Marco Di

Renzo L2s

Harald Haas
email: h.haas@ed.ac.uk

Transmit-Diversity for Spatial Modulation (SM): Towards the Design of High-Rate Spatially-Modulated Space-Time Block Codes

Spatial Modulation (SM) is a recently proposed low-complexity modulation scheme for multiple-antenna wireless systems. Recent works have shown that state-of-the-art SM provides a multiplexing gain with respect to single-antenna systems by avoiding inter-channel interference, but it is inherently unable to achieve transmit-diversity. The aim of this paper is to shed light on the design of multiple-antenna wireless systems that exploit the SM concept and can achieve transmit-diversity gains. More specifically, we propose a novel modulation concept, which in a unique fashion combines the high multiplexing gain offered by SM and the transmit-diversity gain provided by Space-Time Block Codes (STBCs) technology. This modulation scheme is introduced here for the first time and referred to as "Spatially-Modulated STBCs" (SM-STBC). By using analysis and simulation, we show that SM-STBC achieves multiplexing and diversity gains, as well as still retains a single-stream receiver implementation. By using numerical simulations, we show that the proposed code can provide better performance than stateof-the-art Alamouti and rate-3/4 STBCs.

I. INTRODUCTION

Multiple-antenna technology is a rich area of research [START_REF] Mietzner | Multipleantenna techniques for wireless communications -A comprehensive literature survey[END_REF]. In the realm of the many solutions proposed to date, Spatial Modulation (SM) is a recently proposed modulation scheme that promises a low-complexity transmitter and receiver design, along with improved system performance with respect to many state-of-the-art multiple-antenna solutions [START_REF] Yang | Information-guided channel-hopping for high data rate wireless communication[END_REF]- [START_REF] Di Renzo | A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels[END_REF].

The breakthrough idea of SM is to exploit the spatial domain as an additional dimension to convey part of the information bits [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF]. As a result, SM turns out to be a high spectrally-efficient transmission technology with an equivalent code rate greater than one [START_REF] Song | A channel hopping technique I: Theoretical studies on band efficiency and capacity[END_REF]. Recent studies have shown that SM can achieve a higher capacity than multiple-antenna schemes with similar decoding complexity, such as Space-Time Block Codes (STBCs) [START_REF] Yang | Information-guided channel-hopping for high data rate wireless communication[END_REF], along with a smaller bit error probability than Vertical Bell Laboratories Layered Space-Time (V-BLAST) and Alamouti schemes [START_REF] Mesleh | Spatial modulation[END_REF].

Although originally conceived as an interference-free spatial-multiplexing scheme [START_REF] Mesleh | Spatial modulation[END_REF], many researchers have recently started investigating the potential of this technology for receive-and transmit-diversity. In particular, transmitdiversity is receiving a growing interest due to its suitability for downlink applications with single-antenna and lowcomplexity mobile units. More specifically, many researchers have focused their attention on a low complexity implementation of SM, which is known as Space Shift Keying (SSK) modulation [START_REF] Jeganathan | Space shift keying modulation for MIMO channels[END_REF]. Unlike SM, in SSK modulation only the spatialconstellation diagram is used to modulate the information bits, thus trading-off transmitter and receiver complexity for the achievable data rate [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF]. As far receive-and transmitdiversity for SSK modulation is concerned, the following contributions are available in the literature. In [START_REF] Jeganathan | Space shift keying modulation for MIMO channels[END_REF] and [START_REF] Di Renzo | On the performance of space shift keying MIMO systems over correlated Rician fading channels[END_REF], it is shown that SSK modulation can achieve a receive-diversity gain that increases linearly with the number of antennas at the receiver. In [START_REF] Jeganathan | Generalized space shift keying modulation for MIMO channels[END_REF], it is shown that, regardless of the number of simultaneously-active antennas at the transmitter, SSK modulation is unable to provide transmit-diversity gains. In [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF], a simple method is introduced to overcome that limitation. The solution is applicable to a transceiver with two transmit-antenna and one receive-antenna, and neither incurs in any spectral efficiency loss nor requires multiple simultaneously-active antennas at the transmitter. In [START_REF] Sugiura | A unified MIMO architecture subsuming space shift keying, OSTBC, BLAST and LDC[END_REF], transmit-diversity is achieved by sending redundant information in non-overlapping time-slots, thus incurring in a spectral efficiency loss. In [START_REF] Di Renzo | Space shift keying (SSK-) MIMO over correlated Rician fading channels: Performance analysis and a new method for transmitdiversity[END_REF], it is proved that the method in [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF] is unable to provide full-diversity for an arbitrary number of antennas at the transmitter and, in general, it allows us to achieve transmit-diversity only equal to two. In [START_REF] Di Renzo | Space shift keying (SSK) modulation: On the transmitdiversity/multiplexing trade-off[END_REF], the idea in [START_REF] Di Renzo | Space shift keying (SSK-) MIMO over correlated Rician fading channels: Performance analysis and a new method for transmitdiversity[END_REF] is extended and new schemes to achieve transmitdiversity greater than two are proposed. However, all these techniques are mainly concerned with SSK modulation, while, to the best of the authors knowledge, the design of transmitdiversity schemes for the more general SM concept have never been considered so far 1 . Only in [START_REF] Handte | BER analysis and optimization of generalized spatial modulation in correlated fading channels[END_REF], the authors have studied the achievable transmit-diversity of SM and have pointed out that SM cannot achieve transmit-diversity gains. However, no solutions are provided to cope with this issue and it is shown that the absence of transmit-diversity may result, especially for high correlated fading channels, in a substantial performance loss of SM with respect to the Alamouti scheme.

Motivated by all the above, this paper aims at shedding light on the issues related to the design of SM schemes with transmit-diversity. Since SM includes as special cases both SSK modulation and conventional modulation schemes [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF], we show that SM needs to combine conventional transmitdiversity [START_REF] Derryberry | Transmit diversity in 3G CDMA systems[END_REF] methods (e.g., Alamouti [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] and STBCs [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]) and SSK transmit-diversity solutions to be able to get full transmit-diversity gains. Thus, the design of SM schemes with transmit-diversity is a more challenging task than the design of SSK modulation and STBCs alone. More specifically, when adding the spatial-constellation diagram on top of the signalconstellation diagram of conventional modulation we have to make sure that the SSK modulation component of SM does not destroy the orthogonality property of STBCs, thus still allowing the system to i) get transmit-diversity gains, and ii) enable the adoption of a single-stream receiver design. This paper provides some guidelines to construct such modulation/coding schemes. More specifically, we are interested in developing SM-STBCs with diversity order equal to two, which add the SSK modulation principle on top of the Alamouti code. In other words, this paper reports a new code with transmitdiversity equal to two, and, as opposed to classical Alamouti code, rate greater than one. Moreover, it is shown that a simple single-stream decoder can be used at the receiver without loss in optimal decoding performance. Numerical results are provided to complement our analytical investigation and to show that the proposed scheme outperforms state-of-the-art SM, Alamouti, and STBCs for the same spectral efficiency and receiver complexity. Very interestingly, it is shown that the proposed solution can outperform the more complicated rate-3/4 STBCs [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF] with diversity gain three and four.

The reminder of this paper is organized as follows. In Section II, the system model and the main assumptions are briefly summarized. In Section III, we introduce the design of SM-STBCs for multiple-antenna systems, highlight the issues related to the efficient design of such codes, and provide some guidelines for the design of SM-STBCs with transmit-diversity equal to two and low decoding complexity. In Section IV, our claims are substantiated through Monte Carlo simulations. Finally, Section V concludes the paper.

II. SYSTEM MODEL

We consider a Multiple-Input-Single-Output (MISO) communication system with N t antennas at the transmitter and N r = 1 antennas at the receiver. The assumption N r = 1 does not limit the generality of the results derived in this paper since we are mainly interested in studying transmit-diversity schemes. By following the analysis in [START_REF] Di Renzo | On the performance of space shift keying MIMO systems over correlated Rician fading channels[END_REF], it can be proved that the results obtained in this paper can be extended to multiple receive-antenna, and that the overall diversity achieved by the resulting system is simply multiplied by N r . At the receiver, we assume Maximum-Likelihood (ML) decoding with Full Channel State Information (F-CSI) [START_REF] Jeganathan | Spatial modulation: Optimal detection and performance analysis[END_REF], [START_REF] Di Renzo | A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels[END_REF], which can be obtained through the transmission of pilot symbols before data transmission. To simplify the analytical derivation, we consider a Multilevel Phase Shift Keying (M-PSK) modulation scheme (i.e., the signal-constellation diagram [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF]), and frequency-flat, unit-power, and independent and identically distributed (i.i.d.) Rayleigh fading channels over all the wireless links. The complex fading gain over the wireless link from the t-th transmitantenna to the receiver is denoted by {h t } N t t=1 . Finally, perfect time synchronization at the receiver and negligible differences in the propagation delays from the transmit-to the receiveantenna are assumed [START_REF] Di Renzo | A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels[END_REF].

Additional notation used throughout this paper is as follows: i) we adopt a complex-envelope signal representation; ii) (•)

* is the complex-conjugate operator; iii) |•| 2 denotes square absolute value; iv) E {•} is the expectation operator; v) Re {•} denotes the real part operator; vi)

Q (x) = 1 √ 2π +∞ x
exp -t 2 2 dt is the Q-function; vii) N a is the number of simultaneously active antennas at the transmitter, with 1 ≤ N a ≤ N t ; viii) E m is the average total energy transmitted by the N a active antennas that emit a non-zero signal. It is assumed that E m is equally distributed among the active antennas, i.e. each active antenna emits a signal with energy E m /N a ; ix) T m denotes the signaling interval for each information message; x) N 0 is the power spectral density per dimension of the Additive White Gaussian Noise (AWGN) at the receiver input; xi) for ease of notation, we set γ=E m /(4N a N 0 ); xii) S M and S N h denote the signal-and spatial-constellation diagrams with size M and N h , respectively. The generic element of S M is s l = sl E m /N a for l = 1, 2, . . . , M , where sl is a unit-energy complex number, and the generic element of S N h is the N a -tuple of channel gains h t1 , h t2 , . . . , h t Na Nt

t 1 =t 2 =... =t N a =1 ; xiii) {w t (•)} N t
t=1 is the pulse shape used at the t-th transmit-antenna; xiv) • is the floor function; xv) • • is the binomial coefficient; and xvi) Pr {•} denotes probability.

A. On the Need of STBCs to Achieve Transmit-Diversity

The distinguishable feature of SM with respect to other spatial multiplexing methods is the capability of achieving a multiplexing gain without the need to multiplex many data streams. In other words, a single transmit-antenna is active at any time instance [START_REF] Mesleh | Spatial modulation[END_REF]. The fundamental question is to understand whether, by still keeping a single active antenna at the transmitter, we can achieve transmit-diversity gains. As far as SSK modulation is concerned, we have shown in [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF], [START_REF] Di Renzo | Space shift keying (SSK-) MIMO over correlated Rician fading channels: Performance analysis and a new method for transmitdiversity[END_REF] that this is actually possible via suitable pulse shaping design at the transmitter. This operation can be regarded as a form of precoding that operates on the waveform of the transmitted signal. As far as SM is concerned, in [START_REF] Handte | BER analysis and optimization of generalized spatial modulation in correlated fading channels[END_REF] the authors have shown that, regardless of the number of simultaneously-active antennas at the transmitter, no transmit-diversity gains can be achieved if pulse shaping is not adopted at the transmitter. This latter result shows that state-of-the-art SM [START_REF] Basar | Space-time block coding for spatial modulation[END_REF] cannot achieve transmit-diversity even though multiple antennas are simultaneously transmitting. An intuitive explanation of this outcome can be obtained from the analytical model we have recently developed for SM in [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF]. In [6, Eq. ( 11)], we have shown that the Average Bit Error Probability (ABEP) of SM is approximately given by the weighted summation of the individual ABEPs of the SSK modulation and the conventional modulation schemes composing SM. In other words, this framework highlights that a necessary condition for SM to achieve transmit-diversity, Div tx , is that both the SSK modulation and the conventional modulation schemes inherently embedded into SM have transmit-diversity Div tx . If, for example, the conventional modulation scheme embedded into SM has transmit-diversity equal to one, then the resulting SM system will always have diversity equal to one regardless of the transmit-diversity of the SSK modulation scheme. Thus, the result in [START_REF] Handte | BER analysis and optimization of generalized spatial modulation in correlated fading channels[END_REF] can be readily understood: even though multiple antennas are simultaneously active, none of the transmit-diversity methods in [START_REF] Derryberry | Transmit diversity in 3G CDMA systems[END_REF] is adopted in [START_REF] Handte | BER analysis and optimization of generalized spatial modulation in correlated fading channels[END_REF] and, so, the resulting system cannot achieve any transmitdiversity gains. Similar arguments can be used for the SSK sub-system embedded into SM.

From the arguments above, it follows that multiple transmitantenna need to be active to get transmit-diversity gains in SM, as well as that SM should be generalized and integrated into one of the methods in [START_REF] Derryberry | Transmit diversity in 3G CDMA systems[END_REF] to increase the transmitdiversity gain. In this paper, we limit ourselves to designing SM schemes with transmit-diversity equal to two, by using the Alamouti code as the building block to get transmitdiversity gains for the conventional modulation sub-system [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]. Generalization to other STBCs [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF] is possible, but it is not pursued in this paper due to space constraints.

B. Signal Structure

According to Section II-A, we assume two simultaneouslyactive antennas at the transmitter, i.e., N a = 2. Thus, the spatial-constellation diagram might have a maximum number of points equal to N H = 2 log2 ( N t N a ) [START_REF] Jeganathan | Generalized space shift keying modulation for MIMO channels[END_REF], [START_REF] Di Renzo | Space shift keying (SSK) modulation: On the transmitdiversity/multiplexing trade-off[END_REF]. The goal of this paper is to find the sub-set of points 2 N h ≤ N H such that the proposed system has transmit-diversity equal to two.

The working principle of the proposed scheme is as follows. At the transmitter, the bitstream emitted by a binary source is divided into blocks of log 2 (N h ) + 2 log 2 (M ) bits each, with log 2 (N h ) and 2log 2 (M ) being the number of bits needed to identify a point (i.e., a pair of antennas) in the antenna-array and the pair of symbols used by the Alamouti code in the signal-constellation diagram, respectively. Each block is split into two sub-blocks of log 2 (N h ) and 2log 2 (M ) bits. The bits in the first sub-block are used to select the N a = 2 antennas that are simultaneously switched on for data transmission, while all the other antennas are kept silent in the current signaling time interval. The bits in the second sub-block are used to choose a pair of symbols in the signal-constellation diagram, which are modulated by using the Alamouti code and transmitted over two consecutive and orthogonal time slots. The receiver solves a hypothesis detection problem to jointly estimate the antennas at the transmitter that are not idle, and the two signal waveforms that are transmitted from it, which results in the estimation of the unique sequence of log 2 (N h ) + 2 log 2 (M ) bits emitted by the encoder.

Accordingly, let (h x , h y ) for x = y = 1, 2, . . . , N h and (s p , s q ) for p, q = 1, 2, . . . , M be the actual transmitted points of the spatial-and signal-constellation diagrams, respectively, the signals received in the first, r 1 (•), and second, r 2 (•), time slots are as follows, respectively (t ∈ [0, T m )) [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]:

( r 1 (t) = h x s p w x (t) + h y s q w y (t) + η 1 (t) r 2 (t) = h x `-s * q ´wx (t) + h y s * p w y (t) + η 2 (t) (1)
where η 1 (•) and η 2 (•) are the AWGNs in the first and second time slot, respectively. In (1), we can recognize the structure of the Alamouti code [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]. The important difference with the Alamouti scheme is that in SM-STBC the pair of channels (h x , h y ) is not known a priori, but need to be estimated among the sets of points in S N h . We will see in Section III that this might destroy the inherent orthogonality of the Alamouti code. Furthermore, let us emphasize that, for the sake of generality, we have assumed that each transmit-antenna has a different shaping filter 3

{w t (•)} N t
t=1 . The analysis of system setups with different shaping filter is motivated by [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF], [START_REF] Di Renzo | Space shift keying (SSK-) MIMO over correlated Rician fading channels: Performance analysis and a new method for transmitdiversity[END_REF], [START_REF] Di Renzo | Space shift keying (SSK) modulation: On the transmitdiversity/multiplexing trade-off[END_REF], where it has been shown that orthogonal filters can be used to improve the transmit-diversity gain of SSK modulation. Since, as explained in Section II-A, SSK modulation is embedded in SM, we will show in Section III that the shaping filters play an important role in the design of our new modulation scheme.

C. ML Receiver Design

At the destination, the receiver needs to estimate the pair of spatial-and signal-constellation points (h x , h y ) and (s p , s q ), respectively. By assuming a ML-optimum detector, the receiver decides in favor of the pair of points (h x, h ŷ ) and (s p, s q ) that minimize the decision metric [START_REF] Jeganathan | Spatial modulation: Optimal detection and performance analysis[END_REF], [START_REF] Di Renzo | A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels[END_REF], [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]:

D (hx,hŷ),(sp,sq) = Z Tm ˛r1 (t) -ˆhx s pw x (t) + h ŷ s q w ŷ (t) ˜˛2 dt + Z Tm ˛r2 (t) - h -h xs * q w x (t) + h ŷ s * pw ŷ (t) i˛2 dt (2)
where the ˆsymbol is used to denote the generic transmitted signal under test at the destination.

The performance of this receiver can be analyzed in terms of Pairwise Error Probability (PEP), which is the probability that having transmitted (h x , h y ) and (s p , s q ) the detector decides in favor of (h x, h ŷ ) and (s p, s q ) with (h x, h ŷ ) = (h x , h y ) or (s p, s q ) = (s p , s q ). In formulas, we have:

PEP = PEP `(hx, hy; sp, sq) → `hx , h ŷ ; s p, s q ´= Pr n D (hx,hy),(sp,sq) > D (hx,hŷ),(sp,sq) o (3)
The complexity of this receiver depends on how the spatialconstellation diagram is chosen. This is studied in Section III.

III. TOWARDS THE DESIGN OF EFFICIENT SM-STBCS

To understand the main objectives of this section, let us summarize the two distinguishable features of the Alamouti code, and STBCs in general. The structure of these codes allow them to [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF], [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]: i) achieve a transmit-diversity gain equal to the number of active antennas at the transmitter (e.g., Div tx = 2 for the Alamouti code), and ii) use a detector whose complexity increases linearly with the size, M , of the signal-constellation diagram. In this section, we seek to design SM-STBCs that simultaneously achieve these two properties. We prove that there are fundamental limitations when adding the SSK component on top of conventional modulation schemes. Similar to [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF], we use PEP analysis in our analytical derivation. Two scenarios are studied: i) a simple system setup where all the antennas at the transmitter use the same shaping filter, i.e., w 0 (t) = w t (t) for t = 1, 2, . . . , N t , and ii) a setup where the shaping filters can be different.

A. Same Shaping Filters

When w 0 (t) = w t (t) for t = 1, 2, . . . , N t , (2) reduces to:

D (hx,hŷ),(sp,sq) = Z T m ˛r1 (t) -ˆhx s pw 0 (t) + h ŷ s q w 0 (t) ˜˛2 dt + Z T m ˛r2 (t) - h -h xs * q w 0 (t) + h ŷ s * pw 0 (t) i˛2 dt (4)
with r 1 (t) = h x s p w 0 (t) + h y s q w 0 (t) + η 1 (t) and r 2 (t) = -h x s * q w 0 (t) + h y s * p w 0 (t) + η 2 (t). 1) Analysis of Receiver Complexity: Let us first analyze if a single-stream detector can be used in this case. By substituting r 1 (•) and r 2 (•) into (4), and after some algebra, we get:

D (hx,hŷ),(sp,sq) = D (sp) (hx,hŷ),(sp,sq) + D (sq) (hx,hŷ),(sp,sq) (5)
where:

D (sp) (hx,
h ŷ s * q s q - 2Re n h * x h xs q s * q o -2Re ˘h * x h ŷ s * p s q ¯+ 2Re n h * y h xs p s * q o -2Re ˘η * 1 h y s q ¯+ 2Re n η * 2 h x s * q o (7)
with η1 = Tm η 1 (t) w 0 (t) dt, η2 = Tm η 2 (t) w 0 (t) dt. From ( 5)-( 7), we observe that, unlike the Alamouti code, D (s p) (h x,h ŷ ),(s p,s q ) and D (s q ) (h x,h ŷ ),(s p,s q ) depend on both s p and s q . Thus, the multi-stream ML-optimum detector in (4) cannot be decomposed into two single-stream detectors without loss in performance. Only if h x = h x and h ŷ = h y this is possible. In fact, this scenario reduces to the pure Alamouti code where -2Re h * y h xs * q s p + 2Re h * x h ŷ s q s * p = 0 in (6) and -2Re h * x h ŷ s * p s q + 2Re h * y h xs p s * q = 0 in [START_REF] Di Renzo | A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels[END_REF]. In conclusion, the first result of this paper is as follows: whatever the spatial-constellation diagram is, if the shaping filters at the transmitter are all the same, adding the SSK component on top of the Alamouti code destroys its inherent orthogonally. So, no single-stream decoder can be used and the receiver complexity is of the order of N h M N a correlations.

2) Analysis of Transmit-Diversity: Although the receiver in (4) loses an important property of STBCs, it is instructive to analyze its transmit-diversity gain to understand the complexity/performance trade-off with respect to the SM scheme.

Transmit-diversity can be studied by computing the PEP in (3). In particular, by substituting (4) in (3), and after lengthly analytical computations, the following result can be obtained:

PEP = Q r γ h |H 1 | 2 + |H 2 | 2 i ! (8)
where

H 1 = h x s p + h y s q -h xs p -h ŷ s q and H 2 = -h x s * q + h y s * p + h xs * q -h ŷ s * p.
By carefully looking at (8), we can readily conclude that the system under analysis provides a transmit-diversity equal to two if ρ 12 = E {H 1 H * 2 } = 0, i.e., if the generalized fading gains H 1 and H 2 are uncorrelated [START_REF] Veeravalli | On performance analysis for signaling on correlated fading channels[END_REF]. This conclusion follows from the fact that, by conditioning upon the transmitted signal-constellation point, H 1 and H 2 are complex Gaussian random variables, for which uncorrelatedness implies independence. By taking into account that we assume i.i.d. fading with unit-power, and that, by definition, x = y and x = ŷ (and, thus, E h x h * y = E h xh * ŷ = 0), we have:

ρ 12 = sps q E {hxh * x} -sps pE n hxh * ŷ o + sqs q E {hyh * x} -s psq E n hyh * ŷ o + s psq E {h xh * x } -s pspE ˘hx h * y + s q sqE ˘hŷ h * x ¯-s q spE ˘hŷ h * y ¯ (9)
Let us analyze ρ 12 in (9) when h x = h x and h ŷ = h y , which corresponds to the computation of the PEP of the conventional (M-PSK) modulation scheme embedded into SM. In this case, it is easy to see that ρ 12 = 0. Thus, adding SM on top of the Alamouti code does not reduce the transmit-diversity. However, in order to make sure that the transmit-diversity of the system is equal to two we need to prove that all the PEPs have transmit-diversity equal to two.

Let us study two possibilities for the spatial-constellation diagram: i) the constellation points belong to non-overlapping sets, i.e., the pairs (h x , h y ) and (h x, h ŷ ) have no elements in common, and ii) the constellation points belong to overlapping sets, i.e., the pairs (h x , h y ) and (h x, h ŷ ) have one element in common. As an example, consider a system setup with N t = 4 and N h = 2. A system with spatial-constellation diagram having points (h 1 , h 2 ) and (h 3 , h 4 ) belongs to the first case study, while a system having points (h 1 , h 2 ) and (h 1 , h 3 ) belongs to the second case study. From ( 9), we conclude that: i) if the spatial-constellation diagram is composed by non-overlapping sets of points, then ρ 12 = 0 and, thus, the transmit-diversity of the system is always Div tx = 2; on the other hand ii) if the spatial-constellation diagram is composed by overlapping sets, then ρ 12 = 0, and it is not possible to guarantee that the system has transmit-diversity equal to two.

In conclusion, the second result of this paper is as follows: if the shaping filters at the transmitter are all the same, transmitdiversity equal to two can be guaranteed by partitioning the spatial-constellation diagram into non-overlapping sets of antennas. However, a multi-stream receiver is needed at the destination for optimum ML-decoding.

B. Different Shaping Filters

In Section III-A, we have shown that, when the same shaping filters are used at the transmitter, receiver complexity is the price to be paid to get Div tx = 2. In this section, we prove that the adoption of different shaping filters allows us to achieve the desired transmit-diversity gain without the need of a multi-stream decoder. The properties that the filters must satisfy to this end are derived by resorting to PEP analysis.

In this case, the decision metric in (2) with r 1 (•) and r 2 (•) in ( 1) can be still re-written as shown in ( 5), but D (s p) (h x,h ŷ ),(s p,s q ) and D (s q ) (h x,h ŷ ),(s p,s q ) are as follows:

D (sp) (hx,hŷ),(sp,sq) = |h x| 2 ˛sp ˛2 + ˛hŷ ˛2 ˛sp ˛2 -2Re ˘h * x h xs * p s p wx,x - 2Re n h * y h ŷ s p s * p wy,ŷ o -2Re ˘h * y h xs * q s p wy,x + 2Re n h * x h ŷ s q s * p wx,ŷ o -2Re n η * 1,x h x s po -2Re n η * 2,ŷ h y s * po ( 10
)
D (sq) (hx,hŷ),(sp,sq) = |h x| 2 ˛sq ˛2 + ˛hŷ ˛2 ˛sq ˛2 -2Re ˘h * y h ŷ s * q s q wy,ŷ - 2Re n h * x h xs q s * q wx,x o -2Re ˘h * x h ŷ s * p s q wx,ŷ + 2Re n h * y h xs p s * q wy,x o -2Re n η * 1,ŷ h y s q o + 2Re n η * 2,x h x s * q o (11)
where wa,b = T m w a (t) w b (t) dt and ηξ,b =

T m η ξ (t) w b (t) dt for a = {x, y}, b = {x, ŷ}, ξ = {1, 2} .
1) Analysis of Receiver Complexity: From ( 10) and ( 11), we conclude that a single-stream receiver can be obtained if the following two conditions are verified simultaneously:

 wx,ŷ = R T m w x (t) w ŷ (t) dt = 0 wy,x = R T m wy (t) w x (t) dt = 0 (12)
which reduce to designing some shaping filters to be timeorthogonal to some others. In particular, w x (•) must be orthogonal to w ŷ (•), and w y (•) must be orthogonal to w x (•).

By assuming that the orthogonality conditions in ( 12) are both satisfied, the ML-optimum decoder in (2) can be decomposed into two independent ML-optimum single-stream decoders [START_REF] Jeganathan | Generalized space shift keying modulation for MIMO channels[END_REF] and [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF], which depend only on (s p , s p) and (s q , s q ), respectively. In other words, the low-complexity feature of Alamouti/STBCs is preserved. Each decoder has a computational complexity of the order of N h M correlations, and, thus, the overall complexity only increases linearly with M . We note that the decoders in [START_REF] Jeganathan | Generalized space shift keying modulation for MIMO channels[END_REF] and [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF] provide an estimate, (h x, h ŷ ), of (h x , h y ), and these estimates might be different. In our receiver implementation, we randomly choose one of them with equal probability.

In conclusion, the third result of this paper is as follows: ML-optimum low-complexity single-stream decoding can be guaranteed via an adequate choice of the precoding shaping filters at the transmitter. In particular, some pairs of filters (see [START_REF] Sugiura | A unified MIMO architecture subsuming space shift keying, OSTBC, BLAST and LDC[END_REF]) should have zero cross-correlation function.

2) Analysis of Transmit-Diversity: Finally, let us analyze if the low-complexity decoders in [START_REF] Jeganathan | Generalized space shift keying modulation for MIMO channels[END_REF] and [START_REF] Di Renzo | Performance comparison of different spatial modulation schemes in correlated fading channels[END_REF] provide the desired transmit-diversity gain. The study is here performed only for detecting symbol s p , but a similar analytical development can be used to prove the same result for symbol s q . The PEP is as follows:

PEP = PEP `(h x , h y ; s p ) → `hx , h ŷ ; s p´= Pr  D (sp) (hx,hy) (s p ) > D (sp) (hx,hŷ) (s p ) ff (13)
where 10) by taking into account [START_REF] Sugiura | A unified MIMO architecture subsuming space shift keying, OSTBC, BLAST and LDC[END_REF], i.e.:

D (s p) (h x,h ŷ ) (s p ) is (
D (sp) (hx,hŷ) (s p ) = |h x| 2 ˛sp ˛2 + ˛hŷ ˛2 ˛sp ˛2 -2Re ˘h * x h xs * p s p wx,x - 2Re n h * y h ŷ s p s * p wy,ŷ o -2Re n η * 1,x h x s po -2Re n η * 2,ŷ h y s * po (14) and D (s p)
(h x ,h y ) (s p ) can be obtained from ( 14) by replacing (x, ŷ; p) with (x, y; p).

We start by analyzing the transmit-diversity gain when (13) reduces to the computation of the PEP of the conventional modulation scheme embedded into SM. In this case, we have x = x and ŷ = y, and the PEP in ( 13) is, after some algebra:

PEP = Q r γ h" |h x | 2 + |h y | 2 " ˛sp -s p ˛2i ! (15)
from which, we conclude, since h x and h y are i.i.d., that the proposed system offers transmit-diversity equal to Div tx = 2. Thus, with the adoption of the filter design in [START_REF] Sugiura | A unified MIMO architecture subsuming space shift keying, OSTBC, BLAST and LDC[END_REF], adding SSK modulation on top of the Alamouti code does not introduce any reduction in the transmit-diversity gain.

Let us now study the setup when the PEP in [START_REF] Di Renzo | Space shift keying (SSK-) MIMO over correlated Rician fading channels: Performance analysis and a new method for transmitdiversity[END_REF] does not reduce to the computation of the probability of error of the conventional modulation scheme. The analysis of all the possible combinations of shaping filters at the transmitter is quite complicated and, due to space constraints, cannot be entirely included in this paper. So, we present here only a simple case study. Among the possible choices of the shaping filters, we consider, inspired by the analysis in Section III-A.2, the setup where the points in the spatial-constellation diagram belong to non-overlapping sets, i.e., the pairs (h x , h y ) and (h x, h ŷ ) have no elements in common. Furthermore, we assume w x = w y and w x = w ŷ with wx,x = Tm w x (t) w x (t) dt = 0. In other words, the Alamouti code emitted by each pair of antennas is shaped by the same filter, and the filters used by the antennas constituting each point in the spatial-constellation diagram are time-orthogonal. With these assumptions, the PEP reduces to:

PEP = Q r γ h |hx| 2 + |hy| 2 + |h x| 2 + ˛hŷ ˛2i ! (16)
Since all the fading gains in ( 16) are different and i.i.d., we conclude that ( 16) has a transmit-diversity gain equal to Div tx = 4. We emphasize that the transmit-diversity gain of the system proposed in this paper is not four but only two since we have proved in [START_REF] Basar | Space-time block coding for spatial modulation[END_REF] that when the PEP reduces to the analysis of the error performance of the embedded conventional modulation scheme the diversity gain is Div tx = 2. However, the high diversity gain in ( 16) is expected to provide an additional and non-negligible coding gain, which can significantly increase the overall performance of the system.

In conclusion, the fourth result of this paper is as follows: ML-optimum low-complexity single-stream decoding with transmit-diversity of two can be guaranteed via an adequate choice of both the precoding shaping filters and the spatialconstellation diagram at the transmitter. In particular, some pairs of filters (see [START_REF] Sugiura | A unified MIMO architecture subsuming space shift keying, OSTBC, BLAST and LDC[END_REF]) must have zero cross-correlation function, and the spatial-constellation diagram should be a partition 4 of the transmit-antenna array.

IV. NUMERICAL RESULTS

In this section, we show some numerical results to substantiate the claims in the sections above, and to compare the proposed solutions with state-of-the-art transmit-diversity methods. Two case studies are considered: i) a worst-case setup, which achieves Div tx = 1 and needs a multi-stream decoder at the destination. This is obtained by using the same shaping filters in all the antennas at the transmitter along with a spatial-constellation diagram composed by overlapping sets of points (see Section III-A.2). This system is denoted by SM-STBC; and ii) a best-case setup, which achieves Div tx = 2 and needs a single-stream decoder at the destination. This is obtained by using different and time-orthogonal shaping filters at the transmitter along with a spatial-constellation diagram composed by non-overlapping sets of points (see Section III-B.2). This system is denoted by Time-Orthogonal-Signal-Design (TOSD-) assisted SM-STBC (TOSD-SM-STBC). Hermite pulse waveforms are used to guarantee the orthogonality in the time domain [START_REF] Di Renzo | Space shift keying (SSK) modulation: On the transmitdiversity/multiplexing trade-off[END_REF]. The rest of the simulation setup can be found in Section II.

In Fig. 1, a performance comparison of the proposed SM-STBC and TOSD-SM-STBC modulation schemes with SM and the Alamouti code for a rate of 3 and 5 bits/s/Hz is shown. Numerical results confirm the claims in Section III: SM-STBC achieves Div tx = 1, while TOSD-SM-STBC achieves Div tx = 2. In particular, TOSD-SM-STBC provides a substantial performance improvement with respect to the Alamouti code with almost the same receiver complexity. The performance improvement is due to the multiplexing gain introduced by using the spatial-constellation diagram on top of the Alamouti code, and to adequately choosing it to maintain the desired transmit-diversity gain. The price to be paid for this performance improvement is the need to use N h = 4 timeorthogonal shaping filters at the transmitter. Also, we notice that even SM-STBC can outperform the Alamouti code for a rate of 5 bits/s/Hz. The price to be paid in this case is the need of a multi-stream decoder at the receiver. Surprisingly, we notice that SM-STBC provides better performance when the size of the spatial-constellation diagram is smaller than the size of the signal-constellation diagram. This is an opposite behavior with respect to conventional SM [START_REF] Di Renzo | Performance analysis of spatial modulation[END_REF]. The reason is that, in this case, adding the SSK modulation part on top of the Alamouti code seems to destroy most of the advantages of this latter modulation scheme. By reducing the size of the spatial-constellation diagram we minimize this effect.

In Fig. 2, the performance of SM-STBC and TOSD-SM-STBC is compared to the more powerful STBCs introduced in [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF]. Let us emphasize here that STBC-H3 and STBC-H4 codes with rate-3/4 require three and four simultaneouslytransmitting antennas to achieve transmit-diversity three and four, respectively. On the contrary, SM-STBC and TOSD-SM-STBC only require two active antennas at the transmit-

E m /N 0 [dB] Alamouti [M=32] SM [Nt=2, M=16] SM [Nt=8, M=4] SM-STBC [Nt=4, Nh=4, M=16] SM-STBC [Nt=7, Nh=16, M=8] TOSD-SM-STBC [Nt=8, Nh=4, M=16]
Fig. 1. ABEP against E m /N 0 . Performance comparison with SM [START_REF] Mesleh | Spatial modulation[END_REF] and Alamouti [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] schemes for a rate of 3 bits/s/Hz (left) and 5 bits/s/Hz (right).

ter. We notice that for a low rate SM-STBC incurs in a substantial performance loss with respect to STBC-H3 and STBC-H4 codes: this is due to the impossibility of achieving transmit-diversity. On the contrary, TOSD-SM-STBC offers performance similar to the very powerful STBC-H3 and STBC-H4 codes. The ABEP of TOSD-SM-STBC degrades only for very high SNRs, where the higher diversity gain offered by STBC-H3 and STBC-H4 starts becoming more effective. Furthermore, we notice that the performance gain offered by TOSD-SM-STBC considerably increases when the data rate increases. We observe that the proposed lowcomplexity and full-transmit-diversity-achieving TOSD-SM-STBC modulation scheme provides much better performance than STBC-H3 and STBC-H4 codes for a very large range of SNRs. In the low-SNR regime, SM-STBC can outperform STBC-H3 and STBC-H4 codes as well, but this performance gain is achieved with a substantial increase in the receiver complexity, as SM-STBC requires a multi-stream detector.

In conclusion, the numerical examples shown in this section confirm the claims in Section III, and, more important, show that the proposed SM-STBC with optimized spatialconstellation diagram and orthogonal filter design can yield better performance than state-of-the-art multiple-antenna technology with comparable receiver complexity.

V. CONCLUSION

In this paper, we have introduced a new modulation/coding concept, which we have called "Spatially-Modulated STBCs". The new proposal takes advantage of SM and STBC technologies to design transmit-diversity and high-rate modulation schemes, which exploit the location-specific property of the wireless channel as an additional dimension for data transmission. By using PEP analysis, we have proposed some general methods to design low-complexity SM-STBCs. Furthermore, numerical results have shown that SM-STBCs outperform many state-of-the-art multiple-antenna schemes specifically designed to achieve transmit-diversity.

ACKNOWLEDGMENT

We gratefully acknowledge support from the EPSRC (EP/G011788/1) and the European Union (PITN-GA-2010-264759, GREENET project) for this

  [Nt=3, Nh=2, M=16] SM-STBC [Nt=5, Nh=8, M=8] TOSD-SM-STBC [Nt=4, Nh=2, M=16]

Fig. 2 .

 2 Fig.2. ABEP against E m /N 0 . Performance comparison with STBC-H3 and STBC-H4[START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF] codes for a rate of 1.5 bits/s/Hz (left) and 4.5 bits/s/Hz (right).

After the submission of this paper,[START_REF] Basar | Space-time block coding for spatial modulation[END_REF] was published. In that paper, a different approach to design STBCs for SM is proposed. We emphasize that single-stream decoding is sub-optimum for the solution in[START_REF] Basar | Space-time block coding for spatial modulation[END_REF].

In this paper, N h is assumed to be a power of two.

To avoid any misunderstandings with our notation, we note that wx (t) = wy (t) does not imply x = y, but x = y does imply w x (t) = w y (t).

A partition of a set X is a set P of non-empty subsets of X such that every element x in X is in exactly one of these subsets, i.e., the union of the elements of P is equal to X, and the intersection of any two distinct subsets of P is empty.


Nora M Hagelberg
email: nora.hagelberg@tyks.fi

Tuija H Nieminen

Teijo I Saari

Mikko Neuvonen

Pertti J Neuvonen

Kari Laine

Klaus T Olkkola

Interaction of oxycodone and voriconazole-a case series of patients with cancer pain supports the findings of randomised controlled studies with healthy subjects

Keywords: oxycodone, voriconazole, interaction, CYP3A

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

for pain relief prone to drug-drug interactions. Recent studies in healthy subjects show that in the elimination of oxycodone, CYP3A is more significant than CYP2D6. Inhibition of hepatic and/or intestinal CYP3A with voriconazole, itraconazole, ketoconazole, HIV protease inhibitors or grapefruit juice increases exposure to oral oxycodone in healthy subjects and may enhance its analgesic or adverse effects [START_REF] Hagelberg | Voriconazole drastically increases exposure to oral oxycodone[END_REF][START_REF] Grönlund | Exposure to oral oxycodone is increased by concomitant inhibition of CYP2D6 and CYP3A pathways but not by inhibition of CYP2D6 alone[END_REF][START_REF] Nieminen | Oxycodone concentrations are greatly increased by the concomitant use of ritonavir or lopinavir/ritonavir[END_REF][START_REF] Nieminen | Grapefruit juice enhances the exposure to oral oxycodone[END_REF][START_REF] Saari | Effects of itraconazole on the pharmacokinetics and pharmacodynamics of intravenously and orally administered oxycodone[END_REF][START_REF] Samer | The effects of CYP2D6 and CYP3A activities on the pharmacokinetics of immediate release oxycodone[END_REF][START_REF] Kummer | Effect of the inhibition of CYP3A4 or CYP2D6 on the pharmacokinetics and pharmacodynamics of oxycodone[END_REF][START_REF] Samer | Genetic polymorphisms and drug interactions modulating CYP2D6 and CYP3A activities have a major effect on oxycodone analgesic efficacy and safety[END_REF].

In this issue of European Journal of Clinical Pharmacology, Watanabe et al. describe a case report of a male patient with Burkitt's lymphoma who experienced enhanced analgesic and adverse effects of oxycodone during concomitant treatment of a systemic fungal infection with i.v. voriconazole [START_REF] Watanabe | Effects of voriconazole co-administration on oxycodone-induced adverse effectsa case in the retrospective survey[END_REF]. In addition, they present a retrospective case series of eight other patients with cancer pain who were predisposed to an interaction between oxycodone and voriconazole. Six of these patients experienced drowsiness, vomiting or hypopnea within a few days after starting voriconazole treatment and in some patients, the need for additional doses of opioid diminished. In one patient, analgesia was inadequate after discontinuation of voriconazole. The adverse effects observed in patients were more typical for opioids than voriconazole. Based on the pharmacokinetic results of previous interaction studies conducted in healthy subjects, the authors conclude that the observed symptoms in their patients were due to increased oxycodone plasma concentrations caused by CYP3A inhibition by voriconazole.

Polymorphism of CYP2D6 has been suggested to influence the pharmacokinetic profile and analgesic effect of oxycodone. Two of the patients in the case series did not experience alterations in the effects of oxycodone during voriconazole treatment. In ultrarapid CYP2D6 metabolisers, oxycodone concentrations have remained low even during concomitant administration of CYP3A inhibitors [START_REF] Hagelberg | Voriconazole drastically increases exposure to oral oxycodone[END_REF][START_REF] Saari | Effects of itraconazole on the pharmacokinetics and pharmacodynamics of intravenously and orally administered oxycodone[END_REF][START_REF] Samer | The effects of CYP2D6 and CYP3A activities on the pharmacokinetics of immediate release oxycodone[END_REF]. It is thus theoretically possible that one or both of these patients were ultrarapid metabolisers of CYP2D6. Similarly, drugs inhibiting CYP2D6 or the poor metaboliser CYP2D6 genotype, may partly explain the observed findings in patients with exaggerated opioid effects.

Although interaction studies of opioid analgesics in healthy subjects are useful and provide accurate pharmacokinetic data, there are limitations especially when it comes to assessing the impact of the interaction on the analgesic effect of the drug. The doses of opioids in studies in healthy subjects have to be kept substantially lower than what can be seen in clinical reality. On the other hand, experimental pain tests do not always correlate very well with clinical pain for a variety of reasons, and the log-linear relationship between drug concentration and effect may further hamper the interpretation. Because the main focus of studies conducted in healthy subjects is on the pharmacokinetic interaction, these studies are often underpowered in terms of demonstrating significant differences in pharmacodynamic measurements.

It is challenging to study drug-drug interactions in all clinical patient populations, especially in those who are critically ill or require multiple medications. Patient data from case series of clinical drug-drug interactions such as that presented by Watanabe et al. are valuable as they test the clinical applicability of interaction studies conducted in healthy subjects.

Conflict of interest

Dr. Klaus T Olkkola has worked as a pharmacokinetic consultant for Akela Pharma Inc., Turku, Finland. Kari Laine is shareholder of Medbase Ltd. Other authors declare that they have no conflicts of interest.


P Baraldi

M Compare

G Rossetti

E Zio

A Modelling Framework to Assess Maintenance Policy Performance in Electrical Production Plants

A framework to qualitatively assess the performance of maintenance policies in electrical production plants is presented. A Monte Carlo simulation scheme is combined with fuzzy logic for modelling the component degradation. The novelty of the work consists in the modelling of the influence of the actual living conditions on the degradation of the specific component under analysis; this is done by using linguistic fuzzy rules which formalize the expert knowledge on the degradation process. An example of application regarding a water-feeding turbo pump is presented to illustrate the potential of the proposed approach.

1.

Introduction

Since the opening of the electricity market, utilities have been forced to be more competitive by reacting promptly and reliably to the demand/offer dynamics. This entails efficient component maintenance policies, including ordinary maintenance, unexpected maintenance and restoration, which have a large impact on the plant performance and production costs. The establishment of a maintenance policy requires that various options be considered, of the type of maintenance plan (corrective, preventive, opportunistic, etc.), the type and timing of the maintenance tasks (overhaul, monitoring, scheduled replacement, etc.), the maintenance echelon (repair on site or in workshop), etc. The decisions on the maintenance policy must be taken considering various and typically conflicting criteria, e.g., availability, safety and costs. Given the dimension and complexity of the problem, maintenance policy decision making must be supported by system modelling and optimization.

In spite of the many efforts in this direction, it seems fair to say that in many practical maintenance decision making cases the situation is not so brilliant: maintenance policies are in many instances still based on the maintenance schedules recommended by the vendor, which are usually conservative or are only based on qualitative information driven by experience and engineering rationale [START_REF] Zio | Reliability engineering: old problems and new challanges[END_REF]; on the other hand, for their practical use maintenance optimization models should avoid excessive details and strive for a balance with the data available for the estimation of their parameters [START_REF] Van Rijn | Maintenance modeling and applications; lessons learned[END_REF].

A common approach for defining maintenance policy is based on the use of stochastic models of the degradation, failure and repair processes; [START_REF] Valdez-Flores | A survey of preventive maintenance models for stochastically deteriorating single-unit systems[END_REF], [START_REF] Singpurwalla | Survival in dynamic environment[END_REF], [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF] provide detailed surveys on the stochastic models that have been successfully used for maintenance modelling in different domains (e.g., hydraulic structures, dikes (van Noortwijk, 2009), pipelines [START_REF] Hong | Inspections and maintenance planning of pipelines under external corrosion considering generation of new defects[END_REF], cutting tools [START_REF] Jeang | Tool replacement policy for probabilistic tool life and random wear process[END_REF], etc.). These models can be grouped into two main categories:  Stochastic models describing the behaviour of the degradation process affecting a given component. It is assumed that failure occurs either when a critical parameter reaches a fixed or random threshold or when a traumatic event occurs (e.g., a random shock). Some examples of models of stochastic degradation processes belonging to this category can be found in (Barata, Guedes Soares, Marseguerra, & Zio, 2002), (Zille, Bérenguer, Grall, Despujols, & Lonchampt, 2007), [START_REF] Saassouh | Online maintenance policy for a deteriorating system with random change mode[END_REF], [START_REF] Deloux | Predictive maintenance policy for a gradually deteriorating system subject to stress[END_REF].

 Stochastic models describing the evolution of the values of the components failure rates. In particular, stochastic processes like the doubly stochastic Poisson processes, shot noise, etc. [START_REF] Singpurwalla | Survival in dynamic environment[END_REF] and [START_REF] Cox | Point Processes[END_REF], can be used for modelling directly the evolution of the failure rate of a component due to the influence of the dynamic environment. However, the failure rate of a component is an unobservable entity; thus, it is conceptually more direct to consider stochastic processes of covariates (which are observable and, often, measurable variables) for modelling the impact of the living conditions in which a component works on its failure behaviour. In this case, an hazard rate stochastic process is induced via stochastic processes of covariates [START_REF] Singpurwalla | Survival in dynamic environment[END_REF].

An alternative and pragmatic approach, proposed in (MIL-HDBK-217F, 1995) suggests to multiply the base value of the component failure rate by empirical factors to account for the specific living conditions (e.g., environment, working cycles, etc.). Despite its pragmatism, the approach is not directly applicable in support to maintenance decision making, which would require also the knowledge (even qualitative) of the component degradation level in order to model the effects of the maintenance actions.

To address this problem, (Zille, Bérenguer, Grall, Despujols, & Lonchampt, 2007) have proposed a framework for modelling the degradation of a component as a discrete stochastic process using parametric probability density functions (pdfs) to characterize the transition times among the degradation states and accounting for the influence of the component actual living conditions by modulating the pdfs parameters. This approach can effectively model the process of component degradation allowing the assessment of a maintenance policy, but relies on a large number of statistical parameters which may be difficult to estimate in real applications due to lack of experimental data.

Indeed, in practice expert judgement is often the main source of information on the degradation behaviour of components. To effectively deal with such type of information, a fuzzy approach to the estimation of the component degradation state is proposed in this work. Linguistic fuzzy rules elicited from experts are used to describe the influence of the living conditions on the component degradation process. From the knowledge of the degradation state of the component, its failure and repair rates are then determined and used within a Monte Carlo simulation scheme for computing system availability and costs.

The present paper is structured as follows: in Section 2 the simulation framework is presented; Section 3 describes the proposed degradation model; in Section 4 a case study is presented and the results are commented. Finally, come conclusions are proposed in the last Section.

Assessment of Maintenance Policy Performance

The assessment of the performance of a maintenance policy is usually based on the computation of the system availability and costs. When dealing with complicated systems, the use of Monte Carlo simulation is preferable, in order to avoid the introduction of excessively simplifying hypotheses in the representation of the system behaviour (Marseguerra & Zio, 2002).

In the proposed approach, the rates characterizing the failure processes are not fixed a priori, but depend on the component degradation state, which evolves during the stochastic life of the system. To follow the time-dependence of the failure rates, which emerges from the degradation process, the time domain is discretized by dividing the mission time into bins of fixed duration 𝐷𝑡; at the beginning of each time bin, all the Monte Carlo simulation model parameters are updated and kept constant through the bin.

A key issue in this computational framework is the estimation of the Monte Carlo parameters (i.e., failure rates and repair rates) which depend on the degradation state of the components.

In all generality, let us consider a system made up of 𝐶 components. The 𝑖-th component is affected by 𝑀 𝑖 degradation processes, which may be influenced by the failure/degradation behaviour of the other 𝐶 -1 components of the system. The generic 𝑗-th degradation process impacts on a number 𝑁 𝑗 𝑖 of failure modes whose stochastic occurrence in time is described by 𝑁 𝑗 𝑖 different failure rates 𝜆 𝑗 ,𝑘 𝑖 , 𝑘 = 1,2, … , 𝑁 𝑗 𝑖 .

A macro-state of availability 𝜉 𝑖 is assigned to the generic 𝑖-th component; it can take two mutually exclusive values:

 "ON" which indicates that the component is working.

 "OFF" which indicates that the component is not working.

Assuming independence between the degradation mechanisms, the value of the failure rate of the generic 𝑖-th component, whose macro-state is "ON", is:

𝜆 𝑖 = 𝜆 𝑗 ,𝑘 𝑖 𝑁 𝑗 𝑖 𝑘=1 𝑀 𝑖 𝑗 =1
The case of dependence between the degradation mechanisms is not addressed in this work; however, it seems important to remark that unjustifiably ignoring dependence may result in an underestimation of the system reliability [START_REF] Wang | Reliability prediction based on degradation modeling for systems with multiple degradation measures[END_REF]).

The macro-state "OFF" has a number of possible sub-states that specify the condition of the component while not working (e.g., under corrective maintenance, under preventive maintenance, waiting for the availability of the repair team, etc.). A repair state 𝜇 𝑖 is associated to the generic component 𝑖, whose macro-state is "OFF"; the value is assumed to be dependent directly on the type of the maintenance action, the influencing factors (IFs) of the actual living conditions and the degradation state of the component.

Estimation of the Parameters of the Monte Carlo Simulation Model

This section is dedicated to the description of the approach for the estimation of the values of the parameters needed in the Monte Carlo simulation modelling framework (i.e., failure and repair rates) taking into account the influencing factors of the actual living conditions.

Degradation and Failure Processes

The macro-state "ON" of the component 𝑖 can be further specified by a vector ) of the failure rates related to the 𝑁 𝑗 𝑖 failure modes influenced by the 𝑗-th degradation process is univocally associated.

𝑫 𝑖 = (𝐷
The modelling framework proposed appears quite flexible and detailed in the representation of the component degradation and failure behaviour. As such, it offers a general modelling power, which however must be carefully tailored in light of the data and information available for estimating the model parameters. Thus, it is expected that its use in practice be made in a form which is balanced (i.e., more or less detailed) with the robustness of the parameters estimation allowed by the particular case considered.

Fuzzy Parameter Estimation (FPE) model

The FPE model has the goal of estimating the values of the parameters needed by the Monte Carlo simulation model (i.e., failure and repair rates), taking into account the influencing factors of the actual living conditions. The FPE model interacts with the Monte Carlo simulation model whenever an estimation of the parameters is needed (i.e., at the begin of each time bin, when a failure occurs, when a maintenance action is performed).

The degradation process is modelled as a discrete-state process and a failure rate is associated to each degradation state; the problem is then that of linking the living conditions experienced by the component to its degradation state. Unfortunately, the relationships between the IFs and the degradation states are not completely known.

To handle the associated uncertainties, in this work a fuzzy logic approach is proposed, in which the link between the IFs and the degradation states is described by means of Fuzzy Rule Bases (FRBs).

Failure Rate Estimation

The first aim of the FPE model is to estimate the degradation state of the components taking into account the living conditions in which they work. In this respect, three issues have to be addressed: the identification of the IFs, their model description and the evaluation of their effects on each degradation mechanism. Three modules are proposed to tackle these issues:

 Central Module (CM);  Backward Module (BM);  Forward Module (FM).

Central Module

The Central Module is intended to identify the IFs that actually influence the considered degradation mechanism; both expert opinion and physical models from literature can constitute the knowledge base to support this phase. To aid this task, five general Influencing Factors (IFs) are introduced, divided into two groups. The first group includes the re-configurable IFs:

 IF 1 : Environment. It includes the environmental variables (temperature, humidity, vibration, etc.) which are expected to influence the degradation and failure behaviour of the component. It is considered re-configurable because some interventions can be done in order to modify its value; for example, the external temperature or humidity can be controlled, if possible, by setting up heating or air conditioning systems, the vibration level can be reduced by performing maintenance actions to eliminate the cause of vibration, etc.  IF 2 : Operational Mode. The set of variables which influence the stress conditions of the component (e.g. duty cycle, frequency of stops/re-starts, etc.); they can be changed during the life time of the component, depending on the demands and opportunities of operation.  IF 3 : Maintenance Policy. It contains all the variables related to the maintenance characteristics (corrective, preventive, opportunistic, frequency of inspection, etc.). Changes from periodic to condition-based maintenance or changes of the periods between two successive inspections may be economically profitable in different phases of the component life, so that the maintenance policy is often dynamically re-calibrated during the components mission time.

The second group contains the fixed, not re-configurable IFs:

 IF 4 : Age.

 IF 5 : Quality (e.g., of design, materials, manufacturing, etc.).

Backward Module

The Backward Module relates the selected IFs with measurable variables which describe the living conditions, by means of fuzzy logic-based models. The motivation for adopting a fuzzy logic modelling framework is due to the fact that most of the IFs are expected to be more easily represented and assessed by the experts using linguistic statements rather than numeric variables; for example, "the environment is favourable" or "the maintenance is efficient" are clear expert statements of qualitative nature, whereas it could be difficult to establish a numerical scale representing the environment or the maintenance.

The construction of the fuzzy logic model for the assessment of the generic 𝑖-th IF, IF i , is based on the following steps:

 Define the linguistic terms qualitatively representing the levels of the IF i (e.g., Environment = IF 1 ≡ (Soft, Medium, Heavy)).

 Set the range of variability (Universe of Discourse, UoD in fuzzy arithmetic terminology) of the IF i and partition it into the fuzzy sets representative of the defined linguistic terms, with their ambiguity.

 Identify the measurable physical variables that define the IF i (e.g., IF 1 ; measurable variables: "Temperature", "Humidity", "Width of the temperature range in a day", "Wind", etc.).

 Define the linguistic terms qualitatively representing the values that each measurable variable can take (e.g., Temperature ≡ (Low, Medium, High)).

 Partition the physical measurement ranges of the input physical variables into fuzzy sets representative of the linguistic terms (e.g., Temperature ≡ Low; support of the fuzzy set: -5, 20 °𝐶).

 Define the set of fuzzy inference rules (Fuzzy Rule Base, FRB), which relate the linguistic terms (fuzzy sets) of the input physical variables to those of the output level of the IF i (e.g., if "Temperature" is "High" and "Humidity" is "High" then "Environment" is "Heavy") and then the fuzzy inference engine to be used. In this work, a Mamdani-like fuzzy system has been employed [START_REF] Babuska | Fuzzy Modeling for Control[END_REF]).

Once the model has been developed, it can be used for the assessment of the IF i level. Notice that the output of the model is a fuzzy value which accounts for the ambiguity inherent in the assessment.

Forward Module

The Forward Module establishes the functional relationship between the level of the IFs and the failure rates, taking into account the influence of the living conditions on the degradation process. The objective of the FM is to provide a description, in terms of fuzzy rules, of how the IFs impact on the evolution of the degradation process. In particular, given the relationship between the degradation states and the failure rates, the FM is put into practice by assessing the vector of the degradation state, 𝑫 𝑖 , for those components whose macro-state is "ON". This assessment is performed by fuzzy logic models (one for each degradation mechanism of each component) built on FRBs which link the IFs to the vectors 𝑫 𝑖 (Baraldi, Zio, Compare, Rossetti & Despujol 2009).

Once the generic 𝑗-th degradation mechanism of the 𝑖-th component has been estimated, the vector 𝝀 𝑗 𝑖 of the failure rates towards the component failure modes is univocally associated. The numerical values of such failure rates can be obtained either from historical data or from expert opinion.

Obviously, there is no universal method for defining the FRB which models the relations of interest: the knowledge of the physical problem (e.g., literature models, expert judgement and historical data) needs to be formalized for expressing the role played by the various elements affecting the dynamics of the degradation process.

Repair Rate Estimation

The repair rates, 𝜇 𝑖 , of the components under maintenance, i.e. for which 𝜉 𝑖 = "OFF", generally depend on their living conditions and on the type of the maintenance action. The estimation of the repair rates is based on:

 The current time instant, 𝑡, provided by the system clock time.

 The vectors 𝑫 𝑖 , 𝑖 = 1,2, … , 𝐶, of the components degradation states. The maintenance actions, in practice, may depend on the values of these vectors, e.g. a repair action is foreseen if the state achieved by the 𝑗-th degradation mechanism, 𝐷 𝑗 𝑖 , is greater than a threshold 𝑑 𝑗 𝑖 . The repair rates also depend on the values of the vectors 𝑫 𝑖 , e.g., the Mean Time To Repair (MTTR) of the degradation process 𝑗 may be different from that of the degradation process 𝑘.

The knowledge base of this module is the maintenance manager, which univocally associates a maintenance action to a certain component state (i.e., defines the maintenance policy). In the particular case when the maintenance actions have a fixed duration, the MC module is provided directly with the value of duration of such actions.

Case Study

Introduction

In order to illustrate the application of the proposed methodology, an example concerning a Water-Feeding Turbo Pump (WFTP) of a steam generator of a nuclear power plant has been considered. A team of experts has identified the degradation processes affecting the components of the WFTP and the associated IFs and symptoms. A risk-importance analysis has been carried out to identify the components and their degradation processes, whose detailed modelling would improve the accuracy in the representation of the system failure behaviour. Following the obtained results, the case study for the present work has been focused on the contact fatigue degradation mechanism of the seals of the WFTP. No consideration has been given to other components and their degradation processes, although some may lead to an acceleration of the degradation process under consideration.

The degradation of the seals of the WFTP due to contact fatigue is caused by the development of cracks that affect the ability of the seals to avoid leaks. The creation and propagation of these cracks is a complex physical phenomenon, which has been modelled in a number of different ways [START_REF] Marquis | Fatigue design and Reliability[END_REF], [START_REF] Shigley | Standard Handbook of Machine Design[END_REF]). According to these models, the degradation is mainly influenced by the loads applied on the component, its constitutive materials and production process and some geometrical factors characterizing the crack such as its size, notch radius, position with respect to the direction of the loads, etc.

The model presented in this work is based on the assumption that the length of the most critical crack of the component defines its degradation level. Moreover, it is assumed that the length of the crack can only increase in time and preventive maintenance on the component has the effect of decreasing the speed of propagation of the crack (e.g., reducing the fatigue strength by increasing the notch radius of the most critical crack (Marquis & Solin, 1999)), but cannot reduce its length. In the modelling, the following three degradation states are considered:

1. "Good": the component is as new or almost new; no maintenance actions are foreseen if the component is in this state; the failure rate is 𝜆 = 10 -5 ℎ -1 . 2. "Medium": the seals of the WFTP in this state need some actions aimed at decreasing the crack growth rapidity; the failure rate in this state is 𝜆 = 5 • 10 -4 ℎ -1 . 3. "Bad": if the component is in this degradation state it is convenient to replace it; the failure rate in this state is 𝜆 = 10 -3 ℎ -1 .

Notice that the superscripts and subscripts of the failure rate, 𝜆 𝑗 ,𝑘 𝑖 , previously introduced have been omitted since only one component with one degradation mechanism and one failure mode is considered.

In this case study, the following maintenance actions are assumed to be performed on the component:

 Control: periodic overhaul check of the component. It is assumed that the degradation state is easily visible and to be detected it does not need a laboratory test or disassembling of the component. This action is considered to be of negligible duration. Furthermore, this is the only scheduled action.

 Preventive maintenance: maintenance action conditioned by the result of a control. The preventive maintenance action is dependent on the result of a control action. If the component is found to be in state "Good", no action is performed. If the degradation state is "Medium", the component undergoes a repairing action aiming at slowing down the degradation process: this action has a duration of 2 ℎ. Finally, if the component is in state "Bad", it is replaced: this action takes 20 ℎ of time.

 Corrective maintenance: maintenance action following a failure of the component. The corrective action is assumed to be the replacement of the component. Due to the fact that this event is unscheduled, this action brings an additional duration of 10 ℎ, with respect to the replacement after a control, leading to a total duration of 30 ℎ. In particular, the additional time may be caused by the supplementary time needed for performing the procedure of replacement after failure or to the time elapsed between the occurrence of the failure and the start of the replacement actions.

Since in this case study all the considered actions have deterministic duration, it is not necessary to estimate the repair rates.

The values of the parameters characterizing the system are reported in The final objective of this case study is the identification of the optimal control period (i.e., the time interval which leads to the best performance in terms of availability and costs).

Fuzzy Parameter Estimation

Failure Rate Estimation

In this Section, the three modules described in Section 3.2.1 are put to work for estimating the failure rates in the considered case study.

Central Module

The definition of the model of the degradation process requires the identification of the IFs influencing the degradation state evolution of the component. In the present case study, this has been done by resorting to a FMECA analysis performed by safety analysts. The identified IFs are the following:

 IF 1 : Environment. It is assumed that the influence of the environment on the considered degradation mechanism is mainly caused by the vibrations in the location at which the component works. In particular, the measurable variables on which the IF 1 depends are the mean values of the frequency and of the amplitude of the vibration fundamental wave in the time elapsed since the component has started to work. The Universe of Discourse (UoD) of this IF, arbitrarily scaled on [0,1], is partitioned into three Fuzzy Sets: "Soft", "Medium" and "Heavy".  IF 3 : Maintenance. The component is periodically inspected by operators to control its degradation level. The maintenance policy, a priori established, requires that no maintenance action is performed if the degradation state is found "Good" at control whereas a corrective maintenance action is performed if the component is found in state "Medium" and a replacement action is carried out when the component is in degradation state "Bad". A variation of the period between two successive controls causes a modification of the degradation process; in particular, the more frequent are the controls the less is the time in which the degradation advances without any action for reducing its speed. To describe this IF, the three Fuzzy Sets "Frequent", "Medium" and "Rare" are identified by the expert on the UoD 0, 2 • 10 4 ℎ of the control period variable.  IF 4 : Age. This IF measures the time since the component has been working. The UoD of this IF is the interval [0, 𝑇 𝑀𝑖𝑠𝑠 ], with mission time 𝑇 𝑀𝑖𝑠𝑠 = 10 5 ℎ; on this interval, three Fuzzy Sets "Young", "Medium" and "Old" are defined by the expert by means of triangular membership functions. In general, the older the component the higher its degradation level.

The membership functions defining the fuzzy sets of the IFs are defined by experts on the basis of their knowledge and engineering sense of practice (Figure 1Figure 1).

Backward Module

The tailoring of the BM to the considered case study consists in identifying the physical variables on which the IF 1 depends (the IF 3 and the IF 4 are already directly described by the variables control period and time, respectively). The vibration level, whose range of variability has been arbitrarily set to [0, 1], adequately characterizes the defined IF 1 and its value is computed starting from the values of two physical variables measured by means of sensors (e.g., strain gauges): amplitude and frequency of the vibration fundamental wave. In particular, the mean values of these variables in the time elapsed since the system has started to work are given in input to the BM, which links them to the IF 1 by means of a FRB.

Figure 2Figure 2 shows the fuzzy sets, defined by means of triangular membership functions, partitioning the variables in input to the BM:

1. "Low", "Medium" and "High" are the fuzzy sets defined on the UoD [0, 5] 𝑚𝑚 describing the mean value of the amplitude of the fundamental wave; Table 2 Table 2 shows the rules that model the influence of the mean values of the Amplitude and the Frequency on the IF 1 . For example, the bottom-right element of Table 1 represents the rule: if Amplitude is Low and Frequency is Low then IF 1 is Soft.

Mean frequency of the fundamental wave

High Medium Low

Mean amplitude of the fundamental wave

High

Heavy Heavy Medium

Medium

Medium Medium Soft

Low

Medium Soft Soft

Table 2. Fuzzy rules defining the relationship between the inputs and the outputs of the BM tailored to the IF1.

269

Generally speaking, the vibration in the location in which the system of interest works is caused by other components either because they are degrading (e.g., the increase of the eccentricity of the centre of gravity in rotating machines) or because they have been designed in such a way that a periodic load is applied on the other coupled components (e.g., alternating machines discharging loads on the same basement of the system of interest). Since, in general, the behaviour of both the components producing the vibration and the other components of the overall system (which modify the vibration wave) is stochastic, the vibration profile suffered by the components is also stochastic.

For simplicity, but without loss of generality, in the present case study, an arbitrarily chosen vibration profile is assumed in input to the BM, in terms of the mean amplitude and the mean frequency of the fundamental wave (Figure 3Figure 3).

Such profile "lived" by the component influences its degradation behaviour; the intensity of such influence is assessed by means of the dedicated fuzzy logic model built. 2 Table 2 within a Mamdani inference system results in the time profile of the degrees of activation of the Soft, Medium and Heavy levels of IF 1 reported in Figure 5Figure 5. The Medium level is the most activated for large part of the mission time; Soft and Heavy levels are less activated, and in a similar way. In the first part of the mission time, the rule "if Amplitude is High and Frequency is Low then Environment is Medium" has the largest activation degree whereas the rules "if Amplitude is High and Frequency is Medium then Environment is Heavy" and "if Amplitude is Medium and Frequency is Low then Environment is Soft" are those with largest activation degrees among those with "Heavy" and "Soft" consequents, respectively. With the vibration profile of Figure 3Figure 3, the two latter rules increase their activation degrees up to the central part of the mission time as the activation degree of the first rule becomes smaller; this leads to the three levels having almost the same degree of activation of about 0.5 at 𝑡 = 5.5 • 10 4 ℎ: at this time, there is complete uncertainty on the influence of the IF 1 on the degradation level of the component; then, in the central part of the mission time, the activation degree of the set "Medium" starts again to increase because the activation degree of the rule "if Amplitude is Medium and Frequency is Medium then Environment is Medium" becomes larger whereas the degrees of activation of the rules with consequents "Heavy" and "Low" begin to decrease.

Forward Module

The objective of the Forward Module is to provide a description, in terms of fuzzy rules, of how the IFs impact on the evolution of the degradation process. In other words, a FRB is built which links the identified IFs with the component degradation state and thus its failure rate.

In the considered case study, the Forward Module consists in identifying the failure rate of the seals of the WFTP. More precisely, a fuzzy model has been built based on rules as, for example: "if Environment is Soft and Maintenance is Frequent and Age is Young and Previous Degradation State is Good then Degradation State is Good". As before, the rules defining the FRB are obtained from expert knowledge. The antecedent "Previous Degradation State" has been introduced in order to ensure that the degradation state does not decrease as the age of the component increases.

The output fuzzy set "Degradation State" is eventually defuzzyfied to limit the propagation of the uncertainty. Defuzzyfication is done by simply selecting the degradation state with the highest degree of activation.

Figure 6Figure 6 shows the application of the proposed model on the component which lives in the environment previously introduced and inspected every 7000 ℎ, with no failures during the mission time.

Maintenance policy assessment

In the present Section, the results of the Monte Carlo unavailability estimation of the component presented in Section 4.1 are reported and discussed. The computational model has been developed in Fortran.  an increase in the unavailability due to the contribution of those simulated components which have had a failure before 𝑡 = 1.26 • 10 4 ℎ and thus reach the degradation state 2 with a delay;  a decrease of the unavailability due to the reduced failure rate (10 -5 ℎ -1 ) of those simulated components that have undertaken corrective maintenance.

The second effect is prevalent and thus the unavailability decreases. The second peak occurs at 𝑡 = 1.40 • 10 4 ℎ, when the first control occurs after the component has entered in degradation state 2 and thus all the simulated components that did not have a failure before are now unavailable, due to the downtime associated to the preventive maintenance action.

Notice that in the considered case study it is extremely unlikely to achieve the degradation state 3: with a failure rate associated to the degradation state 2 equal to 5 • 10 -4 ℎ -1 and a time interval of 4.98 • 10 4 ℎ between the achievement of the degradation states 2 and 3, the probability of encountering a system in a degradation state 3 is smaller than 𝑒 -5•10 -4 •4.98•10 4 = 1.39 • 10 -11 . This is the reason of the nonappearance of a peak of unavailability at 𝑡 = 6.3 • 10 4 ℎ, at which the component would reach the degradation state 3 (Figure 6Figure 6).

Maintenance Policy Optimization

The proposed framework has been used to optimize the maintenance policy described in Section 4; in particular, the optimization has been performed with respect to the Control Period. The mean unavailability shows an initially decreasing trend, with a first minimum in correspondence of a Control Period equal to 5000 ℎ and another, deeper one in correspondence of a Control Period of 10000 ℎ, after which the trend starts increasing. The maintenance cost has a similar trend, but with only the minimum in correspondence of a Control Period of 10000 ℎ.

One may then conclude that under the considered maintenance policy, the best Control Period is 10000 ℎ, with respect to both availability and costs. On the other hand, the relative flatness of the minimum is such that there is a wide interval of Control Period values in which both the mean unavailability and the maintenance cost are small and with little variations, which gives a margin of operational flexibility for choosing the Control Period value also accounting for other criteria (e.g., opportunistic maintenance).

Conclusions

A novel modelling framework has been proposed for assessing the impact of the adopted maintenance policy and of the specific conditions in which a component works on the performance of the overall system of which the component is part.

Given the lack of experimental evidence on the influence of the living conditions of a component on its degradation, expert judgment is often used. This has suggested the use of a fuzzy approach for representing the expert knowledge in the degradation model. Then, Monte Carlo simulation is used to assess the goodness of the maintenance policy in terms of system availability.

To illustrate the approach, the modelling framework has been applied to the seals of the WFTP and their degradation due to contact fatigue. The proposed modelling approach has allowed the optimization of the control period, within the considered maintenance policy. The example has shown the potential of the approach but some issues remain open:

 The the validation on a multi-component system;

 The the definition of the maintenance policy is done only in terms of the control period and in terms of definition of the thresholds between the degradation levels. It does not account for the effectiveness of the maintenance tasks, the dependence of the effect of the maintenance action on some other factors (e.g., the number of maintenance interventions occurred in the past or the time since the last maintenance), the human errors that can happen when a maintenance action is performed, etc. Inclusion of these aspects could be needed when performing the comparison of different maintenance policies;

 The the operation of defuzzyfication performed on the output of the Forward Module, does not propagate the uncertainties affecting the degradation state reached by the component. This leads to MC simulations which sample from exponential distributions without considering the uncertainty of the parameters of those distributions.;

 The the Mamdani inference limits the activation degrees of the degradation states to values smaller than 1, i.e., it is not guaranteed that the maximum of the activation degree of the degradation state is equal to 1. This problem, which leads to a smaller confidence on the degradation state, may be overcome by considering more sophisticated inference systems.

As concluding remark, it seems important to stress one more time that the practical exploitation of the flexibility offered by the modelling framework proposed needs to be carefully managed with respect to the data available for the estimation of the many model parameters involved; this is fundamental in order to avoid overparameterization of the model.

Figure 1 .

 1 Figure 1. Fuzzy sets partitioning the degradation state (a), the IF 'Environment' (b), the IF "Maintenance" (c) and the IF "Age" (d).

Formattato:

  Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt 2. "Low", "Medium" and "High" are the fuzzy sets defined on the UoD [0, 200] 𝐻𝑧 describing the mean value of the frequency of the fundamental wave.

Figure 2 .

 2 Figure 2. Fuzzy sets of the variables in input to the Backward Module.

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Figure 4Figure 4

 4 shows the activation profile in time of the fuzzy sets Low, Medium and High, representative of the vibration conditions in terms of mean amplitude (top Figure) and frequency (bottom Figure) of the fundamental wave. The combination of these activations by the FRB of Table

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Figure 3 .

 3 Figure 3. Vibration profile applied to the component, in terms of mean amplitude (to the left) and mean frequency (to the right).

Figure 4 .

 4 Figure 4. Degrees of activation of the fuzzy sets partitioning the variables in input to the Backward Module for the given vibration profile.

Figure 5 .

 5 Figure 5. Degrees of activation of the fuzzy sets partitioning the IF1, for the given vibration profile.

Figure 6 .

 6 Figure 6. Activation degree of the IFs fuzzy sets and of the degradation state, failure rate value and defuzzyfied degradation state considering a control period of 7000 h when no failure occurs. The evolution of IF 1 (Figure 6Figure 6(a)) and IF 4 (Figure 6Figure 6(c)) is straightforward until the time instant 𝑡 = 6.3 • 10 4 ℎ, when the component is found to be in the degradation state "Bad" and it is replaced by a new one, whose age is zero and with no accumulated vibration. From that time on, the IF 1 is computed taking into account the vibration suffered by the newly installed component and the IF 4 evolves naturally as its age. The IF 3 (Figure 6Figure 6(b)) is constant, regardless the replacement of the component, since the maintenance policy is the same throughout the mission time.

Figure 6Figure 6

 6 Figure 6Figure 6(e) shows the defuzzyfied degradation state of the component, which directly determines the failure rate value (Figure 6Figure 6(d)).

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  Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt Formattato: Tipo di carattere: 11 pt [Type text] 273 The instantaneous unavailability of the component with the related 68.3% confidence interval (i.e., plus and minus one standard deviation) is shown in Figure 7Figure 7.

Figure 7 .

 7 Figure 7. Instantaneous component unavailability and its standard deviation. Two large peaks appear in the first part of the component mission time. The first, at 𝑡 = 1.26 • 10 4 ℎ, corresponds to the time instant in which the degradation process has a transition from degradation state 1 to 2, with the failure rate of the component worsening from 10 -5 ℎ -1 to 5 • 10 -5 ℎ -1 . After 𝑡 = 1.26 • 10 4 ℎ two different conflicting trends are observed:

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Figure 8 Figure 8 and

 88 Figure 8 Figure 8 and Figure 9Figure 9 show the mean unavailability of the component and related 68.3% confidence interval and the maintenance costs for varying values of the Control Period.

Figure 8 .Figure 9 .

 89 Figure 8. Estimated mean unavailability varying the Control Period, with related 68.3% confidence interval.

Table 1

 1

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Table 1 .

 1 These have been set with arbitrary engineering rationale.

	Parameter	Value
	Mission time	10 5 ℎ
	Control Period	7000 ℎ
	Duration of a repair action	2 ℎ
	Duration of a replacement action	20 ℎ
	Additional time related to a failure	10 ℎ
	Cost of a control action	5 $
	Cost of a repair action	10 $
	Cost of a replacement action	1000 $
	Additional cost related to a failure	1000 $

Table 1 .

 1 Values of the parameters characterizing the system.

  Table 3Table 3 shows the values of the parameters used in the case study:

	Parameter	Value
	𝐷𝑡	100 ℎ
	Number of MC trials	10000
	CPU time (Intel Pentium, 1.6 GHz)	56 𝑠𝑒𝑐

Table 3 .

 3 Monte Carlo parameters.

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