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Sensitivity and turnpike results for the optimal control of PDEs and - - PowerPoint PPT Presentation

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Sensitivity and turnpike results for the optimal control of PDEs and their use for model predictive control Lars Gr une Mathematisches Institut, Universit at Bayreuth based on joint work with Manuel Schaller, Anton Schiela (both

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SLIDE 1

Sensitivity and turnpike results for the

  • ptimal control of PDEs and their use for

model predictive control

Lars Gr¨ une

Mathematisches Institut, Universit¨ at Bayreuth based on joint work with Manuel Schaller, Anton Schiela (both Bayreuth), Marleen Stieler (Ludwigshafen) supported by

ICODE Workshop on numerical solutions of HJB equations Paris, 8-10 January 2020

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SLIDE 2

Outline

Setting and problem formulation What makes model predictive control work? Efficient numerical realization for PDEs A sensitivity result for general linear evolution equations Numerical examples

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 2/40

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SLIDE 3

Setting and problem formulation

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SLIDE 4

Setup

Consider abstract control systems ˙ y(t) = f(y(t), u(t)), y(0) = y0 with y(t) ∈ X, u(t) ∈ U, X, U suitable spaces

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 4/40

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SLIDE 5

Setup

Consider abstract control systems ˙ y(t) = f(y(t), u(t)), y(0) = y0 with y(t) ∈ X, u(t) ∈ U, X, U suitable spaces Problem: infinite horizon optimal control

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 4/40

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SLIDE 6

Setup

Consider abstract control systems ˙ y(t) = f(y(t), u(t)), y(0) = y0 with y(t) ∈ X, u(t) ∈ U, X, U suitable spaces Problem: infinite horizon optimal control Optimality criterion: for a running cost ℓ : X × U → R solve minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 4/40

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SLIDE 7

Setup

Consider abstract control systems ˙ y(t) = f(y(t), u(t)), y(0) = y0 with y(t) ∈ X, u(t) ∈ U, X, U suitable spaces Problem: infinite horizon optimal control Optimality criterion: for a running cost ℓ : X × U → R solve minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt subject to state/control constraints y(t) ∈ Y, u(t) ∈ U

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 4/40

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SLIDE 8

Receding horizon control

minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt Direct solution of the problem is numerically hard

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 5/40

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SLIDE 9

Receding horizon control

minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt Direct solution of the problem is numerically hard Alternative method: receding horizon or model predictive control (MPC)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 5/40

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SLIDE 10

Receding horizon control

minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt Direct solution of the problem is numerically hard Alternative method: receding horizon or model predictive control (MPC) Idea: replace the infinite horizon problem by the iterative solution of finite horizon problems minimize

u(·)

JT(y0, u) = T ℓ(y(t), u(t))dt with fixed T > 0 and y(t) ∈ Y, u(t) ∈ U

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 5/40

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SLIDE 11

Receding horizon control

minimize

u(·)

J∞(y0, u) = ∞ ℓ(y(t), u(t))dt Direct solution of the problem is numerically hard Alternative method: receding horizon or model predictive control (MPC) Idea: replace the infinite horizon problem by the iterative solution of finite horizon problems minimize

u(·)

JT(y0, u) = T ℓ(y(t), u(t))dt with fixed T > 0 and y(t) ∈ Y, u(t) ∈ U We obtain a feedback control by a receding horizon technique

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 5/40

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SLIDE 12

MPC from the trajectory point of view

y t x 2 3 4 5 6

τ τ τ τ τ τ

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 13

MPC from the trajectory point of view

t x 2 3 4 5 6

τ τ τ τ τ τ

y black = predictions (open loop optimization)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 14

MPC from the trajectory point of view

τ

t x y

1

2 3 4 5 6

τ τ τ τ τ

black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-15
SLIDE 15

MPC from the trajectory point of view

τ

t x y

1

2 3 4 5 6

τ τ τ τ τ

black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-16
SLIDE 16

MPC from the trajectory point of view

2

t x 2 3 4 5 6

τ τ τ τ τ τ

y black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 17

MPC from the trajectory point of view

2

t x 2 3 4 5 6

τ τ τ τ τ τ

y black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 18

MPC from the trajectory point of view

3

y t x 2 3 4 5 6

τ τ τ τ τ τ

black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 19

MPC from the trajectory point of view

...

3

t x 2 3 4 5 6

τ τ τ τ τ τ

y black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 20

MPC from the trajectory point of view

4

y t x 2 3 4 5 6

τ τ τ τ τ τ

... black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 21

MPC from the trajectory point of view

...

4

t x 2 3 4 5 6

τ τ τ τ τ τ

... y black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-22
SLIDE 22

MPC from the trajectory point of view

5

y t x 2 3 4 5 6

τ τ τ τ τ τ

... ... black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-23
SLIDE 23

MPC from the trajectory point of view

...

5

t x 2 3 4 5 6

τ τ τ τ τ τ

... ... y black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-24
SLIDE 24

MPC from the trajectory point of view

6

y t x 2 3 4 5 6

τ τ τ τ τ τ

... ... ... black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

slide-25
SLIDE 25

MPC from the trajectory point of view

6

y t x 2 3 4 5 6

τ τ τ τ τ τ

... ... ... black = predictions (open loop optimization) red = MPC closed loop yMPC(t, y0)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 6/40

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SLIDE 26

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

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SLIDE 27

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity

StrobeMediaPlayback.swf Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-28
SLIDE 28

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability

StrobeMediaPlayback.swf Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-29
SLIDE 29

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-30
SLIDE 30

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations applicability to real-world industrial applications

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-31
SLIDE 31

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available

  • nline

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

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SLIDE 32

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available

  • nline

But: The trajectory delivered by MPC can be far from optimal!

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-33
SLIDE 33

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available

  • nline

But: The trajectory delivered by MPC can be far from optimal! Key questions in this talk: When does MPC yield closed loop trajectories with approximately optimal performance?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

slide-34
SLIDE 34

Why use MPC?

What is the advantage of MPC over other methods of solving

  • ptimal control problems?

significantly reduced computational complexity real time capability ability to react to perturbations applicability to real-world industrial applications applicability to problems in which data becomes available

  • nline

But: The trajectory delivered by MPC can be far from optimal! Key questions in this talk: When does MPC yield closed loop trajectories with approximately optimal performance? How can we implement MPC efficiently for PDEs?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 7/40

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SLIDE 35

What makes model predictive control work?

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SLIDE 36

The turnpike property

The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time

[von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83]

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

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SLIDE 37

The turnpike property

The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time

[von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83]

In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

slide-38
SLIDE 38

The turnpike property

The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time

[von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83]

In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist) In this talk we stick to the equilibrium setting

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

slide-39
SLIDE 39

The turnpike property

The turnpike property demands that there exists a particular trajectory — the turnpike —, such that all optimal trajectories (regardless of initial condition and optimization horizon) stay near this trajectory most of the time

[von Neumann ’45, Dorfman/Samuelson/Solow ’57, McKenzie ’83]

In the simplest case, this particular trajectory is an equilibrium (but extensions to periodic orbits or more general time varying trajectories exist) In this talk we stick to the equilibrium setting We illustrate it by a simple discrete time example

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 9/40

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SLIDE 40

Example: a macroeconomic model

Consider a classical 1d macroeconomic model

[Brock/Mirman ’72]

Minimize the finite horizon objective N−1

k=0 ℓ(y(k), u(k)) with

ℓ(y, u) = − ln(Ayα − u), A = 5, α = 0.34 and dynamics y(k + 1) = u(k)

  • n

Y = U = [0, 10]

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

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SLIDE 41

Example: a macroeconomic model

Consider a classical 1d macroeconomic model

[Brock/Mirman ’72]

Minimize the finite horizon objective N−1

k=0 ℓ(y(k), u(k)) with

ℓ(y, u) = − ln(Ayα − u), A = 5, α = 0.34 and dynamics y(k + 1) = u(k)

  • n

Y = U = [0, 10] y = invested capital; u = investment in next time step

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

slide-42
SLIDE 42

Example: a macroeconomic model

Consider a classical 1d macroeconomic model

[Brock/Mirman ’72]

Minimize the finite horizon objective N−1

k=0 ℓ(y(k), u(k)) with

ℓ(y, u) = − ln(Ayα − u), A = 5, α = 0.34 and dynamics y(k + 1) = u(k)

  • n

Y = U = [0, 10] y = invested capital; u = investment in next time step Ayα = capital after one time step

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

slide-43
SLIDE 43

Example: a macroeconomic model

Consider a classical 1d macroeconomic model

[Brock/Mirman ’72]

Minimize the finite horizon objective N−1

k=0 ℓ(y(k), u(k)) with

ℓ(y, u) = − ln(Ayα − u), A = 5, α = 0.34 and dynamics y(k + 1) = u(k)

  • n

Y = U = [0, 10] y = invested capital; u = investment in next time step Ayα = capital after one time step Ayα − u = consumed capital; ln(·) = utility function

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

slide-44
SLIDE 44

Example: a macroeconomic model

Consider a classical 1d macroeconomic model

[Brock/Mirman ’72]

Minimize the finite horizon objective N−1

k=0 ℓ(y(k), u(k)) with

ℓ(y, u) = − ln(Ayα − u), A = 5, α = 0.34 and dynamics y(k + 1) = u(k)

  • n

Y = U = [0, 10] y = invested capital; u = investment in next time step Ayα = capital after one time step Ayα − u = consumed capital; ln(·) = utility function On infinite horizon, it is optimal to stay at the equilibrium ye ≈ 2.2344 with ℓ(ye, ue) ≈ 1.4673

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 10/40

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SLIDE 45

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectory for N = 5

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-46
SLIDE 46

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 7

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-47
SLIDE 47

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 9

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-48
SLIDE 48

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 11

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-49
SLIDE 49

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 13

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-50
SLIDE 50

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 15

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-51
SLIDE 51

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 17

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-52
SLIDE 52

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 19

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-53
SLIDE 53

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 21

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-54
SLIDE 54

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 23

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-55
SLIDE 55

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 25

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-56
SLIDE 56

Illustration of the Turnpike Property

5 10 15 20 25 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Optimal trajectories for N = 5, . . . , 25 The turnpike property makes MPC work...

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 11/40

slide-57
SLIDE 57

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-58
SLIDE 58

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-59
SLIDE 59

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-60
SLIDE 60

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-61
SLIDE 61

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-62
SLIDE 62

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-63
SLIDE 63

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-64
SLIDE 64

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-65
SLIDE 65

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-66
SLIDE 66

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-67
SLIDE 67

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-68
SLIDE 68

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-69
SLIDE 69

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-70
SLIDE 70

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-71
SLIDE 71

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-72
SLIDE 72

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-73
SLIDE 73

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-74
SLIDE 74

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-75
SLIDE 75

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-76
SLIDE 76

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-77
SLIDE 77

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-78
SLIDE 78

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-79
SLIDE 79

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-80
SLIDE 80

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-81
SLIDE 81

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-82
SLIDE 82

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-83
SLIDE 83

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-84
SLIDE 84

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-85
SLIDE 85

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-86
SLIDE 86

MPC and the turnpike property

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n x(n)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 12/40

slide-87
SLIDE 87

Approximation properties of MPC

In order to formalize how good MPC approximates the infinite horizon problem, we define yMPC(t, y0) = solution generated by MPC starting in y0 uMPC(t) = control function generated by MPC JMPC

S

(y0) = S ℓ(yMPC(t, y0), uMPC(t))dt

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 13/40

slide-88
SLIDE 88

Approximation properties of MPC

In order to formalize how good MPC approximates the infinite horizon problem, we define yMPC(t, y0) = solution generated by MPC starting in y0 uMPC(t) = control function generated by MPC JMPC

S

(y0) = S ℓ(yMPC(t, y0), uMPC(t))dt Furthermore, we define KL :=

  • β : R2

≥0 → R≥0

  • continuous,

β(r, t) ր ∞ as r → ∞ β(0, t) = 0, β(r, t) ց 0 as t → ∞

  • β( )

r*, t r t (0, 0) (0, 0) r, t* β( ) Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 13/40

slide-89
SLIDE 89

Approximation properties of MPC

Theorem: If the turnpike property at an optimal equilibrium (ye, ue) and suitable controllability and regularity conditions hold, then there exist ε1(T), ε2(S) → 0 as T → ∞ and S → ∞, such that the following properties hold

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

slide-90
SLIDE 90

Approximation properties of MPC

Theorem: If the turnpike property at an optimal equilibrium (ye, ue) and suitable controllability and regularity conditions hold, then there exist ε1(T), ε2(S) → 0 as T → ∞ and S → ∞, such that the following properties hold (1) Approximate average optimality: lim sup

S→∞

1 S JMPC

S

(y0) ≤ ℓ(ye, ue) + ε1(T)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

slide-91
SLIDE 91

Approximation properties of MPC

Theorem: If the turnpike property at an optimal equilibrium (ye, ue) and suitable controllability and regularity conditions hold, then there exist ε1(T), ε2(S) → 0 as T → ∞ and S → ∞, such that the following properties hold (1) Approximate average optimality: lim sup

S→∞

1 S JMPC

S

(y0) ≤ ℓ(ye, ue) + ε1(T) (2) Practical asymptotic stability: there is β ∈ KL: yMPC(t, y0) − ye ≤ β(y0 − ye, t) + ε1(T) for all k ∈ N

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

slide-92
SLIDE 92

Approximation properties of MPC

Theorem: If the turnpike property at an optimal equilibrium (ye, ue) and suitable controllability and regularity conditions hold, then there exist ε1(T), ε2(S) → 0 as T → ∞ and S → ∞, such that the following properties hold (1) Approximate average optimality: lim sup

S→∞

1 S JMPC

S

(y0) ≤ ℓ(ye, ue) + ε1(T) (2) Practical asymptotic stability: there is β ∈ KL: yMPC(t, y0) − ye ≤ β(y0 − ye, t) + ε1(T) for all k ∈ N (3) Approximate transient optimality: for all S ∈ N: JMPC

S

(y0) ≤ JS(y0, u) + Sε1(T) + ε2(S)

for all admissible u with y(S, y0, u) − ye ≤ β(y0 − ye, S) + ε1(T)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 14/40

slide-93
SLIDE 93

Illustration of (2) and (3)

e

y y t

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-94
SLIDE 94

Illustration of (2) and (3)

(T)

e

y ε y t

1

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-95
SLIDE 95

Illustration of (2) and (3)

(T)

e

y ε y t

1

(2): yMPC(t) converges to the ε1(T)-ball around ye

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-96
SLIDE 96

Illustration of (2) and (3)

(T)

e

y ε y t S

1

(2): yMPC(t) converges to the ε1(T)-ball around ye

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-97
SLIDE 97

Illustration of (2) and (3)

(T)

e

y ε y t S

1

(2): yMPC(t) converges to the ε1(T)-ball around ye

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-98
SLIDE 98

Illustration of (2) and (3)

(T)

e

y ε y t S

1

(2): yMPC(t) converges to the ε1(T)-ball around ye

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-99
SLIDE 99

Illustration of (2) and (3)

(T)

e

y ε y t S

1

(2): yMPC(t) converges to the ε1(T)-ball around ye

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-100
SLIDE 100

Illustration of (2) and (3)

(T)

e

y ε y t S

1

(2): yMPC(t) converges to the ε1(T)-ball around ye (3): cost of all other trajectories reaching the ball at time K is (3): higher than that of yMPC(t) up to the error Sε1(T) + ε2(S)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 15/40

slide-101
SLIDE 101

When does the turnpike property hold?

[Carlson et al. ’91, Gr. ’13, Gr./Stieler/Pirkelmann ’18]: strict

dissipativity implies turnpike property for many system classes

[Gr./M¨ uller ’16]: in discrete time the turnpike property is

equivalent to strict dissipativity for controllable systems

[Gr./Guglielmi ’18f]: the turnpike property is equivalent to

detectability-like characterizations for stabilizable finite dimensional linear quadratic problems

[H¨

  • ger/Gr. ’19]: Input-output-to-state stability (IOSS)

implies strict dissipativity and hence the turnpike property

[Tr´ elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f, Gr./Schaller/ [ Schiela ’19f]: turnpike property is implied by stabilizability and

detectability for PDE governed linear quadratic problems

[Porretta/Zuazua ’13, Gugat/Tr´ elat/Zuazua ’16, Zuazua ’18, [ Gugat/Hante ’18]: turnpike property for various hyperbolic PDEs

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 16/40

slide-102
SLIDE 102

When does the turnpike property hold?

[Carlson et al. ’91, Gr. ’13, Gr./Stieler/Pirkelmann ’18]: strict

dissipativity implies turnpike property for many system classes

[Gr./M¨ uller ’16]: in discrete time the turnpike property is

equivalent to strict dissipativity for controllable systems

[Gr./Guglielmi ’18f]: the turnpike property is equivalent to

detectability-like characterizations for stabilizable finite dimensional linear quadratic problems

[H¨

  • ger/Gr. ’19]: Input-output-to-state stability (IOSS)

implies strict dissipativity and hence the turnpike property

[Tr´ elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f, Gr./Schaller/ [ Schiela ’19f]: turnpike property is implied by stabilizability and

detectability for PDE governed linear quadratic problems

[Porretta/Zuazua ’13, Gugat/Tr´ elat/Zuazua ’16, Zuazua ’18, [ Gugat/Hante ’18]: turnpike property for various hyperbolic PDEs

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 16/40

slide-103
SLIDE 103

Efficient numerical realization for PDEs

slide-104
SLIDE 104

Idea of efficient numerical approach

t = T t = 0

exact solution numerical solution

τ Implemented in MPC-loop

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 18/40

slide-105
SLIDE 105

Idea of efficient numerical approach

t = T t = 0

exact solution numerical solution

τ Implemented in MPC-loop Idea: use a fine discretization for small t and a coarse discretization for large t

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 18/40

slide-106
SLIDE 106

Idea of efficient numerical approach

t = T t = 0

exact solution numerical solution

τ Implemented in MPC-loop Idea: use a fine discretization local discretization error for small t and a coarse

  • ε(t) is small for small t

discretization for large t and large for large t

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 18/40

slide-107
SLIDE 107

Idea of efficient numerical approach

t = T t = 0

exact solution numerical solution

τ Implemented in MPC-loop Idea: use a fine discretization local discretization error for small t and a coarse

  • ε(t) is small for small t

discretization for large t and large for large t What about the global numerical error?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 18/40

slide-108
SLIDE 108

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

slide-109
SLIDE 109

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0 At the second glance, there is a problem

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

slide-110
SLIDE 110

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0 At the second glance, there is a problem: solving the optimal control problem involves the adjoint equation which is solved backwards

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

slide-111
SLIDE 111

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0 At the second glance, there is a problem: solving the optimal control problem involves the adjoint equation which is solved backwards ˙ y(t) = Hλ(y(t), u(t), λ(t)), y(0) = y0 ˙ λ(t) = −Hy(y(t), u(t), λ(t)), λ(T) = 0

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

slide-112
SLIDE 112

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0 At the second glance, there is a problem: solving the optimal control problem involves the adjoint equation which is solved backwards ˙ y(t) = Hλ(y(t), u(t), λ(t)) + ε1(t), y(0) = y0 ˙ λ(t) = −Hy(y(t), u(t), λ(t)) + ε2(t), λ(T) = 0 large εi(t) for t ≈ T can propagate backwards to t ≈ 0

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

slide-113
SLIDE 113

Sensitivity w.r.t. numerical errors

At a first glance, one might conjecture that if the local discretization error ε(t) is large only for t ≈ T, then this should not affect the solution at times t ≈ 0 At the second glance, there is a problem: solving the optimal control problem involves the adjoint equation which is solved backwards ˙ y(t) = Hλ(y(t), u(t), λ(t)) + ε1(t), y(0) = y0 ˙ λ(t) = −Hy(y(t), u(t), λ(t)) + ε2(t), λ(T) = 0 large εi(t) for t ≈ T can propagate backwards to t ≈ 0 Is there a structural property that can save this idea?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 19/40

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SLIDE 114

A sensitivity result for general linear evolution equations

slide-115
SLIDE 115

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-116
SLIDE 116

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-117
SLIDE 117

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-118
SLIDE 118

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-119
SLIDE 119

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term and the optimization objective min

u

1 2 T C(y(t) − yd)2

Y + R(u(t) − ud)2 Udt

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-120
SLIDE 120

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term and the optimization objective min

u

1 2 T C(y(t) − yd)2

Y + R(u(t) − ud)2 Udt

U, Y Hilbert spaces

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-121
SLIDE 121

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term and the optimization objective min

u

1 2 T C(y(t) − yd)2

Y + R(u(t) − ud)2 Udt

U, Y Hilbert spaces; R ∈ L(U, U) elliptic

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-122
SLIDE 122

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term and the optimization objective min

u

1 2 T C(y(t) − yd)2

Y + R(u(t) − ud)2 Udt

U, Y Hilbert spaces; R ∈ L(U, U) elliptic B, C admissible, possibly unbounded

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-123
SLIDE 123

Setting

We consider general linear evolution equations d dty = Ay + Bu + f, y(0) = y0 A : D(A) ⊂ X → X generates a C0-semigroup T (t) X Hilbert space; f ∈ L1(0, T; X) source term and the optimization objective min

u

1 2 T C(y(t) − yd)2

Y + R(u(t) − ud)2 Udt

U, Y Hilbert spaces; R ∈ L(U, U) elliptic B, C admissible, possibly unbounded Solution concept: mild solution

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 21/40

slide-124
SLIDE 124

Optimal control problem

Optimality condition: Pontryagin’s Maximum Principle yields     C∗C − d

dt − A∗

ET

d dt − A

−BQ−1B∗ E0    

  • =: M

y λ

  • =

    C∗Cyd Bud + f y0     , +     ε1 ε2     where E0y := y(0) and ETλ := λ(T)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 22/40

slide-125
SLIDE 125

Optimal control problem

Optimality condition: Pontryagin’s Maximum Principle yields     C∗C − d

dt − A∗

ET

d dt − A

−BQ−1B∗ E0    

  • =: M

˜ y ˜ λ

  • =

    C∗Cyd Bud + f y0     +     ε1 ε2    , where E0y := y(0) and ETλ := λ(T)

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 22/40

slide-126
SLIDE 126

Optimal control problem

Optimality condition: Pontryagin’s Maximum Principle yields     C∗C − d

dt − A∗

ET

d dt − A

−BQ−1B∗ E0    

  • =: M

˜ y ˜ λ

  • =

    C∗Cyd Bud + f y0     +     ε1 ε2    , where E0y := y(0) and ETλ := λ(T) Define δy = ˜ y − y, δλ = ˜ λ − λ

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 22/40

slide-127
SLIDE 127

Optimal control problem

Optimality condition: Pontryagin’s Maximum Principle yields     C∗C − d

dt − A∗

ET

d dt − A

−BQ−1B∗ E0    

  • =: M

˜ y ˜ λ

  • =

    C∗Cyd Bud + f y0     +     ε1 ε2    , where E0y := y(0) and ETλ := λ(T) Define δy = ˜ y − y, δλ = ˜ λ − λ Idea: Use δy + δλ ≤ M −1(ε1, 0, ε2, 0) plus exponential weighting

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 22/40

slide-128
SLIDE 128

Sensitivity result

Theorem: Define ρ := e−µtε1(t)Lp(X) + e−µtε2(t)Lp(X) for p = 1 or p = 2 and assume the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 are bounded independently of T

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 23/40

slide-129
SLIDE 129

Sensitivity result

Theorem: Define ρ := e−µtε1(t)Lp(X) + e−µtε2(t)Lp(X) for p = 1 or p = 2 and assume the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 are bounded independently of T. Then there are µ, c > 0 with e−µtδyL2(X) + e−µtδλL2(X) + e−µtδuL2(U) ≤ cρ e−µtδyC(X) + e−µtδλC(X) ≤ cρ

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 23/40

slide-130
SLIDE 130

Sensitivity result

Theorem: Define ρ := e−µtε1(t)Lp(X) + e−µtε2(t)Lp(X) for p = 1 or p = 2 and assume the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 are bounded independently of T. Then there are µ, c > 0 with e−µtδyL2(X) + e−µtδλL2(X) + e−µtδuL2(U) ≤ cρ e−µtδyC(X) + e−µtδλC(X) ≤ cρ If B is bounded then in addition e−µtδuL∞(U) ≤ cρ holds

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 23/40

slide-131
SLIDE 131

Sensitivity result

Theorem: Define ρ := e−µtε1(t)Lp(X) + e−µtε2(t)Lp(X) for p = 1 or p = 2 and assume the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 are bounded independently of T. Then there are µ, c > 0 with e−µtδyL2(X) + e−µtδλL2(X) + e−µtδuL2(U) ≤ cρ e−µtδyC(X) + e−µtδλC(X) ≤ cρ If B is bounded then in addition e−µtδuL∞(U) ≤ cρ holds For p = 1 and · C(X)-norms this implies y(t) − ˜ y(t) ≤ T ceµ(t−s)(ε1(s) + ε2(s))ds

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 23/40

slide-132
SLIDE 132

Sensitivity result

Theorem: Define ρ := e−µtε1(t)Lp(X) + e−µtε2(t)Lp(X) for p = 1 or p = 2 and assume the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 are bounded independently of T. Then there are µ, c > 0 with e−µtδyL2(X) + e−µtδλL2(X) + e−µtδuL2(U) ≤ cρ e−µtδyC(X) + e−µtδλC(X) ≤ cρ If B is bounded then in addition e−µtδuL∞(U) ≤ cρ holds For p = 1 and · C(X)-norms this implies y(t) − ˜ y(t) ≤ T ceµ(t−s)(ε1(s) + ε2(s))ds Large errors for s ≈ T are exponentially damped at t ≈ 0

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 23/40

slide-133
SLIDE 133

Boundedness of M −1

How do we get a T-independent bound for the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 ?

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 24/40

slide-134
SLIDE 134

Boundedness of M −1

How do we get a T-independent bound for the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 ? Definition: (i) We say that (A, B) is exponentially stabilizable, if there is K ∈ L(X, U) such that the semigroup generated by A + BK is exponentially stable (ii) We say that (A, C) is exponentially detectable if (A∗, C∗) is exponentially stabilizable

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Boundedness of M −1

How do we get a T-independent bound for the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 ? Definition: (i) We say that (A, B) is exponentially stabilizable, if there is K ∈ L(X, U) such that the semigroup generated by A + BK is exponentially stable (ii) We say that (A, C) is exponentially detectable if (A∗, C∗) is exponentially stabilizable Theorem: If the control system is exponentially stabilizable and detectable, then the above norms are bounded independently of T

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SLIDE 136

Boundedness of M −1

How do we get a T-independent bound for the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 ? Definition: (i) We say that (A, B) is exponentially stabilizable, if there is K ∈ L(X, U) such that the semigroup generated by A + BK is exponentially stable (ii) We say that (A, C) is exponentially detectable if (A∗, C∗) is exponentially stabilizable Theorem: If the control system is exponentially stabilizable and detectable, then the above norms are bounded independently of T This is the hard part of the analysis

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 24/40

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SLIDE 137

Boundedness of M −1

How do we get a T-independent bound for the norms M −1(L1(X)×X)2→C(X)2 M −1(L2(X)×X)2→C(X)2 M −1(L1(X)×X)2→L2(X)2 M −1(L2(X)×X)2→L2(X)2 ? Definition: (i) We say that (A, B) is exponentially stabilizable, if there is K ∈ L(X, U) such that the semigroup generated by A + BK is exponentially stable (ii) We say that (A, C) is exponentially detectable if (A∗, C∗) is exponentially stabilizable Theorem: If the control system is exponentially stabilizable and detectable, then the above norms are bounded independently of T This is the hard part of the analysis For details: manuel.schaller@uni-bayreuth.de

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Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup

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Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup Particularly, our results apply to hyperbolic PDEs and boundary control and observations

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SLIDE 140

Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup Particularly, our results apply to hyperbolic PDEs and boundary control and observations Recall: for PDE governed linear quadratic problems, the turnpike property is implied by stabilizability and detectability [Tr´

elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f]

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SLIDE 141

Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup Particularly, our results apply to hyperbolic PDEs and boundary control and observations Recall: for PDE governed linear quadratic problems, the turnpike property is implied by stabilizability and detectability [Tr´

elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f]

The same mechanism that generates the turnpike behaviour damps out errors in backward time

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SLIDE 142

Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup Particularly, our results apply to hyperbolic PDEs and boundary control and observations Recall: for PDE governed linear quadratic problems, the turnpike property is implied by stabilizability and detectability [Tr´

elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f]

The same mechanism that generates the turnpike behaviour damps out errors in backward time Extension to certain PDEs with nonlinearities (semilinear, quasilinear) possible

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SLIDE 143

Discussion

Under the same condition we obtain a turnpike result that generalizes many of the mentioned results in the literature, as we require neither boundedness of B and C nor that A generates an analytic semigroup Particularly, our results apply to hyperbolic PDEs and boundary control and observations Recall: for PDE governed linear quadratic problems, the turnpike property is implied by stabilizability and detectability [Tr´

elat/Zhang/Zuazua ’18, Breiten/Pfeiffer ’18f]

The same mechanism that generates the turnpike behaviour damps out errors in backward time Extension to certain PDEs with nonlinearities (semilinear, quasilinear) possible — work in progress

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SLIDE 144

Numerical examples

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SLIDE 145

What do we expect to see?

We expect to see the following effect: Fine grid for small t

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SLIDE 146

What do we expect to see?

We expect to see the following effect: Fine grid for small t Residual ε small for small t

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SLIDE 147

What do we expect to see?

We expect to see the following effect: Fine grid for small t Residual ε small for small t Absolute error u − ˜ u small for small t

exponential sensitivity

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SLIDE 148

What do we expect to see?

We expect to see the following effect: Fine grid for small t Residual ε small for small t Absolute error u − ˜ u small for small t

exponential sensitivity

However, we do not want to select the grids manually

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SLIDE 149

What do we expect to see?

We expect to see the following effect: Fine grid for small t Residual ε small for small t Absolute error u − ˜ u small for small t

exponential sensitivity

However, we do not want to select the grids manually goal-oriented estimation [Meidner ’08, Meidner/Vexler ’07ff]

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SLIDE 150

Goal oriented error estimation

In goal-oriented error estimation the error of a particular quantity of interest is estimated

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 28/40

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SLIDE 151

Goal oriented error estimation

In goal-oriented error estimation the error of a particular quantity of interest is estimated We use

JT (y, u) := T ℓ(y(t), u(t)) dt and Jτ(y, u) := τ ℓ(y(t), u(t)) dt

with time interval τ = 0.5

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SLIDE 152

Goal oriented error estimation

In goal-oriented error estimation the error of a particular quantity of interest is estimated We use

JT (y, u) := T ℓ(y(t), u(t)) dt and Jτ(y, u) := τ ℓ(y(t), u(t)) dt

with time interval τ = 0.5 For the discontinuous Galerkin discretization in time we can prove: Theorem: Let (A, B), (A∗, C∗) be exponentially stabilizable. Then the error indicators ητ for Jτ satisfy ητ(t) ∼ c(τ)e−µt with c(τ), µ > 0 independent of T

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SLIDE 153

Test problem

min

(y,u)

1 2y − yd2

L2([0,30]×[0,1]2) + α

2 u2

L2([0,30]×[0,1]2)

d dty = −d∆y + µy + u, y(0) = 0, yd =

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SLIDE 154

Optimal solution

Open loop optimal solution t = 0.0

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SLIDE 155

Optimal solution

Open loop optimal solution t = 0.3

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SLIDE 156

Optimal solution

Open loop optimal solution t = 0.6

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SLIDE 157

Optimal solution

Open loop optimal solution t = 0.6, . . . , 2.7

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SLIDE 158

Optimal solution

Open loop optimal solution t = 3.0

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SLIDE 159

Adaptive grid in time

Standard error estimator for JT

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SLIDE 160

Adaptive grid in time

Standard error estimator for JT

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SLIDE 161

Adaptive grid in time

Standard error estimator for JT

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SLIDE 162

Adaptive grid in time

Standard error estimator for JT

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SLIDE 163

Adaptive grid in time

Standard error estimator for JT

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SLIDE 164

Adaptive grid in time

Standard error estimator for JT

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SLIDE 165

Adaptive grid in time

Standard error estimator for JT

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SLIDE 166

Adaptive grid in time

Standard error estimator for JT

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SLIDE 167

Adaptive grid in time

Standard error estimator for JT

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SLIDE 168

Adaptive grid in time

Standard error estimator for JT

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SLIDE 169

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 170

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 171

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 172

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 173

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 174

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 175

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 176

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 177

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 178

Adaptive grid in time

Goal-oriented error estimator for Jτ, focusing on [0, τ]

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SLIDE 179

Adaptive grid in time

Comparison to standard error estimator

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SLIDE 180

Adaptive grid in time

Stability of MPC closed-loop solutions 2 4 6 8 time points 1 2 3 4 11 time points 0.5 1 1.5 2 1 2 3 4 time t 21 time points refined for Jτ refined for JT reference 0.5 1 1.5 20 0.5 1 1.5 2 time t 41 time points

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SLIDE 181

Adaptive grid in time

5 8 11 21 31 41 100 101 102 103 104 number of time grid points Cost of MPC closed-loop solutions refined for Jτ refined for JT reference

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SLIDE 182

Adaptive grid in space

Goal oriented (bottom) vs. standard error estimator (top) JT(y, u) Jτ(y, u) τ = 0.5 T = 9

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SLIDE 183

Adaptive grid in space

572 962 1443 1924 2405 2886 3367 3848 0.5 1 1.5 2 total space DOFs Cost of MPC closed-loop solutions refined for Jτ refined for JT reference

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SLIDE 184

Adaptive grid in space and time

5 8 11 21 0.1 0.2 0.3 0.4 number of time points Cost functional value Cost of MPC closed-loop solutions (5000 total space DOFs) refined for Jτ refined for JT reference

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Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems

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SLIDE 186

Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems (“Model reduction in time”)

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SLIDE 187

Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems (“Model reduction in time”) The existence of the turnpike property is the key structural property to make this approach work

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 39/40

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SLIDE 188

Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems (“Model reduction in time”) The existence of the turnpike property is the key structural property to make this approach work Exponential controllability and detectability imply this property for linear quadratic PDE problems

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 39/40

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SLIDE 189

Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems (“Model reduction in time”) The existence of the turnpike property is the key structural property to make this approach work Exponential controllability and detectability imply this property for linear quadratic PDE problems The same mechanism that leads to the turnpike property also causes an exponential damping of numerical errors in backward time

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 39/40

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SLIDE 190

Summary and outlook

Model Predictive Control can be seen as a method for splitting up an infinite horizon optimal control problem into the iterative solution of finite horizon problems (“Model reduction in time”) The existence of the turnpike property is the key structural property to make this approach work Exponential controllability and detectability imply this property for linear quadratic PDE problems The same mechanism that leads to the turnpike property also causes an exponential damping of numerical errors in backward time This can be exploited by adaptive discretization strategies via goal oriented error estimators

Lars Gr¨ une, Sensitivity and turnpike results for optimal control of PDEs and MPC, p. 39/40

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SLIDE 191

References

  • L. Gr¨

une, Approximation properties of receding horizon optimal control, DMV Jahresbericht, 118, 3–37, 2016

  • L. Gr¨

une, Economic receding horizon control without terminal constraints, Automatica, 49, 725–734, 2013

  • L. Gr¨

une, M. Stieler, Asymptotic stability and transient optimality of economic MPC without terminal conditions, Journal of Process Control, 24 (Special Issue on Economic MPC), 1187–1196, 2014

  • L. Gr¨

une, M. Schaller, A. Schiela, Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by Model Predictive Control, SIAM J. Control Optim., 2019

  • L. Gr¨

une, M. Schaller, A. Schiela, Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations, Journal of Differential Equations, online version appeared 2019

  • L. Gr¨

une, M. Schaller, A. Schiela, Space-time adaptivity for Model Predictive Control of parabolic PDEs, in preparation

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