An Underactuated Drift-Free Left Invariant Control System on a Specific Lie Group

Abstract

The paper presents a geometrical overview on an optimal control problem on a special Lie group. The Hamilton-Poisson realization of the dynamics offers us the possibility to study the system from mechanical geometry point of view.

1. Introduction

Recent work in nonlinear control has drawn attention to drift-free systems with fewer degrees than state variables. These arise naturally in problems of motion planning for wheeled robots subject to nonholonomic controls [1], models of kinematic drift effects in space subjects to appendage vibrations or articulations [1], the molecular dynamics [2], the autonomous underwater vehicle dynamics [3], the car's dynamics [4], and spacecraft dynamics [5]. The purpose of our paper is to study a class of left-invariant, drift-free optimal control problems on a specific Lie group 𝐺. The class of all control-affine left-invariant, drift-free optimal control problems on 𝐺 can be reduced to a class of two typical controllable left-invariant control systems on 𝐺. The left-invariant, drift-free optimal control problems involve finding a trajectory-control pair on 𝐺, which minimizes a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle 𝑇^∗𝐺 using the optimal Hamiltonian on 𝒢^∗, where the maximum principle yields the optimal control. The energy-Casimir method is used to give sufficient conditions for nonlinear stability of the equilibrium states. Around this equilibrium states, we are able to find the periodical orbits using Moser's theorem. In the last paragraph, we have studied the numerical integration via three methods: Lie-Trotter algorithm, Kahan's algorithms, and Runge-Kutta 4th method. Numerical simulations and a comparison between these three methods are presented too.

2. The Geometrical Picture of the Problem

Let 𝐺 be the Lie group given by𝐺=1𝑥²𝑥₄0𝑒^−𝑥1𝑥₃001∈ℳ³(ℝ)∣𝑥₁,𝑥₂,𝑥₃,𝑥₄∈ℝ.(2.1)

Proposition 2.1. The Lie algebra 𝒢 of 𝐺 is generated by 𝐴𝐴₁=000−001000,𝐴₂=000100000₃,=000000010,𝐴₄=000000100,(2.2) and the Lie algebra structure of 𝒢 is given by the following [⋅,⋅]𝐴₁𝐴₂𝐴₃𝐴₄𝐴₁0𝐴₂−𝐴₃0𝐴₂−𝐴₂0𝐴₄0𝐴₃𝐴₃−𝐴₄00𝐴₄0000.(2.3)

Proposition 2.2. The minus-Lie-Poisson structure on 𝒢^∗≃(𝑅⁴)^∗≃𝑅⁴ is generated by the matrix Π₋=⎡⎢⎢⎣0−𝑥²𝑥₂𝑥₃0−𝑥₃0−𝑥₄00𝑥0₄0000⎤⎥⎥⎦.(2.4)
An easy computation leads one to the following.

Proposition 2.3. The following two systems are drift-free-left invariant controllable systems on 𝐺, namely: 𝑋̇=𝑋𝑋̇=𝐴𝑋₁𝐴₁𝑢₁+𝐴₂𝑢₂+𝐴₃𝑢₃,(2.5)𝑢₁+𝐴₂𝑢₂+𝐴₃𝑢₃+𝐴₄𝑢₄,(2.6) where 𝑋∈𝐺,𝐴_𝑖 are the matrix defined above, and 𝑢_𝑖∈𝐶^∞(ℝ,ℝ),𝑖=1,4.

Proof. Since the span of the set of Lie brackets generated by 𝐴₁,𝐴₂, and 𝐴₃ coincides with 𝒢 the proposition is a consequence of a result due to Jurdjevic and Sussmann [6].

3. An Optimal Control Problem for the System (2.5)

Let 𝐽 be the cost function given by𝐽𝑢₁,𝑢₂,𝑢₃=12^𝑡𝑓₀𝑐₁𝑢²¹(𝑡)+𝑐₂𝑢²²(𝑡)+𝑐₃𝑢²³(𝑡)𝑑𝑡𝑐₁>0,𝑐₂>0,𝑐₃>0.(3.1)

Then we have the following.

Proposition 3.1. The controls that minimize 𝐽 and steer the system (2.5) from 𝑋=𝑋₀ at 𝑡=0 to 𝑋=𝑋_𝑓 at 𝑡=𝑡_𝑓 are given by 𝑢₁=1𝑐₁𝑥₁,𝑢₂=1𝑐₂𝑥₂,𝑢₃=1𝑐₃𝑥₃,(3.2) where 𝑥_𝑖’s are solutions of 𝑥̇₁=−1𝑐₂𝑥²²+1𝑐₃𝑥̇₂𝑥²³,=1𝑐₁𝑥₁𝑥₂−1𝑐₃𝑥̇₃𝑥₃𝑥₄,=−1𝑐₁𝑥₁𝑥₃+1𝑐₂𝑥̇₄𝑥₂𝑥₄,=0.(3.3)

Proof. Let us consider the optimal Hamiltonian given by 𝐻𝑥₁,𝑥₂,𝑥₂,𝑥₄=12𝑥𝑐²¹₁+𝑥²²𝑐₂+𝑥²³𝑐₃.(3.4) It is in fact the controlled Hamiltonian 𝐻_opt given by 𝐻_opt=𝑥₁𝑢₁+𝑥₂𝑢₂+𝑥₃𝑢₃−12𝑐¹𝑢²¹+𝑐₂𝑢²²+𝑐₃𝑢²³,(3.5) which is reduced to 𝒢^∗ via Poisson reduction. Then the optimal controls are given by 𝑢₁=1𝑐₁𝑥₁,𝑢₂=1𝑐₂𝑥₂,𝑢₁=1𝑐₃𝑥₃,(3.6) where 𝑥_𝑖’s are solutions of the reduced Hamilton's equations given by 𝑥¹𝑥₂𝑥₃𝑥₄^𝑡=Π₋⋅∇𝐻,(3.7) which is nothing else than the required (3.3).

Remark 3.2. It is easy to see from (3.3) that 𝑥₄=constant, and so the dynamics (3.3) can be put in the equivalent form 𝑥̇₁=−1𝑐₂𝑥²²+1𝑐₃𝑥̇₂𝑥²³,=1𝑐₁𝑥₁𝑥₂−𝑘𝑐₃𝑥̇₃𝑥₃,=−1𝑐₁𝑥₁𝑥₃+𝑘𝑐₂𝑥₂.(3.8)

The goal of our paper is to study some geometrical and dynamical properties for the system (3.8).

Proposition 3.3. The dynamics (3.8) has the following Hamilton-Poisson realization: ℝ³,Π,𝐻,(3.9) where Π=0−𝑥²𝑥₂−𝑥𝑥₂₃0−𝑘𝑘0(3.10) and the Hamiltonian 𝐻𝑥₁,𝑥₂,𝑥₃=12𝑥𝑐²¹₁+𝑥²²𝑐₂+𝑥²³𝑐₃.(3.11)

Proof. Indeed, it is not hard to see that the dynamics (3.8) can be put in the equivalent form 𝑥̇₁,𝑥̇₂,𝑥̇₃^𝑡=Π⋅∇𝐻,(3.12) as required. Moreover, the function 𝐶 given by 𝐶=𝑘𝑥₁+𝑥₂𝑥₃(3.13) is a Casimir of our configuration. Indeed, (∇𝐶)^𝑡Π=0,(3.14) as desired.

Remark 3.4. The phase curves of the dynamics (3.8) are intersections of 𝑥²¹𝑐₁+𝑥²²𝑐₂+𝑥²³𝑐₃=const.,(3.15) with 𝑘𝑥₁+𝑥₂𝑥₃=const.,(3.16) see Figure 1.

Figure 1 

The phase curves of the system (3.8) (𝑐1=1, 𝑐2=2, 𝑐3=3, and 𝑘=10).

Proposition 3.5. The dynamics (3.8) has an infinite number of Hamilton-Poisson realizations.

Proof. An easy computation shows us that the triples ℝ³,{⋅,⋅}_𝑎𝑏,𝐻_𝑐𝑑,(3.17) where 𝐶{𝑓,𝑔}_𝑎𝑏=−∇𝐶_𝑎𝑏⋅(∇𝑓×∇𝑔),(∀)𝑓,𝑔∈𝐶^∞ℝ³,ℝ_𝑎𝑏,=𝑎𝐶+𝑏𝐻,𝐻_𝑐𝑑=𝑐𝐶+𝑑𝐻,𝑎,𝑏,𝑐,𝑑∈ℝ,𝑎𝑑−𝑏𝑐=1,(3.18) define Hamilton-Poisson realizations of the dynamics (3.8), as required.

Remark 3.6. The above proposition tells us in fact that (3.8) is unchanged, so the trajectories of motion in ℝ³ remain the same when 𝐻 and 𝐶 are replaced by linear combinations of 𝐻 and 𝐶 with coefficients which form a real matrix with det one.

4. Stability and Periodical Orbits

It is not hard to see that the equilibrium states of our dynamics (3.8) are𝑒𝑒^𝑀1=(𝑀,0,0),𝑀∈ℝ,_𝑀2=−𝑘𝑐¹√𝑐₂𝑐₃,−𝑐₂𝑐₃𝑒𝑀,𝑀,𝑀∈ℝ,_𝑀3=𝑘𝑐¹√𝑐₂𝑐₃,𝑐₂𝑐₃𝑀,𝑀,𝑀∈ℝ.(4.1)

Let 𝐴 be the matrix of the linear part of the system (3.8), that is,A=⎛⎜⎜⎜⎜⎜⎜⎜⎝0−2𝑐₂𝑥₂2𝑐₃𝑥₃1𝑐₁𝑥₂1𝑐₁𝑥₁−𝑘𝑐₃−1𝑐₁𝑥₃𝑘𝑐₂−1𝑐₁𝑥₁⎞⎟⎟⎟⎟⎟⎟⎟⎠,(4.2) then the characteristic roots of 𝐴(𝑒^𝑀1), respectively, 𝐴(𝑒^𝑀2), respectively, and 𝐴(𝑒^𝑀3) are given by𝜆₁=0,𝜆_2,3=±𝑐₂𝑐₃𝑀²−𝑐₁𝑘²𝑐₁√𝑐₂𝑐₃,(4.3) respectively,𝜆₁=0,𝜆_2,3=±2𝑖√𝑐₁𝑐₃𝑀,(4.4) respectively,𝜆₁=0,𝜆_2,3=±2𝑖√𝑐₁𝑐₃𝑀,(4.5) so we can conclude with the following.

Proposition 4.1. The equilibrium states 𝑒^𝑀1,𝑀∈ℝ, are spectrally stable if 𝑀∈(−(𝑐₁𝑘/√𝑐₂𝑐₃),(𝑐₁𝑘/√𝑐₂𝑐₃)).

Proposition 4.2. The equilibrium states 𝑒^𝑀2 and 𝑒^𝑀3,𝑀∈ℝ, are spectrally stable for any 𝑀∈𝑅.

We can now pass to discuss the nonlinear stability of the equilibrium states 𝑒^𝑀1,𝑒^𝑀2, and 𝑒^𝑀3, 𝑀∈ℝ.

Proposition 4.3. The equilibrium states 𝑒^𝑀1,𝑀∈ℝ^∗, are nonlinearly stable if 𝑀∈(−(𝑐₁𝑘/√𝑐₂𝑐₃),(𝑐₁𝑘/√𝑐₂𝑐₃)).

Proof. We will make the proof using energy-Casimir method (see [7]). Let 𝐻_𝜑=𝐻+𝜑(𝐶)=𝑥²¹2𝑐₁+𝑥²²2𝑐₂+𝑥²³2𝑐₃+𝜑𝑘𝑥₁+𝑥₂𝑥₃(4.6) be the energy-Casimir function, where 𝜑∶𝑅→𝑅 is a smooth real-valued function defined on 𝑅.
Now, the first variation of 𝐻_𝜑 is given by 𝛿𝐻_𝜑=𝑥¹𝑐₁𝛿𝑥₁+𝑥²𝑐₂𝛿𝑥₂+𝑥³𝑐₃𝛿𝑥₃+𝜑̇𝑘𝑥₁+𝑥₂𝑥₃⋅𝑘𝛿𝑥₁+𝑥₂𝛿𝑥₃+𝑥₃𝛿𝑥₂,(4.7) where 𝜑̇=𝜕𝜑𝜕𝑘𝑥₁+𝑥₂𝑥₃.(4.8) This equals zero at the equilibrium of interest if and only if 𝜑̇(𝑘𝑀)=−𝑀𝑘𝑐₁.(4.9) The second variation of 𝐻_𝜑 is given by 𝛿²𝐻_𝜑=1𝑐₁𝛿𝑥₁²+1𝑐₂𝛿𝑥₂²+1𝑐₃𝛿𝑥₃²+𝜑̈⋅𝑘𝛿𝑥₁+𝑥₂𝛿𝑥₃+𝑥₃𝛿𝑥₂²+2𝜑̇⋅𝛿𝑥₂𝛿𝑥₃.(4.10) Since 𝑀∈(−(𝑐₁𝑘/√𝑐₂𝑐₃),(𝑐₁𝑘/√𝑐₂𝑐₃)) and if we choose the function 𝜑 such that 𝜑̇(𝑘𝑀)=−𝑀𝑘𝑐₁𝜑̈(𝑘𝑀)>0,,(4.11) we can conclude that the second variation of 𝐻_𝜑 at the equilibrium of interest is positive definite and, thus, 𝑒^𝑀1 are nonlinearly stable.

Similar arguments lead us to the following result.

Proposition 4.4. The equilibrium states 𝑒^𝑀2,𝑀∈ℝ^∗ and 𝑒^𝑀3,𝑀∈ℝ^∗ are nonlinearly stable for any 𝑀∈ℝ^∗.

In order to find the periodical orbits around the equilibrium states 𝑒^𝑀1, we make use of the property that the dynamics described by a Hamilton-Poisson system takes place on the symplectic leaves of the Poisson configuration manifold, to prove the existence of periodic orbits by looking for periodic orbits of the symplectic Hamiltonian completely integrable system obtained by the restriction of the Lorenz system to the regular coadjoint orbits of 𝐺^∗. This procedure will be implemented around nonlinearly stable equilibrium states. The procedure is the following: we consider the system restricted to a regular coadjoint orbit of 𝐺^∗ that contains a nonlinearly stable equilibrium, and then we will get the existence of periodic solutions for the restricted system. These periodic solutions are periodic solutions also for the unrestricted system.

Proposition 4.5. Near 𝑒^𝑀1=(𝑀,0,0), 𝑀∈(−(𝑐₁𝑘/√𝑐₂𝑐₃),(𝑐₁𝑘/√𝑐₂𝑐₃)), the reduced dynamics has, for each sufficiently small value of the reduced energy, at least 1 periodic solution whose period is close to 2𝜋𝑐₁√𝑐₂𝑐₃𝑘²𝑐²¹−𝑐₂𝑐₃𝑀².(4.12)

Proof. Indeed, we have successively the following:(i) the restriction of our dynamics (3.8) to the coadjoint orbit 𝑘𝑥₁+𝑥₂𝑥₃=𝑘𝑀(4.13) gives rise to a classical Hamiltonian system,(ii) the matrix of the linear part of the reduced dynamics has purely imaginary roots. More exactly 𝜆_2,3=±𝑖𝑘²𝑐²¹−𝑐₂𝑐₃𝑀²𝑐₁√𝑐₂𝑐₃,(4.14)(iii)consider the following: span∇𝐶𝑒^𝑀1=𝑉₀,(4.15) where 𝑉₀=ker𝐴𝑒^𝑀1=span100,(4.16)(iv)the reduced Hamiltonian has a local minimum at the equilibrium state 𝑒^𝑀1 (see the proof of Proposition 4.3).
Then our assertion follows via the Moser-Weinstein theorem with zero eigenvalue; see [8] for details.

Remark 4.6. The existence of the periodical orbits around the equilibrium points 𝑒₂ and 𝑒₃ remains an open problem, the Moser-Weinstein theorem with zero eigenvalue being inconclusive.

5. Lax Formulation and Numerical Integration of the Dynamics (3.8)

Proposition 5.1. The dynamics (3.8) allows a formulation in terms of Lax pairs.

Proof. Let us take the following: 𝐿=0𝑙¹²𝑙₁₃−𝑙₁₂0𝑙₂₃𝑙₁₂−𝑙₁₃−𝑙₂₃0,=2√𝑐₃𝑥₂𝑐₂√𝑐₁+2𝑥³√𝑐₁𝑐₂+√2√𝑐₂−2√2−𝑐₁𝑐₂𝑐₂√𝑐₁𝑘−8√𝑐₃𝑐₂√𝑐₁𝑙₁₃,=−2√𝑐₃𝑥₁√𝑐₁−2𝑐₃2−𝑐₁𝑐₂𝑐₂√𝑐₁𝑥₂−22−𝑐₁𝑐₂𝑐₁𝑐₂𝑥₃+𝑐²√2−𝑐₁𝑐₂−4𝑐₂√𝑐₁𝑐₂𝑘−42𝑐₃2−𝑐₁𝑐₂𝑐₂√𝑐₁𝑙₂₃,=−2𝑐₃2−𝑐₁𝑐₂𝑐₁√𝑐₂𝑥₁+2𝑐₃𝑐₂𝑥₂+√2𝑥₃+42𝑐₃𝑐₂𝐵=0𝑏₁₂−𝑘,𝑏₁₃−𝑏₁₂0𝑏₂₃𝑏₁₂−𝑏₁₃−𝑏₂₃0,=𝑐²2√2𝑥₃+𝑘−2√2−𝑐₁𝑐₂𝑘−4√2𝑐₂𝑐₃2𝑐₂√𝑐₁𝑐₃𝑏₁₃,=−2√2𝑐₂√𝑐₂𝑐₃𝑥₁+2𝑘+𝑐₂2−𝑐₁𝑐₂−4√𝑐32+𝑥₂+2√𝑐₂𝑘4𝑐₂√𝑐₁𝑐₃𝑏₂₃,=−√2−𝑐₁𝑐₂𝑐₁√2𝑥¹+𝑥₂−√𝑐₂2√2𝑐₃𝑘+2,(5.1) then, using MATHEMATICA 7.0, we can put the system (3.8) in the equivalent form 𝐿̇=[𝐿,𝐵],(5.2) as desired.

We will discuss now the numerical integration of the dynamics (3.8) via the Kahan integrator, Lie-Trotter integrator [9], and also via Runge-Kutta 4th steps integrator, and we will point out some properties of Kahan and Lie-Trotter integrators.

It is easy to see that for the equations (3.8), Kahan's integrator can be written in the following form: 𝑥^1𝑛+1−𝑥^𝑛1=−ℎ𝑐₂𝑥^2𝑛+1𝑥^𝑛2+ℎ𝑐₃𝑥_2𝑛+1𝑥^3𝑛+1𝑥^𝑛3,−𝑥^𝑛2=ℎ2𝑐₁𝑥^1𝑛+1𝑥^𝑛2+𝑥^2𝑛+1𝑥^𝑛1−ℎ𝑘2𝑐₃𝑥^𝑛3𝑥_3𝑛+1+𝑥^3𝑛+1,−𝑥^𝑛3=−ℎ2𝑐₁𝑥^1𝑛+1𝑥^𝑛3+𝑥^3𝑛+1𝑥^𝑛1+ℎ𝑘2𝑐₂𝑥^𝑛2+𝑥^2𝑛+1.(5.3)

Using MATHEMATICA 8.0, we can prove the following proposition which shows the incompatibility of the Kahan's integrator with the Poisson structure of the system (3.8).

Proposition 5.2. Kahan's integrator (5.3) has the following properties:(i)it is not Poisson preserving;(ii)it does not preserve the Casimir 𝐶 of the Poisson configuration (ℝ³,Π);(iii)it does not preserve the Hamiltonian 𝐻 of the system (3.8).

We will discuss now the numerical integration of the dynamics (3.8) via the Lie-Trotter integrator.

To begin with, let us observe that the Hamiltonian vector field 𝑋_𝐻 splits as follows:𝑋_𝐻=𝑋_𝐻1+𝑋_𝐻2+𝑋_𝐻3,(5.4) where𝐻₁=12𝑐₁𝑥²¹,𝐻₂=12𝑐₃𝑥²²,𝐻₃=12𝑐₃𝑥²³.(5.5) Their corresponding integral curves are, respectively, given by𝑥¹𝑥(𝑡)₂𝑥(𝑡)₃(𝑡)=𝐴^𝑖𝑥¹𝑥(0)₂𝑥(0)₃(0),𝑖=1,3,(5.6) where𝐴₁=010𝑒00^{(𝑎/𝑐1)𝑡}𝑒00_{−(𝑎/𝑐1)𝑡}𝑎=𝑥₁,𝐴(0),₂=⎡⎢⎢⎢⎢⎣1−𝑏𝑐₂00𝑘1𝑡00𝑐₂𝑏=𝑥𝑡₂1⎤⎥⎥⎥⎥⎦,𝐴(0),₃=⎡⎢⎢⎢⎣10𝑐𝑐₃01−𝑘𝑡𝑐₃0𝑐0=𝑥₃1𝑡⎤⎥⎥⎥⎦,(0).(5.7) Then the Lie-Trotter integrator is given by𝑥^1𝑛+1=𝑥^𝑛1−𝑏𝑐₂𝑡𝑥^𝑛2+𝑐𝑐₃𝑡+𝑘𝑏𝑐₂𝑐₃𝑡²𝑥^𝑛3𝑥_2𝑛+1,=𝑒^{(𝑎/𝑐1)𝑡}𝑥^𝑛2−𝑘𝑐₃𝑥_3𝑛+1𝑡𝑒^{(𝑎/𝑐1)𝑡}𝑥^𝑛3,=𝑘𝑐₂𝑡𝑒^{(−𝑎/𝑐1)𝑡}𝑥^𝑛2−𝑘²𝑐₂𝑐₃𝑡²𝑒^{(−𝑎/𝑐1)𝑡}𝑥^𝑛3.(5.8)

Now, a direct computation or using MATHEMATICA leads us to the following.

Proposition 5.3. The Lie-Trotter integrator (5.8) has the following properties:(i) it preserves the Poisson structure Π;(ii)it preserves the Casimir 𝐶 of the Poisson configuration (ℝ³,Π);(iii)it does not preserve the Hamiltonian 𝐻 of the system (3.8);(iv)its restriction to the coadjoint orbit (𝒪_𝑘,𝜔_𝑘), where 𝒪_𝑘=𝑥₁,𝑥₂,𝑥₃∈ℝ³∣𝑘𝑥₁+𝑥₂𝑥₃=const.(5.9)and 𝜔_𝑘 is the Kirillov-Kostant-Souriau symplectic structure on 𝒪_𝑘, gives rise to a symplectic integrator.

Remark 5.4. If we compare these methods, see Figures 2, 3, and 4, with the 4th-step Runge-Kutta method, we can see that Lie-Trotter integrator and Kahan's integrator give us a weak approximation of our dynamics. However, Kahan's integrator and the Lie-Trotter integrator have the advantage of being more easily implemented.

Figure 2 

The 4th-step Runge-Kutta integrator of the system (3.8) (𝑐1=1, 𝑐2=2, 𝑐3=3, 𝑘=10, and 𝑥1(0)=𝑥2(0)=𝑥3(0)=1).

Figure 3 

The Kahan's integrator of the system (3.8) (𝑐1=1, 𝑐2=2, 𝑐3=3, 𝑘=10, and ℎ=1, 𝑥1(0)=𝑥2(0)=𝑥3(0)=1).

Figure 4 

The Lie-Trotter integrator of the system (3.8) (𝑐1=1, 𝑐2=2, 𝑐3=3, 𝑘=10, and 𝑥1(0)=𝑥2(0)=𝑥3(0)=1).

6. Conclusion

The paper analyses a drift-free left invariant controllable system on a special Lie group. The Hamilton-Poisson realization of the system allows us to study the system from the mechanical geometry point of view. This means that we can use specific tools as energy-Casimir method for nonlinear stability, the Moser's theorem to find the periodical orbits, and Poisson integrators to make the numerical integration of the dynamics. In addition, we use non-Poisson integrators (Kahan's integrator and Runge-Kutta 4th-step integrator) to make a comparison between the obtained results. Numerical simulations via MATHEMATICA 8.0 are presented too. Similar problems have been studied on the Lie groups 𝑆𝑂(3),𝑆𝑂(4) (see [10]), on the Heisenberg Lie groups 𝐻(3) and 𝐻(4), or 𝑆𝐸(2,ℝ)×𝑆𝑂(3) (see [11]).

Acknowledgments

This paper was supported by the project “Development and support of multidisciplinary postdoctoral programmes in major technical areas of national strategy for Research-Development-Innovation” 4D-POSTDOC, Contract no. POSDRU/89/1.5/S/52603, and the project was cofunded by the European Social Fund through Sectorial Operational Programme Human Resources Development 2007–2013.