Climate modification directed by control theory

Transcription

1 Climate modiication directed by control theory Wang Liang (Department o Control cience and Control Engineering, Huazhong University o cience and Technology, WuHan, 3007, P.R.China, wl@smail.hust.edu.cn) [bstract] Climate modiication measures to counteract global warming receive some more new attentions in these years. Most current researches only discuss the impact o these measures to climate, but how to design such a climate regulator is still unknown. This paper shows the control theory could give the systematic direction or climate modiication. But the control analyzing also reveals that climate modiications should only be regarded as a last-ditch measure. 1 Introduction IPCC (Intergovernmental panel on climate change) announce its Fourth ssessment Report in daptation and mitigation options are suggested to avoid all climate change impacts. Besides the mitigation method like aorestation in this report (1), some more ambitious measures are also reconsidered in these years, including putting up space shields that cover billions o square meters, using chemicals to relect sunlight or increase Earth s cloud cover, stimulating massive growth o phytoplankton in the oceans (2-7). ssessment o climate mitigation or manipulation has also been reviewed under various ramings including economics, risk, politics, and environmental ethics (8). But all these work still can t answer these questions: How to adjust the measures according to prediction or actual impact o active manipulation? Can the project be readily reversed i it goes awry? How to evaluate the eect o one measure in conjunction with other methods? Obviously, a systematic strategy to modiy the climate system is still absent. Here we will shows the control theory could meet this requirement. 2 Basic analyses or climate control design Control theory is an interdisciplinary branch o engineering and mathematics, which deals with inluencing the behavior o dynamical systems. The controller is designed to manipulate the inputs to a system to obtain the desired eect on the output o the system. electing the proper model is the irst step or control research. There are three kinds o climate models, simple climate model, earth models o intermediate complexity (EMICs) and global climate models (GCMs) (9). Their complexities increase in turn. Many research shows the results given by most complex GCM can also be captured by much simple model. o we use simple climate model to design the control law and then veriy this law in sophisticated GCM models. Here we select a simple climate model considering the atmosphere and the underlying surace (10,11,12):

2 C C dt dt dts dt = (1 α ) a[1 + (1 a) α ] Q + εδt = (1 α )(1 a)(1 α ) Q δt s 2εT + εδt 3 s + H ( T H ( T s T ) T ) (1) C =.6 10 Wm K C = Wm K = 32Wm 8 2 δ = Wm K, = H = 5.9Wm K. Here a, ε = , α = , α = , s Q, T, T is average air temperature and surace air temperature. The description o other parameters could be ound in (10). This model is zero-dimensional, representing global mean and vertically integrated conditions. In the model, energy is exchanged between the surace, atmosphere and space by short and long-wave radiation and by latent and sensible heat. Then we should determine the appropriate control input or said control measure. The overwhelming majority o climate manipulation proposals aim to alter radiative energy luxes, either by increasing the amount o outgoing inrared radiation through reduction o atmospheric CO2, or by decreasing the amount o absorbed solar radiation through an increase in albedo. Here we will consider this measure as control input. In principle, the use o space-based solar shields has signiicant advantages over other options (Fig.1). Because solar shields eect a clean alteration o the solar constant, their side eects would be both less signiicant and more predictable than or other albedo modiication schemes (13). Fig.1. pace-based albedo modiication (a picture rom Internet) Many current climate control proposals belong to open loop, which only consider the impact o measures. But how to regulate these measures according to the modiication eect is not mentioned, which may result in the system unstable. For example, erosols produced in the lower stratosphere can counteract some o the eects o global warming. But excessive aerosols may overly oset the warming and bring adverse impacts. To avoid the problems o the open-loop, most current controller uses eedback to control states or outputs o a dynamical system. The name close-loop eedback comes rom the inormation path in the system: process inputs have an eect on the process outputs, which is measured with sensors and processed by

3 the controller; the result (the control signal) is used as input to the process, closing the loop. For aerosols measures, the amount o aerosols injected in the air should be adjusted according to its impact to climate. o Here U denotes the control input. s a close-loop control system, it should be the unction T,. o mark it as U T, T ). s an albedo modiication control, it could be added in T ormula (1) as ollows: C C ( dt = (1 α ) a[1 + (1 a) α ] Q[1 U ( T, T )] + εδt 2εTs + H ( Ts T) dt dts = (1 α )(1 a)(1 αs) Q[1 U ( T, T )] δt + εδt H ( T T) dt 3 Here U is the percent o energy luxes that the control implementer could curtain o. particular issue is the requirement or a control system to perorm properly in the presence o input and state constraints. In the physical world every signal is limited. For example, the energy we can manipulate is very limited comparing with climate system. o the designed controller should avoid sending control signals that cannot be ollowed by the physical system. Here set the control constraint as: U [0,3%] (3) Then we need decide a main control aim. For climate system, most related researchers think that manipulations need not be aimed at changing the environment, but rather may aim to maintain a desired environmental state against perturbations either natural or anthropogenic. This aim could be regarded as state regulator problem and constant tracing problem in control theory. The ollowing parts will give the detailed discussion or these two control problems. (2) 3 Climate state regulator design The state regulator is an operator driving the system to equilibrium state by minimum cost. The cost unction could be the energy consumption, total time, etc. We could deine the climate regulator problem as ollows. For given system (1), its equilibrium state is: T = K, T = K Then or some temporary perturbations, the temperature raise to: T = 27K, T = 292K The system structure has no changes. We want to drive the climate system to original state as soon as possible. This problem could be deined as: Initial state: T ( t ), T ( t )] (270.2,288.0) [ 0 0 = Final state: [ T ( t ), T ( t )] = (27,292)

4 Cost unction: J t = t0 1 dt Control aims: J = min(j ) Many dierent design methods are available this problem. The risks posed by climate control are suiciently novel that, in general, the relevant biological and geophysical science is too uncertain to allow quantitative. In control engineering, typically a simpler mathematical model is chosen in order to simpliy calculations; otherwise the true system dynamics can be so complicated that a complete model is impossible, especially or climate system. But no real physical system truly behaves like the series o dierential equations used to represent it mathematically. o the climate control system must always have some robustness property. robust controller is such that its properties do not change much i applied to a system slightly dierent rom the mathematical one used or its synthesis. Now there are several robust design methods like H design. Here we select the cell mapping design method, which is simple, robust and easy to be understood (1-16). This method belongs to Bellman dynamic programming method. We can depict this method in an easy way: it divides the state space into small rectangle regions called cells. The control input is also discretized. Then a search algorithm is applied to assign only control input or each rectangle region/cell to meet the optimal requirement. The main operation o cell mapping design method is described as ollows: 1) elect interesting state region. T [268,276], T [286,29]. 2) Construct cell. We divide the T and T into 2 6 = 6 equal pieces, so there are 6 6 = 096 cells. 3) Determine the control input set U. The input set is divide into 8 equal pieces: U = {0,0.37%,0.75%,1.13%,1.5%,1.87%,2.25%,2.62%,3%}. ) Obtain Mappings rom every cell with the entire set o control inormation U. For each cell, we need calculate N = 9 number o cell transitions. Here select one quarter as the integral time. U The related cost is also obtained in this step. 5) Decide the only control inormation U or each cell. Dynamic programming is employed to identiy this inormation. The search algorithm associates each cell with a control action that maps the cell to a cell trajectory with the optimal cost. Discrete Optimal Control Table (DOC) is obtained to represent the discrete global optimal control solution. ll the optimum trajectories rom every possible initial condition in the cell state space can be generated once this database is built. The detailed algorithm could be ound in Hsu s original paper. In our instance, there are 3271 controllable cells. cell is said to be controllable i there exists a sequence o controls which could bring this cell to the target cell. The controllable region and DOC table or this example is shown in Fig.2:

5 29 Controllable region Uncontrollable region 29 B (K) T (K) T 0.02 U T (K) T (K) 0 Fig.2. () Controllable region, (B) Optimal control table, the control value or each cell/region is represented by dierent color. 6) The optimal sequence o control is readily obtainable rom DOC. Because DOC stores the singe control input or every cell, in operation process, the system only need read the input value rom DOC according to the eedback state. The cell optimal solution o our system is shown in ig.3: 27 ystem response without control C 273 T (K) 272 ystem response under control U B t (quarters) 292 ystem response without control ystem response under control t (quarters) T (K) t (quarters) Fig.3. (), (B) Response o system, (C) Control input In Fig.3., B, we could ind the system need about 60 quarters to resume to equilibrium state without any control. But i we add the control, it only needs about 20 quarters to resume. Global warming oset design Now the main topic or climate modiication is the compensation o CO2 emission. It s a tracing problem. Here assume the CO2 reach 560ppm in 2150, two times o concentration beore industrialization. Then CO2 will not increase anymore. We simulate this scene in a complex

6 GCM model GIⅡ(17,18). The global air surace temperature will increase rom ~286.88K to a new equilibrium ~.3K ater 2090, which is shown in Fig (K) T Fig.. Global temperature rom 2050 to 2150 The temperature in 2065 is ~288.2K, which is similar to current condition. o we use it as the reerence scene. The temperature changes between 2056 and 2100 are shown in Fig.5. Fig.5. Temperature changes between 2056 and 2100 Here the control aim could be deined as reducing the temperature rom equilibrium state ~.3k to the state o 2065 (~288.2K) using least time. We also design the control law by simple climate model (1), where the impact o CO2 is eatured by ε. We could adjust this parameter to simulate this climate system. I ε = , the new equilibrium state is: T = K, T =. 3K, this temperature is close to the equilibrium state o GCM models. Because there is no a corresponding air temperature in GCM model, we simply assume T T in GCM. We begin the control in 2100, its = s T =. 3K, T = K. This problem could be deined as:

7 Initial state: T ( t ), T ( t )] (271.8,.3) [ 0 0 = Final state: [ T ( t ), T ( t )] = (269.7,288.2) Cost unction: J t = t0 1 dt Control aims: J = min(j ) We still use the cell mapping method to design this control law. Here select the interesting area: T [269,273], T [287,291], integral time:1 year, other setting is similar to the operation in paragraph 3. The DOC table o this system is shown in Fig.6: (K) T U T (K) 0 Fig.6. Optimal control table or global warming oset Then we veriy this control law in GCM model. In every year, we choose the control input rom DOC table according to the average temperature o ormer year. The global average temperature reaches the control aim in 2105, which is shown in Fig.7:.5 T (K) Fig.7. Global temperature rom 2050 to 2150 I there is no a control strategy, we may not oset the warming so precisely. The control will end in Unlike the regulator problem, the temperature will rise again in o the control should continue according to the temperature o The DOC table should also be

8 adjusted i CO2 concentration markedly changes. I concentration gets back to a normal level, the control strategy could be converted to state regulator control. The temperature changes between reerence year 2065 and 2105 are shown in Fig.8: Fig.8. Temperature changes between 2056 and 2105 From Fig.8, we could ind the compensation is not very uniorm, though the dierence o averages temperatures between 2056 and 2105 is only 0.03K. To avoid this problem, multiple compensation measures or more precise simple climate model and more complex design method could be considered in the uture. 5 Conclusions The control theory could make the climate modiication more reliable and sae. But real climate system is a complex nonlinear, decentralized, strong coupling, distributed-parameters system, which still can t be well grasped by current control theory. The climate modiication provides a new big challenge or control research. Moreover, the eiciency o control design strongly relies on how well we know the controlled system. But our knowledge about climate is still very limited. o we d better test this technology in re-building the environment o Mars, but not on the Earth. Now the more reasonable and direct method may be still to curtail emissions o greenhouse gases. Reerence 1. IPCC report, Climate Change 2007:ynthesis Report 2. W. K. David, Geoengineering the Climate: history and prospect. nnual Review Energy Environ,25, (2000) 3. National cademy o ciences. Policy implications o greenhouse warming: Mitigation, adaptation and the science base. Washington, D.C: National cademy Press, 1992

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