nonlinear feedback control laws from nonlinear model ... · nonlinear feedback control ws from nmpc...

Nonlinear feedback control laws from NMPCM. Hinze

Nonlinear feedback control laws from nonlinear modelpredictive control

Minneapolis, March 18, 2016

Michael Hinze

Outline

Introduction to model predictive control

The receding horizon control concept

Construction of nonlinear state feedback controllers

Discretization w.r.t. timeOptimal control problem with transition constraintsInexact solution of OCP

Applications

Flow control (Navier-Stokes)Advective ow control (Boussinesq approximation)Multi-phase ow control (Cahn-Hilliard Navier-Stokes)

People involved

Daniel Wachsmuth (RICAM): MPC for Navier-Stokes equations,

Ulrich Matthes (Uni Hamburg): Control of Boussinesq equations,

Christian Kahle (Uni Hamburg): Control of multiphase ows,

Given some initial state x0, nd a control law Bu(t) = K(x(t)) which steers thestate x(t) towards a given trajectory x:

x(t)!−→ x(t) t →∞

Mathematical model:

x(t) + Ax(t) = b(x, t) + Bu(t) state,y(t) = Cx(t) observation,x(0) = x0

x desired stationary state, or

x a reference trajectory obtained from open loop optimal control.

Related research (only PDE control)

Ito & Kunisch 2002-2006: MPC framework for abstract innite dimensional dynamical systems (ESAIM: ControlOptim. Calc. Var., 8:741760, 2002, SIAM J. Cont. Optim., 45(1):207225, 2006).

Altmüller, Grüne & Worthmann 2012-2013: MPC for semilinear parabolic PDEs (GAMM Mitteilungen35(2):131145, 2012).

Benner & Hein 2006-2009: Model predictive control for nonlinear parabolic dierential equations based on a linearquadratic Gaussian design (PAMM 2009, PhD thesis Hein 2009).

Bänsch & Benner 2007-2013: Riccati-based feedback control of ows (SPP 1253).

Krstic & Smyshlyaev 2008-2010: Boundary control of parbolic PDEs using backstepping techniques (SIAM,Philadelphia, 2008; Princeton University Press, Princeton, NJ, 2010).

Choi, Temam, Moin & Kim 1993: Feedback control for unsteady ow and its application to the stochastic Burgersequation (J. Fluid Mech. 253:509543, 1993).

Fursikov 2001-2004: Stabilizability of Navier-Stokes equations with boundary feedback control in 2 and 3d (J.Math. Fluid Mech., 3 (2001), pp. 259301, Discrete Contin. Dyn. Syst., 10 (2004), pp. 289314).

Raymond, Raymond & Dharmatti 2006-2011: Feedback boundary stabilization of the Navier-Stokes equations(SIAM J. Cont. Optim., 45(3):790828, 2006, SIAM J. Cont. Optim., 49(6):23182348, 2011).

Hinze & Volkwein, Hinze 2002-2005: Analysis of instantaneous control (Nonlinear Anal. 50:126, 2002, SIAM J.Control Optim., 44(2):564583, 2005).

System theoretic point of view (H. SICON 44, 2005)

In practice feedback control (closed loop control) based on observations isneeded. If a mathematical model is available, Model Predictive Control (MPC)may be applied.

1 At time tk compute an optimal time discrete control strategyuk+1, . . . , uk+l .

2 Apply uk+1 and proceed to tk+1.

3 Set k = k + 1.

4 Goto 1

Idea: apply suboptimal variant called instantaneous control; solve the optimalcontrol problem only approximately by e.g. applying a few steepest descent steps(for l = 1 proposed by H. Choi, 1995).

MPC schematic (from Grüne & Pannek)

With t = tn , tp = tn+N and tc = tn+1 perform

1 Prediction step: solve optimization problem on [t, t + tp],

2 Control step: apply control on [t, t + tc ],

3 Receding horizon step: t = t + tc , goto 1.

MPC with tc = tp ≡ h, discretization

Discretize the state equation w.r.t. time (your favorite scheme!)

(I + hA)xk+1 = xk + hbk

and minimize at every time step an instantaneous version of the cost

min J(uk+1) = γ2|uk+1|2 + 1

2|C(xk+1 − xk)|2

s.t.(I + hA)xk+1 = xk + hbk + Buk+1.

Time discretization here with implicit Euler.

Feedback oracle

1 Set x0 = φ, k = 0 and t0 = 0.

2 Given an initial control uk0, set

uk+1 = RECIPE(uk0, xk , xk , tk)

3 Solve

(I + hA)xk+1 = xk + hb(xk , tk) + Buk+1.

4 Set tk+1 = tk + h, k = k + 1. If tk < T goto 2.

Instantaneous control

For instantaneous control the oracle RECIPE is given by

u = RECIPE(v , xk , z, tk)

Solve (I + hA)x = xk + hb(xk , tk) + Bv ,solve (I + hA)∗λ = −C∗(Cx − z),

set d = αv + B∗λ.determine ρ > 0,

set RECIPE = v − ρd (= Pad (v − ρd) in case of constraints).

This oracle realizes steepest descent for problem (Pk).

Feedback operators: Instantaneous control

u = RECIPE(0, xk , xk , tk), E := (I + hA)−1.

Instantaneous control rewritten

(I + hA)xk+1 =

xk + hbk−ρBB∗E∗C∗CE(xk − xk)− hρBB∗E∗C∗CE(b(xk)− Axk)︸︷︷︸Buk+1=:Kd

I(xk )

This is the semi-discrete version of

x + Ax = b−ρ

hBB∗E∗C∗CE(x − x)− ρBB∗E∗C∗CE(b(x)− Ax)︸︷︷︸

h=:KI (x)

x(0) = x0.

Feedback operators: Instantaneous control

u = RECIPE(0, xk , xk , tk), E := (I + hA)−1.

Instantaneous control rewritten

(I + hA)xk+1 =

xk + hbk−ρBB∗E∗C∗CE(xk − xk)− hρBB∗E∗C∗CE(b(xk)− Axk)︸︷︷︸Buk+1=:Kd

I(xk )

x + Ax = b−ρ

hBB∗E∗C∗CE(x − x)− ρBB∗E∗C∗CE(b(x)− Ax)︸︷︷︸

h=:KI (x)

x(0) = x0.

Model predictive control

For model predictive control the oracle RECIPE is given by

u = RECIPE(xk , z, tk)

Solve the optimality system for u

(I + hA)x = xk + hb(xk , tk) + Bu(I + hA)∗λ = −C∗(Cx − z)γu + B∗λ = 0 (γu + B∗λ ≥ 0) in case of constraints.

set RECIPE = u

This oracle realizes solution of problem (Pk).

Feedback operators: MPC

With C ≡ Id and B = id let S := γ(E∗E + γI )−1E∗E .Model predictive control rewritten

(I + hA)xk+1 = xk + hb(xk)−1

γS(xk − xk + hb(xk)− hAxk)︸︷︷︸

uk+1=:KdO

x + Ax = b(x)−1

γhS(x − x + hb(x)− hAx)︸︷︷︸

h=:KO (x)

, x(0) = x0.

Feedback operators: MPC

With C ≡ Id and B = id let S := γ(E∗E + γI )−1E∗E .Model predictive control rewritten

(I + hA)xk+1 = xk + hb(xk)−1

γS(xk − xk + hb(xk)− hAxk)︸︷︷︸

uk+1=:KdO

x + Ax = b(x)−1

γhS(x − x + hb(x)− hAx)︸︷︷︸

h=:KO (x)

, x(0) = x0.

MPC schematic revisited

With t = tn , tp = tn+N and tc = tn+1 perform

1 Prediction step: solve optimization problem on [t, t + tp],

2 Control step: apply control on [t, t + tc ],

3 Receding horizon step: t = t + tc , goto 1.

MPC with tc = h, tp = Nh

Discretize the state equation w.r.t. time (your favorite scheme!)

Set x0 := xk , xi ≈ x(tk + ih)(i = 1, . . . ,N).

(∗) T

x1...xp

x0...0

b(x0)...

b(xp−1)

u1...up

and minimize at every time step an instantaneous version of the cost; withX := (x1, . . . , xp)t and U := (u1, . . . , up)t solve

min J(U) = γ

2‖U‖2 + 1

2‖CX − z‖2

s.t.(∗) ≡ transition constraints

Controller construction now along the lines of the previous slides.

Comments on controller construction

Optimal control problem in MPC approach need not admit a (unique)solution. Thus MPC controller need not be well dened.

Transition constraints (often) guarantee a well dened control to statemapping. An instantaneous controller in this case is well dened.

Time marching scheme for the state and discretization of the state inprediction step may dier.

Analysis (stability, decay, length of prediction horizon) of MPC schemes forPDEs is emerging (Altmüller, Grüne & Worthmann). Results forinstantaneous control available in special situations (C = Id ,B = Id).

Promising approach: combine controller construction introduced here withtechniques developed by Altmüller, Grüne, and Worthmann (GAMMMitteilungen 35(2):131145, 2012)

Application: control of Navier Stokes systems

The classical instationary NS system: steer y to y , where

yt − ν∆y + (y∇y) +∇p = Bu in ΩT ,−div y = 0 in ΩT ,

+IC + BC .

The Boussinesq approximation: steer y to y and/or τ to τ

yt + (y · ∇)y − ν∆y +∇p + β τ ~g = Byu in ΩT ,−div y = 0 in ΩT ,

τt + (y · ∇)τ − χ∆τ − f = Bτu in ΩT ,+IC + BC .

The Cahn-Hilliard Navier-Stokes system: steer c to c

yt − 1

Re∆y + y · ∇y +∇p + Kc∇w = Bu in ΩT ,

−div y = 0 in ΩT ,

ct − 1

Pe∆w +∇c · y = 0 in ΩT ,

−γ2∆c + Φ′(c)− w = 0 in ΩT ,+IC + BC .

+IC + BC .

yt − 1

ct − 1

Pe∆w +∇c · y = 0 in ΩT ,

−γ2∆c + Φ′(c)− w = 0 in ΩT ,+IC + BC .

+IC + BC .

yt − 1

ct − 1

Pe∆w +∇c · y = 0 in ΩT ,

−γ2∆c + Φ′(c)− w = 0 in ΩT ,+IC + BC .

Decay, distributed control of classical NSE, H. SICON 2005

A := −ν∆, b(y) := P[(y∇)y ], solenoidal setting. Let ρ denote stepsize instepest descent, h time stepping.

ρ < 2 positive, h small(ρ), b and y smooth and bounded in appropriate norms.

Theorem 1: The iterates y j of instantaneous control strategy satisfy

|y j − y(tj )| ≤ c κj

with some positive κ < 1.

Theorem 2: For the unique solution y of the continuous control law

y + Ay = b(y)−

hB∗B(y − y)− ρB∗B(b(y)− b(y)) + ˙y + Ay − b(y)

y(0) = y0,

|y − y | decays exponentially to zero with order −ρh, i.e.

|y(t)− y(t)| ≤ C exp(−ρ

ht) |y(0)− y(0)|

|y j − y(tj )| ≤ c κj

y + Ay = b(y)−

y(0) = y0,

|y(t)− y(t)| ≤ C exp(−ρ

ht) |y(0)− y(0)|

|y j − y(tj )| ≤ c κj

y + Ay = b(y)−

y(0) = y0,

|y(t)− y(t)| ≤ C exp(−ρ

ht) |y(0)− y(0)|

|y j − y(tj )| ≤ c κj

y + Ay = b(y)−

y(0) = y0,

|y(t)− y(t)| ≤ C exp(−ρ

ht) |y(0)− y(0)|

Exponential decay, laminar cavity ow, Stokes tracking

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

10−12

10−10

10−8

10−6

10−4

10−2

Evolution of |y−z|2

ρ=0.01ρ=0.1 ρ=1 ρ=5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

10−10

10−8

10−6

10−4

10−2

Evolution of |y−z|H

ρ=0.01ρ=0.1 ρ=1 ρ=5

BFS ow, domain

Cost functionals:

J(y , u) = 1

[∫Ωs|y − yst |2 dx + γ

∫Γc

u2 dΓc]dt,

J(y , u) =T∫0

[∫Γs

∂y1∂x2

(∂y1∂x2− |∂y1

∂x2|) dΓs + γ

∫Γc

g2 dΓc]dt.

BFS feedback control, movies

Boundary control with backow observation in volume after the step.

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

Inst. control 1-step MPC 3-step MPC

Boundary control with backow observation at the bottom boundary.

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

Inst. control [2,9] Inst. control [4,9] Inst. control [6,9]

MPC for Boussinesq approximation in the cavity

[−P∆ −Gr~g0 −∆

], b(x, t) = b(y , τ ) :=

[P[(v∇)v ]

(v∇)τ

min J(y , τ, u, uF , uQ) =i+M∑j=i+1

(y j − z j )2dx +c1

(τ j − S j )2dx

u j2dx +

dx +c4

dx) s.t. transition constraints

dtτ j+1 − a∆τ j+1 = cQ u

dtτ j − (y j∇)τ j

a∂ητ j+1 = α(u j+1 − τ j+1 |Γ)1

dty j+1 − ν∆y j+1 +∇pj+1 = cFu

dty j − (y j∇)y j − 1

dtτ j+1~g

−div y j+1 = 0y j+1 |Γ = 0

for j = i , . . . , i + M − 1.

MPC Results

Control target

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Boundary heating

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Distributed heating

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

distributed force

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

As expected: distributed force is more eective than distributed heating, whichis more eective than boundary heating.

MPC results - control horizon

0 50 100 150 200 250 300 350 400

10−1

M=1M=2M=4M=8M=16M=32M=64M=128IC

Cost decreases with increasing the length of the control horizon M (dis-tributed heating)

100 101 102

M=1=ICM=2M=4M=8M=16M=32M=64no control

MPC with boundary heating only works for suciently large control hori-zons.

Tracking perturbed optimal trajectories

0 1 2 3 4 5 6 7 8 9 1010

10−1

0.10.20.4optimal

Instantaneous control with distributed heating tracks perturbed trajectory, theoptimal open loop strategy fails.

Instantaneous control of CHNS

At each time instance tk solve approximately the minimization problem

minu∈U

Jk(c, u) =1

(c − ckd )2 dx +α

∫Ω|u|2 dx (Pk )

y −τ

Re∆y = u + f (1)

c −τ

Pe∆w = cold − τ∇cold y (2)

−γ2∆c + λs(c)− w = cold (3)

f (cold ,wold , yold ) = yold − τcold∇wold − τ (yold∇)yold

λs(c) = s(max(0, c − 1) + min(0, c + 1))

Note: scheme (1)-(3) is not recommended to integrate the CHNS. It only servesthe purpose of controller construction.

The gradient of Jk

Here we have

∇Jk(uk0

) = αuk0

where p3 stems from the solution to the adjoint system

−γ2∆p2 + λ′s(c)p2 − τ∇p1 · y + p1 = c − cd (4)

p2 = −τ

Pe∆p1 (5)

p3 −τ

Re∆p3 = τc∇p1. (6)

Example: Circle2Square

Initial state (circle) and desired state (square)

Morphing: Circle2Square

Circle2Square

Circle2Square: snapshot of controlled states

Circle to Square

MPC: Circle to Square

0 5 10

10−1

||c−cd||

L2(Ω)

H=0.015H=0.025H=0.050

Figure: Deviation of stabilized square from desired square for several predictionhorizons.

Dirichlet boundary control of rising bubble: bottom and side walls

RisingBubbleControl

Dirichlet boundary control of rising bubble: 20 controls at side walls

RisingBubbleControl

Instantaneous control: decay

0 1 2 3 40

||c−cd||

L2(Ω)

Figure: Deviation of controlled bubble from desired bubble.

Dirichlet boundary control of rising bubble: 5 controls at side walls

RisingBubbleControl

Instantaneous control: decay

0 2 4 6 8

||c−cd||

L2(Ω)

Figure: Deviation of controlled bubble from desired bubble.

Thank you very much for your attention

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