5.8 A simple unstable CSTR example
5.8.4 Passivation and loop-shaping of the unstable CSTR: illustra-
is
nowpresent asa secondorder term originatingfrom the autocataliticsecond order
reaction. Ifwesubstitutethenormalizedvariables(5.50)totheconservationbalance
equation (5.49)the followingnormalized state equation is obtained:
dp
FromtheequationabovewecanidentifytheelementsoftheHamiltoniandescription
tobe
Bypartial integration we get:
V
5.8.3 Passivity analysis of the unstable CSTR
The passivity analysis isperformedusing the internalHamiltonianof the system
H
We can see that the above functionis of nodenite sign because of the presence of
the secondand thirdorder terms ofdierentconstant coecients. This meansthat
the system fails tobepassive inthe generalcase.
5.8.4 Passivation and loop-shaping of the unstable CSTR:
illustration of the controller tuning method
System parameters and open-loop response
Let us introduce the normalized concentration variables c
A
. The conservation balance equation(5.49) then takes the form
dc
The parameter values used in the simulations are shown in Table 5.1. There were
two initialconcentrationvalues (c
A
(0)) given for the simulations:
1
respectively. It is easily seen from the data that c
A
is an unstable equilibrium for
the system as it is illustrated in both of the sub-gures in g. 5.2. Nonlinear
proportional feedback controller
Letus apply the following feedback controller
u=k
where k
c
is an appropriately chosen controller gain and w is the new reference
signal. The new reference, w was set to 0 for the simulations. The chosen value
of the controller gain is shown in Table 5.1. The closed loop simulation results in
g. 5.3 show that the proposed control method indeed stabilizes the equilibrium
c
. Here again, the simulation was performed using two dierent initial
conditions asabove as shown inthe two sub-gures of g. 5.3.
Controller tuning method based on stability region analysis
It is an important question for a nonlinear controller to determine its stability
re-gion as a function of the state variables with its parameter(s) xed. For this very
simple case this problem can be solved analytically. Let us consider the nonlinear
proportionalfeedback controller above with its gain xed at k
c
= 10. In fact, it is
easy to show that the resulting closed loop system with the parameters described
above ispassive with respect tothe supply rate wy if
c
A
> 1:9088 kmol
m 3
i:e: c
A
>0:3912
wherey=c
A
. Inordertoshowthis, letustakethesimplestoragefunctionV(c
A
. It can becalculated that
@V
> 1:9088 (5.56)
and equality holds only if c
A
= 0. Since g(c
A
) = 1 in the closed loop state space
modelwe can deduce that
y=L
where L
g
is the Lie-derivative of the simple storage function with respect to the
function g. Therefore it follows that the closed loop system is passive in the given
interval. Thetimederivativeofthe storagefunction(asafunctionofc
A
)isdepicted
ing. 5.4.
5.9 Summary
Usingathermodynamicapproachof constructingand analyzingdynamicmodels of
process plants the simple Hamiltonian model of lumped process systems has been
constructedbasedonmechanicalanalogue. Theconservedextensivequantitiesform
the system. The approach is applicable for systems where Kirchho convective
transport takes place together with transfer and sources of various type. Systems
with constant molar holdup and uniform pressure in every balance volume satisfy
these conditions. The resulted simple Hamiltonianmodelcan be used for passivity
analysis because it contains a storage function together with the nonlinear state
space model of the system in a special canonical form. This type of modelenables
us to design a nonlinear PD feedback controller for passivation and loop shaping.
The general results are illustrated on simple examples of practical importance: on
abilinear heat exchanger celland onan isothermCSTR with nonlinear reaction.
T ci
T hi T (h)
T (c) v c
v h
Figure5.1: The heat exchanger celland its variables
0 200 400 600 800 1000
0.7 0.75 0.8 0.85 0.9 0.95 1
time [s]
outlet concentration [kmol/m 3 ]
0 200 400 600 800 1000
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6
time [s]
outlet concentration [kmol/m 3 ]
Figure5.2: Open loopsimulation results
0 500 1000 1500 2000 1
1.5 2 2.5
time [s]
outlet concentration [kmol/m 3 ]
0 500 1000 1500 2000
2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8
time [s]
outlet concentration [kmol/m 3 ]
Figure 5.3: Closedloopsimulationresults
m 1800 kg
k 510
4 m
3
kmol s
c
Ain
0.4
kmol
m 3
c
A
2.3
kmol
m 3
v
2.5058 kg
s
k
c
10
-Table 5.1: Parameter valuesof the simulatedCSTR
-0.8 -0.6 -0.4 -0.2 0
-2 -1 1 2 3
Figure 5.4: Timederivative of the storage function asa functionof c
A
Conclusions
In conclusion, the main contributions and the proposed theses of this work are
summarizedinthenext section,thenthepublicationsrelatedtothisdissertationare
listed and nally,the possible directions of further researchare given. The relevant
chapterofthe dissertationand thelabelsofthe relatedpublications(enumerated in
section6.2) are indicated inparenthesis.
6.1 Theses
Thesis 1 Model-based fault diagnosis of processsystems (Chapter 2)
([P1], [P5], [P6], [P7], [P11])
A method has been developed for the model-based fault detection and
di-agnosis of nonlinear process systems. Physical model has been used for the
descriptionof the process dynamicsand semi-empiricalmodels have been
ap-plied forfault modeling.
1. It has been shown that the performance of the fault detection and
iso-lation algorithms is improving with the increasing level of detail of the
process models. A method has been worked out for the spatial
localiza-tion of the faults using measured signals belonging to dierent spatial
locationsof the system.
2. It has been shown that safesimultaneous fault detection and isolationis
possible using the grey- or white-box models of the faults together with
the process model.
The results have been illustrated on the example of countercurrent
heat-exchangers. Theknownprocessdynamicshavebeenusedasalterfor
extract-ingcharacteristic fault-relevantinformation fromthe measurement data.
Re-cursiveparameter estimation and signal-changedetection methodshave been
appliedfor faultdetection and diagnosis.
Thesis 2 Nonlinear model analysis of process systems (Chapter 3)
([P2], [P8], [P9], [P12], P[13])
Theanalysis ofnonlinear process systems given innonlinearinput-ane state
sionshavebeendrawnfromthecomparisonofthelinearandnonlinearanalysis
methods onthe example of fermentation processes.
1. Exploiting thespecial structural properties of process models, the
gener-ally highlycomplicated nonlinear reachability analysis becomes
analyti-cally computable. On the example of continuous fermentation processes
ithas beenshown thatthesingularpointsobtainedfromthereachability
analysis have clear physicalmeaning.
2. It has been shown that a large class of isothermfed-batch fermentation
processes is not reachable with the inlet feed ow rate, since the rank of
the reachabilitydistribution is less than the numberof state variablesin
each point of the state space. The coordinates transformation suitable
for transforming the state space model of fed-batch fermentation
pro-cesses tocontrollabilitycanonical formhasbeendetermined. Ithas been
shown that the calculated coordinates transformation is independent of
the source function (that is of the fermentation kinetics) in the model.
Usingthe calculated coordinates-transformationthe minimalstate space
realization offed-batchfermentation processeshas been given. A
dimen-sionally homogenious conserved quantity has been determined which is
a nonlinear combination of the state variables. The results have been
generalized for the temperature-dependent (non-isotherm) case.
3. It has been shown that the zero dynamics of continuous isotherm
fer-mentationprocesses is globallyasymptoticallystable withrespect tothe
substrate concentration as output independently of the source function.
This means continuous fermentation processes are globally
minimum-phase systemswith respect tothe substrate concentration. Furthermore,
if one involves the biomass concentration into the output, it makes the
stabilityregionofthe zerodynamicsnarrower. Thisimpliesthat the
sta-bility regionof a controlled(closed loop) system is alsonarrowerin this
case.
Thesis 3 Analysis based controlstructure selection (Chapter 4)
([P3], [P9], [P14])
The role of the nonlinear analysis results incontrolstructure design has been
investigated. Linear and nonlinearstatic controllers forthe stabilizingcontrol
of continuous fermentation processes have been designed. The operation of
the controllers have been compared based ontheir performance measures and
recommendationsfor their use have been given.
1. A methodforusingthe priorinformationfromthe nonlinearmodel
anal-ysis (stability region,reachability distribution,zero dynamics)inthe
de-sign of nonlinear controllers has been developed.
2. Usingtheoretical analysis and simulationexperiments ithas been shown
thatthe controllersdesigned onthebasisofthe nonlinearanalysisresults
have more advantageous properties than the linear controllers designed
using the locallylinearized modelof the process.
tinuousfermentationprocesseshas beenworked out. The design
parame-ter ofthe controlleristhequadraticLyapunov-functionoftheclosedloop
system.
4. A generallyapplicable methodhas beendeveloped forthe local
stabiliza-tion of locally reachable nonlinear single-input input-ane state space
models. The method creates a link between linear optimal control and
nonlinear systems. Using the method itis possible toselect those linear
outputs that make the system at least locallyminimum-phase.
Thesis 4 Hamiltonian view on process systems (Chapter 5)
([P4])
Dynamicmodelsdescribingalargeclassofprocesssystemscanbetransformed
intotheso-calledsimpleHamiltonianform. Basedonthis simpleHamiltonian
descriptionnonlinear stabilizingand loop-shapingcontrollerscan bedesigned
for process systems.
1. A method has been worked outfor the tuningof Hamiltonianstabilizing
and loop-shapingcontrollersthatisbasedonthe globalstabilityanalysis
of the closedloopsystem.
2. The condition of the simple Hamitonian description of process systems
corresponding to the source function has been given. Based onthis
con-dition it has been shown that process systems not containing source or
containing only one component can always be described in the simple
Hamiltonianframework.