Two-Stage Adaptive Feed-forward

Predictive Control: with Applications to Algal Growth Systemswith Applications to Algal Growth Systems

Michael Buehner

Ph.D. Preliminary Exam

June 26th, 2008

11:00 am

Biofuels From Microalgae

Use CO2, sun energy, nutrients to produce microalgae biomass

Create storage lipids for biofuel Create storage lipids for biofuel production via “stressing”

DOE’s Aquadic Species Program

NREL (1978-1996)

Solix Biofuels

Issues with Current TechnologyWhat are the desired operating conditions?

What are the theoretical limits and how What are the theoretical limits and how do we achieve them?

What is a cost effective solution?

What microalgae strain should be used?

What method should be used for growing and stressing the microalgae?

Topics

Adaptive Feed-forward (FF) Predictive Control

Algal Growth System (AGS) ModelingAlgal Growth System (AGS) Modeling

Understand and improve process

Address control objectives

AGS Control

Reduce cost

Improve performance

Adaptive FF Predictive Control

Motivation

Architecture

LTI MethodsLTI Methods

Examples

Future Work

Neuralmuscular Actuation Systems

Calculate desired path (FF calculation)

Balistic response (FF Control)Balistic response (FF Control)

Dynamic corrections to ballistic response (FB control on small error signals)

Architecture

G = GnoiGi

Desired Closed Loop: GnoiPdes

FF 1: DC Gain = 1

FF 1 and FF 2: Open-loop Stable

LTI Method 1

GK

GKT

+=

142

GK

GT

+=

143 desidesnoi PGTPGTT

1

434241

−+=

LTI Method 1 Cont.

desidesnoi PGTPGTT +=−1

434241

GK

GKT

+=

142

GK

GT

+=

143

desnoi

desnoi

desnoidesnoi

desiinoi

desnoi

desidesnoi

PG

PGGK

GK

PGGK

PGGK

GK

PGGK

GGPG

GK

GK

=

+

+=

++

+=

++

+=

−

1

1

1

1

1

1

)(

1

1

434241

LTI Method 2 (Smith Predictor)

desides PGTPTT1

434241

−+=

KG

GKT

i+=

142

KG

GT

i+=

143

LTI Method 2 Cont.

desides

GGKGG

PGTPTT +=−

)(

1

434241

KG

GKT

i+=

142

KG

GT

i+=

143

desnoi

desnoi

i

i

desnoi

i

desnoi

i

i

desi

i

inoides

i

inoi

PG

PGKG

KG

PGKG

PGKG

KG

PGKG

GGP

KG

KGG

=

+

+=

++

+=

++

+=

−

1

1

1

1

1

11

)( 1

LTI Example Using Standard FB

dse

s

ssG

τ−

+

+=

1

5)(

PI Controller

Kp = 0.1111

Ki = 0.1005

Disturbance

d(t) = 0.1 u(t-20)

Plant

1=τ

e(t) u(t)

r(t)

y(t)

plant output

disturbance

d(t)

Step

s+5

s+1

Plant TF (G_i) Plant Delay (G_noi)

FB error FB control

PID

FB Controller

Add1

1=dτ

LTI Example with FB Only Plots

0.5

1

1.5

Inputs

/ O

utp

ut

Stable Minimum-Phase FOPTD System with τd = 1

r(t)

y(t)

d(t)

0 5 10 15 20 25 30 35 400

time (sec)

0 5 10 15 20 25 30 35 40-0.5

0

0.5

1

time (sec)

FB

err

or

/ C

ontr

olle

r O

utp

ut e(t)

ufb

(t)

LTI Example Using Method 1

ds

noi esGτ−

=)(

1

5)(

+

+=

s

ssGi

dse

s

ssG

τ−

+

+=

1

5)(

Plant

1=dτ

Feed-forward Terms

1

1)(

+=

ssPdes

τ

0≥τ

r_ff(t)

r_ti lde_ff(t)

e(t) u_fb(t) u(t)

r(t)

y(t)

plant output

disturbance

Step

s+5

s+1

Plant TF (G_i) Plant Delay (G_noi)

FF+FB control

s+1

s+5

FF 2 (G_i^{-1})

FF 2

1

tau.s+1

FF 1 TF (P_des)

FF 1 (G_noi)

FF 1

FB error FB control

PID

FB Controller

Disturbance

Add1

Add

Method 1 Plots

0.5

1

1.5

Inputs

/ O

utp

ut

Stable Minimum-Phase FOPTD System with τ = 0.5 and τd = 1

r(t)

rtildef f

(t)

y(t)

0 5 10 15 20 25 300

time (sec)

Inputs

/ O

utp

ut

y(t)

d(t)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

time (sec)

FB

err

or

/ C

ontr

olle

r O

utp

ut e(t)

ufb

(t)

u(t)

LTI Example Using Method 2

ds

noi esGτ−

=)(

1

5)(

+

+=

s

ssGi

dse

s

ssG

τ−

+

+=

1

5)(

Plant

1=dτ

Feed-forward Terms

1

1)(

+=

ssPdes

τ

0≥τ

r_ff(t)r(t) s+11 r_ff(t)

r_ti lde_ff(t)

e(t) u_fb(t) u(t)

r(t)

y(t) plant output

disturbance

Step

Plant Delay (G_noi)

u(t) y (t)

Plant

s+5

s+1

G_i

u(t) y (t)

G

FF+FB control

s+1

s+5

FF 2 (G_i^{-1})

FF 2

1

tau.s+1

FF 1 TF (P_des)

FF 1

FB error FB control

PID

FB Controller

Disturbance

Add4

Add3

Add2

Add

Method 2 Plots

0.5

1

1.5

Inputs

/ O

utp

ut

Stable Minimum-Phase FOPTD System with τ = 0.3 and τd = 1

r(t)

rtildef f

(t)

y(t)

d(t)

0 5 10 15 20 25 300

time (sec)

d(t)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

time (sec)

FB

err

or

/ C

ontr

olle

r O

utp

ut e(t)

ufb

(t)

u(t)

Method 2 Plots (Kp = 2, Ki =1)

0.5

1

1.5

Inputs

/ O

utp

ut

Stable Minimum-Phase FOPTD System with τ = 0.3 and τd = 1

r(t)

rtildef f

(t)

y(t)

0 5 10 15 20 25 300

time (sec)

Inputs

/ O

utp

ut

d(t)

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

time (sec)

FB

err

or

/ C

ontr

olle

r O

utp

ut e(t)

ufb

(t)

u(t)

Future Work – LTI Methods

Optimized designs

Closed-loop GnoiPdes

Feedback controller KFeedback controller K

Robustness analysis and design

Smith predictor

Model uncertainty

Future Work – NLTV Methods

Nonlinear Time-varying (NLTV) Methods

Nonlinear models

Reinforcement learningReinforcement learning

Echo state networks

Gi-1 : well studied

Pdes : research required

AGS Modeling

Motivation

Types of Models

Prior WorkPrior Work

Overall Model

Future Work

Motivation

Understand Process

Feedback Controller SynthesisFeedback Controller Synthesis

FF Control

Prior Work

Reactors

Open Ponds

Airlift ReactorsAirlift Reactors

Flat Panel Reactors

Modeling Techniques

Curve Fits

1st and 2nd order LTI Systems

Monod Kinetics

Types of Models

AGS Testbed at Solix Biofuels

Testbed AGS with 3 PBRs

Each PBR contains 2 panels

Commanded Inputs

media

nutrientsmedia

bubble

Water Chemistry

Subsystem

PBR Volume (L)

PBR Surface Area (m^2)

O2diss

initV_L

Inital Volume (L)

Harvest Rate

intialVolme

mediaRate

harv estRate

v olume

surf aceArea

Dimension

CO2diss

CO2 MFC

Air MFC

Overall Model

Sensor Measurements

Geography

Date and Time

pH

long_deg

longtude

long_std

longitude standard

l ipidFraction

lat_deg

latitude

dayl ight_sav

daylight savings

cellmass

CO2diss

O2diss

CO2vent

O2vent

pH

bubble

pump

CO2in

CO2rate

O2rate

growthRate

sensor_OD

Turbidity Sensor

Temperature

Pump CO2rate

O2rate

growthRate

lipidFraction

harvest

cellDensity

opticalDensity

nutrients

CO2

O2

pH

temp

clight

harvestRate

darkRespiration

ODsensor

Photosynthesis

Subsystem

sensor_PAR

PAR Sensor

OD

O2vent

Air MFC

date_time

geography

sun intesity

cell density

usable PAR

Light Subsystem

Harvested Algae

Growth Rate

date_time_num

Date Number

R

Dark Respiration Rate

CO2vent

Air MFC

Light Subsystem

Physical / Algebraic Model

Photosynthetically Active Radiation (PAR)

43% of incident light energy43% of incident light energy

Number of Available PAR photons

PARalgae = f1(PARsensor, sun position, mixing)

Growth Model Overview

Microalgal growth is function of incident light photons

Exponential for sparse culturesExponential for sparse cultures

Linear for more dense cultures

Microalgae will respirate in the dark (i.e., loss of biomass)

Harvesting

Growth Model Overview Cont’d

Measures of Growth

Biomass produced

CO consumedCO2 consumed

O2 produced

Microalgae are 50% Carbon

1.83 g CO2 / g Microalgae

8 Moles Light / Mole O2 Produced

Growth Model

)geometry,mixing,(

),min(

algaedense

densealgaealgae

algaealgaePARPARalgae

mfm

mmm

uRmmIKm D

=

=

−−=&

malgae < mdense - exponential growth (KPARIPAR – R)

malgae ≥ mdense - linear growth, exponetial Decay

)geometry,mixing,( algaedense mfm =

algaeOO

algaeCOCO

22

22

mKm

mKm

&&

&&

=

=

Growth Phases

time

ma

ss

Exponential

Growth

Linear

Growth

Stationary

Growth Model Sim Without Saturation

0.5

1

1.5

2

2.5

3

Norm

aliz

e M

icro

alg

ae M

ass

Actual Mass

0 10 20 30 40 50 60 70 80 900

0.5

hours

Norm

aliz

e M

icro

alg

ae M

ass

Actual Mass

Modeled Mass without Saturation

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

hours

PA

R (

mol/m

2/h

)

Growth Model Sim With Saturation

0.5

1

1.5

2

Norm

aliz

e M

icro

alg

ae M

ass

Actual Mass

0 10 20 30 40 50 60 70 80 900

hours

Norm

aliz

e M

icro

alg

ae M

ass

Actual Mass

Modeled Mass with Saturation

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

hours

PA

R (

mol/m

2/h

)

Water Chemistry Model

Dynamic physical and empirical model

Mass balances

Input CO2 and O2Input CO2 and O2Dissolve CO2 and O2Consumed CO2 and O2Vented CO2 and O2Nutrients input

Nutrients consumed

Future Work

Define other interactions through experimentation

MixingMixing

Nutrient availability

Use biological and physical based models when possible

Lipid production model

AGS Control

Completed Work

FF + FB pH control

Observer-based FF controllerObserver-based FF controller

Example with FB only control

Ongoing Research

Two-Stage FF Control

Lipid Production (w/ open Q’s)

pH Control

Observer-Based FF Control

Example pH regulation

05/24 05/25 05/26 05/27 05/28 05/291

1.5

2

Solix Data

Dry

Ma

ss

2000

Incid

en

t R

ad

iatio

n

PAR (µ mol /(m2 sec))

PYRA (W/m2)

05/24 05/25 05/26 05/27 05/28 05/290

1000

2000

Incid

en

t R

ad

iatio

n

05/24 05/25 05/26 05/27 05/28 05/297.1

7.2

7.3

pH

05/24 05/25 05/26 05/27 05/28 05/290

2

4

Fe

ed

ba

ck C

on

tro

l

PYRA (W/m )

setpoint

measured

MFC Flow

Two-Stage FF AGS Control

ufb(t) : CO2 flow rate

r(t) : target culture density

rff(t) : growth trajectory

: pH trajectory for given growth)(~ trff

=

mixing

ratenutrient

flowCO

)(

2

tu ff

)(

0

0

)(

)( tu

tu

tu ff

fb

+

=

=

2

2

CO dissolved

O dissolved

density

pH

)(ty

Lipid Production

Microalgae respond to stress by creating lipids

Stress includes nutrient depletion, COStress includes nutrient depletion, CO2limitation, and intense sun energy

Biphasic approach

Grow to a target density

Deplete nutrients

One-Shot Nutrient Limited Growth

Expected ContributionsTwo-Stage Adaptive FF Predictive Control

Theoretical framework

Robust performance analysis and design

Flat Panel AGS Model

Physically based

AGS Control

Improve Resource Utilization

See Appendix for Other Work

Future Directions – FB Control

Optimized Design

Design Pdes and K

Robust Performance Analysis and DesignRobust Performance Analysis and Design

Model Uncertainty

Smith Predictor

Future Directions – FF Control

Nonlinear Adaptive and Predictive FF Control Methodologies

TechniquesTechniques

Impact on Robustness

NLTV Desired Closed-loop (Pdes) Design

NLTV Gi-1 Design

Future Directions – AGS

Improve Growth Model

Develop Lipid Model

Improve Controller PerformanceImprove Controller Performance

Extend Proposed Architecture to AGS Control

Questions ?Questions ?