Vkx
k
s
which can be inside the operating region depending on
the values of k
x and k
s
. The results of the analysis of the zero dynamics show that
thebest choice ofoutput tobe controlled isthesubstrate concentrationand involving
the biomass concentration into the output generally brings singular points into the
zerodynamics andmakes thestabilityregion narrower. Thesetheoreticalissueswill
help inunderstanding the following simulationresults.
3.9 Stability analysis of continuous fermentation
pro-cesses
As localstability analysis shows, in a neighborhood of the desired operating point
thesystem isstable,butbecausethe pointisveryclosetothefoldbifurcationpoint
(X
this stability region is small. This is illustrated in g. 3.4 which shows that the
system moves to the undesired wash-out steady state when it is started fromclose
neighborhoodof the desired operatingpoint(X(0)=4:7907 g
Stability analysis based on local linearization aroundthe operating point
depends uponthe eigenvalues of the linearized state matrix A inEqs. (3.32)-(3.33)
which are complex conjugate values inour case:
12
= 0:60170:5306i (3.106)
We can see that the process is indeed stable around the operating point but the
linear analysis doesnot giveany informationonthe extent of the stability region.
0 20 40 60 80 100 120 0
1 2 3 4 5 6 7 8 9 10
Time [h]
Concentration [g/l]
Biomass concentration Substrate concentration
Figure3.4: Open loopbehaviorof the system
Nonlinear stability analysis is based on Lyapunov technique which aims at
ndingapositivedenitescalar-valuedgeneralizedenergy function V(x)whichhas
negativedenitetimederivativeinthewholeoperatingregion. Mostoftenageneral
quadraticLyapunov functioncandidate is used in the formof
V(x)=x T
Qx
with Q being a positive denite symmetric quadratic matrix, usually of diagonal
form. This function is scalar-valued and positive denite everywhere. The
stabil-ity region of an autonomous nonlinear system is then determined by the negative
deniteness of itstime-derivative:
dV
dt
=
@V
@x _ x=
@V
@x
f(x)
where
f(x)=f(x)in the open loop case (assuming zero input) and
f(x)=f(x)+
g(x)C(x)intheclosedloopcasewhereC(x)isthestaticlinearornonlinearfeedback
law.
The diagonalweighting matrixQinthequadraticLyapunov functionisselected
in a heuristic way: a state variable which does not produce overshoots during the
simulation experiments gets a larger weight than another state variable with
over-shooting behaviour. In the new norm dened by this weighting, a more accurate
estimateofthestabilityregioncanbeobtained. Withthisanalysiswecannot
calcu-late theexact stability regionbut theresults give valuableinformationforselecting
the controller type and tuning its parameters. The nonlinear stability analysis
re-sults inthe time derivativeof the quadratic Lyapunov functionasa function ofthe
Q=I
state variables,whichisa two variate functioninour case seen ing. 3.5. The
sta-bility region of the open-loopsystem is the region on the (x
1
;x
2
) plane over which
thefunctionisnegative. WeremarkthatthisheuristiccandidateLyapunovfunction
selection method serves mainlyfor illustrational purposes and is not considered as
ascientic contribution.
3.10 Summary
The main contributions of this chaptertothe analysis ofnonlinear process systems
are as follows:
Using the special structural properties of process models, the generally
com-putationallycomplexnonlinear reachability analysis can be performedanalytically.
Furthermore, in sections 3.6 and 3.7 it was shown that the singular points of the
reachabilitydistributionhaveclear physicalmeaning. Theseresults wereillustrated
using the models of continuous and fed-batch fermentation processes.
In section3.7rigorousnonlinear analysiswasused foranalyzingthereachability
propertiesofa simplefed-batchfermenter modelandto relatethemtothe
physico-chemical phenomena taking place in the reactor. With a help of this grey-box
approachwehaveshownthatthe knowndicultiesofcontrollingsuchprocessesare
primarilycaused by the factthat the rank ofthe reachabilitydistribution isalways
two which is less than the number of state variables being three. Furthermore, a
istics that shows the nonlinear combination of the state variables that is constant
independently of the input. The coordinates transformation is independent of the
most uncertain part of the state space model: the source () function, too. The
results are extended to the four state variable non-isotherm case, and to nonlinear
fed-batch chemical reactors with generalreaction kinetics. The rank of the
reacha-bilitydistributionremainstwointhis casegivingrise totwoconserved combination
of the state variables independently of the input. The structural properties of the
process models enabling toapply the proposed analytical technique have alsobeen
described.
In section 3.8 it was shown on the example of continuous bioreactors that the
notion of zero dynamics is very useful for output selectionfor control purposes. It
wascalculated thatthe best controlled outputchoice isthe substrate concentration
since the system is minimum-phasewith respect to this output.
Analysis Based Control Structure
Selection
4.1 Motivation
The control of nonlinear process systems is a challenging and emerging
interdis-ciplinary eld of major practical importance. The most common way to control
nonlinear process systems is either use linear techniques on locally linearized
ver-sions of the nonlinear models or use model-based predictivecontrol[76].
At the same time a number of powerful and theoretically well grounded tools
andtechniques areavailablefornonlinear controlinthe eldofsystems and control
theory ([35], [89]) which are applied successfully in other application areas. These
techniques, however, require most often symbolic computation and may be
non-feasible for real process systems. This may be one of the reasons they are not
known and not appliedextensively for process systems.
Fermentationprocessesinparticularexhibitstrongnonlinearcharacteristicsand
areknown tobediculttocontrol. Theinvestigated simplefermentationprocess is
thereforeused asabenchmark problemforadvancednonlinear analysis and control
techniques. Many authors have examined the various approaches of analyzing and
controlling fed-batch ([45], [9], [39], [92]) and continuous fermentation processes
([86], [44], [87]).
Nonlinear controllersof dierent type are designed and compared on the
exam-ple a simple fermenter near an optimal production operating point which is close
toits foldbifurcationpoint. Nonlinearanalysis of stability,controllabilityand zero
dynamics presented in the previous chapter is used to investigate open-loop
sys-tem properties, toexplore the possible controldiculties and to design the system
outputtobeused forcontrol. A widerange ofcontrollersare testedincluding
pole-placement and LQ controllers, feedback and input-output linearization controllers
and anonlinear controller based ondirect passivation. The comparison isbased on
time-domainperformance and oninvestigating the stability region, robustness and
tuningpossibilitiesof the controllers.
ysis
Variouscontrollersofdierenttype: pole-placementcontroller,LQcontrollers,
feed-back linearization based controllers and nonlinear controllers based on direct
pas-sivation are compared here on the example of a simple fermentation process. The
process model used is given by eqs. (3.26)-(3.27)and the model parameters are
shown inTable 3.6.1.
4.2.1 The control problem statement and comparison
view-points
Model analysis aboveshow the following special properties of the controlproblem:
(1) The desired setpoint is close to a bifurcation point which is a local singular
point fromthe viewpoint of controllability.
(2) Thesystemisstableonlyinanarrowneighborhoodofthethedesiredoperating
point.
Moreover, wecannotmeasurethe biomassconcentrationbecauseofphysicalreasons
and modelling uncertainty is present in the source term of the model equations.
Therefore we prefer controllers of the followingtype.
(3) The controller uses only the substrate concentration for feedback, that is it
shouldbea partialstate feedback controller.
(2) The closed-loop system should be robust with respect to uncertainties in the
reactionkinetic function .
The performance analysis of the controllers is performed by extensive simulation
studiesandby investigatingtheoverall behaviourofthe controllers. Thecontrollers
are then compared using the followingperformance criteria:
stability region
FortheroughestimationoftheextentofthestabilityregionasimpleLyapunov
functionin the positivedenite formof V(x)=x T
Qx isdened andthe time
derivative of this function is computed and analyzed as the function of the
state variables.
time-domainperformance
Here the qualitative behaviour of the responses, the presence of overshoots
and the possible spurious steady states are investigated.
tuning possibilities
Besides of the availabilityof guidelines onhowtotune the controller
parame-ters,robustnesswithrespecttothecontrollerparametersarealsoinvestigated.
The detailed simulation results are shown in Appendix A, only the most
inter-esting cases were included intothe main text.
S
x
K x
-u y
Figure4.1: Controlcongurationofstatic linearfeedback (pole-placement,LQ
con-trol)
4.2.2 Pole-placement controller
The purpose of this section is
toprovide asimple controllerdesign approach for later comparison
toexaminethepossibilitiesofstabilizingthesystembypartialfeedback
(prefer-ablyby feeding back the substrate concentration only)
Pole placement by full state feedback
Firstly,a full state feedback is designed such that the poles of the linearized model
of the closed loop system are at [ 1 1:5]
T
. The necessary full state feedback
gain is
K
pp
=[ 0:3747 0:3429] (4.1)
The schematiccontrolcongurationofstatic linearfeedback isshown ing. 4.1. A
simulation run is shown in g. 4.2 starting from the the initial state X(0)= 0:1 g
l
andS(0)=0:5 g
l
. Asitisvisible, theclosedloopnonlinear systemhas anadditional
non-desired stable equilibrium point and the controller drives and stabilizes the
system at this point. It can be easily calculated from the state equations and the
parameters of the closed loop system that this stable undesired operating point is
at X = 3:2152 g
l
, S = 3:5696 g
l
. The time derivative of the Lyapunov function is
shown in g. A.9. This phenomena warns us not to apply linear controllers based
on locally linear models for a nonlinear system without a careful deep preliminary
investigation.
Partial linear feedback
Motivated by the fact that the zero dynamics of the fermenter is globally
asymp-totically stable when the substrate concentration is the output, let usconsider the
0 5 10 15 20 25 30 35 40
−5
−4
−3
−2
−1 0 1 2 3 4
Time [h]
Centered concentration [g/l]
Biomass concentration Substrate concentration
Figure 4.2: Centered state variables and input, full state feedback pole placement
controller,X(0) =0:1 g
l
,S(0) =0:5 g
l
following static partialstate feedback
u=Kx
whereK =[0 k]i.e. weonlyusethe substrateconcentrationforfeedback. The
sta-bilityregion ofthe closed-loopsystem can be investigated usingthe time-derivative
ofthe quadraticLyapunov function. g. A.10shows that thestabilityregion ofthe
closed loop system is quitewide. Furthermore, it can be easily shown that for e.g.
k = 1 the only stable equilibrium point of the closed loop system (except for the
wash-outsteady state) isthe desiredoperatingpoint. The eigenvalues of the closed
loopsystem withk =1 are 0:9741and 2:6746.
4.2.3 LQ control
LQ-controllersare popularandwidely usedforprocess systems. Theyare known to
stabilizeanystabilizablelineartimeinvariantsystemglobally,thatisovertheentire
state space. This type of controller is designed for the locally linearized model of
the process and minimizesthe cost function
J(x(t);u(t))= Z
1
0 x
T
(t)R
x
x(t)+u T
(t)R
u u(t)
dt (4.2)
where R
x
and R
u
(the design parameters) are positive denite weighting matrices
of appropriate dimensions. The optimalinput that minimizes the above functional
is in the form of a linear full state feedback controller u = Kx. The results for
two dierentweighting matrix selections are investigated.
0 0.5 1 1.5
−5
−4
−3
−2
−1 0 1
Time [h]
Centered concentration [g/l]
Biomass concentration Substrate concentration
Figure4.3: Unstablesimulationrun, LQcontroller,expensivecontrol,X(0)=0:1 g
l ,
S(0)=0:5 g
l
Cheap control
In this case the design parameters R
x
and R
u
are selected to be R
x
= I 22
and
R
u
=1. The resulting fullstate feedback gain is K =[ 0:6549 0:5899].
Expensive control
The weighting matricesinthis case were R
x
=10I 22
and R
u
=1. There were no
signicant dierences in terms of controller performance compared to the previous
case. The full state feedback gain in this case was K =[ 1:5635 2:5571].
Thestabilityregionisagaininvestigatedusingthetime-derivativeofaquadratic
Lyapunovfunction. Thetime-derivativefunctionasafunctionof thecentered state
variables forcheap and expensivecontrolis seen ings A.11and A.12respectively.
Unlike the linear case when aLQR always stabilizes the system, it is seen that the
stabilityregiondoesnot coverthe entire operatingregion. Indeed,asimulationrun
ing. 4.3starting withan"unfortunate" initialstate shows unstable behaviour for
the nonlinear fermenter model.
Note that an LQ controller is structurally the same as a linear pole placement
controller (i.e. a static linear full state feedback). Caused by this fact non-desired
stable steady states may appear dependingon the result of the LQ-design.
4.2.4 Local asymptotic stabilization via feedback
lineariza-tion
Anonlineartechnique,feedbacklinearizationisappliednextforchangingthesystem
dynamics into a linear one. Then, dierent linear controllers can be employed on
Exact linearization via state feedback
In order to satisfy the conditions of exact linearization, rst we have to nd an
articialoutput function(x) that is asolution of the PDE [35].
L
g
(x)=0 (4.3)
Note that the above equation are exactly the same as the equation (3.94) used
for determining the coordinate transformation for analyzing zero dynamics of the
system. Letus choose the simplestpossibleoutput function againi.e.
(x)=
Thenthe componentsof statefeedback u= (x)+(x)v for linearizingthe system
are calculated as
(x) =
and the new coordinates are
z
The state space modelof the system in the new coordinates is
_
which islinearand controllable. Theexactlylinearized modelmayseem simplebut
if we have a look at the new coordinates we can see that they are quite
compli-cated functions of x depending on both state variables. Moreover, the second new
coordinate z
2
depends on which indicates that the coordinate transformation is
sensitive with respect to uncertainties in the reactionrate expression. Simulations
showedthatit'shardtonumericallyevaluatethefunctionsandandthepartially
closed loop system produced non-feasibly large inputs. The system can be exactly
linearized theoretically,but the feedback is hard tocompute inpractice. Moreover,
inanengineeringsenseit'snotpracticallyusefultolinearizesuchanoutputfunction
as.
S
x
α(x )
β(x) +
+ u y
v
-o
k
Figure4.4: Controlcongurationofinput-outputlinearizationwithstabilizingouter
feedback
I/O linearization
Here we are looking for more simple and practically useful formsof linearizingthe
input-output behaviour of the system. The static nonlinear full state feedback for
achieving this goalis calculated as
u= (x)+(x)v = L
f h(x)
L
g h(x)
+ 1
L
g h(x)
v (4.9)
provided thatL
g
h(x)6=0ina neighborhoodof theoperatingpointwhere v denotes
thenewreferenceinput. Aswewillsee thekeypointindesigningsuchcontrollersis
the selectionof the output(h) functionwhere the original nonlinear state equation
(3.26) is extended by a nonlinear output equation y = h(x) where y is the output
variable. Thecontrolcongurationofinput-outputlinearizationisshowning. 4.4.
Controlling the biomass concentration In this case h(x)=
X =x
1 and
(x)= L
f h(x)
L
g h(x)
=
V
max (x
2 +S
0 )
K
2 (x
2 +S
0 )
2
+x
2 +S
0 +K
1 F
0
(4.10)
(x)=
V
x
1 +X
0
(4.11)
The outer loopfor stabilizingthe system isthe following
v = kh(x) (4.12)
Itwasfoundthatthestabilizingregionofthiscontrollerisquitewidebutnotglobal.
The time-derivativeof the Lyapunov functionis shown in g. 4.5.
6
;
Figure4.5: TimederivativeoftheLyapunovfunctionasafunctionofcentered state
variables q
1
=1;q
2
=1, linearizingthe biomassconcentration, k =0:5
Controlling the substrate concentration In this case the chosen output is
h(x)=
S =x
2
. The full state feedback is composed of the functions
(x)=
V
max (x
2 +S
0 )(x
1 +X
0 )
Y(K
2 (x
2 +S
0 )
2
+x
2 +S
0 +K
1 )(S
F x
2 S
0 )
+F
0
(4.13)
(x)=
V
S
f x
2 S
0
(4.14)
In the outer loop a negative feedback with the gain k = 0:5 was applied i.e. v =
0:5x
2
The time derivative of the Lyapunov function of the closed loop system as
afunctionof
X and
S isvisibleing. A.13. Notethatforthis case wehaveproved
(seethezerodynamicsanalysis)thattheclosedloopsystemisgloballystableexcept
for the singularpointswhere the biomass concentration is zero.
Controllingthe linearcombinationof the biomass and the substrate
con-centrations In this case the output of the system was chosen as
h(x)=Kx (4.15)
where the row vector K is calculated as the result of the previously desribed LQR
cheap controldesign problem. Then the functions and are given as
(x)=
Kf(x)
Kg(x)
(4.16)
S
x
v p (x)
+
+ u
y v
-k
L g V(x)
SDVVLYH ORVVOHVV V\VWHP
Figure 4.6: Control conguration of passivity based control with stabilizing outer
feedback
(x)= 1
Kg(x)
(4.17)
The value of K was [ 0:6549 0:5899]
T
and in the outer loop a negative feedback
withthe gain k =0:5 wasapplied. Thetime derivativeofthe Lyapunov function of
theclosedloopsystem ising. A.14. Althoughthis outputselectionisnotthe best
choice for continuous fermentation processes (because it is possible to analytically
examinethezerodynamics),it'sbasicideacanbegeneralizedforhigherdimensional
systems (see section 4.3).
4.2.5 Passivity based control
In orderto designa controller whichstabilizesthe system over the entire operating
regionweturnbacktothequadraticLyapunovfunctionusedforclosed-loopstability
regionanalysis before:
V(x)=x T
Qx (4.18)
with apositivedenite diagonalweighting matrix
Q=
q
1 0
0 q
2
(4.19)
Thecontrolstrategyisrsttorenderthesystemlosslesswithaninnerstatefeedback.
Therefore the input is decomposed in the following way:
u=v
p
(x)+v (4.20)
where v
p
is the inner feedback and v is the new external input. The state equation
of the partially closedloopsystem with v =0 isgiven by
_
x=f(x)+g(x)v
p
(x) (4.21)
p
respect tothe storagefunction V inEq. (4.18):
d
Ifwe dene the output as
y=L
then the partially closed loop system has the KYP property (see Denition 3.3.2),
thus it becomes passive (more precisely, lossless) with the storage function V (see
Theorem 3.3.1) with respect tothe supply rate
s(v;y)=vy (4.26)
Therefore the system can bestabilizedwiththe outer feedback (see Theorem 3.3.2)
v = ky; k >0 (4.27)
Usingthe specialquadratic form of V(x)in Eq. (4.18) we obtain
v = k
Theschematicstructure of thepassivity basedcontrolcongurationisshown ing.
4.6.
Inthecaseofthesimplefermentermodelthefollowingsimplequadraticfeedback
isobtained:
v = kx
Theabovedirectpassivationbasedcontrollerstabilizesthefermenterglobally,which
is seen from the time-derivative of the quadratic Lyapunov function corresponding
tothe parametervalues q
1
designparameters.
Note that the positive denite matrix Q need not to be necessarily a diagonal
matrix. In fact, a good starting point for the choice of Q is to design a locally
stabilizing linear controller for the nonlinear system (e.g. an LQ-controller), and
solve the Lyapunov equation
A T
Q+QA= P
for Q using a positive denite matrix P and the state matrix A of the linearized
closed loop system. Then the solution Q can determine the prescribed Lyapunov
function in eq. 4.18. Also note that Q cannot be an arbitrarily chosen positive
denite matrix, because the zero state detectability property (see Denition 3.3.3)
shouldbefullled for the articial outputy=L
g V(x).
The evaluationand comparisonof the controllers isperformed using the predened
evaluationviewpoints introducedbefore.
Stability region has been investigated using the quadratic Lyapunov function.
It has been found that the stabilityregion is
Narrow, hard to determine for full state feedback pole placement controller
and for LQR(expensive control) controllers.
Wide, hard to determine forpartiallinear controller andfor LQR(cheap
con-trol) controller.
Wide, can be estimated using the zero-dynamics analysis for linearization of
the biomass concentration controller and for Linearization of the linear
com-binationof the biomass and substrate concentration.
Global forLinearizationofthe substrateconcentrationandfor passivity based
control.
Time-domainperformance canbecharacterizedfortheinvestigatedcontrollers
asfollows.
Acceptable with the possibility of having non-desired stable steady states for
fullstate feedback pole placement controller and for LQRcontrollers.
Excellent forthe input-outputlinearizationtypecontrollersofeverykind:
Excellent forthe input-outputlinearizationtypecontrollersofeverykind: