Christian Ebeke
email: cebeke@imf.org

Désiré Omgba
email: desire.omgba@supco-amiens.fr

Bd F Mitterrand

Les Auteurs

Désiré Luc

Omgba

Patrick Plane

Catherine Araujo Bonjean

Oil rents, governance quality, and the allocation of talents in developing countries

Keywords: Rent-seeking, occupational choice, oil rents. JEL Classification: D72, J24, Q32

Les commentaires et analyses développés n'engagent que leurs auteurs qui restent seuls responsables des erreurs et insuffisances.

Introduction

Education plays an important role in the debate over the curse of natural resources. The works conducted so far have focused on whether there is a negative correlation between investment in education and the presence of natural resources. [START_REF] Gylfason | Natural Resources, Education and Economic Development[END_REF] and [START_REF] Birdsall | Natural Resources, Human Capital and Growth in Resource Abundance and Economic Development[END_REF] argue that there is a negative correlation between dependence on natural resources and investment in education. Stijn (2006) disputes this result.

This paper proposes to go beyond this binary question (presence of negative correlation or not), and raises the incentive problem on the distribution of talents in human capital in countries with natural resources. Indeed, [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF] propose a framework for analyzing the determination of talent and its importance for economic growth, the rationale being that when the most talented people of a nation become entrepreneurs, they enable innovation and improving technology, leading to increased productivity and to the fostering of economic growth. Conversely, when the most talented people become rent-seekers, this hurts growth because their orientation is not in favour of productive activities. Thus, the authors identify two types of activities which can attract talented people: entrepreneurship and rent-seeking activities. The first, entrepreneurs, are related to training in engineering sciences.

The latter, rent-seekers, are related to training in law schools [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF]. The authors suggest that a larger number of engineers will have a positive impact on growth, while a greater number of law concentrators, tends to reduce economic growth.

What will bring the agents to move towards the acquisition of one specific talent in preference to another are the incentives provided by the country. These incentives are mainly determined by the size of the market, the nature of the contract, and the returns to scale of the activity.

The work of [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF] is fundamental to understanding the allocation of talent in an economy. However, it is limited, not only because it does not test elements that may determine this allocation, but also because it does not include the importance of natural resources in the analysis or the rents that are associated with their presence, as the existence of such rents may condition the talent allocation.

Consequently, this paper contributes to extend this analysis. We empirically examine, for the first time, the impact of oil rents on the allocation of talent in a large sample of developing countries. The question which we address in this paper is whether the existence of natural resource rents conditions the students of the countries concerned to prefer more training in law (oriented to rent-seeking activities) rather than engineering courses (oriented to entrepreneurship activities).

In doing so, the paper is a contribution to the traditional studies of incentives in the presence of natural resources. More specifically, the paper is close to the fundamental contribution of [START_REF] Mehlum | Institutions and the Resource Curse[END_REF] on the impact of natural resources on entrepreneurship. However, it differs substantially from it on a number of points. Indeed, while [START_REF] Mehlum | Institutions and the Resource Curse[END_REF] consider only the population of entrepreneurs in their analysis, this paper focuses on the wide base that is the allocation of human capital as a whole, and shows that this allocation is determined by the presence of natural rent. More specifically, the purpose of this paper is not simply to see the effects of rent-seeking activities on growth, which have been widely reviewed by previous studies. Rather, this paper empirically highlights the elements that determine the choice of these activities in a student population. This, to the best of our knowledge, has not yet been studied.

The main objective of this paper is to measure the effect of oil rents on the allocation of talents in developing countries. We hypothesize that the effect of oil rents is non linear and depends upon the governance quality. The hypothesis that is tested is that resource rich and badly governed countries are more likely to exhibit talents in the rent seeking activities (sectors) while resource rich and well governed countries exhibit lower ratio of talented individuals associated with rent seeking activities. Therefore, this paper goes beyond the traditional analyses of the resource curse in oil rich countries that posit the governance and institutional quality as the main conditional variables of the effect of the dependency upon natural resources. This paper seeks to provide a granular origin of this institutional based resource curse problem. Oil rich and badly governed countries are more likely to fall into a bad governance quality trap because more talented individuals are associated with training correlated with rent seeking activities. The idea is that the bad governance quality environment is regenerated by the misallocation of talented individuals in the society (the vicious circle of the bad governance in oil rich countries).

Two main arguments can be evoked to explain the effect of oil rents on the allocation of talents which is conditional on the governance quality. First, the demand side hypothesis could suggest that individuals may endogenously choose training that are positively with rent seeking when the governance and institutional background of the countries is poor. In such a context, participating in rent seeking activities is less subject to sanctions and gives a high payoff to the rent seekers. 1 Second, the supply side hypothesis suggests that oil rich but badly governed countries may be those with a low provision of training that favor entrepreneurship. 2 From a political economy perspective, this can be a strategy for the government to better control the voters through the instruments of wages and employment. For the individuals, the public employment offer operates like an insurance mechanism in a context of bad governance and high uncertainty. Individuals therefore are incited to choose training that are economically associated with rent seeking activities for several reasons: those training are available, publicly granted, require few number of years of schooling, and allow capturing the natural resource rents owing to the bad governance quality context.

Our analysis, conducted on a sample of 69 developing countries under various econometric methods, shows that oil rents positively affect the proportion of law concentrators (oriented to rent-seeking activities) and negatively affect the proportion of engineers (oriented to productive activities). However, these effects remain non-linear, and in fact they are conditional on the governance quality. While oil rents in well governed countries tend to orient talents towards productive activities, oil rents in badly governed countries tend to orient talents towards rent-seeking activities.

The remainder of the paper is as follows. Section 2 proposes a brief literature review and highlights the theoretical mechanisms. Section 3 provides an empirical test of the hypothesis that talented individuals are mainly observed in rent seeking activities in developing oil rich countries that suffer from governance problems. Section 4 performs robustness checks while Section 5 concludes.

1 Another explanation comes from the fact that individuals living in oil badly governed countries may choose training that will provide them with a high probability to find a job due to the small private sector size. In such a context, individuals choose training that can increase their probability to be hired in the public sector, the main employer in the absence of a strong private sector. 2 A companion argument that also supports this supply side hypothesis is that the lack of governance quality leads the government to deviate from the optimal education policy and the incumbent provides less structures allowing training less associated with rent seeking activities.

Literature review and theoretical discussion

The literature on the resource curse has been discussing the performance of resources-rich countries for nearly two decades. Since the seminal regression of [START_REF] Sachs | Natural Resource Abundance and Economic Growth, Working Paper 5398[END_REF], which shows a negative correlation between dependence on natural resources and economic growth, several channels have been examined through which natural resources could undermine development. [START_REF] Sachs | Natural Resource Abundance and Economic Growth, Working Paper 5398[END_REF] explained their results by the Dutch disease phenomenon. Many authors who have followed [START_REF] Sachs | Natural Resource Abundance and Economic Growth, Working Paper 5398[END_REF] extend the analysis through other mechanisms (than the Dutch disease) and other outcomes (than growth). Thus, [START_REF] Hausmann | An alternative interpretation of the resource curse: theory and policy implications[END_REF] extend the analysis to volatility. [START_REF] Tornell | The Voracity Effect[END_REF] and [START_REF] Mehlum | Institutions and the Resource Curse[END_REF] present the institutional deficit as an explanatory factor for the curse. Other authors extend the curse to different indicators, including democracy (Ross, 2001), armed conflict (Collier and Hoeffler, 2000) and education [START_REF] Gylfason | Natural Resources, Education and Economic Development[END_REF]. [START_REF] Gylfason | Natural Resources, Education and Economic Development[END_REF] is thus the first to empirically focus on the relationship between natural resources and education. The author shows that oil-producing countries tend to have low levels of education. His result appears robust to three indicators of education: public expenditure on education relative to national income, expected years of schooling for girls, and gross secondary-school enrolment. The author explains this result by the fact that the dependence on natural resources leads to the inadvertent or deliberate neglect of progress in human capital.

Oil-rich countries invest the least in education, which leads to poor performance in human capital. [START_REF] Gylfason | Natural Resources, Education and Economic Development[END_REF] concludes that the nations who believe that natural capital is their greatest asset develop a false sense of security and become negligent about the accumulation of human capital. [START_REF] Birdsall | Natural Resources, Human Capital and Growth in Resource Abundance and Economic Development[END_REF] support Gylfason's results. The authors also suggest a hypothesis about the low investment in education in resources-rich countries. This is, according to these authors, the fact that the dependence on natural resources tends to break the virtuous circle of a relationship between education, growth and inequality. For the authors, higher private returns on human capital lead to higher rates of private investment, including among the poorest. This leads to an increase in productivity and less inequality in the future. [START_REF] Birdsall | Natural Resources, Human Capital and Growth in Resource Abundance and Economic Development[END_REF] argue that in a country with natural resources, governments are tempted to deviate from the policies that create this virtuous circle. The Dutch disease effects of natural resources adversely affect areas of high labour intensity such as the agricultural sector in developing countries. This will tend to reduce returns in human capital in these sectors employing the poor. Ultimately, there will be less investment and therefore no increase in productivity or reduction of inequalities. 3The aforementioned papers have a common feature. They essentially discuss the negative relationship between natural resources and human capital. They do not account for the learners' qualifications as an element of the analysis. Indeed, as pointed out by [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF], the fact that a country tends to have a greater population of students in law schools is not neutral for economic growth. The authors note that this allocation of talent could be the cause of stagnation of development in many countries in Africa and Latin America and could explain the development of East Asian countries. Thus if the proportion of entrepreneurs is crumbling, technological progress slows and growth weakens.

Consequently, this paper is interested in what might determine the allocation of talent in developing countries. Specifically, the paper looks at the incentives of students in the oil-rich economies. Indeed, among the natural resources, oil seems to be the resource with the highest probability of occurrence of the resource curse [START_REF] Ross | What do we know about natural resources and civil war?[END_REF]Manzano and Rigobon, 2006). This singularity is primarily due to the importance of oil rents (Manzano and Rigobon, 2006).

It makes sense therefore to ask whether the presence of such rents could lead students to choose courses that lead to rent-seeking activities.

This question of the allocation of talents in resources-rich countries is not novel. However, while one can note an extensive theoretical literature on the allocation of talent in resourcerich countries, little has been done in empirical literature. Among the theoretical literature, one notes the significant contributions of [START_REF] Baland | Rent-Seeking and Resource Booms[END_REF] and [START_REF] Mehlum | Institutions and the Resource Curse[END_REF]. [START_REF] Baland | Rent-Seeking and Resource Booms[END_REF] are interested in the specialization of certain economies in rent-seeking rather than in productive activities in a context of a natural resource boom. The authors suggest that this orientation depends on the relative importance of the sector in productive activities before the boom. If productive activities were far more important than the rent-seeking activities, then this sector will succeed in capturing this rent and there will be a dilution of rent-seeking activities. Otherwise, the activities of rent-seeking will be more important [START_REF] Baland | Rent-Seeking and Resource Booms[END_REF].

However, the authors examine only the boom period (positive external shock). Moreover, in [START_REF] Baland | Rent-Seeking and Resource Booms[END_REF], rents are derived from quantitative restrictions on imports developed by [START_REF] Krueger | The Political Economy of the Rent-Seeking Society[END_REF], in which we are less interested in this paper, which is closer to [START_REF] Mehlum | Institutions and the Resource Curse[END_REF].

These authors focus on the entrepreneurs' allocation between productive and unproductive activities. However, compared to the model of [START_REF] Baland | Rent-Seeking and Resource Booms[END_REF], their model is specifically concerned with the study of rent generated by natural resources. Indeed, it is well known that the exploitation of natural resources causes two opposing effects. On the one hand it increases the income of the country, and on the other hand it causes the displacement of private agents (including entrepreneurs) from the most productive sectors of the economy towards the natural resource sector. It induces, for the occasion, rent-seeking behaviour.

Agents will therefore make a tradeoff between using their resources for productive activities or using these resources to search for and capture rents. The decision to move from one category of activity to the other will depend on the profitability of each sector. [START_REF] Mehlum | Institutions and the Resource Curse[END_REF] suggest that profitability will depend on the quality of the institutions in place. For the authors, if institutions are of good quality, production activities are more profitable than rentseeking activities. Indeed, in the absence of institutions of good quality, the opportunity cost of rent-seeking activities decreases. Consequently, entrepreneurs abandon the productive sectors to engage in rent-seeking activities. This diversion from the productive sector leads to a decline in productivity throughout the economy. This decrease in productivity leads to lower growth. Natural resources will therefore be a blessing or a curse depending on the quality of the home institutions [START_REF] Mehlum | Institutions and the Resource Curse[END_REF].

These studies remain theoretical and predictions in terms of allocation of talent are not subject to empirical test. Our study aims to fill this gap.

Empirical Analysis a) Econometric model of the non linear effect of oil rents on the allocation of talents

This paper tests the hypothesis that the effect of oil rents on the allocation of talents toward rent seeking activities is more positive at high levels of bad governance. We follow [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF] in their theoretical discussion on the determinants of the allocation of talent for the selection of the control variables. In this case, we can specify the following equations:

ܶ ൌ ܺ ᇱ ߚ ߠ ଵ ܮܫܱ ߠ ଶ ܶܵܰܫ ݀ ߝ (1) ܶ ൌ ܺ ᇱ ߚ ሺߙ ߜ ܶܵܰܫ ܮܫܱሻ ܶܵܰܫߛ ݀ ߝ (2)
with ߝ the residual term, ݀ the regional dummies and i stands for the country.

The main hypotheses tested are: ߠ ଵ ൌ 0, ߙ ൏ 0, and ߜ 0. The definition adopted is similar to that used by [START_REF] Mariani | Migration as an antidote to rent-seeking?[END_REF] and is broader than that adopted by [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF]. These authors retain Enrolment in law as a proxy for the choice of careers in rent-seeking activities, and Enrolment in engineering as a proxy for career 4 ߙ identifies the effect of oil rents on the allocation of talents toward rent seeking activities in countries exhibiting an index of governance equal to 0 (this corresponds to the highest governance quality score) and it is therefore expected to be negative. This suggests that more talented people choose entrepreneurship activities in oil rich and well governed countries.

choices in productive activities. [START_REF] Mariani | Migration as an antidote to rent-seeking?[END_REF] extends this definition. Thus, like him, we consider enrolment in engineering and science as a proxy for productive activities, enrolment in the social sciences, business and law as a proxy for rent-seeking activities. In addition to this first cross-section analysis on the allocation of talent, we are more interested in the differences between countries. Indeed, after all, students in different countries might be more focused on studies in social sciences, business and law studies than studies in engineering and hard sciences, simply because the latter requires more effort and probably earns less. In this context, it appears more interesting to examine whether these differences are greater in countries with oil revenues.

All the dependent variables are drawn from the database of the Statistical Yearbook of UNESCO.

represents the oil rents as percentage of country GDP. In the literature, various variables are used to measure the dependence on natural resources. The most common are the percentage of exports of natural resources in total exports, and the percentage of exports in GDP. Papers using either of these variables tend to confirm the hypothesis of the resource curse [START_REF] Sachs | Natural Resource Abundance and Economic Growth, Working Paper 5398[END_REF][START_REF] Leite | Does Mother Nature Corrupt ? Natural Resources, Corruption and Economic Growth, Working Paper 99[END_REF]. However, some studies using other measures, such as level of production (Stijn, 2006) or the percentage of resource revenues in government revenues [START_REF] Herb | No Representation without Taxation? Rents[END_REF], often lead to conclusions against the curse of natural resources. But, as pointed out by [START_REF] Rosser | The Political Economy of the Resource Curse: a Literature Survey[END_REF], these studies on the natural resource curse address the specific behaviours associated with agents in the presence of rents generated by the exploitation of natural resources. Therefore, it appears that a measure in terms of rents from the exploitation of natural resources is the most appropriate [START_REF] Rosser | The Political Economy of the Resource Curse: a Literature Survey[END_REF]. This paper highlights the incentives in the student choices in the presence of oil revenue, and therefore the Oil rents variable is relevant to this purpose. The variable Oil rents is derived from the World Bank Development Indicator online dataset.

is an indicator of governance vulnerability. This variable appears as a key element of the analysis. This is because not only is it established that institutional quality is an important element in the management of revenues from exploitation of natural resources in the resource curse literature (see, e.g., Sala-i-Martin and Subramanian;2003;Melhum et al., 2006), but also, from the work of [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF], it is an important element in the allocation of talents. In this paper we integrate, in the first analysis, the governance vulnerability index as a whole. Thus, in the first approach, we follow the implementation of institutional quality in the growth model (Melhum et al., 2006). In the second approach, we break up the governance vulnerability index into its components. The rationale is that different institutional variables may show different effects on the allocation of talent. The governance variables are drawn from the Worldwide Governance Indicators (WGI) database of the World Bank. This dataset is used given that it provides measures of governance for a large number of countries since 1996. 5 The Worldwide Governance Indicators (WGI) project has reported aggregate and individual governance indicators for 212 countries and territories since 1996. Six dimensions of governance are reported: voice and accountability, political stability and absence of violence, government effectiveness, regulatory quality, rule of law, control of corruption. 6 More recently, the six indicators were defined as:

-Voice and Accountability -measuring the extent to which a country's citizens are able to participate in selecting their government; as well as freedom of expression, freedom of association, and a free media.

-Political stability and Absence of Violence -measuring perceptions of the likelihood that the government will be destabilized or overthrown by unconstitutional or violent means, including political violence or terrorism.

5 Kaufmann, Kraay, and Mastruzzi (2009) construct a meta-indicator that aggregates a host of different measures, from firm, investor, and population surveys to expert and international organization assessments to come to their overall measurements of the quality of governance. Data are available at the Worldwide Governance Indicators (WGI) project website under the following address: http://info.worldbank.org/governance/wgi/index.asp For more details on the construction of the indices, refer to Kaufmann, Kraay and Mastruzzi (2009). "Governance Matters VIII: Aggregate and Individual Governance Indicators, 1996-2008". World Bank Policy Research Working Paper Series, 4978. 6 It should be noted that these governance indicators are all based on data from expert assessments, polls of experts and surveys of government officials and businesses, and therefore capture perceptions of the government process rather than any formal aspects of the actual government structure in any given country. This creates the important problem that perceptions are shaped not just by the government environment, but also by many other aspects of the socio-economic environment, thereby creating its own set of endogeneity and reverse causality issues. There is a large literature critical of the World Governance Dataset (Arndt and Oman, 2006;Kurtz and Shrank, 2006;Kurtz and Shrank, 2007). Kaufmann, Kraay and Mastruzzi have categorized some of these critiques as concerns about the comparability of the indicators across countries and across time; concerns about bias in expert polls or in particular sources; and concerns about the independence of the different data sources and the consequences for the aggregate indicators. (Kaufmann, Kraay and Mastruzzi, 2006). More recently, Thomas (2010) dismisses the Worldwide Governance Indicators (WGI) as an 'elaborate and unsupported hypothesis' because of the failure to demonstrate the 'construct validity' of these indicators. A short answer to Thomas (2010) is provided by Kaufmann et al. (2010). The authors cast doubts on the practical consequences of failure to meet the criteria of construct validity and therefore minimize this critique.

-Government effectiveness -measuring the quality of public services; the quality of the civil service and the degree of its independence from political pressures; the quality of policy formulation and implementation, and the credibility of the government's commitment to such policies.

-Regulatory quality -measuring the ability of the government to formulate and implement sound policies and regulations that permit and promote private sector development.

-Rule of law -measuring the extent to which Law Enforcement agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, the police and courts, as well as the likelihood of crime and violence.

-Control of corruption -measuring the extent to which public power is exercised for private gain (including both petty and grand forms of corruption), as well as "capture" of the state by elites and private interests.

To build the indicators of governance used in the econometric estimations, we reverse all the original indicators of governance quality by the following formula:

ݐݏ݊ܫ ൌ ݐݏ݊ܫ െ ሻݐݏ݊ܫ݉݅݊ሺ ሻݐݏ݊ܫሺݔܽ݉ െ ሻݐݏ݊ܫ݉݅݊ሺ
where ሻݐݏ݊ܫ݉݅݊ሺ and ሻݐݏ݊ܫሺݔܽ݉ represent the minimum and the maximum of each indicator of governance quality, respectively. This transformation ensures that ܶܵܰܫ will have a range between 0 and 1. On this basis, ܶܵܰܫ increases with the deterioration of the quality of governance. Moreover, this ensures the standardization of these variables into new indices which are therefore reasonably comparable. Given the fact that the indices are distributed over the same interval [0, 1], the coefficients of the interactive terms (oil rents crossed with the governance variable) will allow direct comparison across the different equations.

The paper begins by using an indicator of the overall quality of governance which combines all the six separate dimensions into a single index. The principal component analysis method is used to achieve this. The aggregate index of governance is the first principal component of the vector of the six indicators of governance already constructed. Table A1 in Appendix A shows that the first principal component accounts for almost 81% of the overall variance. The table also presents the eigenvectors and the correlation between the synthetic indicator and each variable. Meanwhile, the resulting aggregated indicator of governance has been rescaled to be ranged between 0 and 1. Next, the paper uses separately each indicator of governance quality to assess the impact of the oil rents on the allocation of talents conditional on the level of governance quality.

The other variables in the analysis are included in the vector X. They mainly consist in proxies of the determinants of talent allocation as suggested by [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF]. This includes the size of government, the degree of openness, the cost of registering property, and the access to credit by the private sector. The size of government, the degree of openness and the access to credit are from the World Development Indicators (2009). The cost of registering property is drawn from the World Bank's 'Doing Business'. To these variables, we added regional dummies ( ݀ ) identifying the regional variability in the allocation of talents.

b) Preliminary evidence

Equations 1 and 2 are estimated by the OLS method with a full set of regional dummies. A cross-section of 69 countries is used with data averaged over the period 2000-2008. This period is retained so that we have the most data on our dependent variable (allocation of talents) and on governance quality. It is therefore deemed to constitute the largest possible sample of developing countries.

The results of the estimations of the model 1 do not reject the hypothesis that the linear effect of oil rents on the allocation of talents is, statistically null. This holds whatever the control variables which are included and whatever the dimension of the governance quality which is additively controlled for. 7 Table 1 presents the results of the estimations of model 2, which now allows a nonlinear effect of the oil rents on the allocation of talents. In the first two columns (1 and 2) the dependent variable is ܶ . The Governance vulnerability variable is the overall institutional quality variable obtained from the method of principal component analysis of the different variables of the WGI database. These components are Corruption, Rule of law, Regulatory quality, Government effectiveness, Political stability, and Voice & Accountability.

The negative sign and the significance of the coefficient associated with the variable Oil-rentto-GDP in column 1, reflect a negative impact of oil rents on students' incentives towards orientation to rent-seeking activities when the country has good governance (Governance vulnerability equal to 0). This effect of oil rents on the allocation of talent is not linear. In this case, the significance of the positive coefficient of the multiplicative variable (Oil rent*Governance vulnerability) reflects the fact that the poorer the country's governance quality is (high Governance vulnerability), the more students will be encouraged to choose courses leading to law concentrators (training oriented to rent-seeking activities). This result is not rejected in column 2, even when we use the Governance quality variable in the initial period (2000) instead of taking an average as in the previous case. Using the governance quality observed at the beginning of the period analysis would allow the reduction the potential endogeneity bias associated with the coefficient of the variable oil rents. This is particularly relevant when the endogeneity issue arises from the reverse causality problem between the oil rents and the allocation of talents. The coefficient associated with the Oil rentto-GDP variable is still negative and significant while the coefficient associated with the multiplicative variable (Oil rent * Initial governance vulnerability) is still positive and significant.

To summarize, these first results suggest that, in the presence of oil revenues, most students choose law training (oriented to rent-seeking) if -and only if -the governance quality of the country is poor. Otherwise (if the governance quality is good), they will be tempted by engineering studies (oriented to productive activities).

Robustness checks a) Testing an alternative dependent variable

Thus far, the results showed that the allocation of talent depends on the presence of oil revenues and the governance quality of the countries. However, it can be argued that having a dependent variable, taken as the difference between the enrolment in social sciences, business and law and the enrolment in engineering, manufacturing and construction, would result in forcing the regressors to symmetrically impact both sectors. After all, [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF] note that certain items that affect the number of lawyers may act disproportionately on the All the models include the full set of regional dummies. All the original series of governance drawn from the WGI dataset have been reverted so that high values refer to bad governance quality. The governance vulnerability used here is the aggregation of the WGI indices using the principal component analysis. The composite index is ranged between 0 and 1. *** p<0.01, ** p<0.05, * p<0.1 population of engineers. We take this observation into account by changing the dependent variable in regressions 3 and 4 of Table 1. The dependent variable retained is the proportion of Enrolment in law defined as the enrolment in social sciences, business and law as percentage of the sum of the enrolment in social sciences, business and law and the enrolment in engineering, manufacturing and construction.

The expected econometric results are confirmed. In columns 3 and 4 of Table 1, the coefficient associated with the Oil rent variable is still negative and significant while the coefficient associated with the multiplicative variable remains positive and significant. This suggests that talented individuals are more allocated toward rent seeking activities in oil rich countries exhibiting a bad governance quality, while the opposite holds for those exhibiting a better governance quality score. In addition, we also note that the coefficient of the Cost of registering property variable is also positive and significant, showing that the higher the cost for property rights is, the more rent-seeking activities are present.

b) Breaking-down the aggregate index of governance quality

Previous results indicated that the effect of oil rents on the allocation of talents is conditional to the level of the overall governance quality. This result is confirmed by various measures of the dependent variable (allocation of talents). However, it does not give us any indication of the specific component of governance which acts on the allocation of talent. One of them could well dominate the composite indicator, making it worthwhile therefore to break up this composite indicator and return to specific measures of quality of governance. The goal is to be as specific as possible in order to achieve the most appropriate recommendations.

We consider this point in Table 2. This table shows the estimation results of various governance variables on talent allocation.8 These variables include Corruption, Rule of law, Regulatory quality, Government ineffectiveness, Political instability and Unaccountability.

The coefficients associated with oil rents remain negative. The coefficients associated with multiplicative variables (Oil rents * Governance indicators) are also positive as in the previous case. These coefficients are highly significant on four of these variables: Corruption, Rule of law, Regulatory quality and Government ineffectiveness. The significance of All the models include the full set of regional dummies. All the original series of governance drawn from the WGI dataset have been reverted so that high values refer to bad governance quality. Governance measures are ranged between 0 and 1. *** p<0.01, ** p<0.05, * p<0.1 corruption and the regulatory quality appears to correspond to a theoretical prediction by [START_REF] Murphy | The Allocation of Talent: Implications for Growth[END_REF]. Indeed, in their theoretical analysis, the authors insisted on the definition of property rights and the possibility of easy access to rents as the main determinants of the allocation of talent. In this context, the significance of the multiplicative Oil rent * Corruption reflects the fact that the more corrupt a country is, the easier it is to dispose of the rents. Consequently, more students will be able to choose law training (oriented to rent-seeking). The significance of the multiplicative (Oil rents* Regulatory quality) suggests that when most property rights are poorly defined and defended in an oil country, most agents do dare to start productive activities.

These results are not rejected when we retain as the dependent variable the ratio of enrolment in law as percentage of the sum of the enrolment in law and the enrolment in engineering (Table 3). They are also robust to the use of initial values of the governance variables (Table Table 4).

c) Testing other indicators of corruption

Because subjective, institutional variables are often (more than other variables) subject to errors of measure (Acemoglu et al., 2001), although the use in this paper of various governance variables may lead to the conclusion that these measurement errors do not explain our results, we undertake to evaluate these results with alternative databases. Indeed, until now, we have used indicators from the WGI database. For this evaluation, we use data from Transparency International and ICRG on corruption. We want to know if taking other sources of data corruption will amend our results.9

Table 5 presents results using the new measures of corruption. Whatever the corruption variable that is used, the results do not reject those obtained previously. The coefficient associated with the oil rents is significant and negative while the variable Oil rents* Corruption exhibits a significant and positive coefficient. So, the more corrupt a country is, the easier it is to gain access to rents and therefore more students will choose law training (oriented to rent-seeking).

d) Results from simultaneous regressions

It may happen that the two sectors (law and engineering) could be simultaneously. The models describing the dynamic of these two sectors can be linked via the correlation between the two error terms of the equations. In this context, the SURE method allows us to take this feature into account and provides estimates of the system of the two equations. 10 The two dependent variables are respectively: enrolment in law (column 1) and enrolment in engineering (column 2) in Table 6. The proportion of enrolment in engineering is defined as the enrolment in engineering, manufacturing and construction as percentage of the total enrolment in tertiary education. The enrolment ratio in law is defined as the enrolment in social sciences, business and law as percentage of the total enrolment in tertiary education. 6 highlight a positive effect of oil rents on the number of lawyers conditional on the extent of corruption: resource rich countries that suffer from governance problems exhibit more lawyers than the others. In contrast, there is a negative effect of oil rents on the proportion of engineers conditional on poor performance in terms of control of corruption. Put differently, resource rich countries that suffer from governance problems exhibit less engineers than the others.

Results of Table

To sum up, all the robustness checks do not invalidate the main finding of the present study, notably that oil rents affect the incentives of individuals in favour rent-seeking activities. The effect, however, remains non-linear, and is in fact conditional on the governance quality.

While oil rents in less corrupt countries tend to orient talents towards productive activities, oil rents in highly corrupt countries tend to orient talents towards rent-seeking activities.

Conclusion

This paper has provided the econometric evidence on the interaction between oil rents and the allocation of talents in developing countries. It proposed to go beyond the simple binary relationship linking the level of human capital and petroleum resources.

We investigated the determinants of occupational choices in a population of students in law school and students of engineering. We found that the presence of oil revenues determines the allocation of talent but the result is not linear. This nonlinearity is determined by the governance quality. In other words, while oil rents in less corrupt countries tend to orient talents towards productive activities, oil rents in highly corrupt countries tend to orient talents towards rent-seeking activities.

These results provide another vision of the resource curse. Indeed, a student population that has the possibility of easy access to rents owing to a favorable environment characterized by a high corruption, will move more easily to the training that are correlated with rent seeking activities. This crowds out engineering education, which is capable of bringing innovation, the basis of higher productivity and economic growth in the long term.

  4 In other terms, the effect of oil rents on the allocation of talents ሺܶ ሻ is statistically null ሺߠ ଵ ൌ 0ሻ, but becomes statistically significant once this effect is conditioned upon the governance quality. Therefore, the marginal effect of oil rents ቀ డ் డைூ ൌ ߙ ߜ ൈ ܶܵܰܫ ቁ on the allocation of talents towards rent seeking activities ሺܶ ሻ is more positive in badly governed countries ሺܶܵܰܫ ሻ.The dependent variable ܶ indicates the talent allocation. It is measured as the difference between the share of students enrolled in training correlated with rent seeking activities (social sciences, business and law) and those enrolled in training correlated with productive activities (engineering, manufacturing and construction). More formally, we have: is defined as the enrolment in social sciences, business and law minus the enrolment in engineering, manufacturing and construction, and expressed as percentage of the total enrolment in tertiary education.

Table 1 .Conditional effect of oil rents on the allocation of talents according to the level of an aggregated governance quality index: OLS with regional dummies. Dependent variable: Difference between law & engineering Proportion of the enrolment in law

 1 Robust t-statistics in parentheses.

	(1)	(2)	(3)	(4)

Table 2 . Conditional effect of oil rents on the allocation of talents according to the levels of several dimensions of governance quality: OLS with regional dummies.

 2

	Dependent variable:		Difference between law & engineering
		(1)	(2)	(3)	(4)	(5)	(6)
	Oil rent-to-GDP	-6.412***	-6.836***	-4.880***	-4.908***	-2.073	0.437
		(-3.232)	(-3.174)	(-3.032)	(-3.617)	(-1.425)	(0.457)
	Oil rents * Corruption	8.867***
		(3.290)
	Oil rents * Rule of law		12.07***
			(3.310)
	Oil rents * Regulatory quality			9.716***
				(3.215)
	Oil rents * Government effectiveness				9.012***
					(3.848)
	Oil rents * Political stability					4.171
						(1.578)
	Oil rents * Accountability						-0.195
							(-0.124)
	Corruption	-6.404
		(-0.411)
	Rule of law		-22.80
			(-1.243)
	Regulatory quality			-27.52
				(-1.619)
	Government effectiveness				-19.34
					(-0.907)
	Political stability					-24.70**
						(-2.075)
	Accountability						-30.32***
							(-2.919)
	Real GDP per capita growth	-1.382	-1.318	-0.790	-1.036	-1.254	-0.879
		(-1.024)	(-1.073)	(-0.639)	(-0.810)	(-0.962)	(-0.693)
	Initial real GDP per capita	-0.202	-1.793	-1.251	-0.954	-2.516	-2.762
		(-0.0913)	(-0.787)	(-0.623)	(-0.370)	(-1.097)	(-1.403)
	Trade openness	0.0623	0.0672	0.0481	0.0617	0.00244	0.0583
		(1.153)	(1.205)	(0.923)	(1.161)	(0.0418)	(1.060)
	Foreign direct investment-to-GDP	-0.655**	-0.524*	-0.313	-0.466	-0.247	-0.419
		(-2.227)	(-1.892)	(-1.138)	(-1.651)	(-0.818)	(-1.577)
	Government consumption-to-GDP	-0.373	-0.432	-0.266	-0.367	-0.354	-0.451
		(-1.193)	(-1.432)	(-0.998)	(-1.240)	(-1.304)	(-1.649)
	Private credit-to-GDP	-0.0674	-0.0466	-0.0842	-0.0627	-0.0130	-0.0791
		(-0.721)	(-0.477)	(-0.871)	(-0.684)	(-0.124)	(-0.786)
	Registering property cost	0.0912	0.102**	0.112**	0.109**	0.0874	0.0850
		(1.614)	(2.039)	(2.117)	(2.099)	(1.650)	(1.614)
	Protection of investors index	0.154	0.0853	0.0656	0.116	0.00859	-0.0635
		(1.168)	(0.682)	(0.452)	(0.809)	(0.0610)	(-0.463)
	Constant	31.64	51.80*	45.70*	40.56	60.57**	64.79**
		(1.113)	(1.744)	(1.720)	(1.173)	(2.143)	(2.581)
	Observations	69	69	69	69	69	69
	R-squared	0.299	0.327	0.301	0.317	0.283	0.308

Notes: Robust t-statistics in parentheses.

Table 3 . Conditional effect of oil rents on the allocation of talents according to the levels of several dimensions of governance quality: Testing an alternative dependent variable. OLS with regional dummies. Dependent variable: Proportion of the enrolment in law

 3

	(1)	(2)	(3)	(4)	(5)	(6)

Notes: Robust t-statistics in parentheses. All the models include the full set of regional dummies. All the original series of governance drawn from the WGI dataset have been reverted so that high values refer to bad governance quality. The governance indices are ranged between 0 and 1. *** p<0.01, ** p<0.05, * p<0.1

Table 4 . Conditional effect of oil rents on the allocation of talents according to the initial levels of several dimensions of governance quality: Testing an alternative dependent variable. OLS with regional dummies. Dependent variable: Proportion of the enrolment in law

 4 Robust t-statistics in parentheses. All the models include the full set of regional dummies. All the original series of governance drawn from the WGI dataset have been reverted so that high values refer to bad governance quality. The governance indices are ranged between 0 and 1. *** p<0.01, ** p<0.05, * p<0.1

	(1)	(2)	(3)	(4)	(5)	(6)

Table 5 . Testing alternative variables of corruption.OLS with regional dummies. Dependent variable: Difference between law & engineering Proportion of the enrolment in law

 5 Robust t-statistics in parentheses. *** p<0.01, ** p<0.05, * p<0.1. (a) Transparency International corruption measure; (b) ICRG corruption measure.

		(1)	(2)	(3)	(4)
	Oil rent-to-GDP	-5.760***	-0.902***	-4.958***	-0.496**
		(-3.715)	(-4.004)	(-3.421)	(-2.605)
	Oil rents * Corruption (a)	6.879***		5.826***
		(3.641)		(3.512)
	Oil rents * Corruption (b)		0.477***		0.221**
			(3.276)		(2.023)
	Corruption (a)	-2.781		4.354
		(-0.209)		(0.451)
	Corruption (b)		-3.092		-1.230
			(-1.160)		(-0.690)
	Real GDP per capita growth	-1.066	-1.720	-0.108	0.234
		(-0.880)	(-1.640)	(-0.156)	(0.287)
	Initial real GDP per capita	0.401	-0.500	0.113	-0.307
		(0.191)	(-0.239)	(0.0677)	(-0.193)
	Trade openness	0.0557	-0.0298	0.0257	-0.0536
		(1.148)	(-0.661)	(0.628)	(-1.047)
	Foreign direct investment-to-GDP	-0.461*	-0.130	-0.0283	0.110
		(-1.869)	(-0.581)	(-0.124)	(0.521)
	Government consumption-to-GDP	-0.220	0.451	0.105	0.230
		(-0.783)	(1.660)	(0.451)	(1.027)
	Private credit-to-GDP	-0.0539	-0.0204	-0.0269	-0.0168
		(-0.741)	(-0.221)	(-0.439)	(-0.207)
	Registering property cost	0.104**	0.0901**	0.128***	0.112***
		(2.475)	(2.201)	(4.629)	(3.963)
	Protection of investors index	0.102	-0.0233	0.116	0.0295
		(0.960)	(-0.166)	(1.540)	(0.288)
	Constant	23.54	37.73*	60.04***	73.12***
		(0.819)	(1.779)	(3.167)	(4.349)
	Observations	69	57	69	57
	R-squared	0.272	0.371	0.425	0.482

Table 6 . Results from simultaneous equations. Estimation method : SURE Dependent variable:

 6 Robust t-statistics in parentheses. Regional dummies are included in each equation. *** p<0.01, ** p<0.05, * p<0.1.

	Enrolement	Enrolment
	in Law	in Engineering
	(1)	(2)

Stijns (2006) disputes these results. The author uses abundance in natural resources as a variable, and finds that natural resource abundance positively affects education levels.

The six indicators of governance quality are tested separately. Ideally, the six indicators would be used simultaneously in the models. We tried this but got unreliable results with no statistically significant coefficients. We interpreted this result by the high colinearity between each of the six dimensions of governance.

Again, the governance variables are rescaled to be between 0 and 1, with higher values indicating a bad level of governance quality.

Regional dummies are included in each equation of the SURE model to control for the unobserved regional characteristics which can determine the allocation of talents.

Acknowledgements

We sincerely thank Sébastien Marchand, Jean-Louis Combes, Sylvianne Guillaumont Jeanneney, Gérard Chambas, Michael Goujon, Samuel Guerineau, Mireille Ntsama Etoundi, Kilama Eric, Thierry Yogo, and participants at the Cerdi -ANR seminar for their valuable comments and suggestions.

APPENDICES

Appendix A. Results of the principal component analysis.

Appendix B. Descriptive statistics and list of countries


Guillaume Colin
email: guillaume.colin@univ-orleans.fr

Yann Chamaillard
email: yann.chamaillard@univ-orleans.fr

Benoit Bellicaud
email: benoit.bellicaud@univ-orleans.fr

Robust control for the air path of a downsized engine

Keywords: Engine Control, Robust Control, PID, Uncertainty model, Nonlinear systems

This paper presents a simple method for designing a robust controller that can be used on nonlinear systems, here the air path of a downsized engine. This computationally inexpensive controller is thus obtained faster and more systematically than the usual look-up-table based controllers. The successive steps of the method are: system identification, system normalization and linearization, nominal model design, model uncertainty design, and controller design with property verification. This robust controller is applied to the air path of a downsized spark ignition engine, i.e. intake manifold pressure control and boost pressure control.

Introduction

More stringent standards are being imposed to reduce the fuel consumption and pollutant emissions of spark ignition (SI) engines. Modern automobile engines must therefore meet the challenging, and often conflicting, goals of minimizing pollutant emissions and fuel consumption while satisfying driving performance over a wide range of operating conditions. Engine downsizing is one of the most effective strategies for improving fuel economy while maintaining the advantage of the low emission capability of three-way catalytic systems and combining several well known technologies [START_REF] Lecointe | Downsizing a gasoline engine using turbocharging with direct injection[END_REF]. Downsizing is the use of a smaller capacity engine operating at higher specific engine loads, i.e. at better efficiency points [START_REF] Leduc | Downsizing of gasoline engine -an efficient way to reduce co 2 emissions[END_REF]. Turbocharging is one of the relevant ways to achieve efficient downsizing [START_REF] Guzzella | Ic-engine downsizing and pressurewave supercharging for fuel economy[END_REF]. Moreover, engine control is necessary to achieve an efficient engine torque control [START_REF] Guzzella | Introduction to Modeling and Control of Internal Combustion Engine Systems[END_REF], which includes in particular the control of ignition coils, fuel injectors and air actuators (throttle, turbocharger, etc).

System description

The air intake of a turbocharged SI engine, represented in Figure 1 can be described as follows. The compressor (pressure p boost ) produces a flow from the ambient air (pressure p amb ). The compressor flow is cooled down by an intercooler and goes through the throttle T h into the intake manifold (pressure p man ). At the exhaust side, the exhaust flow is split in two parts: the turbine flow and the wastegate flow. The turbine flow powers up the turbine and through a shaft drives the compressor. Thus, the supercharging pressure p boost can be adjusted by the turbine flow which is controlled by the All of the results presented here were produced with a 0.6 Liter turbocharged 3 cylinder MCC Smart engine. A complete torque control is necessary to test the proposed method (control of ignition coils, injectors, throttle position). This was implemented on the engine test bench through a fast prototyping system based on xPCTarget from Matlab R . It is worth noting that the Air Fuel Ratio (AFR) control must be very accurate to limit overshoot and undershoot of the AFR. The throttle actuator control will not presented here. In this engine configuration, the wastegate valve, which is linked to a membrane, is pneumatically controlled with a three way electrovalve connected between boost pressure and atmospheric pressure as shown in figure 1.

Problem formulation

As cylinder pressure sensor is not available, the intake manifold pressure p man becomes the most measurable image of the engine torque. Hence, the air path control manipulates the intake manifold pressure. For a given intake manifold pressure p man , there is an infinite number of solutions for opening the actuators, but only one is optimal from the energy point of view. To reach maximum efficiency, the throttle T h should be wide open. Engine control should therefore maintain the throttle open when p boost > p amb in order to reduce pumping losses in steady state. It is worth noting that the throttle is open when p boost ≈ p man . This implies that the supercharging pressure target is the same as the intake manifold pressure target. This is a pure fuel consumption minimization choice without regarding the transient performance. Another solution will be to adapt the boost pressure set point function of the desired efficiency loss as in [START_REF] Colin | Compromis consommation et performance pour le contrle d'un moteur turbocompress[END_REF]. In this case an efficiency loss of 0% corresponds to a pure fuel consumption choice.

Therefore, the control is multiobjective: to have the maximum opening of the throttle and to track the intake manifold pressure target. Moreover, this nonlinear system considered has subsystems with very different dynamics: the response time of the subsystem intake/throttle is much smaller than that of the subsystem turbocharger/wastegate. The actuators to be controlled are the throttle and the wastegate, that have saturations which must be taken into account for the controller design. The global control scheme for the air path of a turbocharged SI engine is shown on figure 2. The torque set point is directly linked to the driver's request.

Then, the supervisor provides an air mass set point which is next translated into an intake manifold pressure set point p man sp . For the intake manifold pressure controller, the manipulated variable is the throttle position set point T h and the controlled variable is the intake manifold pressure p man . For the boost pressure controller, the manipulated variable is the wastegate set point W G and the controlled variable is the supercharging pressure p boost .

Air

Main contributions

To control the air path of a spark ignition engine, various solutions have been proposed in the literature. For turbocharger control, in [START_REF] Karnik | Electronic throttle and wastegate control for turbocharged gasoline engine[END_REF], a state feedback control is proposed using a physical model of the turbocharger. However, in this control scheme, which has been tested only in simulation, the fastest response is not produced with a wide open throttle and a fully closed wastegate [START_REF] Eriksson | Control and optimization of turbocharged spark ignited engines[END_REF]. In addition, at a steady speed, the throttle is not wide open; as discussed above, however, this is an important point in reducing fuel consumption. In [START_REF] Lezhnev | Boosted gasoline direct injection engines: Comparison of throttle and vgt controllers for homogeneous charge operation[END_REF], a variable feedforward plus a PI controller (based on exhaust pressure) is proposed for an engine with a Variable Geometry Turbine (VGT). In spite of tuning difficulties / problems, this control scheme, which has only been tested in simulation, gives quite good results. In [START_REF] Colin | Neural control of fast nonlinear systems -application to a turbocharged si engine with VCT[END_REF], an Internal Model Control is used for the intake manifold pressure and Neural Linearized Predictive Control is used for the boost pressure. This control scheme, tested on an engine test bench and compared to the proposed controller, gives excellent performance results, but can pose some problems for the calibration of series vehicles. In [START_REF] Moulin | Modelling and control of the air system of a turbocharged gasoline engine[END_REF], a control based on physical model is used for the boost pressure control and was tested on a vehicle. This scheme can also pose some calibration problems on series vehicles. Finally, as simple PID controllers [START_REF] Wakeman | Closed loop turbocharger control with wastegate functions[END_REF] generate overshoots as shown in [12], gain-scheduling PID controllers, widely used in industry, could be used. Nevertheless, they are not easy to tune and are problematic for calibration and robustness.

In the literature, robustness, fast tuning and low computational cost are not always ensured all together. That is why, in this paper, we propose a method to synthesize a computationally inexpensive robust controller obtained faster and more systematically than the usual look-up-table based controllers, for a downsized Spark Ignition engine. This simple method, which ensures that the control is robust, can have a wide domain of application. In the proposed control scheme, based on linear control theory [13], the system nonlinearities will be represented by an average linear model with uncertainties.

The different steps for the controller design are detailed in the following :

• normalization and linearization (section 2),

• nominal model design (section 3),

• uncertainty representation (section 4),

• controller design (section 5).

Section 2 assumes linearization and normalization of the system, which will have a consequence for system identification and nominal model design.

Hence, the nonlinear system will be excited on all operating points in order to identify the normalized systems (section 3). From these data, a nominal linear model is computed and next the uncertainties are modeled (section 4).

Consequently, the controller can be synthesize with stability proof (section 5). In the last part of the paper, control results are shown first in simulation and next on an engine test bench.

Normalization and linearization

The normalization and linearization step, based on [13], considers a single input single output (SISO) system governed by an ordinary differential equation (ODE):

   dz(t) dt = f (z(t), v(t)) w(t) = g (z(t), v(t)) (1)
where z(t) are the states, v(t) is the input and w(t) the output. This model can be normalized around an operating point z 0 . A nominal constant input v 0 and a nominal constant output w 0 are associated to a nominal point z 0 . The following normalized system can be then considered:

   dx(t) dt = f 0 (x(t), u(t)) y(t) = g 0 (x(t), u(t)) (2)
Three new variables x(t), y(t) and z(t) are so introduced:

         z(t) = z 0 x(t) = z 0 (x 0 + δx(t)) w(t) = w 0 y(t) = w 0 (y 0 + δy(t)) v(t) = v 0 u(t) = v 0 (u 0 + δu(t)) (3)
where x(t) are the states, u(t) is the input and y(t) the output. The hypothesis of only small deviations from the normalized equilibrium points is made.

In equation ( 3), x 0 , y 0 and u 0 can be assumed equal to 1. An advantage of normalization is that it is possible to define easily a small difference, i.e.

δx(t)

1, δy(t) 1 and δu(t) 1 (notion of smallness).

Finally, the following linear model is defined:

   dx(t) dt = Ax(t) + bu(t) y(t) = cx(t) + du(t) (4) with A ∈ R n×n , b ∈ R n×1 ,c ∈ R 1×n ,d ∈ R, where:    A = ∂f 0 ∂x x=x 0 ,u=u 0 b = ∂f 0 ∂u x=x 0 ,u=u 0 c = ∂g 0 ∂x x=x 0 ,u=u 0 d = ∂g 0 ∂u x=x 0 ,u=u 0 (5)
Describing a real dynamic system using a normalized and linearized mathematical model is convenient for the analysis and synthesis of appropriate control systems. However, the controller obtained with this normalized and linearized system cannot be directly applied to the real system. Therefore, normalization and de-normalization blocks have to be added on a classical closed loop control scheme. The interface is shown in figure 3.

Firstly, the normalization is chosen as (from (3)) : The error of the controller is therefore:

δy(t) = w(t) w 0 -y 0 = w(t) w 0 -1 (6) Controller Non linearity description δ 0 R l t S t w 0 v ε C δu w + * desc p o / v 0 y + δy ref =0 Regulator System w + + + u 0 =1 * * y u - δy - y 0 =1 + Normalization De-normalization
c (t) = δy ref -δy(t) (7)
where the reference δy ref is equal to zero due to the normalization and linearization steps. Secondly, the de-normalization is chosen as (from (3)):

v = v 0 (u 0 + δu) = v 0 (1 + δu) ( 8
)
where δu is the output of the controller Hence, the control scheme shown on figure 3 is equivalent to the one shown on figure 4. It can be seen that a gain scheduling controller appears in the proposed control scheme because the output of the regulator is multiplied by the nominal input v 0 . This nominal input v 0 has been previously computed by a non linearity description (here a static look-up-table function of engine speed and intake manifold pressure set point). Moreover, a feedforward is also present because the nominal input v 0 is added to the output of the controller u. As the normalization is done with respect to the set point, the reference must not be equal to zero, which is the case here for the air path of the turbocharged SI engine. A small variable change must be done for the nominal input v 0 to make it always different from zero (otherwise the controller will be inactive as shown in figure 4).

Nominal model design

The step of linearization states that linear models are necessary to synthesize the control. Hence, all operating points will be swept to make the identification of these models. Here, a frequential identification is performed [START_REF] Ljung | System Identification Theory for the User[END_REF] on all the representative operating points: 52 for the throttle and 19

for the wastegate (some operating points are the same). The engine speed N e goes from 1500 rpm to 4500 rpm and torque goes from 0 Nm to 90 Nm (minimum to maximum). Small engine speeds have been removed because no boost pressure is available and because our engine brake is unstable in this domain. A multisinus excitation (sum of sinus at different frequencies) for the system input is performed while system output is acquired. Finally, Fast Fourier Transform (FFT) is used to obtain the frequential representation of the system (figures 5 and 6). Note that the average values of inputs and outputs are saved in order to compute the nonlinearity description of figure 4.

- as the average between the maximum and the minimum system at each frequency (red system on figures 5 and 6). This choice (and not the average of all systems) is directly linked to the norm used to modelize uncertainties (13). This reference system, not physical, is a sequence of values, where each value corresponds to a behavior at one given frequency.

The reference system is then used to identify the average model, called here nominal model. This model will be used for the controller design and will be defined as a continuous transfer function. The transfer function can be found using an optimization method with an appropriate criterion to minimize. The criterion to minimize is the weighted sum of root mean squared error (RMSE) of the magnitude and the RMSE of the phase, where the error is computed between the reference system and the nominal model. As the nominal model must be as simple as possible, a first or a second order transfer function can be chosen:

P 1 (s) = b 0 + b 1 s 1 + a 1 s (9)
P 2 (s) = b 0 + b 1 s + b 2 s 2 1 + a 1 s + a 2 s 2 (10)
For the intake manifold pressure controller, a first order is sufficient, yielding the following transfer function (figure 7):

P T h (s) = -0.005101s + 0.8246 0.09747s + 1 (11)
For the boost pressure controller, a second order is necessary, yielding the following transfer function (figure 7):

P W G (s) = 0.604 0.05723s 2 + 0.3115s + 1 (12)
Here, just by looking at the phases of the Bode diagrams (figure 7) one can see that the throttle T h to intake manifold pressure p man system is a first order and the wastegate W G to boost pressure p boost system is a second order.

Uncertainty model design

A description of the uncertainties is necessary to allow the analysis of the closed loop system, especially the robustness property. In this work, the simplest method is used. The idea is to parameterize the uncertainty by the set:

S =          P (s) (1 + ∆W (s)) ||∆|| ≤ 1 arg{∆} ∈ [-π, π] (13)
where P (s) is the nominal model, W (s) an uncertainty bound and ∆ is the uncertainty generator. Here, the set is therefore defined by a nominal frequency response P (s), a unit disc centered in the origin defined by ∆, and this disc is translated by P (s) and scaled by a factor P (s)W (s). Since for a fixed frequency the phase is not relevant due to the uncertainty circle around the nominal model, the magnitude may be used only to formulate a condition on the uncertainty bound W (s). The uncertainty bound W (s) is considered here as a first or a second order model computed from the measured maximum uncertainty as in [13] (Figures 8 and9). Other uncertainties could be added in this model, e.g. aging of the system, variation of turbine or compressor characteristics, identification uncertainty.

The representation of the uncertainties around the nominal model is given in figure 10 for the throttle to intake manifold pressure system and in figure 11 for the wastegate to boost pressure system.

The uncertainty model for the intake manifold pressure controller is given by the following second order transfer function:

W T h (s) = 0.0005204s

Measured maximal uncertainty 1st order measurements

Controller design

As seen before, the system has been normalized and linearized so that any linear controller, available in the linear control theory literature, could be used in the control scheme of figure 4. The classical PID controller, tuned with a pole placement method in which robustness is verified, is chosen here for its simplicity and in order to prove the effectiveness of the proposed method. To avoid problems with the saturations, an anti-reset windup (ARW) has to be added to the controllers [15].

Assuming a nominal plant P (s), a conservative bound on its uncertainty W (s) and a controller C(s) and assuming the nominal closed loop is asymptotically stable, the controller C(s) will produce an asymptotically stable closed loop system for all the plants in the set S if the following inequality is verified for all frequencies (Nyquist stability theorem):

|1 + C(s)P (s)| ≥ |C(s)P (s)W (s)| (16)
Two cases of pole placement are here considered to find the coefficients K, T i and T d of the PID controller C(s) [START_REF] Pressure | Pa) results vs. time (s) on the engine test bench: set point (blue solid), intake manifold pressure (red dash), boost pressure[END_REF], as shown in table 1.

C(s) = K(1 + sT i + s 2 T i T d ) sT i (17)
In this table, the computation of K, T i and T d is done while identifying 1 + C(s)F (s) = 0 to the denominator of G(s).

For the intake manifold pressure controller, as the nominal model is a first order, the desired (reasonable) system dynamic is chosen as a second order that ensures a robust control. For example, the parameters ω n = 10rad/s and ζ = 1.2 ensure robustness for the closed loop true system and for the closed loop nominal model with modeled uncertainties (i.e. ( 16) satisfied) as shown in figure 12. With this desired system the following coefficients of the PI controller are found K = 1.4731, T i = 0.1350.

Open loop

as+b cs+1 a 3 b 2 s 2 +b 1 s+1 nominal system F (s) Desired closed ω 2 n s 2 +2ζωns+ω 2 n αω 3 n (s 2 +2ζωns+ω 2 n )(s+αωn) loop system G(s) K 2ζcωn-1-aω 2 n c/b b+a 2 ω 2 n /b-2ζaωn (2ζω 2 n α+ω 2 n) b 1 -1 a 3 T i Kb ω 2 n (c+Ka) Ka 3 αω 3 n b 1 T d 0 2ζωnb 1 +αωnb 1 -b 2 Ka 3
Table 1: Pole placement to find the coefficients of the PID controller For the boost pressure controller, as the nominal model is a second order, the desired (reasonable) system dynamic is chosen as a third order that ensures a robust control. For example, the parameters ω n = 3rad/s, ζ = 1.1 and α = 1 ensure robustness for the closed loop true system and for the closed loop nominal model with modeled uncertainties (i.e. ( 16) satisfied) as shown in figure 13. With this desired system the following coefficients of the PID controller are found K = 1.0734, T i = 0.4195, T d = 0.3670. The derivative of the controller has been filtered to avoid the classical high frequency problems. All the figures shown in this paper used the filtered derivate. A good margin of robustness is ensured for the closed loop system as shown in figure 13. Tuning parameter ω n for boost pressure controller is smaller than for the intake manifold pressure controller, because desired dynamic are different. Equation ( 16) is verified for each controller as shown on figure 14. In this figure, |1 + C(s)P (s)| is always greater than |C(s)P (s)W (s)| which proves the stability of the two closed loop systems.

Results

All of the results presented here were produced with a 0.6 Liter turbocharged 3 cylinder MCC Smart Engine (see section 1.1 for more details). In the presented results, both controllers runs together with the control scheme of figure 2.

Firstly, both controllers are validated in simulation on various transients. controllers are quite good. Indeed, only a small overshoot of boost pressure appears (at 85s at the top of figure 16) and there is no steady state error. Figure 17 shows the intake manifold pressure set point p man sp and the intake manifold pressure p man when the torque target changes. Figure 18 shows the actuators response and figure 19 shows the engine speed during the same test.

Set Point Measurement

As the performances are quite correct in terms of intake manifold pressure, torque is well controlled. Nevertheless, some pressure oscillations appears due to throttle oscillations around limp-home position (around 360s) and an overshoot of 70mbar can seen at 360s that could be suppressed while filtering the set point (as used in industry). In this test, the turbocharger controller fixes naturally the wastegate fully opened which correspond to 0%.

Figure 20 shows the intake manifold pressure set point p man sp , the boost pressure p boost (which cannot be less than p amb ≈ 1bar) and the intake manifold pressure p man when the torque target changes. Figure 21 shows the actuators response during the same test and the contributions of the feedforward, which is not negligible. Some boost pressure oscillations appears at 25s because of pumping turbocharger problem (no dump valve on this engine) which have a consequence on intake manifold pressure. As in simulation, small overshoots of boost pressure appear (at 17s and 21s) and there is no steady state error (e.g. at 24s) except between 26s and 30s where boost pressure cannot reach the target because wastegate is already fully opened.

Note that the throttle is opened as wide as possible at steady state (at 24s on figure 21), because p man ≈ p boost so that the objectives are satisfied. A comparison between the proposed control and the control scheme presented in [START_REF] Colin | Neural control of fast nonlinear systems -application to a turbocharged si engine with VCT[END_REF] is shown on figure 20 for intake manifold pressure. On can see that the results with the proposed control scheme gives more longer overshoots (e.g. at 12s) that the results obtained with [START_REF] Colin | Neural control of fast nonlinear systems -application to a turbocharged si engine with VCT[END_REF]. Nevertheless, this overshoot of 35mbar can be considered as acceptable.

Conclusion

In this paper, we have proposed a simple method for designing a robust controller that can be used on nonlinear systems. This computationally inexpensive controller is thus obtained faster and more systematically than the usual look-up-table based controllers. The successive steps of the method are: system identification, system normalization and linearization, nominal model design, model uncertainty design, and controller design with property verification. The system has been normalized and linearized so that any linear controller, available in the linear control theory literature, could be used in the control scheme. The classical PID controller, tuned with a pole placement method in which robustness is verified, is chosen here for its simplicity and in order to prove the effectiveness of the proposed method. This robust control methodology has been applied to the air path of a downsized spark ignition engine, i.e. intake manifold pressure control and boost pressure control. The good control performances of the proposed method have been demonstrated first in simulation and next on an engine test bench.

Figure 1 :

 1 Figure 1: Air intake of a turbocharged SI engine

Figure 2 :

 2 Figure 2: Global scheme for the air path control of the turbocharged SI engine

Figure 3 :

 3 Figure 3: Interface between Linear normalized Controller with the nonlinear system

Figure 4 :

 4 Figure 4: Equivalent proposed control scheme that points out a gain scheduling controller with a feedforward

Figure 5 :Figure 6 :

 56 Figure 5: Bode diagram of the identified systems (black) and average system (red): system input is the normalized throttle set point and system output is the normalized intake manifold pressure

Figure 7 :

 7 Figure 7: Identification results: bode diagram of the the reference system (from T h to p man in red solid, from W G to p boost in green dash dot) and of the identified linear model (from T h to p man in black dash, from W G to p boost in blue dot)

Figure 8 :

 8 Figure 8: Amplitude of uncertainty (dB) versus frequency (rad/s) for the throttle to intake manifold pressure system

Figure 9 :Figure 10 :

 910 Figure 9: Amplitude of uncertainty (dB) versus frequency (rad/s) for the wastegate to boost pressure system

Figure 11 :

 11 Figure 11: Nyquist diagram of the linear model with the modeled uncertainties for the wastegate to boost pressure system

Figure 12 :

 12 Figure 12: Nyquist representation of C(s)P (s) for the intake manifold pressure controller: nominal model (blue), real systems (red), modeled system with circle amplitude of |C(s)P (s)W (s)| around C(s)P (s) (black)

Figures 15 and 16 1 -Figure 13 :Figure 14 :

 1611314 Figures 15 and 16 shows two examples of realized tests at two different engine speeds with variation of torque set point. The dynamic performance of both

Figure 15 :Figure 16 :

 1516 Figure 15: Simulation results at 3500rpm. Intake manifold pressure (Pa) vs. time (s) (top): set point p man sp (blue dash) and measurement p man (green solid) ; Actuator position (%) vs. time (s) (bottom): throttle set point (blue dash), throttle position (green solid), wastegate fixed at 0% (not shown here)

  )

Figure 17 :

 17 Figure 17: Pressure (Pa) results vs. time (s) on the engine test bench: set point (blue solid), intake manifold pressure (red dash), boost pressure (black dot)

Figure 18 :Figure 19 :

 1819 Figure 18: Actuator (%) results vs. time (s) on the engine test bench: throttle set point (blue solid), throttle position (red dash), wastegate closing (black dot)

  of[START_REF] Colin | Neural control of fast nonlinear systems -application to a turbocharged si engine with VCT[END_REF] Intake Manifold pressure for the proposed control Boost pressure for the proposed control

Figure 20 :

 20 Figure 20: Pressure (Pa) results vs. time (s) on the engine test bench: set point (blue solid), p man with control proposed in[START_REF] Colin | Neural control of fast nonlinear systems -application to a turbocharged si engine with VCT[END_REF] (red dash dot), p man with proposed control (green dash), p boost with with proposed control (black dot)

Figure 21 :

 21 Figure 21: Actuator (%) results vs. time (s) on the engine test bench: throttle feedforward (red dash), throttle set point (blue solid), wastegate feedfoward (green dash dot), wastegate closing (black dot)

List of Figures