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The role of state constraints for turnpike behaviour and strict dissipativity of optimal control problems Lars Gr une Mathematisches Institut, Universit at Bayreuth based on joint work with Roberto Guglielmi (GSSI, LAquila) Control of

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SLIDE 1

The role of state constraints for turnpike behaviour and strict dissipativity of

  • ptimal control problems

Lars Gr¨ une

Mathematisches Institut, Universit¨ at Bayreuth based on joint work with Roberto Guglielmi (GSSI, L’Aquila)

Control of state constrained dynamical systems Padova, 25–29 September 2017

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SLIDE 2

Outline

The turnpike property Strict dissipativity Linear quadratic problems Main results

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 2

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SLIDE 3

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 4

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces Brief notation x+ = f(x, u)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 5

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces Brief notation x+ = f(x, u) Interpretation: xu(n) = state of the system at time tn

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 6

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces Brief notation x+ = f(x, u) Interpretation: xu(n) = state of the system at time tn u(n) = control acting from time tn to tn+1

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 7

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces Brief notation x+ = f(x, u) Interpretation: xu(n) = state of the system at time tn u(n) = control acting from time tn to tn+1 f = solution operator of a controlled ODE/PDE

  • r of a discrete time model

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 8

System class

We consider nonlinear discrete time control systems xu(n + 1) = f(xu(n), u(n)), xu(0) = x with xu(n) ∈ X, u(n) ∈ U, X, U normed spaces Brief notation x+ = f(x, u) Interpretation: xu(n) = state of the system at time tn u(n) = control acting from time tn to tn+1 f = solution operator of a controlled ODE/PDE

  • r of a discrete time model (or a numerical

approximation of one of these)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 3

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SLIDE 9

The turnpike property

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SLIDE 10

The turnpike property

The turnpike property describes a behaviour of (approximately)

  • ptimal trajectories for a finite horizon optimal control problem

minimise

u

JN(x, u) =

N−1

  • n=0

ℓ(xu(n), u(n)) with state and input constraints xu(n) ∈ X, u(n) ∈ U

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 5

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SLIDE 11

The turnpike property

The turnpike property describes a behaviour of (approximately)

  • ptimal trajectories for a finite horizon optimal control problem

minimise

u

JN(x, u) =

N−1

  • n=0

ℓ(xu(n), u(n)) with state and input constraints xu(n) ∈ X, u(n) ∈ U Informal description of the turnpike property: any optimal trajectory stays near an equilibrium xe most of the time

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 5

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SLIDE 12

The turnpike property

The turnpike property describes a behaviour of (approximately)

  • ptimal trajectories for a finite horizon optimal control problem

minimise

u

JN(x, u) =

N−1

  • n=0

ℓ(xu(n), u(n)) with state and input constraints xu(n) ∈ X, u(n) ∈ U Informal description of the turnpike property: any optimal trajectory stays near an equilibrium xe most of the time We illustrate the property by two simple examples

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 5

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Example 1: minimum energy control

Example: Keep the state of the system inside a given interval X minimising the quadratic control effort ℓ(x, u) = u2 with dynamics x+ = 2x + u and constraints X = [−2, 2], U = [−3, 3]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 6

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Example 1: minimum energy control

Example: Keep the state of the system inside a given interval X minimising the quadratic control effort ℓ(x, u) = u2 with dynamics x+ = 2x + u and constraints X = [−2, 2], U = [−3, 3] For this example, the closer the state is to xe = 0, the cheaper it is to keep the system inside X

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 6

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SLIDE 15

Example 1: minimum energy control

Example: Keep the state of the system inside a given interval X minimising the quadratic control effort ℓ(x, u) = u2 with dynamics x+ = 2x + u and constraints X = [−2, 2], U = [−3, 3] For this example, the closer the state is to xe = 0, the cheaper it is to keep the system inside X

  • ptimal trajectory should stay near xe = 0

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 6

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

Optimal trajectory for N = 5

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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SLIDE 25

Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 1: optimal trajectories

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

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

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 7

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Example 2: a macroeconomic model

The second example is a 1d macroeconomic model

[Brock/Mirman ’72]

Minimise the finite horizon objective with ℓ(x, u) = − ln(Axα − u), A = 5, α = 0.34 with dynamics x+ = u

  • n

X = U = [0, 10]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

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Example 2: a macroeconomic model

The second example is a 1d macroeconomic model

[Brock/Mirman ’72]

Minimise the finite horizon objective with ℓ(x, u) = − ln(Axα − u), A = 5, α = 0.34 with dynamics x+ = u

  • n

X = U = [0, 10] Here the optimal trajectories are less obvious

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

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Example 2: a macroeconomic model

The second example is a 1d macroeconomic model

[Brock/Mirman ’72]

Minimise the finite horizon objective with ℓ(x, u) = − ln(Axα − u), A = 5, α = 0.34 with dynamics x+ = u

  • n

X = U = [0, 10] Here the optimal trajectories are less obvious On infinite horizon, it is optimal to stay at the equilibrium xe ≈ 2.2344 with ℓ(xe, ue) ≈ 1.4673

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

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Example 2: a macroeconomic model

The second example is a 1d macroeconomic model

[Brock/Mirman ’72]

Minimise the finite horizon objective with ℓ(x, u) = − ln(Axα − u), A = 5, α = 0.34 with dynamics x+ = u

  • n

X = U = [0, 10] Here the optimal trajectories are less obvious On infinite horizon, it is optimal to stay at the equilibrium xe ≈ 2.2344 with ℓ(xe, ue) ≈ 1.4673 One may thus expect that finite horizon optimal trajectories also stay for a long time near that equilibrium

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 8

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Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 33

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 34

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 35

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 36

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 37

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 38

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 39

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 40

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 41

Example 2: optimal trajectories

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 9

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SLIDE 42

How to formalize the turnpike property?

5 10 15 20 25 −0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 n x(n)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

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How to formalize the turnpike property?

n 5 10 15 20 25 x(n)

  • 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

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SLIDE 44

How to formalize the turnpike property?

n 5 10 15 20 25 x(n)

  • 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Number of points outside the blue neighbourhood is bounded by a number independent of N (here: by 8)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 10

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SLIDE 45

The turnpike property: formal definitions

Let xe be an equilibrium, i.e., f(xe, ue) = xe

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

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The turnpike property: formal definitions

Let xe be an equilibrium, i.e., f(xe, ue) = xe Turnpike property: For each ε > 0 and ρ > 0 there is Cρ,ε > 0 such that for all N ∈ N all optimal trajectories x⋆ starting in Bρ(xe) satisfy the inequality #

  • k ∈ {0, . . . , N − 1}
  • x⋆(k) − xe ≥ ε
  • ≤ Cρ,ε

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

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SLIDE 47

The turnpike property: formal definitions

Let xe be an equilibrium, i.e., f(xe, ue) = xe Turnpike property: For each ε > 0 and ρ > 0 there is Cρ,ε > 0 such that for all N ∈ N all optimal trajectories x⋆ starting in Bρ(xe) satisfy the inequality #

  • k ∈ {0, . . . , N − 1}
  • x⋆(k) − xe ≥ ε
  • ≤ Cρ,ε

Near equilibrium turnpike property: For each ε > 0, δ > 0 and ρ > 0 there is Cρ,ε,δ > 0 such that for all x ∈ X and N ∈ N, all trajectories xu with xu(0) = x ∈ Bρ(xe) and JN(x, u) ≤ Nℓ(xe, ue) + δ satisfy the inequality #

  • k ∈ {0, . . . , N − 1}
  • xu(k) − xe ≥ ε
  • ≤ Cρ,ε,δ

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 11

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SLIDE 48

History

Apparently first described by [von Neumann 1945]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 49

History

Apparently first described by [von Neumann 1945] Name “turnpike property” coined by

[Dorfman/Samuelson/Solow 1957]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 50

History

Apparently first described by [von Neumann 1945] Name “turnpike property” coined by

[Dorfman/Samuelson/Solow 1957]

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 51

History

Apparently first described by [von Neumann 1945] Name “turnpike property” coined by

[Dorfman/Samuelson/Solow 1957]

Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 52

History

Apparently first described by [von Neumann 1945] Name “turnpike property” coined by

[Dorfman/Samuelson/Solow 1957]

Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983] Renewed interest in recent years [Zaslavski ’14,

Tr´ elat/Zuazua ’15, Faulwasser et al. ’15, . . . ]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 53

History

Apparently first described by [von Neumann 1945] Name “turnpike property” coined by

[Dorfman/Samuelson/Solow 1957]

Extensively studied in the 1970s in mathematical economy, cf. survey [McKenzie 1983] Renewed interest in recent years [Zaslavski ’14,

Tr´ elat/Zuazua ’15, Faulwasser et al. ’15, . . . ]

Many applications, e.g., structural insight in economic equilibria; synthesis of optimal trajectories

[Anderson/Kokotovic ’87]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 12

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SLIDE 54

Application: Model predictive control

Turnpike properties are also pivotal for analysing economic Model Predictive Control (MPC) schemes

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 13

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SLIDE 55

Application: Model predictive control

Turnpike properties are also pivotal for analysing economic Model Predictive Control (MPC) schemes MPC is a method in which an optimal control problem on an infinite horizon minimise

u

J∞(x, u) =

∞

  • n=0

ℓ(xu(n), u(n)) is approximated by the iterative solution of finite horizon problems minimise

u

JN(x, u) =

N−1

  • n=0

ℓ(xu(k), u(k)) with fixed N ∈ N

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 13

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SLIDE 56

MPC from the trajectory point of view

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 57

MPC from the trajectory point of view

n x 1 2 3 4 5 6 x black = predictions (open loop optimization)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 58

MPC from the trajectory point of view

1

n x 1 2 3 4 5 6 x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 59

MPC from the trajectory point of view

1

n x 1 2 3 4 5 6 x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 60

MPC from the trajectory point of view

2

n x 1 2 3 4 5 6 x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 61

MPC from the trajectory point of view

2

n x 1 2 3 4 5 6 x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 62

MPC from the trajectory point of view

3

n x 1 2 3 4 5 6 x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 63

MPC from the trajectory point of view

3

n x 1 2 3 4 5 6 ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 64

MPC from the trajectory point of view

4

n x 1 2 3 4 5 6 ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 65

MPC from the trajectory point of view

4

n x 1 2 3 4 5 6 ... ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 66

MPC from the trajectory point of view

5

n x 1 2 3 4 5 6 ... ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 67

MPC from the trajectory point of view

5

n x 1 2 3 4 5 6 ... ... ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 68

MPC from the trajectory point of view

6

n x 1 2 3 4 5 6 ... ... ... x black = predictions (open loop optimization) red = MPC closed loop

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 14

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SLIDE 69

Approximation result for MPC

If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

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SLIDE 70

Approximation result for MPC

If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved The result exploits that the red closed loop trajectory approximately follows the first part of the black predictions up to the equilibrium

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

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SLIDE 71

Approximation result for MPC

If the finite horizon problems have the turnpike property, then a rigorous approximation result can be proved The result exploits that the red closed loop trajectory approximately follows the first part of the black predictions up to the equilibrium We illustrate this behaviour by our second example for N = 10

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 15

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SLIDE 72

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 73

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 74

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 75

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 76

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 77

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 78

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 79

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 80

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-81
SLIDE 81

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-82
SLIDE 82

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-83
SLIDE 83

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-84
SLIDE 84

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-85
SLIDE 85

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-86
SLIDE 86

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-87
SLIDE 87

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-88
SLIDE 88

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-89
SLIDE 89

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-90
SLIDE 90

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-91
SLIDE 91

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-92
SLIDE 92

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-93
SLIDE 93

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-94
SLIDE 94

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-95
SLIDE 95

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-96
SLIDE 96

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-97
SLIDE 97

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-98
SLIDE 98

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-99
SLIDE 99

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-100
SLIDE 100

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-101
SLIDE 101

MPC for Example 2

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, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

slide-102
SLIDE 102

MPC for Example 2

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)

Extension to non equilibrium turnpikes possible

[Zanon/Gr./Diehl ’17, Gr./M¨ uller ’17, Gr./Pirkelmann ’17]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 16

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SLIDE 103

Strict dissipativity

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SLIDE 104

Dissipativity

x+ = f(x, u) Introduce functions s : X × U → R and λ : X → R s(x, u) supply rate, measuring the (possibly negative) amount of energy supplied to the system via the input u in the next time step λ(x) storage function, measuring the amount of energy stored inside the system when the system is in state x

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 18

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SLIDE 105

Dissipativity

Definition [cf. Willems ’72] The system is called strictly pre-dissipative if there are xe ∈ X, α ∈ K such that for all x ∈ X, u ∈ U the inequality λ(x+) ≤ λ(x) + s(x, u) − α(x − xe) holds

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 19

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SLIDE 106

Dissipativity

Definition [cf. Willems ’72] The system is called strictly pre-dissipative if there are xe ∈ X, α ∈ K such that for all x ∈ X, u ∈ U the inequality λ(x+) ≤ λ(x) + s(x, u) − α(x − xe) holds α ∈ K: α : R+

0 → R+ 0 , continuous,

strictly increasing, α(0) = 0

r (0, 0) r α( )

The system is called strictly dissipative if it is strictly pre-dissipative with λ bounded from below

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 19

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SLIDE 107

Physical interpretation of dissipativity

λ(x+) ≤ λ(x) + s(x, u) − α(x − xe) physical interpretation of strict dissipativity

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 20

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SLIDE 108

Physical interpretation of dissipativity

λ(x+) ≤ λ(x) + s(x, u) − α(x − xe) physical interpretation of strict dissipativity: λ(x) = energy stored in the system s(x, u) = energy supplied to the system

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 20

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SLIDE 109

Physical interpretation of dissipativity

λ(x+) ≤ λ(x) + s(x, u) − α(x − xe) physical interpretation of strict dissipativity: λ(x) = energy stored in the system s(x, u) = energy supplied to the system strict dissipativity: energy can not be generated inside the system a certain amount of energy α(x − xe) must be dissipated (= given to the environment)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 20

slide-110
SLIDE 110

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-111
SLIDE 111

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-112
SLIDE 112

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-113
SLIDE 113

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

(passivity = dissipativity with s(x, u) = y, u, where y = h(x) is the output of the system)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-114
SLIDE 114

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

(passivity = dissipativity with s(x, u) = y, u, where y = h(x) is the output of the system)

Passivity, in turn, is a classical property of electrical circuits

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-115
SLIDE 115

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

(passivity = dissipativity with s(x, u) = y, u, where y = h(x) is the output of the system)

Passivity, in turn, is a classical property of electrical circuits Strict (or strong) dissipativity is mentioned in [Willems ’72] but is not so often used; strict passivity is more commonly found

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-116
SLIDE 116

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

(passivity = dissipativity with s(x, u) = y, u, where y = h(x) is the output of the system)

Passivity, in turn, is a classical property of electrical circuits Strict (or strong) dissipativity is mentioned in [Willems ’72] but is not so often used; strict passivity is more commonly found Dissipativity has widespread applications in stability theory and

  • ptimal control

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

slide-117
SLIDE 117

History

Dissipativity was defined for continuous time systems in

[Jan C. Willems, Dissipative Dynamical Systems, Part I & II, 1972] (still one of the best readings on the topic)

It was the result of the endeavour to generalise passivity

(passivity = dissipativity with s(x, u) = y, u, where y = h(x) is the output of the system)

Passivity, in turn, is a classical property of electrical circuits Strict (or strong) dissipativity is mentioned in [Willems ’72] but is not so often used; strict passivity is more commonly found Dissipativity has widespread applications in stability theory and

  • ptimal control

Translation to discrete time systems is quite straightforward

[Byrnes/Lin ’94]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 21

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SLIDE 118

Relation between strict dissipativity and turnpike

The relation between strict dissipativity and turnpike behaviour was investigated in detail in [Gr., M¨

uller ’17]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 22

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SLIDE 119

Relation between strict dissipativity and turnpike

The relation between strict dissipativity and turnpike behaviour was investigated in detail in [Gr., M¨

uller ’17]

Typical result: Theorem: Assume X is closed and bounded and U is compact, ℓ is continuous and bounded from below, xe is an equilibrium around which the system is locally controllable and ue ∈ argmin{ℓ(xe, u) | u ∈ U, f(xe, u) = xe}

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 22

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SLIDE 120

Relation between strict dissipativity and turnpike

The relation between strict dissipativity and turnpike behaviour was investigated in detail in [Gr., M¨

uller ’17]

Typical result: Theorem: Assume X is closed and bounded and U is compact, ℓ is continuous and bounded from below, xe is an equilibrium around which the system is locally controllable and ue ∈ argmin{ℓ(xe, u) | u ∈ U, f(xe, u) = xe} Then the following statements are equivalent (a) The system is strictly dissipative with supply rate s(x, u) = ℓ(x, u) − ℓ(xe, ue) and a bounded storage function (b) The near equilibrium turnpike property holds

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 22

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SLIDE 121

Linear quadratic problems

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SLIDE 122

LQ problems

From now on we consider linear quadratic finite dimensional discrete time problems with X = Rn, U = Rm, x+ = Ax + Bu, ℓ(x, u) = xTQx + uTRu + bTx + dTu with Q = CTC and R > 0

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 24

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SLIDE 123

LQ problems

From now on we consider linear quadratic finite dimensional discrete time problems with X = Rn, U = Rm, x+ = Ax + Bu, ℓ(x, u) = xTQx + uTRu + bTx + dTu with Q = CTC and R > 0 We call an eigenvalue µ of A unobservable, if the corresponding eigenvector v satisfies Cv = 0

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 24

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SLIDE 124

LQ problems

From now on we consider linear quadratic finite dimensional discrete time problems with X = Rn, U = Rm, x+ = Ax + Bu, ℓ(x, u) = xTQx + uTRu + bTx + dTu with Q = CTC and R > 0 We call an eigenvalue µ of A unobservable, if the corresponding eigenvector v satisfies Cv = 0 Note: unobservable eigenvectors satisfy vTQv = 0 ⇒ they are not visible in the quadratic part of the cost function

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 24

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SLIDE 125

LQ problems

From now on we consider linear quadratic finite dimensional discrete time problems with X = Rn, U = Rm, x+ = Ax + Bu, ℓ(x, u) = xTQx + uTRu + bTx + dTu with Q = CTC and R > 0 We call an eigenvalue µ of A unobservable, if the corresponding eigenvector v satisfies Cv = 0 Note: unobservable eigenvectors satisfy vTQv = 0 ⇒ they are not visible in the quadratic part of the cost function The same holds for solutions x(t) starting in v with u(t) ≡ 0

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 24

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SLIDE 126

Storage functions for LQ problems

x+ = Ax + Bu, s(x, u) = xTQx + uTRu + bTx + dTu Lemma: For LQ problems, a storage function λ can always be chosen of the form λ(x) = xTPx + qTx with P satisfying Q + P − ATPA > 0

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 25

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SLIDE 127

Storage functions for LQ problems

x+ = Ax + Bu, s(x, u) = xTQx + uTRu + bTx + dTu Lemma: For LQ problems, a storage function λ can always be chosen of the form λ(x) = xTPx + qTx with P satisfying Q + P − ATPA > 0 This matrix inequality has a solution P if and only if all unobservable eigenvalues satisfy |µ| = 1

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 25

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SLIDE 128

Storage functions for LQ problems

x+ = Ax + Bu, s(x, u) = xTQx + uTRu + bTx + dTu Lemma: For LQ problems, a storage function λ can always be chosen of the form λ(x) = xTPx + qTx with P satisfying Q + P − ATPA > 0 This matrix inequality has a solution P if and only if all unobservable eigenvalues satisfy |µ| = 1 Moreover, the solution satisfies P > 0 if and only if all unobservable eigenvalues satisfy |µ| < 1 (“(A, C) detectable”)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 25

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SLIDE 129

Strict dissipativity and pre-dissipativity

Lemma: The LQ-problem is strictly dissipative if and only if P > 0 or X is bounded

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 26

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SLIDE 130

Strict dissipativity and pre-dissipativity

Lemma: The LQ-problem is strictly dissipative if and only if P > 0 or X is bounded Proof: “⇐” is straightforward, since under these conditions λ(x) = xTPx + qTx is obviously bounded from below

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 26

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SLIDE 131

Strict dissipativity and pre-dissipativity

Lemma: The LQ-problem is strictly dissipative if and only if P > 0 or X is bounded Proof: “⇐” is straightforward, since under these conditions λ(x) = xTPx + qTx is obviously bounded from below “⇒” needs some more work in order to show that P ≥ 0,P > 0 contradicts strict dissipativity for unbounded X

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 26

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SLIDE 132

Main results

slide-133
SLIDE 133

Main result without state constraints

Without state constraints: Theorem: Consider the LQ problem with (A, B) stabilizable, Q = CTC and state and control constraint sets X = Rn and U ⊆ Rm. Then the following properties are equivalent

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 28

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SLIDE 134

Main result without state constraints

Without state constraints: Theorem: Consider the LQ problem with (A, B) stabilizable, Q = CTC and state and control constraint sets X = Rn and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly dissipative at an equilibrium (xe, ue) ∈ int (X × U)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 28

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SLIDE 135

Main result without state constraints

Without state constraints: Theorem: Consider the LQ problem with (A, B) stabilizable, Q = CTC and state and control constraint sets X = Rn and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the turnpike property at an equilibrium (xe, ue) ∈ int (X × U)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 28

slide-136
SLIDE 136

Main result without state constraints

Without state constraints: Theorem: Consider the LQ problem with (A, B) stabilizable, Q = CTC and state and control constraint sets X = Rn and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the turnpike property at an equilibrium (xe, ue) ∈ int (X × U) (iii) The pair (A, C) is detectable, i.e., all unobservable eigenvalues µ of A satisfy |µ| < 1

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 28

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SLIDE 137

Main result without state constraints

Without state constraints: Theorem: Consider the LQ problem with (A, B) stabilizable, Q = CTC and state and control constraint sets X = Rn and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the turnpike property at an equilibrium (xe, ue) ∈ int (X × U) (iii) The pair (A, C) is detectable, i.e., all unobservable eigenvalues µ of A satisfy |µ| < 1 Moreover, if one of these properties holds, then the equilibria in (i) and (ii) coincide

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 28

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SLIDE 138

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

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SLIDE 139

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly pre-dissipative at an equilibrium (xe, ue) ∈ int (X × U)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

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SLIDE 140

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly pre-dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the near equilibrium turnpike property at an equilibrium (xe, ue) ∈ int (X × U)

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

slide-141
SLIDE 141

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly pre-dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the near equilibrium turnpike property at an equilibrium (xe, ue) ∈ int (X × U) (iii) All unobservable eigenvalues µ of A satisfy |µ| = 1

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

slide-142
SLIDE 142

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly pre-dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the near equilibrium turnpike property at an equilibrium (xe, ue) ∈ int (X × U) (iii) All unobservable eigenvalues µ of A satisfy |µ| = 1 Moreover, if one of these properties holds, then the equilibria in (i) and (ii) coincide

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

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SLIDE 143

Main result with state constraints

With bounded state constraints: Theorem: Consider the LQ-problem with Q = CTC and state and control constraint sets X ⊂ Rn bounded and U ⊆ Rm. Then the following properties are equivalent (i) The problem is strictly pre-dissipative at an equilibrium (xe, ue) ∈ int (X × U) (ii) The problem has the near equilibrium turnpike property at an equilibrium (xe, ue) ∈ int (X × U) (iii) All unobservable eigenvalues µ of A satisfy |µ| = 1 Moreover, if one of these properties holds, then the equilibria in (i) and (ii) coincide. If, in addition, (A, B) is stabilizable then the turnpike property holds.

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 29

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SLIDE 144

Discussion

Obviously, the conditions in the state constrained case are significantly less restrictive

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 30

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SLIDE 145

Discussion

Obviously, the conditions in the state constrained case are significantly less restrictive: with bounded state constraints, all unobservable eigenvalues µ of A must satisfy |µ| = 1, i.e., all unobservable uncontrolled solutions must converge to 0

  • r diverge to ∞ exponentially fast

without bounded state constraints, all unobservable eigenvalues µ of A must satisfy |µ| < 1, i.e., all unobservable uncontrolled solutions must converge to 0 exponentially fast

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 30

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SLIDE 146

Discussion

Obviously, the conditions in the state constrained case are significantly less restrictive: with bounded state constraints, all unobservable eigenvalues µ of A must satisfy |µ| = 1, i.e., all unobservable uncontrolled solutions must converge to 0

  • r diverge to ∞ exponentially fast

without bounded state constraints, all unobservable eigenvalues µ of A must satisfy |µ| < 1, i.e., all unobservable uncontrolled solutions must converge to 0 exponentially fast Is there an intuitive explanation for this fact?

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 30

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SLIDE 147

Example 1 reloaded

Cost function ℓ(x, u) = u2 Dynamics x+ = 2x + u Constraints X = [−2, 2], U = [−3, 3]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 148

Example 1 reloaded

Cost function ℓ(x, u) = u2 ⇒ Q = 0, C = 0 Dynamics x+ = 2x + u Constraints X = [−2, 2], U = [−3, 3]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 149

Example 1 reloaded

Cost function ℓ(x, u) = u2 ⇒ Q = 0, C = 0 Dynamics x+ = 2x + u ⇒ µ = 2 Constraints X = [−2, 2], U = [−3, 3]

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 150

Example 1 reloaded

Cost function ℓ(x, u) = u2 ⇒ Q = 0, C = 0 Dynamics x+ = 2x + u ⇒ µ = 2 Constraints X = [−2, 2], U = [−3, 3] The dynamics have the (single) eigenvalue µ = 2, which is unobservable

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 151

Example 1 reloaded

Cost function ℓ(x, u) = u2 ⇒ Q = 0, C = 0 Dynamics x+ = 2x + u ⇒ µ = 2 Constraints X = [−2, 2], U = [−3, 3] The dynamics have the (single) eigenvalue µ = 2, which is unobservable Hence, the turnpike property holds for bounded constraints (as we have seen) but it cannot hold for X = R

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 152

Example 1 reloaded

Cost function ℓ(x, u) = u2 ⇒ Q = 0, C = 0 Dynamics x+ = 2x + u ⇒ µ = 2 Constraints X = [−2, 2], U = [−3, 3] The dynamics have the (single) eigenvalue µ = 2, which is unobservable Hence, the turnpike property holds for bounded constraints (as we have seen) but it cannot hold for X = R Indeed, in this case all optimal solutions grow exponentially, because u ≡ 0 is clearly the optimal control

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 31

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SLIDE 153

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-154
SLIDE 154

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action The bounded constraints make the unbounded solutions expensive, because we need to counteract using “expensive” control action

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-155
SLIDE 155

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action The bounded constraints make the unbounded solutions expensive, because we need to counteract using “expensive” control action This forces the optimal trajectories to the turnpike

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-156
SLIDE 156

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action The bounded constraints make the unbounded solutions expensive, because we need to counteract using “expensive” control action This forces the optimal trajectories to the turnpike Hence, state constraints help to enforce the turnpike property, which is in many regards a desirable feature

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-157
SLIDE 157

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action The bounded constraints make the unbounded solutions expensive, because we need to counteract using “expensive” control action This forces the optimal trajectories to the turnpike Hence, state constraints help to enforce the turnpike property, which is in many regards a desirable feature Outlook Analyse these relations for continuous time and infinite dimensional systems

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-158
SLIDE 158

Conclusion

General principle for bounded constraints: The solutions belonging to eigenvalues |µ| > 1 become unbounded without control action The bounded constraints make the unbounded solutions expensive, because we need to counteract using “expensive” control action This forces the optimal trajectories to the turnpike Hence, state constraints help to enforce the turnpike property, which is in many regards a desirable feature Outlook Analyse these relations for continuous time and infinite dimensional systems What happens if xe is at the boundary of X?

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 32

slide-159
SLIDE 159

References

  • L. Gr¨

une and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, submitted

  • 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

  • f economic MPC without terminal conditions, Journal of Process

Control, 24 (Special Issue on Economic MPC), 1187–1196, 2014

  • L. Gr¨

une, C.M. Kellett, S.R. Weller, On a discounted notion of strict dissipativity, Proceedings of NOLCOS 2016, IFAC-PapersOnLine 49, 247–252, 2016

  • L. Gr¨

une, M.A. M¨ uller, On the relation between strict dissipativity and turnpike properties, Systems & Control Letters, 90, 45–53, 2016

Lars Gr¨ une, The role of state constraints for turnpike behaviour and strict dissipativity, p. 33