Leyla Arioua
email: leyla.arioua@yahoo.fr

Bogdan Marinescu
email: bogdan.marinescu@irccyn.ec-nantes.fr

Multivariable Control With Grid Objectives of an HVDC Link Embedded in a Large-Scale AC Grid

The HVDC links are increasingly used not only to interconnect asynchronous AC systems but are also embedded into a same meshed AC power system. Thanks to its speed and flexibility, the HVDC technology is able to provide transmission system advantages as transfer capacity enhancement and power flow control. In addition, studies have shown that the way of controlling the HVDC converters impacts the stability of the AC system. This can be particularly exploited to enhance the dynamic power system performances during transients. In this paper a robust multivariable control design for HVDC link converters is proposed. It is based on the coordination of the control actions of the HVDC converters and the use of a control model which takes into account the dynamics that mostly impact stability of the neighbour zone of the HVDC link. This new methodology was used to synthesize the controller for an actual grid 1000 MW HVDC link reinforcement project called "Midi-Provence" in the southern part of the French grid. The synthesis, implementation and validation processes are presented in detail. The new controller is tested in comparison with the standard vector control. A large-scale dynamic model of the whole European power system, currently used and updated by the European TSO's for the interconnection studies has been used with Eurostag simulation software.

Introduction

The HVDC link is a mean of transmission of electric power based on high power electronics. Thanks to its speed and flexibility, the HVDC technology is able to provide the transmission system advantages as transfer capacity enhancement and power flow control [START_REF] Hingorani | Understanding FACTS: concepts and technology of flexible AC transmission systems[END_REF]. Initially, it was used in power systems to interconnect asynchronous AC systems. The ends of the HVDC link are electrically independent one from each other and this avoids the propagation of perturbations between the two AC grids as it is the case of the England-France interconnection [START_REF] Goodrich | The 2000 MW HVDC link between England and France[END_REF]. Nowadays, HVDC links are increasingly embedded into a same meshed AC system in order to enhance the grid's transmission capability and flexibility of the power system. In this context, the HVDC link co-exists with other AC system elements as for instance AC lines and generators. Several projects of insertion of this type of device in a meshed AC system are underway in Europe, as for instance France-Spain HVDC project [3], France-Italy interconnection and Midi-Provence HVDC project [4]. This latter is our case of study. In fact, a 1000 MW HVDC link will be inserted in France between the two areas Fos and Gaudière, in order to enhance capacity of power transmission and the stability of the Midi-Provence region. The need of such HVDC link is due to the fact that only one single 400 kV AC line interconnects the two areas (see Fig. 1). It was shown that the strategy used to control the HVDC converters can impact the stability of the system in which the link is embedded [START_REF] Hauer | Robustness issues in stability control of large electric power systems[END_REF][START_REF] Vovos | Enhancement of the Transient Stability of Integrated AC/DC Systems Using Active and Reactive Power Modulation[END_REF][START_REF] Smed | Utilizing HVDC to damp power oscillations[END_REF][START_REF] Hammad | Improving the dynamic performance of a complex AC/DC system by HVDC control modifications[END_REF][START_REF] Shun | Influence of VSC HVDC on transient stability: Case study of the Belgian grid[END_REF][START_REF] Taylor | HVDC controls for system dynamic performance[END_REF][START_REF] Latorre | Control of a VSC-HVDC operating in parallel with AC transmission lines[END_REF][START_REF] Henry | Influence of Embedded HVDC Transmission on System Security and AC Network Performance[END_REF]. This conducted us develop a methodology which takes into account, in a control model, the neighbour zone of the HVDC link and the dynamics that most impact transient stability. This control model captures the dynamics that are relevant for the transient stability of the power system. The controller synthesis is done on the base of this control model. As a consequence, power system performances are enhanced in addition to local performances. Preliminary results have been presente in [START_REF] Arioua | Control of HVDC links with Overall Large-Scale Grid Objectives[END_REF]. This approach is fully developed here in order to take into account some key critical fault situations at the synthesis stage using the robust H ∞ theory. Also the methodology is adapted to handle realistic large-scalecases as the European power system. As a mater of fact, in other papers [START_REF] Hu | Improvement of transient stability in AC system by HVDC Light[END_REF][START_REF] Mao | Application of an optimal coordinated control strategy to VSC HVDC[END_REF], the transient stability is considered as an a priori objective for the synthesis of coordinated controllers for HVDC. However, the effectiveness of these methods had not been proved in the case of large-scale power systems. The proposed control is tested in comparison with the standard vector control via simulations performed with Eurostag [START_REF] Meyer | EUROSTAG, a single tool for power system simulation[END_REF]. The rest of the paper is organized as follows: in Section 2, the large-scale case used for this work is described. Then in Section 3, the method for developing the control model for the power system is presented. This control model is used in Section 4 to synthesize a robust multivariable output-feedback controller for the HVDC converters. Section 5 deals with the computation and implementation of the controller. Finally, in Section 6, simulation results are presented to illustrate the performances and the robustness of the new controller. An earlier version of this paper was published in [START_REF] Arioua | Implementation and Validation of Multivariable Control With Grid Objectives of an HVDC Link[END_REF].

The Midi-Provence HVDC Project

Midi-Provence project aims to strengthen the link between the two dynamic regions, Provence-Alpes-Côte d'Azur and Languedoc-Roussillon by inserting an HVDC link of 1000 MW capacity between them. This two regions are now directly connected by a single 400 kV AC line going from Tavel (Est of Gard) to La Gaudière (in the Aude) (see Fig. 1). In case of breakdown of this link, the electric power security of much of the south of France would be weakened. In some worst scenarios, this situation could lead to a major power cut in the south of France, and even in-beyond Europe, to balance supply and demand. This HVDC project responds thus to a need of enhancement of both stability and transport capacity in this region [4]. The Midi-Provence zone is considered as part of the whole European system. For the latter, we used the reference model which is maintained and generally used for interconnection studies by the European TSO's. It is a detailed non-linear model including generators along with their regulations (AVRs, PSS and Governors) and where the high voltage network (225, 400kV ) is modelled. It consists of 1121 generators, 7625 nodes, 10404 lines, 2550 transformers and 458 GVA global apparent power. The Midi-Provence HVDC link is of 230 km length and 1000 MW power capacity. It is placed as shown in Fig. 1. Note that we have considered a 2020 forecast winter situation.

Figure 1: Midi-Provence interconnected power system whole European model. The way to do this it is not a straightforward reduction method, but a specific methodology developed in accordance with the objectives of the control. It is detailed in the rest of the paper.

Structure of the control model

The control model is composed of two parts: the AC power system (grid and machines) and the HVDC link. We retain in the control model the most critical machines, i.e., those which cause the system loss of synchronism after a severe perturbation. All the other machines are skipped. Also, after this operation, the lines which no longer connect machines are considered as irrelevant topology and are also simplified.

Construction of the control model

The study area selection

The study area is the zone which is impacted by the HVDC link and corresponds to the neighbour zone of the HVDC interconnection of which transient stability depends on the HVDC dynamics (see Fig. 1). The extent of this zone is detected by standard stability studies usually run by TSOs.

Critical machines selection strategy

The machines retained in the control model are based on both nonlinear and linear analysis as explained below.

1. Selection on the base of CCT First, a transient stability margin of the power system is estimated by the Critical Clearing Time (CCT) which is defined as the maximal fault duration for which the system remains transiently stable [START_REF] Kundur | Power stability and control[END_REF]. The instability is then manifested by the loss of synchronism of a group of machines. The most critical machines are the ones which have the smallest CCT for a given class of faults. The latter faults are the ones put into evidence by the stability studies for the region usually run by TSOs and mentionned above. Doing so for the case Midi-Provence considered in this paper, 14 machines have been retained for the control model 2. Selection on the base of coupling of distant machines Next, some machines external to the stability study zone may be incorporated to the control model. Indeed, electrical interaction may exist between geographically distant machines. It is well known that these interactions are characterized by inter-area modes which are of electromechanical type, i.e., they are associated to the rotors of the machines (see, e.g., [START_REF] Rogers | Power system oscillations[END_REF]). Another mode which may involve geographically distant machines has been put into evidence. It is of different nature since involves only the D-axes of several distant machines and has been thus called electrical coupling modes. These both types of modes can be put into evidence by systematic modal analysis of the AC grid. The list defined at step 1 for the machines to be retained for the control model in the initial neighbour zone of the HVDC link for which the transient stability analysis (CCT evaluation) has been performed, has to be enriched by the machines outside this zone but which are involved in the coupling modes mentioned above together with the machines found at step 1.

Reduction of the topology

Reduction of the topology of the power system is also performed. The buses to which the retained machines are connected are kept along with the neighbour zone buses. The rest of the buses and branches are replaced by a WARD-PV method [START_REF] Baldwin | Dynamic Ward equivalents for transient stability analysis[END_REF], i.e., by equivalent impedances and injectors.

Perturbed models

The model described above can be considered as nominal in the sense that it corresponds to the no incident situation of the power system. A family of perturbed models was also generated. Each member of this family corresponds to a model under a critical fault situation. The electric system is then considered as a set of perturbed systems. The most severe fault generates the the worst-case situation (see Section 4.3).

Modelling of the VSC-HVDC

A voltage-source converters (VSC) based HVDC is considered as in Fig. 2a. The converters are VSC employing IGBT power semiconductors, one operating as a rectifier and the other as an inverter. The two converters are connected through a DC cable. These converters have the ability to rapidly control the transmitted active power, and also to independently exchange reactive power with the AC system at each end. The high frequency switching operation of the power electronics is neglected and each converter can be considered as an ideal sinusoidal voltage source whose magnitude U c and phase angle θ can be controlled (see [START_REF] Latorre | Active and reactive power control of a VSC-HVDC[END_REF]). The VSC-HVDC is thus modelled as two sources, each one in series with the converter transformer (see Fig. 2b where Ūc1 = U c1i + jU c1j , Ūc2 = U c2i + jU c2j and θ 1 , θ 2 are, respectively, the magnitudes and the phase angles of the voltage sources, P 1 , P 2 , and, Q 1 , Q 2 are, respectively, active and reactive powers exchanged with the power system and X T 1 , X T 2 are the transformer reactances). In this model of HVDC, it is assumed that the DC voltage is kept close to its rated voltage, because its dynamic is much faster than the one of currents. Therefore, the losses of the converters are assumed constant, regardless of the current through the converters. They are represented as a constant active load. However, the losses of the DC cables are neglected [START_REF] Latorre | Active and reactive power control of a VSC-HVDC[END_REF]. This leads to In addition, notice that P 2 = -P 1 . The control variables can be expressed as follows:

P 1 = (U 1 (sin θ 1 U c1i -cos θ 1 U c1j ))/X T 1 Q 1 = (U 2 1 -U 1 (cos θ 1 U c1i -sin θ 1 U c1j ))/X T 1 P 2 = (U 2 (sin θ 2 U c2i -cos θ 2 U c2j ))/X T 2 Q 2 = (U 2 2 -U 2 (cos θ 2 U c2i -sin θ 2 U c2j ))/X T 2 . (1)
U c1i = U c1i0 + ∆U c1i , U c1j = U c1j0 + ∆U c1j , U c2i = U c2i0 + ∆U c2i , U c2j = U c2j0 + ∆U c2j (2)

Validation of the control model

The resulting control model is validated by non-linear simulations comparing responses of the reduced and the full model when a fault occurred near the HVDC link. Theses simulations were done using Eurostag software. Fig. 3a and Fig. 3b for instance, illustrate the speed responses of two generators retained in the control model G 1 and G 2 . We can observe that the two responses of the control model and the full model are closed to each other for this two generators.

Synthesis of the control law

In our previous work [START_REF] Arioua | Control of HVDC links with Overall Large-Scale Grid Objectives[END_REF], an output-feedback controller for HVDC converters has been developed using an LQ pole placement [START_REF] Arioua | Control of HVDC links with Overall Large-Scale Grid Objectives[END_REF]. In this present work an H-infinity control theory is used to take into account not only the nominal model of the system, but also the family of disturbed models presented in Section 3.2.4. This allowed us to improve both performances and robustness which allows a more robust control design.

Design Requirements

The synthesis of the HVDC controller is done such that the performances are ensured for several cases of fault (i.e, the enhancement of transient stability) along with local ones which are tracking of references for active power, reactive power. For these purpose, a robust multivariable control is applied.

To avoid the use of remote variables (e.g., machines speed, angles, ...), an output feedback structure described in Fig. 4 is used. Only variables available at converter stations are used and a coordinated controller is developed for both converters. A linear approximation of the control model around an equilibrium point is used for the synthesis.

Linear approximation of the control model

The control model includes the machines retained in the control model and their regulations (AVRs and governors) be represented by the non linear differential and algebraic equations

ẋ = f (x, U) y = g(x, U) (3)
where x represents the state variables (speeds, angles of machines, AVRs and governors variables ...) of the machines retained in the control model, U the control variables, y the outputs and let y ref be the control references : The model of the European power system is of a very large scale (17020 sate variables and 43940 total variables). Thanks to the control model developed in Section 3, we can reduce the order of the model considered for the control to 310 state variables with a total of 1117 variables. By performing a minimal realization using Robust control toolbox of Matlab, a 30th-order control model is obtained. Notice that dedicated tools (like, e.g, Eurostag [START_REF] Meyer | EUROSTAG, a single tool for power system simulation[END_REF]), provide the linear model (4) for large-scale power systems.

U=[U c1i , U c1j , U c2i , U c2j ] y=[P 1 , Q 1 , Q 2 ]. y ref =[P ref 1 , Q ref 1 , Q ref 2 ].

Mixed sensitivity H-infinity control design

Using the nominal model P (s) developed in Section 3 and the set of the linearised perturbed models described in Section 3.2.4, an uncertain model P ∆ (s) is generated and is expressed as P ∆ (s) = (1 + ∆(s)W t (s))P (s) as shown in Fig. 5 where :

• ∆(s) : is the normalized model deviation with respect to the nominal model.

• W t (s) : is the uncertainty weight and is used to capture how the relative uncertainty varies with frequency.

The Fig. 6 shows the singular values (frequency responses for multivariable systems) of the deviation of all the perturbed models with respect to the nominal one and the weight W t (s).

The uncertain model generated above leads us to develop a controller taking into consideration the worst-case situation. This can be achieved using some methods existing in the control theory (for example µ synthesis, H ∞ control). In this paper the H ∞ control and more precisely the Mixed sensitivity H-infinity control design is used. The H ∞ problem formulation and control law computation is recalled in the Appendix.

Let the sensitivity S(s) = (I + P (s)K(s)) -1 be the closed loop transfer function matrix from the reference inputs to the error and the complementary sensitivity T (s) = P (s)K(s)(I + P (s)K(s)) -1 be the closed-loop transfer function matrix from the references to the measured outputs where P is the nominal model, K the controller, and I identity matrix. The principle of this control design is that if it exists a controller K(s) given by its transfer matrix which ensures W t (s)T (s) ∞ ≤ 1 for all frequencies, the closed-loop system is ensured stable for all plants in P ∆ (see [START_REF] Skogestad | Multivariable feedback control[END_REF]).

The desired performances are achieved if W s (s)S(s) ∞ ≤ 1. W s (s) and W t (s) are the weights used to shape the sensitivity and the complementary sensitivity respectively. The weight W t (s) is used to ensure robustness against model uncertainties. The weight W s (s) is based on shaping the sensitivity for desired tracking performances.

The state space model is augmented with the matrix weights W t (s) and W s (s) and form the augmented system G(s) (described in the Appendix) as shown in Fig. 7.

The objective of the mixed sensitivity problem is to find a stabilizing controller which minimizes.

W s (s)S(s) W t (s)T (s) ∞ (5)
In this work W t (s) is the uncertainty weight and W s (s) is chosen in the form (see [START_REF] Zhou | Essentials of robust control[END_REF])

W s (s) = s/ k √ M s + ω b s + ω b k √ ε k (6)
where ω b is the bandwidth, M s the peak sensitivity , ε the steady state error desired with respect to a step input, and k is the order of the weighting function. Notice that making ε small enforce the integral action (in our case ε = 10 -4 )

Computation and implementation of the controller

The robust multivariable controller K(s) responding to the mixed sensitivity problem formulated in Section 4 is calculated by solving the Ricatti equation as it is described in the Appendix. In practice, a function of Robust control toolbox of Matlab is used to solve the problem. The final controller is given in its state representation (A K , B K , C K , D K ) and it is of the same order as the plant model augmented by the weights W s and W t . The model obtained after reduction and minimal realization is of 30th-order. The controller obtained is of 40th-order which is too high for the implementation.

To deal with this problem, a reduction of the controller order is envisaged. A reduction method based on the singular values [START_REF] Zhou | Essentials of robust control[END_REF] helped us to generate a controller of 5th-order. The truncations has been performed with some existing functions in Robust control design toolbox of Matlab. This resulating controller can be written in its matrix transfer function form:

K(s) = C K (sI -A K ) -1 B K ,
where A K is a state matrix of 5th order.

A coordinated structure

The controller transfer matrix obtained above can be detailed

∆U 1 ∆U 2 = K 11 (s) K 12 (s) K 21 (s) K 22 (s) K(s) ∆y 1 ∆y 2 (7) ∆U 1 = ∆U c1i ∆U c1j , ∆U 2 = ∆U c2i ∆U c2j (8)
and

∆y 1 = ∆P 1 ∆Q 1 , ∆y 2 = ∆Q 2 . ( 9
)
The transfer matrices K 12 (s) and K 21 (s) are the cross-terms of the controller and ensure the coordination between the control of the two HVDC converters. In the standard vector control, HVDC links are controlled with a decentralized structure, which is equivalent to consider a priori that the two HVDC converters don't interact (K 12 (s) = 0 and K 21 (s) = 0). This hypothesis is not valid when the HVDC is used in the new grid context mentioned in the introduction. Each term of the K ij transfer matrix is in the form of :

k ij = α ij s 4 n=1 1 + τ n s 1 + γ n s (10)
In Our case of study the controller is given in the Table 1. One can observe that it is of 5th-order which is due to the reduction of the controller as we have explained above.

Simulation tests

This section illustrates the effectiveness of the H-infinity controller proposed. Simulation tests using Eurostag software [START_REF] Meyer | EUROSTAG, a single tool for power system simulation[END_REF] were performed. The new synthesized controller is tested on the full model introduced in Section 2. An export of 1000 MW is considered from Fos to Gaudière via the DC link.

The robust H-infinity controller is compared with the standard vector controller (see for example [START_REF] Li | Control of HVDC Light System Using Conventional and Direct Current Vector Control Approaches[END_REF]) which has a cascade structure with an inner loop more rapid than the outer one. The proportional and integral gains of this control are synthesized using standard criteria for electrical drives. The two controllers are tuned to satisfy almost the same performance specifications (the usual time setting for HVDC voltage and power control).

Transient stability

In a first scenario, a symmetrical double fault at Tavel-Tamareau, cleared after 100 ms by opening the lines at their both ends is simulated. Fig. 9a and Fig. 9b show the responses of active and reactive powers exported. It can be observed that the dynamic responses for this case of fault are better with the new controller especially for the reactive power. Indeed, more reactive power is needed with the vector control. Moreover, a limitation is reached in this case during the transient. In a second scenario, a symmetrical short-circuit is applied at one of the HVDC terminals (in the Fos 2 shows that the critical clearing time are augmented when the new control is used.

(s 2 +0.022s+4.510 -3 ) (s 2 +0.011s+3.1210 -3 ) k 13 2.56 s (s+0.075) (s+0.06) (s+0.012) (s+0.06)
(s 2 +0.041s+6.210 -3 ) (s 2 +0.011s+3.1210 -3 ) k 33 23 2.6 s (s+0.062) (s+0.06) (s+0.075) (s+0.03)
(s 2 +0.078s+510 -3 ) (s 2 +0.011s+3.1210 -3 ) -5.94 s (s+0.050) (s+0.06) (s+0.02) (s+0.03)

Robustness against variation of the operating point

The proposed controller was synthesized using the linearised model. This lead us to perform some robustness tests against variation of the operating point of the HVDC link. We have considered the situation of a 450 MW import of active power from Gaudière area to Fos. As for the previous tests, a 100 ms symmetrical fault is applied at one of the HVDC terminals (Gaudière area). The active and reactive powers imported are represented in Fig. 10a and Fig. 10b and both are comparable with the ones obtained when the export scenario is considered (Fig. 9a and Fig. 9b). This confirm the good robustness of the performances of the proposed controller against variation of operating conditions.

Conclusion

A new method for synthesizing the controllers of HVDC links has been presented. It is based on the use of a control model which takes into account the region of the power system neighbouring the HVDC link and impacted by it. The main advantage of this new methodology is that the main dynamics of the full power system are taken into account via a control model. It has been shown that this strategy can be applied to a realistic large scale power system.

Moreover, an H-infinity control design has been used. It allowed us to consider at the synthesis stage some faulted situations and this contributed to enhance robustness along with performances of The control model proposed in this methodology is a way to connect the system expertise to the control one. More specially, in more cases, the manufactures do not possess the knowledge and the data to construct such a model and, as a consequence, the control of the DC link is synthesized with a very simplified model of the surrounding system. Most of the time, infinite buses or large power generators are considered connected to the two ends. The control model proposed here is rich enough to improve the performances of the controllers and simple enough to be built and provided by TSO's to the manufactures.

8 Appendix: Solution of the H ∞ control problem

The system's representation

The stationary linear system can be given by its state representation :

ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (11)
The transfer matrix of this system is

P (s) = C(sI -A) -1 B + D (12)
and is noted P = A B C D .

The controller K(s) and the uncertainties of G(s) can be put into evidence according to Figure 11.

where

   ẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) z(t) = C 1 x(t) + D 11 w(t) + D 12 u(t) y(t) = C 2 x(t) + D 21 w(t) + D 22 u(t) (13)
G =   A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22   , (14)
The transfer matrix G(s) of this augmented system can be partitioned according to the inputs/outputs:

G(s) = G 11 (s) G 12 (s) G 21 (s) G 22 (s) . (15)

Problem formulation

The plant G has two inputs, the exogenous input w, that includes reference signal and disturbances, and the control variables u. There are two outputs, the error signals z that we want to minimize, and the measured variables y, that we use to control the system. y is used in K to calculate the control variable u.

It is possible to express the dependency of z on w as: z = F l (P, K)w,

F l (P, K) = G 11 + G 12 K(I -G 22 K) -1 G 21 , (16)
called the lower linear fractional transformation.

The objective of the H ∞ control design is to find a controller K which stabilizes the closed loop for all the plants in the family P ∆ (s) and such that F l (P, K) is minimised according to the H ∞ norm. The infinity norm of the transfer function matrix is defined as: F l (P, K) ∞ = sup ω σ(F l (P, K)(jω)) where σ is the maximum singular value of the matrix F l (P, K)(jω).

Since the H ∞ norm to be minimized is a superior value over ω of the maximum singular value σ of the transfer F l (P, K)(jω), this allows us to develop a controller which takes into consideration the worst-case situation. Notice that the singular values are used for in the case of multivariable systems to represent the model gains over all input and output directions.

Solution of the H ∞ control problem

Consider the generalized state representation described by ( 13), ( 14) and its transfer matrix given by [START_REF] Mao | Application of an optimal coordinated control strategy to VSC HVDC[END_REF]. We assume that D 22 = 0, i.e., it don't exist any direct transmission from the input to the output. The solution of the problem is based on the solution of the algebraic Riccati equation [START_REF] Zhou | Essentials of robust control[END_REF] XE

+ E T X -XW X + Q = 0, (17)
with

W = W T and Q = Q T .
A stabilizing solution X of ( 17), if it exists, is a symmetrical matrix such that (E-W X) is a stable matrix (eigenvalues with strictly negative real parts). This solution is noted

X = Ric E -W -Q -E T (18)
The solution of the Riccati equation : Since R n and Rn are not singular, we define the two Hamiltonian matrices

Let be R n = D T 1 * D 1 * - γ 2 I
H = A 0 -C T 1 C 1 -A T - B -C T 1 D 1 * R -1 n D T 1 * C 1 B T (19)
J = A T 0 -B 1 B T 1 -A - C T -B 1 D T * 1 R-1 n D * 1 B 1 T C . (20)
If we put X = Ric(H) and Y = Ric(J), we can define the two matrices :

F = -R -1 n (D T 1 * C 1 + B T X) = F 1 F 2 =   F 11 F 12 F 2   (21)
L = -(B 1 D T * 1 + Y C T ) R-1 n = L 1 L 2 = L 11 L 12 L 2 , (22)
where F 1 , F , the solution will be given by the following theorem :

Theorem 1 : If a system G(s) verifies the hypotheses 1 to 4 described above then : a. it exists a controller K(s) which stabilizes the closed-loop and such that F l (P, K) ∞ < γ if and only if : b. if the conditions of part a.) are satisfied, then the controllers which stabilise the system, and which verify F l (P, K) ∞ < γ are given by : The controller calculated for φ(s) = 0 is called the Central Corrector, it is generally used under the form :

K(s) = F l (M, φ) (23
K(s) = Â B1 Ĉ1 D11
(25)

( a )Figure 2 :

 a2 Figure 2: VSC-HVDC link

  (a) Speed response of G1 in the two cases of full and reduced models (p.u) (b) Speed response of the G2 in the two cases of full and reduced models

Figure 3 :

 3 Figure 3: Speed responses to a 100 ms short-circuit in the two cases of full and reduced models (p.u)

Figure 4 :

 4 Figure 4: Structure of the output-feedback control

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Plant with multiplicative output uncertainty

2

 2 +0.009s+4.3110 -3 ) (s 2 +0.011s+3.1210 -3 ) area). As for the previous case the transient responses are enhanced in comparison to the standard vector control. This is confirmed by analysis of the transient stability of the zone. In fact, Table

( a )

 a Response of the active power P1 (p.u) (b) Response of the reactive power Q1 (p.u)

Figure 8 :

 8 Figure 8: Responses to a 100 ms Tavel-Tamareau short-circuit

Figure 9 :

 9 Figure 9: Responses to a 100 ms short-circuit near Fos area in case of 1000 MW power export

( a )

 a Response of the active power P2 (p.u) (b) Response of the reactive power Q2 (p.u)

Figure 10 :

 10 Figure 10: Responses to a 100 ms short-circuit near Gaudière area in case of 450 MW power import

Figure 11 :

 11 Figure 11: Standard form representation of a system

D 1 *

 1 = D 11 D 12 and D * 1 = D 11 D 21.

1 .

 1 γ > max(σ[D 1111 D 1112 ], σ[D 1111 D 1121 ])2. it exists solutions X ≥ 0 and Y ≥ 0 which verify the two Riccati equations relating to Hamiltonians matrices H and J, and such that: ρ(XY ) < γ 2 , where ρ(.) design the spectral radius.

  D 1121 D T 1111 (γ 2 I m 1 -P 2 -D 1111 D T 1111 ) -1 D 1112 -D 1122 D12 ∈ R m 2 ×m 2 and D21 ∈ R p 2 ×p 2 are two arbitrary matrices which verify : D12 DT 12 = I m 2 -D 1121 (γ 2 I m 1 -P 2 -D T 1111 D 1111 ) -1 D T 1121 DT 21 D21 = I p 2 -D T 1121 (γ 2 I p 1 -m 2 -D 1111 D T 1111 ) -1 D 1121 B2 = Z(B 2 + L 12 ) D12 and B1 = -ZL 2 + Z(B 2 + L 12 ) D11 Ĉ2 = -D21 (C 2 + F 12 ) and Ĉ1 = F 2 -D11 (C 2 + F 12 ) Z = (I n -γ -2 Y X) -1 Â = A + BF -B1 (C 2 + F 12 )

Table 1 :

 1 Transfer functions of the controller K(s)

						Transfer
						function
	k 11	80.83 s	(s+0.018) (s+0.06)	(s+0.012) (s+0.03)	(s 2 +0.03s+5.2410 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 21	-18.42 s	(s+0.019) (s+0.06)	(s+0.101) (s+0.03)	(s 2 +0.05s+6.310 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 31	-1.0469 s	(s+0.028) (s+0.06)	(s+0.02) (s+0.03)	(s 2 +0.035s+0.58010 -2 ) (s 2 +0.011s+3.1210 -3 )
	k 41	5.74 s	(s+0.051) (s+0.06)	(s+0.042) (s+0.03)	(s 2 +0.079s+6.110 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 12	11.863 s	(s+0.3012) (s+0.06)	(s+0.15) (s+0.03)	(s 2 +0.09s+4.110 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 22	-80.04 s	(s+0.0412) (s+0.06)	(s+0.012) (s+0.03)	(s 2 +0.010s+8.3210 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 32	-0.68032 s	(s+0.01) (s+0.06)	(s+0.0048) (s+0.03)	(s 2 +0.08s+5.1210 -3 ) (s 2 +0.011s+3.1210 -3 )
	k 42	-3.565 s	(s+0.012) (s+0.06)	(s+0.0054) (s+0.03)	(s 2 +0.021s+4.8910 -3 ) (s 2 +0.011s+3.1210 -3 )
	k

Table 2 :

 2 Critical clearing time validation of the two controllers

	Fault	Standard	New
	simulated controller controller
		CCT [ms] CCT [ms]
	f ault1	216	224
	f ault2	192	204
	f ault3	183	195
	f ault4	181	192
	f ault5	194	205

  2 , F 11 and F 12 are formed respectively by m 1 , m 2 , m 1 -p 2 and p 2 lines and L 1 , L 2 , L 11 L 12 are formed respectively by p 1 , p 2 , p 1 -m 2 and m 2 columns . A solution exists if the two following conditions are satisfied : 1. (A, B 2 ) is stabilizable and (C 2 , A) detectable; 2. D 12 = 0 I m2 and D 21 = 0 I p2 are then, respectively, of rank m 2 and p 2 ; 3. ∀ω ∈ R, rang A -jωI n B 2 C 1 D 12 = n + m 2 then this matrix is of full rank in addition P 12 has not invariant zeros on the imaginary axis; 4. ∀ω ∈ R, rang A -jωI n B 1 C 2 D 21 = n + p 2 then this matrix is of full rank in addition P 21 has not invariant zeros on the imaginary axis; Let D 11 = D 1111 D 1112 D 1121 D 1122 with D 1122 is of dimension m 2 × p 2

The large-scale model mentioned in Section 2 cannot be used for the synthesis of the control. As a consequence, a reduced order model called a control model which must preserve the dynamics that most impact the stability of the neighbour region of the HVDC link should be extracted from the

Acknowledgment

The authors would like to thank Prof. Eric Monmasson, University of Cergy-Pontoise and Dr. Alexandre Parisot, head of Integration of New Technology division (INT) of RTE for their helpful suggestions and remarks.


Imed Zidane
email: zidaneimed@hotmail.com

Bogdan Marinescu
email: bogdan.marinescu@satie.ens-cachan.fr

Mohamed Abbas-Turki
email: abbas-turki@satie.ens-cachan.fr

Satie

Linear Time-Varying Control of the Vibrations of Flexible Structures

Keywords: linear time-varying systems, flexible structures, vibrations attenuation

Recent results on pole placement for linear time-varying (LTV) systems are exploited here for the control of flexible structures. The infinite dimensional system is approached as usual by a restricted number of modes of interest, according to the frequency range in which the system is exploited. The difference with the previous approaches (like, e.g., gain scheduling) is that only one finite dimension linear model is used and its parameters (frequency and damping of the modes) are varying according to the operating conditions. The control model is thus LTV and a regulator is analytically synthesized to ensure closed-loop stability and performances. This regulator is also LTV and thus automatically (without any special switching action) tracks the operating * Corresponding author 1 Page 1 of 44 conditions of the system. The case of a flexible beam is studied in simulation as well as in laboratory experimentation. Hence, a suitable application of the proposed LTV synthesis is done to find a compromise between the complexity of interpolation and its efficiency. Furthermore, the bad dissipation factor of flexible systems leads them to be good candidates to proof the efficiency of the proposed gain scheduling strategy. This work validates on a real system the LTV pole placement approach and opens the way to a new gain scheduling strategy for the control of LTV, nonlinear or infinite dimensional systems.

Introduction

Flexible structures are usually encountered in industry. A lot of studies have been thus devoted to the attenuation of their vibrations (see, e.g., [START_REF] Gerardin | Théorie des vibrations, application la dynamique des structures[END_REF] and [START_REF] Preumont | Vibration control of active structures: An introduction[END_REF]). The main difficulty comes from the infinite dimension of such systems ( [START_REF] Lasiecka | Galerkin approximations of infinite-dimensional compensators for flexible structures with unbounded control action[END_REF] and [4]). Indeed, it is reminded in Section 2 how the behavior of such a system is given by a class of oscillatory modes. For the control synthesis, a restricted number of modes is retained according to the frequency range in which the system is operated [START_REF] Schilders | Model order reduction: Theory, research aspects and applications[END_REF]. However, on the one hand, the characteristics (frequency and damping) of the members of this subclass of modes can vary in time [START_REF] Jeanneau | A semi-adaptive frequency control law for flexible structures[END_REF]. On the other hand, the number of excited modes can change according to the operating band [START_REF] Mohammadpour | Control of Linear Parameter Varying Systems with Applications[END_REF]. Adaptive and gain scheduling controls have been used to overcome these difficulties (see, e.g., [START_REF] Shojaei | Adaptive robust time-varying control of uncertain non-holonomic robotic systems[END_REF]). How-2 ever, in these approaches, it is difficult to guarantee the stability of the overall control scheme in case of hidden coupling modes (see, e.g., [START_REF] Leith | Input-output linearization velocitybased gain scheduling[END_REF]). Moreover, according to [START_REF] Bérard | Gain scheduled flight control law for a flexible aircraft : A practical approach[END_REF], a low damping factor induces sensitivity in the model identification. Indeed, due to a fast change in their phase inducing a low stability margin, flexible systems are very sensitive. For this reason, most of the previous research related to flexible systems deal with their robustness and the most used criteria are H ∞ [START_REF] Balas | Robustness and Performance Tradeoff s in Control Design for Flexible structures[END_REF]12,[START_REF] Cavallo | Robust Control of Flexible Structures With Stable Bandpass Controllers[END_REF] and sliding mode [START_REF] Gu | Active Vibration Supression of a Flexible Beam with Piezoceramic Patches using Robust Model Reference Control[END_REF]. Therefore, the control performance of flexible systems is relegated to a less important target.

The knowledge of the different operating bands (cycle of operation of the machine, for example, the launch phase of a space launcher and other cycles) led us to use for the system a linear time-varying (LTV) model in order to capture the aforementioned variations of the system. The latter is still considered as infinite-dimensional described by a partial differential equation. As in the classic approaches, it is modeled by a restricted number of modes of interest but, in our approach, only one finite dimension linear model is used and its parameters (frequency and damping of the modes) are varying according to the operating conditions.

In [START_REF] Gielen | Stabilisation of linear delay difference inclusions via time-varying control Lyapunov functions[END_REF] it has already been shown that necessary conditions for exponential stability can be provided if time-varying Lyapunov functions are used. Such stability conditions can be more directly obtained if the structure (poles) of the LTV system is directly considered. More specifically, in [START_REF] Adami | Time-varying notch filters for control of flexible structures and vehicles[END_REF] and [START_REF] Adami | Control of a flexible, hypersonic scramjet vehicle using a differential algebraic approach[END_REF] LTV notch filters have been used to damp vibrations of flexi-3 ble structures and vehicles. They are placed on the control error path and their dynamics are tuned using the PD-eigenvalue theory introduced in [START_REF] Zhu | Series and parallel d-spectra for multi-input-multi-output linear time-varying systems[END_REF] for differential operators with time-varying coefficients. The latter algebraic approach was extended and fully developed in [19] and related references to define poles and provide stability conditions for open as well as closed-loop LTV systems. In comparison to the PD eigenspectrum mentioned above, the poles are also defined for LTV systems given by differential operators which do not have roots over the field of definition of their coefficients. The field extensions needed in this case to find the poles allowed us to characterize stability and to solve the pole placement problem for such general LTV systems. The output feedback pole placement method introduced in [START_REF] Marinescu | Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems[END_REF] is used here to analytically synthesize a regulator to directly ensure closedloop stability and performances. This overcome the common difficulty to ensure the closed-loop stability when adapting the regulator to the evolution of the operating conditions. This latter difficulty is faced, for example, in the classical gain scheduling where several linear time-invariant (LTI) models are constructed around each operating point and an LTI regulator is synthesized for each model; the final regulator is obtained by switching among the members of the class of regulators according to the evolution of the operating conditions. The regulator which results from our approach is also LTV and thus automatically (without any special switching action) tracks the operating conditions of the system.

The paper is organized as follows. In Section 2, the standard modelization of flexible structures is briefly recalled. Section 3 gives the basics of the LTV output feedback control. In Section 4 it is first shown how an LTV model for a smart beam can be obtained if some parameters are assumed varying with the operating conditions. The mathematical background presented in Section 3 is used next for the synthesis of an LTV regulator which attenuates the vibrations of the beam. Section 5 presents simulation as well as laboratory results for the control of the smart beam while Section 6 is devoted to concluding remarks.

2 Flexible structures modeling

Dynamic equations

The purpose of modeling is to know the resulting displacement field at any point of the structure. In the general case, the equations of the dynamics of the structure lead to partial differential equations (PDE), which do not admit simple analytical solutions, with the exception of the case of simple structures (for example, beams and plates simply supported on their contour). It is then necessary to perform a discretization of the problem where the deformation

-→ ω (x, y, z, t) ∈ R n is approached by -→ ω (x, y, z, t) = N i=1 -→ η i (x, y, z)q i (t), (1)
where, (x, y, z) ∈ R 3 are spatial coordinates, t is time variable, q i ∈ R are the general coordinates, -→ η i ∈ R n the approximation functions and N the number of degrees of freedom (DOF). For the beam, the -→ η i are given by their proper modes, as shown in Fig 1 . In our study, we used SDTools software to identify proper modes. In the case of the beam, these modes can 5 be found analytically [START_REF] Leleu | Amortissement actif des vibrations d'une structure flexible de type plaque à l'aide de transducteurs piézoélectriques[END_REF]. The Lagrange formalism is used to provide the dynamic equations. Let T denote the kinetic energy and U the potential energy of the structure. Using the system description in equation [START_REF] Gerardin | Théorie des vibrations, application la dynamique des structures[END_REF], Lagrange equations can be written [START_REF] Gerardin | Théorie des vibrations, application la dynamique des structures[END_REF]:

∂ ∂t ∂T ∂ qi - ∂T ∂q i + ∂U ∂q i = F i (2)
where F i ∈ R are the generalized efforts on each DOF.

This leads straightforward to the following dynamic equation

Mq + C q + Kq = F (3)
where F = (F 1 , F 2 , . . . , F N ) is the vector of generalized efforts, M ∈ R n×n and K ∈ R n×n are the mass and the stiffness matrices, respectively,

6

C ∈ R n×n is the damping matrix. M is symmetric positive definite and K is symmetric semipositive definite. C is determined a posteriori because it is difficult to be modeled using structure characteristics.

We define the modal coordinates v ∈ R n by projecting the generalized displacement on the modal basis:

q(t) = Φv(t); v(t) + Γ v(t) + Ω 2 v(t) = Φ T F(t) (4)
Φ ∈ R n×n is the modal basis matrix, Ω = diag{ω 1 , ..., ω N } is the modal frequencies matrix. If we assume also that Γ = diag{2ω 1 ξ 1 , ..., 2ω N ξ N } (Basile's hypothesis [START_REF] Leleu | Amortissement actif des vibrations d'une structure flexible de type plaque à l'aide de transducteurs piézoélectriques[END_REF]) which is generally verified in such systems [START_REF] Gerardin | Théorie des vibrations, application la dynamique des structures[END_REF], we get N independent equations of second order and every one defines a vibration mode

vi (t) + 2ξ i ω i vi (t) + ω 2 i v i (t) = Φ T F i (t) (5)
where ξ i , ω i are, respectively, the damping and the frequency of the i th vibration mode. F i is the ith element of the vector F.

The parameter identification is done through the minimization of J which is the quadratic criterion of log modules:

J = log|G(ω)| -log| Ĝ(ω)| 2 (6)
where G and Ĝ are respectively the transfer functions of the system and of the model and ω is the pulsation in rad/sec. Since the problem is nonconvex, the proper pulsations are initialized visually for damping factors as well. Although the algorithm corrects sufficiently good pulsations, the very low value of the damping factor does not allow a good quality of its 7 identification even when using the log scale. As shown in the criterion, the most energizing modes are those selected by the model and they are at low frequencies. To minimize J, we use "fmincon" Matlab function on the frequency domain of interest. We point out that G does not vanish on the chosen set of frequencies.

State-space and transfer representations

The choice of the state vector

x = (ω 1 v 1 , v1 , ω 2 v 2 , v2 , . . . , ω N v N , vN ) T (7)
[22] leads to the following state-space representation

ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (8)
where

A = diag{A i }, A i =    0 ω i -ω i -2ξ i ω i    B = [0, b 1 , 0, b 2 , . . . , 0, b N ] T C = [c p 1 , c v 1 , c p 2 , c v 2 , , . . . , c pN , c vN ] (9)
b i and u (control input vector) depend on the type of the actuators used, c pi , c vi and y depend on the sensors type (position or velocity sensors). D is generally zero [START_REF] Leleu | Amortissement actif des vibrations d'une structure flexible de type plaque à l'aide de transducteurs piézoélectriques[END_REF]. The choice (7) of the state vector gives the same range of values to its components which leads to a well conditioned problem.

The transfer function of a such structure is composed of the sum of the transfers of every subsystem. To simplify the exposal, consider a SISO system. Its state representation (8) leads to the transfer function

H(s) = N i=1 k i s 2 + 2ξ i ω i s + ω 2 i (10)
where N → +∞ and k i characterize the static gain of each mode.

The case of a clamped-free (cf. Fig. 4) beam was treated in this work.

Its transfer is given in Fig. 2.

Flexible structure LTV model

For a control model, only a very restricted number of modes of a flexible structure model ( 10) must be considered. The most poorly damped modes in the frequency range in which the system is excited are usually chosen.

However, the excitation usually varies with the operation conditions of the system, for example: aircraft taking off sequence. To capture this, a fixed control model (fixed number of modes) with time-varying characteristics (frequency and damping of the oscillatory modes) has been adopted in the sequel. In this case, N in (10) is finite and ξ i , ω i are time-varying. This 9 leads to an LTV transfer matrix. A specific formalism to handle this kind of mathematical objects is recalled in the next section.

LTV systems and control

3.1 LTV systems

Linear systems as modules

In the algebraic framework initiated by Malgrange [START_REF] Malgrange | Systèmes différentiels à coefficients constants[END_REF] and popularized in systems theory by several authors, a linear system is a finitely presented module M over the ring R = K[s] of differential operators in s = d/dt with coefficients in an ordinary differential field K. If K does not exclusively contains constants (i.e., elements of derivative zero), M is an LTV system. R = K[s] is an Euclidean domain thus a two-sided Ore domain and it has a field of left fractions and a field of right fractions which coincide (see [START_REF] Cohn | Free rings and their relations[END_REF], [19]); this field is denoted by Q = K(s). A transfer matrix of an input-output LTV system (M, u, y) is a matrix with entries in Q [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF]. The following notations are used in the sequel to denote matrices: H (or H(t)) is a matrix with entries in K, H(s) a matrix with entries in R (a polynomial matrix) and H(s) a matrix with entries in Q (a transfer matrix). The greatest common divisors lead to the following factorizations of a given LTV transfer matrix:

Proposition 1 [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF]: Any LTV transfer matrix H(s) has:

(i) left-coprime factorizations: H(s) = A L (s) -1 B L (s), i.e., A L (s)y = B L (s)u 10 (ii) right -coprime factorizations: H(s) = B R (s)A R (s) -1 , i.e., y = B R (s)ξ and u = A R (s)ξ related by the diophantine equation 1 A L (s)B R (s) -B L (s)A R (s) = 0. (11)

Autonomous systems

The autonomous part of the system corresponds to the torsion module

T ∼ = M/[u] R .
The can be given by a state representation ẋ = Ax (where ẋ = dx dt ),

A ∈ K n×n or, equivalently, by a scalar differential equation

P (s)y = 0, P(s) = s n + n-1 i=0 a i s i ( 12
)
where y is a generator of T .

Stability of LTV systems

Let ∆ K (α) = {α c = α + ċc -1 , c ∈ K, c = 0} be the conjugacy class of α over K. A skew polynomial has an infinite number of (right) roots. They belong to at most n conjugacy classes, where n is the degree of the polynomial [START_REF] Lam | Wedderburn polynomials over division rings, I[END_REF].

A fundamental set of roots of a skew polynomial P (s) in ( 12) is a set {γ 1 , ..., γ n } of n P-independent (in the sense of definition in [START_REF] Lam | Wedderburn polynomials over division rings, I[END_REF]) roots. It is directly linked to a fundamental set of solutions {y 1 , ..., y n } of the differential equation (12) [START_REF] Marinescu | An intrinsic algebraic setting for poles and zeros of linear time-varying systems[END_REF]: ẏi /y i = γ i , i = 1, ..., n. Notice that such a set may not 1 When dealing with LTV systems, polynomials in [START_REF] Balas | Robustness and Performance Tradeoff s in Control Design for Flexible structures[END_REF] are skew, i.e., belongs to the noncommutative ring R = K[s] equipped with the commutation rule sa = as + ȧ, ∀a ∈ K, which is the Leibniz rule of derivation of a product.

11

exist over K, the initial field of definition of the coefficients of P (s), but over a field extension Ǩ ⊃ K.

Definition 1: [19]

• A real analytic function f such that lim

t→+∞ f (t) = +∞, ḟ (t) > 0, lim t→+∞ ḟ (t) f (t) = 0 is called an Ore function. • If f is an Ore function, C(f ) is an Ore field.
Definition 2 [START_REF] Marinescu | An intrinsic algebraic setting for poles and zeros of linear time-varying systems[END_REF]: Re{γ i (t)} < 0, i = 1, ..., n.

Let γ i ∈ Ǩ, i = 1, ...,

Closed-loop stability and pole placement

Proposition 2 [START_REF] Marinescu | Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems[END_REF]: It is assumed that the plant of the classical closed-loop in Fig. 3 is controllable and observable, i.e., completely described by its transfer matrix H(s). Consider factorizations (i) and (ii) for the plant

Q p×m H(s) = A L (s) -1 B L (s) = B R (s)A R (s) -1
and for the regulator

G(s) = S L (s) -1 R L (s) = R R (s)S R (s) -1 . Then: 1. Polynomials A cl (s) = A L (s)S R (s) + B L (s)R R (s) (13)
12 and

A cl (s) = S L (s)A R (s) + R L (s)B R (s) (14)
have the same conjugacy classes of roots.

2. If the roots of A cl (s) (or, equivalently, of A cl (s)) belong to an Ore field, then the poles of the closed-loop system are the conjugacy classes of a fundamental set of roots of A cl (s) (or A cl (s)).

3. If G(s) is chosen such that the poles of the closed-loop system satisfy the condition of Theorem 1, then the control-loop system is internally exponentially stable, i.e., all transfers r → y, r → v, d → y and d → v in Fig. 3 are exponentially stable.

The output-feedback pole placement problem consists thus in solving the Bézout equation ( 13) (or, equivalently, ( 14)) for S R (s) and R R (s) (or, equivalently, for S L (s) and R L (s)).

Description of the system

The structure studied is an aluminum beam clamped-free, of which characteristics are given in Table 1. The experimental control-loop is presented in Fig. 4. It is equipped with a piezoelectric actuator and sensor, glued on each side of the beam, at the clamped side. The beam is fixed on a shaker which generate vibrations. This flexible structure has its modes at low frequency (16Hz for the first mode), the same frequency range as the aviation structures. The input and output signals are the voltages measured by the sensor and, respectively, applied to the actuator. The sensor is connected to a charge amplifier. The aim of this amplifier is to apply, via the operational amplifier, a zero potential difference between the two electrodes of the sensor, in order to cancel the influence of capacity (wiring or sensor). The control law is computed and applied via a dSpace T M card DS1103. For a significant deformation of the beam, the voltage amplifier is used with gain of 13. The magnitude of the Bode plot of the transfer function between the above input and output is given in Fig. 2. A photo of the experiment set-up is given in Fig. 4.

Length
f (t) = 1 + 120t 1 + t . ( 15
)
This frequency profile excites the first two modes (at f 1 = 16Hz and f 2 = 100Hz) and captures the variation of the excitation band.

Parameter

k 1 ξ 1 ω 1 k 2 ξ 2 ω 2
Value 2500 0.01 104 rad/s 10 5 0.5+0.01t 1+t 600 rad/s Table 2: LTV model parameters.

To attenuate the vibrations of the structure, the classic LTI solution, called Positive Position Feedback (PPF) controller [START_REF] Preumont | Vibration control of active structures: An introduction[END_REF], consists in taking the two modes into consideration for the controller design whatever the frequency profile of the exciting signal is. As a result, even if stability is ensured, performances are low. An LTV model is used to improve these results. 15

LTV model of the beam

A second order LTV model of type [START_REF] Adami | Time-varying notch filters for control of flexible structures and vehicles[END_REF] for which the most energetic modes have been considered is used for the beam:

H(s) = k 1 P 1 (s) + k 2 P 2 (s) (16)
where

     P 1 (s) = s 2 + 2ξ 1 ω 1 s + ω 2 1 P 2 (s) = s 2 + 2ξ 2 ω 2 + ξ2 ξ 2 1-ξ 2 2 s + ω 2 2 -ξ2 ω 2 1-ξ 2 2 ( 17
)
and the expression of P 2 (s) is such that H(s) has the following fondamental set of poles:

     λ 1,2 = -ξ 1 ω 1 ± jω 1 1 -ξ 2 1 λ 3,4 = -ξ 2 (t)ω 2 ± jω 2 1 -ξ 2 (t) 2 . ( 18
)
ξ 2 (t) is chosen to have the same dynamic as the frequency profile as shown in Table 2 along with the other parameters of the model. It ranges from 0.5 at the begining -to model the absence of excitation of the second mode to 0.01 when the second mode is excited. In order not to change the phase of the first mode, ξ 2 (t) does not start at 1.

LTV controller design

To design the controller, a left-coprime factorization (i) is written for ( 16)

H(s) = A L (s) -1 B L (s) (19) 16
where A L (s) is the (left) minimal polynomial of P 1 (s) and P 2 (s) (see, e.g., [19]):

A L (s) = (P 1 (s), P 2 (s)) l ; B L (s) = k 1 B 1 (s) + k 2 B 2 (s) (20)
B 1 (s), B 2 (s) are such that A L (s) = B 1 (s)P 1 (s) = B 2 (s)P 2 (s).
As the damping of the modes characterize the amplitude of the vibrations of the structure, the purpose of the active control of vibrations is to assign desired values ξ i to the dampings. The control system in Fig. 4 leads to the closed loop in Fig. 3, where:

• the reference signal r = 0;

• the disturbance signal d represents the contribution of the shaker in the sensor output signal;

• the control signal u = -v calculated by the controller gives an output y that "ideally" neutralize the shaker effect.

The controller is synthesized using the procedure described in Section 3 to reach the following objectives:

• Ensure the stability of the closed-loop system;

• Increase the damping of the two modes: ξ1 = ξ2 = 0.3;

• Keep the same natural frequencies;

• Ensure the causality of the controller;

• Do not excite the neglected modes.

For this, the poles of the closed-loop system are placed to:

     λ 1,2 = -ξ 1 ω 1 ± jω 1 1 -ξ 2 1 λ 3,4 = -ξ 2 ω 2 ± jω 2 1 -ξ 2 2 . (21)
As a consequence, A cl is chosen to be:

A cl (s) = (s 2 + 2 ξ1 ω 1 s + ω 2 1 )(s 2 + 2 ξ2 ω 2 s + ω 2 2 )Q(s) (22)
where Q(s) represents additional exponentially stable poles to ensure the causality of the controller and not affect the remaining modes. Giving the forms of H(s) and A cl (s), the factors S R (s) and R R (s) of the controller G(s) in equation ( 13) must have the following form:

     S R (s) = s 3 + s 2 (t)s 2 + s 1 (t)s + s 0 (t) R R (s) = r 3 (t)s 3 + r 2 (t)s 2 + r 1 (t)s + r 0 (t) (23)
Moreover, in order to ensure causal solutions G(s) in ( 13),

Q(s) = (s + 150)(s + 200)(s + 300). (24)

LTV controller computation and implementation

Next, the controller coefficients in ( 23) must be found by solving the Bézout equation ( 13) by identifying the coefficients of the two terms.

The computation of S R (s) and R R (s) is done using a symbolic computation tool. As the obtained coefficients of the two polynomials were too complicated, we proceeded to their first order Padé-type simplification in order to be easily used in simulations and real-time tests (Table III in Appendix).

First, notice that the obtained simplified expressions of the coefficients provide trajectories close the ones of the exacte expressions as shown in Fig. 12 of Appendix.

Next, the simplified controller provides a closed-loop dynamic very close to the desired one. Fig. 5 shows that the closed-loop poles obtained using the simplified controller are quite close to the desired poles. Notice that this kind of simplification can be carried out also when working with large systems (polynomial A L of higher degree) and leads to simple expressions of the coefficients of the regulator (as the ones in Table III of Appendix) which can be easily implemented to obtain a higher order regulator. This opens the way to applications which need regulators of higher order. To provide a causal implementation of the regulator, one canonical LTV state-space form is computed from the factors (2) [START_REF] Marinescu | An intrinsic algebraic setting for poles and zeros of linear time-varying systems[END_REF]. In our case the rightform is :

       0 1 0 0 0 1 -s 0 (t) -s 1 (t) -s 2 (t)        , B ctrl =        0 0 1        C ctrl = r 0 (t) -r 3 (t)s 0 (t) r 1 (t) -r 3 (t)s 1 (t) r 2 (t) -r 3 (t)s 2 (t) (25)

Control results

The designed LTV controller was both tested in simulation and on the real system.

Simulation results

The simulation results in Fig. 6 (a)-(b) show that the controller damps the two vibration modes at the level expected from the design. It has also been studied the effect of actuator saturation: when this is taken into account (Fig. 6 (a)), the damping action is less efficient. Robustness against high order dynamics has also been tested by injecting a white noise of frequency 150Hz on the measurement y. Fig. 7 shows that this noise is correctly filtered (at least not amplified) by the closed-loop.

Experimental results

Four comparisons were performed on the experimental benchmark presented in Section 4 (for which the saturation of the actuator cannot be avoided):

• the open-loop compared with the LTV closed-loop to illustrate the effect of such an active control (Fig. 8 • the proposed LTV controller compared with the two LTI controllers designed from its parameters frozen for t → 0 and t → +∞ (Fig. 9 Fig. 8 (a) shows that the control system ensures stability and good damping of the vibrations. However, we notice that the second mode is more damped than the first one. This is mainly due to physical constraints inherent to the system (independent of the controller used). Indeed, the static gains k 1 and k 2 reported in Table . 2 show that the actuator has more effect on the second mode than the first one. This makes the control of the first mode difficult, knowing the limitation of the actuator in practice. Of course, this is not the case in simulation with no actuator constraints as shown in the above section. Notice also that, in order to control the two modes with a classic control (like LQG, H ∞ ), the resulting controller must be at least of order 4 to have the same performance as our LTV controller of order 3. Notice also that the good results obtained in experimentation prove, in addition to performances, robustness against unmodeled dynamics. Indeed, on the one hand, the regulator is synthesized for only 2 modes of the system and, on the other hand, it is used on the real physical in its simplified form presented in Section 4.4 and Appendix.

Conclusions

This work validates on a real physical system the LTV pole placement approach [START_REF] Marinescu | Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems[END_REF]. Moreover, hints for the practical implementation like, e.g., the regulator truncation, have been proposed. This opens the way to industrial applications for which the LTV modelization and control were long time avoided because both of the lack of theoretical results and the complexity of 24 solutions and computations when such results exist. This LTV approach opens also the way to a new scheduling technique. Indeed, large operating domains can be covered with an LTV model for which some parameters are varied. This stands both for nonlinear systems (for which the linearization around given trajectories give LTV models) and for infinite dimensional ones as for the case treated here. As for this approach stability is directly ensured, the forthcoming effort will be concentrated on the way to choose/construct the scheduling variables in order to optimally capture the variations of the operating conditions.

The case of flexible structures will be further investigated by considering higher order systems like the multidimensional structures, e.g., vibrating plates.

Figure 1 :

 1 Figure 1: Modal deformations of clamped-free beam: -Mode 1, --Mode 2; • • • Mode 3; -. -Mode 4

Figure 2 :

 2 Figure 2: Clamped-free beam modeling: input-output transfer.

Figure 3 :

 3 Figure 3: LTV multi-input/multi-output closed-loop

Figure 4 :

 4 Figure 4: Experimental control-loop

Figure 5 :

 5 Figure 5: Closed-loop poles.(+ : desired poles, o : variation in time of the poles obtained with the simplified controller).

  loop [Balck] -LTV control [Grey] (a) Control with saturation.

  loop [Balck] -LTV control [Grey] (b) Control without saturation.

Figure 6 :

 6 Figure 6: Simulation results: Control with and without saturation.

21

 21

Figure 7 :

 7 Figure 7: Simulation results: robustness test (Up: Output and Noised Output; Down: Output with noised feedback and Output without noised feedback).

Fig. 8 (

 8 Fig. 8 (b) illustrates the advantage of the LTV controller over a conventional controller. Both techniques have similar effects on the first mode, but the PPF control amplifies the second mode instead of damping it. This is quite logical, because the PPF control was designed to control only the first

  Balck] -LTV control [Grey] (b) LTV controller Vs PPF controller.

Figure 8 :

 8 Figure 8: Experimental results: Comparaison LTV controller Vs Open-loop and PPF controller.

Fig. 9 (

 9 Fig. 9 (a) and (b) put into evidence the advantage of variation of the parameters of the LTV controller to continuously adapt to the disturbance.The same figures show that the first LTI controller mitigate well the first mode, but deteriorates progressively when the damping of the second one is decreasing. The second LTI controller does the opposite. The LTV controller combines the two advantages and ensures stability in a simple and direct way.

23

 23

  Balck] -LTV control[Grey] (a) LTV controller Vs LTI controller (LTV at t → 0).

4

  Sensor voltage [volt] Time [sec] LTI control [Balck] -LTV control [Grey] (b) LTV controller Vs LTI controller (LTV at t → ∞).

Figure 9 :

 9 Figure 9: Experimental results: Comparaison LTV controller Vs LTI controllers (LTV at t → 0) and (LTV at t → ∞).

Table 1 :

 1 beam cracteristics.

		Width Thikness Material Young module	Density
	30 cm	1 cm	1.56 mm	AU4G	78 GPa	2970 kg.m -3
	The following scenario is considered to reproduce the control problem
	with this benchmark. Assume that the shaker excites the beam with a
				14

93x25mm (300 x 300 DPI)

136x97mm (300 x 300 DPI)

39x28mm (300 x 300 DPI)

157x125mm (300 x 300 DPI)

141x180mm (300 x 300 DPI)

113x138mm (300 x 300 DPI)

197x112mm (300 x 300 DPI)

191x108mm (300 x 300 DPI)

194x111mm (300 x 300 DPI)

139x186mm (300 x 300 DPI)

162x125mm (300 x 300 DPI)