Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of a Climate Dynamic Model

Chemical Engineering: An International Journal (CEIJ), Vol. 1, No.2, 2024
11
BIFURCATION ANALYSIS AND MULTIOBJECTIVE
NONLINEAR MODEL PREDICTIVE CONTROL OF A
CLIMATE DYNAMIC MODEL
Lakshmi N Sridhar
Chemical Engineering Dept. University of Puerto Rico Mayaguez PR 00681
ABSTRACT
Rigorous bifurcation analysis and multi-objective nonlinear Model Predictive Control
Calculations are performed on a climate model involving atmosphere and ocean dynamics. The
Bifurcation Analysis showed the existence of unwanted oscillation causing Hopf Bifurcations
while the Multiobjective Nonlinear Model Predictive Control calculations resulted in the control
profiles exhibiting spikes. Both the Hopf bifurcations and the spikes were eliminated using an
activation factor involving the tanh function. Bifurcation analysis was performed using the
MATLAB software MATCONT while the multi-objective nonlinear model predictive control was
performed with the optimization language PYOMO. Numerical results are presented and
explained.
KEYWORDS
Climate, Atmosphere, Optimization, Control
1. INTRODUCTION
Climate dynamics is extremely complex and several models have been developed to understand
and control the highly nonlinear behavior of the air motion in and temperature gradients in the
atmosphere and the ocean. In this work, multiobjective nonlinear model predictive control and
bifurcation analysis are performed on a climate model that involves convective motion and
temperature gradient in a climate model involving atmosphere and ocean dynamics. This paper is
organized as follows. The background material is first presented followed by the model
description and details of the bifurcation analysis and the multiobjective nonlinear model
predictive control calculation procedures. The results and discussion are then presented.
2. BACKGROUND
The increase in global warming and the devastating effect of Hurricanes has motivated research
involving strategies to control and even attempt to modify the climate. Several books (Shepherd
et al 2009; Trenberth 1992; Washington, 2005 Olver and Bridgman 2014; Barry and Hall-
McKim, 2014; Weart, 2014; Dennis, 1980; Fleming 2010, Dijkstra, 2013, Summerhayes, 2015 )
have been published discussing the uncertain nonlinear patterns and the need to control the global
climate. Hoffmann (2002) discussed strategies to control the global weather. Curic et al (2007)
used the cloud-resolving mesoscale model to study cloud seeding impact on precipitation.
Mitchell and Finnegan (2009) investigated the possibility of modifying cirrus clouds to reduce
global warming. Significant research on climate change was performed by Garstang et al (2005),

12
Bengtsson(2006), Crutzen (2006), Wigley (2006), MacCracken(2009) Robock et al (2009),
McClellan et al (2012)and Guo et al (2015). Soldatenko and co-workers (2014, 2015, 2017) have
studied nonlinear dynamics and performed optimal control studies for mathematical models of
climate manipulation.
3. MOTIVATION AND OBJECTIVES
Almost all optimization and optimal control of mathematical models of climate dynamics involve
single-objective optimization. This work aims to perform multiobjective nonlinear model
predictive control in conjunction with bifurcation analysis of a mathematical model involving
climate dynamics. The model used is described in Soldatenko(2017) where the Earth’s Climate
System (ECS) considers both and atmosphere.
4. MODEL DESCRIPTION
The coupled nonlinear model, Soldatenko (2017) consists of the atmosphere and ocean
components. , ,
A A A
x y z represent the intensity of convective motion and horizontal and vertical
temperature gradients in the atmosphere and , ,
B B B
x y z represent the same variables in the ocean.
The dynamic model equations are
( ) ( )
( )
( ) ( )
( ) ( )
( )
A
A A B
A
A A A A B
A
A A A B
B
B B A
B
B B B B A
B
B B B A
dx
y x c ax k
dt
dy
rx y x z c ay k
dt
dz
x y bz cz
dt
dx
y x c ax k
dt
dy
rx y ax z c ay k
dt
dz
ax y bz cz
dt




   
    
  
   
    
  
(1)
The parameter values are 1; 0; 8 / 3; 10; 28; 0.1
a k b r
 
     
5. BIFURCATION ANALYSIS
There has been a lot of work in chemical engineering involving bifurcation analysis throughout
the years. The existence of multiple steady-states and oscillatory behavior in chemical processes
has led to a lot of computational and analytical work to explain the causes for these nonlinear
phenomena. Multiple steady states are caused by the existence of branch and limit points while
oscillatory behavior is caused by the existence of Hopf bifurcations points.
One of the most commonly used software to locate limit points, branch points, and Hopf
bifurcation points is MATCONT(Dhooge et al (2003¸2004) Govearts(2000), and Kuznetsov,
[1998)] This software detects Limit points(LP), branch points(BP) and Hopf bifurcation
points(HB). Consider an ODE system

13
( , )
x f x 

& (2)
n
x R
 Let the tangent plane at any point x be 1 2 3 4 1
[ , , , ,.... ]
n
v v v v v  . Define matrix A given
by
1 1 1 1 1 1
1 2 3 4
2 2 2 2 2 2
1 2 3 4
..........
..........
...........................................................
.................................................
n
n
f f f f f f
x x x x x
f f f f f f
x x x x x
A


     
     
     
     

1 2 3 4
..........
..........
n n n n n n
n
f f f f f f
x x x x x 
 
 
 
 
 
 
 
 
 
 
     
 
     
 
 
 
(3)
 is the bifurcation parameter. The matrix A can be written in a compact form as
[ | ]
f
A B




(4)
The tangent surface must satisfy
0
Av  (5)
For both limit and branch points the matrix B must be singular. For a limit point (LP) the n+1 th
component of the tangent vector 1
n
v  = 0 and for a branch point (BP) the matrix
T
A
v
 
 
 
must be
singular. For a Hopf bifurcation, the function det(2 ( , )@ )
x n
f x I
 should be zero. @ indicates
the bialternate product while n
I is the n-square identity matrix. More details can be found in
Kuznetsov ( 1998) and Govaerts (2000) . Sridhar [2011] used Matcont to perform bifurcation
analysis on chemical engineering problems.
6. MULTIOBJECTIVE NONLINEAR MODEL PREDICTIVE CONTROL
ALGORITHM
The multiobjective nonlinear model predictive control (MNLMPC) method was first proposed by
Flores Tlacuahuaz(2012) and used by Sridhar(2019). This method does not involve weighting
functions, nor does it impose additional constraints on the problem unlike the weighted function
or the epsilon correction method(Miettinen, 1999). In a problem involving a set of ODE
( , )
( , ) 0 ;
L U L U
dx
F x u
dt
h x u x x x u u u

    
(6)

14
the MNLMPC method first solves dynamic optimization problems independently
minimizing/maximizing each variable i
p individually. The minimization/maximization of i
p
will lead to the values
*
i
p . Then the optimization problem that will be solved is
* 2
min { }
( , ); ( , ) 0
;
i i
i
L U L U
p p
dx
subject to F x u h x u
dt
x x x u u u

 
   

(7)
This will provide the control values for various times. The first obtained control value is
implemented and the remaining is discarded. This procedure is repeated until the implemented
and the first obtained control values are the same. The optimization package in Python, Pyomo
(Hart et al, 2017), where the differential equations are automatically converted to a Nonlinear
Program (NLP) using the orthogonal collocation method will be used. The resulting nonlinear
optimization problem was solved using the solvers IPOPT (Wächter And Biegler, 2006) and
confirmed as a global solution with Baron (Tawarmalani, M. and N. V. Sahinidis 2005). To
summarize the steps of the algorithm are as follows
1. Minimize/maximize i
p subject to the differential and algebraic equations that govern the process
using Pyomo with IPOPT and Baron. This will lead to the value
*
i
p at various time intervals ti. The
subscript i is the index for each time step.
2. Minimize
* 2
{ }
i i
i
p p

 subject to the differential and algebraic equations that govern the process
using Pyomo with IPOPT and Baron. This will provide the control values for various times.
3. Implement the first obtained control values and discard the remaining.
Repeat steps 1 to 3 until there is an insignificant difference between the implemented and the first
obtained value of the control variables or if the Utopia point is achieved. The Utopia point is when
*
i i
p p
 for all
7. RESULTS AND DISCUSSION
7.1 Bifurcation Oscillations and Control Profile Spikes
Bifurcation analysis of some models reveals the existence of oscillation causing Hopf
bifurcations. Optimal control of some nonlinear ODE models produces spikes in the control
profile. Both oscillations and control profile spikes are inconvenient as they impede optimization
and make the implementation of control necessary control variables difficult. The tanh activation
factor is used in neural networks (Szandała, 2020; Kamalov et al (2021) ; Dubey et al, 2022; .
and in optimal control problems to eliminate spikes in the optimal control profile (Sridhar; 2023a,
2023b, 2023c).
Oscillations are similar to spikes and in the case of this climate model problem, oscillation
causing Hopf bifurcations and spikes in the control profiles are obtained when bifurcation
analysis and MNLMPC calculations are performed on the climate model equations (Eq. 1) . An
activation factor involving the tanh function is used to eliminate both control profile spikes and

15
Hopf bifurcation oscillations. When the variable c (Eq. 1), which is both the bifurcation
parameter and control variable was modified to (c tanh(c)/500) the oscillation causing Hopf
bifurcation and the spikes in the control profile disappeared.
7.2 Bifurcation Analysis of Climate Model
The bifurcation analysis reveals 2 Hopf bifurcation points the co-ordinates of which are
[ , , , , , , ]
A A A B B B
x y z x y z c = ( 8.085944, 8.297009, 27.274397, 10.623396, 12.229902,
28.400484 0.198679 ) and ( 4.362063 9.939176 42.764834 27.027898 36.028865 34.255114
2.063465 ). When the bifurcation parameter c (Eq. 1) is modified to (c tanh(c)/500) both Hopf
bifurcations disappear Fig. 1 shows both the bifurcation diagrams without and with the activation
factor. The two Hopf bifurcation points when the activation factor was not used are indicated by
the letter H in one of the curves.
7.3 MNLMC for Climate Model
First, the variables
0 0 0 0 0 0
, , , , ,
f f f f f f
t t t t t t
A A A B B B
x y z x y z
      are individually minimized in both
the cases, with and without the use of the tanh activation factor. In both the cases, each of the
mimized values was 0. The multiobjective nonlinear model predictive control involved the
minimization of
2 2 2 2 2 2
0 0 0 0 0 0
( 0) ( 0) ( 0) ( 0) ( 0) ,( 0)
f f f f f f
t t t t t t
A A A B B B
x y z x y z
         
      . In both
the cases the multiobjective optimal control problem resulted in an optimal value of 0 (Utopia
point). When no activation factor was used, the MNLMPC control value of c was
0.03896989876805711. When c was modified to (c tanh(c)/500) the MNLMPC control value of
c obtained was 0.7495817434219589.
Figures 2a, 2b, and 2c show the variables and control profiles when the activation factor was not
used. Figures 3a, 3b, and 3c show the same profiles when the activation factor was used. Fig. 2c
shows distinct spikes in the control profile. The spikes disappeared when the activation factor
was implemented (Fig. 3c).
The numerical results indicate that unwanted oscillation causing Hopf bifurcations and spikes in
the control profiles were effectively eliminated when the tanh function activation factor was
implemented.
8. CONCLUSIONS
The results of this work demonstrate the existence of oscillation causing Hopf bifurcation points
in climate models considering atmospheric and ocean dynamics. When the multiobjective
nonlinear model predictive control(MNLMPC) of this model was performed, spikes were
observed in the control profile. Both the Hopf bifurcations and spikes were eliminated when the
activation factor involving the tanh function was implemented. Bifurcation analysis and
MNLMPC calculations for climate models with time delay would be future work.
Data Availability Statement
All data used is presented in the paper

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Conflict of Interest
The author, Dr. Lakshmi N Sridhar has no conflict of interest.
ACKNOWLEDGMENT.
Dr. Sridhar thanks Dr. Biegler for introducing him to the tanh function and Dr. Carlos Ramirez
for encouraging him to write single-author papers.
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Fig. 1 (bifurcation curves without (two Hopf bifurcation points c =0.198679 ; c= 2.063465) and with
activation factor(no Hopf Bifurcations)
Fig. 2a (xa, ya,za without activation factor)

19
Fig. 2b (xb, yb,zb without activation factor)
Fig. 2c ( c vs t without activation factor; note the spikes;
MNLMPC value of c= 0.03896989876805711)

20
Fig. 3a (xa, ya,za with activation factor)

21
Fig. 3b (xb, yb,zb with activation factor)
Fig. 3c (c .vs. t with activation factor (no spikes))
MNLMPC value of c= 0.7495817434219589

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