Zilong Zhao
email: z.zhao-8@tudelft.nl

Bogdan Robu

Ioan Landau

Luc Dugard

Nicolas Marchand

Louis Job
email: louis.job@sciencespo-grenoble.fr

Modelling and Optimal Control of MIMO System -France Macroeconomic Model Case

Keywords: MIMO, Optimal control, LQR, Macroeconomic data

Applying control theory to economic problems has been studied since 1970s. Early stage studies focus on policy analysis and optimizations. With the development of theory, recent researches touch on more applications such as (1) monetary policy, (2) fiscal policy, and (3) resource allocation problems using PID control and optimal control. Even the problems change, the studies that applied optimal control follow the same idea: finding a control law to optimize an objective function for a dynamic system under certain constraints.

In this paper, we focus on french macroeconomic model. Variables such as Gross Domestic Production, Exportation, Importation, Household Consumption, Gross Fixed Capital Formation and Public expenditure are included in the analysis. Our objective is to maintain a constant economic growth rate according to the available resources. We implement a one-ofa-kind optimal control via Linear-Quadratic regulator (LQR) to achieve that. Since we aim to maintain a constant growth rate instead of a constant value, a specific control system is designed for this purpose. We implement three experiments: (1) variation on parameter of LQR, (2) constraints on inputs, and (3) perturbations on outputs. Results show that our designed control system can guide the output to the desired growth rate. Varying parameters of LQR can change the convergence speed of the system. Constraints on specific inputs can delay the system convergence and stimulate other inputs to compensate for the constraints. Perturbations on outputs experiment show that our control law can help the system to recover to the original statespace.

I. INTRODUCTION

Control theory is commonly applied to mechanical and physical systems. But it is a strong tool to solve optimization problem in various areas [START_REF] Ghiassi | Robust (deep) learning framework against dirty labels and beyond[END_REF], [START_REF] Younesian | Qactor: On-line active learning for noisy labeled stream data[END_REF]- [START_REF] Zhao | Event-based control for online training of neural networks[END_REF]. Applying control theory for economic problems has been studied since 1970s. [START_REF] Chow | Development of Control Theory in Macroeconomics[END_REF] summarizes the development of stochastic control theory in macroeconomic policy analysis in three periods. The first is pre-1970 when the major ideas of policy analysis and of optimization were formed [START_REF] Dow | The optimum rate of saving[END_REF]. The second is the early and middle 1970s when formal stochastic control theory was rapidly developed for and applied to the study of macroeconomic policy [START_REF] Chow | Analysis and Control of Dynamic Economic Systems[END_REF], [START_REF] Friedman | Economic Stabilization Policy: Methods in Optimization[END_REF]. The third period, beginning in the late 1970s, was stimulated by the introduction of the idea of rational expectations in economic analysis [START_REF] Blanchard | The production and inventory behavior of the american automobile industry[END_REF].

Recent works focus more on applications. [START_REF] Hawkins | Monetary policy and PID control[END_REF] proposes a general class of PID-based monetary policy rules, the feedback rules let the model use a control signal (e.g. central bank's policy interest rate) responds to movements in a small number of macroeconomic factors, such as the current amount of labor market slack and the deviation of the rate of inflation from its target. Under an optimal control monetary policy [START_REF] Brayton | Optimal-control monetary policy in the frb/us model[END_REF], the current and expected future path of the policy is instead typically calculated with a procedure that minimizes a cost function subject to certain constraints. For fiscal policy [START_REF] Léonard | Optimal Control Theory and Static Optimization in Economics[END_REF] and resource allocation [START_REF] Clark | Mathematical bioeconomics : the optimal management of renewable resources[END_REF], [START_REF] Intriligator | The applications of control theory to economics[END_REF] problems, they follow the same ideas of optimal control, differences are the cost function and the constraints. To estimate the asset holdings of a portfolio, [START_REF] Georges | A Discrete-Time State Observer Approach to Discovering Portfolio Holdings[END_REF] uses algorithms applied to nonlinear dynamic systems to estimate the state with a discrete-time observer.

In this paper, we will apply control techniques to a French macroeconomic model using real economic data (available as time series) starting from 1980s. Our objective would be to design a meaningful control policy that would allow a constant growth rate of the Gross Domestic Product. After choosing the appropriate variables as inputs and outputs, we will first model this system as Multiple-Input and Multiple-Output (MIMO) system. After data pre-processing to make the time series stationary, the orders and parameters of MIMO system are estimated and validated before transforming it into a state-space model. The model we estimate in this paper is an autoregressive model with external inputs. To satisfy our objective, an optimal control solution: Linear-Quadratic regulator (LQR) is designed and implemented. By varying parameters of LQR, we can control the converging speed of output to state-space. A further simulation result shows that if we put constraints on the level of one of input signals, we can observe the compensation effect from other inputs. But even there is no limitation on the amplitude of inputs variables, the level of output and its converging speed to state-space is still slightly worse than the case without constraints. Perturbations on outputs are also studied, which simulates a scenario of an economic crisis. The experiments are also developed with and without constraints on input signals. Results show that our system can quickly recover from the disturbance, and constraints on input signals delay the recovery.

II. SYSTEM IDENTIFICATION

In this section, the process of estimating the economic model is given. After the data preparation (to make is stationary), a MIMO model is computed. The parameters of the model are then estimated and validated.

A. Data Preparation

All the data used throughout this paper are obtained from the INSEE (Institut National de la Statistique et des Études Économiques) which regroups the official economic French Data1 . A plethora of data is available, nevertheless for a nonexpert it is hard to decide which makes sense in our case. After several discussions with experts on these macroeconomic issues, we decide to study 6 time series namely: Gross Domestic Production (GDP), Exportation (EXP), Importation (IMP), Household Consumption (HC), Gross Fixed Capital Formation (GFCF) and Public Expenditure (PE). In the INSEE database we find that all this data are quarterly, ranged from the first quarter of 1980 to the fourth quarter of 2018, 1980T1 to 2018T4, where T1 and T4 denote first and fourth quarter of the year 2 . Following a similar analysis as the one performed in [?] we decide the inputs and outputs of the system. The inputs of the model would be variables on which the appropriate governmental structures can act and incite their modification, namely Household Consumption, Gross Fixed Capital Formation and Public Expenditure. On the other side, the outputs of the model are variables on which we suppose that, on a regular basis, are only measured but the government can not directly act on: Gross Domestic Production, Exportation, Importation.

The reader should note that the original data from INSEE are presented on the values of current price and make no adjustment for inflation. This is problematic as the current price measure measures for example GDP, inflation or asset prices using the actual prices we notice in the economy not the real, deflated, one. Before anything else, we therefore need to deflate it by using France GDP deflator (base year: 2014) obtained from the World Bank. Deflated time series are showed in Fig. 1. As we deal with time series having an economic meaning, we prefer to use natural logarithm to better linearize them. The intuition behind this choice comes from macroeconomics, where it is quite common to consider it as it eases the analysis (see for example [START_REF] Gelman | Data analysis using regression and multilevel/hierarchical models, volume Analytical methods for social research[END_REF]). Moreover, as we will use these time series to do linear regression, we must ensure that all these series are stationary. Augmented Dickey-Fuller (ADF) test cheung1995lag is a commonly used statistical unit root test to examine whether a given time series is stationary or not. In our case, we would test the stationary for the natural logarithm of our original time series as well as the first difference of natural logarithm. Results are shown in Table I where LGDP, LEXP, LIMP, LHC, LGFCF and LPE denote the natural logarithm of our 6 time series introduced before: GDP, EXP, IMP, HC, GFCF and PE. From TableI we can therefore observe that the test, in the case of the natural logarithm of the original data, cannot reject the null hypothesis that the variable contains a unit root, which implies that the original time series are therefore not stationary. Nevertheless, we can notice that the results for the first difference of natural logarithm of all the series rejects the null hypothesis at 99% (i.e., note the * * * following the values as indicator). We can therefore infer that these time series using the first difference of natural logarithm are stationary. Moreover, we will use the first difference of natural logarithm of GDP, EXP, IMP, HC, GFCF and PE in later analysis (hereinafter referred to as DLGDP -for difference logarithm GDP, DLEXP, DLIMP, DLHC, DLGFCF and DLPE).

B. Selection of model order

In order to introduce our economic model, we first define the input and output vectors as follows:

y =   y 1 y 2 y 3   =   DLGDP DLEXP DLIMP   u =   u 1 u 2 u 3   =   DLHC DLGFCF DLPE   (1)
where y is the output (endogenous variables) of our model, u is the input (exogenous variables) of our model. Note that the selections of y and u are done according to their economic interpretation and attributes. Input (exogenous) variables are therefore the factors we could manipulate in an economic system (e.g., increasing public expenditure for example) while output (endogenous) variables are the consequences that we could only observe in our case but not directly interfere with (e.g., the variation of importations for example).

In the general case, consider an "m-input-p-output" system represented by a canonical input-output representation [START_REF] Guidorzi | Invariants and canonical forms for systems structural and parametric identification[END_REF], for i = 1, 2, ...p:

y i (k) = p j=1 nij q=1 a ijq y j (k + q -n i -1) + m j=1 ni q=1 b ijq u j (k + q -n j -1) + e i (k) (2)
where p and m is the numbers of outputs and inputs, y i (k) denotes the value of output y i at time k, a ijq and b ijq are the coefficients of y i (k + q + n i -1) and u i (k + q -n j -1), e i (k) is the white noise at time k. Moreover, n i are the observability indices and n ij are given by:

n ij = min{n i , n j }, if i ≤ j (3)
and

n ij = min{n i + 1, n j }, if i > j (4)
We apply this method for each output using the technique of "instrumental variables" (see for example [START_REF] Landau | Digital Control Systems: Design, Identification and Implementation (Communications and Control Engineering)[END_REF]), and implementing a criterion which penalizes the model complexity as we want to estimate models of reduced order (see for example [START_REF] Duong | On a criterion for estimating the structure of linear mimo systems[END_REF], [START_REF] Landau | Digital Control Systems: Design, Identification and Implementation (Communications and Control Engineering)[END_REF] among others). Fig. 2 shows the order selection process for y 1 , y 2 and y 3 while Criteria variable calculates the average of estimation errors. The Criteria is defined as J(n) as follows:

J(n) = min θ 1 N Y (t) -R(n) θ 2 (5)
where n is the estimated system order, N is the number of data, θ is the estimation of parameters. Y (t) is the real value and R(n) θ is the estimated value. S(n, N) is the part served as penalty and defines as follows:

S(n, N ) = 2nlog(N ) N (6)
According to [START_REF] Landau | Digital Control Systems: Design, Identification and Implementation (Communications and Control Engineering)[END_REF], J(n) goes towards 0 as the estimated order approaches the true one. Therefore, as order increases, we can see from Fig. 2 that criteria decreases to 0. But S(n, N ) increases with the increase of the chosen order.

As it can be seen from Fig. 2, the estimated model between the inputs and each of the three outputs is: 5 for y 1 , 4 for y 2 and 5 for y 3 . One should notice that these estimated order are not definitive as we still need to pass the validation process to decide the final order.

C. Estimation and Validation of parameters

After finding the model order, Least Squares method is used to estimate the parameters of the 3 equations for y1, y2 and y3. After each estimation, a whiteness test (autocorrelation test) will be applied to make sure the residuals from the estimated equation are white noise, which means the estimated model have already extracted all the knowledge from training data (this problem is well studied in both economic [START_REF] Araujo | Econométrie. Amphi Économie. Bréal[END_REF] and control [START_REF] Landau | Digital Control Systems: Design, Identification and Implementation (Communications and Control Engineering)[END_REF] domain). The algorithms of calculation vary through these studies, but the goal remains the same: testing if values are mutually uncorrelated. In this paper, we choose to use the implementation of autocorrelation test function in [START_REF] Baiocchi | GRETL: Econometric software for the GNU generation[END_REF].

Consider T as the total number of data points in the dataset, we can therefore conclude that all the autocorrelation values should be in the range (0 ± 1.96 T 0.5 ). In our case, from 1980T1 to 2018T4 we have 156 data points, so the limit is ±0.157 here. Fig. 3 shows the final autocorrelation test for estimation residuals of y 1 , y 2 and y 3 where the limit of ±0.157 is manifested by the horizontal blue lines.

Please recall that our objective is to find the minimal model, which means a model with the lowest order (minimum number of variables), but still valid. Therefore, once we find a valid candidate model we would still try to reduce the order by eliminating the variables whose coefficient is much smaller than the others by comparing their absolute value. Nevertheless this can pose problems concerning the validity of the mode (for example the residual not being a white noise anymore). Therefore, every time we delete one variable we need to reestimate all the parameters then re-do the whiteness test on residuals in order to check the validity of the new model. We continue to remove variables until none can be removed from the equation without compromising the validity of the model.

The minimal model can be written as follows:

y 1 (k) = a 115 y 1 (k -1) + a 114 y 1 (k -2) + a 113 y 1 (k -3) + a 112 y 1 (k -4) + a 111 y 1 (k -5) + a 124 y 2 (k -2) + a 123 y 2 (k -3) + a 122 y 2 (k -4) + a 135 y 3 (k -1) + a 134 y 3 (k -2) + a 133 y 3 (k -3) + b 111 u 1 (k -1) + b 131 u 3 (k -1) + e 1 (k) (7)
y 2 (k) = a 214 y 1 (k -1) + a 221 y 2 (k -4) + a 222 y 2 (k -3) + a 223 y 2 (k -2) + a 224 y 2 (k -1) + a 234 y 3 (k -1) + b 211 u 1 (k -1) + b 221 u 2 (k -1) + e 2 (k) (8)
y 3 (k) = a 314 y 1 (k -1) + a 313 y 1 (k -2) + a 324 y 2 (k -1) + a 323 y 2 (k -2) + a 334 y 3 (k -1) + a 333 y 3 (k -2) + a 332 y 3 (k -3) + a 331 y 3 (k -4) + b 321 u 2 (k -1) + e 3 (k) (9)
As we can see, the final orders of the reduced model are 5 for y 1 , 4 for y 2 and 4 for y 3 . The numerical values of the parameters are detailed in below If we define our state vector X as: 7), ( 8), ( 9) can be written as a discrete-time statespace model in the following form:

X(k) = [y 1 (k -1) y 1 (k -2) y 1 (k -3) y 1 (k -4) y 1 (k -5) y 2 (k -1) y 2 (k -2) y 2 (k -3) y 2 (k -4) y 3 (k -1) y 3 (k -2) y 3 (k -3) y 3 (k -4)] T (10)
X(k+1) = A • X(k) + B • u(k) Y(k) = C • X(k) + D • u(k) (11)
where k ∈ Z + , output y and control u vectors are given by equation 1,

A ∈ R 13×13 ; B ∈ R 13×3 ; C ∈ R 1×13 ; D ∈ R 1×3 .
The values of A, B, C, D are showed in equation ( 12) below, where O i×j is the zero matrix of size i × j.

Eigenvalues of the state matrix A are checked and all are within the unit circle which means the open-loop model is stable as we expected. Controllability and observability of the system are also tested, the rank of the observability and controllability matrices are equal to the number of states, which suggests that our system is controllable and observable.

III. OPTIMAL CONTROL POLICY

The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem [START_REF] Anderson | Optimal Control: Linear Quadratic Methods[END_REF].

In this section, we will first introduce Linear-Quadratic Regulator (LQR) and then, according to the physical nature of our input and output, we develop several modifications of the method.

First, consider a discrete-time linear system given by:

X(k+1) = A • X(k) + B • u(k)
the cost function of discrete time LQR in finite horizon can be presented as follows:

J = X(N ) T QX(N ) + N -1 k=0 (X(k) T QX(k) + u(k) T Ru(k) + 2x(k) T N u(k)) (14)
where Q, R are the weight matrices for state and input and the cross term matrix N is set to 0 in our case as the states and input vectors do not have the same economic meaning in order to be multiplied together.

The feedback control law that minimizes the cost function J is:

u(k) = -K • X(k) ( 15
)
where K is given by:

K = (R + BP T B) T (B T P A + N T ) ( 16
)
and P is the unique positive definite solution to the discrete time algebraic Riccati equation (DARE):

P = A T P A -(A T P B + N )(R + B T P B) -1 (B T P A + N T ) + Q (17)
More details about the practical implementation of LQR in our case are given in Sec. IV-A below.

A. Reference Input

As we will focus on controlling GDP, then we need to implement our designed reference to let output reach the desirable value. Recall that y 1 from Eq. 1 is the first difference of natural logarithm of GDP, illustrated below: ; C = 1 0 0 0 0 1 0 0 0 1 0 0 0 ; D = 0 0 0 (12)

y 1 (k) = ln(GDP(k)) -ln(GDP(k-1)) = ln( GDP(k) GDP(k-1) ) (18) A =                0.
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0                B =         0.
If we want our GDP to have constant p percent (%) increasing, we have:

GDP(k) GDP(k-1) = 1 + p 100
and then the quarterly GDP increasing ratio p can be interpreted as:

p = (e y1(k) -1) × 100 ( 19
)
where e is the base of the natural logarithm. One can remark that as we want a constant increasing ratio, i.e. p constant, y 1 needs to be constant.

B. Overall control system

In the state-space, the state and input vectors are constant. If we can define the optimal state-space X r , the corresponding input vector as u r and the desired output as Y r , at equilibrium all these variables are satisfying the following equations according to Eq. ( 11):

X r (k) = A • X r (k) + B • u r (k) Y r (k) = C • X r (k) + D • u r (k) (20)
We see from Fig. 4 that, different from general control method (e.g. PID), the Reference in LQR does not directly apply feedback from the output but reacts on feedback control law u = -K • X. Nevertheless, as our state vector contains only the previous values of the output we do not need an observer before implementing our control law. To drive the error between X(k) and X r (k) to 0, we make a change on Eq. ( 11), namely:

X(k) -X r (k) = A • X(k) + B • u(k) -X r (k) = A • X(k) + B • u(k) -A • X r (k) -B • u r (k) = A • (X(k) -X r (k)) + B • (u(k) -u r (k)) (21)
∆X(k) = A • (∆X(k)) + B • (∆u(k)) (22)
and re-write the cost function of LQR (13) as:

J r = N -1 k=0 ((∆X(k)) T Q(∆X(k)) + (∆u(k)) T R(∆u(k)) +2(∆X(k)) T N (∆u(k))) + (∆X(N )) T Q(∆X(N )) (23)
the feedback control law that minimizes J r can be written as:

∆u(k) = -K • ∆X(k) (24)
where K = (R + BP T B) T (B T P A + N T ) is independent of state and input vectors. We can notice therefore that the K in ( 15) and ( 24) does not change.

The control law [START_REF] Zhao | Robust anomaly detection on unreliable data[END_REF] can also be written as:

u(k) = -K • (X(k) -X r (k)) + u r (k) (25)
According to Eq. ( 25)we can not directly apply designed output into the feedback, we will need a pre-processing function to transfer the desired output Y r to the desired state vector X r and input vector u r when the system reaches the state-space. One thing to notice that there will not be only one pair of X r and u r to satisfy the pre-processing function condition, it will be a range for both value, we will let experts to choose the values which make more sense in real world.

Recall the cost function of LQR [START_REF] Georges | A Discrete-Time State Observer Approach to Discovering Portfolio Holdings[END_REF].where N is set to 0. In this case, we set the weight matrix

Q = C T Q C, since Y (k) = C • X(k)
, and the auxiliary matrix Q weights the plant output [START_REF] Triantafyllou | Maneuvering and control of surface and underwater vehicles[END_REF]. We find therefore the usual conclusions: when R >> C T Q C, the cost function is dominated by the control effort u, and so the controller minimizes the control action itself, this control strategy is used when the control signal is constrained; when R << C T Q C, the cost function is dominated by the output Y , and there is less penalty for using large u.

IV. SYSTEM EVALUATION

A. Setting-up the evaluation

According to the reality, a yearly GDP growth ratio of 3.2% is interesting to study3 . To reach this level, the quarterly GDP increasing ratio p is around 0.8% (i.e. (1.008) 4 ≈ 1.032).

From [START_REF] Léonard | Optimal Control Theory and Static Optimization in Economics[END_REF] we know:

y 1r = ln(1 + p 100 ) = ln(1.008) = 0.007968 ≈ 0.008
Following the same calculation process as above, we set y 2r = 0.0175 and y 3r = 0.0086. In the end, Y r = [y 1r y 2r y 3r ] is the reference.

Recall the relations between Y r , X r and u r in [START_REF] Murray | Control and dynamical systems[END_REF], in our experiment, since the output is identical to the first entry of the state vector, the matrix D will be 0 in our case. These equations can be re-written as:

X r (k) = (I -A) -1 • B • u r (k) Y r (k) = C • X r (k) ( 26
)
where I is the identity matrix of suitable dimensions. As we explained in the end of Sec. III-B, there is not only one pair of X r and u r that satisfy [START_REF] Zhao | Event-based control for online training of neural networks[END_REF]. After our selection, one reasonable pair of X r and u r is: Knowing about the constraint of u, diagonal weights [START_REF] Murray | Control and dynamical systems[END_REF] of Q and R are used.

X r (k) = [0.
Q =    q 1 . . . q nq    ; R = ρ    r 1 . . . r nr    (27)
where n q = rank(A) = 13, n r = rank(B) = 3. For the sake of simplicity, we will let q i = 1 for all i ∈ [1, 13], and r j = 1 for all j ∈ [1, 3], we will use ρ to adjust the input/state balance. We choose ρ ∈ {1, 10, 100} to implement the experiments to compare the converging speech and observe the input signal range.

B. Experimental Evaluation

For the evaluation of the proposed control strategy, we implement the system detailed in Fig. 4.

The initial state vector X 0 of the state-space model is set by a linear regression over the real data only from 1980T1 to 1981T1. The resulting values are given below: X 0 = [0.001, 0.012, -0.008, -0.004, -0.004, 0.036, 0.032, 0.013, -0.03, 0.007, 0.017, 0.006, -0.002] T Moreover, due to the meaning of our variables we have the following constraints regarding the three inputs: u 1 ∈ [0, 0.3], u 2 ≤ [0, 0.008], u 3 ≤ [0, 0.008]. Details about these limits are given below.

For the sake of better presenting our results, throughout all the evaluation procedure we will consider the same simulation time, namely 50 time points (i.e. quarters).

1) Variation of ρ without Constraints on Input Signals: Let us first discard the limitations on the control signal in order to observe the behavior of the control system as well as it's limitations for different values of ρ. The output (i.e. y 1 ) is illustrated in Fig. 5a for ρ ∈ {1, 10, 100}. A more detailed result is showed in Tab. II where the settling time as well as the variation range of the output are given. For the time being, the reader can just ignore the result of "1 with constraint" which will be detailed later on. From these results, we can clearly conclude that increasing the ρ value will also increase the convergence time to the state-space. Nevertheless, this comes with a significant impact on the inputs.

As we expected, the benefits of decreasing ρ come with cost, translated by the fact that the input exceeds the maximum allowed limits. Fig. 6a, 6b and 6c show the input signals evolution during the control. From the figures and table, we can see that when ρ = 1, the input signal ranges are much larger than the other two comparisons, and the maximum values are also always higher. In this paper, to apply optimal control, French macroeconomic quarterly data from 1980T1 to 2018T4 are used. It consists of 6 variables: GDP (Gross Domestic Production), EXP (Exportation), IMP (Importation), HC (Household Consumption), GFCF (Gross Fixed Capital Formation) and PE (public expenditure). Three canonical input-output models are estimated, which first difference of natural logarithm of GDP, EXP and IMP are endogenous variables (outputs), and first difference of natural logarithm of HC, GFCF and PE are exogenous variables (inputs). We first estimate the order of each equation, then the parameters. Once the residuals of models pass the whiteness test, we transfer the models to a state-space model.

After estimating the model, an optimal control solution: (Linear-Quadratic regulator) LQR is designed for our problem which is to maintain a constant GDP increasing ratio at a certain level. The algorithm is developed on Simulink. The experiments with or without constraints on input signals are both implemented: 1) The results without constraints on input signals show that a lower parameter ρ in LQR will lead to a faster convergence to state-space; 2) For experiment with constraints on inputs, we impose a limitation on the maximum value that first difference of natural logarithm of HC can reach. The result shows that when one input cannot reach the level it used to reach, other inputs will compensate for this absence. This action will force other input signals to reach an even higher level, and the final result of output is still slightly worse than the experiment without constraint as we expected.

Perturbations on outputs are also studied, and the experiments are also developed with and without constraints on input signals. Results show that our system can quickly recover from the disturbance. And constraints on input signals can delay the recovery.

The control structure designed in this paper has good applicability and extensibility in economics. This work can be further extended with more variables, the constraints are also not limited to only the range of input signals, it can impose the constraints on inputs and outputs subject to a specific model of the dynamics of the macro-economy.

Fig. 1 :

 1 Fig. 1: Original data (Unit: Billion Euro).

Fig. 2 :Fig. 3 :

 23 Fig. 2: Order Selection

Fig. 4 :

 4 Fig. 4: Modified LQR Control problem

(a) u 1 (Fig. 6 :

 16 Fig. 6: Input traces under different ρ values

TABLE I :

 I ADF test for unit root

	Variables	ADF Level	ADF First Difference
	LGDP	-1.10548(1)	-7.66072(0)***
	LEXP	-3.28665(2)	-6.29252(3)***
	LIMP	-2.87331(4)	-6.20372(3)***
	LHC	-1.45805(3)	-5.59492(2)***
	LGFCF	-3.60257(2)	-4.01801(1)***
	LPE	-1.92627(1)	-5.69958(1)***
	Null hypotheses: Variable contains an unit root
	Notes: (1) ***, ** and * denote significance at the 1%, 5%,
	and 10% levels, respectively.
	(2) Figures in parentheses are the number of lags (delays)
		used.

TABLE II :

 II Output behavior. Please note that the state-space error is always 0 due to the type of the controller we consider.

	ρ	time to state-space (unit in quarter)	output variation range
	ρ = 1	18	(0.001, 0.008)
	ρ = 10	32	(-0.001, 0.008)
	ρ = 100	37	(-0.002, 0.008)
	ρ = 1 and constraint on input	20	(0.001, 0.008)
	008, 0.008, 0.008, 0.008, 0.008, 0.0173, 0.0173,
	0.0173, 0.0173, 0.0086, 0.0086, 0.0086, 0.0086] T
	u r (k) = [0.024, 0.003, 0.003] T

https://www.insee.fr/fr/accueil

The reader must note that especially due to the arrival Covid-19 pandemics, data from 2019 were not taken into account in this study.

Actually, the recent 10 years (2010-2019) average GDP growth ratio of France is 1.38%, but if we look back 25 years ago, the highest GDP growth ratio are showing during 1998-2001, which the average ratio is around 3.2%. Therefore, we want to study what measurements should be implemented to sustain this growth ratio.

ACKNOWLEDGMENTS

This work has been funded by the French LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01). This work has been partially supported by MIAI@Grenoble Alpes, (ANR-19-P3IA-0003)

2) Constraints on Input Signals: In our first experiments, we do not impose constraints on input signals, but in reality, there are some levels that input signals cannot reach. Therefore, in this experiment, we set ρ = 1 as well as a maximum limit 0.03 on signal u 1 : u 1 ≤ 0.03 (which is strictly lower than the maximum value of u 1 in the non-constrained case). Input signals results of "ρ = 1 with constraint on u 1 " are showed in Fig. 6a, 6b and 6c and Tab. III. Comparing to only ρ = 1 result, we can see for signal u 1 , the minimum value of the range is still 0.009, but the maximum value of the range becomes 0.03 (which is the limit) instead of 0.038. For u 2 and u 3 , the maximum value increases s they are being used to compensate for the insufficiency of u 1 . The reader should also note that for the sake of readability, all the input behavior is also detailed in Table III.

From Fig. 5a, 6a, 6b and 6c, we can observe that even u 2 and u 3 react to compensate limitation of u 1 , at 1 st and 4 th quarter, new results are slightly lower than the result without constraint. As for the converging speed, from Tab. II, we can also notice that experiment with constraint converges slower than the experiment without constraint. But still better than the results of ρ = 10 or 100.

Moreover, one thing to note is that the curves of "ρ = 1 with perturbation" and "ρ = 1 with perturbation and constraint" are overlapped during the quarter from 0 to 25. And the curves of ρ = 1, 10, 100 and "ρ = 1 with constraint" are overlapped during the quarter from 25 to 50. The signal ranges are showed in Tab. III.

3) Perturbation on Output Signals: In this experiment, we still keep ρ = 1, and we add perturbation on output signals to simulate economic crisis, to see how the system will react. From the reference signals setting in Sec. IV-A, we know that when the system is stable, y 1 , y 2 , y 3 should equal to 0.008, 0.0173 and 0.0086. And u 1 , u 2 , u 3 should equal 0.024, 0.003 and 0.003.

From Fig. 5b, 6a, 6b and 6c, we can see that at 24th quarter, the system has converged to a stable state. Then we add a negative perturbation pulse signal -0.16 on y 1 , y 2 and y 3 at 25th quarter. The "ρ = 1 with perturbation and constraint on u i " curves are the scenario where we not only implement the perturbation, but also implement constraints on all the input signals, for u 1 limited between 0 and 0.03, u 2 , u 3 are limited between 0 and 0.008: u 1 ∈ [0, 0.3], u 2 ≤ [0, 0.008], u 3 ≤ [0, 0.008]. "ρ = 1 with perturbation" experiment implements the perturbation, but has no constraints on input signals.

Fig. 5b shows that after about 20 quarters after 25th quarter, the two systems totally recover from the perturbation, apparently the curve without constraint recovers faster than the other. From Fig. 6a 6b and 6c, we can observe that the amplitude of all inputs for the experiment of "ρ = 1 with perturbation" is higher than "ρ = 1 with perturbation and constraint on u i " (for each input). The input signal ranges shown in Tab. III also confirm that. One interesting point in Fig. 6b and 6c reveal that if we do not impose constraint on input signals, inputs u 2 and u 3 can be negative, recall that u 2 represents the first difference of logarithm of Gross Fixed Capital Formation (also called investment). A negative signal means instead of investing during the crisis, we should sell our assets. As u 3 represents the first difference of logarithm of Public Expenditure, negative means we need to reduce government spending. Nevertheless, all these conclusions, although correct from the engineering point of view, need to be coordinated with expert's advice.


Shifeng Cheng

Michael Melkonian

Stephen A Smith

Samuel Brockington

John M Archibald

Pierre-Marc Delaux

Fay-Wei Li

Barbara Melkonian

Evgeny V Mavrodiev

Wenjing Sun

Yuan Fu

Huanming Yang

Douglas E Soltis

Sean W Graham

Pamela S Soltis

Xin Liu

Xun Xu
email: xuxun@genomics.cn

Ka-Shu Wong

Xun Xu

Ka-Shu Gane
email: gane@ualberta.ca

Wong

0KP: A phylodiverse genome sequencing plan

Keywords: 10KP, plants, samples, genome sequencing, genomics, biodiversity, phylogenomics, open community, MGISEQ

Understanding plant evolution and diversity in a phylogenomic context is an enormous challenge due, in part, to limited availability of genome-scale data across phylodiverse species. The 10KP (10,000 Plants) Genome Sequencing Project will sequence and characterize representative genomes from every major clade of embryophytes, green algae, and protists (excluding fungi) within the next 5 years. By implementing and continuously improving leading-edge sequencing technologies and bioinformatics tools, 10KP will catalogue the genome content of plant and protist diversity and make these data freely available as an enduring foundation for future scientific discoveries and applications. 10KP is structured as an international consortium, open to the global community, including botanical gardens, plant research institutes, universities, and private industry. Our immediate goal is to establish a policy framework for this endeavor, the principles of which are outlined here.

Introduction

Based on the success of the 1KP (1000 Plants) Initiative [START_REF] Kp | Plants) Initiative[END_REF][START_REF] Wickett | Phylotranscriptomic analysis of the origin and early diversification of land plants[END_REF], an international multidisciplinary consortium that sequenced and analyzed transcriptomes from more than 1000 species of green plants representing most of the known diversity within Viridiplantae, we are now aiming to sequence complete genomes from more than 10,000 plants and protists. 10KP will address fundamental questions in plant evolution and diversity, providing data on more than 10,000 species representing every major clade of embryophytes (land plants), green algae (chlorophytes and streptophytes), and protists (photosynthetic and heterotrophic). For embryophytes, we will sequence nonflowering plants (bryophytes, lycophytes, ferns, and gymnosperms) and flowering plants (angiosperms). In addition to green algae, we will also sequence diverse clades of photosynthetic and heterotrophic protists, representing some of the most enigmatic and unexplored eukaryotic microbes. This project was launched at the XIXth International Botanical Congress (2017, Shenzhen, China) and was covered by a Science News release in July 2017 [START_REF] Normile | Plant scientists plan massive effort to sequence 10,000 genomes[END_REF].

Following the Bermuda Principles and the Fort Lauderdale Agreement, this project will make the resulting genomics data freely available. In the spirit of the Toronto Data Release Workshop and Statement [START_REF] Birney | Prepublication data sharing[END_REF] recommendation, which encourages large-scale sequencing projects to produce a citable statement for their data and intentions for downstream analyses and publications, we present this marker paper to outline our overall plans and explain how interested parties can get involved.

The basic goal of 10KP is to build an annotated reference genome for a member of every genus of the Viridiplantae (land plants and green algae), as well as a phylodiverse set of species representing both photosynthetic and heterotrophic protists. These data will provide a wealth of information to address fundamental questions across the plant/eukaryotic tree of life, e.g., enabling studies of phylogeny, origin/acquisition and diversification of specific traits, gene and genome duplication, correlation between genomic and morphological changes, and convergent evolution of important genetic networks. The scope and quantity of data produced by 10KP will allow researchers to develop new techniques that address fundamental questions in evolution and comparative genomics.

We will complete this project over the next 5 years (2018-2023), including sample acquisition, sequencing, genome assembly, analyses, and (initial) publications. Major supporters include BGI-Research, the nonprofit division of BGI-Shenzhen, and China National GeneBank (CNGB), an open nonprofit scientific platform that is managed by BGI-Shenzhen.

Through 10KP, we hope to foster imaginative and highquality research that addresses major questions in plant and protist biology while also, indirectly, demonstrating the value of both preserving and investigating biodiversity. Our effort is meant to complement, not replace, research programs supported through other funding agencies worldwide. We are open to collaborations with all interested research groups.

Species Lists and Phylogenetic Diversity

The number of extant species of Viridiplantae and protists is unknown. An estimate for embryophytes, gleaned from various databases (Table 1), is that there are at least 380,000 known species representing approximately 23,562 genera in 667 families; however, Govaerts [START_REF] Govaerts | How many species of seed plants are there?: A response[END_REF] estimated a higher number. For green algae and photosynthetic protists, approximately 40,000 species have been described, and predictions for species yet to be described range from 25,000 to 100,000 [START_REF] Guiry | How many species of algae are there?[END_REF]. In the case of heterotrophic eukaryotic microbes, estimates are much less clear; approximately 440,000 species have been described, most of which (approximately 400,000) are fungi [START_REF] Adl | Diversity, nomenclature, and taxonomy of protists[END_REF]. The reality is that much of eukaryotic microbial diversity remains unexplored and the number of genera and species is essentially unknown.

Fewer than 300 green plants and protists have had their genomes sequenced and published. The vast majority of the tree of life remains unexplored at the level of complete genomes. 10KP will fill this gap. Specifically, we will sequence genomes from at least 8000 seed-plant genera (Figure 1), at least 1000 nonseed plant genera (Figure 2), at least 1000 green algae, and at least 3000 photosynthetic and heterotrophic protists (Figure 3). The total number of species sequenced will exceed 10,000-the moniker 10KP is essentially a milestone toward a larger goal. We anticipate sequencing a proportionately larger number of eudicot genomes, given that this clade comprises nearly 75% of all angiosperm species (Figure 4). Many of the genomes to be sequenced are cornerstones for addressing important and longstanding questions in biology and evolution, while others represent unexplored potential for medicinal compounds and/or the discovery of high-value natural products. Some representative species from diverse clades are shown (Figures 567).

Sequencing Priority and Data Release

For embryophytes, we are expecting a community effort with sample submission and processing online (Figure 8). The species sampling will be coordinated by Douglas E. Soltis and Pamela S. Soltis (flowering plants) and Sean W. Graham (nonflowering plants). For this part, we will prioritize the sequencing in 2 stages: Stage 1. Create family-level high-quality reference genomes, ideally with chromosome-scale assemblies to facilitate comparative and evolutionary genomics research across the green tree of life. Stage 2. Increase the sample density to the genus level, while recognizing that many genera are likely not monophyletic.

For some of the larger genera, we may sequence 2 or more distantly related species. Note that we will accept samples for genus-level sequencing even during the first phase of the project when we are focused on family-level sequencing, but these samples may be not processed immediately. We will, however, conduct appropriate quality controls before freezing the samples for later sequencing.

For green algae (chlorophytes and streptophytes) and photosynthetic protists, all samples will be channelled through public culture collections, specifically the Culture Collection of Algae at the University of Cologne (http://www. ccac.uni-koeln.de/) managed by Michael Melkonian and Barbara Melkonian, to ensure uniform quality control. For heterotrophic protists, project coordination will be led by researchers at the Centre for Comparative Genomics & Evolutionary Bioinformatics at Dalhousie University, Halifax, Canada (J. M. Archibald).

Downloaded from https://academic.oup.com/gigascience/article/7/3/giy013/4880447 by Institut National Polytechnique de Toulouse user on 06 April 2022 The numbers were combined and integrated from the Open Tree of Life and the Plant List (2013) [START_REF]The Plant List, a working list of all plant species[END_REF], especially for flowering plants, with a particular focus on the "accepted species" for the nonflowering plants due to its classification difficulties. Some numbers are not the same or inconsistent across different databases. The aim of this table is simply to give a rough estimate of the species distribution in the major clades and the 3 largest families within each clade. The MGISEQ platform, a BGI-developed next-generation sequencing technology based on DNA nanoballs and combinatorial probe anchor synthesis, will be combined with evolving methodologies for improving long-range contiguity, including linked-read technologies from 10X genomics or single-tube long fragment reads from Complete Genomics (a division of BGI). HiC libraries may also be used to construct chromosome-level genome reference assemblies for a member of every family or for other critical species selected by the consortium. Generally speaking, sample providers must prepare either (a) a minimum amount of DNA >50 μg and DNA fragments >20 kb or (b) a minimum amount of DNA >5 ng and DNA fragments >50 kb or preferably >100 kb. For (a), we will use the traditional hierarchical shotgun strategy by combining paired-end libraries with a series of mate-pair libraries. For (b), we will use the strategy of MGISEQ + 10X. The details of these platforms and the library/sequencing strategies are summarized in Table 2. ). This shows the phylogenetic relationships for the 5 major categories of seed-free plants, including bryophytes (hornworts, liverworts, mosses), lycophytes, and ferns.

Equisetidae

Annotated genome sequences will be released through the CNGB website (http://db.cngb.org/cnsa) and accompanied by regular submissions of peer-reviewed GigaScience "data release" publications that provide independent quality assessment and give credit/authorships to the appropriate sample providers. Data releases will occur on a regular basis (e.g., quarterly) once the data satisfy one of a series of tiered quality assessments (e.g., gene-sized contigs, sufficient for synteny analysis, chromosome-scale assembly). Additional increases in sample density may be coordinated with the larger Earth Biogenome Project [START_REF] Pennisi | Biologists propose to sequence the DNA of all life on Earth[END_REF], which will likely encompass at least another half decade of effort. A brief workflow for the 10KP is described (Figure 9).

Sample Requirements and Sequencing Technologies

It is essential that proper sample documentation be provided. This includes vouchers for taxonomic verification (embryophytes and eukaryotic microbes are dealt with differently), provenance data (detailing the source, origin, and geo-location of the species for sequencing), and prior informed consent for genome sequencing and data release (from appropriate authorities in compliance with the Nagoya protocol). For green algae and photosynthetic protists, the biological specimen is the strain (with unique numerical identifier) kept in a public repository (culture collection); no strain will be sequenced in 10KP that has not previously been deposited in a public culture collection. For heterotrophic protists, the situation is more difficult as many are difficult to grow and protist collections often lack the resources to keep these difficult strains in a living state. Good communication within the 10KP consortium and with external partners will be critical.

All of the sequencing will be conducted on BGI's low-cost high-accuracy high-throughput MGISEQ platform, in combination with newly developed linked-read technologies (e.g., 10X genomics or BGI's single-tube long fragment reads phasing technology). These approaches require much smaller amounts of DNA (only 2 ng/library) than traditional methods but they also require high-quality high-molecular-weight (>50 kb) extractions. The dominant alternative is the traditional "hierarchical shotgun" strategy with a series of mate-pair largeinsert libraries. However, this approach requires a much larger amount of DNA (e.g., >100 μg), which is prohibitive for many plant species. We note that while our stated goal is to build [START_REF] Simpson | Protist Diversity and Eukaryote Phylogeny[END_REF]. Lineages with 1 or more photosynthetic/plastid-bearing groups are highlighted with a square. The Archaeplastida are the eukaryotic "supergroup" to which green algae and embryophytes belong. Protist genomes sequenced as part of 10KP will come from diverse lineages but exclude true fungi and animals. high-quality reference genomes, transcriptome data are important for genome annotation; if live tissues are available, we will also sequence transcriptomes.

Basal Rosids

Basal Asterids

Gymnosperms

Collaborative Proposals

10KP will also consider collaborative proposals that build on the existing dataset and generate new data using the MGISEQ platform. The aim of these projects should be to go beyond covering the diversity of species and, instead, to address important questions in basic and applied science. Up to 20% of the 10KP sequencing capacity will be devoted to these collaborations. Potential collaborators who wish to launch major subprojects within 10KP should provide a brief (maximum 5 pages) proposal as examplified in the supplementary template (proposal template for subprojects included as supplementary file). For stage 1, we anticipate more active involvement of highly motivated and skilled laboratories, whereas for stage 2, we anticipate more of a community effort to supply the majority of the remaining genera. For algae and protists, all samples will come from the public collections and channelled through the Culture Collection of Algae at the University of Cologne, where quality controls will be enforced.

Figure 1 :

 1 Figure 1: A phylogeny of seed plants (dated phylogeny based on Smith and Brown, in press). Colors correspond to the number of species in the subtending lineage (red = lower to blue = higher). Some larger clades are highlighted around the phylogeny along with the estimated number of genomes to be sequenced by 10KP in bold below the name. Smaller numbers and arrows inside the phylogeny indicate estimates of some of the already available genomes within the identified clade.

Figure 2 :

 2 Figure 2: Summary tree of nonseed plants (based on the 1KP Capstone Analysis, in press). This shows the phylogenetic relationships for the 5 major categories of seed-free plants, including bryophytes (hornworts, liverworts, mosses), lycophytes, and ferns.

Figure 3 :

 3 Figure3: Summary tree of eukaryotes. Schematic diagram shows the known or predicted relationships among the major eukaryotic groups, based on multi-gene analyses, featuring diverse eukaryotic microbes (algae and protists)[START_REF] Simpson | Protist Diversity and Eukaryote Phylogeny[END_REF]. Lineages with 1 or more photosynthetic/plastid-bearing groups are highlighted with a square. The Archaeplastida are the eukaryotic "supergroup" to which green algae and embryophytes belong. Protist genomes sequenced as part of 10KP will come from diverse lineages but exclude true fungi and animals.

Figure 4 :

 4 Figure 4: Distribution of species/genus/family abundance across the major clades of embryophytes. Most species belong to the eudicot clade, for which the largest families include Asteraceae, Orchidaceae, Fabaceae, Rubiaceae, and Poaceae.

Figure 5 :

 5 Figure 5: Representative images of species from different clades/families of flowering plants. The species names presented here are: a. Canella winterana (L.) Gaertn. (Angiosperms, Magnoliids, Canellales, Canellaceae). Flowers and inflorescence. Photo credit: Walter Judd. b. Austrobaileya scandens C.T. White (Angiosperms, Austrobaileyales, Austrobaileyaceae). Flower. Photo credit: Walter Judd. c. Ceratophyllum demersum L. (Angiosperms, Ceratophyllales, Ceratophyllaceae). Habit and inflorescence. Photo credit adapted from Christian Fischer, CC BY SA 3.0 Wikimedia Commons. d. Illicium floridanum J.Ellis (Angiosperms, Austrobaileyales, Schisandraceae). Flower. Photo credit: Walter Judd. e. Piper neesianum C. DC. (Angiosperms, Magnoliids, Piperales, Piperaceae). Habit and inflorescence. Photo credit: Walter Judd. f. Myrothamnus flabellifolius Welw. (Angiosperms, Eudicots, Gunnerales, Myrothamnaceae). Habit and leaves. Photo credit adapted from JMK, CC BY SA 3.0 Wikimedia Commons. g. Dillenia indica L. (Angiosperms, Eudicots, Dilleniales, Dilleniaceae). Photo credit: Walter Judd.

Figure 6 :

 6 Figure 6: Representative images of species from various clades/families of nonflowering plants. The species names presented here are: a. Picea abies (L.) H. Karst. (Pinophyta, Pinales, Pinaceae). Shoots and female cones. Photo credit adapted from Magnus Manske (CC BY SA 3.0 Wikimedia Commons). b. Ginkgo biloba L. (Ginkgophyta, Ginkgoales, Ginkgoaceae). Leaves and male inflorescence. Photo credit adapted from Sten, CC-BY SA 3.0 Wikimedia Commons. c. Cibotium barometz (L.) J.Sm. (Polypodiopsida, Cyatheales, Cibotiaceae). Fronds (leaves). Photo credit: Pi-Fong Lu. d. Adiantum caudatum Klotzsch (Polypodiopsida, Polypodiales, Pteridaceae). Fronds (leaves) and habit. Photo credit: Pi-Fong Lu. e. Marsilea crenata C.Presl (Polypodiopsida, Salviniales, Marsileaceae). Fronds (leaves) and habit. Photo credit: Pi-Fong Lu. f. Asplenium viride Huds. (Polypodiopsida, Polypodiales, Aspleniaceae). Fronds (leaves) and habit. Photo credit: Pi-Fong Lu. g. Diphasiastrum complanatum (L.) Holub. (Lycopodiopsida, Lycopodiales, Lycopodiaceae. Habit. Photo credit: Pi-Fong Lu. h. Bryum capilare Hedwig (Bryopsida, Bryales, Bryaceae). Gametophyte and Sporophyte. Photo Credit adapted from Lairich Rig (CC BY SA 2.0 Wikimedia Commons). i. Marchantia polymorpha L. (Marchantiopsida, Marchantiales, Marchantiaceae). Thalli with gemmae (asexual reproductive structures). Photo Credit adapted from Holger Casselmann (CC-BY SA 3.0 Wikimedia Commons).

Figure 7 :

 7 Figure 7: Light micrographs of diverse protists, including members of different eukaryotic "supergroups." a. Stylonema (Archaeplastida [Plantae], red alga). b. Cyanoptyche (Archaeplastida [Plantae], glaucophyte). c. Scherffelia (Archaeplastida [Plantae], Viridiplantae, chlorophyte). d. Stephanosphaera (Archaeplastida [Plantae], Viridiplantae, chlorophyte). e. Chaetosphaeridium (Archaeplastida [Plantae], Viridiplantae, streptophyte). f. Mallomonas (stramenopiles, chrysophyte). g. Coscinodiscus (stramenopiles, diatom). h. Synedra (stramenopiles, diatom). i. Sphacelaria (stramenopiles, brown alga). j. Trithigmostoma (alveolates, ciliate). k. Cryptomonas (Cryptista). l. Paramoeba (Amoebozoa). Micrographs courtesy of Gerd G ünther (http://www.mikroskopia.de/index.html), Sebastian Hess (Halifax; Scherffelia), and Ivan Fiala (Czech Republic; Paramoeba).

Figure 8 :

 8 Figure 8: Sample submission portal from the CNGB/10KP website. Figure shows the sample submission portal (as well as the underlying database management) on the 10KP website https://db.cngb.org/10kp/; this website (version 1.0) is still evovling. This sample submission portal is prepared specifically for land plants; most samples will come from botanical gardens or botany research centers/laboratories worldwide. A global community effort is crucial to help supply all families and all genera. For stage 1, we anticipate more active involvement of highly motivated and skilled laboratories, whereas for stage 2, we anticipate more of a community effort to supply the majority of the remaining genera. For algae and protists, all samples will come from the public collections and channelled through the Culture Collection of Algae at the University of Cologne, where quality controls will be enforced.

Figure 9 :

 9 Figure 9: An overview of the 10KP strategy.

Table 1 :

 1 Statistics of the described species of embryophytes (land plants) distributed among major clades

	Clade	Family	Genus	Species		Three largest families
	Hornworts	5	11	130	Anthocerotaceae (161)	Dendrocerotaceae (25)	Notothyladaceae (16)
	Liverworts	87	387	7356	Lejeuneaceae (2270)	Jungermanniaceae(725)	Lepidoziaceae (580)
	Mosses	111	874	13,000	Pottiaceae (3223)	Hypnaceae (2520)	Bryaceae (2108)
	Lycophytes	3	18	1338	Lycopodiaceae (475)	Selaginellaceae (404)	Isoetaceae (51)
	Ferns	48	319	10,578	Dryopteridaceae (1871)	Polypodiaceae (1601)	Pteridaceae (1226)
	Gymnosperms	12	88	1104	Pinaceae (255)	Zamiaceae (216)	Cupressaceae (166)
	ANA grade	6	20	193	Nymphaeaceae (88)	Schisandraceae (84)	Hydatellaceae (12)
	Monocots	78	3505	76,119	Orchidaceae (28,576)	Poaceae (12,397)	Cyperaceae (6311)
	Magnoliids	20	450	9528	Lauraceae (3106)	Piperaceae (2770)	Annonaceae (2174)
	Asterids	143	9763	135,213	Asteraceae (38,700)	Rubiaceae (14,412)	Lamiaceae (8671)
	Rosids	154	6582	101,245	Fabaceae (26,245)	Euphorbiaceae (6904)	Rosaceae (6626)
	Basal eudicots	17	464	7535	Ranunculaceae (3119)	Proteaceae (1492)	Papaveraceae (1062)
	Others	NA	NA	1579
	Embryophyta	667	23,562	381,425

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Table 2 :

 2 Technologies/platforms for DNA library construction and sequencing

					Minimum	Minimum
				Quality requirement	requirement for	requirement for
	Platform/strategy	Insert size	Assemblers	for tissue samples	DNA quantity	DNA fragments	Assembly results
	MGISEQ (hierarchical	170/350 bp SOAP de novo	Net weight of fresh	2 μg	>20 k	construct contigs
	shotgun)			plant tissue ≥4 g
		2-5 K			20 μg		construct scaffolds
		10 k			30 μg
		20 k			50 μg
		40 k			60 μg
	MGISEQ + 10X		Supernova2	50-75 mg dried or	2 ng	>50/100 k	construct scaffolds
				100-200 mg fresh leaf
				tissue
	MGISEQ + long		To be developed	50-75 mg dried or	1 ng	>50/100 k	construct contigs
	fragment reads			100-200 mg fresh leaf
				tissue
	MGISEQ + 10X/long			500 mg, living		>50/100 k	chromosome-scales
	fragment reads + Hi-C			tissues/cells

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Acknowledgements

We acknowledge many individuals who provided support in various aspects. Min Liu, Fengzhen Chen, Lei Chen, Xiaofeng Wei, and Sanjie Jiang from BGI contributed to the construction of the 10KP website. This project was supported by the Shenzhen Municipal Government of China (JCYJ20150529150409546, JCYJ20150529150505656).

Additional file

Supplementary file, proposal template.docx.

Abbreviations

10KP: 10,000 Plants Genome Sequencing Project; CNGB: China National Gene Bank.

Competing interests

S.C., W.S., Y.F., H.Y., X.L., and X.X. are employees of BGI Shenzhen. The authors otherwise declare that they have no competing interests.