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On Optimal Input Design in System Identification for Model - - PowerPoint PPT Presentation

Nov 30, 2022 220 likes •506 views

On Optimal Input Design in System Identification for Model Predictive Control Mariette Annergren Joint work with Christian A. Larsson and Hkan Hjalmarsson ACCESS and Automatic Control Lab KTH Royal Institute of Technology, Stockholm,

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On Optimal Input Design in System Identification for Model Predictive Control

Mariette Annergren

Joint work with Christian A. Larsson and Håkan Hjalmarsson

ACCESS and Automatic Control Lab KTH Royal Institute of Technology, Stockholm, Sweden

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Outline

  • 1. Theory
  • 2. Identification Algorithm
  • 3. MPC Example
  • 4. Conclusions
  • 5. Future Work

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Introduction

Definition

Framework for experiment design in system identification for control, specifically MPC.

  • Objective:

Find an input signal that minimizes the cost related to the system identification experiment.

  • Constraint:

A specified control performance is guaranteed when using the estimated model in the control design.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Notation

The model structure is parametrized by θ.

  • True system is given by θ0.
  • Estimated model is given by ˆ

θ.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Application Set

  • Application cost: Vapp(θ,θ0), for example

Vapp(θ,θ0) = 1 T

T

∑

t=1

yt(θ0)−yt(θ)2

2.

  • Application specification:

Vapp(θ,θ0) ≤ 1 γ , γ > 0.

  • Acceptable parameter set:

Θapp(γ) =

  • θ |Vapp(θ,θ0) ≤ 1

γ

  • .

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Application Set (cont.)

Ellipsoidal approximation: Θapp(γ) ≈ Eapp(γ) =

  • θ | (θ −θ0)TV ′′

app(θ0,θ0)(θ −θ0) ≤ 2

γ

  • .

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Application Set (cont.)

Scenario approach: Θapp(γ) ≈

  • θi, i = 1...M < ∞|Vapp(θi,θ0) ≤ 1

γ

  • .

More on scenario approach:

  • G. C. Calafiore and M. C. Campi, 2006.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

System Identification Set

Asymptotic quality property: ˆ θ ∈ ESI(α) =

  • θ | (θ −θ0)TIF(θ −θ0) ≤ χ2

α(n)

N

  • .

(Key result from prediction error/maximum likelihood system identification.)

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Optimal Input Signal Design

  • Estimated parameters:

ˆ θ ∈ ESI(α) =

  • θ | (θ −θ0)TIF(θ −θ0) ≤ χ2

α(n)

N

  • .
  • Acceptable parameters in application:

ˆ θ ∈ Θapp(γ) =

  • θ |Vapp(θ,θ0) ≤ 1

γ

  • .
  • Experiment cost:

fcost(Φu).

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Optimal Input Signal Design (cont.)

Optimization Problem

minimize

Φu

fcost(Φu) subject to ESI(α) ⊆ Θapp(γ) 0 ≤ Φu(ω), ∀ ω

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Optimal Input Signal Design (cont.)

Optimization Problem

minimize

Φu

fcost(Φu) subject to ESI(α) ⊆ Θapp(γ) 0 ≤ Φu(ω), ∀ ω

Can be approximated as a convex problem!

Using:

  • ellipsoidal approximation ⇒ LMI,
  • scenario approach ⇒ scalar linear inequalities,
  • finite dimensional parametrization ⇒ LMI.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Identification Algorithm

Issues:

  • θ0 is unknown.
  • Evaluation of Vapp(θ,θ0).

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Identification Algorithm

Issues:

  • θ0 is unknown.
  • Evaluation of Vapp(θ,θ0).

Solutions:

  • Use estimates instead of θ0.
  • Evaluate Vapp(θ,θ0) in simulation.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Identification Algorithm (cont)

Proposed algorithm:

  • 1. Find an initial estimate of θ0.
  • 2. Evaluate Vapp(θ,θ0) in simulation.
  • 3. Design the optimal input signal.
  • 4. Find a new estimate of θ0.

Discussion on iterative approach:

  • L. Gerencsér, H. Hjalmarsson, J. Mårtensson, 2009.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

MPC Example

Control objective: Reference tracking of the lower tank levels using MPC.

u1 x1 x2 x3 x4 u2 γ1 γ2

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

MPC Example (cont)

  • Application cost:

Vapp(θ,θ0) = 1 T

T

∑

t=1

yt(θ0)−yt(θ)2

2.

  • Experiment cost: Input power,

fcost(Φu) = trace 1 2π

π

−π φu(ω)dω

  • .

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

MPC Example (cont)

MPC(ˆ θ) M (ˆ θ) MPC(θ) M (ˆ θ) Vapp(θ,θ0) ≈ Vapp(θ, ˆ θ) = 1

T ∑T t=1 yt(ˆ

θ, ˆ θ)−yt(θ, ˆ θ)2

2

u u y(ˆ θ, ˆ θ) y(θ, ˆ θ) r Simulation MPC

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

  • Optimal:

91 % success.

  • White:

15 % success.

Water level [cm] Time [s] Water level [cm] Time [s] 100 150 200 250 300 100 150 200 250 300 14 15 16 14 15 16

White: 8×N gives same success rate as optimal.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Conclusions

  • Identification algorithm for MPC.
  • Increased control performance.
  • Linear framework applicable on nonlinear systems.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Future Work

  • How to choose Vapp.
  • Realistic MPC applications.
  • Closed-loop identification.
  • Toolbox for optimal input design, MOOSE.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

MOOSE, www.ee.kth.se/moose

Definition

MOOSE is a model based optimal input design toolbox developed for Matlab. It features

  • optimal input design,
  • easy-to-use text interface,
  • compatibility with Matlab Control System Toolbox.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Thank you!

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Additional slides

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Real Water Process

Optimal:

test . 20 . 40 . 60 . 80 . 100 . 120 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20 . Time(s) . Waterlevels(cm)

For details:

  • C. A. Larsson, Licentiate Thesis, 2011.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Real Water Process (cont)

White:

. . 20 . 40 . 60 . 80 . 100 . 120 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20 . Time(s) . Waterlevels(cm)

For details:

  • C. A. Larsson, Licentiate Thesis, 2011.

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

  • High

power: 0 % success.

  • Low

power: 85 % success.

Water level [cm] Time [s] Water level [cm] Time [s] 100 150 200 250 300 100 150 200 250 300 14 15 16 14 15 16

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Introduction Notation Application Set System Identification Set Optimal Input Signal Design Identification Algorithm Identification Algorithm Identification Algorithm MPC Example MPC Example MPC Example MPC Example Conclusions

Optimal Input Signal Design (cont.)

Geometric interpretation

  • Cost function:

minimize input energy ⇔ maximize ESI.

  • Region constraint:

ESI(α) ⊆ Θapp(γ) → ESI(α) ⊆ Eapp(γ). Figure: ESI (blue) and Eapp (black).

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