Wassim Hamidouche

Mickael Raulet

Olivier Déforges

MULTI-CORE SOFTWARE ARCHITECTURE FOR THE SCALABLE HEVC DECODER

Keywords: HEVC, Scalable HEVC, High level parallel processing and wavefront parallel processing

The scalable high efficiency video coding (SHVC) standard aims to provide features of temporal, spatial and quality scalability. In this paper we investigate a pipeline and parallel software architecture for the SHVC decoder. The proposed architecture is based on the OpenHEVC software which implements the high efficiency video coding (HEVC) decoder. The architecture of the SHVC decoder enables two levels of parallelism. The first level decodes the base layer and the enhancement layers in parallel. The second level of parallelism performs the decoding of both the base layer and enhancement layers in parallel through the HEVC high level parallel processing solutions, including tile and wavefront. Up to the best of our knowledge, it is the first real time and parallel software implementation of the SHVC decoder. On an Intel Xeon processor running at 3.2 GHz, the SHVC decoder reaches the decoding of 1600p enhancement layer at 40 fps for x1.5 spatial scalability with using six concurent threads.

INTRODUCTION

The high efficiency video coding (HEVC) standard was finalized in January 2013 by the Joint Collaborative Team on Video Coding (JCT-VC) as joint effort between ITU-T and ISO/IEC [START_REF] Sullivan | Overview of the high efficiency video coding standard[END_REF]. HEVC standard can reach the same subjective video quality as its predecessor H.264/AVC at about a half bitrate [START_REF] Ohm | Comparaison of the Coding Efficiency of Video Coding standards including High Efficiency Video coding (HEVC)[END_REF]. This gain is obtained thanks to new tools adopted in the HEVC standard, such as quadtree-based block partitioning, large transform and prediction blocks, accurate intra/inter predictions and the in-loop sample adaptive offset (SAO) filter. Moreover, HEVC standard was designed with a particular attention to complexity, where several steps can be easily peformed in parallel [START_REF] Chi | Parallel Scalability and Efficiency of HEVC Parallelization Approaches[END_REF][START_REF] Bossen | Hevc complexity and implemnation analysis[END_REF]. These tools allow to leverage multicore processors and achieve a real time encoding/decoding of high resolution videos (HD, 4K2K and 8K4K). The JCT-VC is currently developing the scalable extension of the HEVC standard (SHVC). The objective behind SHVC is to provide features of temporal, spatial and quality (SNR) scalability with a simple and efficient coding architecture [START_REF]Joint Call for Proposals on Scalable Video Coding Extensions of High Efficiency Video Coding (HEVC)[END_REF]. Since the temporal scalability is enabled in HEVC with a hierarchical temporal prediction structure, SHVC concentrates on spatial and SNR scalability. Several scalable solutions [START_REF] Shi | Spatially Scalable Video Coding for HEVC[END_REF][START_REF] Helle | A Scalable Video Coding Extension of HEVC[END_REF][START_REF] Chen | Scalable Video Coding Extension for HEVC[END_REF][START_REF] Zhao | Inter-layer Intra Mode Coding for the Scalable Extension of HEVC[END_REF] were proposed as a response to the SHVC call for proposal [START_REF]Joint Call for Proposals on Scalable Video Coding Extensions of High Efficiency Video Coding (HEVC)[END_REF]. The approved approach is based on multi-loop decoding structure (i.e. all intermediate layers need to be decoded) and uses the same technologies of HEVC with an inter-layer prediction to improve the coding efficiency. This solution allows a gain of 15%-30% in terms of rate-distortion compared to a simulcast coding solution. The HEVC standard defines three main concepts, including tile, slice and wavefront, that enable a high level parallel processing of both encoding and decoding [START_REF] Chi | Parallel Scalability and Efficiency of HEVC Parallelization Approaches[END_REF]. Tile and slice concepts break at their boundaries the dependencies of both intra predictions and the probabilities of the context-adaptive binary arithmetic coding (CABAC). This allows to encode/decode slices or tiles of one frame on separate cores. However, the intra prediction limitation and the initialization of the CABAC context decrease the coding performance in terms of rate distortion, especially for large number of tiles/slices per frame. Moreover, the in-loop filters cannot be performed in parallel at the tile/slice edges without extra control mecanisme. The wavefront parallel processing (WPP) solution was proposed to the HEVC standard in [START_REF] Clare | Wavefront parellel processing for HEVC Encoding and Decoding[END_REF]. The WPP concept enables the decoding of several coding tree block (CTB) rows in parallel. This is possible by the initialization of the CABAC context at the start of each CTB row. The overhead caused by this initialization is limited since the CABAC context at each CTB row is initialized by the CABAC context state at the second CTB of the previous CTB row. Therefore, the decoding of each CTU row can be carried out on separate threads with a minimum delay of two CTBs between adjacent CTB rows. Authors in [START_REF] Gudumasu | Real time SHVC software decoding with multi-threaded parallel processing[END_REF] proposed a real time and parallel implementation of the SHVC decoder. This decoder is CTB groups based parallelism and performs a real time decoding of 1080p50 enhancement layer (EL) for x1.5 spatial scalability on an Intel i7 processor with using 8 concurent threads.

In this paper we propose a pipeline and parallel architecture for the SHVC decoder [START_REF] Hamidouche | Pipeline and parallel architecture for the SHVC decoder[END_REF]. This architecture enables two levels of parallelism where the base layer (BL) and the EL frames are decoded in parallel thanks to the pipeline architecture. The parallel architecture, as a a second level of parallelism, performs the decoding of each frame in parallel with WPP solution. The SHVC decoder is based on the Open-HEVC software [START_REF]Open source HEVC decoder (OpenHEVC)[END_REF] which implements a conforming HEVC decoder. On an Intel Xeon processor running at 3.2 GHz, the SHVC decoder reaches the decoding of 1600p EL at 40 frames per second (fps) for x1.5 spatial scalability with using 6 concurent threads. This paper is organized as follows. Section 2 describes the architecture of the OpenHEVC decoder enabling wavefront parallel processing solution. The pipeline and parallel architecture of the SHVC decoder is investigated in Section 3. The performance of the SHVC decoder is assessed and discussed in Section 4, and finally Section 5 concludes this paper.

PARALLEL SINGLE LAYER HEVC DECODER

WPP solution under the OpenHEVC decoder

The architecture of the OpenHEVC software is quite simple and is based on a coding tree unit (CTU) decoding. The proposed WPP implementation performs all decoding steps at the level of the CTU in a single pass. Figure 1 shows an overview of the OpenHEVC architecture. The hls decode row function decodes all CTUs of one row in the slice. It browses in raster scan the CTUs within the row and calls the recursive function hls coding tree to decode each CTU. There are specific functions that handle the prediction and the transform of the prediction and the transform units, namely hls prediction unit and hls transform unit, respectively. Once all coding units (CU) within a CTU are decoded, the deblocking filter (DF) and then the SAO filter are performed on the decoded CTU. However, when performing the DF of the current CTB, the right and the down CTB neighborhoods are not available (ie. not yet decoded). Therefore, the right and down edges of the current CTB are filtered when its right and down CTBs are being filtered, respectively. In this solution, the DF and the SAO filters are delayed with one CTU and one CTU row for only the right and the down edges of a CTB, respectively. The WPP extension in the OpenHEVC architecture is straight forward. This is possible by running the hls decode row function on separate threads to decode several adjacent CTU rows in parallel. The delay in terms of CTU, noted d, required by the wavefront solution between two adjacent CTU rows is managed by an integer type array shared by all threads. The i th value of the array is used to count the number of decoded CTUs within the i th CTU row. Thus, the hls decode row function increments the related array value for each decoded CTU and decodes a new CTU only if the d next CTUs of the previous CTU row are decoded.

Analytical performance of the WPP solution

The analytical speedup of the WPP solution represents the upper bound of its experimental performance. It also gives an idea on the parameters that control the performance of the wavefront solution. Let us consider x the number of CTB columns, y the number of CTB rows and d the delay in terms

n = min nb cpu threads, x d (1)
where d ∈ N + and nb cpu threads is the number of threads selected to decode the video sequence.

The analytical speedup γ is derived as follows:

γ =    xy xy n +d(n-1) , if y mod n = 0 xy x y n +d((x mod n)-1) , if y mod n = 0 (2)
where x, y, n ∈ N + . We can notice from Equation 2 that the analytical speedup depends on three main parameters: the video resolution (x and y), the effective number of threads (n) and the CTB delay between two adjacent CTB rows (d). In addition, a large division remainder between the number of CTB rows and the effective number of threads (y mod n) also decreases the performance of the wavefront solution. The division remainder corresponds to the inactive threads waiting for the decoding of the last CTB rows of the frame.

PIPELINE AND PARALLEL SHVC DECODER

Overview of the SHVC standard

In the case of spatial scalability with two layers, the SHVC encoder consists of two encoders, one for each layer. The BL HEVC encoder encodes the downsampled version of the original video and feeds the second encoder with the decoded picture and the corresponding motion vectors (MVs). The EL Fig. 2. Pipeline and parallel architecture of the SHVC decoder for spatial scalability with two layers SHVC encoder encodes the original video with using the upsampled BL picture and its upscaled MVs for inter-layer prediction. Concerning the SNR scalability, the encoding process remains unchanged except that the BL picture and its MVs are not upsampled and upscaled, respectively, at the EL encoder. The SHVC standard also supports a BL coded with H.264/AVC encoder. In this case, only the decoded BL, without its MVs, is provided to the SHVC EL encoder.

Real time and pipeline SHVC decoder

In this section we introduce a pipeline and parallel architecture for the SHVC decoder. The first step consists in extending the OpenHEVC software to support the new operations introduced in the SHVC standard, namely upsampling of the decoded BL frame, scaling its MVs and managing the upsampled BL picture as an additional reference picture in the EL decoder. Thus, the SHVC decoder consists of l instances of the OpenHEVC decoder, one HEVC decoder for each layer, with l = 1, ..., L the number of layers. The SHVC decoder enables two levels of parallelism. The first level performs the decoding of the BL and the EL frames simultaneously on separate threads. For each decoder, the second level of parallelism carries out the decoding of both the BL and the EL frames in parallel through the HEVC high level parallel processing solutions. Moreover, the upsampling of the BL and the upscaling its MVs are also carried out in parallel. In fact, when parallel decoding of the EL frame is enabled, the CTB rows of the BL picture and the corresponding MVs are upsampled and upscaled in parallel. Figure 2 summarizes the proposed pipeline and parallel architecture of the SHVC decoder for spatial scalability with two layers (l = 2).

As illustrated in Figure 2, two instances of OpenHEVC decoder are created and run on separate threads. These two decoders correspond to the HEVC BL decoder and SHVC EL decoder, respectively. The parser running on the main thread parses the SHVC bitstream and feeds the two decoders with the corresponding frame (access units). For the first decoded frame, the EL decoder waits until the BL frame has been decoded and then the EL frame i is simultaneously decoded with the next BL frame (frame i + 1). To limit the BL buffer to one frame, the BL decoder might not decode the frame i + 2 since the EL frame i has not yet been decoded.

RESULTS AND DISCUSSIONS

Experimental configuration

We run the SHVC decoder on a computer fitted with 6 cores Intel Xeon processor. Table 1 summarizes the system and software configurations used to carry out the experiments.

Concerning the video coding configuration, the common test conditions defined in the HEVC standard [START_REF] Bossen | Scalable high efficiency video coding test model 3 (SHM 3)[END_REF] were considered. In order to show the performance for larger resolution video, we added to the set of conformance video sequences two 3840 × 2160 video sequences from the STV High Definition Multi Format Test Set. All the selected video sequences were encoded with SHVC reference software [START_REF]SHVC Reference Software[END_REF] in two layers (l = 2) and two scalability configurations were considered: x2 and x1.5. The SHVC video sequences were coded in low delay coding configuration with enabling the wavefront feature where the delay d = 2. The quantization parameter (QP) of the BL was set to 27 and 32, while the QP of the EL is equal to the BL QP minus 2. The performance of the proposed pipeline and parallel SHVC decoder is compared to the sequential SHVC decoder: the decoding of the BL and the EL frames are carried out in sequential order.

The number of threads used to decode the BL and the EL are noted n and m, respectively. In addition to the single thread configuration (m = n = 1), the number of threads in the sequential SHVC decoding configuration is set as follows n ∈ {2, 3, 4, 5, 6} with m = n. The corresponding decoding configuration in the pipeline SHVC architecture is the follow-

Results

Figure 3 illustrates the speedup of the WPP implementation under the single layer OpenHEVC decoder for different video resolutions. The experimental speedup is compared to the upper bound performance of the wavefront solution computed with Equation 2. We can notice that the experimental speedup is close to the optimal speedup, especially when the number of threads is below 5. With using 6 threads, the performance of the proposed implementation decreases and reaches an accelerating factor of 4.5 for 3840x2160 resolution videos, instead of the upper bound value of 5.5. This is because we use the maximum number of CPU cores including the one used by the operating system. Figure 4 shows the performance of the pipeline SHVC architecture in terms of decoding time per frame for the three main decoding steps: BL decoding, upsampling the BL picture and the EL decoding. We can notice that the decoding time of the three steps remain constant with using one thread and two threads (two decoders in parallel and n = m = 1). In these two decoding configurations each decoding step is performed on a single thread. However, the whole decoding time, in the later configuration, decreases by the BL decoding time since the decoding of the BL and the EL are performed in parallel in the pipeline architecture. For number of threads between 3 and 5, we only increases the number of threads for the EL, since the decoding time of the EL is higher than the decoding time of the BL and the WPP solution is more efficient with large video resolutions. Figures 5 compares the performance of sequential and pipeline SHVC decoders in terms of decoding frame rate for different video resolutions. We can notice that the pipeline architecture is more efficient for videos of low resolutions (EL 1080p). In fact, the pipeline parallelism enables adjusting the number of threads for each layer by providing more threads to the EL for which the WPP solution is more efficient. However, for high resolution videos (EL 1600p and 2160p) the performance of the pipeline and sequential architecture is similar. The performance of the pipeline architecture decreases when the num-Fig. 4. Decoding time performance of the SHVC decoder ber threads is equal to 2 (1,1). Indeed, the decoding time of the EL including the upsampling is much higher than the decoding time of the BL. Therefore, running in parallel the BL and the EL decoders both on a single thread is less efficient than the WPP with 2 threads for each step in sequence. The pipeline solution could provide better performance for SHVC bitstreams with more than one EL since the decoding time of the ELs is similar.

CONCLUSION

In this paper we proposed a multiple threads architecture for the SHVC decoder. This decoder enables two levels of parallelism where the decoding of the SHVC layers is pipelined, and each layer is decoded in parallel based on the HEVC high level parallel processing solutions. The first end-to-end video demonstration using the proposed real time SHVC decoder within the GPAC player was presented in the 106 th MPEG meeting [START_REF] Hamidouche | A scalable HEVC demonstration within GPAC player[END_REF].

Fig. 1 .

 1 Fig. 1. Blocks diagram of the OpenHEVC decoder

Fig. 3 .

 3 Fig. 3. Speedup performance of the WPP solution (QP=32)

Fig. 5 .

 5 Fig. 5. Decoding frame rate performance of the SHVC decoder (QP=32)

Table 1 .

 1 Configuration of the experiments

	System		Software
	Processor	Intel Xeon Compiler	GCC-4.6
		E5-1650	OS	Ubuntu 12.04
	ISA	X86-64	Kernel	3.5.0-34
	Clock frequency 3.2 GHz	OpenHEVC cff4b48a94
	Level 3 cache	12 MB	release	(based on HM11.0)
	Cores	6


Arthur Fleig
email: arthur.fleig@uni-bayreuth.de

Lars Grüne
email: lars.gruene@uni-bayreuth.de

Roberto Guglielmi
email: &roberto.guglielmi@uni-bayreuth.de

Some results on Model Predictive Control for the Fokker-Planck equation* Extended Abstract

A Model Predictive Control scheme is applied to track the solution of a Fokker-Planck equation over a fixed time horizon. We analyse the dependence of the total cost functional on several parameters of the algorithm, in particular on the prediction horizon, on the regularization parameter, and on the sampling time. Comparison among different numerical simulations show valuable improvements by properly tuning the scheme's parameters. Our numerical study is complemented by a theoretical controllability analysis explaining the superior performance of controls with time and space dependence.

I. INTRODUCTION A. Model Predictive Control

Model Predictive Control (MPC), also known as Receding Horizon Control (RHC), is a control method that computes a feedback law by iteratively solving optimal control problems on finite time horizon, instead of coping directly with an optimal control problem on an infinite time interval (see the monographs [START_REF] Rawlings | Model Predictive Control: Theory and Design[END_REF] and [START_REF] Grüne | Nonlinear Model Predictive Control. Theory and Algorithms[END_REF] for an introduction to MPC). MPC can be briefly described as follows: consider a discrete time control system of the form z(k + 1) = g(z(k), u(k)) , z(0) = z 0 [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF] with k ∈ N 0 , state z(k) ∈ X and control u(k) ∈ U for suitable state and control constraint sets X ⊂ Z and U ⊂ U, where the state space Z and the control space U are metric spaces.

The MPC scheme constructs a feedback law µ : X → U for the closed loop system

z µ (k + 1) = g(z µ (k), µ(z µ (k))) (2)
through the following steps: 0. Given an initial value z µ (0) ∈ X, fix the length of the receding horizon N and set n = 0. 1. Initialize the state z 0 = z µ (n) and minimize the functional

J N (z 0 , u) := N-1 ∑ k=0 l(k, z(k), u(k)) (3)
subject to [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF]. Let u * ∈ U N be the resulting optimal control and set µ(z µ (n)) := u * (0).

The application of MPC to infinite dimensional systems governed by Partial Differential Equations (PDEs) goes back to the work [START_REF] Ito | Receding horizon optimal control for infinite dimensional systems[END_REF], where terminal constraints and control Lyapunov functionals were added as terminal costs to garantee the stability of the closed loop solution to the finite horizon problem. Furthermore, MPC schemes were applied to parabolic PDEs with either distributed or boundary control in [START_REF] Dubljevic | Boundary Predictive Control of Parabolic PDEs[END_REF] and [START_REF] Dubljevic | Predictive control of parabolic PDEs with state and control constraints[END_REF]. However, the construction of suitable terminal regions and costs is in general a challenging task; for this reason, in most industrial applications an MPC scheme without terminal constraints is adopted [START_REF] Qin | A survey of industrial model predictive control technology[END_REF]. Indeed, under suitable conditions the optimization objective will force the optimal trajectories to end up in a suitable terminal region (see, for example, [START_REF] Grüne | NMPC without terminal constraints[END_REF]). Several theoretical results on the MPC setting without stabilizing terminal constraints have been collected in [START_REF] Grüne | Nonlinear Model Predictive Control. Theory and Algorithms[END_REF]. In the case of a linear parabolic equation with either distributed or boundary control, a rigorous analysis of the dependence of the receding horizon on the cost functional and the system parameters has been developed in [START_REF] Altmüller | Distributed and boundary model predictive control for the heat equation[END_REF]. Furthermore, a comparison between the qualitative behaviour of an MPC scheme for a heat equation for different types of boundary control has been carried out in [START_REF] Altmüller | A comparative stabiliy analysis of Neumann and Dirichlet boundary MPC for the heat equation[END_REF].

In a PDE context, the solution of the equation is, of course, defined in continuous time. In a continuous time setting, the discrete times k from Section I-A indicate the re-optimization times and the map g from ( 1) can be obtained by sampling the PDE model in time. The state z(k) in the discrete time model then represents the infinite dimensional state of the PDE model at time t = t k ; for details see, e.g., [2, Section 3].

II. OPTIMAL CONTROL AND THE FOKKER-PLANCK EQUATION

Our interest for studying the Fokker-Planck equation mainly stems from its connection with the optimal control of the Probability Density Function (PDF) associated to stochastic processes. For explaning this connection, let us consider the continuous time stochastic process described by the (Itô) stochastic differential equation

dX t = b(X t ,t; u)dt + σ (X t ,t)dW t , (4)
where t ∈ [0, T E ] for a fixed terminal time T E > 0 and the state variable X t ∈ R is subject to deterministic infinitesimal increments of the drift term b and to random infinitesimal increments dW t of a Wiener process. In the setting considered here the control function acts through the term b and may either depend on space and time or may be merely time dependent as in [START_REF] Annunziato | Optimal control of probability density functions of stochastic processes[END_REF], [START_REF] Annunziato | A Fokker-Planck-based control framework for multidimensional stochastic processes[END_REF].

In deterministic dynamics, the optimal control is achieved by finding the control law u that minimizes a given objective given by a cost functional J(X, u).

In the non-deterministic case of (4), the state evolution X t represents a random variable. Therefore, when dealing with stochastic optimal control, usually the average of the cost function is considered [START_REF] Fleming | Deterministic and Stochastic Optimal Control[END_REF]. In particular, the cost functional usually is of the form

J(X, u) = E T E 0 L(t, X t , u(t))dt + ψ(X T E ) ,
for suitable running cost L and terminal cost ψ.

On the other hand, the state of a stochastic process can be characterized by the shape of its statistical distribution which is represented by the Probability Density Function (PDF). Therefore, a control methodology defined via the PDF would provide an accurate and flexible control strategy that could accommodate a wide class of objectives, cf. also [START_REF] Brockett | New issues in the mathematics of control[END_REF]Section 4]. For this reason, in [START_REF] Forbes | Control design for first-order processes: shaping the probability density of the process state[END_REF], [START_REF] Jumarie | Tracking control of nonlinear stochastic systems by using path cross-entropy and Fokker-Planck equation[END_REF], [START_REF] Karny | Towards Fully Probabilistic Control Design[END_REF], [START_REF] Wang | Robust Control of the Output Probability Density Functions for Multivariable Stochastic Systems with Guaranteed Stability[END_REF] probability density function control schemes were proposed, where the cost functional depends on the PDF of the stochastic state variable. In this way, a deterministic objective results and no average is needed.

As shown in [START_REF] Annunziato | Optimal control of probability density functions of stochastic processes[END_REF], [START_REF] Annunziato | A Fokker-Planck-based control framework for multidimensional stochastic processes[END_REF], the PDF associated to the stochastic process (4) satisfies a Fokker-Planck equation [START_REF] Risken | The Fokker-Planck Equation: Methods of Solution and Applications[END_REF] with a control acting through the divergence term. This is a partial differential equation of parabolic type with Cauchy data given by the initial PDF distribution. It can be expressed as follows

∂ t f (x,t) -1 2 ∂ 2 x σ (x,t) 2 f (x,t) + ∂ x (b(x,t; u) f (x,t)) = 0 , f (x,t k ) = ρ k (x) , (5)
on the domain Q k := Ω × (t k ,t k + T ), where Ω ⊂ R d , d ∈ N, t k := kT with a sampling time T > 0, and some given initial distribution ρ k . Notice that in general, the space domain in (5) is R d instead of Ω. However, if localized SDEs are under consideration, or if the objective is to keep the PDF within a given compact set of Ω and the probability to find X t outside of Ω is negligible, then it is reasonable to consider bounded Ω ⊂ R d with homogeneous Dirichlet boundary conditions. Given a desired distribution f d : Ω × R → R, a minimization problem of type (3) subject to (5) that attempts to keep f as close as possible to f d using a control that is only time dependent can be posed as

min u J( f , u) := 1 2 N-1 ∑ k=0 f -f d 2 L 2 (Q k ) + λ |u(t k )| 2 (6)
for some positive constant λ , also called Tychonov regularization. For a control depending on time and space, the objective becomes

min u J( f , u) := 1 2 N-1 ∑ k=0 α f -f d 2 L 2 (Q k ) + λ u 2 L 2 (Q k ) , (7)
where the weight α > 0 is introduced solely for numerical reasons.

Setting z(k) = f (•,t k ), ( 6) and ( 7) can be rewritten as (3) with

l(k, z, u) = 1 2 f z -f d 2 L 2 (Q k ) + λ 2 |u(t k )| 2 and l(k, z, u) = α 2 f z -f d 2 L 2 (Q k ) + λ 2 u 2 L 2 (Q k ) ,
respectively. Here, f z denotes the solution of ( 5) with initial time t k and initial distribution ρ k = z.

III. MPC FOR THE FOKKER-PLANCK EQUATION

In this talk we are going to present numerical results as well as first steps towards a theoretical analysis of MPC applied to the Fokker-Planck equation. Probably the first papers which applied MPC to the Fokker-Planck equation were [START_REF] Annunziato | Optimal control of probability density functions of stochastic processes[END_REF] in the one dimensional case and [START_REF] Annunziato | A Fokker-Planck-based control framework for multidimensional stochastic processes[END_REF] in higher dimensions, both with the purpose to track a (smooth) target trajectory f d . The particular type of MPC scheme in these references uses the horizon N = 2 and the functional (3) with

J(z, u) = 1 2 z(1) -f d (•,t 1 ) 2 L 2 (Ω) + λ 2 |u| 2 .
The numerical results we are going to present in this talk extend these results in various ways: on the one hand, we use the cost functionals ( 6) and [START_REF] Dubljevic | Boundary Predictive Control of Parabolic PDEs[END_REF] for implementing longer prediction horizons N > 2 in MPC. Particularly, we investigate the interplay between N and the sampling time T and its impact on the quality of the solution. for the tracking of the PDF of the one dimensional Ornstein-Uhlenbeck process with b(x,t, u) = ux and σ (x,t) = 0.8 in equation ( 4). The initial and target PDF used are

ρ 0 (x) = 1 √ 2π • 0.1 2 exp - x 2 2 • 0.1 2 and f d (x,t) = exp -[x-2 sin(πt/5)] 2 2•0.2 2 √ 2π • 0.2 2
, respectively. Furthermore, the parameters in the objective functionals ( 6) and ( 7) are given by λ = 0.1 and α = 100. Note that Ω is chosen large enough such that the error made from disregarding R \ Ω is negligible. Obviously, for this example the controlled PDF (in solid blue) tracks the desired PDF (in dashed red) much better for larger prediction horizon N.

On the other hand, we investigate the improvements which can be achieved when the control u is chosen time and space dependent. From a control point of view, this corresponds to a control structure which has both state dependent (i.e., feedback) character but may also vary with time. However, since MPC yields a (discrete time) feedback law [12, Section 3], the time dependence of u is actually induced via the dependence on the evolution of the PDF f , i.e., via a dependence on the state of the Fokker-Planck equation ( 5). The additional dependence on the state x of (4) allows for a significant increase of the quality of the tracking of the MPC feedback, which becomes visible by comparing Figure 5 with state and time dependent u = u(t, x) with the merely time dependent u = u(t) in Figure 3. The simulations shown in the figures have been obtained by using space discretization as in [START_REF] Annunziato | Optimal control of probability density functions of stochastic processes[END_REF] and explicit Euler discretization in time. The optimization problem has been solved with the projected gradient method and the Newton method with BFGS Hessian approximation.

On the theoretical side, in the talk we explain the role of controllability properties of the Fokker-Planck equation for obtaining good tracking results. We explain how the theoretical controllability results for the Fokker-Planck equation [START_REF] Blaquiere | Controllability of a Fokker-Planck equation, the Schrödinger system, and a related stochastic optimal control[END_REF], [START_REF] Porretta | On the planning problem for a class of mean field games[END_REF] imply that a control dependent both on time and space allows for the highly precise tracking via MPC feedback laws illustrated in Figure 5.

Fig. 1 :

 1 Fig. 1: Ornstein-Uhlenbeck (state) Sampling time T = 0.05, N = 2, u = u(t), t = 1.2

  ,

Fig. 2 :Fig. 3 : 2 Fig. 4 :

 2324 Fig. 2: Ornstein-Uhlenbeck (control) Sampling time T = 0.05, N = 2, u = u(t)

Fig. 5 :

 5 Fig. 5: Ornstein-Uhlenbeck Sampling time T = 0.01, N = 16, u = u(x,t), t = 1.2, α = 100

*This work was partially supported by the EU under the 7th Framework Program, Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN SADCO, GA number 264735-SADCO.


Lars Grüne

Mario Zanon

Periodic optimal control, dissipativity and MPC -Extended Abstract

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

Economic MPC is a variant of model predictive control (MPC) in which the objective consists in directly optimizing a given performance index as opposed to tracking a given reference.

Proving stability of economic MPC schemes is hard, as the stage cost (x, u) does in general not have a minimum on the trajectory the system converges to. The idea of rotating the cost using the Lagrange multipliers λ has been proposed in [4] in order to prove stability. The proof relies on an equivalent auxiliary MPC scheme with a rotated stage cost that has a stationary point at the optimal steady state. The rotated stage cost is obtained by adding the term λ T x -λ T f (x, u) to the stage cost. In [START_REF] Amrit | Economic optimization using model predictive control with a terminal cost[END_REF] this idea has been extended to a nonlinear rotation, given by a function λ(x). This generalization is equivalent to the systems theoretic notion of strict dissipativity [START_REF] Willems | Dissipative dynamical systems. I. General theory[END_REF], [START_REF]Dissipative dynamical systems. II. Linear systems with quadratic supply rates[END_REF] and allows one to rotate and convexify the stage cost of the auxiliary MPC scheme. For a given system and stage cost, if there exists a function λ(x) that satisfies a strict dissipativity property, then stability of the MPC scheme follows.

A first extension of this framework to periodic systems has been proposed in [START_REF] Zanon | A Lyapunov function for periodic economic optimizing model predictive control[END_REF], where the Lagrange multipliers λ k of a periodic optimal trajectory have been used to rotate the cost with a linear term. In this paper, we propose different ways of extending the notion of dissipativity to the periodic case in order to both rotate and convexify the stage cost of the auxiliary MPC scheme, thus proving stability of periodic economic MPC schemes for a more general class of systems.

II. SETTING

We consider discrete time nonlinear systems governed by the dynamics

x(k + 1) = f (x(k), u(k)) (1)
Research partially supported by the EU under the 7th Framework Program, Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN, GA number 264735-SADCO. The paper was written while M. Zanon visited the University of Bayreuth during his SADCO secondment.

L. Grüne is with the Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany, e-mail: lars.gruene@uni-bayreuth.de.

M. Zanon is with the Optimization in Engineering Center (OPTEC), KU Leuven, Belgium, e-mail: mario.zanon@esat.kuleuven.be.

with f : X × U → X. Solutions for initial value x 0 and control sequence u are denoted by x u (k, x 0 ).

For given state and control constraints sets X ⊂ X, U ⊂ U , each initial value x 0 ∈ X and any N ≥ 1 we denote the admissible control sequences by

U N (x 0 ) := {u(•) ∈ U N | x u (k + 1, x 0 ) ∈ X, u(k) ∈ U ∀ k = 0, . . . , N -1}.
For a stage cost : X × U → R, we consider the finite horizon functional

J N (x, u(•)) := N -1 k=0 (x u (k, x), u(k))
and the infinite horizon averaged functional

J ∞ (x, u(•)) := lim sup K→∞ 1 K J K (x, u(•)).
which are well defined for all u(•) ∈ U N (x) or u(•) ∈ U ∞ (x), respectively. Given an initial value x MPC (0) ∈ X, the basic model predictive control (MPC) scheme works as follows:

(i) set n := 0 (ii) minimize J N (x MPC (n), u(•)) over all control sequences u(•) ∈ U N (x MPC (n)) and denote the optimal sequence by u (•) (iii) set x MPC (n + 1) := f (x MPC (n), u (0)), u MPC (n) := u (0), n := n + 1 and go to (ii) Since the stage cost is not of tracking type (i.e., does not necessarily penalize the distance to a pre-specified equilibrium) this MPC scheme is often termed economic MPC [START_REF] Amrit | Economic optimization using model predictive control with a terminal cost[END_REF], [START_REF] Angeli | On average performance and stability of economic model predictive control[END_REF]. The scheme presented here does not use terminal constraints or costs. Often, such additional devices are added. In this talk, we either consider the scheme without terminal constraints or the scheme in which the minimization in (ii) is performed under the additional terminal point constraint. For other ways of chosing terminal conditions see [START_REF] Amrit | Economic optimization using model predictive control with a terminal cost[END_REF], [START_REF] Angeli | On average performance and stability of economic model predictive control[END_REF].

The classical notion of (strict) dissipativity [START_REF] Willems | Dissipative dynamical systems. I. General theory[END_REF], [START_REF]Dissipative dynamical systems. II. Linear systems with quadratic supply rates[END_REF] has recently gained renewed interest in the context of economic MPC.

Definition 2.1: The system (1) is called strictly dissipative with respect to a steady state (x s , u s ) ∈ X × U of (1) for supply rate (x, u) -(x s , u s ) if there exists a storage function λ : X → R and a function ρ ∈ K ∞ such that the inequality

(x, u) -(x s , u s ) + λ(x) -λ(f (x, u)) ≥ ρ( x -x s )
holds for all x ∈ X and u ∈ U .

If a system together with a stage cost is strictly dissipative, then this has several consequences:

• The system is optimally operated at steady state [6].

This means that for all initial values x ∈ X feasible control sequences the inequality

lim inf K→∞ 1 K J K (x, u) ≥ (x s , u s )
holds.

• For economic MPC with terminal constraint, the averaged performance J ∞ (x MPC (0), u MPC ) equals (x s , u s ) and the steady state x s is asymptotically stable for the closed loop solutions [4], [START_REF] Angeli | On average performance and stability of economic model predictive control[END_REF]. • For economic MPC without terminal constraint, the averaged performance J ∞ (x MPC (0), u MPC ) equals (x s , u s ) + ε(N ) and -under an exponential turnpike property which in turn is implied by dissipativity and suitable controllability properties [START_REF] Damm | An exponential turnpike theorem for averaged optimal control[END_REF] -the closed loop solutions converge to a neighborhood of x s with radius ε(N ), [5]. Here ε(N ) is an error term satisfying

ε(N ) → 0 as N → ∞.
It is well known that the optimal value is not necessarily attained at an equilibrium. Particularly, it may happen that periodic orbits exhibit smaller average values than any feasible equilibrium, see, e.g., [START_REF] Angeli | On average performance and stability of economic model predictive control[END_REF]Section VII] or our examples, below. For this reason, in the next section we discuss two variants of the dissipativity notion which are adapted to characterizing periodic orbits.

III. PERIODIC DISSIPATIVITY NOTIONS

We first define what we mean by a periodic orbit. Definition 3.1: A set of points Π = {x p 0 , . . . x p P }, P ≥ 1, is called a feasible periodic orbit with control sequence u p 1 , . . . , u p P -1 if x p k ∈ X, k = 1, . . . , P , u p k ∈ U, k = 1, . . . , P -1, x p 0 = x p P and

x p k+1 = f (x p k , u p k ) for k = 0, . . . , P -1.
The number P is called the period of the orbit Π and if (x k , u k ) = (x l , u l ) for all k, l = 0, . . . , P -1 with k = l then P is called the minimal period of Π. Note that in our terminology an equilibrium is a periodic orbit with period P = 1.

The first extension of strict dissipativity to periodic orbits is a generalization of the periodic strong duality from [START_REF] Zanon | A Lyapunov function for periodic economic optimizing model predictive control[END_REF]. Definition 3.2: A periodic orbit Π with period P is called strictly dissipative with periodic storage function, if there exist storage functions λ 0 , . . . , λ P -1 : X → R and a function ρ ∈ K ∞ such that the inequalities

L(x, u) := (x, u) -(x p k , u p k ) + λ k (x) -λ k+1 (f (x, u)) ≥ ρ( x -x p k ) (2)
hold for all x ∈ X, all u ∈ U and all k = 0, . . . , P -1.

It is easily seen that this definition is equivalent to Definition 2.1 for P = 1. Example 3.3: Consider the 1d nonlinear dynamics

f (x, u) = 0.9 x + x |x| u,
and the stage cost

(x, u) = (x -2)(x -1)(x + 1)(x + 2) + (u -10) 2 .
The optimal trajectory is periodic with period P = 2 and can be computed numerically: Π = {x p 0 , -x p 0 } = {1.6715, -1.6715} and u p 1 = u p 0 = 0.1x p 0 . Using λ 1 (x) = -λ 0 (x) = γx, γ = 19.6502, we obtain

L 0 (x, u) := (x, u) + λ 0 (x) -λ 1 (f (x, u)) = (x -2)(x -1)(x + 1)(x + 2) + u 2 + 0.1γx -γu, L 1 (x, u) := (x, u) + λ 1 (x) -λ 0 (f (x, u)) = (x -2)(x -1)(x + 1)(x + 2) + u 2 -0.1γx + γu.
The functions L k (x, u) satisfy the strict dissipation inequalities [START_REF] Angeli | On average performance and stability of economic model predictive control[END_REF].

Another definition can be obtained by considering the Pstep system with dynamics defined by

f P (x, ũ) :=      x ũ(P, x 1 ) x ũ(1, x 1 )
. . .

x ũ(P -1, x 1 )      , (3)
for x = (x 1 , . . . , x P ) ∈ X P and ũ = (u 0 , . . . , u P -1 ) ∈ U P . Then for every a periodic orbit Π of (1) and every k ∈ {1, . . . , p} the point

xp [k] = (x p k , . . . , x p P , x p 1 , . . . , x p k-1 ) ∈ X P (4)
is an equilibrium of (3) for the control ũp

[k] = (u p k , . . . , u p P , u p 1 , . . . , u p k-1 ) ∈ U P . Definition 3.4: A periodic orbit Π with period P is called P -step strictly dissipative, if there exist a storage function λ : X P → R and a function ρ ∈ K ∞ such that for some k ∈ {1, . . . , P } the quantity

L(x, ũ) := ˜ (x, ũ) -˜ (x p [k] , ũp [k] ) + λ(x) -λ(f P (x, ũ)),
with stage cost ˜ (x, ũ) = P -1 k=0 (x k , u k ), and equivalently ˜ (x p , ũp ) =

P -1 k=0 (x p k , u p k ), satisfies L(x, ũ) ≥ ρ( x -xp [k] ),
for all x ∈ X P and all feasible u ∈ U P . Note that this definition is exactly Definition 2.1 applied to the system (3).

The cost function associated with this definition has a special structure, i.e. it is the sum of P terms (x k , u k ). This implies that also function λ(x, ũ) must have the same structure, i.e. λ(x, ũ) = P -1 k=0 λ k (x k , u k ). Thus, in many cases, Definition 3.4 is equivalent to checking Definition 3.2 on a full period. The two definitions are however not equivalent, as shown by the following example.

Example 3.5: Consider the system x + = -x and the cost

(x) = x 2 for x ≥ 0 0 otherwise .
This system satisfies Definition 3.4 with P = 2 and Π = {0, 0} for λ ≡ 0 but not Definition 3.2 for λ k ≡ 0. This example also shows that Definition 3.4 can be useful to check dissipativity in the steady state case, as this system does not satisfy the standard steady state Definition 2.1.

A more general definition can be obtained if, for a periodic orbit Π, we define dist(x, Π) := min k=0,...,P -1 x -x p k . Moreover, dist(x, Π) := P -1 k=0 dist(x k , Π). Definition 3.6: A periodic orbit Π with period P is called P -step strictly dissipative with respect to a set, if there exist a storage function λ : X P → R and a function ρ ∈ K ∞ such that, for some k ∈ {1, . . . , P }, the quantity

L(x, ũ) := ˜ (x, ũ) -˜ (x p [k] , ũp [k] ) + λ(x) -λ(f P (x, ũ)), with stage cost ˜ (x, ũ) = P -1 k=0 (x k , u k ), and equivalently ˜ (x p , ũp ) = P -1 k=0 (x p k , u p k ), satisfies L(x, ũ) ≥ ρ(dist(x, Π)),
for all x ∈ X P and all feasible u ∈ U P . Note that this definition coincides with the one given in [START_REF] Amrit | Economic optimization using model predictive control with a terminal cost[END_REF].

Example 3.7: Consider the 1d dynamics

f (x, u) = -x + u
and the stage cost

(x, u) = (x -2)(x -1)(x + 1)(x + 2) + u 2 -2x + u.
The periodic optimal trajectory is given by Π = {x p 0 , x p 1 } = { √ 10/2, -√ 10/2} and u p 0 = u p 1 = 0. Using Definition 3.6, with x = [x 0 , x 1 ], ũ = [u 0 , u 1 ], λ(x) = x 0 + x 1 , and using x 1 = f (x 0 , u 0 ), we obtain

L(x, ũ) = 2x 4 0 -10x 2 0 -4u 2 0 + 25 2 + u 4 0 -4u 3 0 x 0 + 6u 2 0 x 2 0 -4u 0 x 3 0 + 10x 0 u 0 + u 2 1 .
Obviously, this expression becomes minimal in u 1 for u 1 = 0. In (x 0 , u 0 ), an analysis with MAPLE reveals that the expression has two global minima at (x 0 , u 0 ) = (± √ 10/2, 0) at which the value 0 is attained. This implies the desired existence of ρ.

In the previous example, the generalization of the dissipativity concept in Definition 3.6 allowed us to characterize dissipativity of an optimal periodic trajectory which would not be covered by Definition 3.2 or 3.4. The following example will illustrate that this definition also allows to define dissipativity for a set of optimal trajectories. Example 3.8: Consider the previously defined 1d dynamics f (x, u) = -x + u but now with a different stage cost

(x, u) = (x -2)(x -1)(x + 1)(x + 2).
This function has two minimizers at x = ± √ 10/2, hence the periodic orbit jumping between these two states is a candidate for a minimizing periodic orbit. However, also staying in one of the two minima is an equivalently good candidate.

Using λ(x) = 0 we obtain

L(x, ũ) := 1 k=0 L k (x k , u k ) = 1 k=0 (x k , u k ) -(x p k , u p k ).
Expanding this expression, one obtains

L 0 (x 0 , u 0 ) + L 1 (f (x 0 , u 0 ), u 1 ) = 2x 4 0 -10x 2 0 + u 4 0 -4u 3 0 x 0 + 6u 2 0 x 2 0 -5u 2 0 -4u 0 x 3 0 + 10u 0 x 0 + 25 2 .
The variable u 1 does not enter the equation and is thus free. The variables (x 0 , u 0 ) have the following solutions:

x 0 = ± √ 10/2 and u 0 = 0, or u 0 = ± √ 10. Solutions with u 0 = 0 correspond to a periodic trajectory oscillating between the two minima of function (x, u), while solutions with u 1 = √ 10 or u 1 = -√ 10 correspond to steady state trajectories that stay in one of the two minima of (x, u). This characterizes infinitely many trajectories, as both the periodic and the steady state trajectories are globally optimal. Indeed, all trajectories which can be described as (x, u) ∈ {(± √ 10/2, 0), ( √ 10/2, √ 10), (-√ 10/2, -√ 10)} have the same minimal averaged value. This implies that, when the system is in one of the two minima of (x, u), it is optimal both to stay in that minimum or to jump to the other minimum. For this reason, it is not possible to use a dissipativity concept which would only render one of the trajectories dissipative.

IV. RESULTS

In this talk we will present the following results with the appropriate conditions:

• Periodic dissipativity with periodic storage function implies that the system is optimally operated at a periodic orbit with period P • Periodic dissipativity with fixed storage function implies that the system is optimally operated at a periodic orbit with period P • Periodic dissipativity implies that MPC finds the optimal periodic orbit and yields (approximate) optimal average performance The last point is illustrated by the following numerical examples.

Consider Example 3.3 and an MPC scheme with horizon N = 5. Starting from the initial condition x(0) = 3, the obtained trajectory is displayed in Figure 1, left graph. The same simulation was also run for Example 3.7. The resulting trajectory is displayed in Figure 1, right graph. Example 3.8 does not have a unique optimal trajectory. All possible trajectories are displayed in Figure 2. Starting from any point, reaching either x p 0 or x p 1 are both optimal moves. Thus, also when one of the two steady states is reached, the system can freely jump to the other steady state. The periodic optimal states are displayed in dotted line.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Left graph: trajectory obtained for Example 3.3 starting from x(0) = 3. Right graph: trajectory obtained for Example 3.7 starting from x(0) = 3. The periodic optimal states for both examples are displayed in dotted line.


Alioune Camara
email: aliounec@gmail.com

N M Baldé

S Enoru

J S Bangoura

E Sobngwi

F Bonnet

Prevalence of anxiety and depression among diabetic Africans patients in Guinea: association with HbA1c levels

Keywords: Anxiety, depression, type 2 diabetes, risk factors

Aim.-The prevalence and risk factors associated with symptoms of anxiety and depression were determined in African people with diabetes.

Methods.-This cross-sectional study involving 491 out-patients with type 2 diabetes (T2D) recruited from four diabetes clinics (Conakry, Labé, Boké, Kankan) in Guinea was carried out. The Hospital Anxiety and Depression Scale was used to evaluate the symptoms of anxiety and depression. Logistic regression analysis stratified by sex was performed to identify the associated risk factors.

Results.-Anxiety and depression symptoms were present in 58.7% and 34.4% of the 491 people with T2D (62.7% women, mean ± SD age: 57.9 ± 10.2 years). The odds ratios (95% CI) of risk factors independently associated with anxiety were urban residence 2.98 (1.81 to 4.89) in women, low level of socio-economic status 0.19 (0.05 to 0.70), HbA1c ≥9.0% 2.61 (1.07 to 6.39)] in men. Factors associated with depression were urban residence 2.13 (1.27 to 3.58)], older age 1.03 (1.01 to 1.06)], socio-economic status 2.21 (1.34 to 3.66), no previous measurement of HbA1c 12.45 (1.54 to 100.34) in women and insulin therapy 2.28 (1.05 to 4.92), HbA1c ≥9.0% 3.85 (1.02 to 14.48) in men.

Conclusions.-Anxiety and depression symptoms in people with T2D are common in Guinea.

Urban residence, level of socio-economic status and high levels of HbA1c were significantly associated with a higher risk of anxiety or depression, highlighting the psychological burden related to diabetes in Africa.

Introduction

According to the World Health Organization, we are facing an epidemic of diabetes in developing countries. Currently, over 80% of people with diabetes live in low and middle income countries (LMICs) [START_REF] Maher | Health transition in Africa: practical policy proposals for primary care[END_REF]. This clinical situation relates also to Guinea where the agestandardized prevalence of diabetes according the Guinean census was 6.5% (95% CI: 5.3-7.7%) [START_REF] Baldé | Diabetes and impaired fasting glucose in rural and urban populations in Futa Jallon (Guinea): prevalence and associated risk factors[END_REF].

A large body of evidence has highlighted that both anxiety and depression are more common in people with diabetes than in the general population [START_REF] Van Dooren | Depression and Risk of Mortality in People with Diabetes Mellitus: A Systematic Review and Meta-Analysis[END_REF][START_REF] Egede | Diabetes and depression: Global perspectives[END_REF]. Furthermore, it has been consistently shown that depression is associated with an increased risk of morbidity and mortality in people with diabetes [START_REF] Van Dooren | Depression and Risk of Mortality in People with Diabetes Mellitus: A Systematic Review and Meta-Analysis[END_REF][START_REF] Egede | Diabetes and depression: Global perspectives[END_REF] and that depression may have a deleterious impact on adherence to glucose lowering treatments [START_REF] Gonzalez | Depression and Diabetes Treatment Nonadherence: A Meta-Analysis[END_REF]. However, recent studies suggest that psychological disorders remain often undiagnosed and therefore not appropriately treated among people with diabetes [START_REF] Pouwer | Should we screen for emotional distress in type 2 diabetes mellitus?[END_REF][START_REF] Li | Undertreatment of Mental Health Problems in Adults With Diagnosed Diabetes and Serious Psychological Distress The Behavioral Risk Factor Surveillance System, 2007[END_REF]. This issue is particularly relevant in Africa where healthcare infrastructures have mainly focused on infectious diseases rather than on type 2 diabetes (T2D). Yet, the combination of depression and diabetes among the poor populations in LMICs might favor the development of diabetic complications and ultimately greater morbidity and mortality [START_REF] Mendenhall | Depression and type 2 diabetes in low-and middle-income countries: A systematic review[END_REF].

Epidemiological data on the characteristics of T2D patients who are most affected by anxiety/depression in LMICs remain sparse but are essential to implement public health programmes in these countries [START_REF] Mendenhall | Depression and type 2 diabetes in low-and middle-income countries: A systematic review[END_REF]. The goal of the present study is to assess the prevalence of both anxiety and depression in people with T2D in Guinea and to identify the factors associated. Furthermore, we examined whether the association between psychological distress and diabetes is affected by the level of glycaemic control in these people from a low-income country who have limited access to regular healthcare.

Materials and Methods

Study setting and design

This cross-sectional multicentre study was conducted in four institutions in Guinea: the Endocrinology, Diabetology and Metabolic Disease Unit of the University Teaching Hospital (UTH) of Conakry, and the Diabetes Unites of regional Hospitals at Boké, Kankan and Labé.

Population

Between August 2009 and October 2010, a study to improve access to glycated haemoglobin (HbA1c) in diabetic patients was carried out in Cameroon and Guinea. The study included patients aged ≥16 years and having T2D for at least 12 months. They were contacted at the outpatient clinic of the four study sites and invited to participate in the study. Patients who had lost a family member (parent, brother, sister, husband, children) and/or lost their job in the month preceding the study were excluded from the study. In total, 491 patients with T2D were included in this study.

Methods

A semi-structured questionnaire and a standardized interview were completed for all patients.

Data involving socioeconomic status (SES), history of diabetes and levels of anxiety and depression were collected.

1 Socio-demographic profile

The Socio-demographic data collected included age, sex, zone of residence (rural or urban), marital status (single or married) and unemployment status (yes or no). The level of education was divided into two classes (less than 7 years of school, at least 7 years of school). SES was assessed from unemployment status and level of education and dichotomized into low (lower education and unemployment) or high (higher education or/and employment).

2 History of diabetes and clinical data

Other information collected included type of glucose lowering treatment, duration of diabetes and previous measurement of HbA1c (yes or no). HbA1c was classified into: < 7.0%, 7.0 to 8.9% and ≥ 9.0%.

The clinical data collected including current tobacco smoking (yes or no) and known hypertension (yes or no). Alcohol consumption during the previous month was self-reported.

Body mass index (BMI) was calculated.

3 Level of Anxiety and depression:

Symptoms of anxiety and depression were evaluated using the Hospital Anxiety and Depression Scale (HADS) [START_REF] Zigmond | The hospital anxiety and depression scale[END_REF]. The French version that we used had previously been validated, both in family medicine and in hospital settings [START_REF] Lepine | Anxiety and depression in inpatients[END_REF]. The HADS measures levels of symptoms in the last week. There is a medium-to-strong correlation between the HADS score and other instruments used to measure anxiety and depression, including the Beck's Depression Inventory, Spielberger's State-Trait Anxiety Inventory, the Symptom Checklist 90 Scale, and the Montgomery-Asberg Depression Rating Scale [START_REF] Bjelland | The validity of the Hospital Anxiety and Depression Scale. An updated literature review[END_REF]. The main characteristic of HADS is that items that could be attributed to physical illnesses, such as insomnia, fatigue, headaches, dizziness, sleep, and appetite disturbance, have been omitted to avoid falsepositive cases among individuals with somatic diseases. HADS has 14 questions: seven related to depression (HADS-D) and seven related to anxiety (HADS-A). Each question has four possible responses from zero (no symptoms) to three (maximum symptoms). The severity of the symptoms is determined by the total score obtained in each sub-scale (HADS-A and HADS-D) and is classified as: 0-7 (normal), 8-10 (mild disorder) and 11-21 (marked disorder). HADS-D covers mainly anhedonia and loss of interest, which are core depressive symptoms, while HADS-A covers the core anxiety features of worry and tenseness.

Ethical considerations:

The ethics committee of the Ministry of public health of Guinea approved the study.

Only patients who signed the informed consent form were included in the study.

Statistical analysis:

The prevalence of symptoms of anxiety and depression was determined using the HADS scale and classified as normal, mild, and marked disorder. For the logistic regression analysis, the HADS scores were dichotomized as normal (score 0 -7) and presence of anxiety / depression (score 8 -21), to include all possible cases of anxiety and depression as suggested by Zigmond [START_REF] Zigmond | The hospital anxiety and depression scale[END_REF]. Univariate logistic regression analyses evaluated the relationships between anxiety and depression and the associated factors, and results are presented as odd ratios (OR) and 95 % confidence intervals. Interactions were tested with sex, and as there was a signification interaction for some variables, all results are presented stratified on sex.

Variables with p values <0.20 in the univariate tests were selected as covariates for the multivariable models. A p value <0.05 was considered statistically significant. Analysis used SAS (version 9.3; SAS Institute, Cary, NC).

Results

General patients' description

General characteristics of the 491 people with T2D included in this study are presented in Table 1. The majority was female (62.7%), married (76.4%), with a job (54.2%) and high level of education (50.5%). No patient was treating with antidepressant drugs. Mean age was 57.9 ± 10.2 years, and the men were older (59.8 ± 9.7) than the women (56.7 ± 10.3). Only 6.7% of the patients had a previous measurement of HbA1c. Out of all the study patients, 15.7% had a good glycaemic control (HbA1c <7.0%), and 29.1% (n=143) were being treated with insulin, with the remainder taking oral glucose-lowering drugs.

Prevalence of anxiety and depression

On the HADS, a mean score of 8.5 ± 3.2, with a median of 8 for anxiety was recorded.

For depression, the mean score was 6.3 ± 3.3 with a median of 6. Table 2 presents the prevalence of anxiety and depression as classified by the HADS according to gender. Symptoms of anxiety were present in 58.7% of patients while 34.4% had symptoms of depression. The prevalence of marked anxiety was 27.5% and was more common in women (36.1%) when in men (13.1%). Marked depression was present in 11.4% of the population, and again was more common in women (14.0%) as compared to men (7.1%). Of our 491 diabetic patients, 127 (25.9%) had both anxiety and depression and neither anxiety nor depression was related to overweight/obese status.

Factors associated with anxiety

As shown in Table 3

Discussion

This study showed that both anxiety and depression are common in people with T2D atending out-patients clinics in Guinea. The prevalence of depression and anxiety in the present study is higher than that observed previously in Caucasian type 2 diabetic populations [START_REF] Anderson | The Prevalence of Comorbid Depression in Adults With Diabetes A meta-analysis[END_REF][START_REF] Collins | Anxiety and depression symptoms in patients with diabetes[END_REF].

However, a recent large cross-sectional multinational study did not find an association between diabetes and the prevalence of depressive symptoms in Africa, in contrast to other continents [START_REF] Mommersteeg | The association between diabetes and an episode of depressive symptoms in the 2002 World Health Survey: an analysis of 231,797 individuals from 47 countries[END_REF].

Other studies reported a higher prevalence of depression for diabetes in developing countries with results similar to our results in Pakistan 57.9% had anxiety and 43.5% had depression [START_REF] Khuwaja | Anxiety and depression among outpatients with type 2 diabetes: A multi-centre study of prevalence and associated factors[END_REF]. In a recent review, the percentage of people with depression among those with diabetes was 45.9% in South Africa and between 15 and 30% in Nigeria [START_REF] Mendenhall | Depression and type 2 diabetes in low-and middle-income countries: A systematic review[END_REF]. This high prevalence of anxiety and depression in LMICs such as Guinea could be explained by gender inequality, social insecurity, low educational levels and poverty [START_REF] Stein | Depression and anxiety in the developing world: is it time to medicalise the suffering?[END_REF].

In the present study, we observed that poor glucose control was independently associated with both anxiety and depression in men. Some previous reports have shown a positive association between HbA1c levels, fasting blood glucose and the level of anxiety [START_REF] Egede | Diabetes and depression: Global perspectives[END_REF][START_REF] Pouwer | Should we screen for emotional distress in type 2 diabetes mellitus?[END_REF][START_REF] Lustman | Depression in diabetic patients: the relationship between mood and glycemic control[END_REF]. Depressive mood has been also associated with glucose levels in T2D [START_REF] Engum | Depression and Diabetes A large population-based study of sociodemographic, lifestyle, and clinical factors associated with depression in type 1 and type 2 diabetes[END_REF]. In a recent study in Netherlands, several individual depressive symptoms were related to higher HbA1c in outpatients with T2D and these associations persisted over time [START_REF] Bot | Differential associations between depressive symptoms and glycaemic control in outpatients with diabetes[END_REF]. Underlying mechanisms proposed to explain the increase in glycaemia are enhanced inflammation [START_REF] Doyle | Diabetes, depressive symptoms, and inflammation in older adults: results from the health, aging, and body composition study[END_REF],

insulin resistance [START_REF] Silva | A review of the association between depression and insulin resistance: pitfalls of secondary analyses or a promising new approach to prevention of type 2 diabetes?[END_REF], alterations in insulin secretion [START_REF] Bot | Depressive symptoms, insulin sensitivity and insulin secretion in the RISC cohort study[END_REF] and activation of the hypothalamicpituitary-adrenal axis [START_REF] Penninx | Understanding the somatic consequences of depression: biological mechanisms and the role of depression symptom profile[END_REF]. Furthermore, depression and anxiety are also linked with poorer behavioral management of diabetes and glycemic control [START_REF] Kendzor | The association of depression and anxiety with glycemic control among Mexican Americans with diabetes living near the U.S.-Mexico border[END_REF].

The use of insulin therapy was independently associated with the symptoms of depression. This is in agreement with previous reports [START_REF] Papelbaum | Depression, glycemic control and type 2 diabetes[END_REF][START_REF] Joseph | Proportion of depression and its determinants among type 2 diabetes mellitus patients in various tertiary care hospitals in Mangalore city of South India[END_REF]. Insulin therapy could be associated with negative beliefs about the future and the risk of death for the patient.

However, the need for insulin therapy often indicates a more severe stage of the disease, which is characterized by age, poorer glycaemic control, and a higher rate of complications. Nevertheless, in our study, this association persisted after adjusting for age and HbA1c, suggesting that the perceptions surround insulin therapy itself, irrespective of the metabolic context, could favour the development of depressive symptoms. Only a prospective study with a longitudinal follow-up could adequately test this hypothesis.

The lack of previous measurement of HbA1c was independently associated with the presence of depression in women. A number of studies showed that depression is associated with poor perceived control of diabetes and poor self-care behaviors [START_REF] Egede | Role of Motivation in the Relationship Between Depression, Self-care, and Glycemic Control in Adults With Type 2 Diabetes[END_REF][START_REF] Gonzalez | Depression, Self-Care, and Medication Adherence in Type 2 Diabetes Relationships across the full range of symptom severity[END_REF]. In addition, it may be speculated that previous measurement of HbA1c might be an indirect indicator of the patient's participation in a structured diabetes medical programme, which may have contribute to giving more reassurance to the patient, thereby explaining why these patients were less depressed. The lack of information on glycaemia status might also be potentially worrying to the patient. In addition, depression may prevent efforts dedicated to health, resulting in a lower probability of having an HbA1c measurement. Nevertheless, reverse causality, by which poor glycaemic control may induce greater psychological distress, cannot be excluded.

It was observed that age was independently associated with symptoms of depression whereas duration of diabetes was not an independent risk factor after accounting for age.

Findings for the relationship between age and depression in diabetes have been conflicting, with some studies reporting age as a risk factor for depression [START_REF] Khuwaja | Anxiety and depression among outpatients with type 2 diabetes: A multi-centre study of prevalence and associated factors[END_REF][START_REF] Leone | Diabetes and depression comorbidity and socio-economic status in low and middle income countries (LMICs): a mapping of the evidence[END_REF] whereas in other studies, younger age was related to depressive symptoms [START_REF] Egede | Independent Factors Associated With Major Depressive Disorder in a National Sample of Individuals With Diabetes[END_REF].

Our study provides new findings concerning the relationship between socioenvironmental factors and the presence of anxiety/depression in people with T2D in a developing African country. Our results for women showed that those of low SES were twice as likely to be depressed compared with of high SES. This is in agreement with other studies showing that the risk of depression is higher for diabetic patients with lower SES [START_REF] Leone | Diabetes and depression comorbidity and socio-economic status in low and middle income countries (LMICs): a mapping of the evidence[END_REF][START_REF] Everson | Epidemiologic evidence for the relation between socioeconomic status and depression, obesity, and diabetes[END_REF]. In contrast, however a higher SES was independently found to be associated with the symptoms of anxiety. Unemployment is also a consistent risk factor for psychological disorders, suggesting the importance to taking into consideration the presence of depression among patients with diabetes who are unemployed or who have less education, which is common in Africa [START_REF] Patel | Poverty and common mental disorders in developing countries[END_REF]. It has been shown that depression is more commonly seen among those with a low family income, non-professional/administrative employment, not current employment and so are dependent, and those living alone and with less social support [START_REF] Leone | Diabetes and depression comorbidity and socio-economic status in low and middle income countries (LMICs): a mapping of the evidence[END_REF]. Indeed, it has long been recognized that individuals with lower SES suffer a disproportionate share of the burden and consequences of numerous diseases than those with higher SES [START_REF] Adler | Socioeconomic Disparities In Health: Pathways And Policies[END_REF].

It was also revealed that an urban area of residence was independently associated with symptoms of both anxiety and depression in women, a relationship that remains controversial in the developing countries. A study in Pakistan showed a greater prevalence of mental disorders in urban areas than in rural areas [START_REF] Rahim | Epidemiology of mental disorders in young adults of a newly urbanized area in Khartoum, Sudan[END_REF]. In contrast, no significant association between depression comorbidity and place of residence was found in a study from Nigeria [START_REF] Agbir | Depression among medical outpatients with diabetes: A cross-sectional study at Jos University Teaching Hospital, Jos, Nigeria[END_REF].

Our present study has several intrinsic limitations. First, as symptoms of depression and anxiety were only measured at one time point, this study cannot directly evaluate the long-term impact of diabetes on the incidence of anxiety /depression. Thus the observational nature of the study allows no conclusions to be drawn on the causality of the link between depression and poor glycaemic control. Second, the study population was not randomly sampled, which limits attempts to generalize the results to all populations with T2D in Guinea. Third, the HADS-D score predominantly reflects melancholic depressive symptoms over the past week. Thus, levels of lifetime depression and the proportion of subjects with atypical depressive symptoms might have been underestimated in our cohort. Also, the study may have slightly underestimated the prevalence of anxiety and depression because patients who had lost a family member and/or lost their job in the month preceding the study were excluded. Finally, it was not possible to adjust for risk factors such as a previous or family history of depression, childhood experiences, life experiences and sickle cell disease. The prevalence of the latter is high in Guinea, and its presence is known to affect measurements of HbA1c [START_REF] Ama | Would sickle cell trait influence the metabolic control in sub-Saharan individuals with type 2 diabetes?: Sickle cell trait and diabetes in Africa[END_REF].

Our findings show that people with T2D in Sub-Saharan Africa are at risk of anxiety and depression, just as reported in high-income countries [START_REF] Petrak | Psychosocial factors and diabetes mellitus: evidence-based treatment guidelines[END_REF]. The high prevalence of anxiety and depression in Guinea is an important additional public-health burden, as the country faces an alarming increase in the prevalence of T2D in Africa [START_REF] Mbanya | Diabetes in sub-Saharan Africa[END_REF].

These findings also suggest that the healthcare infrastructure, which has traditionally focused on infectious diseases in Guinea, needs to evolve to take better account of the psychological burden associated with diabetes, particularly in urban areas. The screening and monitoring of psychological disorders in people with diabetes are still neglected in Africa, and the treatment of diabetes-related depression is rare in these countries, too [START_REF] Crabb | Attitudes towards mental illness in Malawi: a cross-sectional survey[END_REF][START_REF] Barke | The stigma of mental illness in Southern Ghana: attitudes of the urban population and patients' views[END_REF].

Medico-economic studies have shown that the coexistence of depression and diabetes is associated with greater healthcare services and medical costs [START_REF] Egede | Diabetes and depression: Global perspectives[END_REF].

In conclusion, our results show that both anxiety and depression are common in people with T2D living in Guinea, irrespective of overweight/obesity status. Poor control of glycaemia, residence in an urban area, the absence of previous measurement of HbA1c and use of insulin therapy appear to be risk factors for depression in this population. These findings suggest that depression in T2D patients in Africa needs to be screened for and taken into consideration in their medical care.

  , anxiety in men was significantly associated with low SES [Odds

	ratio: 0.19 (95% CI: 0.05 to 0.70)], HbA1c 7-8.9% [2.80 (1.13 to 6.93)] and HbA1c ≥9.0%
	[2.61 (1.07 to 6.39)] in the multivariable model. In women, only residence in urban area [2.98
	(1.81 to 4.89)] was associated with anxiety in the multivariable model.
	3.4 Factors associated with depression
	In a multivariate model (Table 4), depression in men was significantly associated with
	insulin therapy [2.28 (1.05 to 4.92)], and HbA1c ≥9.0% [3.85 (1.02 to 14.48)]. In women, age
	[1.03 (1.01 to 1.06)], residence in an urban area [2.13 (1.27 to 3.58)], low level of SES [2.21
	(1.34 to 3.66)], and no previous measurement of HbA1c [12.45 (1.54 to 100.34)] were
	independently associated with depression.

Table 1

 1 Characteristics of the 491 people with type 2 diabetes in outpatient clinics in Guinea.

	Men (n=183)	Women (n=308)	P value

Table 2

 2 Prevalence according to severity of symptoms of anxiety and depression as measured by Hospital Anxiety and Depression Scale (HADS) in 491 diabetic outpatients in Guinea by gender.

	Anxiety, n (%)	Depression, n (%)

Table 3

 3 Factors associated with symptoms of anxiety (Score HADS-A ≥ 8) in outpatients with Type 2 diabetes in Guinea.

					Anxiety (Scale HADS-A≥8-21 vs. 0-7)
				Men (n=183)			Women (n=308)
			OR crude		OR adjusted		OR crude		OR adjusted
	Characteristics		(95% CI)	P value	(95% CI)	P value	(95% CI)	P value	(95% CI)	P value
	Ages (years)		0.98 (0.96-1.02)	0.34			0.98 (0.96-1.00)	0.09	0.98 (0.96-1.00)	0.06
	Zone of Residence	Urban vs. Rural 1.17 (0.64-2.13)	0.61			2.98 (1.81-4.89)	<0.001	2.98 (1.81-4.89)	<0.001
	Married	Yes vs. No	0.75 (0.18-3.09)	0.69			1.09 (0.66-1.79)	0.74
	Socio-Economic Status	Low vs. High	0.22 (0.06-0.77)	0.02	0.19 (0.05-0.70)	0.01	0.91 (0.56-1.47)	0.70
	Insulin therapy use	Yes vs. No	1.06 (0.53-2.10)	0.88			1.43 (0.85-2.42)	0.18	1.17 (0.67-2.05)	0.58
	Duration of diabetes (years)		0.99 (0.94-1.03)	0.57			1.05 (1.01-1.11)	0.03	1.05 (0.99-1.10)	0.05
	Previous measurement of HbA1c No vs. Yes	1.29 (0.45-3.72)	0.63			1.51 (0.56-4.10)	0.41
	HbA1c (versus <7.0%)	7.0 to 8.9%	2.55 (1.05-6.23)	0.04	2.80 (1.13-6.93)	0.02	1.09 (0.51-2.33)	0.82
		≥9.0%	2.38 (1.00-5.75)	0.04	2.61 (1.07-6.39)	0.03	1.19 (0.59-2.41)	0.62
	Known HTA	Yes vs. No	0.57 (0.27-1.22)	0.14	0.55 (0.25-1.22)	0.14	0.60 (0.27-1.31)	0.20
	BMI (kg/m²)		0.97 (0.89-1.05)	0.49			1.00 (0.95-1.05)	0.91
	Current smoker	Yes vs. No	1.63 (0.48-5.54)	0.43			-	-
	Alcohol consumption	Yes vs. No	1.30 (0.61-2.77)	0.49			0.71 (0.12-4.29)	0.70

HADS-A: Hospital Anxiety and Depression Scale for anxiety ; 95% CI: 95% confidence interval

Table 4

 4 Factors associated with symptoms of depression (Scale HADS-D ≥ 8) in outpatients with Type 2 diabetes in Guinea. Hospital Anxiety and Depression Scale for depression.; 95% CI: 95% confidence interval

					Depression (Scale HADS-D≥8-21 vs. 0-7)
				Men (n=183)			Women (n=308)
			OR crude		OR adjusted		OR crude		OR adjusted
	Characteristics		(95% IC)	P value	(95% IC)	P value	(95% IC)	P value	(95% IC)	P value
	Ages (years)		1.03 (0.99-1.07)	0.10	1.04 (0.99-1.08)	0.07	1.04 (1.01-1.06)	0.003	1.03 (1.01-1.06)	0.02
	Zone of Residence	Urban vs. Rural	0.96 (0.49-1.88)	0.90			1.69 (1.04-2.75)	0.03	2.13 (1.27-3.58)	0.004
	Married	Yes vs. No	0.57 (0.13-2.51)	0.46			0.81 (0.50-1.30)	0.38
	Socio-Economic Status	Low vs. High	1.34 (0.48-3.75)	0.58			2.61 (1.63-4.18)	<0.001	2.21 (1.34-3.66)	0.002
	Insulin therapy use	Yes vs. No	3.03 (1.46-6.27)	0.002	2.28 (1.05-4.92)	0.04	1.42 (0.87-2.31)	0.15	1.27 (0.75-2.16)	0.37
	Duration of diabetes (years)		1.03 (0.97-1.08)	0.26			0.99 (0.95-1.04)	0.80
	Previous measurement of HbA1c No vs. Yes	5.87 (0.75-45.73)	0.09	6.65 (0.81-54.94)	0.08	11.22 (1.47-85.75)	0.02	12.45 (1.54-100.34)	0.02
	HbA1c (versus <7.0%)	7.0 to 8.9%	3.51 (0.96-12.88)	0.06	2.99 (0.80-11.11)	0.10	1.34 (0.63-2.87)	0.44
		≥9.0%	5.47 (1.53-19.55)	0.008	3.85 (1.02-14.48)	0.04	1.44 (0.71-2.91)	0.32
	Known HTA	Yes vs. No	0.78 (0.34-1.78)	0.56			0.64 (0.33-1.26)	0.19	0.57 (0.27-1.18)	0.19
	BMI (kg/m²)		0.91 (0.83-1.00)	0.05	0.92 (0.83-1.02)	0.12	0.94 (0.90-0.99)	0.02	0.97 (0.92-1.92)	0.29
	Current smoker	Yes vs. No	0.61 (0.13-2.92)	0.53			-	-
	Alcohol consumption	Yes vs. No	1.07 (0.46-2.49)	0.88			0.38 (0.04-3.45)	0.39
	HADS-D:

Acknowledgment

This project is supported by a grant from the BRIDGES International Diabetes Federation, a project of the International Diabetes Federation, supported by an educational grant from Eli Lilly and Company.

We thank the French Society of Diabetes (SFD) for the award to Dr. Alioune Camara in 2009.

We thank warmly the health professionals who contributed to the recruitment and monitoring of patients, and those who participated in the study.

Competing interests

No potential conflict of interest relevant to this article was reported.