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Bifurcation Control of Sample Chemical Reaction Systems
Copyright @ IF AC Advanced Control of Chemical Processes, Pisa, Italy, 2000
BIFURCATION CONTROL OF SAMPLE CHEMICAL REACTION SYSTEMS Bodil Recke >,1 Britta R0nde Andersen »,2 Sten Bay J0rgensen »>,3
FLS-Automation, H0ffdingsvej 34, DK2500 Valby, Denmark Haldor Tops0e A/S, Nym0llevej 55, DK2800 Lyngby, Denmark »> Dept. Chem. Eng. , DTU Build. 229, DK2800 Lyngby, Denmark
>
»
Abstract: Bifurcation control of Hopf bifurcations is applied to two systems exhibiting a subcritical Hopf bifurcation . A subcritical Hopf bifurcation is characterized by the emergence of an unstable periodic solution, quite often coexisting with a stable periodic solution in a hysteresis scenario. This type of scenario can lead to large amplitude oscillations if the Hopf bifurcation point is crossed. Therefore it is desirable to change this bifurcation scenario, which can be achieved with the bifurcation control described in this paper. The purpose of this type of control is to convert a subcritical Hopf bifurcation to a supercritical one. An additional benefit is that the resulting stable periodic solution will have a relatively small amplitude at least close to the Hopf bifurcation point . The method is first successfully applied to a small example system, to illustrate the calculations and subsequently to an industrial size heat integrated ammonia reactor . For the ammonia reactor the existence region of the sustained oscillation is reduced dramatically and the maximum amplitude is reduced from approximately 200°C to about 60°C. Furthermore the nominal operating point no longer coexist with stable periodic solution. Copyright © 2000 IFAC Keywords: Partial Differential Equations , Oscillation, Nonlinear analysis , Nonlinear control, Reactor Control
1. INTRODUCTION
(Recke, 1998; Andersen, 1999). In this paper specific attention is paid to Hopf bifurcations . These bifurcations often occur in recycle systems where either energy or mass is recycled. An industrial case resulting in oscillation with amplitudes in the order of 200°C was reported by (Naess et al. , 1993) . This quite damaging event was investigated by (Morud and Skogestad, 1998) and further elaborated on by (Andersen , 1999) . The purpose of this paper is to show how knowledge of the nonlinear dynamic behaviour can be utilized in designing a control for the process converting the dynamic behaviour of the closed loop system to a more desirable one. The chemical system investigated initially in the present paper is the simple Salnikov model. That
Complex dynamic behaviour is often experienced in chemical processes. This behaviour is associated with the nonlinearities in the process. These nonlinearities have the most pronounced effect around bifurcation points i.e. points where the system solutions change stability and/or number of possible solutions. This type of behaviour have been seen in a host of chemical systems as diverse as azeotropic distillation systems (Esbjerg et al. , 1998) and fixed bed reactors with recycle I
bre@fisautomation .com [email protected] Corresponding author, sbj@kt .dtu .dk
569
has the possibility of exhibiting periodic solutions. These oscillating solutions arise as a consequence of a supercritical Hopf-bifurcation resulting in a stable periodic solution and an unstable steady state. The periodic solution undergoes a cyclic fold bifurcation and finally terminates in a subcritical Hopf-bifurcation, where the steady state regains stability. The purpose of the bifurcation control used in this article is to convert the subcritical Hopf-bifurcation into a supercritical one. Additionally the bifurcation control should stabilize the bifurcation point it self. Furthermore the control applied is capable of diminishing the amplitude of the stable periodic solution obtained in the controlled system. The advantage of this is that the rather 'nasty' behaviour of an unstable periodic solution can be converted to a stable small amplitude periodic solution without changing the static behaviour of the system. The same bifurcation control is then applied to an industrial scale ammonia reactor with similar bifurcation behaviour. The results demonstrate that the dynamics of the industrial reactor can be changed significantly resulting in more desirable properties around optimal productivity operating conditions.
2. SIMPLE MODEL
The model investigated in this section is the so called Salnikov model proposed by (Salnikov, 1948 j Scott, 1991). This set of equations describes a system which exhibits oscillations for an exothermic reaction in a closed chemical system. It is included to illustrate the technic which will be applied to an industrial scale ammonia reactor in section 3. The simple reaction scheme involved is P-+A rate = kopj Eo = 0 A-+B+Q rate = kIa, kI = Al exp( -Ed RT) where it is assumed that P is a 'pool chemical' i.e. the concentration does not change as a function of time or has infinite capacity. A model in dim ensionless variables is given by. (1)
-d8 =
(2)
1 + 108
dT
o.exp
(8) + -1 108
- (8 - 8a )
•
50
o o
o
o
10
•••
I
o ~-o 0005
0 .01
0.01S
2.1 Open Loop Bifurcation Behaviour A brief review of the bifurcation behaviour of the Salnikov model can be found in (Scott, 1991). The bifurcation parameter shown in figure 1 is the initial (and constant) concentration of the precursor in the reaction (J1.) . The figure shows that two Hopf bifurcations exist in the present
0.02 0.025 003 Initial Precur80f Concentration
0 .035
0.04
0 .045
M
Fig. 1. Bifurcation behaviour showing the maximum temperature 8 with J1. as bifurcation parameter. (--) stable steady state, (- --) unstable steady state, (.) stable periodic solution, (0) unstable periodic solution . system. The one occurring at the lowest value of J1. is a supercritical Hopf leaving the formerly stable steady state unstable, and marking the initiation of a stable periodic solution. The Hopf bifurcation occurring at the higher value of J1. is a subcritical one, where an unstable periodic solution and an unstable steady state merges resulting in a stable steady state. In the bifurcation diagram the cyclic fold marking the exchange of stability of the periodic solution can also be seen.
2.2 Bifurcation Control
The purpose of bifurcation control is to transform the subcritical Hopf bifurcation occurring in the open loop system, to a supercritical one, and at the same time stabilize the bifurcation point itself. This effect can be accomplished with a non linear controller consisting of quadratic and cubic terms according to (Abed and Fu, 1986). The notation used in theorem 1 can be explained from a standard dynamical system given by: ;i;
-do. = J1. - Ko.exp ( -8- ) dT
• • •• •
= Jp. (x, u(x))
(3)
For the system described by equation (3) , at some parameter value of J.I. a bifurcation occurs. At this value the left and right eigenvectors ([' and r' respectively) can be calculated from.
= Lo Lor' = >..r' I'Lo = I' >.. ojp.
ox
(4) (5) (6)
Where>.. is the critical eigenvalue at the bifurcation point. The eigenvectors are then normalized by setting the first component of r' to 1 and then choosing /' such that Ir = 1. The vector notation
0 .05
Table 1 Minimum gains necessary to stabilize the bifurcation point for different w-values minimum stabilizing gain w-value 0.5 2e 3 2 0.059
with no I refers to the normalized version of the eigenvectors. "Y is defined as. "Y = !!j;; Theorem 1. Let the hypothesis (H) hold, (H) The matrix Ao = (8 f / 8x )(0,0) has a pair of simple, complex conjugate eigenvalues >'1 = iw c, >'2 = iwc on the imaginary axis, where Wc i- O. Moreover, all other eigenvalues of Ao have negative real part. Additionally assume that h i- O. That is, the critical eigenvalues are controllable for the linearized system. Then there is a feedback u(x) with u(O) = 0 which solves the local smooth feedback stabilization problem for x = f(x , u) and the local Hopf bifurcation control problem for equation (3) . Moreover, this can be accomplished with only third order terms in u(x) , leaving the critical eigenvalues unaffected. The condition is that h i- 0 is checked, and the results is that at the subcritical Hopf bifurcation point the value of h : : : 0 + 3.6737e- 3 i i- 0 (numerical values can be found in (Recke, 1998)) . Consequently it is possible to convert the subcritical Hopf bifurcation to a supercritical one by using a nonlinear feedback with only cubic terms. The intended manipulated variable in the system is Ba, and the following feedback control will be applied Ba = -GB3 .
the Hopf bifurcation, and one well after the transformation. By increasing the gain in the controller it is possible to not only transform the subcritical Hopf bifurcation into a supercritical one, but also to diminish the amplitude of the resulting periodic solution. The latter result implies that it is possible to reduce the effect of the periodic solution since small amplitude oscillations possibly will be acceptable in most chemical systems.
3. AMMONIA REACTOR 3.1 Open Loop Bifurcation Behaviour
Ammonia reactors are industrially important reactors. (Naess et al. , 1993) reports an incident with large amplitude temperature oscillations in an industrial ammonia reactor with three catalyst beds, quench cooling between the beds and a feed-effluent heat exchanger. (Morud and Skogestad, 1998) studied this incident and explained the temperature oscillations with a Hopf bifurcation. (Andersen, 1999) studied a simpler ammonia reactor with only one catalyst bed and a feedeffluent heat exchanger. The occurrence of periodic solutions was investigated using bifurcation analysis. Also for the simpler reactor sustained temperature oscillations can occur due to a Hopf bifurcation. However, the periodic solution arising in the Hopf bifurcation point Hw is unstable due to a subcritical Hopf bifurcation. Continuation of the unstable periodic solution revealed a cyclic fold bifurcation elL turning the periodic solution stable, figure 3. The subcritical Hopf bifurcation followed by a cyclic fold bifurcation explains the sudden large amplitude oscillations experienced in the industrial reactor. Thrning the sub critical Hopf bifurcation supercritical must be of interest to industry, not only because smaller amplitude oscillations are expected but also because hysteresis phenomena are avoided. Figure 4 shows a sketch of the reactor system consisting of a single catalyst bed with a feed-effluent heat exchanger. The model of this reactor is described in (Andersen, 1999) . The mass and energy balances are
2.3 Results and discussion Adding a washout filter to the system, as suggested by (Abed et al. , 1995) restricts the nonlinear control to changing the dynamic behaviour (i.e. the stability of the periodic solutions) , without changing the location and stability properties of the open loop steady states of the system. A washout filter is in essence just a stable high-pass filter. The reSUlting extended control system can be written as. dz dT = B - wzBa = -G (B - WZ)3 (7) Two parameters needs to be set now, are the gain in the controller (G) and w in the washout filter. The value of w determines the cutoff frequency. The obvious constraint on w, is that it must be positive. Furthermore it should be chosen below the characteristic frequency of the periodic solution . The required gain for stabilization of the bifurcation point and transformation of the bifurcation depends on the value of w. As a rule of thumb it can be said that the higher w the higher gain is necessary. In this application w = 0.5 is used. The result of two different gains are shown in figure 2, corresponding to one just before transformation of
571
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40
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15
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10
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~~-0~,~~-0~,0~,~0,~0175~0~ ,02~~0,=~~0,0~3~0~=~~0~ , ~--~0,~~~o~·
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Initial Preanot ConcenIration (Ja)
Initial Precursor Concentration ijL)
Fig. 2. Closed loop bifurcation behaviour, depicting the maximum temperature of the periodic solution (B) with J.t as bifurcation parameter and w = 0.5 and gains left: 0.01 Oust before the Hopf bifurcation is transformed), right: 5 (well after transformation). (--) stable steady state, (---) unstable steady state, (e) stable periodic solution, (0) unstable periodic solution. with the boundary conditions 400
IV :V
III
....············8 OO~.lL
380
c,J00J'i '
o
~
00:
0:
o
00
",," :0
0_ -
:
.. J.(,I,.0
~280
,"
,
: 0
F"~
240
•: ... .
~
~
•••
0
0
0
C:'L
ooooo~
Feed Flow Rate [%]
Fig. 3. I-Parameter bifurcation diagram for open loop system, continuation of feed flow rate. (--) stable steady state, (- - -) unstable steady state, Hopf bifurcations (Hw,Hld, fold bifurcation points (Fu,FL), (e) stable periodic solution, (0) unstable periodic solution and cyclic fold bifurcation points
dBin Ctube dT
Heat exchanger
Bad outlet
= Bleed -
, Bm + NTUtubeflBtube (14)
out - Bbed - B. hell + NTUshellflBshell
In this work it was chosen to use the feed temperature (Bleed) as actuator. As measurement the inlet temperature to the catalytic bed is used (Bin)such that the delay from the control action to the measurement is relatively small. It can be seen from figure 3 that the bifurcation point Hw is a subcritical Hopf bifurcation and thus an eligible point for bifurcation control. The purpose of the control is to convert the sub critical Hopf bifurcation into a supercritical one. This conversion will eliminate region IV where the upper stable steady state coexists with a stable
8 ( -n- 8X = - x8Xi - + --i) + Dar 8z 8z PeM 8z (8)
8B 8B = _x + i. (_1_8B) + /3Dar 8T 8z 8z PeH 8z
In K
z=l
3.2 Results and Discussion
Fig. 4. Sketch of the ammonia reactor system.
= kl (PN2 ( ~~2 PNH3
- 8z
(15)
Catalyst bed
r'
0- 8BI
dB~h!ll _ out
C.hell~
(C w , C1L).
TM ,,8Xi -HTH 8T
_ _ 1_8BI xPeH 8z z=o
The ammonia reaction is reversible. The work by (Andersen, 1999) indicates that the reversible nature of the reaction causes the Hopf bifurcation to be subcritical. Earlier work with bifurcation analysis of irreversible reactions has found supercritical Hopf bifurcations (Recke, 1998; Kienle et al., 1995). The feed effluent heat exchanger is of the shell and tube type. Each side is approximated with one well-mixed tank.
220L-~60:--~---8~0~----~10~0~----1~2~0~
Reactor outlet
z=o
0= 8Xi I 8z z=l (12)
(13)
0 .0 : :
= B/
/ n 8Xil z=o - xPeM 8z z=o
:
OOOoooo !!; ·····. o
Bin
U
.
:..:. •
= Xi
.
c,j
' 0
~300
0
0
.
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:
0
in Xi
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2340 f!! ~320
~260
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~ 2.K (P~H3) PH, _
(9)
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(10)
= blln T + ~ + b3 + b4 T + b5 T2 + b6 T3 (11)
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~490
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,,,
~350
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100
150
200
VI
,
250
300
350
100
50
x (%)
Region I
Fold (FLl
57.38
Hopf (HIL)
69.15
Cyclic Fold (Cw)
98 .04
Hopf (Hw)
102.1
Fold (Fu)
104.7
Cyclic Fold (CILl
128.0
II III
IV V
Static S U 1 2 2 2 1
VI
1
I
1
150
200
III
250
300
350
Dimensionless Time
Dimensionless Time Bifurcation
III
V
III
Periodic S U
Region
x (%)
Bifurcation
Fold (FLl
57.38
Hopf (HILl
69.15
Hopf (Hw)
102.1
Cyclic Fold (CILl
103.0
Fold (Fu)
104.7
1 1 1 2
1 1 1 1
2 1
I
Static S U 1
II
2
1
III
2
1
V
1
2
VII
1
2
I
1
1
Periodic S U
1 1
1
Fig. 6. Feed flow rate step change simulations for open and closed loop ammonia converter. left: Open (with hysteresis) , right: Closed (without hysteresis) . The number and stability of the solutions in the different regions can be seen from the tables. and an unstable periodic solution (and of cause a stable and an unstable steady state) . This region is potentially dangerous since a disturbance could result in the system settling to the stable periodic solution even if no flow rate changes occur. With some numerical manipulations it can be seen that this Hopf point (Hw) satisfy the conditions of theorem 1. The remaining work is to choose the gain and the cutoff frequency in the washout filter. It was chosen to use the same frequency as with the simple system investigated initially i.e. w = 0.5 . The gain has to be sufficiently large to ensure that the Hopf bifurcation of the closed loop is supercritical. The result of the one parameter bifurcation analysis for the closed loop , with x used as bifurcation parameter is depicted in figure 5. Comparing the two I-parameter bifurcation diagrams in figure 3 and 5 it is obvious that the region labeled IV is non-existing in the closed loop system. This result demonstrates the desired feature, that the nominal operating point formerly located in region IV is no longer coexisting with a stable periodic solution. Thus the possible hysteresis scenario is eliminated in the closed loop. In figure 6 a dynamic simulation sequence of subsequent step changes in feed flow
400 380 §:s60 ~
.2340 ~
8.320 E
~300
~280
H,,§_-;'~
0
0 0
0
0
"0
~260
240 220L-~60--~---8~0------1~0~0~----12~0--
Feed Flow Rate [%]
Fig. 5. I-Parameter bifurcation diagram for closed loop system, continuation of feed flow rate. (--) stable steady state, (- - -) unstable steady state, Hopf bifurcations (Hw , H 1L ) , fold bifurcation points (Fu ,Fd, (.) stable periodic solution, (0) unstable periodic solution and cyclic fold bifurcation points (CW , C1L) .
rate is performed for both the open and the closed loop. Initially the system is at the upper stable
573
steady state with )( = 1.0 (in open loop this is region IV, in closed loop it is region Ill) . At time T = 0 the flow rate is increased beyond the Hopf bifurcation point to )( 1.025 resulting in sustained oscillations for both systems, but as can be seen the amplitude of the closed loop is significantly smaller. The flow rate is then reduced back to )( = 1.0 here the hysteresis of the open loop system is seen since the system stays on the stable periodic solution. For the closed loop the system settles back on the upper stable steady state i.e. the hysteresis is eliminated. Next the flow rate is reduced further to )( = 0.95 i.e. below the upper cyclic fold of the open loop system (into region Ill) and now both systems settles to the upper stable steady state. The locations and types of solutions for open and closed loop are shown in figure 6, where the existence range of the stable periodic solutions is dramatically decreased in the closed loop. Figure 7 shows the control action necessary to stabilize the periodic solution after a step change in the feed flow rate from )( = 1.0 to )( = 1.025. From this figure it can be seen that the control requires changes of approximately 10 QC which is not unrealistic. Note that the feed temperature has to oscillate with a period time close to the thermal residence time of the bed.
=
4. CONCLUSIONS
In summary, bifurcation control is a very easy method to apply. Only a few conditions needs to be checked in order to assure that the method is applicable. The inclusion of washout filters further makes it possible to preserve the open loop stability scenario of the steady states and only affect the dynamics and thereby the stability 240.-----~----~------~----~--~
v
III
()
~235
:::l
~ ~ E
Q)
f~ ~;jIJ~---... Q)
u.
50
100
150
Dimensionless Time
Fig. 7. Control action (Bleed) necessary to transform the Hopf bifurcation from sub- to supercritical. Effect of feed flow rate change )( = 1.0 to )( = 1.025.
574
of the periodic solutions. The main advantage of the bifurcation control is that unstable periodic solutions can be transformed into stable small amplitude ones, which often can be considered as acceptable behaviour. On the specific example of an industrial ammonia converter it has been shown that the hysteresis scenario of the open loop can be eliminated with the bifurcation control investigated in this paper. This bifurcation control converts the subcritical Hopf bifurcation into a supercritical one. This control action additionally reduces the maximum amplitude or the stable oscillation drastically. All in all the dynamic behaviour is improved significantly.
5. REFERENCES
Abed , Eyad H. and Jyun-Horng Fu (1986). Local feedback stabilization and bifurcation control , I. hopf bifurcation. Systems & Control Letters 7,11-17. Abed , Eyad H., Hua O. Wang and Alberto Tesi (1995). Control of bifurcation and chaos. In: The Control Handbook (W. S. Levine, Ed.). eRC Press. Boca Raton, FL. Andersen , Britta R0nde (1999) . Nonlinear Dynamics of Catalytic Ammonia Reactors. PhD thesis. Technical University of Denmark. Department of Chemical Engineering. AnEsbjerg, Klavs , Torben Ravn dersen , Dirk Mller, Wolfgang Marquardt and Sten Bay J0rgensen (1998). Multiple steady states in heterogeneous azeotropic distillation sequences. Ind. Eng. Chem. Res. 37, 44344452. Kienle, A., G. Lauschke, V. Gehrke and E.D. Gilles (1995). On the dynamics of the circulation loop reactor- numerical methods and analysis. Chemical Engineering Science 50, 2361-2375. Morud, John C. and Sigurd Skogestad (1998). Analysis of instability in an industrial ammonia reactor. AIChE Journal 44, 888-895. Naess , L., A. Mjaavatten and J-O. Li (1993). Using dynamic process simulation from conception to normal operation of process plants. Computers chem. Engng. 17(5/6) , 585-600. Recke, Bodil (1998) . Nonlinear Dynamics and Control of Chemical Processes. PhD thesis. Technical University of Denmark. Department of Chemical Engineering. in Press. Salnikov, 1. Ye. (1948) . Themokinetic model of a homogeneous periodic reaction. Dokl. Akad. Nauk SSSR 60, 405-8. Scott, S. K. (1991). Chemical Chaos. number 24 In: The International Series of Monographs on Chemistry . Oxford University Press.
BIFURCATION CONTROL OF SAMPLE CHEMICAL REACTION SYSTEMS Bodil Recke >,1 Britta R0nde Andersen »,2 Sten Bay J0rgensen »>,3
FLS-Automation, H0ffdingsvej 34, DK2500 Valby, Denmark Haldor Tops0e A/S, Nym0llevej 55, DK2800 Lyngby, Denmark »> Dept. Chem. Eng. , DTU Build. 229, DK2800 Lyngby, Denmark
>
»
Abstract: Bifurcation control of Hopf bifurcations is applied to two systems exhibiting a subcritical Hopf bifurcation . A subcritical Hopf bifurcation is characterized by the emergence of an unstable periodic solution, quite often coexisting with a stable periodic solution in a hysteresis scenario. This type of scenario can lead to large amplitude oscillations if the Hopf bifurcation point is crossed. Therefore it is desirable to change this bifurcation scenario, which can be achieved with the bifurcation control described in this paper. The purpose of this type of control is to convert a subcritical Hopf bifurcation to a supercritical one. An additional benefit is that the resulting stable periodic solution will have a relatively small amplitude at least close to the Hopf bifurcation point . The method is first successfully applied to a small example system, to illustrate the calculations and subsequently to an industrial size heat integrated ammonia reactor . For the ammonia reactor the existence region of the sustained oscillation is reduced dramatically and the maximum amplitude is reduced from approximately 200°C to about 60°C. Furthermore the nominal operating point no longer coexist with stable periodic solution. Copyright © 2000 IFAC Keywords: Partial Differential Equations , Oscillation, Nonlinear analysis , Nonlinear control, Reactor Control
1. INTRODUCTION
(Recke, 1998; Andersen, 1999). In this paper specific attention is paid to Hopf bifurcations . These bifurcations often occur in recycle systems where either energy or mass is recycled. An industrial case resulting in oscillation with amplitudes in the order of 200°C was reported by (Naess et al. , 1993) . This quite damaging event was investigated by (Morud and Skogestad, 1998) and further elaborated on by (Andersen , 1999) . The purpose of this paper is to show how knowledge of the nonlinear dynamic behaviour can be utilized in designing a control for the process converting the dynamic behaviour of the closed loop system to a more desirable one. The chemical system investigated initially in the present paper is the simple Salnikov model. That
Complex dynamic behaviour is often experienced in chemical processes. This behaviour is associated with the nonlinearities in the process. These nonlinearities have the most pronounced effect around bifurcation points i.e. points where the system solutions change stability and/or number of possible solutions. This type of behaviour have been seen in a host of chemical systems as diverse as azeotropic distillation systems (Esbjerg et al. , 1998) and fixed bed reactors with recycle I
bre@fisautomation .com [email protected] Corresponding author, sbj@kt .dtu .dk
569
has the possibility of exhibiting periodic solutions. These oscillating solutions arise as a consequence of a supercritical Hopf-bifurcation resulting in a stable periodic solution and an unstable steady state. The periodic solution undergoes a cyclic fold bifurcation and finally terminates in a subcritical Hopf-bifurcation, where the steady state regains stability. The purpose of the bifurcation control used in this article is to convert the subcritical Hopf-bifurcation into a supercritical one. Additionally the bifurcation control should stabilize the bifurcation point it self. Furthermore the control applied is capable of diminishing the amplitude of the stable periodic solution obtained in the controlled system. The advantage of this is that the rather 'nasty' behaviour of an unstable periodic solution can be converted to a stable small amplitude periodic solution without changing the static behaviour of the system. The same bifurcation control is then applied to an industrial scale ammonia reactor with similar bifurcation behaviour. The results demonstrate that the dynamics of the industrial reactor can be changed significantly resulting in more desirable properties around optimal productivity operating conditions.
2. SIMPLE MODEL
The model investigated in this section is the so called Salnikov model proposed by (Salnikov, 1948 j Scott, 1991). This set of equations describes a system which exhibits oscillations for an exothermic reaction in a closed chemical system. It is included to illustrate the technic which will be applied to an industrial scale ammonia reactor in section 3. The simple reaction scheme involved is P-+A rate = kopj Eo = 0 A-+B+Q rate = kIa, kI = Al exp( -Ed RT) where it is assumed that P is a 'pool chemical' i.e. the concentration does not change as a function of time or has infinite capacity. A model in dim ensionless variables is given by. (1)
-d8 =
(2)
1 + 108
dT
o.exp
(8) + -1 108
- (8 - 8a )
•
50
o o
o
o
10
•••
I
o ~-o 0005
0 .01
0.01S
2.1 Open Loop Bifurcation Behaviour A brief review of the bifurcation behaviour of the Salnikov model can be found in (Scott, 1991). The bifurcation parameter shown in figure 1 is the initial (and constant) concentration of the precursor in the reaction (J1.) . The figure shows that two Hopf bifurcations exist in the present
0.02 0.025 003 Initial Precur80f Concentration
0 .035
0.04
0 .045
M
Fig. 1. Bifurcation behaviour showing the maximum temperature 8 with J1. as bifurcation parameter. (--) stable steady state, (- --) unstable steady state, (.) stable periodic solution, (0) unstable periodic solution . system. The one occurring at the lowest value of J1. is a supercritical Hopf leaving the formerly stable steady state unstable, and marking the initiation of a stable periodic solution. The Hopf bifurcation occurring at the higher value of J1. is a subcritical one, where an unstable periodic solution and an unstable steady state merges resulting in a stable steady state. In the bifurcation diagram the cyclic fold marking the exchange of stability of the periodic solution can also be seen.
2.2 Bifurcation Control
The purpose of bifurcation control is to transform the subcritical Hopf bifurcation occurring in the open loop system, to a supercritical one, and at the same time stabilize the bifurcation point itself. This effect can be accomplished with a non linear controller consisting of quadratic and cubic terms according to (Abed and Fu, 1986). The notation used in theorem 1 can be explained from a standard dynamical system given by: ;i;
-do. = J1. - Ko.exp ( -8- ) dT
• • •• •
= Jp. (x, u(x))
(3)
For the system described by equation (3) , at some parameter value of J.I. a bifurcation occurs. At this value the left and right eigenvectors ([' and r' respectively) can be calculated from.
= Lo Lor' = >..r' I'Lo = I' >.. ojp.
ox
(4) (5) (6)
Where>.. is the critical eigenvalue at the bifurcation point. The eigenvectors are then normalized by setting the first component of r' to 1 and then choosing /' such that Ir = 1. The vector notation
0 .05
Table 1 Minimum gains necessary to stabilize the bifurcation point for different w-values minimum stabilizing gain w-value 0.5 2e 3 2 0.059
with no I refers to the normalized version of the eigenvectors. "Y is defined as. "Y = !!j;; Theorem 1. Let the hypothesis (H) hold, (H) The matrix Ao = (8 f / 8x )(0,0) has a pair of simple, complex conjugate eigenvalues >'1 = iw c, >'2 = iwc on the imaginary axis, where Wc i- O. Moreover, all other eigenvalues of Ao have negative real part. Additionally assume that h i- O. That is, the critical eigenvalues are controllable for the linearized system. Then there is a feedback u(x) with u(O) = 0 which solves the local smooth feedback stabilization problem for x = f(x , u) and the local Hopf bifurcation control problem for equation (3) . Moreover, this can be accomplished with only third order terms in u(x) , leaving the critical eigenvalues unaffected. The condition is that h i- 0 is checked, and the results is that at the subcritical Hopf bifurcation point the value of h : : : 0 + 3.6737e- 3 i i- 0 (numerical values can be found in (Recke, 1998)) . Consequently it is possible to convert the subcritical Hopf bifurcation to a supercritical one by using a nonlinear feedback with only cubic terms. The intended manipulated variable in the system is Ba, and the following feedback control will be applied Ba = -GB3 .
the Hopf bifurcation, and one well after the transformation. By increasing the gain in the controller it is possible to not only transform the subcritical Hopf bifurcation into a supercritical one, but also to diminish the amplitude of the resulting periodic solution. The latter result implies that it is possible to reduce the effect of the periodic solution since small amplitude oscillations possibly will be acceptable in most chemical systems.
3. AMMONIA REACTOR 3.1 Open Loop Bifurcation Behaviour
Ammonia reactors are industrially important reactors. (Naess et al. , 1993) reports an incident with large amplitude temperature oscillations in an industrial ammonia reactor with three catalyst beds, quench cooling between the beds and a feed-effluent heat exchanger. (Morud and Skogestad, 1998) studied this incident and explained the temperature oscillations with a Hopf bifurcation. (Andersen, 1999) studied a simpler ammonia reactor with only one catalyst bed and a feedeffluent heat exchanger. The occurrence of periodic solutions was investigated using bifurcation analysis. Also for the simpler reactor sustained temperature oscillations can occur due to a Hopf bifurcation. However, the periodic solution arising in the Hopf bifurcation point Hw is unstable due to a subcritical Hopf bifurcation. Continuation of the unstable periodic solution revealed a cyclic fold bifurcation elL turning the periodic solution stable, figure 3. The subcritical Hopf bifurcation followed by a cyclic fold bifurcation explains the sudden large amplitude oscillations experienced in the industrial reactor. Thrning the sub critical Hopf bifurcation supercritical must be of interest to industry, not only because smaller amplitude oscillations are expected but also because hysteresis phenomena are avoided. Figure 4 shows a sketch of the reactor system consisting of a single catalyst bed with a feed-effluent heat exchanger. The model of this reactor is described in (Andersen, 1999) . The mass and energy balances are
2.3 Results and discussion Adding a washout filter to the system, as suggested by (Abed et al. , 1995) restricts the nonlinear control to changing the dynamic behaviour (i.e. the stability of the periodic solutions) , without changing the location and stability properties of the open loop steady states of the system. A washout filter is in essence just a stable high-pass filter. The reSUlting extended control system can be written as. dz dT = B - wzBa = -G (B - WZ)3 (7) Two parameters needs to be set now, are the gain in the controller (G) and w in the washout filter. The value of w determines the cutoff frequency. The obvious constraint on w, is that it must be positive. Furthermore it should be chosen below the characteristic frequency of the periodic solution . The required gain for stabilization of the bifurcation point and transformation of the bifurcation depends on the value of w. As a rule of thumb it can be said that the higher w the higher gain is necessary. In this application w = 0.5 is used. The result of two different gains are shown in figure 2, corresponding to one just before transformation of
571
50
40
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15
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10
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14
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•
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•
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2~,"
~~-0~,~~-0~,0~,~0,~0175~0~ ,02~~0,=~~0,0~3~0~=~~0~ , ~--~0,~~~o~·
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Initial Preanot ConcenIration (Ja)
Initial Precursor Concentration ijL)
Fig. 2. Closed loop bifurcation behaviour, depicting the maximum temperature of the periodic solution (B) with J.t as bifurcation parameter and w = 0.5 and gains left: 0.01 Oust before the Hopf bifurcation is transformed), right: 5 (well after transformation). (--) stable steady state, (---) unstable steady state, (e) stable periodic solution, (0) unstable periodic solution. with the boundary conditions 400
IV :V
III
....············8 OO~.lL
380
c,J00J'i '
o
~
00:
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00
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:
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,
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240
•: ... .
~
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•••
0
0
0
C:'L
ooooo~
Feed Flow Rate [%]
Fig. 3. I-Parameter bifurcation diagram for open loop system, continuation of feed flow rate. (--) stable steady state, (- - -) unstable steady state, Hopf bifurcations (Hw,Hld, fold bifurcation points (Fu,FL), (e) stable periodic solution, (0) unstable periodic solution and cyclic fold bifurcation points
dBin Ctube dT
Heat exchanger
Bad outlet
= Bleed -
, Bm + NTUtubeflBtube (14)
out - Bbed - B. hell + NTUshellflBshell
In this work it was chosen to use the feed temperature (Bleed) as actuator. As measurement the inlet temperature to the catalytic bed is used (Bin)such that the delay from the control action to the measurement is relatively small. It can be seen from figure 3 that the bifurcation point Hw is a subcritical Hopf bifurcation and thus an eligible point for bifurcation control. The purpose of the control is to convert the sub critical Hopf bifurcation into a supercritical one. This conversion will eliminate region IV where the upper stable steady state coexists with a stable
8 ( -n- 8X = - x8Xi - + --i) + Dar 8z 8z PeM 8z (8)
8B 8B = _x + i. (_1_8B) + /3Dar 8T 8z 8z PeH 8z
In K
z=l
3.2 Results and Discussion
Fig. 4. Sketch of the ammonia reactor system.
= kl (PN2 ( ~~2 PNH3
- 8z
(15)
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r'
0- 8BI
dB~h!ll _ out
C.hell~
(C w , C1L).
TM ,,8Xi -HTH 8T
_ _ 1_8BI xPeH 8z z=o
The ammonia reaction is reversible. The work by (Andersen, 1999) indicates that the reversible nature of the reaction causes the Hopf bifurcation to be subcritical. Earlier work with bifurcation analysis of irreversible reactions has found supercritical Hopf bifurcations (Recke, 1998; Kienle et al., 1995). The feed effluent heat exchanger is of the shell and tube type. Each side is approximated with one well-mixed tank.
220L-~60:--~---8~0~----~10~0~----1~2~0~
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z=o
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(13)
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= B/
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.
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100
150
200
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,
250
300
350
100
50
x (%)
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Fold (FLl
57.38
Hopf (HIL)
69.15
Cyclic Fold (Cw)
98 .04
Hopf (Hw)
102.1
Fold (Fu)
104.7
Cyclic Fold (CILl
128.0
II III
IV V
Static S U 1 2 2 2 1
VI
1
I
1
150
200
III
250
300
350
Dimensionless Time
Dimensionless Time Bifurcation
III
V
III
Periodic S U
Region
x (%)
Bifurcation
Fold (FLl
57.38
Hopf (HILl
69.15
Hopf (Hw)
102.1
Cyclic Fold (CILl
103.0
Fold (Fu)
104.7
1 1 1 2
1 1 1 1
2 1
I
Static S U 1
II
2
1
III
2
1
V
1
2
VII
1
2
I
1
1
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1
Fig. 6. Feed flow rate step change simulations for open and closed loop ammonia converter. left: Open (with hysteresis) , right: Closed (without hysteresis) . The number and stability of the solutions in the different regions can be seen from the tables. and an unstable periodic solution (and of cause a stable and an unstable steady state) . This region is potentially dangerous since a disturbance could result in the system settling to the stable periodic solution even if no flow rate changes occur. With some numerical manipulations it can be seen that this Hopf point (Hw) satisfy the conditions of theorem 1. The remaining work is to choose the gain and the cutoff frequency in the washout filter. It was chosen to use the same frequency as with the simple system investigated initially i.e. w = 0.5 . The gain has to be sufficiently large to ensure that the Hopf bifurcation of the closed loop is supercritical. The result of the one parameter bifurcation analysis for the closed loop , with x used as bifurcation parameter is depicted in figure 5. Comparing the two I-parameter bifurcation diagrams in figure 3 and 5 it is obvious that the region labeled IV is non-existing in the closed loop system. This result demonstrates the desired feature, that the nominal operating point formerly located in region IV is no longer coexisting with a stable periodic solution. Thus the possible hysteresis scenario is eliminated in the closed loop. In figure 6 a dynamic simulation sequence of subsequent step changes in feed flow
400 380 §:s60 ~
.2340 ~
8.320 E
~300
~280
H,,§_-;'~
0
0 0
0
0
"0
~260
240 220L-~60--~---8~0------1~0~0~----12~0--
Feed Flow Rate [%]
Fig. 5. I-Parameter bifurcation diagram for closed loop system, continuation of feed flow rate. (--) stable steady state, (- - -) unstable steady state, Hopf bifurcations (Hw , H 1L ) , fold bifurcation points (Fu ,Fd, (.) stable periodic solution, (0) unstable periodic solution and cyclic fold bifurcation points (CW , C1L) .
rate is performed for both the open and the closed loop. Initially the system is at the upper stable
573
steady state with )( = 1.0 (in open loop this is region IV, in closed loop it is region Ill) . At time T = 0 the flow rate is increased beyond the Hopf bifurcation point to )( 1.025 resulting in sustained oscillations for both systems, but as can be seen the amplitude of the closed loop is significantly smaller. The flow rate is then reduced back to )( = 1.0 here the hysteresis of the open loop system is seen since the system stays on the stable periodic solution. For the closed loop the system settles back on the upper stable steady state i.e. the hysteresis is eliminated. Next the flow rate is reduced further to )( = 0.95 i.e. below the upper cyclic fold of the open loop system (into region Ill) and now both systems settles to the upper stable steady state. The locations and types of solutions for open and closed loop are shown in figure 6, where the existence range of the stable periodic solutions is dramatically decreased in the closed loop. Figure 7 shows the control action necessary to stabilize the periodic solution after a step change in the feed flow rate from )( = 1.0 to )( = 1.025. From this figure it can be seen that the control requires changes of approximately 10 QC which is not unrealistic. Note that the feed temperature has to oscillate with a period time close to the thermal residence time of the bed.
=
4. CONCLUSIONS
In summary, bifurcation control is a very easy method to apply. Only a few conditions needs to be checked in order to assure that the method is applicable. The inclusion of washout filters further makes it possible to preserve the open loop stability scenario of the steady states and only affect the dynamics and thereby the stability 240.-----~----~------~----~--~
v
III
()
~235
:::l
~ ~ E
Q)
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u.
50
100
150
Dimensionless Time
Fig. 7. Control action (Bleed) necessary to transform the Hopf bifurcation from sub- to supercritical. Effect of feed flow rate change )( = 1.0 to )( = 1.025.
574
of the periodic solutions. The main advantage of the bifurcation control is that unstable periodic solutions can be transformed into stable small amplitude ones, which often can be considered as acceptable behaviour. On the specific example of an industrial ammonia converter it has been shown that the hysteresis scenario of the open loop can be eliminated with the bifurcation control investigated in this paper. This bifurcation control converts the subcritical Hopf bifurcation into a supercritical one. This control action additionally reduces the maximum amplitude or the stable oscillation drastically. All in all the dynamic behaviour is improved significantly.
5. REFERENCES
Abed , Eyad H. and Jyun-Horng Fu (1986). Local feedback stabilization and bifurcation control , I. hopf bifurcation. Systems & Control Letters 7,11-17. Abed , Eyad H., Hua O. Wang and Alberto Tesi (1995). Control of bifurcation and chaos. In: The Control Handbook (W. S. Levine, Ed.). eRC Press. Boca Raton, FL. Andersen , Britta R0nde (1999) . Nonlinear Dynamics of Catalytic Ammonia Reactors. PhD thesis. Technical University of Denmark. Department of Chemical Engineering. AnEsbjerg, Klavs , Torben Ravn dersen , Dirk Mller, Wolfgang Marquardt and Sten Bay J0rgensen (1998). Multiple steady states in heterogeneous azeotropic distillation sequences. Ind. Eng. Chem. Res. 37, 44344452. Kienle, A., G. Lauschke, V. Gehrke and E.D. Gilles (1995). On the dynamics of the circulation loop reactor- numerical methods and analysis. Chemical Engineering Science 50, 2361-2375. Morud, John C. and Sigurd Skogestad (1998). Analysis of instability in an industrial ammonia reactor. AIChE Journal 44, 888-895. Naess , L., A. Mjaavatten and J-O. Li (1993). Using dynamic process simulation from conception to normal operation of process plants. Computers chem. Engng. 17(5/6) , 585-600. Recke, Bodil (1998) . Nonlinear Dynamics and Control of Chemical Processes. PhD thesis. Technical University of Denmark. Department of Chemical Engineering. in Press. Salnikov, 1. Ye. (1948) . Themokinetic model of a homogeneous periodic reaction. Dokl. Akad. Nauk SSSR 60, 405-8. Scott, S. K. (1991). Chemical Chaos. number 24 In: The International Series of Monographs on Chemistry . Oxford University Press.