Climatic Change DOI 10.1007/s10584-007-9298-4
A robust sequential CO2 emissions strategy based on optimal control of atmospheric CO2 concentrations Andrew J. Jarvis & Peter C. Young & David T. Leedal & Arun Chotai
Received: 28 July 2005 / Accepted: 7 May 2007 # Springer Science + Business Media B.V. 2007
Abstract This paper formally introduces the concept of mitigation as a stochastic control problem. This is illustrated by applying a digital state variable feedback control approach known as Non-Minimum State Space (NMSS) control to the problem of specifying carbon emissions to control atmospheric CO2 concentrations in the presence of uncertainty. It is shown that the control approach naturally lends itself to integrating both anticipatory and reflexive mitigation strategies within a single unified framework. The framework explicitly considers the closed-loop nature of climate mitigation, and employs a policy orientated optimisation procedure to specify the properties of this closed-loop system. The product of this exercise is a control law that is suitably conditioned to regulate atmospheric CO2 concentrations through assimilating online information within a 25-year review cycle framework. It is shown that the optimal control law is also robust when faced with significant levels of uncertainty about the functioning of the global carbon cycle.
1 Introduction Any action that regulates an aspect of global climate is, by definition, attributable to a global scale control mechanism. Most commonly, these mechanisms are manifest as the natural feedback processes which act to oppose disturbance, hence contributing to the generally equable state of the surface conditions on Earth. Recently however, when faced with the prospect of a self imposed disturbance on global climate, humans have begun considering an active regulation of the climate system through the judicious release of greenhouse gases. If implemented, this behaviour will also be a global scale control mechanism, although it will differ from its natural counterparts in that the outcome would (ideally) be achieved more by way of design. A. J. Jarvis (*) : D. T. Leedal : A. Chotai Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK e-mail:
[email protected] P. C. Young Centre for Research on Environmental Systems and Statistics, Lancaster University, Lancaster, UK
Climatic Change
Through linking climate responses to the anthropogenic greenhouse gas emissions that caused them, climate mitigation measures effectively set up closed-loop systems comprised of the environmental sub-system(s) linking emissions to climate and the socio-economic feedback sub-system(s) that link climate to emissions (see Fig. 1). As in all feedback control, these closed-loop systems will have a set of emergent dynamic properties that differ significantly from the dynamic properties of their component parts. Therefore, the properties of the socio-economic sub-system(s) need to be specified such that the closedloop system behaves as required. For example, an integrated assessment of this closed-loop system might attempt to specify levels of carbon taxation to modulate emissions in response to, or avoidance of, climate induced damages, such that the closed-loop system behaves in some sense optimally (Nordhaus 1991). Although not explicitly coined in these terms, this is in effect a form of optimal control system design, a branch of systems engineering that is in fairly widespread use in many areas of engineering, natural and social science, including economics, under the general heading of control theory (e.g. Richardson 1991). Indeed, Fiddaman (2002) acknowledged this fact and utilised simple empirical control elements within his integrated assessment of the climate–economy system. More generally, control systems design focuses on the design and analysis of feedback control systems from a dynamic systems perspective. This discipline has a long and distinguished pedigree and impacts on contemporary society in many ways through it being embedded in the majority of current technologies. Because of the obvious parallel between climate mitigation and feedback control, it appears logical that some of the numerous tools developed to design and evaluate feedback control systems could be brought to bear on this important area of application. In this paper we start to explore some of these possibilities paying particular attention to the issue of anticipatory and reflexive decision making in mitigation. Currently, the design and analysis of climate mitigation strategies appears to fall into one of two categories; being either anticipatory or reflexive. Anticipatory methods emphasise the predictability of the climate system and hence use models of that system to project future emissions scenarios required to deliver certain desired outcome(s). Notable methods in this category would include model inversion (e.g Wigley 1991) and model optimisation (e.g. Nordhaus 1991). Reflexive strategies on the other hand emphasise the uncertainty associated with the climate system and hence the role played by online observations when applying sequential corrections to mitigation strategies in order to compensate for the inability to predict the behaviour of the climate system accurately (e.g. Hammitt et al. 1992). Obviously, both paradigms are important, with the anticipatory methods providing valuable lead time for planning and the reflexive methods ensuring ultimate compliance to policy objectives in the face of uncertainty. One potential benefit of analysing climate mitigation from a control systems perspective is that it naturally lends itself to integrating model based anticipatory methods with reflexive online strategies through the explicit consideration of the associated feedback dynamics. This is because feedback control is Fig. 1 A schematic block diagram of the feedback relationship between climate responses and their mitigation
Climatic Change
generally comprised of three stages: an off-line model-based design followed by evaluation and refinement of this design on a simulated model of the system and eventual on-line implementation and evaluation on the real system. The off-line design invariably uses a control design model which, as with anticipatory methods such as inversion and optimisation, acts as a concise, contemporary summary of the dominant dynamics of the system to be controlled. However, in control systems design this information is used to construct a control law1 which is specified to anticipate the resultant closed-loop dynamics it will produce when assimilating and responding to the online information required to drive the real system along its desired trajectory. As a result, this control law could act as a concise description of the properties required by a climate mitigation response needed to deliver a particular closed-loop system. It could be argued that, in practice, if any mitigation strategy were to be implemented in accordance with the prevailing scientific view, then it is likely to be a hybrid of the anticipatory and reflexive approaches (Houghton et al. 1997). Initially, an anticipatory strategy could be specified using contemporary models and then, in future review cycles, corrections to this strategy would be made as new information became available, with these corrections probably taking the form of a data assimilation, model updating and anticipatory re-evaluation exercise. Although such an approach at first sight appears reasonable, it is not clear what the closed-loop properties of this type of hybrid approach will be because the dynamic nature of the correction mechanism is not explicitly considered. In contrast, with control systems design, the initial design explicitly includes the consideration of model error such that the resultant control law has, embedded within it, an appropriate compensation mechanism in preparation for the corrective steps that will inevitably be required when the control system and its associated control law is implemented online. In addition, it would be more convincing from a policy perspective to not have to rely on a strategy based on the ultimate elimination of model error, given that a certain proportion of the associated uncertainty may remain unresolved (Lempert and Schlesinger 2000). Again, from a control systems perspective the compensation action in a control law is invariably set up to eliminate such errors without the need for any re-design of the closed-loop system. Using the stabilization of atmospheric CO2 concentrations as an example, we will attempt to show that, provided a reasonable description of the dynamics of the global carbon cycle can be identified, feedback control could provide a useful framework for both specifying and analysing carbon emissions targets that explicitly account for the dynamic feedback nature implicit in both integrated assessment and sequential decision making methodologies. Although this paper is intended as an introductory exploration in order to stimulate further research in this area, certain interesting conclusions can be drawn. For example, the notions of optimal and robust mitigation strategies being mutually exclusive (Lempert and Schlesinger 2000) is challenged. This paper will be written from a control systems perspective and, therefore, requires some introduction to, and use of, the language of this discipline, although every attempt has been made to keep this to a minimum. Readers who are interested in exploring this further are initially pointed to the numerous texts on the subject (e.g. Franklin et al. 2002; Nise 2004). As mentioned above, feedback control is generally comprised of three stages; the model-based design, followed by 1
The control law is the numerical expression of the feedback to be placed around the system we wish to control. Amongst other things, it takes the output of that system and processes it in an appropriate manner to generate the input to that system. Because the form of this law can be subject to a design process, it can be specified such that the closed-loop system it produces has the required set of dynamic properties.
Climatic Change
evaluation on a simulated synthetic system and eventual the implementation on the real system. We have structured the remainder of the paper around the first two stages.
2 The feedback control design There are many methods of model-based feedback control system design but the main objectives of the associated design process are universal; to specify a control law (here for the level of anthropogenic CO2 emissions) such that the dynamics of the closed-loop system are consistent with the desired design criteria. For atmospheric CO2, these criteria could include: 1. The closed-loop system should attain its policy target in a stable, well behaved manner that attempts to reconcile certain value judgements attached to the application when doing so: i.e. given the available information, the anticipated closed-loop response is considered in some sense optimal. 2. The controlled system should be suitably insensitive to disturbances, modelling inaccuracies and observational noise i.e. the reflexive closed-loop response is considered in some sense robust. In climate mitigation, the concepts of optimality and robustness appear juxtaposed given the former relies on knowing a system, whilst the latter rightly emphasises the unknowable (Lempert and Schlesinger 2000). However, optimal control designs are often associated with robustness due to the properties control optimisation confers on the online performance of a controlled system, in conjunction with the inherent insensitivity to model uncertainty that is an important aspect of feedback control systems. By using an appropriate model-based control system design strategy that satisfies the two design criteria above, the resultant control law this design specifies for carbon emissions should reflect a sequential carbon emissions strategy that is both optimal and robust.2 Here, we will use the digital Non-Minimal State Space (NMSS) design approach of Young et al. (1987) because it appears particularly well suited to this climate application (Young and Garnier 2006; Jarvis et al. 2005) although no doubt other well established control design methods will also have much to offer. The key advantage of the NMSS approach is its ability to explicitly handle high order control design models when designing for the anticipated response. As a result, NMSS control can be seen as an important opportunity to embed a contemporary scientific view of the carbon cycle into the sequential decision making process. A full account of the NMSS control methodology lies beyond the scope of this paper so, in addition to the numerous excellent textbooks on digital control that will provide the necessary background, the reader is directed to the tutorial paper of Taylor et al. (2000), and the prior references therein, which provides both the theory behind NMSS control and convenient worked examples. In what follows, the NMSS control design process is developed within the specific context of controlling global atmospheric CO2 concentration. Note that, in this paper, we are using the term ‘robust’ in a very general sense. In particular disciplines however, it can have a more specific meaning. For example, in the systems and control literature it is connected with methods such as H-infinity control systems design, or its relative ‘exponential of quadratic’, which has been used in the NMSS system design (Taylor et al. 1996). Essentially, these are risk averse methods of design intended to minimise the sensitivity of the control system to uncertainty in the control design model parameters.
2
Climatic Change
Figure 2 shows the block diagram representation of the NMSS control structure to be utilised here. xD(k) is the long-term policy target3 for the atmospheric CO2, for example a particular atmospheric CO2 burden that has been agreed on in line with the definitions laid out in United Nations Framework Convention on Climate Change (UNFCCC). We have elected to make this equilibrium policy target external to the design process, as opposed to including it as a part of the optimisation to be conducted in Section 2.3. The aim here was to reflect the possibility that such a target is likely to be based on a political interpretation of what constitutes “dangerous anthropogenic interference” with the climate system. Hence, this pragmatic approach avoids having to quantify the very uncertain long term, global costs and benefits implicit in any interpretation of ‘dangerous’. x(k) is the anticipated atmospheric CO2 burden, not to be confused with the observed burden y(k) to be used later in the synthetic ‘online’ evaluation of the sequential decision making properties of the feedback control law. u(k) are the global CO2 emissions comprised of the target(s) set by the control law in addition to any exogenous emissions disturbances arising from unlegislated sources, u′(k). Here time is discrete reflecting the digital nature of the control and the discrete nature of any online sequential decision making process framed within a review cycle. As a result, a suitable uniform sampling interval, Δt, for the control design has to be chosen. The output of the control design will be a control law that will provide the one-step-ahead (i.e. Δt years) trajectory of CO2 emissions thereby giving a Δt years lead time for implementation of these emissions. Because of the long lead times required to plan the associated infrastructure for a given level of emissions, this sample interval needs to be quite coarse. Here, we have assumed a sampling interval of Δt=25 years. In Fig. 2 the Global Carbon Cycle (GCC) system is represented by a model of the zero dimension dominant mode dynamics of the active GCC. This is the element we wish to control both offline in the design and ultimately online. This is achieved through the action of a feedback control regulator and compensator. The regulator is designed such that the closed-loop is controllable4 and has dynamic properties in line with design criteria 1 above. The compensator will act to eliminate the error e(k)=xD(k)−y(k) by adjusting u(k) in the online implementation, where uncertainty exists, and is designed such that the closed-loop dynamics have properties in line with design criteria 2 above. Before considering the control system design, however, it is necessary to develop a suitable control design model that can be used within this context. 2.1 The control design model The concept of a control design model derives from the control and systems literature and is linked strongly with the ‘dominant mode’ behaviour of dynamic systems. It is well known (see e.g. Young 1999) that, extreme nonlinearity aside, the dynamic characteristics and
3
The script k denotes discrete-time variables sampled at the kth sample instant, as opposed to continuous time variables which are denoted by the script t following the convention in systems literature. For a uniform sampling interval of Δt time units, t=kΔt. 4 In order for the closed-loop to be deemed controllable the control law needs to take account of all states associated with the dominant mode behaviour of the system in question. This fact has lead to the development of stochastic State Variable Feedback (SVF) control to which NMSS belongs. In satisfying this important dynamic condition, the design process is at liberty to allocate the behaviour of the closed-loop as it sees fit. Failure to satisfy this limiting condition will result in restrictions when assigning the closed-loop behaviour.
Climatic Change
Fig. 2 The block diagram representation of the carbon cycle control design model SYSTEM in relation to its NMSS feedback controller comprised of both REGULATOR and COMPENSATOR actions. z−n is the backward shift operator such that x(k−n)=z−nx(k) which is used in order to express the single input, single output relationships between discrete variables. The values of the control model parameters a and b used in the control design are given in Table 1 along with the associated control gains g1−5
response of dynamic systems are often dominated by a relatively small number of dominant modes, as defined by the eigenvalues of the system and their relative importance in determining the output response of that system. Good examples of this are the impulse response functions used to summarise the dynamics of more complex GCC and climate models (see e.g. Joos et al. 1996). These dominant modes are clearly of importance, therefore, in the control system design since the control law needs to be designed such that it acts in light of the properties of these dominant modes and their associated states. Appropriate dominant mode models can be obtained either by estimation against real input– output data or data generated from planned experiments on a simulation model (Young 1999), as is the case with impulse response carbon cycle models (e.g. Maier-Reimer and Hasselmann 1987; Joos et al. 1996). For controlling atmospheric CO2 burden, we need to specify a suitable dominant mode relationship between the emissions input u(k) and the anticipated output perturbation in the atmospheric CO2 burden x(k). For this we have used the impulse response function of Joos et al. (1996) (hereafter referred to as the Joos model). From this we have derived a discretetime transfer function representation suitable for the control system design process. For clarity, we have simplified the Joos model slightly by merging the similar ‘fast’ modes so that it is comprised of just three dominant dynamic modes, one fast (years timescale), one intermediate (centuries timescale) and one slow (infinite timescale associated with the conservativity of the GCC) i.e. xðk Þ ¼ Gf þ Gi þ Gs uðk Þ
ð1Þ
where the G’s are first order discrete time transfer functions (see Table 1). Systems comprised of such a broad spectrum of response times are referred to as ‘stiff’ in control terms. In discretizing the Joos model, we have assumed the input u(k) changes linearly over
Climatic Change Table 1 The parameter values a and b of the third order carbon cycle control design model Eq. 1 and the associated control gains gi (see Fig. 2)
Gf Gi Gs Gg
−25% control design model
Standard control design model
+25% control design model
af=0.0 bf =7.8058 ai =0.8143 bi =5.5727 as =1.0 bs =2.8644 g1 =0.1415 g2 =−0.0909 g3 =−0.9278 g4 =−0.1730 g5 =0.0125
af =0.0 bf =9.2259 ai =0.8530 bi =5.9355 as =1.0 bs =3.8427 g1 =0.1321 g2 =−0.0868 g3 =−1.0131 g4 =−0.1367 g5 =0.0119
af =0.0 bf =10.5100 ai =0.8754 bi =6.1986 as =1.0 bs =4.8470 g1 =0.1253 g2 =−0.0835 g3 =−1.0819 g4 =−0.1184 g5 =0.0116
The control design model has been derived from the fourth order impulse response function of Joos et al. (1996) by a re-estimation against the response of the Joos model to a ramp (rather than impulse) input, sub-sampled at intervals of 25 years and assuming a first order hold on the inputs over the sample interval. The ±25% parameter values used for the model mismatch evaluations shown in Figs. 4 and 6 are also given. The control law gains gi have been derived using these ‘wrong’ control model parameters whilst minimising the objective function (Eq. 5) with respect to gi for a 1,000-year unit step increase of xD(k) using a simplex search procedure.
the 25-year sample interval so that, when we come to the evaluation, we can apply the 25year emissions trajectory on an annual time-step with emissions changing linearly over a 25-year period, as opposed to being held constant. Future control designs could explore more exotic discretisations, although it is felt that the linear approximation is a useful starting point. The discrete-time parameters of the first order elements Gf, Gi and Gs are given in Table 1. 2.2 NMSS control system structure Because the control design model Eq. 1 possess third order dynamics, any method of State Variable Feedback (SVF) control for this GCC system needs, in some explicit or implicit manner, to take 2account of the following five continuous time states associated with this 3 2 model; d dxtð3tÞ , d dxtð2tÞ , d xdðttÞ; , d dut2ðtÞ and d duðttÞ . Young et al. (1987) recognised that this information was contained in present and passed values of the discrete-time input and output variables u(k) and x(k). As a result, the discrete time NMSS regulator control action can be designed on present and past values of u(k) and x(k) because these are always available for direct measurement in digital systems, hence avoiding the need for any state variable estimation or ‘observer’ procedure, such as the Kalman Filter (Kalman 1960). For the discretized version of Eq. 1 we have adopted, the appropriate regulator emissions uR(k) are given by the following NMSS control law; uR ðk þ $tÞ ¼ g1 xðk Þ þ g2 xðk $t Þ þ g3 uðk Þ þ g4 uðk $tÞ
ð2Þ
Because the values of the control gains gi ; i=1,2,...,4 can be specified by the designer, this provides the opportunity to transpose the anticipated dynamic properties of the active GCC, as expressed in the control design model, into the dynamic properties consistent with
Climatic Change
the design criteria 1 above, using an optimization procedure5 (see Section 2.3 below). However, satisfying design criteria 2 requires an additional compensator action in the feedback control. This action is based on eliminating the difference between the desired policy xD(k) and the anticipated response x(k) (or the observed response y(k) when implemented online). This can be achieved using the ubiquitous ‘integral-of-error’ feedback control action, given by the following compensator control law, uC ðk þ $tÞ ¼ uC ðk Þ þ g5 ½xD ðk Þ xðk Þ
ð3Þ
where g5 is referred to as the ‘integral action control gain’. Despite being included as a compensator action, the feedback effects associated with the integral-of-error control introduces important additional dynamics into the closed-loop response and, hence, it must be considered in the design at the same time as specifying the properties of the regulator. This can be seen from Eq. 3 where it is clear that the integral action control gain g5 fundamentally relates deviations from the policy target to the first difference of the compensator emissions. Fortunately, the NMSS control system design procedure, as discussed in the next Section 2.3, handles both the regulatory and compensatory aspects of the control within a single design computation. The NMSS feedback control law, which has been termed the ‘Proportional-Integral-Plus’ (PIP) controller (Young et al. 1987), is then given by uðk þ $t Þ ¼ uC ðk þ $tÞ uR ðk þ $t Þ (see Fig. 2) or, uðk þ $t Þ ¼ g1 xðk Þ þ ðg1 g2 Þxðk $tÞ þ g2 xðk 2$t Þ þð1 g3 Þuðk Þ þ ðg3 g4 Þuðk $t Þ þ g4 uðk 2$t Þ þg5 ðxD ðk Þ xðk ÞÞ
ð4Þ
2.3 Optimization of the NMSS control system design Having identified the appropriate control law, we are now in a position to assign values to the control parameters gi that satisfy the design criteria 1 and 2. In particular, this section will focus on reflecting certain mitigation tradeoffs in the dynamics of the closed-loop by finding the values of the gi that minimises an application specific objective function. The choice of the objectives obviously depends on one’s perspective, as does the choice of the relative weights to be placed on each term to be traded. We have attempted to keep the objectives simple by avoiding quantities like ‘climate damages’, which appear very uncertain, whilst including quantities like ‘policy adherence’ that are in sympathy with the control approach. Further development of the objectives of the optimal control for this application will no doubt be a subject for later studies. However, the key point to note is that control optimization will allow for further objectives to be considered, providing the states one includes in the objective function relate to the closed-loop dynamics being conditioned by the optimization.
5
Optimization is not the only method by which the closed-loop dynamics of the controlled system can be obtained, although it appears a logical choice in the current context because of its obvious parallels with integrated assessment studies in this area. Alternative methodologies for designing the closed-loop system could include pole placement, where the values of the control gains gi are identified by first assigning the required values of the poles of the closed-loop and then analytically deriving the associated values of the control gains from these.
Climatic Change
We have elected to minimise the following policy-orientated quadratic objective function. J fgi g ¼
40 h i2 h i2 X w1 xD ðk Þ xðk Þ jxðk Þ< xD ðk Þ þ w2 xðk Þ xD ðk Þ jxðk Þ>xD ðk Þ þ ½w3 $uðk Þ2 : k¼1
ð5Þ Minimising this objective function attempts to reflect the trade off between three objectives; not undershooting the policy xD(k); not overshooting this policy; and minimising the changes in the emissions level u(k) in order to bring this about. Minimising the total amount of deviations from the policy, xD(k)−x(k), is required in the objective function to condition the integral-of-error compensator response. We have separated this into its overshoot and undershoot components in order to reflect the differing values that may be attached to these two conditions. Minimising undershoot ensures maximum use of the agreed available headspace created by the atmospheric CO2 level policy. The rate of consumption of this headspace depends on the relative value of the weights w1 to w3 and determines the time preference in the use of the available environmental capacity. Minimising the amount of policy overshoot as an objective reflects a desire to avoid unnecessary climate risks (Hare and Meinshausen 2006). Minimising the amount of change in u(k) limits the costs associated with the socio-economic disturbance these changes imply. No distinction between rates of increase and rates of decrease in u(k) are made here. The quadratic terms in Eq. 5 are to provide definition to the response surface of J in relation to the control parameters gi, hence aiding the optimisation. Because the weights w1, w2 and w3 convert the terms in Eq. 5 into a common unit, they are central to defining the relative tradeoffs between the three stated objectives. For w3 we have assumed a value of 1.0 for monetary units of $1012 and emission units of GtC/a. Given the rather novel nature of the policy based objectives associated with w1 and w2, defining monetary conversions for these weights becomes somewhat more problematic. For atmospheric burden units of GtC, we have assumed w1 =0.02 simply because this value delivers an anticipated closed-loop response that consumes the environmental headspace within two centuries (see Fig. 3). For w2 we have assumed the penalty for overshoot is five times that of undershoot i.e. w2 =0.1. Of course, we must stress that these assumed weights are by no means definitive and obviously merit further research, but they will suffice for the present illustration. Using these weights, the control law (Eq. 4) is coupled with the control design model Eq. 1 to form the closed-loop. The cost function (Eq. 5) is computed from this for a 1,000year (40 sample) unit step increase of xD(k), and this objective is optimized with respect to gi using a simplex search procedure.6 The values of gi for the standard and ±25% parameterizations of the control design model are given in Table 1. These are to be used in the evaluation of the control design considered in the next section. For the standard
6
This optimisation has been adapted from, but differs significantly to, the more standard optimal control design approach for SVF control, which generally exploits analytical solutions of state space linear quadratic objective functions using the appropriate matrix Riccatti equation. Such an approach could yield a very similar closed-loop response but we have utilized the alternative approach here so that the cost function has a clearer meaning within the climate context.
Climatic Change
3 Atmospheric burden (GtC)
Fig. 3 a The anticipated optimal closed-loop response of atmospheric CO2 burden (squares) to a unit step increase in the policy xD(k) (dashed-black). This response is comprised of two components, a first order response with a pole of 0.4039 (inverted triangles) and a slower, mildly oscillatory second order response (triangles) with poles of 0.5928± 0.2229i. The anticipated emissions for this closed-loop response are shown in b
2 1 0 -1 -2 -3 0
100
200 300 Time (years)
400
200 300 Time (years)
400
500
a
25
Emissions (MtC/a)
20 15 10 5 0 0
100
500
b
parameterization of the control system design model, the optimal feedback control law is computed as: uðk þ $tÞ ¼ 0:1321xðk Þ þ 0:2189xðk $tÞ 0:0868xðk 2$tÞ þ2:0131uðk Þ 0:8764uðk $tÞ 0:1367uðk 2$t Þ þ0:0119ðxD ðk Þ xðk ÞÞ
ð6Þ
. 3 Evaluation of the control design The anticipated closed-loop responses of x(k) and u(k) to a unit step increase in the atmospheric CO2 policy xD(k) are shown in Fig. 3. From this, we see that the closed-loop reaches its desired policy level after approximately 200 years with no appreciable overshoot. The characteristics of this response are represented by the closed-loop poles of which there are five; 0.0; 0.4039; 0.5928±0.2229i; and 0.9475. These equate to four component responses in the closed-loop. The two responses associated with the poles of 0.0 and 0.9475 play a relatively trivial role and the closed-loop dynamics are dominated by the effects of the three poles 0.4039 and 0.5928±0.2229i (see Fig. 3a). The two responses associated with these three poles are the emergent, dominant closed-loop dynamics resulting from the optimal feedback control law (Eq. 6) being placed around our anticipated
Climatic Change
GCC system (Eq. 1). Note how these dominant closed-loop poles differ markedly from those of the control design model which are 0.0; 0.8530; and 1.0 in the standard case (see Table 1). This demonstrates how the feedback design affords the opportunity to specify dynamic properties of the closed-loop which differ significantly from those of the system we wish to control. In order to evaluate our control system design further, the control law can be tested as if it is being used online by applying it to a surrogate model for the real GCC system. Here, we use the full Joos model as the surrogate system and the evaluation considers two scenarios. Firstly, that the control design is based on the ‘wrong’ control design model, i.e. the anticipated response and, hence, the control law is parameterised incorrectly, a condition otherwise known as model-mismatch. Secondly, that the closed-loop experiences unpredicted exogenous disturbances in the form of additional unlegislated inputs of CO2. For both of these simulations, the simulation time-step is annual, whilst the control law only updates u(k+1) to u(k+25) every 25 years. This provides the 25-year-ahead emissions policy into which the simulation is locked until the full 25 years has elapsed and a new review cycle commences. Figure 4 shows the results from the model mismatch evaluation. Here, the control system design model used to derive the control law is parameterised either 25% below or above the full Joos model response (see Fig. 4 for details). From this we see that, despite this level of mismatch between the actual and anticipated system dynamics, the control law reproduces closed-loop dynamics for y(k) that are close to the anticipated closed-loop behaviour for x (k), demonstrating the robustness of the design. Figure 4 also shows what happens if the anticipated response x(k) rather than y(k) is fed-back under the same conditions of model mismatch. This is, in effect, an open-loop response analogous to model inversion. Here, the lack of online compensation action based on y(k) causes the policy target to be missed by a considerable margin, highlighting the effectiveness of the assimilation of y(k) by the control compensator and also the danger of such open-loop strategies. Note that there is no re-evaluation of the control parameters gi in the model-mismatch evaluation. Instead, the control design remains unchanged and the online information in the form of the observed response y(k) of the surrogate system is assimilated through the control law, with the compensator action generating the necessary corrective measures. Indeed, in normal control systems design, significant online revisions to the control law properties are only performed if it is likely that the closed-loop dynamics will become severely compromised otherwise. Usually, the ability of the feedback control law to make the closed-loop behaviour relatively insensitive to model uncertainty and change proves sufficient and results in acceptable robustness in this regard. Of course, if the changes in the GCC system dynamics happened to be sufficient to cause deleterious changes in the closed-loop performance, it would be necessary to introduce an adaptive control feature, where online recursive parameter estimation tracks significant changes in the model of the GCC system dynamics and the control system design is updated on the basis of these estimated parametric changes (see e.g. Young 1984). This procedure is analogous to the data assimilation and model updating discussed in the introduction. This situation could arise from significant nonlinear events, such as reorganisation of the thermo-haline circulation, dramatically changing the closed-loop properties (Keller et al. 2004). Given the relatively long timescales involved in this particular climate application when compared with say, controlling electronic devices, coupled with the inevitable advances in the understanding of the GCC between review cycles, we envisage that an adaptive control framework would probably be unavoidable. However, extreme nonlinearity aside, the success of the control system performance with
Climatic Change Fig. 4 a The evaluation of the closed-loop response of atmospheric CO2 burden under conditions of model uncertainty. The simulation uses the impulse response function of Joos et al. (1996) as the surrogate system run on an annual timestep. The control law driving the closed loop provides the 25-year step ahead emissions targets shown in b (squares). This control law has been optimised using the ‘correct’ control design model parameters (solid black line in a and b) or the ‘incorrect’ ±25% parameters (grey envelope in a and b; see Table 1 for parameter values). In each case the closed loop output is being required to respond to a unit step increase in the policy xD(k) (dashed-black line). Also shown for comparison is the open-loop response (dotted lines) which is analogous to the anticipated response where only the control design model output is used
respect to CO2 stabilisation does not appear to be dependent on this happening, indicating that even a relatively uncertain knowledge of the GCC is still sufficient to specify an appropriate control for this system. To further assess the ability of the control law to deliver the policy target in the face of unpredicted future events, we can introduce significant exogenous disturbances into the online simulation. Here, this disturbance takes the form of an additional input of carbon into the atmosphere that is independent of the control input (see u′(k) in Fig. 2). This could be viewed either as failure to meet an anthropogenic emission target; additional emissions arising from land use change; and/or natural emissions arising from climate change induced carbon sources/sinks falling outside the domain of any anticipated GCC response. Figure 5 shows the closed-loop response to a one GtC disturbance added to the atmosphere in one year. Without any control, this produces the familiar (unmitigated) impulse response of the Joos model, whereas the controller mitigates this disturbance returning the atmospheric CO2 burden to its initial condition by removing exactly one GtC over the following 300 years. Note that the peak in emissions mitigation occurs some 75 years after the initial disturbance, highlighting the degree of lag built into the mitigation and, hence, the rather conservative nature of this particular parameterization of the control law in relation to input
Climatic Change
1 Atmospheric burden (GtC)
Fig. 5 a The closed-loop evaluation of atmospheric CO2 burden (solid-black) to a unit impulse input disturbance of carbon added to the atmosphere. The associated mitigation of this emissions disturbance is shown in b. The simulations as in Fig. 3 uses the Joos model as the surrogate system run on an annual time-step, whilst the control law is run on a 25-year time-step providing the 25-year ahead emissions trajectory (b squares). The emissions disturbance has also been put into the Joos model with no feedback mitigation for comparison (a dashed)
0.8 0.6 0.4 0.2 0 0
100
200 300 Time (years)
400
200 300 Time (years)
400
500
a
Emissions (MtC/a)
0 -2 -4 -6 -8 -10 0
100
500
b
disturbances. Also note the response to this disturbance is delayed by one review cycle (25 years) as it is a reflexive response. Finally, we have attempted to construct a set of scenarios based on the current conditions and leading to the universal adoption of various atmospheric CO2 stabilisations under conditions of model mismatch. Under these conditions, a significant initial condition is developed, with unregulated anthropogenic emission occurring until 2010, after which time the control law is used to drive the system to stabilisation objectives of 450, 550 and 650 ppmv. This leads to the need to prime the control law correctly when ‘switched on’ in 2010 in order to avoid spurious start-up dynamics. Again, the GCC system follows the full Joos model, whilst the control system design model is anticipated to be either 25% above or below this response when deriving the control law. Figure 6 shows that the transition to the controlled state appears reasonable and that the control law does, indeed, drive the atmospheric CO2 towards the three chosen policy levels despite the model mismatch in the control system design. The effects of the model mismatch are somewhat amplified here when compared to the results in Fig. 4 because the initial conditions in 2010 are also anticipated incorrectly, such that the level of commitment in the closed-loop is either over or under predicted. For the 25% under prediction in the control system design model, this leads to a transient overshoot of the policy target because the control law is over sensitive in this case and also incorrectly accounts for the dynamic
Climatic Change Fig. 6 The closed-loop evaluation of atmospheric CO2 concentration to the imposition of policy targets of xD(k)=450, 550 and 650 ppmv in the year 2010 (dashed lines in a). Prior to this, the atmospheric CO2 concentration is unregulated and responds to the historic anthropogenic CO2 emissions. As in Figs. 3 and 4 the simulations uses the Joos model as the surrogate system run on an annual time-step, whilst the control law is run on a 25-year timestep providing the 25-year ahead emissions trajectory (b). This control law has been optimised using the ‘incorrect’ parameter sets shown in Table 1 in order to recreate conditions of significant model uncertainty
commitments made pre-2010. In the case of the 25% over prediction in the control design model, the opposite is the case, except for the 450 ppmv stabilisation, where the level of commitment in 2010 is such that significant transient undershoot of the 450 ppmv target appears to be no longer possible. Interestingly, Fig. 6b shows that setting the policy level above 550 ppmv tends to promote increases in the rates of change of emissions post 2010 in response to the magnitude of the environmental headspace this creates, whilst setting the level below 550 ppmv has the reverse effect. This suggests that, in terms of this present analysis, imposing a 550 ppmv policy target in 2010 would be consistent with the current dynamic conditions.
4 Discussion and conclusions The feedback control law is, at first site, a somewhat unusual device which might appear to be associated with a purely reactive ‘wait, see and act’ strategy. However, given its role in establishing the optimized closed-loop system, the control law needs to be seen as a concise summary of the dynamic properties required in a socio-economic mitigation response in
Climatic Change
order to achieve a particular set of closed-loop dynamics. Only in terms of the corrective compensatory response can it be viewed as reflexive and, even then, it must be appreciated that the corrective steps this action generates are mindful of the closed-loop consequences they cause, such that these too satisfy the tradeoffs built into the design via the control optimization. In combining the anticipatory and reflexive paradigms, the control approach appears to provide a framework where these two important considerations are no longer juxtaposed. It is clear when comparing Figs. 1 and 2 that the control law acts as a proxy for a socioeconomic mitigation system. Currently, these are represented more explicitly using economic models (e.g. Nordhaus 1991), although econometric models often assume similar forms to control laws (e.g. Engsted and Haldrup 1999). The parallel between the control law and its socio-economic model counterpart could be used to shed fresh light on the closed-loop dynamic properties associated with different mitigation strategies. For example, the dynamics of the controller, considered in isolation (which might then be considered as the socio-economic response postulated by the control law (Eq. 6)), is dominated by poles of 1.0131 and 1.0. The unity pole is associated with the integral-of-error compensator required for the online correction mechanism; while the pole of 1.1337 is associated with the regulator and represents a powerful tendency in the controller to move emissions in the opposite direction to the atmospheric CO2 concentration. Obviously, the controller does not work in isolation and is part of the complete closed-loop system. Consequently, this effect is offset in the closed-loop by the fact that it operates in conjunction with the GCC via the negative feedback. What the parallel between control and socio-economics models will inevitably draw attention to is the methods used to parameterise both approaches. In this paper, we have opted for a policy orientated optimization in order to illustrate the methodology, and we identify this as a key area requiring further research if the links between control and say, integrated assessment studies, are to be resolved. The current framework does not take into account any hard constraints that need to be considered. For example, Yohe and Toth (2000) have discussed these hard constraints within the context of ‘tolerable windows’ in the closed-loop behaviour. In control terms, hard constraints, such as limits on the rates of change or levels of emissions, atmospheric CO2 concentrations or temperature, would be accommodated using adaptive and predictive control methods which attempt to anticipate the approach of constraint boundaries and modify the control behaviour accordingly (Clarke 1994; Wang and Young 2006). Similar strategies could be used to build in hedging strategies within the control that attempt to minimise risk by taking pre-emptive action (Collings et al. 1994). It would also be interesting to explore the effects on the controller design and performance of different implementation dynamics over the review cycle sampling interval. Here, we have assumed a linear, 25 year first order hold and, hence, imply simple implementation dynamics similar to those recently discussed by Pacala and Socolow (2004). However, significant delay on implementation may impact on the closed-loop dynamics in such a way that requires an alternative discretization scheme for the control system design model. An obvious extension to the current analysis would be to consider the more demanding goal of controlling the global mean temperature, where observation noise becomes an important consideration in the control system design and implementation (Leedal 2006). Also, given that there is no need to restrict the control law to the specification of emissions, one could use this approach to analyse a system for specifying control of economic measures (Fiddaman 2002). These analyses would not need to be specified solely at the level of global aggregation because multivariate control frameworks could be used to account for regional differences operating to achieve a common climate goal.
Climatic Change
We conclude that the generality of the feedback control systems approach appears to open up several opportunities for climate mitigation studies. This paper is intended to highlight the relevance of control systems analysis and design to climate mitigation studies, particularly as regards the provision of a framework that integrates anticipatory and reflexive strategies. In identifying some of these opportunities, we hope to stimulate further research in this area. Acknowledgements The authors would like to thank Malte Meinhausen, Robert Lempert and James Taylor for insightful comments made at various stages during the preparation of this manuscript. D.T. Leedal was supported by an NERC studentship number NER/S/A/2002/10392.
References Clarke DW (1994) Advances in model-based predictive control. In: Clarke DW (ed) Advances in ModelBased predictive Control. Oxford University Press, Oxford, pp 3–21 Collings IB, James MR and Moore JB (1994). An information-state approach to linear risk-sensitive quadratic Gaussian control. Proc 33rd IEEE Conf Decis Control 4:3802–3807, vol.4 Engsted T, Haldrup N (1999) Multicointegration in stock-flow models. Oxf Bull Econ Stat 61:237–254 Fiddaman TS (2002) Exploring policy options with a behavioral climate–economy model. Syst Dyn Rev 18 Franklin GF, Powell JD, Emami-Naeini A (2002) Feedback control of dynamic systems. Addison-Wesley Hammitt JK, Lempert RJ, Schlesinger ME (1992). A sequential-decision strategy for abating climate change. Nature 357:315–318 Hare B, Meinshausen M (2006) How much warming are we committed to and how much can be avoided? Clim Change 75:111–149 Houghton JT, Meira Filho LG, Griggs DJ, Maskell K (1997) Stabilization of atmospheric greenhouse gases: physical, biological and socio-economic implications. IPCC Technical Paper III Jarvis AJ, Young PC, Leedal DT, Chotai AC (2005) An incremental CO2 emissions strategy based on a state variable feedback control design. International Symposium on Stabilisation of Greenhouse Gases, Exeter, UK Joos F, Bruno M, Stocker TF, Siegenthaler U, Le Quere C, Sarmiento JL (1996) An efficient and accurate representation of complex oceanic and biosphere models of anthropogenic carbon uptake. Tellus 48B:397–417 Kalman RE (1960) A new approach to linear filtering and prediction problems. Tranactions American Society of Mechanical Engineers, J Basic Eng 82-D:35–45 Keller K, Bolker BM, Bradford DF (2004). Uncertain climate thresholds and optimal economic growth. J. Environ Econ Manage 48:723–741 Leedal DT (2007) Sequential decision making in climate change mitigation: a control systems perspective. PhD thesis. Lancaster University Lempert R, Schlesinger M (2000) Robust strategies for abating climate change. Clim Change 45:387–401 Maier-Reimer E, Hasselmann K (1987) Transport and storage of CO2 in the ocean – an inorganic oceancirculation carbon cycle model. Clim Dyn 2:63–90 Nise NS (2004) Control systems engineering. Wiley, New York Nordaus WD (1991) To slow or not to slow – the economics of the greenhouse effect. Econ J 101:920–937 Pacala S, Socolow R (2004) Stabilization wedges: solving the climate problem for the next 50 years with current technologies. Science 305:968–972 Richardson GP (1991) Feedback thought in social science and systems theory. University of Pennsylvania Press, Philadelphia Taylor CJ, Chotai A, Young PC (2000) State space control system design based on non-minimal state variable feedback: further generalization and unification results. Int J Control 73:1329–1345 Taylor JT, Young PC, Chotai A (1996) PIP optimal control with a risk sensitive criterion. Control ’96, UKAC International Conference on Control, Exeter Wang L, Young PC (2006) An improved structure for model predictive control using non-minimal state space realisation. J Process Control 16:355–371 Wigley TML (1991) A simple inverse carbon cycle model. Glob Biogeochem Cycles 5:373–382 Yohe G, Toth FL (2000) Adaptation and the guardrail approach to tolerable climate change. Clim Change 45:103–128
Climatic Change Young PC (1984). Recursive estimation and time series analysis. Springer, Berlin Young PC (1999) Data-based mechanistic modelling, generalised sensitivity and dominant mode analysis. Comput Phys Commun 115:1–17 Young PC, Garnier H (2006) Identification and estimation of continuous-time, data-based mechanistic models for environmental systems. Environ Model Softw 21:1055–1072 Young PC, Behzhadi MA, Wang C, Chotai A (1987) Direct digital and adaptive control by input–output, state variable feedback. Int J Control 46:1867–1881
