Embedded predictive control for fast mechatronic systems · 2 Application study oward embedded MHE...

Embedded predictive control for fast mechatronic

systems

Mohammad Abdollahpouri

Slovak University of Technology in Bratislava

[email protected]

Supervisor and co-supervisor:Prof. Boris Rohal-Ilkiv and Prof. Tor-arne Johansen

November 28, 2015

Mohammad Abdollahpouri (STUBA) MTR Presentation November 28, 2015 1 / 17

Overview

1 Introduction

2 Application studyToward embedded MHE and NMPCMHE vs. EKF for a vibrating cantilever beamNonlinear vibration systemPIL simulation for vibration energy harvesting

3 Theoretical studyTransforming non-convex problem to convexMulti-stage global estimation

4 Future career


Introduction

Motivation

Title: Embedded MPC for Control of Fast Mechatronic Systems

Real-time implementation using embedded platformsSolving non-convex problemsImplement the method for systems with fast dynamics


Introduction

Embedded systems

Micro-processor/controller

Low cost design

Reliability

InaccessibleSafety reasonsCosts a lot

Small size

Low energy consumption

Design for speci�c tasks

Real-time constraint

Possibility to become mobile

SoC with peripherals �DAC,ADC,USB,I2C,SPI,. . .


Application study Toward embedded MHE and NMPC

NMPC+MHE for Pendubot system

The system dynamics is nonlinear with fast dynamics

So-called RTI scheme was employed

minxp.q,up.q

» T0

�}xptq � xref ptq}

2

Q � }uptq � uref ptq}

2

R

dt

� }xpT q � xref pT q}2

P

s.t. 9xptq � fpxptq, uptq, b̂q,

xptq P X, xpT q P XT ,uptq P U, xp0q � x̂0.

where b̂ is the viscous friction coe�cients estimates

l1

x

y

0

g

lc1

l 2

l c2

q1

q2

τ1

m1,I1

m2,I2

active jointpassive jointcenter of m

The NMPC+MHE approach has been veri�ed in real time usingACADO Code Generation Tool (presented at 5th IFAC NMPC1)

1�Real-Time MHE-Based Nonlinear MPC of a Pendubot System�, Gulan, M.,Salaj, M., Abdollahpouri, M. and Roha©-Ilkiv, B., In Proceedings of the 5th IFACNonlinear Model Predictive Control Conference, Seville, Spain,2015.


Application study Toward embedded MHE and NMPC


Application study MHE vs. EKF for a vibrating cantilever beam

Cantilever vibrating beam setup

For small de�ection, dynamics are considered linear

:qptq � 2ζωn 9qptq � ω2nqptq � αuptq,

Parameters are assumed constant.The joint parameter and state estimation deals with nonlinear dynamics.



Joint state and parameter MHE 2

Real-time implementation of MHE on Raspberry pi model B

Comparison study with standard EKF

Fault/Noise tolerant investigation of estimatorsEmulated structural change by adding weightExecution time analysis

xPC Cantilever beam

PCF8591RP

Measure

ment

PRBS

AI

DO

AI

DI

Fault

em

ula

tion

2Abdollahpouri, M., Takács, G., Roha©-Ilkiv, B. (2015). Real-time movinghorizon parameter estimation for a vibrating active cantilever. Submitted toMechatronics.



Fault analysis on parameter estimation0 10 20 30 40 50

−3

−2

−1

0

1

2

3x 10

−3

Posi

tion (

m)

Time (sec)

0 1 2 3 4 50

10

20

30

40

50

60

70

Input

(V)

Time (sec)

0 10 20 30 40 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Vel

oci

ty (

m/s

ec)

Calculated

MHE

EKF

0 10 20 30 40 500

20

40

60

80

100

120

140

❫✦♥

✭r❛❞✴s❡❝✮

Time (sec)

0 10 20 30 40 500

1

2

3

4

5

6

Time (sec)

❫ ✾✭✲✮

0 10 20 30 40 50−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (sec)

❫ ✱

✭◆

✴❱

❦❣

✮

MHE

EKF

Measured

MHE

Measured

MHE

EKF

MHE

EKF

MHE

EKF

Figure: E�ects of sensor faults



Noise analysis on parameter estimation

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

❫✦♥

✭r❛❞✴s❡❝✮

Time (sec)

0 5 10 15 20 25 30 35 40 45 500.001

1

2

3

❫ ✾✭✲✮

Time (sec)

0 5 10 15 20 25 30 35 40 45 50−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

❫ ✱

✭◆

✴❱

❦❣

✮

Time (sec)

MHE with Gaussian noise

EKF with Gaussian noise

MHE with uniform noise

EKF with uniform noise

Figure: Parameter estimation with additive measurement noise


Application study Nonlinear vibration system

Why nonlinear vibration system?

Needs to be implemented on embedded platform

Double applications � vibration control and energy harvestingState estimation and Nonlinear MPC

Nonlinear vibration attenuationVibration energy harvesting


Application study Nonlinear vibration system

Vibration energy harvesting

Mechanical strain energy Ñ electrical charge

Power is on the order of milliwatts

Battery-less wireless sensors

System dynamics represented by

rC1ppq � C2ppqq2s:q � C2ppqq 9q � rC3ppq � C4q

2sq�

C5v � C6q2v � C7ppq :u,

C8ppq 9v � C9ppqv � C5 9q � C6q29q � 0,

where p is time varying parameter, v is voltageacross the piezoelectric layer and q is displace-ment of the free end.

Chaotic behavior (Du�ng oscillator)

Find the varying parameters to maximize theharvested energy


Application study PIL simulation for vibration energy harvesting

Toward nonlinear vibration system 3

Processor in the loop simulationChaotic behaviorTime-varying parameter need to be estimated

TCP/IP

PCF8591

Raspberry Pi

target PChost PC

Simulink Real-Time REAL-TIME SIMULATION

EMBEDDED COMPUTING PLATFORM

AI/AO

DI/

DO

Figure: Estimated and true phase plane

3�Towards the Implementation of Embedded MHE for a Nonlinear VibrationEnergy Harvesting System�,Abdollahpouri, M. and et. al., submitted to ECC16.


Theoretical study Transforming non-convex problem to convex

A convexity-based homotopy for NMPC 4

Original control-a�ne process model for t P r0, T s

OCP : minxp.q,up.q

» T0

}xptq � xref ptq}2

Qdt� }xpT q � xref pT q}

2

QT:� }x� xref }

2

J

s.t. 9xptq � fpxptqq � gpxptqquptq,

xptq P X, xpT q P XT ,uptq P U, xp0q � x0.

A homotopy formulation with a pseudo state xc and a parameterλ P p0, 1q

P pλq : minxp.q,up.q,xcp.q

1

λ}x� xref }

2

J �1

1� λ}xc � x}

2J

s.t. 9xcptq � fpxptqq � gpxptqquptq,

xcptq P X, xcpT q P XT ,uptq P U, xcp0q � x0.

4Bonilla, J., Diehl, M., Logist, F., De Moor, B., and Van Impe, J. F. (2010). Aconvexity-based homotopy method for nonlinear optimization in model predictive control.Optimal Control Applications and Methods, 31(5), 393-414.


Theoretical study Transforming non-convex problem to convex

Cont.

λÑ 1, P pλq Ñ OCP and λÑ 0, P pλq Ñ CVX

CVXpλq : minup.q,xcp.q

}xc � xref }

2

J

s.t. 9xcptq � fpxref ptqq � gpxref ptqquptq,

xcptq P X, xcpT q P XT ,uptq P U, xcp0q � x0.

Extension of the method to PEP

PEP : minxp.q,p

}x� xm}2J

s.t. 9xptq � fpxptq, umptqq � gpxptq, umptqqp,

xptq P X,p P P.


Theoretical study Multi-stage global estimation

Redundant measurement can leads to LTV

Algebraic nonlinear transformation Ñ LTV model

Example: Let the nonlinear dynamics be 9x � v, 9v � �v2 � u and letd � v2 Ñ 9x � v, 9v � �d� u, 9d � �2 9vv � �2av

1st estimation is based on LTV model Ñ sub-optimal estimate2nd estimation using linearized model Ñ more accurate estimate


Future career

Impact of TEMPO on future

What we have learned/will learn...

Summer/Spring schoolsInternational conferences � Industrial sessionAutomotive Embedded Control Workshop in RenaultProfessional development and complementary skills workshop

International cooperation and building relationsBeing introduced to companiesDoes TEMPO have credit worldwide?

�give a man a �sh and you feed him for a day;

teach a man to �sh and you feed him for a lifetime�


IntroductionApplication studyToward embedded MHE and NMPCMHE vs. EKF for a vibrating cantilever beamNonlinear vibration systemPIL simulation for vibration energy harvesting

Theoretical studyTransforming non-convex problem to convexMulti-stage global estimation

Future career

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Embedded predictive control for fast mechatronic systems · 2 Application study oward embedded MHE...

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