application of lqr techniques to the adaptive control of ... · optimal cooling of steel pro les -...

Optimal Cooling of Steel Profiles - A Model ProblemInterpretation of the Model Problem as MPC

Future Research

Application of LQR Techniques to the AdaptiveControl of Quasilinear Parabolic PDEs

Jens Saakjoint work with

Peter Benner (MiIT)

Professur Mathematik in Industrie und Technik (MiIT)Fakultat fur Mathematik

Technische Universitat Chemnitz

ICIAM July 16, 2007

1/15 [email protected] Jens Saak LQR-Adaptive Control of nonlinear parabolic PDEs

Future Research

Outline

1 Optimal Cooling of Steel Profiles - A Model Problem

2 Interpretation of the Model Problem as MPC

3 Future Research

Future Research

Physical ModelMathematical ModelLQR Control for the Linear SystemLQR Adaptive Control for the Nonlinear Model

Optimal Cooling of Steel Profiles - A Model Problem

1 Optimal Cooling of Steel Profiles - A Model ProblemPhysical ModelMathematical ModelLQR Control for the Linear SystemLQR Adaptive Control for the Nonlinear Model

2 Interpretation of the Model Problem as MPC

3 Future Research

Future Research

Optimal Cooling of Steel Profiles - A Model ProblemPhysical Model

Physical Model: Cooling of steel profiles in arolling mill.Υ := Ω× (0,T ) and Σi := Γi × (0,T ).

c(x)ρ(x) ∂∂t

x(ξ, t) = ∇.(λ(x)∇x(ξ, t)) in Υ,−λ(x)∂νx(ξ, t) = gi (x , u, ξ, t) on Σi ,

x(ξ, 0) = x0(ξ) in Ω,(heat eq.)

state x temperature

c(x) specific heat capacity, ρ(x) density, λ(x)heat conductivity

T ∈ R ∪ ∞ final time. We assume T =∞here for simplicity.

Source: Physical model: courtesy of Mannesmann/Demag.

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Optimal Cooling of Steel Profiles - A Model ProblemMathematical Model

Mathematical model: boundary control forlinearized 2D heat equation.

c · ρ ∂∂t

x = λ∆x , ξ ∈ Ωλ ∂∂n

x = κ(uk − x), ξ ∈ Γk , 1 ≤ k ≤ 7,∂∂n

x = 0, ξ ∈ Γ7.(lin. heat eq.)

FEM discretization, different models forinitial mesh (n = 371),1, 2, 3, 4 steps of global mesh refinement⇒ n = 1357, 5177, 20209, 79841.

Source: Math. model: [Troltzsch/Unger 1999/2001, Penzl 1999, S. 2003]

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Control problem

control heat distribution in Ω

other examples: heating/cooling processes, air conditioning

Main interests here:

use state feedback control, i.e.:

u(t) := F(ξ, t)x(ξ, t)

extend the closed loop Riccati approach to nonlinear systems

Future Research

Control problem

control heat distribution in Ω

other examples: heating/cooling processes, air conditioning

Main interests here:

use state feedback control, i.e.:

u(t) := F(ξ, t)x(ξ, t)

extend the closed loop Riccati approach to nonlinear systems

Future Research

Optimal Cooling of Steel Profiles - A Model ProblemLQR Control for the Linear System

abstract Cauchy problem

x(t) = Ax(t) + Bu(t) x(0) = x0 ∈ X .(Cauchy)

output equation

y(t) = Cx(t)(output)

cost function

J (u) =1

∞∫0

< Qy, y > + < Ru,u > dt (cost)

and the linear quadratic regulator problem is

LQR problem

Minimize the quadratic (cost) with respect to the linear constraints(Cauchy),(output).

Future Research

Optimal Cooling of Steel Profiles - A Model ProblemLQR Control for the Linear System

abstract Cauchy problem

x(t) = Ax(t) + Bu(t) x(0) = x0 ∈ X .(Cauchy)

output equation

y(t) = Cx(t)(output)

cost function

J (u) =1

∞∫0

< Qx, x > + < Ru,u > dt (cost)

and the linear quadratic regulator problem is

LQR problem

Minimize the quadratic (cost) with respect to the linear constraints(Cauchy),(output).

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Optimal Cooling of Steel Profiles - A Model ProblemLQR Control for the Linear System (Losung)

Well understood in the open literature:Analogously to the finite dimensional case for T=∞ we find the

optimal feedback control

u = −R−1B∗X∞x.

where X∞ is the minimal, positive semidefinite, selfadjoint solution of the

algebraic operator Riccati equation

0 = R(X) := Q + A∗X + XA− XBR−1B∗X. (ARE)

e.g. [Lions ‘71; Lasiecka/Triggiani ‘00; Bensoussan et al. ‘92;Pritchard/Salamon ‘87; Curtain/Zwart ’95]

Future Research

Optimal Cooling of Steel Profiles - A Model ProblemLQR Control for the Linear System (Losung)

Well understood in the open literature:Analogously to the finite dimensional case for T<∞ we find the

optimal feedback control

u = −R−1B∗X∞(t)x.

where X∞ is the minimal, positive semidefinite, selfadjoint solution of the

differential operator Riccati equation

X = −R(X) := −Q− A∗X− XA + XBR−1B∗X. (DRE)

See talk by H. Mena on the solution of the DREs for details

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Optimal Cooling of Steel Profiles - A Model ProblemLQR Adaptive Control for the Nonlinear Model

Question:

How can we extend this approach to nonlinear systems?

1 Linearize the system

2 Apply the theory above to compute the control

3 Update the nonlinearity/-ies as the state changes.

4 Restart at 1

Here: Linearization → Freeze the coefficients/material parameters oncertain time intervals

[Troltzsch/Unger 1999/2001] applied this idea successfully in the open loop case.

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Question:

4 Restart at 1

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Question:

4 Restart at 1

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Question:

4 Restart at 1

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Question:

4 Restart at 1

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Question:

4 Restart at 1

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Properties and Ingredients of General MPC SchemesBasic Idea of MPCIdentification of Main IngredientsIdentification of Time Intervals

Interpretation of the Model Problem as MPC

1 Optimal Cooling of Steel Profiles - A Model Problem

2 Interpretation of the Model Problem as MPCProperties and Ingredients of General MPC SchemesBasic Idea of MPCIdentification of Main IngredientsIdentification of Time Intervals

3 Future Research

Future Research

Interpretation of the Model Problem as MPCProperties and Ingredients of General MPC Schemes

Properties:

Model Predictive Control (MPC): class of methods rather than a controltechnique

great acceptance in industrial applications

yields high performance control systems capable of running without expertintervention

applicable to nonlinear systems

Ingredients

MPC consists of 3 major parts:

1 prediction model

2 cost function

3 way to compute the control

e.g. [Garcia/Prett/Morari ’89; Camacho/Bordons ’04; Chen/Allgower ’97/’98]

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Interpretation of the Model Problem as MPCBasic Idea of MPC

Future Research

Interpretation of the Model Problem as MPCIdentification of Main Ingredients

Following the idea of [Garcia/Prett/Morari ’89] nonlinear MPC →linearized optimal control.

Prediction Model

nonlinear heat equation (heat eq.)

Cost Function

obviously given by the (cost)

Way to compute the control

Riccati operator based feedback control for the linearized model

(→ example: [Garcia ’84] batch reactor application produced excellent results)

Future Research

Prediction Model

Cost Function

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Prediction Model

Cost Function

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Prediction Model

Cost Function

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Interpretation of the Model Problem as MPCIdentification of Time Intervals

TC =∞.

TP = δ = time stepsize on the discrete numerical level

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Compare with the DRE case (T <∞) which gives more flexibility inthe MPC horizon choices (i.e. TC ≤ T − t and TP = δ possiblyequal to TC .) (with H.Mena (EPN Quito))

Stabilization proof for Steel cooling from MPC proofs

Sub-optimality error bounds

Step size control for time discretization schemes

Interpretation as instantaneous control?

Thank you for your attention!

Future Research

Compare with the DRE case (T <∞) which gives more flexibility inthe MPC horizon choices (i.e. TC ≤ T − t and TP = δ possiblyequal to TC .) (with H.Mena (EPN Quito))

Stabilization proof for Steel cooling from MPC proofs

Sub-optimality error bounds

Step size control for time discretization schemes

Interpretation as instantaneous control?

Thank you for your attention!15/15 [email protected] Jens Saak LQR-Adaptive Control of nonlinear parabolic PDEs

application of lqr techniques to the adaptive control of ... · optimal cooling of steel pro les -...

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