed on [t, t + T ]: the system reaches state(z(x , t + ∆t),Q(x , t + ∆t)), ∀x ∈ [0, L].

3 Go to 1), with t + ∆t → t and get zt(x) = z(x , t + ∆t),Qt(x) = Q(x , t + ∆t).

Some simulation results

I Simulation based on Preissman numerical scheme;

I A 5 km long canal divided into 10 sections of 500 m each;

I Regulation around a uniform equilibrium state correspondingto a constant relative water level z0 of 1.05 m, along the pool;

I Simulation starting from a uniform equilibrium of 1 m;

I At each time step: complexity of the two-point boundaryvalue problem: N = 11 (spatial discretization) and M = 6(temporal discretization, ∆t = 100s): 4N ×M + 2M = 288equations for 4N ×M + 2M = 288 unknown variables:computation time < 60 s on a Pentium 1,8 Mhz, 512 MoLaptop: real-time control is possible.

Simulations results

0 10 20 30 40 50 60 70 80 903.5

3.6

3.7

3.8

3.9

Time in mns

in m

3/sWater flow rate

0 10 20 30 40 50 60 70 80 900.96

0.98

1

1.02

1.04

1.06

1.08

Time in mns

in m

eter

s

Water level

Simulations results

The multi-pool case

A two-pool case: (easy extension to n pools) :

−

= 1Lx+

= 1Lx

0=x

2Lx =

Regulator gate

Pool 1

Pool 2

I Dynamics of the 2 pools: 2 X 2 PDE coupled by a gatemodel: Q1(L−1 , t) = Q2(L+

1 , t) = Q(t) and Q such that

Q2 = K 2α2 × 2g(z1(L−1 , t)− z2(L+1 , t))

⇔ F (Q(t), z1(L−1 , t), z2(L+1 , t), α(t)) = 0

Two-pool modelling

Spool1 :

B ∂z1

∂t + ∂Q1∂x = q1

∂Q1∂t + ∂

∂x (Q2

1Bz1

) + gBz1∂z1∂x − gBz1(I − J) = kq1

Q1Bz1

µ = u1

(29)

B.C .

{Q1(0, t) = µ(t)Q1(L−1 , t) = Q(t)

(30)

Spool2 :

B ∂z2

∂t + ∂Q2∂x = q2

∂Q2∂t + ∂

∂x (Q2

2Bz2

) + gBz2∂z2∂x − gBz2(I − J) = kq2

Q2Bz2

α = u2

(31)

B.C .

{Q2(L+

1 , t) = Q(t)z2(L2, t) = v(t)

(32)

Q(t)2 = K 2α2 × 2g(z1(L−1 , t)− z2(L+1 , t)), (33)

Optimal control formulation

minu

2∑i=1

[

∫ T

0mi (ui )dt +

∫ T

0

∫ bi

ai

li (zi (x , t),Qi (x , t))dxdt] (34)

s.t. Spool1 + B.C. + I.C.+ T.C. and Spool2 + B.C + I.C.+ T.C.

and the additional algebraic constraint defined on [0,T ]:

Q2 = K 2α2 × 2g(z1(L−1 , t)− z2(L+1 , t)).

Main issues

I How to reduce the computational complexity?

I How to take advantage of distributed control architecture(supervisory control and data acquisition: SCADA) used inlarge-scale water distribution systems?

I Here-proposed solution: use of adecomposition-coordination algorithm based onLagrangian relaxation (D. Georges, Workshop NMPC’06)

An augmented lagrangian formulation

Lc(Q1, z1,Q2, z2,Q, λ11, λ

12, λ

13, λ

21, λ

22, λ

23, p, u1, u2)

=

∫ T

0{[m1(u1) + λ1

3[Q1(0, t)− u1(t)]}dt

+

∫ T

0

∫ L−1

0{l1(z1,Q1) + λ1

1S11 + λ1

2S12}dxdt

+

∫ T

0{[m2(u2) + λ2

3[α− u2(t)]}dt

+

∫ T

0

∫ L2

L+1

{l2(z2,Q2) + λ21S2

1 + λ22S2

2}dxdt

(35)

+

∫ T

0[(p +

c

2F (Q(t), z1(L−1 , t), z2(L+

1 , t)))

× F (Q(t), z1(L−1 , t), z2(L+1 , t), α)]dt

(36)

p is the dual variable associated to the gate constraint F

an additional quadratic term c2‖F‖

2 is introduced

⇒ An augmented Lagrangian Lc

Main improvement of augmented Lagrangians:For ”sufficiently” large c > 0, the augmented Lagrangian admits atleast one local saddle-point (no duality gap) even for non convexproblems.

⇒ Convergence of relaxation algorithms is guaranteed

Some backgrounds on price-decomposition

minu=(u1,u2)

J(u1, u2) sous θ(u1, u2) = 0. (37)

Augmented Lagrangian:

Lc(u, p) = J(u1, u2)+ < p, θ(u1, u2) > +c

2‖θ(u1, u2)‖2

Duality approach:Computation of a saddle-point, solution of:

maxp

minu

Lc(u, p)

A price-decomposition algorithm (UZAWA):1) Solve minu Lc(u, pk)⇒ uk+1

2) pk+1 = pk + ρθ(uk+11 , uk+1

2 )

If both J(u1, u2) = J1(u1) + J2(u2) and θ(u1, u2) = θ1(u1) + θ2(u2)(separable case) and if c = 0 (ordinary lagrangian formulation),then price decomposition-coordination:

1. Decomposition: Solve two independant subproblemsminui Ji (ui )+ < pk , θi (ui ) >⇒ uk+1

i , i = 1, 2

2. Coordination: pk+1 = pk + ρθ(uk+11 , uk+1

2 )

If θ is not separable and c 6= 0, separability via linearization ofc‖.‖2:ALGORITHM (Cohen, 84)

1. Decomposition :

minui

Ji (ui )+ < pk + cθ(uk1 , u

k2 ), θ′i (uk

1 , uk2 ).ui > +

1

2ε‖ui −uk

i ‖2,

=⇒ uk+1i ,

where θ′i is the gradient of θ w.r.t. ui .

2. Coordination:

pk+1 = pk + ρθ(uk+11 , uk+1

2 )

with 0 < ρ ≤ c et 0 < ε < 1/2c (for convex problems).

Application to the 2-pool AOCP1. AOCP of pool 1:

minu1

∫ T

0m1(u1)dt +

∫ T

0

∫ L−1

0l1(z1(x , t),Q1(x , t))dxdt +∫ T

0[< pk + cF k ,F ′ku1

.u1 > +1

2ε(u1 − uk

1 )2]dt s.t. Spool1 +

B.C. + I.C.+ T.C., =⇒ uk+11 (.),

2. AOCP of pool 2:

minu2

∫ T

0m2(u2)dt +

∫ T

0

∫ L2

L+1

li (z2(x , t),Q2(x , t))dxdt +

∫ T

0[<

pk + cF k ,F ′ku2.u2 > +

1

2ε(u2 − uk

2 )2]dt, s.t. Spool1 + B.C. +

I.C.+ T.C., =⇒ uk+12 (.),

3. Coordination:

minQ

∫ T

0[< pk + cF k ,F ′kQ .Q > +

1

2ε(Q − Qk)2]dt,

=⇒ Qk+1(.),

pk+1 = pk + ρF (Qk+1, z1(uk+11 ), z2(uk+1

2 ), α(uk+12 ))

Some potential advantages

I Complexity reduction thanks to the decomposition insub-problems and parallel computation:In our case study, 2 problems of complexity 4(N + 1)×Mversus one problem of complexity 8(N + 1)×M ;

I Well suited for distributed control application (NetworkedControl Systems).

A distributed predictive control scheme

At each instant dt, parallel computation of the two-point boundary value sub-problems

POOL 1

Computation of a two-point boundary value problem (similar to a single-pool problem)

The local variables z1 and u1 at iteration k are sent through the network

POOL 2

Computation of a two-point boundary value problem (similar to a single-pool problem)

The local variables z2 and u2 at iteration k are sent through the network

Coordination: The coordination variables are the Lagrangian multiplier p associated to the gate constraint and the flow rate Q

Update of p and Q ; p and Q are sent through the network

k+1 -> k

Broadcast of variables through the communication network until convergence

NETWORK

Conclusions and perspectives

1. Practical extension of finite-dimensional predictive controltechniques to an infinite-dimensional problem;

2. Computation of the related two-point boundary value problemusing a 2D discretization method based on Preissmanintegration scheme;

3. Theoretical analysis still to be performed (see M. Herty’spresentation);

4. Extension to the multi-pool case via a Lagrangian relaxationtechnique (Decomposition-Coordination) is possible, but hasto be validated;

5. For practical implementation: need of a state observer (can bederived by using variational calculus as a ”dual controlproblem”).