Robust Resource Management Control for CO2 Emission and Reduction of Greenhouse Effect: Stochastic Game Approach

With the increasingly severe global warming, investments in clean technology, reforestation and political action have been studied to reduce CO2 emission. In this study, a nonlinear stochastic model is proposed to describe the dynamics of CO2 emission with control inputs: clean technology, reforestation and carbon tax, under stochastic uncertainties. For the efficient resources management, a robust tracking control is designed to force resources tracking a desired reference output. The worst-case effect of stochastic parametric fluctuations, external disturbances and uncertain initial conditions on the tracking performance is considered and minimized from the dynamic game theory perspective. This stochastic game problem, in which one player (stochastic uncertainty) maximizes the tracking error and another player (control input) minimizes the tracking error, could be equivalent to a robust minimax tracking problem. To avoid solving the HJI, a fuzzy model is proposed to approximate the nonlinear CO2 emission model. Then the nonlinear stochastic game problem could be easily solved by fuzzy stochastic game approach via LMI technique.


Introduction
In recent years, the world has attracted much attention to environmental issues such as atmospheric pollution, conservation of water reserves and the reduction of tropical forests cover.For example, people feel concern about global warming, caused by greenhouse gases (GHG) such as carbon dioxide (CO 2 ), methane, nitrous oxide, sulfur hexafluoride, hydrofluorocarbons and perfluorocarbons, which leading to ecological destruction, climatic anomalies and sea level rise [1][2].However, despite the increasing environmental awareness, global economic success heavily relies on the industrial throughput.People have gained a better life following the expansion of industrial sector and the number of job positions.This has been achieved following the expresses of urban environmental quality, significant increase in pollution, and loss of natural habitats [3].In order to reduce the emissions of GHG, especially CO 2 , without limiting economic growth, substantial investments should target the development of clean technology, expansion of forested areas and some political actions [4][5][6].
A major problem associated with economic growth is the need for the energy, for which fossil fuel is the primary source.Such economic growth resulted in an increase of atmospheric emission of CO 2 , as shown in Table 1 [7].From Figure 1 [8], it is seen that from 1900, the global CO 2 emission increased year by year except in the European Union (EU) that decreased by 2% in the later period (1990)(1991)(1992)(1993)(1994)(1995)(1996), but it is still very high elsewhere.According to United Nations Environment Programme (UNEP) in 2007 [7], this decrease was possible due to many initiatives taken by Germany such as investing in renewable energy, solar power, new technology for car production, reforestation and political actions creating laws requiring 5% reduction of carbon emission.Recently, an indicator called the Ecological Footprint (EF) was concerned by UNEP to relate the 'pressure' exerted by human pollutions on the global ecosystems (Table 2) [7].The EF is expressed in terms of area, and according to the definition provided by WWF [9], it represents "how much productive land and sea is needed to provide the resources such as energy, water and raw materials used everyday.It also calculates   the emission generated from the oil, coal and gas burnt, and determines how much land is required to absorb the waste".This indicator is very useful in establishing how far the present situation is from the ideal condition in terms of emission of CO 2 .The worst EF indicators are found in North America and Western Europe.In order to mitigate the threat of an escalating greenhouse effect, it is necessary to establish a rigorous management process of the available resources to reduce CO 2 emission.These should include direct government incentive to promote pro-environment actions by the private sector and the establishment of stricter pollution regulations.To meet the CO 2 emission limitations and combat global warming, 193 parties (192 states and the EU) have signed and ratified the Kyoto Protocol to the United Nations.
Framework Convention on Climate Change (UNFCCC) [10].The cost estimated for the industrialized countries to implement the Kyoto Protocol ranges from 0.1 to 0.2% of their gross domestic product (GDP) [3].Based on mathematical dynamic models, these costs can be efficiently optimized through control theory methods.
In order to manage the resources commitment to achieve the desired control of CO 2 concentration, mathematical models are required.In previous studies, Nordbous (1991) [11,12] presented a mathematical model to describe the effect of GHG in the economy and to maximize a social welfare function, subject to dynamic constraints for the global temperature and atmospheric concentration of CO 2 .He carried out a study considering low, medium and high level of damages as a function of the concentration of CO 2 .In another study, Nordhous (1993) [13] used the same mathematical model to evaluate optimal taxation policies to stabilize climate and carbon emissions, i.e. enforcing political actions about taxes on the CO 2 emissions from burning coal, petroleum based products and natural gas.Poterba (1993) [14] has discussed the relationship between global warming and GDP growth and considered the influence of certain macroeconomic initiatives on the decrease of the atmospheric emission of CO 2 .For example, a consumption-linked carbon tax to reduce CO 2 emissions by 50% would reduce GDP by 4% in North America, 1% in Europe, and by 19% in some oil exporting countries.In the study of Stollery (1998) [15], an optimal CO 2 emission tax could be initially high, but it would eventually be lowered as emission decline due to energy resource depletion.He also showed that to sustain consumption in the face of both energy resource depletion and economic damage from global warming, it suffices to reinvest the sum of carbon tax revenues and the net energy rents.Caetano et al. (2008) [3] follows the ideas in Stollery [15] and offers a quantitative tool for the efficient allocation of resources to reduce the greenhouse effect caused by CO 2 emission.Their approach was developed by a mathematical model to describe the dynamic relation of CO 2 emission with investment in reforestation and clean technology, and propose a method to efficiently manage the available resources by casting an optimal control problem.Also an optimal tracking control of CO 2 emission was addressed to achieve the emission targets proposed in the Kyoto Protocol for European countries by numerically solving a Hamiltonian function [16].
In the above methods, ordinary nonlinear differential equations with time-invariant parameters are used to describe the deterministic dynamics among CO 2 emissions, forest area expansion and GDP growth.However, these intrinsic parameters may fluctuate area to area and time to time for different regional development or unpredictable situations, like sub-prime crash that initiated in 2007, which may lead to the necessity of estimating new parameters as time or let the control strategies be limited in some specific area.Further, the external disturbances, due to modeling error and environmental noise, should also be considered in order to mimic the real dynamics of CO 2 emission system.Therefore, the dynamical model of CO 2 emission system should be described by stochastic differential equations (SDEs).A differential equation containing a deterministic part and an additional random fluctuation term is called a stochastic differential equation, which has been frequently used to model diverse phenomena in physics, biology and finance [17].In this study, a nonlinear stochastic model is proposed to describe the dynamic system with model uncertainties from intrinsic parametric fluctuations, for the CO 2 emission with investments in reforestation, clean technology and political action about carbon tax.In addition to the intrinsic parametric fluctuations, external disturbances from modeling error and environmental noise are also included in the nonlinear stochastic model of CO 2 emission system, thus the generalized dynamic model could be widely applied to different area and time.Then a reference model is developed to generate the desired dynamics of CO 2 emission system.Finally, a robust model reference tracking control is proposed to manage these available resources, so that the nonlinear stochastic CO 2 emission system can track the desired output of the reference model, in spite of parametric fluctuations and external disturbances [18,19].Since the statistical knowledge of the parametric fluctuation, external disturbance and uncertain initial condition is always unavailable, based on robust H ∞ control theory, the worst-case effect of parametric fluctuations, external disturbances and uncertain initial conditions on the tracking error should be minimized by the control efforts, so that all possible effects on the desired reference tracking, due to these uncertainties, could be attenuated as small as possible.
The parametric fluctuations, external disturbances and uncertain initial conditions are considered as one player to maximize the tracking error, while the control of resource management is considered as another player to minimize the tracking error, from the dynamic game (minimax) theory perspective.This stochastic game problem could be equivalent to a robust minimax tracking problem, to achieve a prescribed reference output, in spite of the worst-case effect of parametric fluctuations, external disturbances and uncertain initial condition.Thus, solving the stochastic game problem for nonlinear stochastic CO 2 emission system will need to solve the Hamilton Jacobi inequality (HJI).At present, there is no analytic or numerical solution for the HJI except simple cases.To avoid solving the HJI for the nonlinear stochastic game problem, a Takagi-Sugeno (T-S) fuzzy model [20] is proposed to interpolate several linearized stochastic systems at different operation points, to approximate the nonlinear dynamics of CO 2 emission system.With the help of fuzzy approximation method to simplify the nonlinear stochastic game problem, it can be easily solved by the proposed fuzzy stochastic game approach via linear matrix inequality (LMI) technique with the help of Robust Control Toolbox in Matlab.Finally, some simulation results are given to confirm the robust minimax tracking performance of the proposed stochastic game approach for reducing the CO 2 emissions and greenhouse effect.
Mathematical Preliminaries: Before the further analysis of the stochastic CO 2 emission system, some definition and lemma of SDE are given in the following for the convenience of problem description and control design: Definition 1 (Ito SDE) [17]: For a given stochastic differential equation is the standard white noise with zero mean and unit variance to denote the random fluctuation, which can be considered as the derivative of Wiener process (or the Brownian motion).
is also a stochastic process satisfied with the following dynamic equation

Stochastic Model of CO 2 Emission under Parametric Fluctuation and External Disturbance
For the convenience of illustration, some ordinary nonlinear differential equations [3,16] has been proposed to model the dynamics of CO 2 emission.Taking account the political actions mentioned in the previous section, the modified equations by introducing a carbon tax control term are used [23,24].The more general deterministic model deals only with a few parameters to represent the dynamics of atmospheric CO 2 , forest area   z t and GDP   y t as follows [3,16]

r t z t u y t s z t u y t hz t y t y t u t
The first Equation in (1) cluding the effects of carbon tax and "virtual tax"-all the effects that are similar to carbon tax, i.e. energy cost rise, consumer prices rise, real wages fall and output and employment fall [25] that can also be directly or indirectly controlled by government order.In (1), 1 , 2 and 3 are control variables to be specified, so that the state variables

 
y t can achieve their desired reference outputs.
The above model has some limitations such as 1) the deterministic nature of economic growth (as expressed by GDP), 2) difficulty in limiting the geographic area, as one country, in political sense, can effect a neighboring state, 3) absence of time-varying parameters to adapt the model to changing situation.Further, the model is too simple and some factors may be neglected, i.e. there exist some un-modeled dynamics.In order to mimic the stochastic dynamics of CO 2 emission, the parameter fluctuations and external disturbances should be considered in the following stochastic model where i  denotes the standard deviation of stochastic parametric fluctuation, and denotes a standard white noise with unit variance, i.e.

 
and so on, i.e. the stochastic property of parametric fluctuations is absorbed by a white noise , and the amplitudes of parametric fluctuations are determined by their standard deviations i  respectively.Then the stochastic model for dynamics of CO 2 emission could be represented by ) For the convenience of analysis and design, the above stochastic CO 2 emission system can be represented by the following Ito stochastic system [17,26]     dw t n t dt  denotes a standard Wiener process or Brownian motion.Actually, the stochastic system for CO 2 emission in (5) can be extended to a more general stochastic CO 2 emission system as follows where denote the state vector, control input vector and external disturbance vector respectively.
denotes the nonlinear interaction vector among the state variables of the CO 2 emission system.
denotes the control input matrix.
denotes the noise dependent parameter fluctuation vector.In the more general model of ( 6), let and so on.Consider a reference model of the stochastic CO 2 emission system in ( 6) with a desired state output as follows where is the reference state vector,

 
are specified beforehand by designer, so that r x t can represent a desired system's state output for the stochastic system of CO 2 emission in (6) to follow.Then, the robust model reference tracking control is to design to make t must be as small as possible, in spite of the influence of stochastic parametric fluctuations, external disturbances and the uncertain ini- , and should be achieved simultaneously as following stochastic game problem where denotes the expectation, and the weighting matrices and are assumed diagonal as follow The diagonal element of denotes the punishii q Q ment on the corresponding tracking error and the diagonal ii of denotes the relative control cost.r R 2


denotes the upper bound of the stochastic game problem in (8).Since the worst-case effect of   v t ,   r t and on the tracking error x t  is minimized by control effort , from the energy point of view, the stochastic game problem in ( 8) is suitable for a robust minimax tracking problem in which the statistics of , and are unknown or uncertain, that are always met in practical control design case, for example, in the stochastic CO 2 emission system.

Remark 1:
If and are all deterministic, then the expectation in the denominator of ( 8) can be neglected.E Because it is not easy to solve the robust minimax tracking problem in (8) subject to ( 6) and ( 7) directly, an upper bound 2   of the minimax tracking problem is proposed to formulate a sub-optimal minimax tracking problem.After that, the sub-optimal minimax tracking problem is solved firstly, then the upper bound 2  is decreased as small as possible to approximate the real robust minimax tracking problem of the stochastic CO 2 emission system.
Since the denominator in ( 8) is independent of   u t t t and is not zero, equation ( 8) is equivalent to [27]   mi Let us denote From the above analysis, the stochastic game problem in (9) or ( 10) is equivalent to finding the worst-case disturbance and reference signal which maximize If there exist and such that the robust minima m solved, then they can satisfy the stochastic game problem in (8 well.Therefore, the first step of robust minimax tracking control design of stochastic CO 2 emission system is to so ng mi ax tr prob ) as lve the followi nim acking lem subject to the CO 2 emission system (6) and the desired reference model (7).
After that, the next step is to check whether the condition To solve the minimax tracking problem in (12), it is convenience to transform the problem into an equivalent minimax regulation problem.
Let us denote thus an augmented stochastic system of ( 6) and ( 7) is obtained as follows where 0 0 Then the minimax tracking problem in (12) can writen as the following minimax regulation problem be re- ) subject to (12) where Then the robust minimax tracking problem in ( 11) is equivalent to the following constrained minimax regulation problem where

Theorem 1:
The stochastic game problem in ( 16) for robust trackochastic CO 2 emission system could be llowing minimax tracking control ing control of st solved by the fo u  and the worst- where is the upper bound of minimax tracking problem (8), based on the analysis above, the minimax rol and the worst-case disturbance tracking subject to (19) In this case, the optimal tracking control   given by ( 17), i.e. 17) should be replaced via so following HJI lving the

Robust Minimax Tracking Control via Fuzzy Interpolation Method
In general, there is no analytic or num cal solution for the HJI in (19) to solve the constrained optimization problem in (21), for robust minim s not ust mi eri ax tracking control of the ate the eral linearized of robust minimax tracking control of stochastic CO 2 the stochastic CO 2 emission system in (6).Recently, T-S fuzzy model has been widely applied to approxim nonlinear system via interpolating sev systems at different operation points [18][19][20].By the fuzzy interpolation method, the nonlinear stochastic game problem could be transformed to a fuzzy stochastic game problem so that the HJI in (19) could be replaced by a set of linear matrix inequalities (LMI).In this situation, the nonlinear stochastic game problem in (8) could be easily solved by fuzzy stochastic game approach for the design t minimax tracking control to achieve a de emission system.
Suppose the nonlinear stochastic CO 2 emission system in (6) can be represented by T-S fuzzy model [20].The T-S model is a piecewise interpolation of several linearized models through membership functions.The fuzzy model is described by fuzzy if-then rules and will be employed to deal with the nonlinear stochastic game problem for robus sired CO 2 emission, under stochastic fluctuations, external disturbances and uncertain initial conditions.The i-th rule of fuzzy model for nonlinear stochastic system in ( 6) is of the following form [18,19] If   where ij F is the fuzzy set, i A , i B and i D are linearized system matrices, g is the number of premise variables and   1 z t , …   g z t are the premise variables.The fuzzy system in ( 24) is inferred as fol ws [18][19][20] lo easily in (25), so that

Th
zzy model in ( 25) is to interpolat li-L func and ar stochastic syst 4) to a the nontion in (6), respectively, by mark 3 her interpolatio ethods suc n be also empl ed to interp several linear stochastic systems to approximate the nonlinear stoch bases

    h x t
Fuzzy identification method [20] Re : Actually, in (25) 25), the augm oximated by t ented system in ( 14) can be also approximated by the following fuzzy system where 0 0 0 , , , 0 0 0 00 After the nonlinear augmented stochastic system in ( 14) is approximated by the T-S fuzzy system in (27), the nonlinear stochastic game problem in ( 14) and ( 16) replaced by solving the fuzzy stochastic game problem in (27) and (16).
Theorem 2: minimax tracking control and worst-se disturbance for fuzzy stochastic game problem in (16) subject to is The ca (27) are solved respectively as follows where P is the positive definite symmetric solution of th ccati-like inequalities e following Ri 1 1 0; f: the HJI in ( 19) is approximated by a set of algebraic inequalities in (29) and the inequality in (20) is also equivalent to the second inequality in (29) subject to (29) In order to solve the above constrained optimization problem in (30) by the conventional LMI method, the inequalities in (29) can be rewritten as following relaxed conditions [30] 2 Stochastic Game Approach Then, we let , and the inequalities in (31) can be equivale By the schur complement [27].The constrained optimization problem in ( 30) is equivalent to the following LMI-constrained optimization problem 1) The fuzzy basis function in ( 25) can be replaced by other interpolati n, for example, cubic spline function.
2) By fuzzy approximation, the HJI in ( 19) of nonlinear stochastic game problem for the robust minimax tracking of nonlinear stochastic CO 2 emission system is replaced by a set of inequalities in (29), which can be easily solved by LMI-constrained optimization in (33 rained optimization to solve 3) The const 0  and in ed by W (32), (33), can be easily solv decreasing 2   until there exists no 0 W  solution in ( 32), (33).4) After solving W an or the following relaxed conditions [30] 1 In order to solve the optimal tracking problem by LMI s equivalent to technique, the optimal tracking control i solving a common or following LMIs, i.e., if is solved from (36), then the optimal fuzzy t control can be obtained as According to the analysis above, the robust minimax tracking control of CO 2 emission system via fuzzy interpolation method is summarized as follows.
Design Procedure: Step 1. Give a desired reference model in (7) for the c CO 2 and construct fuzzy plant rules in (24) and (25).
Step 3. Give the weighting matrices and of minimax tracking problem in (8).

4.
C m ers are n to fit th to em e influence of disturbances on the CO 2 em m, the bounded standard deviations are as   t in (28).u Remark 5: The software package such as Robust Matlab can be employed to solve the LMI-constrained optimization problem in (32), (33) easi

Computational Simulation
onsider the stochastic CO 2 emission system in (5).The values of syste paramet given i Table 3 e actual CO 2 emission in Western Europe [3].In order phasize th ission syste sumed that 1 1 r  2).But these con rol efforts would limit the system behavior too rigid for actual performance hich could not guarantee the control ability of CO 2 emission under dist an (Figur ).In order to attenuate the effect of stochactic disturbance on CO 2 emission system and make a flexible control desi r actual demand immediately, the robu ax tracking control method will be applied after 20 For the robust minimax tracking control purpose, the reference model design requests a prescribed trajectory C .Thus, the system matrix r demand, w urb ces e 3 gn fo st minim 10.
behavior for O 2 emission system stant cont A and reference signal   r t should be specified, based on some standards in prior, to determine the transient response and steady state of the reference model, so that the desired reference signal can perform as a guideline for the tracking control system, for example, if the real parts of eigenvalues of r A are more negative, the tracking control system follow a trajectory prescribed by   r t sooner.In Europe, consider the historical data ge m 1 starting at 1960, it is reasonable to assume an average growth rate of GDP around 3.5%, and the present growth rate of GDP is around 4% for Europe.Moreover, the chan in total forest cover fro 990 to 2000 was positive due to reforestation, but corresponding to only 0.3%  per year [3].Thus, for the purpose of robust resource management control for CO 2 emission and reduction of greenhouse effect, the reference model is set via and the initial state value in 2010 as 3514.14 of clean techno to simulate the desired progressive process logy improvement, forest expansion and GDP increase after 2010.Therefore, based on the reference model, the CO 2 concentration could be decreased to the value in 1960, and GDP could reach a desired steady state that is prescribed without limiting the growth of GDP (Figure 4), i.e. the GDP growth can not be less than the original GDP growth rate 4% in Europe.And the expansion rate of forested area can also be higher than 0.3% until reach an appropriate value.
To avoid solving the HJI in Theorem 1, the T-S fuzzy model is employed to approximate the nonlinear stochastic system described in above section.For the conve-nience of control design, each state is taken with 3 operation points respectively, and triangle type membership functions are taken for the 27 Rules (Figure 5).In order to accomplish the robust minimax tracking performance of the desired reference signal, in spite of the worst influence of stochastic parametric fluctuation, environmental noise, and minimize the control efforts, a set of weighting matrices and are tuned up as follows Q R i.e. with a heavy penalty on the control effort and a light penalty on the tracking error in (8).
to track the desired reference signal to the end (Figure 6).In Figure 7, it shows the responses of the controlled CO 2 emission system with the robust minimax tracking control.As the CO 2 emission target is approached, both investments in reforestation and clean technology tend to decrease, and a positive carbon tax revenue could be achieved in the end.
From the simulation results, it is seen that the effect of intrinsic parametric fluctuations and external disturba erformance of the robust minimax tracking control via T-S fuzzy interpolation is quite satisfactory.

Discussion
From the computer simulation, it is shown that the CO 2 f 6 rm disturbances (Figure 3).To achieve actual demands, i.e. the system can track an appropriate reference model as soon as possible without limiting GDP growth, forest area increase and guarantee CO 2 emission decrease under nces on the reference model tracking of CO 2 emission system can be overcome efficiently by the proposed robust minimax tracking control design.Thus, the tracking p emission system with invariant control efforts can fit the actual data perfectly rom 19 0 to 2010, but could not guarantee its perfo ance under intrinsic or external disturbances or modeling errors, the robust model reference tracking control is proposed from a dynamic game theory perspective, and then can be efficiently solved by fuzzy stochastic game approach.
By employing the robust minimax tracking controls    ly by an acute descent to negative and then climb to positive gently, which means to ensure the unlimited GDP growth and following a desired reference model, the government should provide a financial subsidy to improve industrial throughput in early years, even it creates more pollution, until the scale enterprises can bear the loss of carbon tax.
By tuning the weighting matrices of error punishment and control cost , m en account, becau invariant when t obust minimax tracking control starts.In this study shown that the tracking error is nished by a low and a high control cost , which means to guarantee the robust minimax tracking control performance, the control strategy can endure more tracking error by using less control, thus making the control method efficiently and viably.
If the CO emission model is free of external disturbance, i.e.
, Although international cooperation from tradable quotas and permits can reduce CO 2 emission efficiently, uncertainties about compliance costs have caused countries to withdraw from negotiations.Without tuning any system parameters, these time-invariant control efforts could make the CO 2 emission system too rigid to respond for an immediate need or lead the CO 2 emission system toward an uncontrollable circumstance under disturbance, which may finally lose its control ability for actual dynamics of CO 2 emission system.Optimal control method guarantee the control performance.If the more flexible CO 2 emission targets can be made to incorporate optimum choices of investments with minimum impact on the GDP growth, i.e. taking account of the stochastic disturbances with respect to minimax tracking control problem, then climate agreements for reducing greenhouse effect may become more attractive and efficient [25,37].
In this study, the fuzzy interpolation technique is employed to approximate the nonlinear CO 2 emission system, so that LMIs technique is used to efficiently solve the nonlinear minimax optimization problem in our robust minimax tracking design procedure.Since the proposed robust minimax tracking control design can efficiently control the CO 2 emission in real time to protect environment from the global warming and reduce greenhouse effect, in the future, the applications of robust minimax tracking control design for environmental resource conservation and pollution control under stochastic disturbance would be potential in ecological and economic field.

Conclusions
If current GHG concentrations remai nstant, the eral centuries of inres and sea level rise.Slowing such climate change requires overcoming inertia in political, technological, and geophysical systems.To efficiently manage the resources commitment for decreasing the atmospheric CO 2 , mathematical methods have been proposed to help people make decision.However, how to ensure the desired CO 2 emission performance under stochastic disturbances is still important and infancy.In this study, based on robust control theory and dynamic game theory, a nonlinear stochastic game problem is equivalent to a nonlinear robust minimax tracking problem, for controlling the CO 2 emission system to achieve a desired time response under the influence of parametric fluctuations, environmental noises and unknown initial conditions.
To solve the nonlinear HJI-constrained problem for the robust minimax tracking control design is generally difficult.Instead of solving the HJI-constrained problem, a fuzzy stochastic game approach is proposed to transform this nonlinear robust minimax tracking control problem into a set of equivalent linear robust minimax problems.Such transformation allows us to solve an equivalent LMI-constrained problem for this robust minimax tracking control design in an easier way with the help of Robust Control Toolbox i Matlab.but also guarantees the tracking performance suboptimal con n.And the unknown stem also be considered as a random factor, thus this od can be used to control the CO 2 emission system tracking aro any feasible reference model whenever the control of this system starts.Although this theoretical method rests on the conservative suboptimal method, this fact doesn't frustrate its potential as a government policy guideline and the power of prediction in public decisionmaking.Once these obstacles have been surmounted, i.e. more rapid response by real time monitor via e-government implementation, this method would be powerful to control and manage the economic and ecological resource.What is more is that for its convenient and efficient control design for nonlinear systems with parametric fluctuation stochastic uncertainties, this dynamic game approach can be applied in other fields with similar demands.

Appendix A: Proof of Theorem 1 Equation Section (Next)
For the augmented stochastic system in ( de-note a Lyapunov energy function x  with   0 0. V  Then the regulation problem in ( 15) is equivalent to the following minimax problem Substituting (A2) to (A1) and by the fact that Then the minimax solution is given as follows If Equation ( 19) holds, then From the inequality in ( 16), the minimax solution should be less than For the fuzzy system in (27), let us denote a Lyapunov function , 0 Then the minimax regulation problem lent to the following in ( 16)

V x t dw t x t P h z t A x t Bu t Cv t D h z t h z t x t D PD x t t d t
Substituting (B2) to (B1) and by the fact that

Px t h z t h z t x t PB R B Px t h z t h z t x t D PD x t
In order to simplify the above equation, suppose the inequality in (29) hold, then

Figure 1 .
Figure 1.Global carbon emission.The CO 2 emission has an increasing trend in the world, except for an about 2% decreasing in recent period (1990~1996) in EU.
asymptotically stable matrix and   r t is a desired reference signal.Based on the model reference tracking control, r A and   r t . Since the parametric fluctuations are stochastic, external disturbance and iniuncertain, and reference signal   r t could be arbitrarily assigned, the robust model reference tracking control design should be specified, so that the worst-case effect of three uncertainties , .both the minimax tracking and robustness against uncertainties Control Toolbox in Matlab efficiently.5) If the conventional optimal tracking control in(22) is considered, i.e. the effect of disturbance   v t is not considered in the control design problem, then the optim to letting al tracking control problem is equivalen 2  t  in(8).The optimal fuzzy tracking control desi gn in(29) with   , i.e. solving a common positive definite symmetric matrix 0 P from the following inequalities[27]

Figure 2 .
Figure 2. Simulation and comparison between model and actual data.To fit the actual data, the invariant control efforts: reforestation u 1 , clean technology u 2 and CO 2 tax u 3 , are assumed to be 0.012%, 0.08% and 0 respectively [3].

Figure 3 .
Figure 3.The CO 2 emission system with invariant control efforts under stochastic disturbance.It is seen that the control ability would not be guaranteed under parametric fluctuations and environmental noises.

6 )Figure 4 .Figure 5 .
Figure 4.The desired trajectory of reference model for CO 2 e to decrease CO 2 to the value in 1960, i.e. χ(t) = 398 million t are both higher than 4% in each year.

Figure 6 .
Figure 6.The robust minimax tracking control in the simulation example.

Figure 7 .
Figure 7.The tr nimax tracking control, under the acking performance of CO 2 emission system to a desired reference model by the robust mi influence of parametric fluctuations and environmental noises.
etric fluctuation and environmental noise meas n co world would be committed to sev creasing global mean temperatu without take account of the effect of intrinsic and external disturbances in the design procedure could even not This robust minimax tracking method not only con-

Table 2 . Ecological footprints. Region Hectares/ per capita
The Ito type stochastic differential equation (Ito SDE) of If   in (8), i.e. the effect of   v t ,   is the grade of membership of . Since  