Ana Rivero
email: ana.rivero@ed.ac.uk

S A West

The physiological costs of being small in a parasitic wasp

Keywords: lipids, parasitoids, resource allocation, size-fitness relationship

Knowledge of the relationship between body size and fitness is essential in many models of parasitoid evolutionary ecology. One clear point emerging from previous studies is that the relationship between size and fitness can depend strongly (qualitatively and quantitatively) on the environment in which it is measured. Our results show that, in the parasitoid wasp Nasonia vitripennis, the relationship between parasitoid size and fitness is strongly dependent on the nutritional status of the female; being small is more costly when food is not available. Lipid availability is the main factor mediating the relationship between size and fitness. Small females suffer disproportionately the costs of not feeding because they emerge with small lipid reserves and thus have to rely more heavily on carbohydrates for survival and reproduction. In other words, feeding is more important for small females. Our results (1) provide a physiological explanation for why laboratory estimates of the size-fitness relationship are not as steep as field estimates and (2) suggest that females are likely to show size-dependent foraging and feeding strategies.

INTRODUCTION

Parasitoid wasps have proved to be excellent experimental models for testing a variety of theories on the evolution of life-history strategies [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF]. Female wasps search the environment for hosts. Upon finding a host, each female must make a series of decisions, such as whether to feed or oviposit on the host, how many eggs to lay (clutch size) and what sex the offspring should be (sex ratio) (Charnov andSkinner, 1984, 1985). In many cases, theory has been shown to be able to predict reasonably well the reproductive strategies (behaviour) of females. This is due to the simple link between reproductive behaviour and fitness in these animals; the fitness consequences of, for example, feeding or laying an egg in a host are immediate and relatively clear.

However, many hypotheses about life-history evolution in parasitoids have been tested qualitatively but not quantitatively [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF]. The reason for this is that quantitative tests require accurate estimates of the relationship between parasitoid size and fitness. For example, application of clutch size theory provides qualitative explanations for variation in clutch size across and within species, but quantitative predictions require accurate estimates of the size-fitness relationship [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF]. Consequently, in recent years there has been a considerable increase in attempts to estimate the size-fitness relationship in parasitoids, both in the laboratory and in the field [START_REF] Visser | The importance of being large: the relationship beween size and fitness in females of the parasitoid Aphaereta minuta (Hymenoptera: Braconidae)[END_REF][START_REF] Kazmer | Field tests of the size-fitness hypothesis in the egg parasitoid Trichogramma pretiosum[END_REF][START_REF] West | The relationship between parasitoid size and fitness in the field, a study of Achrysocharoides zwoelferi (Hymenoptera: Eulophidae)[END_REF][START_REF] Ellers | A field study of size-fitness relationships in the parasitoid Asobara tabida[END_REF][START_REF] Ellers | Seasonal changes in female size and its relation to reproduction in the parasitoid Asobara tabida[END_REF].

One clear point to emerge from these studies is that the size-fitness relationship can depend strongly (qualitatively and quantitatively) on the environment in which it is measured (e.g. whether a food source or hosts are provided) and the fitness measure used (e.g. egg production or longevity) [START_REF] King | Length of life of the sexes in Nasonia vitripennis (Walker) (Hymenoptera: Pteromalidae) under conditions of starvation[END_REF][START_REF] Benson | Intraspecific competiton in the population dynamics of Bracon hebetor Say (Hymenoptera: Braconidae)[END_REF][START_REF] Tagaki | The reproductive strategy of the gregarious parasitoid, Pteromalus puparium (Hymenoptera: Pteromalidae) 1. Optimal number of eggs in a single host[END_REF][START_REF] Hardy | Clutch size in a parasitoid wasp: a manipulation experiment[END_REF][START_REF] Bai | Effect of host availability on reproduction and survival of the parasitoid wasp Trichogramma minutum[END_REF][START_REF] Visser | The importance of being large: the relationship beween size and fitness in females of the parasitoid Aphaereta minuta (Hymenoptera: Braconidae)[END_REF][START_REF] West | The relationship between parasitoid size and fitness in the field, a study of Achrysocharoides zwoelferi (Hymenoptera: Eulophidae)[END_REF]. A possible explanation for this is that wasps of different sizes are differentially allocating resources to reproduction and survival according to environmental conditions, such as the availability of food. This could involve a number of interactions between environmental conditions and wasp size, mediated by, and depending heavily upon, physiological mechanisms. For example, when food is abundant, can it enable small females to make up for a lack of stored reserves relative to large females? Also, does female size influence how the different energy resources, such as lipids and sugars, are stored and utilized?

Here we examine whether environmental conditions determine the size-fitness relationship in Nasonia vitripennis, a parasitoid wasp that matures eggs throughout its lifetime. Specifically, we focus upon the influence and interaction between two factors: (1) the amount of food (honey solution) given to adults (females were either fed ad libitum or starved) and (2) body size. We measure the influence (and interaction) of these variables upon two main fitness traits -egg production and longevity -as well as on the underlying physiological resources: ovarian proteins and lipid, glycogen and sugar reserves. We provide a physiological explanation for why field estimates of size-fitness relationships are steeper than laboratory estimates and we discuss the implications for life-history evolution.

MATERIALS AND METHODS

Parasitoid biology

Nasonia vitripennis (Hymenoptera: Pteromalidae) is a gregarious, host-feeding ectoparasitoid of dipteran pupae in bird nests. Females are synovigenic: they are born with a very limited number of mature eggs and egg production and maturation are continuous throughout the life of the female (in contrast to 'proovigenic' parasitoids, which are born with their lifetime complement of eggs; [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF].

In the field, hosts occur in small isolated patches of between 1 and 39 hosts per nest. The number of females ovipositing in each patch can vary from 1 to 9 [START_REF] Molbo | Mating structure and sex ratio variation in a naural population of Nasonia vitripennis[END_REF]. As a result, the number of adult parasitoids that can be reared successfully from a single host pupa varies enormously (anything between 10 and 50; [START_REF] Werren | Sex ratio evolution under local mate competition in a parasitic wasp[END_REF][START_REF] Molbo | Mating structure and sex ratio variation in a naural population of Nasonia vitripennis[END_REF], which leads to a large variation in adult body size [START_REF] Charnov | Complementary approaches to the understanding of parasitoid oviposition decisions[END_REF][START_REF] Molbo | Mating structure and sex ratio variation in a naural population of Nasonia vitripennis[END_REF][START_REF] Flanagan | Local mate competition, variable fecundity and information use in a parasitoid[END_REF]. The flightless males emerge before the females and mate with them as they emerge. Females are winged and need to disperse after mating to find new, suitable hosts in which to feed and lay eggs [START_REF] Werren | Sex ratio evolution under local mate competition in a parasitic wasp[END_REF][START_REF] Molbo | Mating structure and sex ratio variation in a naural population of Nasonia vitripennis[END_REF]. Until a food source is found, females run 'on empty', relying on the resources accumulated as a larva. Temporal starvation is thus a likely scenario in the field, and its duration will greatly depend on the quality of the environment in which the females forage.

Experimental set-up

Parasitoids were reared by placing 10-15 female parasitoids with about 30 fly pupae (Calliphora vicina, Diptera: Calliphoridae) in specimen tubes (7.5 cm long, 2.5 cm in diameter). Under these high-density conditions, the resulting parasitoids show a wide variation in body size (personal observation). The parasitoid cultures and the experiment were carried out at the same temperature (25ЊC) and under the same light : dark photoperiod (16 : 8). Experimental females were obtained by dissecting parasitized hosts and isolating 12-dayold female parasitoid pupae individually in glass tubes (7.5 cm long, 8 mm in diameter). Tubes were kept under standard temperature and light conditions and checked every 2 h. On emergence, females were immediately allocated at random to one of the treatment combinations detailed below. Females were kept in the tubes for 24, 48 or 72 h. During this time, the females were deprived of hosts to avoid the confounding effects of ovipositing behaviour, as in many parasitoids egg production, and hence the pattern of resource allocation, depends on the rate of oviposition (e.g. [START_REF] Rivero-Lynch | The dynamics of egg production, oviposition and resorption in a parasitoid wasp[END_REF].

Since host-feeding and egg-laying cannot be decoupled, females in this experiment had no access to a host meal. However, an important alternative source of nutrients for many parasitoids in the field is sugar-rich sources such as nectar or honeydew [START_REF] Leius | Attractiveness of different foods and flowers to the adults of some hymenopterous parasites[END_REF][START_REF] Jervis | Flower-visiting by hymenopteran parasitoids[END_REF][START_REF] Gilbert | Functional, evolutionary and ecological aspects of feeding-related mouthpart specializations in parasitoid flies[END_REF][START_REF] Jervis | Functional and evolutionary aspects of mouthpart structure in parasitoid wasps[END_REF]. To determine the effect of food on resource allocation, half of the females in each treatment were fed ad libitum on a rich honey solution. For this purpose, a piece of filter paper (∼30 mm 2 ) soaked in a solution of honey in water (∼30% honey by volume) was placed inside the tube on a daily basis (where appropriate). The other half were given a similar piece of paper soaked in water (starved). After the specified experimental period had elapsed, females were killed and stored by deep freezing them at -80ЊC for subsequent dissection and biochemical analysis (see below).

In addition to these six treatment combinations (replication as follows: 24 h unfed, n = 21; 24 h fed, n = 19; 48 h unfed, n = 15; 48 h fed, n = 15; 72 h unfed, n = 15; 72 h fed, n = 16), three other treatments were set up. To determine the reserves at emergence (teneral reserves), a randomly chosen group of females was killed and stored at -80ЊC immediately after emergence (0 h) (n = 14). Finally, to determine longevity and the resources at death (assumed to be the minimum resources possible in an organism), two additional groups of females, one fed on honey solution as above (n = 14) and one starved (n = 17), were kept in the tubes until they died. Longevity was recorded and female bodies were then frozen as above for subsequent biochemical analysis.

Dissections and sample preparations

Females of all treatments were removed from the freezer and allowed to thaw for about 1 h. One of the legs was pulled out and the length of the hind tibia (a standard measurement of size in parasitoids; [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF] was measured with a micrometer. Females were then dissected under a binocular microscope in 50 µl 2% sodium sulphate to extract the ovarioles. The ovarioles were then placed on a slide and the number of mature eggs determined under the optical microscope. Egg maturity was assessed following the method of [START_REF] King | Oogenesis in Nasonia vitripennis (Walker) (Hymenoptera: Pteromalidae)[END_REF]. Ovarioles were placed in a 1.5 ml Eppendorf tube, crushed with a pin and 100 µl of a solution of physiologic water (0.15  NaCl) and 0.001% Triton X-100 (Sigma-Aldrich) was added. Controls consisted of the physiologic water and Triton solution but no ovaries. Tubes were left for 5 days at 4ЊC for the Triton to dissolve the proteins in the ovaries.

To prepare the dissected body of the female for biochemical analysis [START_REF] Van Handel | Assay of lipids, glycogen and sugars in individual mosquitoes: correlations with wing length in field-collected Aedes vexans[END_REF], it was placed in a Pyrex glass tube (7.3 cm long, 1 cm in diameter) and crushed with a glass rod. The sodium sulphate in which it was dissected was recovered with a pipette and added to the tube. An extra 50 µl sodium sulphate and 750 µl of a chloroform-methanol solution (1 : 2, v : v) were added to the tube, which was then left at room temperature for 24 h. The same procedure was followed for the controls but without the female's body.

Biochemical analyses

Protein analysis of the ovaries was carried out using the Bradford dye-binding microassay procedure [START_REF] Bradford | A rapid and sensitive method for the quantitation of microgram quantities of protein utilizing the principle of protein-dye binding[END_REF]. For this purpose, 80 µl of the sample were extracted into a plastic tube (7.5 cm long, 1 cm in diameter), to which 720 µl physiologic water and 200 µl Bradford reagent (Bio-Rad Laboratories, Munich, Germany) were added. After 15 min, samples were read in a spectrophotometer (Jenway-6300) at OD 595 . Protein concentrations were obtained from a standard curve based on bovine serum albumin (Sigma-Aldrich).

Quantification of the body reserves of the females was carried out using a modification of the colorimetric technique developed by [START_REF] Van Handel | Assay of lipids, glycogen and sugars in individual mosquitoes: correlations with wing length in field-collected Aedes vexans[END_REF] for mosquito analysis. This technique allows the determination of lipids, glycogen and sugars (glucose) on the same specimen. Tubes containing the sample with the sodium sulphate and chloroform-methanol mixture (see above) were centrifuged at 1500 rev • min -1 for 15 min. Two 350 µl fractions of the supernatant were extracted into separate Pyrex glass tubes, one for lipid analysis and the other for sugar analysis. The solvent in all tubes was evaporated completely in a heating block at 95ЊC.

For lipid determination, 40 µl of sulphuric acid were added to the tubes and heated at 95ЊC for 2 min; then, 960 µl of vanillin reagent were added (for details of reagent preparation, see van Handel, 1985b), left for 10 min to allow the colour to develop and read in the spectrophotometer at OD 525 against the control. Lipid concentrations were obtained from a standard curve done with commercial vegetable oil (1 mg • ml -1 in chloroform; van Handel, 1985b). For sugar determination, 1000 µl anthrone reagent (for details, see van Handel, 1985a) were added to the tubes, heated at 95ЊC for 15 min and read at OD 625 against the control. Sugar concentrations were obtained from a standard curve done with glucose (1 mg • ml -1 in 25% ethanol; van Handel, 1985a).

The precipitate in the original tube containing all the glycogen was washed with 2000 µl 80% methanol to get rid of any remaining sugars. Tubes were recentrifuged at 1500 rev • min -1 and the supernatant was disposed of. Glycogen concentrations were determined by adding 1000 µl anthrone reagent to the washed precipitate and heating at 95ЊC for 15 min. Since the glycogen fraction contains all the fractured fragments of the female exoskeleton, the samples had to be transferred into 1.5 ml Eppendorf tubes and recentrifugued at 14,000 rev • min -1 for 10 min to precipitate the fragments before reading at OD 625 . As for the sugars, concentrations were obtained from a standard glucose curve.

Statistical analyses

The effects of size and food availability (starved or fed ad libitum) were analysed with generalized linear modelling techniques available in the GLMStat (v.5.3, 2000) statistical package. The significance of each explanatory variable and their interactions was assessed by removing it from the model and analysing the resulting change in deviance (see [START_REF] Crawley | GLIM for Ecologists[END_REF]. After the minimal model (the model including only significant terms and interactions) was obtained, its appropriateness was tested by inspecting a plot of the residuals against the fitted values. Bonferroni-corrected contrasts were carried out using the SPSS statistical package (SPSS Inc., 1995). The significance of Bonferroni-corrected contrasts was determined by establishing whether the 95% confidence intervals included 0 (Norus ˇis, 1995).

RESULTS

Egg production and longevity

All females, irrespective of their size, emerged with no mature eggs in their ovarioles. To determine the pattern of egg production with time, an initial analysis was carried out using time and feeding status as factors and size as a covariate. In this analysis, the three-way interaction was significant (F 2,99 = 8.25, P < 0.01; Fig. 1), indicating that the slope of the size-eggs relationship is dependent on both feeding status and time since emergence (Fig. 1). Subsequent analyses were carried out separately for each of the time intervals. The number of mature eggs present 24 h after emergence was highly dependent on size (F 1,38 = 27.14, r 2 = 0.43, P < 0.001), but independent of whether the females were fed or not (F 1,38 = 0.00, ..) (Fig 1a). The next two time intervals showed a similar pattern, whereby egg load has highly size-dependent. Here, however, the slope of the size-egg relationship was significantly steeper for unfed than for fed females (F 3,29 = 34.52, r 2 = 0.71, P < 0.001 and F 3,30 = 18.15, r 2 = 0.67, P < 0.001 for the minimal model containing the interaction between size and feeding status at 48 h and 72 h respectively) (Fig. 1b,c).

Fed females lived about five times longer than unfed ones (15.71 ± 2.51 days and 3.94 ± 0.23 days, respectively; mean ± standard error). The interaction between size and feeding status was significant (F 1,30 = 13.60, P < 0.001). Female longevity when not fed was very highly significantly dependent on size (t-test for the significance of the slope, t = 3.90, P < 0.005) (Fig. 2a). However, although there was a trend in the same direction for fed females, the slope of the line was not statistically significantly different from zero (t = 1.91) (Fig. 2b).

Protein reserves

Protein is one of the main components of the egg yolk of insects [START_REF] Nijhout | Insect Hormones[END_REF] and parental nutritional experience has been shown to influence egg protein levels in some species [START_REF] Rossiter | Initiation of maternal effects in Lymantria dispar: genetic and ecological components of egg provisioning[END_REF]. Size had a significant effect on the total protein concentration in ovaries (F 1,112 = 17.86, r 2 = 0.13, P < 0.001), and females fed on honey solution for 72 h had significantly higher protein concentrations in their ovaries than females in the other treatment combinations (F 5,112 = 3.32, r 2 = 0.14, P < 0.001). The interaction between size and the time or sugar regime was, however, not significant (F 5,112 = 1.10, ..). The amount of protein invested per mature egg was estimated by dividing the total amount of proteins by the number of mature eggs. This, however, was not correlated with female size in either fed (F 2,46 = 0.48, ..) or unfed (F 2,47 = 0.20, ..) females.

Lipid reserves

Females were born with a high amount of lipid deposits (45.37 ± 3.33 vs 15.03 ± 2.83 µg of sugars and 17.43 ± 1.58 µg of glycogen). The amount of teneral lipids was strongly correlated with female size (F 1,12 = 36.02, r 2 = 0.75, P < 0.001), with the largest females having twice as many teneral lipids as the smallest ones (Fig. 3a,b). Lipid reserves decreased rapidly with time so that by 72 h only half of the original reserves remained (21.37 ± 2.03 µg for fed females and 21.34 ± 3.18 µg for unfed females).

To establish the pattern of lipid use with time, we carried out analyses of covariance for the effects of size and time (0, 24, 48 and 72 h) separately for fed and unfed females. In both cases, the slope of the lipid-size relationship changed significantly with time (fed: 3a,b). Contrast analysis, however, revealed differences between fed and unfed females. Unfed females show a drastic decrease in the slope of lipids between birth (0 h) and the first 24 h (99% joint Bonferroni confidence intervals: lower = -6.15, upper = -1.03) (Fig. 3b). When starved, large females thus use disproportionately more lipids in the first 24 h than smaller females. In contrast, in fed females, the decrease in lipid reserves in the first 24 h was proportional to female size (95% joint Bonferroni confidence intervals between 0 and 24 h: lower = -1.82, upper = 2.82, ..) (Fig. 3a). Thereafter, however, we found no drastic, statistically significant, changes in the slope of successive time intervals in either fed (between 24 and 48 h: lower = -0.39, upper = 3.55, ..; between 48 and 72 h: lower = -1.83, upper = 2.44, ..) or unfed females (between 24 and 48 h: lower = -1.92, upper = 1.27, ..; between 48 and 72 h: lower = -1.74, upper = 1.00, ..).

Glycogen and sugar reserves

Neither glycogen nor sugar reserves at birth was significantly correlated with female size (F 1,11 = 0.03 and F 1,12 = 0.03, respectively; ..). Although glycogen reserves at birth were comparatively low when compared with the lipid reserves (see above), they were significantly higher than the resources at death in both fed (1.32 ± 0.53 µg; F 1,27 = 64.73, r = 0.54, P < 0.001) and unfed (4.37 ± 0.59 µg; F 1,30 = 56.49, r 2 = 0.58, P < 0.001) females (Fig. 4a,b).

While in fed females glycogen reserves did not change with time (F 3,62 = 1.45, ..) (Fig. 4a), a significant interaction between size and time was seen in unfed females (F 3,64 = 5.3, r 2 = 0.15, P < 0.005). Contrast analysis revealed that this significant interaction was due to the relationship between glycogen stores and size being significantly steeper at 24 h than at 0 h (95% joint Bonferroni confidence intervals: lower = 0.93, upper = 3.35). Despite being unfed, large females thus appear to accumulate glycogen reserves in the first 24 h after birth (Fig. 4b).

Females emerged with a very small amount of sugar reserves relative to lipids (see above). These sugar reserves at birth were indeed statistically indistinguishable from the resources at death (8.35 ± 3.18 µg; F 1,28 = 1.00, ..). Sugar reserves were independent of size in both fed (F 1,61 = 0.03, ..) and unfed females (F 1,63 = 2.55, ..). Fed females, however, accumulated significant amounts of sugars with time (F 3,61 = 14.38, r 2 = 0.43, P < 0.0001).

DISCUSSION

Our results show that the relationship between parasitoid size and fitness is strongly dependent on the nutritional status of the female; females pay a greater cost for being small when they are starved. Specifically: (a) from 48 h onwards, the slope of the fecundity-size relationship is significantly steeper when the female is unfed (Fig. 1b,c) and (b) the effect of size on longevity is only statistically significant in unfed females (Fig. 2a). In the remainder of this discussion, we examine the pattern of resource allocation that leads to these relationships and its implications for our understanding of life-history strategies.

To understand the role of feeding in small and large females, we carried out a detailed physiological analysis of resource allocation. For this purpose, we quantified the three main energetic resources in the insect body: sugars, glycogen and lipids [START_REF] Rivero | Incorporating physiology into parasitoid behavioral ecology: the allocation of nutritional resources[END_REF]. Of these, sugars are largely due to feeding, while glycogen and lipids are storage compounds and are therefore regulated by physiological mechanisms [START_REF] Van Handel | Assay of lipids, glycogen and sugars in individual mosquitoes: correlations with wing length in field-collected Aedes vexans[END_REF]. All three resources can be burned to fulfil the various energetic requirements of the insect body but, so far as is known, only lipids are directly involved in egg production [START_REF] Nijhout | Insect Hormones[END_REF]. From an adaptive point of view, lipids are a key physiological currency in parasitoids, as they appear to be the common resource underlying the trade-off between reproduction and survival [START_REF] Ellers | Fat and eggs: an alternative method to measure the trade-off between survival and reproduction in insect parasitoids[END_REF][START_REF] Ellers | Life history evolution in Asobara tabida: plasticity in allocation of fat reserves to survival and reproduction[END_REF].

In N. vitripennis, the largest females emerge with twice as much lipid as the smallest females (Fig. 3a,b). Lipid reserves appear irreplaceable through feeding, as they decrease with time even in females fed ad libitum. Interestingly, these results are in line with those of two previous studies, where, in contrast with other insect taxa [START_REF] Nayar | The fuel for sustained mosquito flight[END_REF][START_REF] Brown | Migration of the monarch butterfly, Danaus plexippus: energy sources[END_REF][START_REF] Van Handel | Metabolism of nutrients in the adult mosquito[END_REF][START_REF] Warburg | Effects of energetic reserves on behavioural patterns of Mediterranean fruit flies (Diptera: Tephritidae)[END_REF], parasitoids also seemed unable to synthesize lipids de novo [START_REF] Ellers | Fat and eggs: an alternative method to measure the trade-off between survival and reproduction in insect parasitoids[END_REF][START_REF] Olson | Effects of sugar-feeding on carbohydrate and lipid metabolism in a parasitoid wasp[END_REF]. Further detailed studies need to be carried out to determine how widespread this characteristic is in parasitoids. The adaptive consequences of this lack of lipogenesis are immediately obvious: teneral lipids would represent a non-replaceable resource and thus are potentially limiting, particularly for small females.

The pattern of use of lipid reserves in the first few days after birth indeed suggests that large females rely more heavily on lipid reserves than small females (Fig. 3a,b). The pattern of lipid use is, however, dependent on the nutritional status of the females. While in fed females lipid reserves decrease progressively with time (Fig. 3a), large unfed females appear to make use of a disproportionate amount of their lipid reserves in their first 24 h of life (Fig. 3b). The largest females, when starved, lose on average over 30 µg of fat (roughly 50% of their reserves) during the first day. There is no straightforward explanation for this observation, given that the number of eggs produced in the first 24 h interval is similar for fed and unfed females (Fig. 1a) and that there is no evidence of differences in egg quality between females of different sizes. Yet, the drastic decrease in lipids concomitant with a drastic increase in egg production suggests that these two phenomena are likely to be associated. Moreover, a higher metabolic rate associated with increasing body size, a wellknown phenomenon in animals [START_REF] West | A general model for the origin of allometric scaling laws in biology[END_REF], cannot explain why the disproportionate use of lipids only occurs in the first 24 h. One potential explanation is that the energetic costs associated with egg production are partly met by sugars in fed females, but exclusively by lipids in unfed ones, as is thought to be the case in some insects [START_REF] Engelmann | The Physiology of Insect Reproduction[END_REF][START_REF] Clements | The Biology of Mosquitoes[END_REF], although the energy costs incurred by egg production require further study [START_REF] Wheeler | The role of nourishment in oogenesis[END_REF]. Large unfed females would thus have to burn more lipids to fulfil the energetic requirements associated with producing up to 15 eggs in the first 24 h of life (Fig. 1a).

The decrease in lipid reserves in starved females appears to be associated with an increase in glycogen reserves taking place in the same period of time. Despite being unfed, the largest females gained on average 10 µg of glycogen over the first 24 h, increasing their overall reserves by more than 50%. In insects, glycogen is an important resource for flight [START_REF] Nayar | The fuel for sustained mosquito flight[END_REF][START_REF] Mason | Influence of carbohydrate deprivation and tethered flight on stored lipid, fecundity and survivorship of the soybean looper (Lepidoptera: Noctuidae)[END_REF][START_REF] Wheeler | The role of nourishment in oogenesis[END_REF][START_REF] Zera | Biochemical aspects of flight and flightlessness in Gryllus: flight fuels, enzyme activities and electrophoretic profiles of flight muscles from flight-capable and flightless morphs[END_REF]. An attractive explanation would be that, faced with a harsh local environment, large females can afford to transform some of the lipids into fuel for flight. This explanation is tentative and requires further investigation. Despite the importance of dispersal in models of parasitoid behaviour and population dynamics, there have so far been no studies on the physiology of flight in parasitoids.

Irrespective of the destiny of the lipids burned in the first 24 h, thereafter all unfed females, irrespective of their size, start their lives with a similar amount of lipid reserves. How can we explain, then, why the largest females live longer? Starved females may rely on the other source of stored energy, glycogen, which from 24 h on is correlated with size (Fig. 4a). N. vitripennis, however, has at its disposal another important source of nutrients: the eggs themselves [START_REF] King | The rate of egg resorption in Nasonia vitripennis (Walker) (Hymenoptera: Pteromalidae) deprived of hosts[END_REF][START_REF] Hopkins | Egg resorption in Nasonia vitripennis (Walker) (Hymenoptera: Pteromalidae)[END_REF]. Although egg resorption is widely thought to provide nutrients for metabolic maintenance in parasitoids (Leius, 1961a,b;[START_REF] King | Length of life of the sexes in Nasonia vitripennis (Walker) (Hymenoptera: Pteromalidae) under conditions of starvation[END_REF][START_REF] Van Lenteren | The parasite-host relationship between Encarsia formosa Gahan (Hymenoptera: Aphelinidae) and Trialeurodes vaporariorum (Westwood) (Homoptera: Aleyrodidae). XVI. Food effects on oogenesis, oviposition, life-span and fecundity of Encarsia formosa and other hymenopterous parasites[END_REF], we are unaware of any formal studies that quantify its role in extending female longevity.

In summary, we suggest that lipid availability is largely responsible for the nutritiondependent relationship between size and fitness. Periods of temporal starvation are likely to occur in the field, where newly emerged females need to disperse from their original patches to find new hosts in which to feed and lay eggs [START_REF] Werren | Sex ratio evolution under local mate competition in a parasitic wasp[END_REF][START_REF] Molbo | Mating structure and sex ratio variation in a naural population of Nasonia vitripennis[END_REF]. Small females suffer disproportionately the costs of not feeding because they are born with small lipid reserves and thus need to rely more heavily on carbohydrates for survival and reproduction. Large unfed females, on the other hand, rely heavily on their large lipid reserves, which provide roughly twice the amount of calories per unit as carbohydrates [START_REF] Nijhout | Insect Hormones[END_REF].

The consequences of these observations for adaptive behaviour should be explored. In particular, because the fitness cost of not feeding is greater for small females, we might expect size-dependent foraging and feeding strategies. In the field, parasitoids can obtain their adult nutritional resources either from the host itself (host feeding) or from non-host sugary sources such as nectar or honeydew. Feeding incurs a cost for the female parasitoid, both in terms of time and energy investment (when hosts and food are found in different parts of the environment; [START_REF] Sirot | Time sharing between host searching and food searching in parasitoids: state-dependent optimal strategies[END_REF] and in terms of lost reproductive opportunities (when an egg cannot be laid in the same host where host-feeding took place; [START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF]. Since the fitness costs of starvation are higher for small females, they may be predicted to spend more time foraging for food and be more likely to host-feed (possibly increasing further the slope of the size-fitness relationship under field conditions). This suggestion could be examined theoretically and tested empirically.

Many hypotheses about parasitoid ecology and life-history evolution require reliable estimates about the relationship between female size and fitness (Charnov andSkinner, 1984, 1985;[START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF]. Our results provide a physiological explanation for why sizefitness relationships are steeper in the field [START_REF] Hardy | Clutch size in a parasitoid wasp: a manipulation experiment[END_REF][START_REF] Godfray | Parasitoids: Behavioural and Evolutionary Ecology[END_REF][START_REF] West | The relationship between parasitoid size and fitness in the field, a study of Achrysocharoides zwoelferi (Hymenoptera: Eulophidae)[END_REF][START_REF] West | Sex allocation and clutch size in parasitoid wasps that produce single-sex broods[END_REF], although other mechanisms may also be at work [START_REF] West | The relationship between parasitoid size and fitness in the field, a study of Achrysocharoides zwoelferi (Hymenoptera: Eulophidae)[END_REF]. This study adds to the glowing number of areas in evolutionary ecology in which an understanding of the physiological basis of resource allocation is required for further progress [START_REF] Jervis | Incorporating physiological realism into models of parasitoid feeding behaviour[END_REF][START_REF] Rivero | Incorporating physiology into parasitoid behavioral ecology: the allocation of nutritional resources[END_REF].

Fig. 1 .

 1 Fig. 1. Regression of number of mature eggs against female size at different time intervals. Solid line and solid circles are fed females; dashed line and empty circles are unfed females.

Fig. 2 .

 2 Fig. 2. Regression of female longevity against size in (a) unfed and (b) fed females.

F

  3,63 = 3.31, r 2 = 0.25, P < 0.05; unfed: F 3,64 = 7.52, r 2 = 0.13, P < 0.001) (Fig.

Fig. 3 .

 3 Fig. 3. Regression of lipid reserves against female size at different time intervals for fed females (a) and unfed females (b). Line for lipid reserves at birth is given as reference and is the same for (a) and (b).

Fig. 4 .

 4 Fig. 4. Regression of glycogen reserves against female size at different time intervals for fed females (a) and unfed females (b). Line for glycogen reserves at birth is given as reference and is the same for (a) and (b).

Rivero and West

Physiological costs of being small

ACKNOWLEDGEMENTS

We would like to thank J. Casas, H. Ferguson, D. Giron and M. Jervis for very useful comments on the manuscript and A. Graham for help with the experiments. This study was funded by the Leverhulme Trust and the BBSRC.


X Bombois

F Morelli

H Hjalmarsson

L Bako

K Colin

Robust optimal identification experiment design for multisine excitation

Keywords:

In least costly experiment design, the optimal spectrum of an identification experiment is determined in such a way that the cost of the experiment is minimized under some accuracy constraint on the identified parameter vector. Like all optimal experiment design problems, this optimization problem depends on the unknown true system, which is generally replaced by an initial estimate. One important consequence of this is that we can underestimate the actual cost of the experiment and that the accuracy of the identified model can be lower than desired. Here, based on an a-priori uncertainty set for the true system, we propose a convex optimization approach that allows to prevent these issues from happening. We do this when the to-be-determined spectrum is the one of a multisine signal.

Introduction

We consider in this paper the problem of optimally designing the spectrum Φ u of the excitation signal u of an open-loop identification experiment. By optimal spectrum, we here mean the spectrum yielding the smallest experiment cost while guaranteeing that the accuracy of the identified parameter vector of the plant transfer function is larger than a given threshold. We thus consider the least costly experiment design framework [START_REF] Bombois | Least costly identification experiment for control[END_REF], but the approach can easily be adapted to other (dual) frameworks [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF][START_REF] Rojas | On the equivalence of least costly and traditional experiment design for control[END_REF][START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF]. The experiment cost J can be defined as a linear combination of the power of the excitation signal u and of the power of the part of the output signal induced by u. The experiment cost will therefore be a function of the spectrum Φ u , but also of the unknown true parameter vector θ 0 (we therefore denote the cost as J (θ 0 , Φ u )). Likewise, the accuracy constraint will also depend on θ 0 and on Φ u since the classical accuracy constraints are of the type P -1 (θ 0 , Φ u ) ≥ R adm where P (θ 0 , Φ u ) is the covariance matrix of the to-be-identified parameter vector (which depends on θ 0 and Φ u ) and R adm a matrix reflecting the desired accuracy. The dependency of the optimal spectrum Φ u,opt on the unknown true parameter vector θ 0 is the so-called chickenand-egg issue encountered in optimal experiment design. This issue is generally circumvented by replacing θ 0 by an initial estimate θinit of θ 0 since, in this case, the optimal experiment design problem boils down to a convex optimization problem (see e.g. [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF][START_REF] Bombois | Least costly identification experiment for control[END_REF]). However, this approach has the drawback that the optimal spectrum is not guaranteed to yield the desired accuracy and that the experiment cost computed with θinit and Φ u,opt can underestimate the actual experiment cost. These observations are at the root of the research area on robust optimal experiment design (see [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF] for a good survey).

In robust experiment design, different lines of research have been considered. In [START_REF] Rojas | Robustness in experiment design[END_REF], a spectrum that yields good accuracy for a very broad set of systems (also of different orders) is discussed. However, in the engineering literature, the most widely used approach is the one that consists in considering an uncertainty set U containing the unknown true parameter vector θ 0 (the socalled min-max design [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF][START_REF] Pronzato | Robust experiment design via optimization[END_REF]). The optimal experiment design problem can then be formulated as determining the spectrum Φ u minimizing the value of a scalar γ under the constraints that J (θ, Φ u ) ≤ γ ∀θ ∈ U and that P -1 (θ, Φ u ) ≥ R adm ∀θ ∈ U . If we denote by γ opt and Φ u,opt the solution of this optimization problem, we have the guarantee that P -1 (θ 0 , Φ u,opt ) ≥ R adm and that γ opt is an upper bound for the actual experiment cost J (θ 0 , Φ u,opt ). However, finding a tractable approach to deal with such a robustified optimal experiment design Preprint submitted to Automatica problem is still an open research question. While this optimization problem can be exactly solved in very particular and simple situations (see e.g. [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF][START_REF] Abrashov | Simple and Robust Experiment Design for System Identification Using Fractional Models[END_REF]), the general approach when facing more complex systems is to replace the initial uncertainty set (containing an infinite number of elements) by a number n of grid points of this uncertainty set U (see e.g. [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF][START_REF] Bombois | Least costly identification experiment for control[END_REF][START_REF] Rojas | Robust optimal experiment design for system identification[END_REF]). Consequently, the cost constraint and the accuracy constraint over the set U in the robustified optimal experiment design problem are replaced each by n constraints (one for each grid point). Even though it is obviously better from a robustification point-of-view than just replacing θ 0 by one grid point i.e. θinit , this relaxation of the original robustified optimal experiment design problem cannot yield the guarantees linked to the original problem and it can become computationally heavy for large values of n.

In [START_REF] Kumar | Robust plant friendly optimal input design[END_REF][START_REF] Forgione | Data-driven model improvement for model-based control[END_REF][START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF], approaches are presented to uniquely tackle the robustified cost constraint J (θ, Φ u ) ≤ γ ∀θ ∈ U (i.e. in the accuracy constraint, θ 0 is replaced by θinit ). However, these approaches all entail some approximation: a first-order approximation in [START_REF] Forgione | Data-driven model improvement for model-based control[END_REF], a second-order approximation in [START_REF] Kumar | Robust plant friendly optimal input design[END_REF] and an approximation based on the unscented transform in [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF].

Our main contribution in this paper is to present an approach in order to tackle the robust optimal experiment design problem without approximation. For this purpose, we observe that, except for its dependence on the to-be-determined spectrum, the robustified cost constraint and the robustified accuracy constraint are similar to constraints that are treated in robustness analysis. Based on this observation and on the separation of graph framework [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF][START_REF] Goh | Robust analysis, sectors and quadratic functionals[END_REF][START_REF] Megretski | System analysis via integral quadratic constraints[END_REF], we derive constraints that are linear in the decision variables of the optimal experiment design problem and that imply the original robustified cost and accuracy constraints. We do that for one of the most commonly used parametrization of the to-be-determined spectrum Φ u i.e. the one corresponding to a multsine [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF]. We however restrict attention to Box-Jenkins (BJ) model structures and to accuracy constraints on the parameters of the plant transfer functions (we can indeed not robustify the noise part of the covariance matrix using the tools presented in this paper).

Notations. The matrix

     X1 0 0 0 . . . 0 0 0 XN     
will be denoted diag(X 1 , ..., X N ) if the elements X i (i = 1, ..., N ) are scalar quantities while it will be denoted bdiag(X 1 , ..., X N ) if the elements X i (i = 1, ..., N ) are matrices. In addition, the Fourier transform of a signal x(t) is denoted x(e jω ), I n represents the identity matrix of dimension n and ⊗, the Kronecker product. Finally, 0 represents a vector or a matrix containing only zeros and A * is the conjugate transpose of the complex matrix A.

Identification

We consider a single-input single-output true system with input u and output y:

y(t) = G 0 (z)u(t) + H 0 (z)e(t) =v(t) (1)
where v(t) = H 0 (z)e(t) is the disturbance acting on the system. In (1), e(t) is a white noise with variance σ 2 e and G 0 (z) and H 0 (z) are stable transfer functions. In addition, H 0 (z) is also assumed to be inversely stable and monic.

The true system (1) will be identified in a BJ model structure i.e., {G(z,

θ) = G(z, ρ), H(z, θ) = H(z, ζ) | θ = (ρ T , ζ T ) T ∈ R k }.
The orders of G(z, θ) and H(z, θ) are chosen in such a way that there exists

θ 0 = (ρ T 0 , ζ T 0 ) T such that G 0 (z) = G(z, θ 0 ) = G(z, ρ 0 ) and H 0 (z) = H(z, θ 0 ) = H(z, ζ 0
). We will denote by k G (resp. k H ) the dimension of ρ 0 (resp. ζ 0 ) and we have thus

k = k G + k H .
If we apply a sequence {u(t) | t = 1, ..., N } of spectrum Φ u to (1) and collect the corresponding output {y(t) | t = 1, ..., N }, an estimate θN of θ 0 can be deduced using prediction error identification [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF]:

θN = arg min θ 1 N N t=1 2 (t, θ) (t, θ) = H -1 (z, θ) (y(t) -G(z, θ)u(t)) (2)
The estimate θN is (asymptotically) normally distributed around θ 0 with a covariance matrix P θ (θ 0 , Φ u ) whose known expression is a function of θ 0 and of the input spectrum Φ u used during the identification experiment [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF]. It is important to note that P -1 θ (θ 0 , Φ u ) is a measure of the accuracy of θN [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF]. In general, we are mainly interested in the accuracy of the part ρN of the vector θN = (ρ T N , ζT N ) T (the part that defines the model G(z, ρN ) of G(z, ρ 0 )). The covariance matrix P ρ of ρN can be deduced from P θ as follows:

P ρ = (I k G 0) P θ (I k G 0) T ,
and its inverse has the following expression as a function of Φ u and θ 0 [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF][START_REF] Bombois | Open-loop vs. closed-loop identification of box-jenkins systems in a least costly context[END_REF]:

P -1 ρ (θ 0 , Φ u ) = N σ 2 e 1 2π π -π F u (e jω , θ 0 )F * u (e jω , θ 0 )Φ u (ω) dω (3) with F u (z, θ) = H -1 (z, ζ) ∂G(z, ρ) ∂ρ (4)
In the sequel, we will suppose that an initial identification experiment has delivered an initial estimate θinit = (ρ T init , ζT init ) T with covariance matrix P θ,init .

Consequently, the following ellipsoid U init is a η%confidence region for the unknown true parameter vector θ 0 :

U init = θ | (θ -θinit ) T P -1 θ,init (θ -θinit ) ≤ χ (5)
with χ such that P r(χ 2 (k) ≤ χ) = η (say 95 %). The significance of this set is that, if the initial experiment and the estimation of θinit is repeated, the true parameter θ 0 will belong to U init in the fraction η of these essays. For this reason, U init can be used as a description of the uncertainty of the initial estimate. From now on, we will assume that θ 0 indeed belongs to the ellipsoid U init constructed based on the initial estimate θinit and its covariance matrix P θ,init . We will suppose that the accuracy P -1 ρ,init of ρinit is not sufficient for the purpose of the identified model (P ρ,init = (I k G 0)P θ,init (I k G 0) T ). In the sequel, we will deem an estimate of ρ 0 sufficiently accurate when the inverse of its covariance matrix satisfies

P -1 ρ ≥ R adm for a given positive-definite matrix R adm ∈ R k G ×k G .
In order to obtain a sufficiently accurate estimate of θ 0 , we need to perform a second experiment that yields an estimate θN = (ρ T N , ζT N ) T having the property that the covariance matrix P ρ (θ 0 , Φ u ) of ρT N is such that P -1 ρ (θ 0 , Φ u ) ≥ R adm (Φ u is the spectrum of the input signal used during the second experiment). Note that the initial and the second identification experiments can also be combined [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF] and the inverse P -1 ρ (θ 0 , Φ u ) of the covariance matrix corresponding to the second experiment must then satisfy: P -1 ρ (θ 0 , Φ u ) + P -1 ρ,init ≥ R adm . For the sequel, it will be important to make the following assumptions:

Assumption 1 We restrict attention to parametrizations G(z, θ) and H(z, θ) that are rational functions of the parameter vector θ:

G(z, θ) = Z N (z)θ 1 + Z D (z)θ (6) H(z, θ) = 1 + Z N,H (z)θ 1 + Z D,H (z)θ ( 7
)
where Z N (z), Z D (z), Z N,H (z) and Z D,H (z) are row vectors of transfer functions.

Assumption 2

The uncertainty U init defined in ( 5) is small enough to guarantee that, like G(z, θ 0 ) and H -1 (z, θ 0 ), G(z, θ) and H -1 (z, θ) are stable transfer functions for all θ ∈ U init . Due to (4), this also implies that F u (z, θ) is a vector of stable transfer functions for all θ ∈ U init .

Note that the classical BJ parametrization used in prediction error identification satisfy Assumption 1 (see Appendix A). Moreover, we can easily verify whether a given U init satisfies the property mentioned in Assumption 2 using the results in [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF].

Optimal experiment design

As done in the least costly experiment design framework [START_REF] Bombois | Least costly identification experiment for control[END_REF], we will design the spectrum Φ u of the excitation signal u of the second experiment in such a way that the accuracy constraint is met with the least perturbation on the system (i.e. with the least identification cost). The perturbation on the system induced by u will be here measured by a linear combination of the power of the input signal and of the power of y(t) = G(z, θ 0 )u(t):

J (θ 0 , Φ u ) = 1 2π π -π 1 + β |G(e jω , θ 0 )| 2 Φ u (ω) dω (8)
where β is an user-chosen constant that weighs the two terms in J . We observe that the cost J (θ 0 , Φ u ) is a function of the unknown θ 0 and of the chosen spectrum Φ u . Based on the above expression, the optimal experiment design can be formulated as the problem of determining the power spectrum Φ u of the second experiment which guarantees that P -1 ρ (θ 0 , Φ u ) + P -1 ρ,init ≥ R adm with the smallest cost J (θ 0 , Φ u ). Like all other optimal experiment design problems, the above optimization problem unfortunately depends on the unknown true parameter vector. As mentioned in the introduction, we will here robustify this optimal experiment design problem using U init . The robustified optimal experiment design problem is therefore: min

Φu, γ γ (9)
such that J (θ, Φ u ) ≤ γ ∀ θ ∈ U init [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] and

P -1 ρ (θ, Φ u ) + P -1 ρ,init ≥ R adm ∀ θ ∈ U init (11)
If we denote by Φ orig u,opt and γ orig opt the solution of this optimization problem, we have that γ orig opt = sup θ∈Uinit J (θ, Φ orig u,opt ). Moreover, the spectrum Φ orig u,opt is, by construction, the spectrum Φ u leading to the smallest value of sup θ∈Uinit J (θ, Φ u ) while guaranteeing the robustified accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. Since we assume that θ 0 ∈ U init , this robustified formulation ensures 1) that the a-priori unknown cost J (θ 0 , Φ orig u,opt ) is smaller than γ orig opt and 2) that P ρ (θ 0 , Φ orig u,opt ) is guaranteed to satisfy P -1 ρ (θ 0 , Φ orig u,opt ) + P -1 ρ,init ≥ R adm . The above optimization problem will be a convex optimization problem if [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] and ( 11) can be transformed into two constraints that are linear in the decision variables Φ u and γ. In the sequel, we will show that, as very often in robustness analysis theory, we cannot find tractable linear constraints that are equivalent to [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] and ( 11), but we can find one that implies [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] and another one that implies [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. This entails a certain conservatism. However, if we solve the optimization problem with these alternative constraints and if we denote its solution by γ opt and Φ u,opt , we still have the guarantee that (1)

P -1 ρ (θ, Φ u,opt ) + P -1 ρ,init ≥ R adm ∀ θ ∈ U init and thus P -1 ρ (θ 0 , Φ u,opt ) + P -1 ρ,init ≥ R adm (2) J (θ 0 , Φ u,opt ) ≤ sup θ∈Uinit J (θ, Φ u,opt ) ≤ γ opt .
In addition, we have also that γ opt is an upper bound for γ orig opt , the solution of the original optimization problem ( 9)- [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF].

We will derive the tractable alternative constraints discussed in the previous paragraph in the case of a commonly used parametrization of the to-be-determined spectrum Φ u [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF]. This spectrum parametrization corresponds to the spectrum of a multisine signal at fixed frequencies ω m (m = 1, ..., L) but with arbitrary amplitudes1 :

Φ u (ω) = π L m=1 c m (δ(ω -ω m ) + δ(ω + ω m )) ≥ 0 ∀ω (12)
The positivity of Φ u (ω) for all ω can be imposed by the constraints c m ≥ 0 (m = 1, . . . , L). Using [START_REF] Kumar | Robust plant friendly optimal input design[END_REF], the constraint (10) can be rewritten successively as follows: [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF] with G(θ) = (G(e jω1 , θ), G(e jω2 , θ), ..., G(e jω L , θ)) T and C = diag(c 1 , c 2 , ..., c L ). Using (3) and ( 12), the term P -1 ρ (θ, Φ u ) in [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] can also be rewritten as follows:

L m=1 c m 1 + β |G(e jωm , θ)| 2 ≤ γ ∀ θ ∈ U init (13) L m=1 c m + β G * (θ) C G(θ) ≤ γ ∀ θ ∈ U init
P -1 ρ (θ, Φu) = N 2 σ 2 e L m=1 cm
Fu(e jωm , θ)F * u (e jωm , θ) + Fu(e -jωm , θ)F * u (e -jωm , θ)

P -1 ρ (θ, Φu) = N 2 σ 2 e k G i=1 k G j=1 (ei ⊗ Fi(θ)) * C(ej ⊗ Fj(θ)) + (ei ⊗ Fj(θ)) * C(ej ⊗ Fi(θ)) (15)
where e i (i = 1, ..., k G ) is a unit vector of dimension 1 × k G whose entries are all zero except the i th entry which is equal to 1 and where F i (θ) (i = 1, ..., k G ) is a complex vector of dimension L × 1 defined as:

Fi(θ) = (Fu,i(e jω 1 , θ), Fu,i(e jω 2 , θ), ..., Fu,i(e jω L , θ)) T [START_REF] Pronzato | Robust experiment design via optimization[END_REF] with F u,i (z, θ) the i th entry of the vector F u (z, θ).

Remark. In [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF], the actual value of the variance σ 2 e of the white noise e (see [START_REF] Abrashov | Simple and Robust Experiment Design for System Identification Using Fractional Models[END_REF]) is generally unknown. However, we can also robustify the optimal experiment design problem against this uncertainty. For this purpose, we can replace σ 2 e in (15) by σ 2 e,max where σ 2 e,max is the maximal value of the η%-confidence interval [σ 2 e,min , σ 2 e,max ] for σ 2 e that can be constructed using the initial identification experiment (the one yielding θinit ) [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF].

In the next two sections, we will derive, using robustness analysis tools, tractable alternatives for both [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] and [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] when the parametrization ( 12) is used for Φ u . These tractable alternatives will be under the form of Linear Matrix Inequality (LMI) constraints [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF] that are linear in γ and in the spectrum coefficients c m (m = 1, . . . , L).

4 Tackling the robustified cost constraint using robustness analysis tools 4.1 Introduction

We will start by deriving a tractable alternative for ( 13)-( 14). The constraints ( 13)-( 14) are indeed equivalent to [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] when the parametrization ( 12) is used for Φ u . In the paper [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF], we have proposed the following tractable alternative for ( 13)-( 14):

L m=0 c m (1 + β α G (ω m )) ≤ γ (17)
with α G (ω m ) = sup θ∈Uinit |G(e jωm , θ)| 2 (a computable quantity for each ω m [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF]). It is clear that [START_REF] Rojas | On the equivalence of least costly and traditional experiment design for control[END_REF] implies [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF] and that it is linear in γ and in c m (m = 1, . . . , L). However, the constraint ( 17) is rather conservative since α G (ω m ) = sup θ∈Uinit |G(e jωm , θ)| 2 can be obtained for different θ at different frequencies ω m (m = 1, . . . , L). In this paper, we propose an alternative approach which, as will be shown in the example section, will generally be less conservative since it will explicitly take into account the dependency on θ of the frequency response elements G(e jωm , θ) in G(θ) (see ( 14)).

Linear Fractional Transformation

An important step towards the developments of this result is to rewrite G(θ) in the Linear Fractional Transformation (LFT) framework [START_REF] Zhou | Essentials of Robust Control[END_REF]. For this purpose, let us first observe that, due to Assumption 1, y(t) = G(z, θ)u(t) can be written as the following LFT in θ involving the internal scalar signal q and the internal vector of signals p: p = θ q and q y = -Z D (z) 1

Z N (z) 0 M G (z) p u (18)
Recall now that the Fourier transform y(e jω ) of y(t) = G(z, θ)u(t) is equal to G(e jω , θ) when u(t) is equal to a pulse signal δ(t) (i.e. u(e jω ) = 1). Consequently, the frequency response G(e jω , θ) of G(z, θ) at one given frequency ω can also be deduced by solving for y(e jω ) in the following system of equations:

p(e jω ) = θ q(e jω ) and q(e jω ) y(e jω ) = M G (e jω ) p(e jω ) 1 [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF] Note that, in this system of equations, all Fourier transforms are well defined for all θ ∈ U init due to Assumption 2.

Using the same reasoning, the vector G(θ), containing the frequency response of G(z, θ) at the frequencies present in the spectrum [START_REF] Kumar | Robust plant friendly optimal input design[END_REF], can be determined by solving for ȳ in the system of equations ( 20) derived using [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF]:

p = (I L ⊗ θ) q q ȳ = M11,G M12,G M21,G M22,G MG p 1 ( 20
)
with p = (p T (e jω1 ), ..., p T (e jω L )) T , q = (q(e jω1 ), ... ...., q(e jω L )) T Remark. It is important to note that the relation between p and q in (20) is in I L ⊗ θ = bdiag(θ, ..., θ) i.e. a repetition of the same θ. This is due to the dependency on θ of the frequency response elements G(e jωm , θ) in G(θ). The approach in [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF] (see Section 4.1) would in fact correspond to a relation between p and q of the form diag(θ(ω 1 ), θ(ω 2 ), ..., θ(ω L )) where θ(ω m ) ∈ U init for all ω m (m = 1, ..., L), but can be different for each frequency ω m (m = 1, ..., L). Indeed α G (ω m ) can be rewritten as sup θ(ωm)∈Uinit |G(e jωm , θ(ω m ))|2 (m = 1, ..., L).

Set of multipliers related to the uncertainty set U init

Since we consider here ( 14), the parameter vector θ in the LFT for G(θ) is restricted to be in the uncertainty set U init (see [START_REF] Bombois | Least costly identification experiment for control[END_REF]). In our approach, a necessary ingredient to find a tractable alternative for ( 14) is to associate, with the set U init , a so-called set of multipliers. In a nutshell, the set of multipliers A n that we will consider in this paper is an explicit and affine parametrization of the quadratic constraint satisfied by the graphs of the signals q n and p n when p n (t) = (I n ⊗ θ)q n (t) with θ ∈ U init (n is an arbitrary integer such that n ≥ 1) [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF][START_REF] Goh | Robust analysis, sectors and quadratic functionals[END_REF][START_REF] Megretski | System analysis via integral quadratic constraints[END_REF]. Definition 1 Consider the set U init defined in (5) satisfying Assumption 2 and an arbitrary integer n ≥ 1. For each value of n, we define the set of multipliers A n as a set of affinely parametrized Hermitian matrices A n (of dimension n(k + 1) × n(k + 1)) that all have the following property:

  In In ⊗ θ   T An   In In ⊗ θ   ≥ 0 ∀θ ∈ Uinit (21)
In other words, A n ∈ A n =⇒ [START_REF] Zhou | Essentials of Robust Control[END_REF].

It is important to stress that the more extensive the parametrization of the set of multipliers, the smaller the conservatism discussed in Section 3 will be [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF][START_REF] Goh | Robust analysis, sectors and quadratic functionals[END_REF][START_REF] Megretski | System analysis via integral quadratic constraints[END_REF]. The set of multipliers A n corresponding to U init can be easily derived 2 from our previous contribution (see Proposition 2 of [START_REF] Barenthin | Identification for control of multivariable systems: controller validation and experiment design via LMIs[END_REF]).

In [START_REF] Barenthin | Identification for control of multivariable systems: controller validation and experiment design via LMIs[END_REF], the set of multipliers A n is in fact developed for an uncertainty set of the form U init = δθ | δθ T P -1 δθ ≤ χ . As shown in these papers, the use of this set of multipliers therefore entails the straightforward transformation of the LFT (18) (i.e. an LFT in θ) into an LFT in δθ = θ -θinit . Another option is to adapt the multipliers of [START_REF] Barenthin | Identification for control of multivariable systems: controller validation and experiment design via LMIs[END_REF] to an uncertainty set U init that is not centered at zero.

Remark.

It is also to be noted that the set of multipliers in Definition 1 can also be derived for uncertainty sets U init that are not ellipsoidal. In other words, the approach presented in this paper is valid for all uncertainty sets U init for which the sets of multipliers A n of Definition 1 can be constructed.

Robustified cost contraint

Taking a set of multipliers A n with n equal to the number of frequencies in the to-be-determined spectrum (i.e., n = L) and considering the LFT representation [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF] of G(θ), we have now all the ingredients to derive a tractable alternative constraint for [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF].

Proposition 1 Consider an initial uncertainty set U init (see ( 5)) satisfying Assumption 2 and the robust cost constraint [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF] obtained when the spectrum Φ u is parametrized as in [START_REF] Kumar | Robust plant friendly optimal input design[END_REF]. Consider the LFT representation [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF] for G(θ) as well as the set of multipliers A L associated with U init (see Definition 1 with n = L). Then, the constraint ( 14) holds for a given γ if we can find a matrix A L ∈ A L such that

V * A L V + L * C L ≤   0 0 0 γ- L m=1 cm β   ( 22
)
where L = M21,G M22,G and

V =   M11,G M12,G I kL 0  
We observe that the matrix inequality (22) is linear in γ, A L and in the coefficients c m (m = 1, ..., L) present in C.

Proof. Let us consider [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF] for a given θ ∈ U init and let us consider the corresponding signals p, q and ȳ = G(θ). Let us then pre-and post-multiply the LMI constraint ( 22) with (p * , 1) and (p T , 1) T , respectively. Using [START_REF] Safonov | Stability and Robustness of Multivariable Feedback Systems[END_REF] and ȳ = G(θ), this yields:

q p * A L q p + G * (θ) CG(θ) ≤ γ - L m=1 c m β ( 23
)
Since p = (I L ⊗ θ)q, we can rewrite (23) as follows:

q *   IL IL ⊗ θ   T AL   IL IL ⊗ θ   q+G * (θ) CG(θ) ≤ γ -L m=1 cm β (24)
The above reasoning can be done for any value of θ ∈ U init . In other words, for the matrix A L ∈ A L found by the optimization problem, (24) holds true for all θ ∈ U init . Consequently, using Definition 1 with n = L, we have therefore also that

G * (θ) CG(θ) ≤ γ- L m=1 cm β
for each θ ∈ U init ; which is the desired result.

In Proposition 1 (but also later in this paper), when we speak of finding a matrix A L ∈ A L , we more precisely mean finding the free parameters in the affine structure of the matrix A L .

Robustified accuracy constraint

Taking inspiration from what has been done for the robustified cost constraint, we will now derive a tractable alternative for the accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] when the parametrization ( 12) is used for Φ u . Recall the expression [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF] for P -1 ρ (θ, Φ u ) and let us observe that F u (z, θ) (see ( 4)) is a rational function of θ due to Assumption 1. Consequently, we can find signals p F and q F such that s(t) = F u (z, θ)u(t) can be expressed as:

p F = (I f ⊗ θ) q F q F s = M 11,F M 12,F M 21,F M 22,F M F (z) p F u ( 25
)
where f = 3 as shown in Appendix A. Note that f = 2 in the case where H(z, θ) = 1 (OE model structure).

Using a similar reasoning as in Section 4.2, we can derive from (25) an LFT expression for si = F i (θ) (i = 1, ..., k G ) defined in [START_REF] Pronzato | Robust experiment design via optimization[END_REF]

M21,F = (H T 1 , ..., H T k G ) T M22,F = (K T 1 , ..., K T k G ) T H i = bdiag(M i:
21,F (e jω1 ), ..., M i:

21,F (e jω L )) (i = 1, ..., k G ) K i = (M i 22,F (e jω1 ), ..., M i 22,F (e jω L )) T (i = 1, ..., k G )
where M i: 21,F (resp. M i 22,F ) denotes the i th line of M 21,F (resp. the i th entry of M 22,F ). Note that, in this system of equations, all elements are well defined for all θ ∈ U init due to Assumption 2.

We have then the following result that gives a tractable alternative for the robustified accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF].

Proposition 2 Consider an initial uncertainty set U init (see ( 5)) satisfying Assumption 2 and the robustified accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] when the spectrum Φ u is parametrized as in [START_REF] Kumar | Robust plant friendly optimal input design[END_REF] and where, for this reason, P -1 ρ (θ, Φ u ) has the expression given in [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF]. Consider the LFT (26) in I f L ⊗ θ which is the LFT representation for s = (F 1 (θ) T , F 2 (θ) T , ..., F k G (θ) T ) T and consider the set of multipliers A k G f L associated with U init (see Definition 1 with n = k G f L). Then, the constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] holds for a given R adm and a given P ρ,init if we can find a matrix

A k G f L ∈ A k G f L such that 3 N 2 σ 2 e k G i=1 k G j=1 (ei ⊗ Xi) * C(ej ⊗ Xj) + (ei ⊗ Xj) * C(ej ⊗ Xi) +   (P -1 ρ,init -R adm ) ⊗   0 0 0 1     -M * A k G f L M ≥ 0 ( 27
)
where

X i = H i K i (i = 1, ...k G ) and M =   I k G ⊗ M11,F M12,F I k G ⊗ I kf L 0  
We observe that the matrix inequality

(27) is linear in A k G f L and in the coefficients c m (m = 1, ..., L) present in C.
Proof. Let us consider (26) for a given θ ∈ U init and let us consider the corresponding signals pF , qF and s = (F 1 (θ) T , F 2 (θ) T , ..., F k G (θ) T ) T . Let us then pre-and post-multiply the LMI constraint ( 27) with (I k G ⊗ (p T F , 1) T ) * and I k G ⊗ (p T F , 1) T , respectively. Using (26), [START_REF] Morelli | Optimal Experiment Design for the Identification of One Module in the Interconnection of Locally Controlled Systems[END_REF] and the lemma in Appendix B , this yields:

P -1 ρ (θ, Φu) + P -1 ρ,init -R adm -X(θ) ≥ 0 ( 28
)
where P -1 ρ (θ, Φ u ) is given by ( 15) and X(θ) is a matrix given by

X(θ) =   I k G ⊗ qF I k G ⊗ pF   * A k G f L   I k G ⊗ qF I k G ⊗ pF  
Since pF = (I f L ⊗ θ) qF , X(θ) can be rewritten as:

X(θ) = (I k G ⊗ qF ) * Y (θ) (I k G ⊗ qF ) Y (θ) =   I k G f L I k G f L ⊗ θ   T A k G f L   I k G f L I k G f L ⊗ θ  
The above reasoning can be done for any value of θ ∈ U init . In other words, for the matrix

A k G f L ∈ A k G f L
found by the optimization problem, (28) holds true for all θ ∈ U init . Using Definition 1 with n = k G f L and the Lemma in Appendix B, note also that:

X(θ) ≥ 0 ∀θ ∈ U init
Consequently, we have that P -1 ρ (θ, Φ u )+P -1 ρ,init ≥ R adm for each θ ∈ U init ; which is the desired result.

6 Convex formulation of the optimal experiment design problem Using Propositions 1 and 2, we can now straightforwardly derive a convex formulation for the robust optimal experiment design problem ( 9)- [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. This formulation will be under the form of a LMI optimization problem [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF].

LMI formulation Consider the parametrization [START_REF] Kumar | Robust plant friendly optimal input design[END_REF] for the to-be-designed spectrum Φ u . The LMI optimization problem has as decision variables a scalar γ > 0, Definition 1) and consists in determining the smallest value of γ for which the LMI constraints ( 22) and (27) hold.

coefficients c m ≥ 0 (m = 1, ..., L), a matrix A L ∈ A L and a matrix A k G f L ∈ A k G f L (see
As mentioned in Section 3, if we denote the solution of the above LMI optimization problem by γ opt and Φ u,opt , γ opt is an upper bound for the solution γ orig opt of the original robustified optimization problem (9)- [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF] i.e. the minimal value of the cost that is required to guarantee the robust accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. Moreover, γ opt is an upper bound for sup θ∈Uinit J (θ, Φ u,opt ) and, since we assume θ 0 ∈ U init , we have the guarantees 1) that P -1 ρ (θ 0 , Φ u,opt ) + P -1 ρ,init ≥ R adm and 2) that J (θ 0 , Φ u,opt ) ≤ γ opt .

Remark. In this paper, we have considered the least costly optimal experiment framework and we have supposed that the desired accuracy is represented by a given matrix R adm . As mentioned in the introduction, the results of this paper can also be used for other optimal experiment design frameworks. For example, let us consider the classical E-optimality framework that, when robustified, consists in:

max Φu, ε ε such that 1 2π π -π Φ u (ω) dω ≤ γ and P -1 ρ (θ, Φ u ) + P -1 ρ,init ≥ εI k G ∀ θ ∈ U init ( 29
)
where ε is a scalar decision variable and γ is a given bound on the power of the input signal. It is clear that (29) can be tackled with the tools of Section 5. Note that, here, R adm = εI k G is not fixed, but a decision variable. This however does not pose a problem since the constraint ( 27) is also linear in R adm .

Numerical illustration

In this section we present simulation results in order to show the effectiveness of our approach. We consider the following system y(t) = G 0 (z)u(t) + e(t) with G 0 (z) = 1 z -1 1-0.7 z -1 and e(t) a white noise having variance σ 2 e = 1. The true parameter vector θ 0 is thus θ 0 = ρ 0 = (1, -0.7)

T . An initial estimate θinit of θ 0 and its covariance matrix P θ,init have been obtained using an experiment of duration N = 1000 with a white noise input signal of variance 0.1: θinit = (0.904, -0.7161)

T . This allows to build the initial uncertainty U init (see [START_REF] Bombois | Least costly identification experiment for control[END_REF]) that is a confidence region for θ 0 with probability level η = 95% (χ = 5.99). This initial estimate does not satisfy the desired accuracy which is here that the standard deviation of the two parameters is smaller than two percents of their exact value. Based on this requirement, R adm is chosen as the inverse of the following diagonal matrix diag((0.02) 2 , (0.014) 2 ) (see e.g., [START_REF] Ghosh | Optimal identification experiment design for LPV systems using the local approach[END_REF]).

We use the LMI formulation of Section 6 to determine the spectrum Φ u of a second identification experiment of duration N = 1000 that will minimize a robustified version of the cost [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF] under the robust accuracy constraint [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. Note that we here choose β = 1 in the expression of the cost [START_REF] Ghosh | Optimal identification experiment design for LPV systems using the local approach[END_REF]. In [START_REF] Kumar | Robust plant friendly optimal input design[END_REF], we choose L = 9 with ω m (m = 1, ..., 9) covering the interval [0. 1 3]. With these settings, the LMI optimization problem of Section 6 yields γ opt = 5.4948 and an optimal multisine of spectrum Φ u,opt for which all the amplitudes are negligible except the ones at ω = 0.5 and ω = 1. By construction, we know that, if an excitation signal of spectrum Φ u,opt is used in the second identification experiment, the obtained accuracy will be satisfactory (i.e. P -1 ρ (θ 0 , Φ u,opt ) + P -1 ρ,init ≥ R adm ) and the cost J (θ 0 , Φ u,opt ) of this second identification experiment will be such that J (θ 0 , Φ u,opt ) ≤ γ opt = 5.4948. The value γ opt is indeed an upper bound for sup θ∈Uinit J (θ, Φ u,opt ) as discussed in Section 3 and thus also an upper bound for the a-priori unknown cost J (θ 0 , Φ u,opt ) which is here equal to 4.7.

In order to check the conservatism linked to the proposed LMI formulation, we will compare the above result with the one that is obtained using the gridding approach for robust optimal experiment design. This approach considers the following optimization problem: min Φu,g, γg

γ g such that J (θ i , Φ u,g ) ≤ γ g ∀ θ i ∈ Θ n and P -1 ρ (θ i , Φ u,g ) + P -1 ρ,init ≥ R adm ∀ θ i ∈ Θ n where Θ n is a set containing n grid points θ i (i = 1, ..., n) such that θ i ∈ U init .
We have solved the above optimization problem for n = 25 using the same spectrum parametrization as above (same number L of frequencies and same frequencies ω m ) and we have obtained γ g,opt = 5.4866 and a spectrum Φ u,g,opt which has also contributions at two frequencies ω = 0.5 and ω = 1, but with (slightly) different amplitudes.

As mentioned in Sections 3 and 6, γ opt (obtained with the approach proposed in this paper) is an upper bound for the solution γ orig opt of the original robustified optimization problem ( 9)- [START_REF] Kumar | Optimal input signal design for plant friendly identification of process systems[END_REF]. It is also clear that the value γ g,opt obtained with the gridding approach is a lower bound for the same quantity. We thus observe that, in this example, the upper bound γ opt = 5.4948 is almost equal to the lower bound γ g,opt = 5.4866. The conservatism is thus very limited in this example. It is also important to note that, unlike the gridding approach, the approach proposed in this paper gives the guarantee that, with Φ u,opt , the robustified accuracy constraint (11) will be respected and also gives a guaranteed upper bound (i.e., γ opt ) on both sup θ∈Uinit J (θ, Φ u,opt ) and the actual cost J (θ 0 , Φ u,opt ) of the second experiment.

To verify this property, we have generated 1000 grid points θ i in U init and we have computed J (θ i , Φ u,opt ) and P -1 ρ (θ i , Φ u,opt ) for these 1000 grid points. For all these grid points, we have indeed observed that P -1 ρ (θ i , Φ u,opt ) + P -1 ρ,init ≥ R adm . We have also observed that the smallest eigenvalue of P -1 ρ (θ i , Φ u,opt ) + P -1 ρ,init -R adm is, for one of these grid points θ i , equal to 0.0004 (and thus very close to zero) and that the cost J (θ i , Φ u,opt ) is equal to 5.4948 for one of these grid points. This once again confirms that the conservatism of our approach is very limited in this example. Let us now compute, for these 1000 grid points, P -1 ρ (θ i , Φ u,g,opt ) with the spectrum Φ u,g,opt obtained with the gridding approach. Here, 82 of the θ i led to a matrix P -1 ρ (θ i , Φ u,g,opt ) for which P -1 ρ (θ i , Φ u,g,opt ) + P -1 ρ,init ≥ R adm is not satisfied. This shows the clear advantage of the approach of this paper upon the gridding approach.

Finally, let us illustrate the discussion in Section 4.1. For this purpose, we will compare the upper bound for sup θ∈Uinit J (θ, Φ u ) given by the lefthand side of ( 17) and the one corresponding to the LMI formulation proposed in this paper. We will do that for Φ u,opt . We know that the upper bound for sup θ∈Uinit J (θ, Φ u,opt ) obtained using the tools proposed in this paper is γ opt = 5.4948. Let us now compute the upper bound

γ α = L m=1 c m,opt (1 + α G (ω m ))
for sup θ∈Uinit J (θ, Φ u,opt ) (see [START_REF] Rojas | On the equivalence of least costly and traditional experiment design for control[END_REF]). This yields γ α = 5.7408, which shows that the approach discussed in Section 4.1 is more conservative.

Conclusion

In this paper, we have presented a convex relaxation that allows to robustify the least costly optimal experiment design problem using an initial uncertainty set for the unknown true parameter vector θ 0 . This robustification is obtained using tools from robustness analysis. In this paper, we have restricted attention to multisine excitation signals. In the future, we will investigate whether the least costly optimal experiment design problem can also be robustified when the to-be-determined excitation spectrum is the one of a filtered white noise. This spectrum parametrization is indeed also a commonly used parametrization in optimal experiment design [START_REF] Jansson | Input design via LMIs admitting frequency-wise model specifications in confidence regions[END_REF]. We will also investigate how the results presented in this paper can be extended to tackle the robustification of more complex accuracy constraints (such as the ones presented in [START_REF] Bombois | Least costly identification experiment for control[END_REF]).

A LFT representation of F u (z, θ)

The classical BJ paramatetrization is as follows: Moreover, we have also that s(t) = F u (z, θ)u(t) is given by:

G(z, θ) = z -n k b 0 + b 1 z -1 + ... + b n b z -n b 1 + f 1 z -1 + ... + f n f z -n f
s(t) =             s 1 (t) ... s n b +1 s n b +2 ... s k G (t)             =             z -n k 1+Z D (z)θ ... z -(n k +n b ) 1+Z D (z)θ -z -1 Z N (z)θ (1+Z D (z)θ) 2
...

    p 1 p 2 p 3     =p F = (I 3 ⊗ θ)     q 1 q 2 q 3     =q F             q 1 q 2 q 3 s 1 ... s k G             =             -Z N,H 0 0 1 Z ∆ -Z D 0 1 0 -Z N -Z D 0 z -n k Z ∆ -z -n k Z D 0 z -n k ... ... .... .... 0 -z -n f Z N -z -n f Z D 0             M F (z)        p 1 p 2 p 3 u       
with Z ∆ (z) = Z D,H (z) -Z N,H (z). When H(z, θ) = 1, the above LFT can be simplified and f = 2.

B Useful lemma for the proof of Proposition 2

Lemma 1 Consider an Hermitian matrix A = A * of dimension n × n and a (complex) matrix B of dimension n × ñ. Then, we have that

A ≥ 0 =⇒ B * AB ≥ 0
Proof. B * AB ≥ 0 is equivalent to the fact that, for all complex vector x of dimension ñ,

x * B * ABx ≥ 0 Denoting y the complex vector Bx of dimension n, the latter matrix inequality is equivalent to: y * Ay ≥ 0 which always holds since A ≥ 0.

  and M11,G = -bdiag(Z D (e jω1 ), ..., Z D (e jω L )) M12,G = (1, ..., 1) T M21,G = bdiag(Z N (e jω1 ), ..., Z N (e jω L )) M22,G = 0

  . If we denote by s = (s T 1 , sT 2 , ..., sT k G ) T , we have indeed pF = (I f L ⊗ θ) = bdiag(M 11,F (e jω1 ), ..., M 11,F (e jω L )) M12,F = (M 12,F (e jω1 ), ..., M 12,F (e jω L )) T

H(z, θ) = 1 +

 1 c 1 z -1 + ... + c nc z -nc 1 + d 1 z -1 + ... + d n d z -n d with θ = (ρ T , ζ T ) T ∈ R k with ρ = (b 0 , ..., b n b , f 1 , ..., f n f ) T and ζ = (c 1 , ..., c nc , d 1 , ..., d n d ) T (k G = n b + n f + 1,

Fig. A. 1 .

 1 Fig. A.1. Representation of the vector Fu(z, θ) given in (A.1)k H = n c + n d , k = k G + k H ). This parametrization satisfies Assumption 1. As an example, if n k = n f = n c = n d = 1 and n b = 0 (i.e., θ = (b 0 , f 1 , c 1 , d 1 ) T ), we have Z N (z) = (z -1 0 0 0), Z D (z) = (0 z -1 0 0), Z N,H(z) = (0 0 z -1 0) and Z D,H (z) = (0 0 0 z -1 ). Moreover, we have also that s(t) = F u (z, θ)u(t) is given by:

1 +

 1 -z -n f Z N (z)θ (1+Z D (z)θ) Z D,H (z)θ 1 + Z N,H (z)θ u(t) (A.1) which is a rational function in θ. This rational function can be expressed as in (25) with f = 3 (see Figure A.1):

The amplitude Am of the sinusoid at ωm is given by Am = √

2cm (m = 1, ..., L)

Note that, in[START_REF] Barenthin | Identification for control of multivariable systems: controller validation and experiment design via LMIs[END_REF], the notations A (resp. ñ) are used instead of An (resp. n).

The matrix made of zeros and a one in the second line of (27) has dimension (f Lk + 1) × (f Lk + 1) .


F Menager

S Percelay

T Freret

M Boulouard

J M Billard

V Bouet

Altering glutamate transmission in combination with an early post-natal stress to mimic schizophrenia in male and female mice

Keywords: positive, negative, cognitive Debilitating disease  heavy costs Behavioural testing

Treatments: not efficient on negative and cognitive symptoms, multiple side-effects Crucial need to refine treatments  Improving validity of animal models of schizophrenia  Combining genetic x environmental factors Maternal separation PND9-24h

Serine racemase deletion induces an increase in activity, promotes social memory troubles, and contributes to a higher sensitivity to maternal separation. Maternal separation mostly contributes to sensorimotor gating deficits. 2-hit model differentially affect males and females. Because some deficits (working memory, object recognition, social recognition) appears only in 2-hit mice, combining factors may help in improving validity of animal models of schizophrenia. Moreover, combining factors reveals differences between males and females, probably accounting for gender vulnerability/resilience differences.



  Locomotor activity in all experiments  Place recognition  Social recognition  Place recognition  Prepulse Inhibition Maternal separation SR KO  Activity  Social recognition in male mice  Working memory in female mice  PPI in female mice  Anxiety-related behavior  Place recognition performances  Object recognition performances in female mice  Neophobia in males

  animal models gathers several deficits considered as hallmarks of schizophrenia.



  Place 05 : Test U de Mann-Whitney n=10-16 # univariate t-test ≠ 0 n=10-16

5 min Short-term memory Spontaneous alternation 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Control MS SR MSxSR Control MS SR MSxSR Male Female

	#	#	#	#	#	#	#
				# univariate t-test >50% n=10-16

MSxSR combination induced working memory deficit in female but not in male mice


Mozammel Bin Motalab

Muktadir Bin Motalab
email: muktadir.ca@gmail.com

Analysis the Impact of Game theory (Pay-off Matrix, Nash Equilibrium) in the High Tech Industry Artificial Intelligence -Digital Assistant

Keywords: Artificial Intelligence, Game Theory, Nash Equilibrium, Digital Assistant, Decision Analysis I

In this paper we are going to discuss and analysis the formal application of game theory which require the identity of independent actors, their preferences, their knowledge, strategic acts they are allowed to make. Each independent actor is assumed to be coherent. This game theory is not limited to case analysis rather psychology, tag of war, business, economy etc.

INTRODUCTION

The basic tool of game theory is the payoff matrix which is a matrix gives the possible outcome of zero-sum game when player A has possible moves and player B moves. The analysis of the matrix in order to determine optimal strategies is the aim of game theory. [START_REF] Robert | International Economicssecond edition[END_REF] After payoff matrix analysis we need to make a decision. Payoff matrix helps to take the decision by Nash Equilibrium [3] [START_REF] Robert | International Economicssecond edition[END_REF]by which each firm must make its own best decision, taking as given each possible action of the rival firm. When each firm is acting that way, the outcome of the game is a Nash equilibrium. The action of each player is the best possible response to the action of the other player.

IT services can be adapted to a new culture in a variety of ways ranging from simple build changes to total redesign in the services. Some need for change becomes obvious with relatively little analysis. For example, a cursory analysis of a country will uncover the need to require AI based digital assistance if it uses a different customer service system, or to indicate product simplification when the local level of technologies is not high. [START_REF] Kiner | Approaches towards Building a Banking Assistant[END_REF][2] Most such superficial changes can be spotted by looking at service use patterns, the economy, and other related culture elements. [START_REF] Shawar | Different measurement metrics to evaluate a chatbot system, Bridging the Gap: Academic and Industrial Research in Dialog Technologies[END_REF]

II. PROBLEM OVERVIEW

In 2017 both company focuses on Digital Assistants Solution one of the specific area of Artificial Intelligence. This is a high-tech solution for recent trends. The core feature of the system are designed to deliver an intuitive, automated experience for all digital channels by engaging customers in natural conversations using voice or text [START_REF] Mhatre | Article: Donna Interactive Chat-bot acting as a Personal Assistant[END_REF] [START_REF] Gayatri Srivathsan | Dialogue Based Assistant for Career Counseling[END_REF].To add a virtual agent to your website, app, or text messaging application, this system offers customers a consistent experience in any digital channel [START_REF] Berry | PTIME: Personalized Assistance for Calendaring[END_REF]. A familiar voice will answer their request whether it's typed into a computer, tapped on a screen or spoken into a device. For this Digital Assistant, M Cube Corporation has developed its own platform and want to serve its customer through this. The cost of RnD is huge for the platform development and already exist in the market. Whereas LeadSoft as a partner of MicroSoft Corporation, one of the leaders in IT sector an American multinational technology company is using their platform to develop the solution.

III. FACTS

In recent years a new breed of Bangladeshi IT marketers is posing an increasingly credible challenge to go global, following in the footsteps of Japanese and European startup IT to become formidable global competitors. HI-tech IT solutions among the growing list of international brands making an impact. Here we define hi-tech as Artificial Intelligence, Block Chain, Internet of Things, Business Intelligence and Cloud Computing but not the contemporary software solution like MIS, ERP and IT support services. Bangladesh governments subsidize high-technology industries (largely focuses on IT) because they may create benefits that spill over to other firms in the economy. That is, governments believe that high-tech industry produces a positive externality. A strategic advantage to export firms that are competing with rivals in international markets is subsidies. This subsidies might give to capture the strategic decision making of two firms, we use game theory, the modeling of strategic interactions (games) between firms as they choose actions that will maximize their returns.

IV. PAYOFF MATRIX AND NASH EQUILIBRIUM

A. Payoff Matrix between Two Firms

The lower-left number in each quadrant shows the profits of M Cube, and the upper-right number shows the profits of LeadSoft. Each firm must decide whether to deploy a Digital Assistant. A Nash equilibrium occurs when each firm is making its best decision, given the action of the other. For this pattern of payoffs, there are two Nash equilibrium, in the upper-right and lower-left quadrants. It means that if LeadSoft produce then M Cube Corporation does not and if M Cube Corporation does Deploy LeadSoft does not. When both the farm are to Deploy then both are going to lose BDT 2,000,000/-. If any of them decided not to deploy the system, then the deploying firm will make a profit of 21,000,000/-and other will not gain anything. If both the farm decided not to deploy, then both will gain no profit.

LeadSoft

Best strategy for M Cube If LeadSoft produces, then M Cube is better off not producing. This finding proves that having both firms produce is not a Nash equilibrium. M Cube would never stay in production, since it prefers to drop out of the market whenever LeadSoft produces. Again, Best Strategy for LeadSoft the decision illustrated in the lower-left quadrant, with LeadSoft producing and M Cube not producing, is a Nash equilibrium because each firm is making its best decision given what the other is doing.

B. Effect of a Subsidy/Support/Fund to LeadSoft

As we know LeadSoft is a local company only which can avail any Subsidy or Fund or support by the LICT (Leveraging ICT) Project of Bangladesh Computer Council (BCC) under the Ministry of Post, Telecom and Information Technology of Bangladesh Government. Here is to remember that, M Cube Corporation is registered and headquartered in Singapore.

When the LICT provide a subsidy of BDT 3,000,000/-to LeadSoft, its profits increase by that much when it develop the system. Now there is only one Nash equilibrium, in the lowerleft quadrant, with LeadSoft deploying but M Cube not deploying. The profits for LeadSoft have increased from BDT 0/-to 24,000,000/-while the subsidy cost only BDT 3,000,000/-, so there is a net gain of BDT 21,000,000/-in IT industry. Best strategy for LeadSoft with the subsidy, LeadSoft now earns BDT 1,000,000/-by producing instead of losing BDT 2,000,000/-. Best strategy for M Cube will want to drop out of the market. Once M Cube makes the decision not to produce, LeadSoft decision doesn't change. The lower-left quadrant is a Nash equilibrium where each firm is making its best decision along with the action of the other is exposed or known. Furthermore, it is the only Nash equilibrium.

LeadSoft M Cube

Deploy

C. Cost Advantage for M Cube

M Cube Corporation has their own platform for AI with Digital Assistant which give them a cost advantage of Producing, integrating and deploying the service. Their system is so individual and no dependency on other systems. This platform provide a faster development which M Cube presume and a technology know how, easily customizable and extendable as well. If M Cube has a cost advantage in the production of Digital Assistant, the payoffs are as shown here. M Cube earns profits of BDT 2,000,000 when both firms are producing and profits of BDT 24,000,000/-when LeadSoft does not produce. Now there is only one Nash equilibrium, in the upper-right quadrant, where M Cube produces and LeadSoft does not.

LeadSoft M Cube

Deploy

D. Subsidy for LeadSoft and Cost Advantages for M Cube

When the Bangladesh provide a subsidy of BDT 3,000,000/-to LeadSoft, its profits increase by that much when it produces. Now the only Nash equilibrium is in the upper-left quadrant, where both firms produce. The profits for LeadSoft have increased from 0 to BDT 1,000,000/-, but the subsidy costs BDT 3,000,000/-million, so there is a net loss of BDT 2,000,000/-million in IT industry welfare Again, M Cube Corporation has cost advantages of BDT 2,000,000/-as they has their own platform for AI with Digital Assistant. M Cube earns profits of BDT 2,000,000 when both firms are producing and profits of BDT 24,000,000/-when LeadSoft does not produce.

LeadSoft

The Profit can be maximized if any of the organization do not lunch their product and other does. Total welfare of the industry in terms of monitory value will be highest in this situation. Nevertheless, Lunching by both organization will not regret or face any potential business loss.

E. Analysis and Findings

 As LeadSoft is availing the LICT fund then of course they are going to deploy the service.

 M Cube Corporation has a leverage of cost effectiveness which convert them to go into the market.

 When there are two Nash equilibrium, there must be some force from outside the model that determines in which equilibrium we are. An example of one such force is the first mover advantage, which means that one firm is able to decide whether or not to produce before the other firm [START_REF] Robert | International Economicssecond edition[END_REF]. Both the company can resolve the issues by present their service firstly in front of customer. This is a healthy competition and both firm can avoid the tag of war.

F. Conclusion:

For a High-tech industry Nash equilibrium is a best way to take decision. As M Cube and LeadSoft both are in the same industry, they both need technical support from each other. IT industry touted as crucial First-mover advantage, although now there is a growing backlash against it. First-mover advantage can be instrumental in building market share. In a duopoly Market Strategic decision are required to capture the market. Time to market is also very important. To capture the conflict resolution with strategic decision making of two firms, we use game theory, the modeling of strategic interactions (games) between firms as they choose actions that will maximize their returns.

Fig. 1 .

 1 Fig. 1. Payoff Matrix -LEVEL PLAYING FIELD FOR BOTH FIRM

Fig. 2 .

 2 Fig. 2. Payoff Matrix -EFFECT OF SUBSIDY/SUPPORT/FUND

Fig. 3 .

 3 Fig. 3. Payoff Matrix --COST ADVANTAGE Whereas LeadSoft using Microsoft Bot Framework for their service development. Dependency on Bot Framework may cost in production, integration and dependency.

Fig. 4 .

 4 Fig. 4. Payoff Matrix -SUBSIDY WITH COST ADVANTAGE

ACKNOWLEDGMENT (Heading 5)

Thanks to Dr. Chowdhury Saima Ferdous, Associate Professor, Department of International Business, Faculty of Business Studies, University of Dhaka for her support to us and patronage us to make this write up bring to a different level to make this publishable.