Keywords: Runoff modelling, lake modelling, urban lake, cyanobacteria

In urban regions the increasing imperviousness of surfaces is responsible for rising runoff volume and speed reflecting in a greater capacity to drag nutrients into lakes. In order to study the impacts of catchment changes on cyanobacteria dynamics in urban lakes, a modelling approach in which a hydrological model is connected to an ecological lake model is proposed for Lake Pampulha (Brazil). In this paper we present the methodology used to prepare and to link both models.The rainfall-runoff model results show good agreement with measurements, the Nash coefficient ranging from 0.71 to 0.90. The lake ecological model fed with runoff water volume calculated by the previous model successful represents water temperature and cyanobacteria dynamics, the root mean square error value being respectively, 0.71°C and 7.20 μg chlaL -1 . Collecting new dataset will allow us to calibrate the water quality module of the hydrological model and to simulate different scenarios of catchment changes.

INTRODUCTION

Estimation of nutrient loading is a key information for environmental management and planning e.g. for the choice of the best strategies for improving and preserving water quality in receiving freshwaters. Among the many pollutants whose origin is the urban runoff, nitrogen and phosphorus are of particular concern for lakes which are especially vulnerable to nutrient enrichment because of their relatively lower retentions time [START_REF] Zhang | Modelling of point and non-point nutrient loadings from a catchment[END_REF]. Eutrophication is responsible for decreasing the ecosystem biodiversity and disrupting water uses such as drinking water supply and fishing. Moreover, eutrophic lakes are propitious environments for cyanobacteria blooms, including potential toxic species [START_REF] Hans | Harmful Freshwater Algal Blooms, With an Emphasis on Cyanobacteria[END_REF].

The most reliable method to quantify nutrient loads from runoff is to establish a comprehensive monitoring of the catchment area; however this is rarely possible due to high involved costs. Furthermore, when it is necessary to perform prognostic studies, measurements are generally insufficient and modelling approaches are frequently applied [START_REF] Chow | Modelling runoff quantity and quality in tropical urban catchments using Storm Water Management Model[END_REF][START_REF] Wu | A Water Quality-Based Approach for Catchment Wide BMP Strategies[END_REF]. Numerical modelling has also been extensively applied to simulate thermal and algal dynamics in lake ecosystems [START_REF] Huang | Hydrodynamic-phytoplankton model for short-term forecasts of phytoplankton in Lake Taihu, China[END_REF][START_REF] Reynolds | The ecological basis for simulating phytoplakton responses to environmental change (PROTECH)[END_REF], especially in view of the complexity of the mechanisms involved in algae growth and in their toxin production.

Despite the interconnection between the catchment runoff and the nutrient load into the receiving water body, only recently these processes have been addressed in an integrated way. The link between the hydrological and the lake model consists in using the runoff output from the first model as forcing drive for the second one [START_REF] Wu | A Water Quality-Based Approach for Catchment Wide BMP Strategies[END_REF][START_REF] Xu | The hydrological calibration and validation of a complexlylinked catchment-reservoir model for the Occoquan catchment, Virginia[END_REF][START_REF] Norton | Using models to bridge the gap between land use and algal blooms: an example from the Loweswater catchment, UK[END_REF]. The objective is to improve the reliability of the results and above all, to apply different scenarios of catchment changes and simulate the response of aquatic ecosystem. Because of the rapid and important evolutions of urban environments, namely, intensification of urbanization, climate change and increasing concern about water quality, this kind of integrated modelling approach offers numerous possibilities to be explored. [START_REF] Xu | The hydrological calibration and validation of a complexlylinked catchment-reservoir model for the Occoquan catchment, Virginia[END_REF] and [START_REF] Norton | Using models to bridge the gap between land use and algal blooms: an example from the Loweswater catchment, UK[END_REF] highlighted that such a coupled modelling approach is more easily accepted by managers and by the general public because it can be more readily seen as a representation of a natural linked system. The researches mentioned above have dealt mainly with rural catchment areas while lakes and reservoirs in urban catchments are understudied. In urban regions, the increasing imperviousness of surfaces is responsible for rising runoff volume and speed, which is reflected in a greater capacity to drag nutrients into receiving water bodies [START_REF] Zhang | Modelling of point and non-point nutrient loadings from a catchment[END_REF]. Actually, there is a strong need to investigate the links between the ecological lake functioning and urban catchment changes such as urbanization and improvements or degradation on sewerage systems and wastewater treatment facilities. This paper presents a modelling approach in which a hydrological model is connected to an ecological lake model which in a further stage will be applied to study the impacts of future catchment changes on cyanobacteria dynamics in an urban lake in Brazil.

MATERIAL AND METHODS

Study site

Lake Pampulha is a small and hypereutrophic reservoir in Belo Horizonte city, Brazil (19°55'S, 43°56'W, Figure 1). The climate in the region is a tropical mountain climate, with a dry season between April and September and a wet season from October to March when 90% of the total annual rainfall occurs (mean rainfall = 1 500 mm year -1 ). Air temperature shows relatively small amplitude over the year with a minimum monthly mean in July of 18°C and a maximum monthly mean in February of 23°C [START_REF] Nascimento | Long-term uncertainties and potential risks to urban waters in Belo Horizonte[END_REF]. Lake Pampulha is fed by eight small creeks (Figure 1), Sarandi and Ressaca creeks being the most important (70% of the inflow rate) and also the most polluted ones [START_REF] Tôrres | Mass balance estimation of nitrogen, carbon, phosphorus and total suspended solids in the urban eutrophic, Pampulha reservoir, Brazil[END_REF]. The main physical and chemical characteristics of Lake Pampulha are listed in Table 1.

Originally, the reservoir was built to supply drinking water to the city, however, since the 1970s, the water quality has degraded as a consequence of the rapid catchment urbanization with neither sanitation infrastructure nor erosion control. Nowadays, lake silting and the reduction of its storage capacity, water eutrophication and the consequent increase of primary production with episodes of cyanobacterial blooms and excessive growth of macrophytes are the main problems to be tackled in Lake Pampulha. Despite its poor water quality, Lake Pampulha is an important tourist spot, the area around the lake is used for recreational and sportive activities by inhabitants and it helps to reduce flood risk in the neighbourhood.

Data collection

Catchment monitoring

Pampulha catchment is monitored by the municipality of Belo Horizonte and within the Stormwater Management Research Project -MAPLU 2. Rainfall data are provided by seven precipitation stations located in or around the catchment (Figure 1). Water level data in Sarandi and Ressaca creeks are measured by two stream discharge stations installed in their outlet. Rainfall depth and flow level are automatically measured every 10 minutes since early October 2011 for almost all stations. Water quality data are not available for the 2011/2012 rainy season and therefore modelling is still restricted to rainfall-runoff simulations.

By December 2012 sensors will be installed in the Lake Pampulha main inlet and in its outlet ( Figure 1 ) to perform continuous high frequency measurements of water temperature and turbidity. At the same spots, two automatic samplers will collect water samples for laboratory analysis (TSS, NH3, NO3, Ptot, PO4 -3 ) during rain events.

Lake monitoring

From mid-September to early November 2011, bi-weekly vertical profiles (eleven profiles in total) of water temperature and algae fluorescence were performed with a spectrofluorometer probe in three different points of the lake (Figure 1). Monthly samples were collected under the surface for nitrate, nitrite, ammonium, phosphate and total phosphorus analysis. Since flow measurements in the tributaries were not yet performed, inflows and outflows could be estimated from a water balance based on the measured water level of Lake Pampulha. Inflow water quality (nitrate, phosphate and ammonium concentration) has been assessed only from February 2012 by bimonthly punctual sampling. Meteorological variables (solar irradiation, wind speed, air temperature and atmospheric pressure) were provided at hourly timestep by a weather station of the Brazilian National Institute of Meteorology located 3 km from the lake.

By December 2012, in a mid-point inside the lake (point P2 in Figure 1), a buoy equipped with sensors to measure water temperature, dissolved oxygen concentration, conductivity and algal fluorescence will be installed. Measurements will be done every 30 min and transmitted daily through GPRS to a database.

Modelling approach

Hydrological model

The hydrological catchment model here adopted is the Storm Water Management Model -SWMM 5 [START_REF] Rossman | Storm Water Management Model. User's Manual[END_REF], a dynamic rainfall-runoff model that computes flows and associated wet weather non-point source pollutant processes. Briefly, the processes taken into account can be described as follows: (1) during dry periods, pollutants (nutrients in our case) are deposited over the land surface; (2 ) during rain events, rain falls over the catchment area and washes off nutrients; (3) according to the characteristics of the catchment area (e.g. imperviousness, slope), it generates outflows in form of infiltration to the groundwater and/or in form of surface runoff; (4) this runoff and part of the sub-surface flow reach the stormwater network and are drained to an outfall (Figure 2, left hand side block).

SWMM represents the catchment in the form of three main objects: subcatchments, nodes and conduits. Subcatchments are the elements where runoff is generated. The rainfall-runoff transformation is modelled by a non-linear reservoir following the kinematic wave model [START_REF] Singh | Hydrologic systems: rainfall-runoff modelling[END_REF]. The outflow of a subcatchment can be routed to another subcatchment or to a node which is the entry point to the conduit network. The flow in the conduits, open or closed, is computed by Saint-Venant equations whose simplification degree can be selected by the user (dynamic wave, kinematic wave or steady flow). More details about SWMM can be found in [START_REF] Rossman | Storm Water Management Model. User's Manual[END_REF].

Since outflow was measured in two different points, i.e. Ressaca and Sarandi outlet, these subcatchments were simulated separately while the others were not represented in the model. Concerning model setup, infiltration was computed using the Curve Number method and dynamic wave was chosen as the flow routing model. Ressaca subcatchment is entirely in Belo Horizonte territory and detailed cadastre of the drains is available. In turn, most of the Sarandi subcatchment area is located in the neighbouring city, Contagem and available data of the storm water network only allowed a much less detailed spatial subdivision of the catchment. Thus Ressaca and Sarandi creek catchments were divided respectively into 54 and 10 subcatchments according to their land features and channel locations which were obtained from topographic maps and storm drain system cadastre [START_REF] Bonnary | Modélisation d'un bassin hydrologique : application au bassin de Pampulha[END_REF]. Maps were also used to provide subcatchment information, such as slope or impervious cover. Thiessen polygons method was applied to spatially distribute rainfall data from available raingages.

An automatic model calibration procedure based on a genetic algorithm was used to calibrate the most difficult to assess parameters. Genetic algorithms can be used for solving optimization problems inspired by natural selection. They have been used for hydrologic models calibration or optimization during the last 20 years and they are now well diffused in hydroinformatics applications [START_REF] Savic | Evolutionary computing in hydrological sciences[END_REF]. The main interest of these algorithms is their efficiency in finding close-to-optimal solutions of high-dimensional optimization problems, without strong hypotheses on the problem setting (e.g. linearity or continuity of the objective function).

The optimisation function adopted was Nash criterion (equation 1) and to achieve a compromise between model performance and computing time the maximum number of iterations was set equal to 100 for Ressaca subcatchment and 50 for Sarandi subcatchment. Equation 1Where Fsim is the calculated flow at timestep i (m 3 s -1 ) ; Fmes is the observed flow at timestep i (m 3 s -1 ) and is the mean measured flow over the simulation period (m 3 s -1 ).

Lake ecological model

Hydrodynamic models are frequently coupled to ecological models: the former describes the physical processes of transport and mixing in the water column, while the latter represents the main chemical and biological processes that affect phytoplankton and higher trophic levels. The deterministic coupled model DYRESM-CAEDYM (hereafter DYCD) is used in order to simulate water temperature and cyanobacteria dynamics in Lake Pampulha. DYRESM (DYnamic REServoir Simulation Model) is a one-dimensional hydrodynamic model based on variable depth layers used to predict the vertical distribution of temperature, salinity and density in lakes and reservoirs. DYRESM can be coupled to CAEDYM -(Computational Aquatic Ecosystem Dynamics Model) for investigating biological and/or chemical processes such as nutrient cycling studies and algal succession [START_REF] Hamilton | Prediction of water quality in lakes and reservoirs. Part I -Model description[END_REF].The input data required to DYCD are (Figure 2, right hand side block) : lake morphometry, inflows (water temperature and nutrient concentrations), outflows, meteorological forcing (wind speed, air temperature, solar radiation, rainfall, cloud cover and vapour pressure) and the initial conditions for cyanobacteria biomass, nutrient and dissolved oxygen concentrations and water temperature.

In our approach, cyanobacteria represented by the equivalent chl-a concentration are not submitted to vertical movements such as migration and settling. Zooplankton groups and higher trophic levels were not included in the simulation. Nitrate and phosphate limitations were simulated through the Michaelis-Menten equation. Half-saturation constants for nitrates and phosphates are set to 55 and 5 µg.L -1 , respectively. Chl-a concentration was converted into carbon by using a constant rate of 40 mg C.mg chla -1 . To assess model performance, the root mean squared error (rmse, Equation 2) between measurements and the simulated water temperature and cyanobacteria biomass at each timestep and each depth was computed where field data were available. Equation 2Where Θsim (ti,,dj) is the water temperature (°C) or the cyanobacteria biomass (μg chla L -1 ) simulated by the model at timestep i and depth j and Θmes (ti,, dj) is the water temperature (°C) or the cyanobacteria biomass (μg chla L -1 ) measured at timestep i and depth j.

Integrated modelling approach

Once calibrated and validated separately the rainfall-runoff and the lake models will be coupled by using the outflow volume and water quality simulated by the hydrological model as input to the hydrodynamic and ecological lake model, which in turn will simulate cyanobacteria dynamics (Figure 2 ). This coupled model could be used to evaluate different scenarios of catchment changes: (i) meteorological changes such as air temperature increase, changes in rainfall regime and in wind speed; (ii) inflow rate and quality degradation due to the intensification of land-use, the expansion of impervious areas, or the opposite, improvements in the sanitary system in the catchment.

As mentioned in paragraph 2.2.2, water quality data from the tributaries of Lake Pampulha are not yet available, only bimonthly punctual sampling were conducted in order to estimate the order of magnitude of nutrient and suspended solids concentrations entering the lake. However, an initial test of the coupled model functioning was undertaken by using outflow simulated by SWMM to replace inflow data in DYCD modelling.

RESULTS AND DISCUSSION

Rainfall-runoff modelling

Runoff from Sarandi catchment was firstly calibrated from 18 th November 2011 to 13 th January 2012 and then validated between 14 th January and 13 th March 2012. In a next step, runoff was calibrated during the latter period and validated over the former one in order to perform a cross-calibration. This approach was applied to confirm the results obtained by the model since available data correspond to only one rainy season. The same procedure was applied for Ressaca catchment from 1 rst February to 18 th March 2012 and from 19 th March to 02 nd May 2012, period in which data were available. For each catchment, eight parameters were calibrated (Table 2): (i) the Manning coefficient of conduits (n-cond ), (ii) the time necessary for a fully saturated soil to completely dry (Dry time), (iii) characteristic width of the overland flow path for sheet flow runoff (w), (iv) Manning coefficient for overland flow over the impervious areas (n-imp), (v) Manning coefficient for overland flow over the pervious areas (n-perv), (vi) depth of depression storage on the impervious areas (s-imp), (vii) depth of depression storage on the pervious areas (s-perv) and (viii) the Curve Number parameter (CN). All parameters were calibrated to obtain a mean value representing each catchment, except CN parameter which was calibrated for each subcatchment in Ressaca and Sarandi catchment.

Runoff modelling from Sarandi catchment showed excellent results and its simulation periods comprised most of the rainy season (total rainfall during calibration and simulation = 1 348.8 mm). Nash coefficient ranges from 0.74 to 0.90 (Figure 3). The model successfully reproduced flow peaks intensity as well as the time they occurred (see example in Figure 3). The rainfall-runoff model of Ressaca catchment was less effective regarding the previous model, but still Nash coefficient values were higher than 0.70 which is a good agreement between measurements and simulation [START_REF] Bennis | New Runoff Simulation Model for Small Urban Catchments[END_REF]. In view of the results obtained for both catchments and taking into account that Sarandi and Ressaca catchments represents respectively 50% and 20% of Lake Pampulha contributing area, the models can be considered calibrated, validated and ready to be used in a coupled approach with Lake Pampulha model. To take into account the runoff produced by the others subcatchments which compose Lake Pampulha catchment, the sum of the daily water volume from Sarandi and Ressaca creeks was increased by 30% according to estimations carried out by [START_REF] Tôrres | Mass balance estimation of nitrogen, carbon, phosphorus and total suspended solids in the urban eutrophic, Pampulha reservoir, Brazil[END_REF].

Water temperature and cyanobacteria biomass modelling

The dataset obtained in 2011 was used to calibrate DYCD for simulating water temperature and cyanobacteria dynamics from 19 th September to 10 th November in the deepest point of Lake Pampulha (point P3 in Figure 1). DYCD was manually calibrated by adjusting layer thickness to ensure a good resolution of model results and then a multiplier factor of the wind speed was fit to take into account different conditions on the lake and at the weather station. In a next step, the most sensitive parameters to cyanobacteria growth were calibrated; their values are shown in Table 4. A retro-calibration of layer thickness and the wind factor was then carried out. Parameters related to phosphorus and nitrogen uptake by cyanobacteria were set as the model default values since nutrients are not limiting-growth to algae in Lake Pampulha during the simulation period.

Model results show good agreement with measurements for both water temperature (mean rmse = 0.61°C, mean temperature over simulation period = 21°C) and cyanobacteria biomass (mean rmse = 6.63 μg chl-a L -1 , mean biomass over simulation period = 30 μg chl-a L -1 ). Figure 4 shows measured and simulated vertical profiles of water temperature. During the first days, water column was rather mixed. From 4 th October, as the temperature on the lake surface rises, the water column becomes more stratified. DYCD model successfully represents this evolution except on 18 th October which coincides with a failure in a water jet located near the sampling point. The water jet has an aesthetic function but it also plays a role on water mixing in this lake area.

Cyanobacteria dynamics follows the water temperature trend: vertical distribution is quite homogenous during the first two weeks and the biomass increases from early October as a result of favourable meteorological conditions (high light intensity and water temperature). Limitation functions calculated by the model for phosphorus and nitrogen are always greater than 0.97; this indicates that cyanobacteria growth is not limited by nutrient concentrations. The ecological model was able to catch these variations. Not surprisingly, the model fails to represent cyanobacterial biomass for the week of October 18 th for the same reason described above for water temperature. Figure 5 shows measured and simulated vertical profiles of cyanobacteria biomass.

Preliminary coupled model

The calibrated and validated rainfall-runoff model was used to simulate runoff volume in Ressaca and Sarandi catchments from 04 th October to 10 th November 2011, which corresponds to a part of the calibration period for Lake Pampulha model. The inflow data obtained from the water balance computation were replaced by the runoff model output and variations in the simulated water temperature and cyanobacteria biomass were checked to assess the coupled model performance.

The change in inflow data has led only to a small degradation in the lake model performance since vertical profiles for water temperature and cyanobacteria biomass have remained rather unchanged.

For water temperature mean rmse = 0.71°C and for cyanobacteria dynamics mean rmse = 7.20 μg chl-a L -1 . The lack of significant variation in the results is encouraging about the two models' coupling. However, it could also stand for a lack of sensitivity of the lake model to the inflow volume model during this period. The study of this "inter-sensitivity" will be the object of further analyses on the coupled model.

CONCLUSION AND PERSPECTIVES

An integrated modelling approach is proposed in which the urban runoff simulated by a hydrological model is used as input for a hydrodynamic-ecological model simulating cyanobacteria dynamics. Both models, SWMM and DYCD revealed a good ability to reproduce processes of the urban water cycle in the catchment and into the receiving water body. A first attempt to couple SWMM and DYCD for the moment concerning only water volume was promising. In the next steps, the lake model will be validated with another dataset and the SWMM water quality block will be used to simulate runoff water quality.

This integrated approach of the catchment and the lake will make possible to study Lake Pampulha as a water body that integrates and responds to changes that occur in its surroundings. The results obtained will help to determine in advance the impacts of a likely intensification of urbanization on this water body which has a great touristic, cultural and landscaping relevance to Belo Horizonte city. The assessment of positive impacts resulting from improvements in the sanitation system will be possible and thus management strategies for the lake and its catchment will be likely better guided. This integrated approach in Lake Pampulha may not only help to guide its restoration and protection, but also may be replicated to other water bodies located in urban areas.

Figure 1 :

 1 Figure 1 : Lake Pampulha (on the right) and its catchment (on the left). Triangles: green -rainfall stations; pinkrainfall and water level stations; blue -automatic samplers and sensors of temperature and turbidity. Red circles: points of sampling in Lake Pampulha (satellite imagery: © 2012 Google Earth, © 2012Digital Globe, © 2012 Geoeye)

Figure 2 :

 2 Figure 2 : Integrated modelling diagram: rainfall-runoff model steps and the lake model inputs and output.

Figure 3 :

 3 Figure 3 : Rainfall-runoff model results (blue line) and measurements (red dots) for Sarandi catchment during calibration period.

Figure 4 :Figure 5 :

 45 Figure 4 : Model results (blue line) and measurements (red dots) for water temperature at point P3. RMSE values are in °C units

Table 1 : Lake Pampulha main features

 1

		Population (inhabitants)	~350 000
		Total area (km 2 )	98
	Pampulha	Ressaca catchment area (km 2 )	20
	Catchment	Sarandi catchment area (km 2 )	49
		Ressaca+Sarandi mean flow dry weather (1) (m 3 s -1 )	0.651
		Ressaca+Sarandi mean flow wet weather (1) (m 3 s -1 )	1.632
		Mean depth (m)	5.1
	Lake Pampulha (2)	Maximum depth (m) Area (m 2 )	16.2 1.97 x 10 6
		Volume (m 3 )	9.9 x 10 6
	Water quality (3)	Ptotal (μg P L -1 )	58 -925 (207)
		PO4 (μg P L -1 )	1.7 -113.1 (22.1)
			3

Table 2 :

 2 Rainfall-runoff model calibrated parameters

	Parameter	Range value	Assigned value Sarandi Ressaca
	n-cond	0.009 -0.1	0.010	0.045
	Dry time	2 -7	2.02	6.12
	Width	40 -3000	2910	1398
	n-imp	0.01 -0.06	0.059	0.011
	n-perv	0.009 -0.05	0.02	0.0093
	s-imp	0.1 -3	2.98	0.44
	s-perv	0.1 -3	1.22	2.88
	CN	50 -98	Mean: 67.4	Mean: 88.8

Table 3 :

 3 Rainfall-runoff model performance

Sarandi catchment Ressaca catchment Simulation Period (numbers of days) Total rainfall (mm) Nash coefficient Period (numbers of days) Total rainfall (mm) Nash coefficient

	Calibration	18 th November to 13 th January (56)	1134.63	0.90	1 rst February to 18 th March (46)	169.65	0.74
	Validation	14 th January to 14 th March (60)	214.13	0.74	19 th March to 2 nd May (46)	169.79	0.74
	Cross-calibration	14 th January to 14 th March (60)	214.13	0.87	19 th March to 2 nd May (46)	169.79	0.79
	Cross-Validation	18 th November to 13 th January (56)	1134.63	0.79	1 rst February to 18 th March (46)	169.65	0.71

Table 4 :

 4 Lake ecological model calibrated parameters

	Parameter	Unit	Range value	Assigned value
	Minimum layer thickness	m	0.2 -1.0	0.6
	Maximum layer thickness	m	0.4 -5.0	4.0
	Wind factor	-	1 -3.0	2.36
	Maximum growth rate	day -1	0.5 -0.7	0.64
	Optimum temperature	°C	31 -33	33
	Maximum temperature	°C	39 -41	39
	Light saturation	μEs -1 m -2	100 -600	290

ACKOWLEDGEMENTS

This work is part of MAPLU 2 project funded by the Brazilian agency FINEP. It is conducted within the CAPES-COFECUB project Ph 704/11. We thank CNPq/Brazil and Région Rhônes-Alpes/France for student scholarships and FAPEMIG for funding complementary equipment. Université Paris-Est and Ecole des Ponts ParisTech are acknowledged for the doctoral grant delivered to the first author. The authors are grateful to the Belo Horizonte municipality for its support and for providing data. Daniel Zuim, Kláudia Silva and Samuel Cardoso are thanked for their help with laboratory analysis and field sampling.


Fr Éd Éric Mazenc
email: fred-eric.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

Corina Barbalata
email: cbarbalata@lsu.edu

Zhong-Ping Jiang
email: zjiang@nyu.edu

Michael Malisoff Mazenc

Z.-P Jiang

Event-Triggered Control for Discrete-Time Systems Using a Positive Systems Approach

Keywords:

We provide an output feedback eventtriggered controller for discrete-time linear systems. We make novel use of positive systems, interval observers, an event-triggered state estimator, and triggering times that are computed from estimator values. This provides a discrete-time analog of our recent positive systems approach for continuous-time systems. A key novel ingredient in our discrete-time event triggers is their use of vectors of absolute values, instead of the usual Euclidean norm. We illustrate the benefits of our method using a model for event-triggered BlueROV2 underwater vehicles.

the positive systems literature is interval observers, which produce intervals containing values of unknown states when the inequalities are interpreted to be componentwise; see [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF]. Control theory based on interval observers and positive systems led to significant novel methods in aerospace engineering, mathematical biology, and other applications.

This motivates the event-triggered controls with outputs in this paper, which we believe are the first use of positive systems and interval observers for event-triggered control of discrete-time systems. The problems in this work cannot be solved by discretizing continuous-time event-triggered works (e.g., [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF]), because of the continuous-time measurements in the event triggering rule in works like [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF]. Instead, we use interval observer and positive systems methods for discretetime systems, using new synergies of small-gain methods [START_REF] Jiang | Nonlinear small-gain theorems for discrete-time feedback systems and applications[END_REF] and discrete-time systems [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF], to prove input-to-state stability (or ISS). A key novel feature is our use of vectors of absolute values instead of standard Euclidean norms; see Section IV.

However, our stability proof uses interval observers for comparison systems, and so is inspired by works such as [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF] insofar that it uses linear Lyapunov functions. One of the positive systems is the dynamics for ( X, -X) where X and X are the upper and lower bounds for the observer state in the interval observer, respectively. The negative sign in front of the X will be required in order to produce dynamics with nonnegative coefficient matrices. On the other hand, our eventtriggered controlled closed loop systems are not assumed to be, or transformed into, positive systems. Instead, we transform interval observers into positive systems. Hence, our work is not overlapping with notable works such as [START_REF] Abate | Improving the fidelity of mixed-monotone reachable set approximations via state transformations[END_REF] and [START_REF] Khajenejad | Tight remainder-form decomposition functions with applications to constrained reachability and interval observer design[END_REF] on estimating reachability sets for (not necessarily positive) dynamics that are transformed into systems that are positive. Also, [START_REF] Abate | Improving the fidelity of mixed-monotone reachable set approximations via state transformations[END_REF], [START_REF] Khajenejad | Tight remainder-form decomposition functions with applications to constrained reachability and interval observer design[END_REF], and [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF] made no use of event-triggering. Hence, this work has the potential to reduce the computational burden as compared with discrete-time controls that are not event triggered. After stating and proving our theorem in Section III, Section IV has an application to a model of the BlueROV2 underwater vehicle that can be used for environmental surveys. This illustrates a benefit of our work in a example that is of compelling interest in robotics.

II. PRELIMINARIES

We review notation that we also used in continuous-time cases in [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF]. The dimensions of our Euclidean spaces are arbitrary unless we indicate otherwise. We omit arguments of functions when no confusion would arise. We set

Z 0 = {0, 1, 2, ...}. For a matrix G = [g ij ] ∈ R r×s , we set |G| = [|g ij |]
, so the entries of |G| are the absolute values of the corresponding entries g ij of the matrix G. We also set G + = [max{g ij , 0}] and G -= G + -G. For a bounded sequence M k ∈ R r×s where M k has (i, j) entry m ijk for i = 1, . . . , r and j = 1, . . . , s, we define the matrix sup k |M k | to be the matrix whose (i, j) entry is s ij where s ij = sup k |m ijk | for i = 1, . . . , r and j = 1, . . . , s. A square matrix is called Schur stable provided all of its eigenvalues are inside the open unit disk in the complex plane. For matrices D = [d ij ] and E = [e ij ] of the same size, we write D < E (resp., D ≤ E) provided d ij < e ij (resp., d ij ≤ e ij ) for all i and j. We then also use D E to mean that there is a pair (i, j) such that d ij > e ij . We adopt similar notation for vectors. We then call a matrix S positive (resp., nonnegative) provided 0 < S (resp., 0 ≤ S), where 0 is the zero matrix. We also use standard notions of input-to-state stability (or ISS) for discretetime systems [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF]. We use || • || to denote the usual Euclidean norm. Also, I is the identity matrix. We consider the system

X k+1 = (A + δ A k )X k + (B + δ B k )u k + δ D k Y k = CX k (1)
where the sequence X k of state vectors is valued in R n , the input sequence k ∈ R n×p represent model uncertainty, and the unknown sequence δ D k ∈ R n represents a disturbance. The uncertainties δ A and δ B represent uncertain model parameters, which are prevalent in underwater marine robotics. We assume:

u k is valued in R p , Y k ∈ R m are the measured variables, A ∈ R n×n , B ∈ R n×p
Assumption 1: There is a matrix K ∈ R p×n such that

H = A + BK (2)
is nonnegative. Assumption 2: There is a matrix Γ ∈ R n×n such that Γ > 0 and such that the matrix

H + |BK|Γ (3)
is Schur stable. Assumption 3: The pair (A, C) is observable. Assumptions 1-2 are not restrictive, insofar that they can be satisfied after a change of coordinates under standard controllability conditions on the pair (A, B), as follows.

When a pair (A a , B a ) is controllable, we can choose a matrix K a such that A a +B a K a is Schur stable, having distinct real eigenvalues on the interval (0, 1). This follows from the Pole Shifting Theorem (e.g., [19, p.186]), which ensures that for any complex values λ 1 , . . . , λ n and any controllable pair (A a , B a ) ∈ R n×n ×R n×p , we can construct a matrix K a such that these λ i 's form the set of all eigenvalues of A a + B a K a . Then, there is a P ∈ R n×n such that P (A a + B a K a )P -1 is both Schur stable and nonnegative (by diagonalizing A a + B a K a ). Hence, if we consider Z k+1 = A a Z k + B a u + δ a,k , then the change of coordinates X k = P Z k gives

X k+1 = AX k + Bu + P δ a,k (4)
with A = P A a P -1 and B = P B a . Then, in this case, (4) satisfies Assumptions 1-2, because

K = K a P -1 gives A + BK = P A a P -1 +P B a K a P -1 = P (A a +B a K a )P -1 which
is Schur stable and nonnegative, and because when a matrix H ≥ 0 is Schur stable, there always exists a positive matrix Γ > 0 such that H + |BK|Γ is Schur stable and nonnegative. This follows because the continuity of eigenvalues as functions of the entries of a matrix ensures that, if all eigenvalues of H are in the open unit circle in the complex plane, then the same is true for H + E when E ∈ R n×n has small enough entries, and because we can choose the entries of Γ to be small enough so that E = |BK|Γ has small enough entries. Hence, we can find our required K by combining the Pole-Shifting Theorem with a similarity transformation. Nonnegativity of H is essential for showing that our interval observers provide the required bounds for proving our exponential stability theorem. However, we will not assume that H is Schur stable. Assumption 2 ensures that there are a positive vector V ∈ R n and a constant p ∈ (0, 1) such that

V (H + |BK|Γ) ≤ pV (5)
holds; see [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]Lemma 2.7,p.79]. By the same reasoning, Assumption 3 provides a matrix L ∈ R n×m and an invertible matrix Q ∈ R n×n such that, with the choices

R = A + LC and QRQ -1 = M, (6)
the matrix R is Schur stable and M ≥ 0. Since M ≥ 0 is Schur stable, we can again apply [4, Lemma 2.7] to find a vector W > 0 in R n and a constant q ∈ (0, 1) such that

W M ≤ qW . (7)
We fix K, L, Γ, p, q, V , and W satisfying the preceding requirements. Then in terms of the sequence

M k = |Q(δ A k + δ B k K)| + |Qδ B k K|Γ, (8)
our last assumption is: Assumption 4: The inequalities

sup k W M + Qδ A k Q -1 ≤ qW and V |LC| |Q -1 | sup k W M k < 1 (9)
are satisfied. Assumption 4 holds when the suprema of the sequences δ A k and δ B k are small enough. For instance, since

|Qδ A k Q -1 | ≤ ||Qδ A k Q -1 ||1 n×n ≤ ||Q|| ||Q -1 || ||δ A k || 1 n×n , and ||M k || ≤ M * max{||δ A k ||, δ B k ||} hold for all k when we choose M * = ||Q||(1 + ||K|| + |||K|Γ||
) and when we define 1 r×s ∈ R r×s to be the matrix each of whose entries is 1, a set of sufficient conditions for Assumption 4 to hold is that [START_REF] Krause | Positive Dynamical Systems in Discrete Time[END_REF] hold for all k, when the denominators in [START_REF] Krause | Positive Dynamical Systems in Discrete Time[END_REF] are nonzero.

||δ A k ||1 1×n ≤ W (qI-M ) n||W || ||Q|| ||Q -1 || and max{||δ A k ||, δ B k ||} < 1 M * ||W || ||V |LC| |Q -1 | ||

III. MAIN THEOREM

A. Statement of Result and Remarks

We next provide our interval observer based output feedback control theorem. A key novel feature is its event trigger rule, which is expressed using vectors of absolute values instead of the standard Euclidean norm.

Theorem 1: Let Assumptions 1-4 hold, and K, L, and Γ satisfy the requirements of Section II. Consider the system

X k+1 = (A + δ A k )X k + (B + δ B k )K Xσ(k) + δ D k Xk+1 = A Xk + BK Xσ(k) + L[C Xk -Y k ] (11)
where σ : Z 0 → Z 0 is defined by the event trigger rule

σ(0) = 0 σ(j + 1) = j + 1 if | Xj+1 -Xσ(j) | Γ| Xj+1 | σ(j + 1) = σ(j) if | Xj+1 -Xσ(j) | ≤ Γ| Xj+1 |. (12)
Then ( 11) is globally exponentially ISS to 0 on R 2n with respect to δ D . Remark 1: Theorem 1 uses u k = K Xσ(k) , where ( 12) can be equivalently defined in the following recursive way. Setting σ(0) = 0, we check whether | X1 -Xσ(0) | ≤ Γ| X1 | holds. If it holds, then we set σ(1) = σ(0) = 0 and no change in the control value is made at time k = 1; otherwise, we set σ(1) = 1 and we update the control value to u 1 = K Xσ(1) = K X1 . Then, we repeat this process with j = 0 replaced by j = 1, and argue inductively to define the control values and σ(j) for all j ≥ 0. The motivation for defining σ in this way is that it will ensure that | Xj -Xσ(j) | ≤ Γ| Xj | holds for all j ∈ Z 0 , as we will show in our proof of the theorem. See Remark 2 for discussions on global exponential ISS, and the special case where X is available for measurement. See also the end of the proof of our theorem, for an approach that can be used to find comparison functions for the ISS estimate, which show how to explicitly treat the uncertainties δ D .

B. Proof of Theorem

The proof has three parts. First, we build interval observers for Xk and Q Xk , where Xk = X k -Xk and Q is from Section II. In the second part, we provide stability analyses for both interval observers. In the third part, we combine the stability analyses to prove the ISS conclusion. First note that direct calculations give

Xk+1 = AX k -A Xk -L[C Xk -CX k ] + δ k = R Xk + δ k where δ k = δ A k X k + δ B k K Xσ(k) + δ D
k , and R = A + LC as before. Therefore, with the notation

ŵk = Xσ(k) -Xk and δ E k = (δ A k + δ B k K) Xk + δ B k K ŵk + δ A k Xk + δ D k (13)
and H = A + BK as defined before, we obtain the dynamics

Xk+1 = H Xk + BK ŵk -LC Xk Xk+1 = R Xk + δ E k . ( 14
)
First Part: We first introduce the interval observer

         X k+1 = H X k + (BK) + w + k + (BK) -w - k + (-LC Xk ) - X k+1 = H X k -(BK) + w - k -(BK) -w + k -(-LC Xk ) + (15)
where for any k 0 ∈ Z 0 , we choose initial conditions satisfying

X k0 ≤ X k0 ≤ X k0 , X k0 ≤ 0, and X k0 ≥ 0. (16)
We next prove that for all k ≥ k 0 , we have

X k ≤ X k ≤ X k , X k ≥ 0, and X k ≤ 0. ( 17
)
To this end, notice that since H is nonnegative, and since (BK

) + ŵ+ k + (BK) -ŵ- k + (-LC Xk ) -≥ 0 and -(BK) + ŵ- k -(BK) -ŵ+ k -(-LC Xk ) + ≤ 0 hold for all k ≥ k 0 , we straightforwardly deduce that Xk ≤ 0 and Xk ≥ 0 for all k ≥ k 0 , (18)
e.g., by induction on k. This makes essential use of the nonnegativity of H. In particular, the Schur stability property of H is not sufficient. Also, the Xk dynamics in ( 14) can be rewritten as

Xk+1 = H Xk +(BK) + ŵ+ k +(BK) -ŵ- k -(BK) + ŵ- k -(BK) -ŵ+ k -(-LC Xk ) + + (-LC Xk ) -. (19)
By combining ( 15) and ( 19), we get

         Xk+1 -Xk+1 = H( Xk -Xk ) + (-LC Xk ) + + (BK) + ŵ- k +(BK) -ŵ+ k Xk+1 -Xk+1 = H( Xk -Xk ) + (-LC Xk ) - + (BK) + ŵ+ k +(BK) -ŵ- k (20)
Since H ≥ 0, (BK) + ŵk + (BK) -ŵ+ k + (-LC Xk ) + ≥ 0, and (BK) + ŵ+ k + (BK) -ŵk + (-LC Xk ) -≥ 0 all hold for all k ≥ k 0 , one can prove by induction that Xk -Xk ≥ 0 and Xk -Xk ≥ 0

for all k ≥ k 0 , which proves the first two inequalities of ( 17), and which also uses the nonnegativity of H in an essential way. Hence, ( 17) holds for all k ≥ k 0 , by [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF]. To derive our interval observer for Q Xk , first note that ( 14) and the formula

Q -1 M Q = R from (6) give Q Xk+1 = M Q Xk + Qδ E k .
We now introduce the interval observer

ξ k+1 = M ξ k + (Qδ E k ) -, ξ k+1 = M ξ k -(Qδ E k ) + (22) with the initial conditions ξ k0 = (Q Xk0 ) + and ξ k0 = -(Q Xk0 ) -. (23)
Since M ≥ 0, we can argue as we did for (15) (with Xk replaced by ξ k , H replaced by M , ŵk replaced by 0, and

-LC Xk replaced by Qδ E k ) to prove that ξ k ≤ Q Xk ≤ ξ k , ξ k ≤ 0, and ξ k ≥ 0 (24)
for all k ≥ k 0 . Second Part: Stability Analysis for Interval Observers ( 15) and (22). Using [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF], elementary calculations give

X k+1 -X k+1 = H X k -X k + (BK) + w + k + (BK) + w - k + (BK) -w - k +(BK) -w + k + (-LC Xk ) + +(-LC Xk ) - = H X k -X k +|BK|| w k |+|LC Xk |. (25)
Here and in the sequel, all equalities and inequalities should be understood to hold for all k ≥ k 0 unless otherwise indicated. Also, by separately considering the cases σ(k) = k and σ(k) = k, we can use [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF] (with the choice j = k -1) to conclude that the variable ŵk defined in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] satisfies

| ŵk | ≤ Γ| Xk | (26)
for all k ≥ k 0 . By left multiplying (25) by V , this gives

V X k+1 -X k+1 ≤ V H X k -X k +V |BK|Γ| Xk |+V |LC Xk |. (27)
Therefore, it follows from ( 17) and ( 5) that

V X k+1 -X k+1 ≤ V (H + |BK|Γ) X k -X k +V |LC Xk | ≤ pV X k -X k + V |LC|| Xk | ≤ pV X k -X k + V |LC||Q -1 ||Q Xk |, ( 28
) because | Xk | ≤ Xk -Xk . Since (24) gives |Q Xk | ≤ ξ k -ξ k for all k ≥ k 0 , we can use (28) to get V X k+1 -X k+1 ≤ pV X k -X k +V |LC||Q -1 |(ξ k -ξ k ), (29)
which is our desired estimate for the first interval observer.

To obtain our stability estimate for the second interval observer ( 22), notice that (22) gives

ξ k+1 -ξ k+1 = M (ξ k -ξ k ) + |Qδ E k |. (30)
Also, since our conditions ( 17) and ( 24) give | Xk | ≤ Xk -Xk and |Q Xk | ≤ ξk -ξ k for all k ≥ k 0 , we can combine ( 13), [START_REF] Peralez | Event-triggered output feedback stabilization via dynamic high-gain scaling[END_REF], and (26) to obtain

|Qδ E k | ≤ |Q(δ A k +δ B k K)|+|Qδ B k K|Γ Xk -Xk +|Qδ D k | + |Qδ A k Q -1 |( ξk -ξ k ). (31)
From Assumption 4, it follows from (30)-(31) that

W (ξ k+1 -ξ k+1 ) = W M (ξ k -ξ k ) + W |Qδ E k | ≤ qW (ξ k -ξ k )+W |Qδ D k | + W M k Xk -Xk (32)
for all k ≥ k 0 , where M k is from [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF].

Third Part. Setting V a (q) = V q, V B (q) = W q, q a = X -X, andc B = 1 -q, we can combine ( 29) and (32) to get

q B = ξ -ξ, γ a (s) = ||V |LC||Q -1 |||s, γ B (s) = sup k ||W M k ||s, γ D (s) = ||W || ||Q||s, c a = 1 -p,
V a (q a k+1 ) -V a (q a k ) ≤ -c a V a (q a k ) + γ a (||q B k ||) and V B (q B k+1 ) -V B (q B k ) ≤ -c b V B (q B k ) + γ B (||q a k ||) + γ D (||δ D k ||) (33)
for all k ≥ k 0 . It now follows from the second inequality in [START_REF] Khajenejad | Tight remainder-form decomposition functions with applications to constrained reachability and interval observer design[END_REF] (which implies that the small-gain condition γ a • γ B (s) < s holds for all s > 0) and the proof of [6, Theorem 1] that the dynamics for the joint variable

X = X -X, ξ -ξ (34)
is globally exponentially ISS on R n with respect to δ D . If we now assume that the initial conditions are

X k0 = X + k0 and X k0 = -X - k0 , (35)
then we can use ( 17), (23), and (24) to get | Xk | ≤ Xk -Xk and |Q Xk | ≤ ξk -ξ k for all k ≥ k 0 , and

Xk0 -Xk0 = | Xk0 | and ξk0 -ξ k0 = |Q Xk0 |, (36)
and therefore also

||( Xk , X k )|| ≤ || Xk -Xk || + || Xk + Xk || ≤ 2|| Xk -Xk || + ||Q -1 ||||Q Xk || ≤ 2|| Xk -Xk || + ||Q -1 |||| ξk -ξ k || ≤ 2 max{2, ||Q -1 ||}||X k || (37)
for all k ≥ k 0 and

||X k0 || ≤ || Xk0 || + ||Q Xk0 || ≤ || Xk0 || + ||Q||(||X k0 || + || Xk0 ||) = (1 + ||Q||)|| Xk0 || + ||Q|| ||X k0 || ≤ (1 + 2||Q||)||(X k0 , Xk0 )|| (38)
for all k 0 ≥ 0. We can use (37)-(38) to convert the ISS estimate for X into the required ISS estimate of the theorem. Remark 2: The preceding proof shows that in the special case where the state X is available for measurement, the theorem remains true if we replace the estimator X by X in the event trigger. Its global exponential ISS conclusion means that there are positive constants c 1 and c 2 such that the following estimate holds for all solutions of (11), all choices of δ A and δ B satisfying Assumption 4, all choices of δ D , and all initial times

k 0 ∈ Z 0 : ||(X k , Xk )|| ≤ c 1 e -c2(k-k0) ||(X k0 , Xk0 )|| + c 1 sup{||δ D r || : k 0 ≤ r ≤ k} for all k ≥ k 0 .
See also Remark 3 for a comparison with event triggers that instead use the usual Euclidean norm.

Remark 3: The triggering rule (12) calls for choosing the event triggering times such that | Xj -Xσ(j) | ≤ Γ| Xj | holds for all indices j; see (26). Due to our | • | notation, this is equivalent to requiring that

| Xj,i -Xσ(j),i | ≤ n k=1 Γ ik | Xj,k | for i = 1, . . . , n, (39)
where Γ = [Γ ij ] and Xr,s is the sth component of Xr for all r ∈ Z 0 and s = 1, . . . , n. By contrast, if we had instead used the standard Euclidean norm to define our triggering criterion (as one would do, e.g., when using quadratic Lyapunov functions), then we would have selected a constant σ * > 0 such that with the σ notation from Theorem 1, we have

|| Xj -Xσ(j) || ≤ σ * || Xj || (40)
for all j ∈ Z 0 . To better understand the effects of using vectors of absolute values in (12) instead of the Euclidean norm in the event triggers, it is therefore useful to find the largest possible σ * such that the requirement from (39) holds for all X and j choices such that (40) holds. Let us now show that this largest value is σ * = min ij Γ ij , i.e., the smallest of the entries of the positive matrix Γ. To this end, first note that with this choice of σ * , and for any v and w in R n such that ||v -w|| ≤ σ * ||v|| and any i ∈ {1, . . . , n}, we can use the subadditivity of the square root to get

|v i -w i | ≤ ||v -w|| ≤ σ * ||v|| ≤ n k=1 Γ ik |v k |. (41)
Hence, with this σ * , the condition from (40) implies (39). Next choose a pair (p, q) such that σ * = Γ pq . First consider the case where p = q, and any constant * > σ * and the choices of v ∈ R n and w ∈ R n whose components are defined by v i = 1 if i = q and v i = 0 if i = q, and by

w i =    1, if i = q * , if i = p 0, if i = p and i = q (42)
for i = 1, . . . , n. With the preceding choices, we have ||v -w|| = * = * ||v||. The preceding choices also give

|v p -w p | = * > σ * = n k=1 Γ pk |v k | (43)
so the condition from (39) would not be satisfied for the index i = p. The same reasoning applies in the case where p = q, by instead defining the components of v and w to be v p = 1, w p = 1 + * , and v i = w i = 0 for all i = p. This shows how the largest (i.e., least conservative) σ * we could have allowed if we had instead used the standard Euclidean norm would have been σ * = min ij Γ ij . In Section IV, we illustrate an advantage of using our triggers based on vectors of absolute values instead of using Euclidean norms.

IV. ILLUSTRATION

We consider a dynamics for the control of the depth and pitch degrees-of-freedom (or DOF) of an autonomous underwater vehicle (or AUV), e.g., the BlueROV2 vehicle which is used for environmental surveys such as the study of corals. The vehicle is assumed to be equipped with a Doppler Velocity Logger (or DVL) that estimates its velocity. When working close to the sea floor, the DVL experiences bottom lock, which makes it impractical to continuously change the control values. Hence, we design a control system for the depth plane, using a more practical sample data approach.

As shown in [18, Equation (9.28)], after linearization and assuming that the vehicle is neutrally buoyant, the linearized dynamics in the depth plane are given by

(m -X ẇ(t) ) ẇ(t) -(mx g + Z q ) q(t) -Z w w(t) -(mU (t) + z q )q(t) = Z γs u Z and (mx g + M ẇ(t)) ẇ(t) + (I yy -M q ) q(t) -M w w(t) + (mx g U -M q )q(t) -M θ θ = M γs u M (44)
whose parameters were experimentally computed and presented in [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF]. We assume that the surge nominal velocity is U = 0.1m/s. The states are the depth and pitch velocity x = [w, q] and the control inputs u Z and u M represent the force and moment required to produce motion of the vehicle. With the parameter values and controller from [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF], the system (44) has the form ẋ(t) = F x(t) + Gu with

which satisfied the requirements of Theorem 1 with the preceding choices of K and L. For our simulations for the time horizon of [0, 10] seconds, the event trigger from Theorem 1 produced an average of 8 trigger times t i . By contrast, using the standard Euclidean norm in the event trigger as in Remark 3 with the same initial conditions, over the same time horizon and with σ * = 0.015 being the smallest element of Γ, the event triggered an average of 12 times in our simulations. The settling times from using our event triggers from Theorem 1 and from using the one based on the Euclidean norm, and from using periodic sampling with period 1, were similar. Since the control from Theorem 1 resulted in a 33% average reduction in the number of triggering times without increasing the settling times, this helps illustrate the potential benefits of our method for reducing the numbers of control recomputation times. We found similar reductions in the numbers of event trigger times under larger uncertainties, for many uncertainty values. For instance, in Fig. 2, we present MATLAB simulations using the same parameter values we used for Fig. 1, except under larger uncertainties δ A and δ B in the state and input matrices. The uncertainty was modeled using a Gaussian representation, with a standard deviation of 0.25. The event was triggered 14 times using the control from Theorem 1, while with the standard 2-norm, it was triggered 19 times. Even under these large uncertainties, the system stabilizes, although oscillations are present. We also found that the settling times and oscillations using the controls from Theorem 1 were similar to what we obtained from instead using periodic controls (i.e., control recomputations at the each time k). Moreover, although the controls from Theorem 1 produced trigger times even in the absence of disturbances, and so do not satisfy the consistency condition from works such as [START_REF] Antunes | Consistent event-triggered control for discrete-time linear systems with partial state information[END_REF], our novel event triggers provide benefits by reducing triggering instances.

V. CONCLUSION

We used interval observer and positive systems methods to prove global exponential ISS of event-triggered output feedback controlled linear discrete-time systems. A key feature was our new class of discrete-time event triggers, using matrices of absolute values instead of the usual Euclidean norm. We illustrated the benefits of using our approach, in an example that has implications for underwater environmental surveys. In future work, we aim to prove generalizations to time-varying systems with state, input, or sensor delays.

  , and C ∈ R m×n are known constant matrices, the unknown sequences δ A k ∈ R n×n and δ B

Fig. 1 .

 1 Fig. 1. Depth-pitch control simulation results Our simulations in Fig. 1 used the K and L from the preceding paragraphs, and the choice Γ = 0.015 0.045 0.15 0.15 ,(47)

Fig. 2 .

 2 Fig. 2. Depth-pitch control simulations for large uncertainties

The work was supported in part by NSF grants 2009659 (Malisoff and Barbalata) and 2009644 (Jiang). Recommended by Senior Editor J. Daafouz.

Assuming that the control is piecewise constant with a constant sample rate s, we now convert the dynamics into a discretetime system of the form Z k+1 = A a Z k + B a u with

by using a variation of parameters on [is, (i+1)s) for each i ∈ Z 0 with a constant control on each interval. This conversion to a discrete time system is strongly motivated by the fact that when implementing robotic controllers using the Robotic Operating System (ROS) for any robot, the implementation must be done in discrete time. It also illustrates the applicability of our method to continuous time plants, after a discretization. We next show how the assumptions of Theorem 1 can be satisfied after a change of coordinates, where we first assume that C = [1, 0]. We choose s = 0.5, and a matrix K a such that H a = A a + B a K a has the eigenvalues 0.25 and 0.5 to get the required Schur stability condition on H a , using the command StateFeedbackGains in Mathematica. Then we diagonalize H a to obtain a new matrix P (A a + B a K a )P -1 = H that is Schur stable and nonnegative. Then, with Moreover, we found that our use of vectors of absolute values in our triggering rule [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF] instead of the usual norm had the desired benefit of reducing the number of triggering times in our numerical experiments. For instance, in the special case where the entire state is available for measurement, and with the initial state X 0 = [1, -2] and the constant perturbation δ D k = [0.2, 0.1] and the time horizon [0, 5] (with A, B, K, Γ, and s chosen as in the previous paragraphs, and with δ A and δ B both being zero), our event trigger (12) (with X replaced by X in ( 12)) produced σ(1) = 1, σ(2) = 2, and σ(3) = σ(4) = σ(5) = 2 and therefore only 2 triggering times for the preceding underwater dynamics. This compares favorably with standard zero-order hold controls which would have required 5 changes of control values on this time horizon. On the other hand, the corresponding standard event triggering criterion ||X j -X σ(j) || ≤ σ * ||X j || based on the usual Euclidean norm, with σ * = 0.25 being the smallest element of Γ (and all other parameters the same as in the preceding paragraph), produced 3 event trigger times on the interval [0, 5]. This illustrates the reduction in computational burden from our novel event trigger [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF]. In this example, we ended at time k = 5 because ||X(k)|| remained below 0.1 after time k = 5 with both triggers, while subjected to the preceding constant external disturbance, so k = 5 was a settling time.

We also found that the event triggers from our theorem provided fewer control recomputation times t i (as compared with the event-trigger || Xj -Xσ(j) || ≤ σ * || Xj || with σ * being the smallest entry of Γ that is based on the usual Euclidean norm) in cases where C = [1, 0], i.e., when only the first component is available for measurement. We illustrate this point in Fig. 1, which shows our MATLAB simulations for the depth-pitch controller of the AUV using the positive system approach from our theorem. Our figure shows the results for three different cases for the initial states, and with the uncertainties δ A , δ B , and δ D taken to be zero.


Frédéric Mazenc
email: fred-eric.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

ISS Inequalities for Vector Versions of Halanay's Inequality and of the Trajectory-Based Approach

Keywords: Stability, delay, positive systems, Halanay's inequality. F

Halanay's inequality is a powerful and widely used tool to prove asymptotic convergence properties of functions that arise in the study of systems with delays or continuousdiscrete features. In its standard form, it applies to scalar valued functions that satisfy decay conditions, with overshoots depending on suprema of the functions over suitable intervals.

Then it provides an exponential decay estimate on the scalar functions. Here, we provide vector versions of Halanay's inequality, and of the so-called trajectory based approach, both yielding input-to-state stability (or ISS) inequalities. Our proofs of the inequalities use the theory of positive systems. We apply our results to prove ISS for interval observers and other cases.

I. INTRODUCTION

Halanay's inequality result, which was introduced in [START_REF] Halanay | Differential Equations: Stability, Oscillation, Time Lags[END_REF], is a well-known powerful tool for the stability analysis of systems with delays [4, Chapter 4]. It has been used in many papers. In the last three decades, it has been revisited in some works, notably in the context of time-varying inequalities; see for instance [START_REF] Baker | Exponential stability in pth mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations[END_REF], [START_REF] Hien | New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF], [START_REF] Katz | Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement[END_REF], [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF], [START_REF] Pepe | A nonlinear version of Halanay's inequality for the uniform convergence to the origin[END_REF], [START_REF] Selivanov | Distributed event-triggered control of transport-reaction systems[END_REF], and [START_REF] Wen | Generalized Halanay inequalities for dissipativity of Volterra functional differential equations[END_REF]. In its basic form, it assumes that w is a nonnegative valued C 1 real valued scalar function that admits constants T > 0, c > 0 (called a decay rate), and d ∈ (0, c) (called a gain) such that ẇ(t) ≤ -cw(t) + d sup t-T ≤ ≤t w( ) holds for all t ≥ 0. Then the usual conclusion is that w satisfies an exponential decay condition. More general versions with time dependent functions c and d allow values t where d(t) > c(t).

The recent 'trajectory based approach' is another stability analysis technique. It is efficient for analyzing complex interconnected systems, notably systems with delay and continuous-discrete features. It was developed in works such as [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory based approach[END_REF], and [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF]. The trajectory based approach applies to continuous nonnegative valued scalar valued functions w that are assumed to admit constants ρ ∈ (0, 1) (called the contractivity constant) and T > 0 such that w(t) ≤ ρ sup ∈[t-T,t] w( ) for all t ≥ T . Then the usual conclusion is that w exponentially converges to zero as t → +∞.

On the other hand, it is well-known that the notion of input-to-state-stability (or ISS), introduced by E. Sontag, has been exceptionally fruitful for control design and stability analysis of dynamical systems; see, e.g., [START_REF] Sontag | The ISS philosophy as a unifying framework for stabilitylike behavior[END_REF]. The ISS property was developed in many contributions and has been used to solve many theoretical as well as applied problems.

ISS inequalities are especially useful when one analyzes the stability of interconnected systems [START_REF] Mazenc | Stability and robustness analysis for switched systems with time-varying delays[END_REF] and when one wants to estimate the impact of disturbances.

Only a few papers on Halanay's inequality and the trajectory based approach propose results of ISS type. In [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory based approach[END_REF] and [START_REF] Pepe | A nonlinear version of Halanay's inequality for the uniform convergence to the origin[END_REF], ISS inequalities are established, but these results do not directly extend to vector inequalities. In [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], we provided necessary and sufficient stability conditions for functions satisfying vector inequalities of Halanay's type. However, the proofs in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] do not lead to ISS inequalities when additive disturbances are present. In [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF], a vector version of the trajectory-based approach is developed, but no ISS inequality is given under additive disturbances. This is a shortcoming of these theories because vector inequalities arise in many circumstances, so there is a strong motivation to cover them.

These remarks motivate the present paper. In a first part, we determine an ISS inequality for functions satisfying a vector version of Halanay's inequality. The first part uses methods from the second part, where we consider vector inequalities enabling us to apply the trajectory based approach, and where we determine ISS inequalities when additive disturbances are present. These results complement [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF], [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] and [16, Lemma 1]. Our proofs of the main results use the theory of positive systems, to obtain explicit constructions of the comparison functions in the ISS estimates.

We use standard notation, which is simplified when no confusion would arise, and where the dimensions of our Euclidean spaces are arbitrary, and the entries of our matrices are real values. Let || • || be the usual Euclidean norm of matrices and vectors, 0 is the zero matrix, and I is the identity matrix. Inequalities between matrices are meant entry-wise, so M 1 ≤ M 2 (resp. M 1 < M 2 ) for matrices M 1 and M 2 of the same size means each entry of M 1 is at most (resp. strictly less than) the corresponding entry of M 2 . Let µ(M ) denote the spectral radius of a square matrix M . A square matrix M is called Schur stable provided µ(M ) < 1. A square matrix is called Metzler provided all of its offdiagonal entries are nonnegative. A function f : R → R n is called nondecreasing provided each of its components f i for i = 1, . . . , n are nondecreasing. We write diag{s 1 , . . . , s n } to denote an n × n diagonal matrix whose arguments give its diagonal elements starting from its upper left entry s 1 .

In the next section, we state our main theorems that produce ISS results under vector Halanany's and vector trajectory-based inequalities. We defer their proofs to Section IV. Their proofs are based on key lemmas from Section III. We illustrate the value of our new approaches in Section V. Possible extensions are suggested in Section VI.

II. STATEMENT OF MAIN RESULTS

A. Vector Halanay's Inequality with ISS

Given real constants τ > 0 and h > 0, a diagonal matrix D ∈ R n×n whose diagonal entries are all positive, a matrix P ∈ R n×n such that P > 0, and δ : [-τ -h, +∞) → [0, +∞) n whose components δ i are piecewise continuous, let W : [-τ, +∞) → [0, +∞) n be a C 1 function such that Ẇ (t) ≤ -DW (t) + δ(t)

+ P sup ∈[t-τ,t] w 1 ( ), . . . , sup ∈[t-τ,t] w n ( ) (1)
for all t ≥ 0, where w i is the ith component of W for all i (but see Section V-B for an illustration of how we can relax the requirement that the matrix D is diagonal). In [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF] and other places in this work, we use the sup notation instead of the S(w t ) notation from [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] to indicate the interval over which the sup is computed (which will sometimes be different intervals later in this paper). Also, for our fixed choice of h, we assume without loss of generality that the δ i 's are nondecreasing and such that δ(t) = δ(0) for all t ∈ [-τ -h, 0]. We consider the case where the matrix -D + P is Hurwitz. This condition is imposed because it is necessary for the GAS of (1) when δ = 0 and when (1) holds with equality; see [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF]Theorem 1]. When -D + P is Hurwitz, the main result of [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] ensures that W exponentially converges to the origin when δ is not present. We also use

M = e -Dh + 0 -h e D d P. (2)
Since the matrix M satisfies M > 0, the Perron-Frobenius theorem [START_REF] Lemmens | Nonlinear Perron-Frobenius Theory[END_REF] provides a vector U ∈ [1, +∞) n such that

M U = cU (3)
where c = µ(M ). In the next section, we will prove that M is Schur stable. Using this fact, it follows that c ∈ (0, 1). Let us introduce the function χ : [-τ, +∞) → R n defined by

χ(t) = -(-I +M ) -1 D -1 I -e -Dh      sup ∈[t-h,t] δ 1 ( ) . . . sup ∈[t-h,t] δ n ( )      (4)
which is well-defined because the entries of D are positive and M is Schur stable; see below. Our first main result is: Theorem 1: With the preceding definitions and notation,

W (t) ≤ e -ln(c) τ +h h n j=1 sup ∈[-τ,h]
w j ( )e ln(c)

τ +h t U + χ(t) (5)
is satisfied for all t ≥ h. For a proof of the preceding theorem, see Section IV-A. Remark 1: Note that (5) provides an N ∈ R n×n such that

W (t) ≤ e -ln(c) τ +h h n j=1 sup ∈[-τ,h] w j ( )e ln(c) τ +h t U +N sup ∈[0,t] δ 1 ( ), . . . , sup ∈[0,t] (6)
for all t ≥ h. From this inequality, we can deduce that

||W (t)|| ≤ e -ln(c) τ +h h n||U ||e ln(c) τ +h t sup ∈[-τ,h] ||W ( )|| +||N ||n sup ∈[0,t] ||δ( )|| (7)
holds for all t ≥ 0, which is the desired ISS property.

B. Vector Trajectory-Based ISS Result

Our proof of Theorem 1 in Section IV-A uses the theorem of this section, which is of independent interest. It provides a vector trajectory-based ISS estimate for a continuous function ω : [-T, +∞) → [0, +∞) n that we assume satisfies

ω(t) ≤ S      sup ∈[t-T,t] ω 1 ( ) . . . sup ∈[t-T,t] ω n ( )      + ∆(t), (8)
for all t ≥ 0, where T > 0 is a given constant, each component ∆ i for i = 1, . . . , n of ∆ is piecewise continuous, and the matrix S > 0 is Schur stable. Without loss of generality, we assume that each function ∆ i (t) is nondecreasing. Moreover, we let ∆ be defined over

[-T, +∞) by ∆(t) = ∆(0) for all t ∈ [-T, 0].
By again using the Perron-Frobenius Theorem [START_REF] Lemmens | Nonlinear Perron-Frobenius Theory[END_REF], we can fix a vector V ∈ R n and a constant q ∈ (0, 1) such that

SV = qV, (9)
with each entry of V in [1, +∞), and we will see below that -I + S is invertible. In terms of the vector valued function

ρ(t) = -(-I + S) -1 ∆(t), (10)
our vector valued ISS trajectory-based result is: Theorem 2: With the preceding definitions and notation,

ω(t) ≤ n j=1 sup ∈[-T,0] ω j ( )e ln(q) T t V + ρ(t) (11)
holds for all t ≥ 0.

For a proof of the preceding theorem, see Section IV.

Remark 2: The inequality [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF] implies that

||ω(t)|| ≤ n||V || sup ∈[-T,0] ||ω( )||e ln(q) T t +||(-I + S) -1 ||||∆(t)|| (12)
which gives

||ω(t)|| ≤ n||V || sup ∈[-T,0] ||ω( )||e ln(q) T t +||(-I + S) -1 || sup ∈[0,t] ||∆( )||, (13)
which is our desired ISS estimate.

In the next section, we provide key lemmas that we will use to prove our theorems.

III. KEY LEMMAS

We first recall the following well known equivalences, e.g, from [20, Proposition 1]: Lemma 1: Let M ∈ R n×n be a Metzler matrix. Then the following three statements are equivalent: (i) M is Hurwitz, (ii) MV < 0 for some vector V > 0, and (iii) M is invertible and M -1 ≤ 0.

We also use: Lemma 2: Let P ∈ R n×n satisfy P > 0, and D ∈ R n×n be a diagonal matrix whose diagonal entries are all positive. Let h > 0 be a given positive real value. If the matrix

-D + P (14)
is Hurwitz, then the matrix ( 2) is Schur stable.

Proof: Assume that -D + P is Hurwitz. Let d > 0 be such that dI -D + P > 0. The Perron-Frobenius theorem [START_REF] Lemmens | Nonlinear Perron-Frobenius Theory[END_REF] ensures that there are V > 0 and k > 0 such that

(dI -D + P )V = kV. ( 15
)
It follows that with the choice c = d -k, we have

(-D + P )V = -cV. (16)
Since -D + P is Hurwitz, necessarily c > 0. Since the diagonal entries of D are positive, one can prove easily that the matrix M in (2) can be written as

M = I + D -1 I -e -Dh (-D + P ). (17)
This equality and [START_REF] Mazenc | Stability and control design for time-varying systems with time-varying delays using a trajectory based approach[END_REF] give

M V = I -cD -1 I -e -Dh V. (18)
Since -cD -1 I -e -Dh is diagonal with negative diagonal entries, there is a constant p ∈ (0, 1) such that M V ≤ pV .

Since the matrix M is nonnegative, the lemma now follows from [6, Chapter 2, Theorem 2.22].

The next lemma follows from (9), because q ∈ (0, 1), and is included for reference because it is key for our analysis:

Lemma 3: Let T > 0 be a given real value, and the matrix S, the vector V , and the constant q ∈ (0, 1) be such that (9) holds. Then e ln(q)

T t V = e ln(q) T (t-T ) SV = S sup m∈[t-T,t] e ln(q) T m V (19)
holds for all t ≥ 0.

We also use: Lemma 4: Let S > 0 be a Schur stable matrix in R n×n and T > 0 be a constant. Let ∆ : [-T, +∞) → [0, +∞) n be such that each of its components ∆ i for i = 1, . . . , n is piecewise continuous. Then the function ρ : [-T, +∞) → R n given by ( 10) is nonnegative valued and nondecreasing, and it satisfies

ρ(t) = Sρ(t) + ∆(t) (20)
for all t ≥ -T . Proof: Since S > 0, the matrix -I +S is Metzler. Also, by the Perron-Frobenius theorem, there are V > 0 and q ∈ (0, 1) such that SV ≤ qV because S is positive and Schur stable. It follows that (-I + S)V ≤ (q -1)V . Since -I + S is Metzler, this inequality implies that -I + S is Hurwitz, by Lemma 1. Then, according to Lemma 1, (-I + S) -1 ≤ 0. Consequently, the function ρ is well-defined nonnegative and nondecreasing. Then, one can prove easily that (20) holds.

Our final lemma is as follows: Lemma 5: Consider a Schur stable matrix S > 0, a constant T > 0, and a continuous function Ω : [-T, +∞) → [0, +∞) n such that

Ω(t) ≤ S      sup ∈[t-T,t] Ω 1 ( ) . . . sup ∈[t-T,t] Ω n ( )      , (21)
for all t ≥ 0. Let q ∈ (0, 1) and V ∈ [1, +∞) n be such that SV = qV . Choose the functions

β 0 (t) = κ 0 e ln(q) T t V (22)
and

κ 0 = n j=1 sup ∈[-T,0] Ω j ( ). (23)
Then

Ω(t) ≤ β 0 (t) (24)
holds for all t ≥ -T .

Proof: Let us introduce the functions

β (t) = κ e ln(q) T t V, (25)
where

κ = κ 0 + , (26)
and where > 0 is any positive constant. Since ln(q) < 0,

β (m) ≥ (κ 0 + )V (27)
for all m ∈ [-T, 0]. We deduce that the inequality

Ω(m) ≤ k 0 V < β (m) (28)
holds for all m ∈ [-T, 0], since V ∈ [1, ∞) n . Bearing in mind that Ω is continuous, we proceed by contradiction: we assume that there is a t c > 0 such that

Ω(t) < β (t) (29)
for all t ∈ [0, t c ) and Ω(t c ) ≮ β (t c ). Then the continuity of Ω ensures that there is an i ∈ {1, ..., n} such that

Ω i (t c ) = β ,i (t c ). ( 30
)
From Lemma 3, we deduce that

β (t) = S      sup ∈[t-T,t] β ,1 ( ) . . . sup ∈[t-T,t] β ,n ( )      (
β ,j ( ) = Ω i (t c ) ≤ n j=1 s i,j sup ∈[tc-T,tc] Ω j ( ). (32)
The last inequality is equivalent to

n j=1 s i,j sup ∈[tc-T,tc] β ,j ( ) - sup ∈[tc-T,tc] Ω j ( ) ≤ 0. (33)
On the other hand, the definition of t c implies that for all j ∈ {1, ..., n}, we have sup

∈[tc-T,tc] β ,j ( ) - sup ∈[tc-T,tc] Ω j ( ) ≥ 0. (34)
Since S > 0, we deduce from (33)-(34) that sup

∈[tc-T,tc] β ,j ( ) = sup ∈[tc-T,tc] Ω j ( ) (35)
for j = 1, . . . , n. Since each function β ,j is nonincreasing, these equalities are equivalent to

β ,j (t c -T ) = sup ∈[tc-T,tc] Ω j ( ) (36)
for all j ∈ {1, ..., n}. Since Ω is continuous, for any j ∈ {1, ..., n}, there is

t j, ∈ [t c -T, t c ] such that sup ∈[tc-T,tc] Ω j ( ) = Ω j (t j, ). (37)
It follows that

β ,j (t c -T ) = Ω j (t j, ). (38)
According to the definition of t c , the inequality β ,j (t c -T ) > Ω j (t c -T ) is satisfied. Therefore (38) implies that t j, ∈ (t c -T, t c ]. We deduce from (38) that β ,j (t j, ) < β ,j (t c -T ) = Ω j (t j, ).

However according to the definition of t c and the fact that t j, ∈ [t c -T, t c ], the inequality β ,j (t j, ) ≥ Ω j (t j, ) is satisfied. This yields a contradiction. We conclude that Ω(t) < β (t) for all t ≥ -T . By continuity of β as a function of , we deduce that (24) holds for all t ≥ -T . Remark 3: Notice that [1, Lemma 1] ensures that under the assumptions of the preceding lemma, Ω exponentially goes to the origin. The preceding lemma complements [START_REF] Ahmed | Dynamic output feedback stabilization of switched linear systems with delay via a trajectory based approach[END_REF] by determining an upper bounding function which exponentially converges to the origin.

IV. PROOFS OF THEOREMS 1-2 A. Proof of Theorem 1

A variation of parameters argument and the nonnegtivity of e -Dt give

W (t) ≤ e -Dh W (t -h) + t t-h e -D(t-) d P      sup ∈[t-τ -h,t] w 1 ( ) . . . sup ∈[t-τ -h,t] w n ( )      + t t-h e -D(t-) d      sup ∈[t-h,t] δ 1 ( ) . . . sup ∈[t-h,t] δ n ( )      , (40)
for all t ≥ h. It follows from our choice of M in (2) that

W (t) ≤ M      sup ∈[t-τ -h,t] w 1 ( ) . . . sup ∈[t-τ -h,t] w n ( )      +D -1 I -e -Dh      sup ∈[t-h,t] δ 1 ( ) . . . sup ∈[t-h,t] δ n ( )      (41)
for all t ≥ h. Let W (s) = W (s + h). Then (41) ensures that for all s ≥ 0,

W (s) ≤ M      sup ∈[s-τ -h,s] w ,1 ( ) . . . sup ∈[s-τ -h,s] w ,n ( )      +D -1 I -e -Dh      sup ∈[s,s+h] δ 1 ( ) . . . sup ∈[s,s+h] δ n ( )      . ( 42
)
By Lemma 2, M is Schur stable. Moreover, M > 0. Then we can apply Theorem 2 to W and obtain:

W (s) ≤ n j=1 sup ∈[-τ -h,0] ω ,j ( )e ln(c) τ +h s U + χ(s + h). (43) Thus W (s + h) ≤ n j=1 sup ∈[-τ,h] ω j ( )e ln(c) τ +h s U + χ(s + h). (44) Taking t = s + h, W (t) ≤ n j=1 sup ∈[-τ,h] ω j ( )e ln(c) τ +h (t-h) U + χ(t) (45)
which is [START_REF] Haddad | Stability theory for nonnegative and compartmental dynamical systems with time delay[END_REF].

B. Proof of Theorem 2

We use the function ω : [-T, +∞) → [0, +∞) n that is defined by

ω(t) =    max{0, ω 1 (t) -ρ 1 (t)} . . . max{0, ω n (t) -ρ n (t)}    . ( 46
)
Let Ψ : R → R be the function defined by

Ψ(m) = 1 if m > 0 and Ψ(m) = 0 if m ≤ 0. Then ω(t) = Λ(t)    ω 1 (t) -ρ 1 (t) . . . ω n (t) -ρ n (t)    (47)
with

Λ(t) = diag Ψ(ω 1 (t)-ρ 1 (t)), ...., Ψ(ω n (t)-ρ n (t)) . ( 48
)
On the other hand, [START_REF] Hien | New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF] and Lemma 4 imply that

ω(t) -ρ(t) ≤ S      sup ∈[t-T,t] ω 1 ( ) . . . sup ∈[t-T,t] ω n ( )      + ∆(t) -Sρ(t) -∆(t) = S      sup ∈[t-T,t] ω 1 ( ) . . . sup ∈[t-T,t] ω n ( )      -Sρ(t) (49)
for all t ≥ 0. Since S ≥ 0, it follows that

ω(t) -ρ(t) ≤ S      sup ∈[t-T,t] (ω 1 ( ) -ρ 1 ( ) + ρ 1 ( )) -ρ 1 (t) . . . sup ∈[t-T,t] (ω n ( ) -ρ n ( ) + ρ n ( )) -ρ n (t)      ≤ S      sup ∈[t-T,t]
(ω 1 ( )-ρ 1 ( ))+ sup

∈[t-T,t] ρ 1 ( )-ρ 1 (t) . . . sup ∈[t-T,t] (ω n ( )-ρ n ( ))+ sup ∈[t-T,t] ρ n ( ) -ρ n (t)     
for all t ≥ 0. Since Lemma 4 implies that ρ is nondecreasing, we deduce that

ω(t) -ρ(t) ≤ S      sup ∈[t-T,t] (ω 1 ( ) -ρ 1 ( )) . . . sup ∈[t-T,t] (ω n ( ) -ρ n ( ))      . ( 50
)
Since Λ is nonnegative, bearing in mind (47), we can left multiply (50) through by Λ(t) and obtain

ω(t) ≤ Λ(t)S      sup ∈[t-T,t] (ω 1 ( ) -ρ 1 ( )) . . . sup ∈[t-T,t] (ω n ( ) -ρ n ( ))      (51)
for all t ≥ 0. As an immediate consequence of (46), we get

ω(t) ≤ Λ(t)S ω (t) (52)
for all t ≥ 0, where the R n -valued function ω is defined by

ω (t) = [sup ∈[t-T,t] ω1 ( ), . . . , sup ∈[t-T,t] ωn ( )]
. Since Λ(t) ≤ I holds for all t ≥ -T , we deduce that

ω(t) ≤ S ω (t). ( 53
)
Since S is Schur stable and positive and since ω(t) ≥ 0 for all t ≥ -T , Lemma 5 implies that with the choice

ν = n j=1 sup ∈[-T,0] max{0, ω j ( ) -ρ j ( )}, (54)
we have ω(t) ≤ νe ln(q)

T t V (55) for all t ≥ -T . This inequality can be rewritten as

   max{0, ω 1 (t) -ρ 1 (t)} . . . max{0, ω n (t) -ρ n (t)}    ≤ νe ln(q) T t V. ( 56
)
As an immediate consequence,

   ω 1 (t) -ρ 1 (t) . . . ω n (t) -ρ n (t)    ≤ νe ln(q) T t V. (57)
This inequality can be rewritten as:

ω(t) ≤ ρ(t) + n j=1 sup ∈[-T,0] max{0, ω j ( )-ρ j ( )}e ln(q) T t V.
Next, note that max{0, ω j ( ) -ρ j ( )} ≤ max{0, ω j ( )} = ω j ( ) because both ρ and ω are nonnegative, by Lemma 4. We deduce that

ω(t) ≤ ρ(t) + n j=1 sup ∈[-T,0] ω j ( )e ln(q) T t V. (58)
This concludes the proof of Theorem 2.

V. ILLUSTRATIONS

We next apply the new approach from the preceding sections to the two main examples from [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF]. Unlike [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], where the vector Halanay's inequalities did not lend themselves to proving ISS estimates, here we establish ISS results and therefore improve on previously reported results.

A. Dynamics from [START_REF] Bekiaris-Liberis | Stabilization of linear strictfeedback systems with delayed integrators[END_REF] We revisit the first illustration of the contribution [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] by addressing the case where an additive disturbance is present. We consider the n-dimensional dynamics

                     ż1 (t) = -r 1 z 1 (t) + z 2 (t -τ 1 ) + δ 1 (t) ż2 (t) = -r 2 z 2 (t) + z 3 (t -τ 2 ) + δ 2 (t) . . . żn-1 (t) = -r n-1 z n-1 (t) + z n (t -τ n-1 ) + δ n-1 (t) żn (t) = -r n z n (t) + χ(t) + δ n (t) (59)
with the input χ, which occurs in [3, Lemma 2] (in the context of stabilizing linear strict-feedback systems having delayed integrators), where the constants r i and τ i are positive for i = 1, 2, . . . , n -1. We have added nonnegative and nondecreasing functions δ i , which may represent disturbances, and which are assumed to be piecewise continuous.

As in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], we assume that χ takes the form

χ(t) = g 1 z 1 (t -τ n1 ) + . . . + g n z n (t -τ nn ) (60)
for constants τ ij ≥ 0 and g i > 0 for i = 1, 2, . . . , n.

For such a function χ, the system (59) is nonnegative (by [5, Proposition 3.1]). Hence, we focus our attention on the nonnegative solutions of (59), but an extension of the result below to the general case can easily be obtained. Choosing any τ > max{τ 1 , . . . , τ n-1 , τ n1 , . . . , τ nn }, it follows that all C 1 solutions Z : [-τ, +∞) → [0, +∞) n of (59) satisfy (1) with D = diag{r 1 , . . . , r n } and

P =        0 1 0 . . . 0 0 0 0 1 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 1 g 1 g 2 g 3 . . . g n-1 g n        (61)
with w i = z i for all i. Then we can find conditions on the r i 's and g i 's to ensure that the corresponding matrix -D +P is Hurwitz. Then Theorem 1 can be applied.

For the sake of illustration, let us consider the special case where n = 2 with g 1 = 1, g 2 = 2, r 1 = 4, and r 2 = 6. Then one can easily check that the matrix -D + P is Hurwitz. Next, for any h > 0, the corresponding matrix M is .

M = I + D -1 I -e -Dh (-D + P ) = I + diag 1-e -4h 4 , 1-e -6h 6 -4 1 1 -4 = e -4h
(64)

Then Theorem 1 ensures that

Z(t) ≤ χ(t) +e ln 1 3 + √ 79 48 τ + ln(2) 2 (t-h) 2 j=1 sup ∈[-τ, ln (2) 2 ]
z j ( )

  1 4+ √ 79 9   (65)
holds for all t ≥ h, where

χ(t) = -(-I + M ) -1 D -1 I -e -Dh δ (t) = - -3 4 3 16 7 48 -7 12 -1 diag 1-e -4h 4 , 1-e -6h 6 δ (t) = 1 15 4 1 1 4 δ (t)
and

δ (t) = sup ∈[t-h,t] δ 1 ( ), sup ∈[t-h,t] δ 2 ( ) . (66)
B. Interval Observer Design Consider the system

Ẋ(t) = AX(t) + p i=1 B i X(t -τ i (t)) + δ(t) (67)
whose state X is valued in R n , where A and the B i 's are constant matrices, p is any positive integer, A is Hurwitz (but not necessarily Metzler), and τ i : [0, +∞) → [0, τ ] for all i are piecewise continuous delays having a known bound τ > 0 for which solutions of (67) are uniquely defined on [0, +∞) for each initial condition. This agrees with the dynamics from [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], except we added the unknown bounded function δ having piecewise continuous components and representing uncertainty. This provides challenges because the stability conditions for (67) in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] were not based on standard Lyapunov function conditions. We provide conditions that are independent of the delays τ i 's which ensure that one can build an interval observer whose existence implies that (67) is ISS; see [START_REF] Mazenc | ISS interval observers for nonlinear systems transformed into triangular systems[END_REF] for background on interval observers. Our analysis is more complex than standard treatments of linear systems, and is called for because of the mildness of our requirements on the coefficient matrices and on the delays. However, as in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], a key ingredient will be the proof of [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]Theorem 2] which provides a time-varying change of variables formula for linear systems having Hurwitz matrices A in their drift term, which is defined by a bounded C 1 function Q : [0, +∞) → R n×n having a bounded inverse and a constant Metzler matrix M 0 such that

Q(t)Q(t) -1 + Q(t)AQ(t) -1 = M 0 (68)
for all t ≥ 0, and such that all diagonal elements of M 0 are negative real values. Fixing a Q that satisfies the requirements of the preceding paragraph, it follows that Z(t) = Q(t)X(t) satisfies

Ż(t) = M 0 Z(t) + p i=1 L i (t)Z(t -τ i (t)) + Q(t)δ(t), (69)
where L i (t) = Q(t)B i Q(t -τ i (t)) -1 for all i and t ≥ τ . Our analysis will use the interval observer

                       Ż(t) = M 0 Z(t) + p i=1 (L i (t)) + Z(t -τ i (t)) - p i=1 (L i (t)) -Z(t -τ i (t)) + (Q(t)δ(t)) + Ż(t) = M 0 Z(t) + p i=1 (L i (t)) + Z(t -τ i (t)) - p i=1 (L i (t)) -Z(t -τ i (t)) -(Q(t)δ(t)) - (70)
where

C + = [max{c i,j , 0}] and C -= C + -C for matrices C = [c i,j ]. The change of coordinates Z ‡ (t) = -Z(t) yields                        Ż(t) = M 0 Z(t) + p i=1 (L i (t)) + Z(t -τ i (t)) + p i=1 (L i (t)) -Z ‡ (t-τ i (t))+(Q(t)δ(t)) + Ż ‡ (t) = M 0 Z ‡ (t) + p i=1 (L i (t)) + Z ‡ (t -τ i (t)) + p i=1 (L i (t)) -Z(t-τ i (t))+(Q(t)δ(t)) -. (71)
Then (71) is cooperative, because M 0 is Metzler, by the reasoning used in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], which also implies ISS of (71) if all positive valued solutions of (71) satisfy an ISS estimate.

We therefore focus on positive solutions of (71). To this end, notice that the variable

Z(t) = Z(t) + Z ‡ (t) (72)
satisfies

Ż(t) = M 0 Z(t) + p i=1 [L i (t) + +L i (t) -] Z(t-τ i (t)) + |Q(t)δ(t)|. (73)
Setting S τ ( Zt ) = sup ∈[t-τ ,t] Z( ), we then have

Ż(t) ≤ M 0 Z(t) + p i=1 [L i (t) + + L i (t) -] S τ ( Zt ) +|Q(t)δ(t)| ≤ D M0 Z(t) + LS τ ( Zt ) + |Q(t)δ(t)| (74)
where D M0 is diagonal matrix whose entries are the diagonal elements of M 0 , and the matrix L > 0 is such that

p i=1 [L i (t) + + L i (t) -] + M 0 -D M0 ≤ L (75)
for all t ≥ 0. By choosing the diagonal matrix D = -D M0 (whose entries are all positive, by our choice of M 0 ), it follows that, if we can choose L such that D M0 + L is Hurwitz, then we can apply Theorem 1 to conclude that the solutions of (73) satisfy an ISS estimate. Using the facts that Z and Z ‡ are nonnegative valued, it now follows from (72) that ( 71) is also ISS, so the origin of ( 70) is also ISS. Moreover, the formulas L i = L + i -L - i for each i and the argument we used in our proof of cooperativity of (71) imply that cooperativity of the dynamics for (Z + , Z -) = (Z -Z, Z -Z). This implies that Z(t) ≥ Z(t) ≥ Z(t) for all t ≥ 0 provided we choose our initial functions for Z and Z such that Z(t) ≥ Z(t) ≥ Z(t) for all t ∈ [-τ , 0]. Hence, (70) provides an interval observer for (69) which gives the ISS property of the Z(t) dynamics. Finally, the bounds |X(t)| ≤ |Q(t) -1 ||Z(t)| for all t ≥ 0 and the boundedness of Q(t) -1 let us conclude that the X dynamics satisfy ISS.

Remark 4: Our sufficient condition for the ISS of (67) is the same as the sufficient condition for global exponential stability to 0 for the δ = 0 case in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF]. As noted in [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], we can often take Q to be constant, e.g., when all eigenvalues of A are real, by choosing Q such that QAQ -1 = M 0 is A's Jordan canonical form. When A is Metzler, we can choose Q = I and then our sufficient condition says that A + p i=1 [B + i + B - i ] is Hurwitz. See [START_REF] Mazenc | Interval observers for linear time-invariant systems with disturbances[END_REF]Section 4.3] for an example of a Hurwitz matrix A having a conjugate pair of complex (nonreal) eigenvalues that calls for a nonconstant choice of the matrix valued function Q.

VI. CONCLUSION

We developed the theories of the Halanay's inequality and the trajectory based approaches, by providing ISS inequalities when vector inequalities are satisfied and additive disturbances are present. This overcame an obstacle to proving ISS analogs of works like [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] with respect to additive uncertainties. Our main tools entailed positive systems methods, based on novel applications of Schur stable and positive matrices.

  31) for all t ≥ 0. Then, with S = [s ij ], condition (21) and (30)-(31) give n j=1 s i,j sup ∈[tc-T,tc]

We illustrated our method in an interval observer design with unknown delays. Extensions to cases where the coefficient matrices in our inequalities are time varying are expected.

Supported by NSF Grants 1711299 and 2009659.