Efficient Numerical Methods for Moving Horizon...

Katholieke Universiteit Leuven

Efficient Numerical Methods for Moving Horizon Estimation

Niels Haverbeke

Doctoral presentation – public defense

Promotor

Prof Dr. Ir. Bart De Moor

Co-promotor

Prof Dr. Moritz Diehl

Chairman

Prof. Dr. Ir. Yves Willems

Jury

Prof. Dr. Ir. Johan Suykens Prof. Dr. Ir. Jan Willems Prof. Dr. Ir. Wim Michiels Prof. Dr. Michel Kinnaert (U.L.B.) Prof. Dr. Michel Verhaegen (T.U.Delft) Prof. Dr. Ir. Lieven Vandenberghe (U.C.L.A.)

Niels Haverbeke – Public defense, 16th September 2011

Overview

Motivation Closed loop control scheme

Principle and situation of MHE

Driven by applications

2

1

3

Optimization problem and KKT system

Huber penalty MHE

4

7

Joint estimation of piecewise changing inputs 8 Convex and nonlinear MHE

Structure exploiting

MHE algorithms

Nonlinear MHE and MPC 9

Interior point methods 5

Active-set methods 6

CONCLUSIONS


MOTIVATION

MOTIVATION


Driven by applications

Recursive techniques, e.g. Kalman filter

Advanced dynamic optimization techniques, e.g. parameter estimation, MPC

Applied to fast systems

Applied to slow systems

Dynamic optimization for fast real-time systems

MOTIVATION


States

Angle of attack Pitch rate

Pitch angle Roll rate

Roll angle

Controls

Elevators Ailerons Rudders

Outputs

Pitch angle Path angle

Example

MOTIVATION


Past Future

Estimate the states and model from past data

States



Roll angle

Controls


Outputs


Example

MOTIVATION


Past Future

Estimate the states and model from past data

States



Roll angle

Controls


Outputs


Example

MOTIVATION


t

t

t

FILTERING:

SMOOTHING:

PREDICTION:

= span of available measurements

TIME

Filtering, smoothing and prediction

Recursive estimation Batch estimation

Window of one time step

Typically online state estimation

Large window

Typically offline optimization

Parameter fitting Kalman filter and extensions

MOTIVATION


MHE principle

Dynamic model

Constraints

Subject to

MOTIVATION


The role of constraints

What can go wrong?

nonlinear model may give rise to multiple optima

Contours of (rescaled) true conditional probability density

EKF tries to fit

MHE retains dominant characteristics: multiple optima

* Source: Haseltine and Rawlings, 2004‏

MOTIVATION


Waste water treatment process

Fifth order system

-0,5

0

0,5

1

1,5

2

2,5

3

3,5

equal. Tank Tank 1 Tank 2 Tank 3 Waste

MHE

Kalman

True losses

Mean losses Mean squared errors

0

1

2

3

4

5

6

equal. Tank

Tank 1 Tank 2 Tank 3 Waste

MOTIVATION


The closed loop control scheme

Controller

Process

Sensor

Estimator

MOTIVATION


The closed loop control scheme

MOTIVATION

MHE MPC

Free initial state

Positive semidefinite Hessian

Changing arrival cost

Control dimension ≈ state dimension

Few active constraints


STRUCTURE EXPLOITING MHE ALGORITHMS



The MHE optimization problem

Linear MHE: a quadratic (sub)problem

Writing down the optimality conditions (KKT system), and

Ordering the block rows,

… yields a highly structured linear system of equations

which can be solved with Riccati and vector recursions



A highly structured KKT system

1. A priori information

Every time step represents one block in the KKT matrix

Information is translated in three steps

2. Model forwarding

3. Measurement updating



Decomposing the KKT system

= .

M L U

LU decomposition yields the normal Riccati recursion

FORWARD VECTOR SOLVE

RICCATI MATRIX RECURSION

BACKWARD VECTOR SOLVE

Updated initial condition

Final state covariance


Niels Haverbeke – Preliminary defense, 1 July 2011

Decomposing the KKT system

LDLT decomposition yields the square-root Riccati recursion

=

M L D

. .

LT

Measurement update and time forwarding via Q-less QR factorizations

Fully exploits symmetry

Yields increased numerical stability



Riccati based MHE

Computation times for 5th order systems

0 20 40 60 80

Time step

Growing horizon Moving horizon

0 20 40 60 80

Time step

Square-root Riccati

Normal Riccati

x 10-4

0

2

4

6

8

2

4

6

8

10

10

x 10-5

N=10

N=5

N=2

N=1



Riccati based MHE

Accuracy

Normal

Square-root

10-4

10-2

10-1

10-3

100

2P PError

n=m=p=2 n=m=p=5 n=m=p=10



Structured QR factorization

10-16

10-14

10-13

Average error 2

QR A

10-15

0 10 20 30 40 50 10-6

Dimension n

10-5

10-4

10-3

0 10 20 30 40 50

Dimension n

A =

Computation time (sec)

Givens

Householder

Structured Givens

Structured Householder


10-2


Interior-point MHE

Primal barrier method

Newton method

With



Interior-point MHE

With and

Modified Riccati recursion

100

102

104

Condition number (Mk)

Condition number (Dk) 101

103

100

102

104

101

103

2

iteration

4 6 8 10 12 14 2

iteration

4 6 8 10 12 14

1 / √κ

Normal update

Square-root update



Interior-point MHE

Computation times

Finite number of iterations with decreasing

Example - second order system

10 iterations yield satisfying results in this case

True state

MHE 10 iterations

MHE converged



Interior-point MHE

Hot starting



Interior-point MHE

Hot starting

A good initialization is necessary for fast convergence

Hot starting with the previous solution or the proposed strategy

Yields convergence improvement for first iterations



A Schur-complement active-set method

Method outline

Compute unconstrained MHE solution

Add to working set

Check constraints

Solve QP in projected space

Fast Riccati based computation

Violated constraints

Searching for such that

Define

Update

Non-negativity constrained QP

Remove inactive constraints and terminate




Gradient projection method for non-negativity constrained QP

x1

x2

1. Cauchy calculation step

2. Projected Newton step

Without projected Newton step

With projected Newton step

Convergence


Principle



1. Use semidefinite Cholesky factorization of M

2. Set

3. Keep adding non-positive constraints to working set

4. Delete rows and colums of (new) working set constraints

5. Continue until all components non-negative

Gradient projection method for non-negativity constrained QP

Projected Newton step

Between outer active-set iterations: Cholesky block downdating (constraints added)

Upon termination: Cholesky block updating (constraints removed)




Convergence

Time step

0

10

20

30

40

0 100 200 300 400 500

1

2

3

4

# final working set constraints

# active constraints

# active-set iterations


Computational burden

uMHE asetMHE Total

Riccati 1 1

Fsolve 1 1

Partial Fsolve

nA nA

Bsolve 1 nit 1+nit

Red. QP

nit

nit



0 100 200 300 400 500

Time step

0

0.5

1

1.5

x 10-3

Interior-point method 10 iterations

Schur-complement active-set method

Computation time



CONVEX AND NONLINEAR MHE



Huber penalty MHE

-M M 0

The Huber penalty

Preserves LS performance

Increases robustness to outliers

Univariate QP

Multivariate SOCP



Huber penalty MHE

Computation

time (sec)

Componentwise univariate Huber

Multivariate Huber

Univariate vs multivariate


n x 2 variables

1 x n+1 variables


Huber penalty MHE

Output

disturbance

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

L2 L1 Huber Huber + L2

Simulation

Prediction


Batch estimation

Mean squared error

(50 repetitions)

500 time steps

50 time steps

2,5 %

contaminated

with N(0,5)


Joint estimation with piecewise inputs


F16 example – linearized longitudinal model

4 states: velocity, angle-of-attack, pitch angle, pitch rate

2 outputs: pitch angle, flight path angle

1 input: elevator deflection

Input: Elevator deflection (deg) State and output - Pitch angle (rad)

Time step Time step


Joint estimation with piecewise inputs

Joint MHE: quality of input estimates

L1 MHE

Cardinality MHE with polishing step


L2 MHE

Low output disturbance level R=0.001 High output disturbance level R=0.01


Nonlinear MHE


Nonlinear MHE =

Structured NLP

Structured QP with inequality constraints

SQP Gauss Newton with globalization

Active-set or interior-point method

Linear algebra

Structured equality constrained QP

Structured KKT system


Nonlinear MHE

Estimation and control of glycemia in critically-ill patients

Controlled variable: glycemia level (G)

Known input: carbohydrate calories flow (FG)

Unkown input: medication (FM)

Manipulated variable: exogenous insulin (FI)

Regulate glycemia to normoglycemic range (80-110 mg/dl)



Nonlinear MHE

Estimation and control of glycemia in critically-ill patients



CONCLUSIONS

CONCLUSIONS


KKT conditions reveal symmetry and structure

Block diagonal structure is preserved in interior-point methods

Decomposition yields Riccati methods

Proposed and demonstrated square-root Riccati method using QR factorizations

Proposed and demonstrated modified square-root Riccati method

Block diagonal structure is NOT preserved in active-set methods

Proposed and demonstrated a dedicated Schur-complement active-set method

Huber penalty increases robustness to outliers

Demonstrated Huber penalty MHE

Joint input estimation with piecewise inputs has finite number of break points

Proposed and demonstrated cardinality MHE yielding a sequence of L1-type MHE

Nonlinear MHE can be solved by SQP Gauss-Newton method

Demonstrated NMHE on a biomedical application

Conclusions

CONCLUSIONS


Future research

Applications

Automotive

Power electronics

Algorithms

Ultra-fast nonlinear MHE: fast simulation

Distributed MHE

Adaptive control: interaction between MHE and MPC

CONCLUSIONS

Intensive Care Unit

Niels Haverbeke – Preliminary defense, 1 July 2011

Battery

Auxiliaries

DC/DC converter

Charger

Infrastructure

DC/DC converter

Inv

MG

Inv

MG

ECU

Torque vectoring control

Look ahead active suspension

Battery parameter estimation and battery management

Power converter control and fast charging power control

Future research

CONCLUSIONS


Promotor

Acknowledgements

Co-promotor

Bart De Moor

Moritz Diehl

Co-authors

Tom Van Herpe, Greet Van den Berghe, Toni Barjas-Blanco, Steven Gillijns, Bert Pluymers, Marcello Espinoza, Joachim Ferreau, Patrick Willems, Jean Berlamont, Po-Kuan Chiang

Financial support

IWT, FWO-Vlaanderen

CONCLUSIONS

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