Pierre-Yves Gicquel
email: pierre-yves.gicquel@univ-lemans.fr

Sebastien George
email: sebastien.george@univ-lemans.fr

Pierre Laforcade
email: pierre.laforcade@univ-lemans.fr

Iza Marfisi-Schottman

Design of a Component-Based Mobile Learning Game Authoring Tool

Keywords: Mobile Learning, Serious Games, Authoring Tool, Situated Learning

Mobile learning games (MLGs) have great potential in education, especially in fields requiring outdoors activities such as botany or cultural heritage education. However, the number of mobile learning games actually used for outdoor education remains very low. The absence of dedicated applications allowing to build MLGs, without technical expertise, is certainly one of the most important factors. To overcome this limitation, we propose a design method for MLG authoring tools, based on reusable components. We describe in this paper the design and first results of Moggle-Designer, a full-web MLG authoring tool requiring no technical expertise. We detail how a model-based approach, combined with component-oriented programming, adequately allow the representation and manipulation of didactic expert knowledge. Finally, we present Moggle-Player, an application for running the designed MLGs on any mobile browser. Moggle-Designer was used to design several mobile games that have been tested during a pilot study in a botanical park.

Introduction

Mobile devices, such as mobile phones or tablets, have proven to be interesting mediums for computer supported learning. These devices are particularly interesting out of the classroom, for authentic learning activities. Mobile Learning Games (MLGs) present great pedagogical interest because they favor environment exploration and self-regulated mediation through conversation between learners [START_REF] Lee | Cooperation begins: Encouraging critical thinking skills through cooperative reciprocity using a mobile learning game[END_REF]. These exploratory and conversational activities allow sustainable learning and appropriation through collaborative knowledge building [START_REF] Sharples | In: Mobile Learning[END_REF].

However, the cost for producing usable MLGs remains prohibitive for most educational structures. The few available authoring tools are either very generic, and require a strong technical expertise, or only allow the design of overly simple and specific learning scenarios [START_REF] Rosyid | Deploying learning materials to game content for serious education game development: A case study[END_REF]. This raises the question of how to design a simple and generic MLG authoring tool. In this paper, we propose the use of a component-based approach. First, we define our research context and questions, then we present the methodology for collecting the needs of teachers in terms of games scenarios and for formalizing these scenarios through a model driven approach. We will then present the Moggle-Designer authoring tool, along with Moggle-Player, which allows users to play the authored MLGs on any mobile browser. Finally, we present the results of a first experimentation in real conditions, in which subjects played various gaming scenarios conceived with Moggle-Designer in a botanical park.

Research Context and Questions

The research context of this paper is the ReVeRIES project 1 (French acronym that translates to dreams and that stands for Educational, Interactive and Fun Plant Recognition on Smartphones). This project aims to introduce mobile applications in botanical education. It relies on the assumption that the urbanization of society has drastically reduced common botanical knowledge, leading to a gradual increase in so-called plant blindness, defined by [START_REF] Wandersee | Preventing plant blindness[END_REF] as the inability to recognize the importance of plants in the biosphere and in human affairs.

In order to motivate children and botanical neophytes to learn about plants and explore their natural environment, we intend to design MLGs that will help to learn about plant characteristics. We define a MLG as a mobile application, combining educational content with a playful scenario to favor learning. This kind of application relies on mobile devices' features (e.g. location, orientation and proximity sensors, media capturing and recording, augmented reality) to contextualize activities. One of the project goals is to propose an MLG authoring tool that does not require any programming knowledge. This authoring tool has a generic base but specific features could be added for enriching MLG with domain-specific activities. In the case of botany, we propose activities based on automatic image analysis to recognize plants from photos of their organs (e.g. leaves, barks, flowers). This aspect is detailed in [START_REF] Bertrand | Bark Recognition to Improve Leaf-based Classification in Didactic Tree Species Identification[END_REF].

Conceiving an authoring tool in the context of mobile learning games raises research questions we address in this paper:

-How to collect and formalize designers' needs, expressed in natural language, in an appropriate formalism for conception? (section 3) -How to facilitate the creation of MLGs based on this formalism and deploy them on the players smartphones? (section 4)

Modeling Mobile Learning Games

This section is dedicated to the modeling issues related to the complexity of the MLG authoring tool. Indeed, the authoring tool should allow the design of a large variety of MLGs. Such learning scenarios require a domain-specific modeling language, at least as a conceptual language to drive the design and development of the authoring-tool. Designing a generic but still botany-oriented modeling language relevant for any botanical and mobile learning scenario is not an achievable objective. Our modeling approach aims to capture, explicit and validate a broadly-usable instructional language. This language being then extended with botany specific concept for the design of botany MLGs scenarios. The next sub-sections detail our global approach, the meta-model we propose, and some feedback about its validation for designing botanical MLG scenarios.

Overview of the Modeling Process

Our modeling process follows several steps:

1. Collecting informal (textual, drawings...) scenarios of botanical MLGs: collaborative sessions with various actors involved in the ReVeR-IES project (including botanical park managers, botanical scientists, and teachers) were conducted. Participants were split into groups and had to propose a botanical MLG for the Echologia natural park in Laval, France (a map was available about the tree species present in the park). The main objective was to identify and collect a large range of MLG activities. Six scenarios were produced and discussed during a collective retrospective [START_REF] Marfisi-Schottman | Meta Serious Game: Supporting Creativity Sessions for Mobile Serious Games[END_REF]. 2. Describing a conceptual model for botanical MLG scenarios: it offers high-level concepts, similar to the natural concepts used by teachers, and is rich enough to adapt to various types of MLGs [START_REF] Marfisi-Schottman | From Idea to Reality: Extensive and Executable Modeling Language for Mobile Learning Games[END_REF]. Without going into detail, the model formalizes a MLG as a series of SituatedGameUnits each related to a POI (Point Of Interest), i.e. a geographical zone involving zero or several objects of interest (mainly plants although not restricted to them). Each SituatedGameUnits can be described in terms of optional elements: triggers (to start the unit), clues (to help the player find the POI), guidance features to the POI (marker on a map, GPS, vibrations, etc.), validation actions (to validate the presence of the player on the POI), OnSiteActivities (activities to do around the POI), rewards (points/items/... in relation to the arrival on the POI and/or according to the success of the OnSiteActivities), pedagogical information (to debrief with the player the knowledge acquired in relation to this unit). 3. Formalizing a specific meta-model for botanical MLG scenarios.

Because the first two points have already been detailed in previous publications ([6] and [START_REF] Marfisi-Schottman | From Idea to Reality: Extensive and Executable Modeling Language for Mobile Learning Games[END_REF]), we detail here the third point: the botanical MLG meta-model.

Meta-modeling Approach

The meta-model we propose is a formal and explicit description of all the concepts, properties and relations required to model a botanical MLG scenario. MLG scenarios are the models, and they are conform to this meta-model. This meta-model has several objectives:

to verify that the MLG scenarios can be formally specified (i.e. machinereadable with no ambiguity) in order to be exploited by the MLG player; to drive the design of the MLG authoring tool: we can check the consistency between the scenario elements (i.e. the visual information depicted in the mock-up screens and the interactions between screens); to drive the specifications and the development of the components-based MLG player engine.

One of the main issues to tackle is to ensure the expressiveness, completeness, consistency and the non-reducibility of the meta-model. This will help reduce the design and development complexity of the authoring and player tools. In order to obtain such a meta-model, we progressively merged all the models of MLGs it needs to cover. Indeed, during the creativity session with the ReVeRIES members, we realized that the designed MLGs actually had three different types of scenarios:

linear scenario (e.g. treasure hunt): game units are chained up (use of guidances to drive the player to the POI); emergent scenario (e.g. interactive walk): game units start when players are physically close to a POI; hub scenario (e.g. geocaching): all game units and clues are available and depicted on a map, players decide in which order they want to play them.

For each of these scenario types, we therefore proposed a dedicated meta-model, and merged it to the meta-model. The merge relies on a (meta-)model composition in the meaning of the Model-Driven Engineering (MDE) principles. Briefly, we concretely merge the meta-concepts sharing a same semantics, gathering their potential different meta-attributes and meta-relations. Concepts that cannot be aligned are added, while elements that can be mapped to existent ones are removed. We also integrated game ingredients (e.g. rewards, points, inventories and virtual items) in order to cover all the elements of the designed MLGs. Concretely speaking, this meta-model was specified using the Eclipse Modeling Framework [START_REF] Steinberg | EMF: Eclipse Modeling Framework. 2 edn. Eclipse Series[END_REF] included in the Obeo designer community IDE. Specifying the MLG meta-model allowed to describe formally the cardinality and restriction between the elements of the meta-model (such as OnSiteActivities).

Moggle-Designer

Presentation

Moggle-Designer (MObile Geolocated Games for Learning) allows non-programmers to create MLGs with a bottom-up design process: learning and gaming components, such as media (e.g. images, videos, formatted text) or point of interest, are first defined as basic building blocks.

These blocks can then be aggregated to construct more complex components. For instance, a guidance activity is composed of a point of interest (the destination) and a medium (feedback) that is displayed when the user reaches this destination. Finally, these building blocks are linked together to create a game unit, which can be seen as a self-contained, independent and complete MLG [START_REF] Marfisi-Schottman | From Idea to Reality: Extensive and Executable Modeling Language for Mobile Learning Games[END_REF].

Game units Game units follows a similar pattern, organized in three phases. This first phase is guiding the player to the place of interest (POI). The designer can chose between two guidance modes, either the POI and user position are displayed on a classic map on the player phone, or a more playful mode where a circle grow as the user get closer to the POI.

Second phase is triggered by the player reaching the POI. The player is then proposed a learning task where she has either to answer questions related to the plant at this POI, or to correctly identify a specie by scanning the correct QR-code (various QR-code being placed on three at the POI). When the player correctly complete a learning task, she is rewarded by a bonus item stored in a personal inventory. For instance the bonus recipe of rozenbottel jam is acquired when the user correctly identify a Rosa Canina.

The third phase consist of proposing a feedback media that gives complementary information about the learning task (e.g information such as an easy way of identifying Rosa Canina from its fruit color).

Design Principles

Reusable and Shareable Building Blocks One common limitation of authoring tools in Educational Game Authoring tools, is the lack of non-trivial working examples that teachers can use/reuse to guide them [START_REF] Torrente | Instructororiented authoring tools for educational videogames[END_REF]. To reduce this issue and favor appropriation of Moggle-Designer by instructors, it encourage, through its design, sharing and reusing game components at different levels of granularity (from simple media component to game unit and complete scenario).

Figure 1 illustrates the relation of composition between shareable building blocks. When an instructor uploads an image file, it appears as a shareable component. The instructor can then create a document containing the image accompanied by informative text. This newly created document, is then used as part of a Multiple Choice Question.

Ownership and Security of Building Block

In order to favor sharing while keeping a personal space for the user, we designed the system such as unidentified users could not access any building blocks. Moreover, users have the liberty to easily share or unshare each of their own contributions. Each block has a 1-1 relation with its creators user-id: the blocks being stored in a document-oriented database, are stored, as a whole, with each of their properties. This allows to easily create unique hash from blocks and track their use. [START_REF] Nielsen | Usability Engineering[END_REF] underlines the critical importance of clear terminology and guidance in a user interface which should speak the user's language. In the case of an authoring tool using building blocks, these block must be easily apprehended by the user.

Terminology and Usability

By proposing different granularities in the building blocks, with complex blocks being created from more simple blocks, we aim at easing the learning curve. An unexperienced user will easily be able to apprehend the principles of Fig. 1. Creating a situated question by combining building blocks simple building blocks (e.g. image and text). Having acquired confidence and effectiveness in the conception of simple building block, the process of creating more complex building block by composition of simple ones will be more intuitive. Moreover, in order to encourage designers to explore and experiment, a direct visual feedback (WYSIWYG) is displayed during the block creation.

Advantages and Technical Challenges of Web Applications

We decided to build Moggle-Designer as a web application. Various reasons guided this choice: first, web applications are platform independent. Moreover, the distribution of a web application is straightforward, there is no installation process for the users and the URL always point to the last stable version. From a technical point of view, we used the Web Component paradigm2 , a recent evolution in Web Application promoted by the W3C. The fundamental idea behind Web Components is to create easily reusable pieces of HTML and JavaScript code. Web Components provide a way to encapsulate the HTML/CSS and encapsulate the JavaScript within the context of execution of the component. This means one can use someone else's components with the complete guarantee that there will be not be any side effects on his/her own components.

Moggle-Player

Moggle-Player is a tool that allows users to play scenarios designed with Moggle-Designer. It is also a full web application, thus it is usable on any mobile device with a browser (Iphone, Android, Windows phone...). The game player makes full use of mobile HTML5 capacities (Figure 2): use of GPS for guiding the user to the different games steps, use of the phone camera for activities based on scanning a QR-code to identify a tree and use of websockets to permit chat between users.

Fig. 2. Use of GPS and phone camera

One particular challenge we faced was to keep the web application running in absence of Internet connection. We set up a cache system (a JavaScript service worker) that preloads everything resources needed to keep the current MLG running when there is a disconnection and synchronizes back with the server when the network connection is restored.

Qualitative Pilot Study

We conducted a pilot study in a botanical park to evaluate how MLGs designed with Moggle-Designer were used and how the activities were perceived. We used three of the MLG scenarios designed during the ReVeRIES design sessions.

Participants and Apparatus

Twelve users took part in the experimentation (four on each MLG scenario) but only nine completed the feedback form, because some users actually shared the same smartphone. The nine that answered the feedback form were aged between 18 and 38 years old (mean: 28, SD: 8), and included four women and five men. Users where from three different backgrounds: four where fellow researchers from our laboratory, five were students in first year of computer science studies and three were working at the botanical park. User's personal smartphones where used for the experiment (these included iPhones and Android devices). All the devices were connected to a wireless mobile network, with variable connectivity.

Organization

We voluntarily gave little information to the players in order to assess the clarity of the MLGs. Three animators from the project team were present in case of technical problems, but they were actually solicited very rarely. The main reasons of solicitations were problem concerning geolocation, some smartphones having poor GPS accuracy.

The sessions lasted approximately 45 minutes. The MLGs were composed of six situated unit games each, implying a short walk through the park to reach the points of interest and progress in the game.

Main Results

We collected direct verbal feedback during and after the activities and also asked users to complete a questionnaire to assess their satisfaction and their interest in MLGs in the context of a botanical park. The verbal feedback was very positive: users enjoyed the fun aspect of having to reach a specific zone and having to observe the plants in the surroundings in order to answer questions.

The analysis of the questionnaire confirmed the verbal feedback: user found the MLGs motivating for the visit and useful to acquire knowledge in that botany. The questionnaire consisted mostly of Likert based unique choices questions with four possible items. A synthesis of the answers is presented in the figure 3.

Related Works

In this section, we compare two well established authoring tools for mobile learning applications to Moggle-Designer. A complete state of the art can be found in [START_REF] Karoui | Mobile learning game authoring tools: Assessment, synthesis and proposals[END_REF].

ARIS

ARIS is a Web based authoring tool that generates IOS Mobile Learning Games. While generating native mobile applications has some benefits, in terms of performance for instance, it greatly limits the possible users. Nevertheless, ARIS is particularly powerful, it allows the creation of different kinds of MLGs, from treasure hunts to location-based storytelling [START_REF] Holden | Aris: augmented reality for interactive storytelling[END_REF]. However, this expressive power comes with a steep learning curve. [START_REF] Perry | Gamifying french language learning: a case study examining a questbased, augmented reality mobile learning-tool[END_REF] argues about ARIS that the challenge is learning the user interface and game logic [...] many who admire the possibilities of this platform find creating and understanding the logic of requirements for objects in a game scenario challenging. Finally, one key difference between ARIS authoring tool and Moggle-Designer is the inability to share resources in ARIS. This design strongly limits re-usability of learning objects.

MLearn4Web

MLearn4Web [START_REF] Zbick | mlearn4web: a web-based framework to design and deploy cross-platform mobile applications[END_REF] is a full web authoring tool allowing teacher to create web learning application. The tool is very easy to apprehend, applications are built as a serie of pages, each page being constructed visually by drag-and-dropping items on the page from a component pallet. First study shows a very good degree of acceptance among teachers.

However, MLearn4Web presents limitations in terms of design possibilities. Applications can only be designed and shared as a whole, that is to say its not possible to design subpart of an application or to reuse part of someone else application. Moogle design principles in contrast allows sharing at different level of granularity, which makes produced games modular and easily reusable. Moreover, MLearn4Web do not allow to implement game mechanics, such as challenges or rewards for instance. This limit the learner experience to be more passive than active. In contrast, the application designed with Moogle incorporate game mechanism, requiring the learner to actively investigate her environment to answer questions.

Conclusion and Future Work

We present, in this paper, a complete design process for MLG (Mobile Learning Game) authoring tools, from the collection of users' needs to implementation details. The MLG scenarios were formalized through a model-based approach while web-components structurally and semantically conform to the model elements were implemented.

This process led to Moggle-Designer, an authoring tool powerful enough to design the types of scenarios expressed by future users, while needing no technical expertise to build these scenarios, and Moggle-Player, capable of playing these MLGs on any mobile device. An experimental validation in the real setting of a botanical park allowed us to validate the expressiveness of the authoring tool. We were able to implement scenarios that were proposed during the collection of users needs phase. We validated that these scenarios were playable and first results suggested a strong interest and engagement of volunteers that played the three proposed MLGs.

The next step is now the opening of the authoring tool to the staff of the botanical part. This step will be eased by having complete MLG scenarios corresponding to their need available as working examples.

Fig. 3 .

 3 Fig. 3. Results of feedback form

http://reveries-project.fr

https://www.webcomponents.org/specs


E Chambon

L Burlion

P Apkarian

Time-response shaping using Output to Input Saturation Transformation

Keywords: constrained control, linear systems, unknown disturbance, interval constraint, time-domain constraint, anti-windup

For linear systems, the control law design is often performed so that the resulting closed-loop meets specific frequency-domain requirements. However, in many cases, it may be observed that the obtained controller does not enforce time-domain requirements amongst which the objective of keeping an output variable in a given interval. In this article, a transformation is proposed to convert prescribed bounds on an output variable into time-varying saturations on the synthesized linear control law. This transformation uses some well-chosen time-varying coefficients so that the resulting time-varying saturations do not overlap in the presence of disturbances. Using an anti-windup approach, it is demonstrated that the origin of the resulting closed-loop is globally asymptotically stable and that the regulated variable satisfies the time-domain constraints in the presence of an unknown finite-energy bounded disturbance. An application to a linear ball and beam model is presented.

Introduction

To stabilise a given system, many techniques exist to obtain a control law satisfying to specified constraints. As far as MIMO systems are concerned, H ∞ loop-shaping can for example be used to enforce frequency-domain requirements. However it is possible that, using such control law, time-domain requirements on a so-called regulated variable α = C α y ∈ R are not fulfilled. This is illustrated on Fig. 1 where α time-response violates expected bounds [α(t), α(t)]. In practice, a good knowledge of the studied system is often sufficient to shape its time-response. However, designing controllers satisfying to such prescribed time-domain requirements remains tedious and relies on numerous trial-and-errors involving simulations. Consequently, for more complex systems, and from the theoretical point of view, dedicated methods are often required to enforce both stability and time-domain constraints.

Amongst existing strategies to enforce time-domain requirements like response-time or overshoot limitation, it is possible to mention the work presented in [START_REF] Gevers | A decade of progress in iterative process control design: from theory to practice[END_REF] which introduced the notion of Iterative Feedback Tuning (IFT). The idea is to shape the closed-loop in response to specific input signals so as to satisfy time-domain constraints. In the PID-tuning case, a comparison with practitioners methods was performed in [START_REF] Mossberg | A comparison of iterative feedback tuning and classical PID tuning schemes[END_REF] which gives a hint on how to achieve time-domain requirements using this method. Time-domain specifications are also treated through optimal control strategies as extensively presented in [START_REF] Goodwin | Constrained control and estimation[END_REF]. These approaches include model predictive control (MPC) in which the optimisation problem can take constraints into account, see for example [START_REF] Chen | A quasi-infinite horizon predictive control scheme for constrained nonlinear systems[END_REF] or [START_REF] Chen | Nonlinear model predictive control schemes with guaranteed stability[END_REF]. Computationally effective methods close to MPC are reported in [START_REF] Ghaemi | Robust control of constrained linear systems with bounded disturbances[END_REF]. The notion of reference-governor to adjust the reference trajectory so that the constraints on the system are satisfied is also noticeable. It was presented in [START_REF] Gilbert | Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor[END_REF] with an application to aerospace systems in [START_REF] Polóni | Disturbance canceling control based on simple input observers with constraint enforcement for aerospace applications[END_REF]. The combination of frequency-domain and time-domain constraints has been explored in [START_REF] Apkarian | Time domain constrained H ∞ -synthesis[END_REF] and references therein. This method makes use of non-smooth bundle optimization methods and is referred to as "constrained structured H ∞ -synthesis". It combines simulation optimization with H ∞ synthesis to enforce both frequencyand time-domain requirements. These strategies often include the control law design, especially when an optimising scheme is used. Alternative schemes including anti-windup systems were proposed for example in [START_REF] Turner | Output violation compensation for systems with output constraints[END_REF] and [START_REF] Rojas | A simple anti-windup strategy for state constrained linear control[END_REF]. Compared to the aforementioned results, the anti-windup design is interesting because the nominal control law remains unchanged when acting far from the constraints. Also, an extensive literature on the subject is available, see [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF] or [START_REF] Galeani | A tutorial on modern anti-windup design[END_REF] for instance. However, there is not necessarily a guarantee on the fact that the time-domain constraints will actually be satisfied. In this article, the approach presented in [START_REF] Burlion | A new Saturation function to convert an output constraint into an input constraint[END_REF] and applied for example in [START_REF] Burlion | Keeping a Ground Point in the Camera Field of View of a Landing UAV[END_REF] is presented in-depth for state-feedback minimum-phase linear systems subject to disturbances. The output-to input-saturation transformation (OIST) theory proposes to reformulate prescribed bounds on the regulated variable α into state-dependent saturations on the control input u. This approach is illustrated on Fig. 2 where an ad hoc saturating block is inserted before the system control input. As indicated in this figure, additional information may be required to express these saturations. Using this method along with some assumptions, it is possible to obtain guarantees on the fulfilment of the time-domain constraints when an unsatisfactory control law has already been designed. This article is an extension of the work presented in Chambon, Burlion, and Apkarian (2015a). It gives a comprehensive description of the method for minimum-phase linear systems with unknown finite-energy bounded disturbances. It also provides a stability proof when inserting the obtained saturations in the closed-loop. Other cases of application were presented in [START_REF] Chambon | Robust output interval constraint using O/I saturation transformation with application to uncertain linear launch vehicle[END_REF].

The article is organised as follows: the ball and beam example is introduced in Sect. 2 as a case study to introduce the OIST problem formulation. Notations and definitions along with some functions properties are also presented in this section as well as the assumptions satisfied by the considered class of systems. Then, the output to input saturations transformation proposed to solve this problem is presented in Sect. 3. Due to the conservatism introduced by the bounds on the disturbances, special attention is paid in the selection of the design parameters so that the resulting time-varying saturations do not overlap1 . Stability of the system in closed-loop with the obtained saturated control is established in Sect. 4. Finally, the whole approach is applied to the linear ball and beam model in Sect. 5. Conclusions and perspectives are then presented in Sect. 6.

Problem formulation

Definitions and notations

Before introducing the problem in Sect. 2 and for the sake of consistency, some definitions and notations are presented or recalled here.

Acronyms

The acronyms listed in Tab. 1 are used throughout the article.

Notations

If not stated otherwise, the state-space representation of a LTI system (G) is denoted

G = (A, B, C, D) where A ∈ R n×n , B = B d B u B e ∈ R n×l , C ∈ R m×n and D = D d D u D e ∈
R m×l . The state vector is then denoted x ∈ R n and the measurements vector is denoted y ∈ R m with m ≤ n. The input vector lying in R l is divided between the control inputs denoted u and the unknown inputs (disturbances) which are denoted d (state disturbance) and e (measurements disturbance). The transfer function from an input u to an output y is denoted T u→y (s) where s is the Laplace variable.

Inequalities involving matrices of identical dimensions are understood component-wise: let

(A, B) ∈ R n×m , then, A ≤ B ⇔ ∀1 ≤ i ≤ n, ∀1 ≤ j ≤ m, A ij ≤ B ij .
For a given bounded vector x(t), if the bounds are known they are denoted (x(t), x(t)), i.e. ∀t, x(t) ≤ x(t) ≤ x(t).

The saturation and deadzone functions applied to a bounded variable x are respectively denoted sat x

x (x) and Dz x x (x). They are related to each other by

sat x x (x) = x -Dz x x (x) (1)
The standard Euclidean norm of a given signal x(t) defined for t ≥ 0 is denoted x . The L 2 -norm of the same signal is denoted x 2 and is given by

x 2 = ∞ 0 |x(t)| 2 dt
(2)

Definitions

Some definitions are now introduced, starting with the notion of the "overlap" of two signals:

Definition 1: In the article, the term "overlap" is used to refer to two signals taking the same value at a given instant and possibly changing order. Given two unidimensional signals s(t) and s(t), a definition of overlapping would thus be:

∃t 1 > 0, δ > 0 such that ∀t < t 1 , s(t) ≤ s(t) and ∀t ∈ [t 1 , t 1 + δ[ , s(t) > s(t).
The Lambert function will be used to define constants. It is defined as follows:

Definition 2: Let ∀x ∈ R, F (x) = xe x . The inverse function of F is the Lambert function denoted W 0 (y) which fulfils ∀y, F (W 0 (y)) = W 0 (y)e W0(y) = y.
This definition is used to define constants which in turn will help define differentiable approximates of some non-differentiable functions:

Definition 3: Let define the constants ξ := 1 2 W 0 1 e + 1 2 and Ξ := ξtanh(ξ)ξ > 0. Using these notations, let also define the following functions, ∀(x, y) ∈ R 2 :

f abs (x) := tanh(x)x + Ξ f max (x, y) := 1 2 [x + y + f abs (x -y)] g(x, y) := f max (f abs (x) , f abs (y)) (3)
which definition is extended to vectors (x, y) ∈ R n×2 in a component-wise manner. This definition triggers some remarks:

Remark 1: Note that ξ is the solution to the equation (2x -1)e 2x = 1. Also note that f abs , f max and g are continuous differentiable over R or R 2 . Moreover,

• ∀x ∈ R, f abs (x) ≥ |x|; • ∀(x, y) ∈ R 2 , f max (x, y) ≥ max(x, y). Proof. First, let f (x) = (2x -1)e 2x . Then, f (ξ) = W 0 1 e e W0
( 1 e ) e = 1 e × e = 1 by definition of the Lambert function. Second, using the basics of functional analysis on the function h(x) defined by ∀x, h(x) = f abs (x) -|x|, the inequalities in Remark 1 are achieved. As far as the function f max is concerned, the same study is performed using the following equality: ∀(x, y), max(x, y) =

1 2 [x + y + |x -y|].
The following definitions are directly related to the implementation of the OIST method.

Definition 4: Let k ∈ N. Considering a LTI system (G), the regulated output variable α(t) = C α y(t) is said to be of relative degree k with respect to u if and only if

x -3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5
Figure 3. Representation of the absolute value function (in black) and its differentiable approximate f abs (in blue) over the interval [-3, 3].

∀0 ≤ i < k -1, C α A i B u = 0 and C α A k-1 B u = 0 (4) Definition 5: Let S = s 0 . . . s k ∈ R k+1 . The function σ(S) = σ(s 0 ) . . . σ(s k ) (5)
where

σ(s i ) := s i-1 if 1 ≤ i ≤ k s k if i = 0 (6)
is called the cyclic permutation of length k + 1 on the elements of S.

Case study

In this section, an example is introduced where the control synthesis problem has been solved. In this case, there is a violation of the expected time-domain performance of the considered regulated variable α which motivates the formulation of a new problem in Sect. 2.3. The solution to this problem is the main result proposed in this article. Note that in this example, a more thorough control design study may be sufficient to enforce the time-domain requirement. However, this is not considered in this article for two reasons:

• the OIST method was proposed to enforce time-domain requirements when the controller is not able to do so; • other criteria often enter in the control design. Enforcing the time-domain criterion may degrade nominal performance from other points of view.

The OIST method proposes to achieve a time-domain requirement while ensuring the nominal2 performance. This case study is dedicated to the position control of a ball on a beam. The physical system and the notations are represented in Fig. 4. The beam is actuated using a lever arm. An unknown disturbance force d is eventually applied to the ball acceleration and an unknown bias3 e is introduced on the measure of the ball position r. Both disturbances are represented on Fig. 5. The state vector of the system is defined by x = r ṙ and the measurements vector by y = x + D e e where D e = 1 0 . The regulated variable is defined as:

α(t) = C α y(t) = C α x(t) + C α D e e(t) = C α x(t) + E α e(t) (7)
with C α = 1 0 and E α = 1. It corresponds to the measured disturbed ball position.

The reason for monitoring this variable is quite obvious. The beam length is limited to L = 1 m which means that even a theoretically stabilizing control law can result in the ball falling off the beam. The objective -i.e. the time-domain requirement -is thus to satisfy ∀t, 0.1 ≤ α(t) ≤ 0.9 (in meters) while driving the system from r 0 = 0.5 m to the setpoint r s = 0.6 m. The system state-space representation is given by:

A = 0 1 0 0 , B = 0 0 0 1 -0.21 0 , C = I 2 , D = 0 0 1 0 0 0 (8)
where the inputs are respectively d, u and e. As far as the nominal control design is concerned, it is decided to implement a state-feedback controller with integral action to achieve steady-state accuracy. Let consider the following augmented system

A a = A 0 -C int 0 , B a = B u 0 0 1 (9)
where B u = 0 -0.21 and C int x = ṙ. Using LQR design on this augmented system with R = diag 10 4 and Q = diag 1 10 10 , the following state-feedback gain K p and integral action gain K i are obtained:

K p = -0.3156 -2.0937 0.0322 -0.7661 , K i = 0.0644 1.5779 (10)
which state-space representation is given by

(K) ẋK (t) = -K i (2)x K (t) + 0 1 {y s -y(t)} y K (t) = -K i x K (t) + K p {y s -y(t)} (11)
where y s = r s 0 . This dynamic controller is stable and yields good results on r s -setpoint tracking. The control signal simulation is shown on Fig. 6. However, violation of the time-domain requirement 0.1 ≤ α(t) ≤ 0.9 occurs as one can see on Fig. 7.

Problem formulation

Let consider a LTI system (G) with the following state-space representation: (G)

   ẋ(t) = Ax(t) + B u u(t) + B d d(t) State y(t) = x(t) + D e e(t) Measurements α(t) = C α x(t) + E α e(t) Regulated variable (12)
where u ∈ R is the control input, d ∈ R is an unknown disturbance and e ∈ R is an unknown signal. The time-domain constraint consists in ensuring α(t) ∈ [α(t), α(t)] , ∀t where α(t) and α(t) are design parameters. This is expressed in the following problem -which was introduced in Burlion ( 2012) and translated in the linear framework in Chambon et al. (2015a):

Problem 1: Find [u(t), u(t)] and C 0 such that α(t) ∈ [α(t), α(t)] , ∀t (13)
for the system described as

           ẋ(t) = Ax(t) + B u u(t) + B d d(t) y(t) = x(t) + D e e(t) α(t) = C α x(t) + E α e(t) u(t) ∈ [u(t), u(t)] x 0 ∈ C 0 (14) where x ∈ R n , y ∈ R m , u ∈ R is the command input, d ∈ R is an unknown disturbance and e ∈ R is an unknown signal with E α ∈ R.
Remark 2: Note that d and e are chosen as unidimensional signals. This is non-limiting as long as the hypotheses are satisfied on each component in case of multidimensional signals.

Finding a solution to this problem motivates the Output to Input Saturation Transformation (OIST) where a constraint on the output α is transformed into saturations [u(t), u(t)] on the control input u.

October 9, 2015 International Journal of Control ijc˙OIST

Assumptions

Prior to the application of the transformation which will be presented in Sect. 3, the considered system has to fulfil some assumptions which are recalled here. First, the time-domain requirements needs to be meaningful:

Assumption 1: It is assumed that the expected bounds on the regulated variable α are chosen consistently:

∀t, α(t) ≥ α(t) (15)
Assumption 2: The time-domain requirement signals are supposed to converge towards constant values satisfying Assum. 1:

lim t→∞ α(t) = α , lim t→∞ α(t) = α (16)
Second, the relative degrees of the regulated variable with respect to the inputs is detailed:

Assumption 3: Let (k, l) ∈ N 2 such that 1 ≤ l ≤ k. It is supposed the regulated variable α is of relative degree k (resp. l) with respect to u (resp. d). Let D(t) = d ḋ . . . d (k-l-1) ∈ R k-l and E(t) = e ė . . . e (k)
∈ R k+1 . The next assumption makes sure that these quantities are bounded by known time-varying matrices.

Assumption 4: Continuous time-varying bounds D(t), D(t) on the unknown disturbance and its derivatives D(t) are supposed to be known, that is

D(t) ≤ D(t) ≤ D(t), ∀t (17)
or, more precisely,

∀0 ≤ i ≤ k -l -1, d (i) (t) ≤ d (i) (t) ≤ d (i) (t), ∀t (18)
The same holds for E(t) with time-varying bounds E(t), E(t) . Also:

lim t→∞ D(t) = D , lim t→∞ D(t) = D , lim t→∞ E(t) = E and lim t→∞ E(t) = E ( 19
)
where D = d 0 . . . 0 , etc.

As conservative as this assumption may be, it is not so different from supposing the disturbances follow some theoretical model. The following assumptions will be used in the proof of the system state convergence in closed-loop:

Assumption 5: The disturbance d is supposed to be of finite energy d 2 < ∞. The same holds for e with energy e 2 < ∞.

Assumption 6: Let K(s) = (A K , B K , C K , D K ) a
strictly stable controller with state vector dimension n K . It is supposed the control law u = K(s)y asymptotically (resp. exponentially) stabilizes system (G) to the origin x = 0 (without loss of generality), under Assum. 5 (resp. d = 0 and e = 0). The controller state at the equilibrium is denoted x K = 0.

In this article, only minimum-phase systems are considered. In the non-minimum-phase case, additional analysis is required to ensure stability, which is considered as a perspective for future works.

Assumption 7: The zeros of the transfer function T u→α (s) from the control input u to the regulated variable α are supposed to be of strictly negative real part.

To simplify the formulation of iterative expressions for the saturations components, the following assumption is made to ensure some terms will not re-appear upon derivation of the components:

Assumption 8: The relative degrees in Assum. 3 satisfy to the relation 2l > k.

For any system fulfilling these hypotheses, the Output to Input Saturation Transformation (OIST) is proposed as a solution to Pb. 1. This transformation is detailed in the following section while closed-loop stability is established in Sect. 4.

Output to Input Saturation Transformation

The Output to Input Saturation Transformation is presented in this section in the case of known LTI systems fulfilling Assum. 3 to 8. In presence of unknown disturbances d and e, time-varying coefficients are used in the obtained saturations so as to avoid overlap. Their expressions are derived from the known bounds on D(t) and E(t).

Regulated variable differentiation and relative degree

Using Assum. 3 and 8, the k-th derivative of the regulated variable α in function of u, d and e (k) has the following expression:

α (k) (t) = C α A k x(t) + E α e (k) (t) + C α A k-1 B u u(t) + k j=l C α A j-1 B d d (k-j) (t) (20)
By definition of the relative degree (Def. 4), the k-th derivative of the regulated variable α thus depends on the control input signal u(t). In the next section, a lemma will be formulated so that properly differentiated design bounds on α can lead to saturations on u, using Eq. (20).

Fulfilling the time-domain requirement

Considering Pb. 1, the objective is to ensure α(t) ∈ [α(t), α(t)], ∀t. In this section, it is shown how adequate constraints on the successive derivatives of α can be used to fulfil this requirement. Let consider a vector of known positive time-varying signals

κ(t) = κ 1 (t) . . . κ k (t) ∈ R k + (21)
which will act as "design parameters". Let α (0) (t) = α(t) and α (0) (t) = α(t). Also consider the vectors

A(t) = α(t) . . . α (k) (t) , Ω(t) = α (0) (t) . . . α (k) (t)
and

Ω(t) = α (0) (t) . . . α (k) (t)
. Using these notations, the following lemma is proposed:

Lemma 1: Let A(0) ∈ Ω(0), Ω(0) . Let define ∀1 ≤ j ≤ k, ∀t: α (j) (t) = κ j (t) α (j-1) (t) -α (j-1) (t) + ˙ α (j-1) (t) α (j) (t) = κ j (t) α (j-1) (t) -α (j-1) (t) + ˙ α (j-1) (t) (22) Then, α (k) (t) ∈ α (k) (t), α (k) (t) , ∀t ⇒ α(t) ∈ [α(t), α(t)] , ∀t Proof. See Appendix A.
Remark 3: As shown in Sect. 3.3, the OIST method consists in transforming output constraints into time-varying input saturations. It will be proved in Th. 1 that the introduction of the timevarying coefficients κ(t) in Eq. ( 21) is crucial to avoid the time-varying input constraints to overlap. Note that in the absence of unknown disturbances, these coefficients could be chosen positive constants, see [START_REF] Burlion | A new Saturation function to convert an output constraint into an input constraint[END_REF].

Remark 4: Considering Eq. ( 22), the notations are determining. For example, it should be clear that ∀j, ∀t, α (j) (t) = (α) (j) (t). More generally, and using notations defined later4 , ∀t, Ω(t) = A(t). Moreover, while A(t) can be determined quite easily, this is much more difficult for Ω(t) since it depends on possibly unknown variables.

The main purpose of this lemma is to be able to satisfy the time-domain requirement on α(t) while setting constraints on α (k) (t). Since the k-th derivative α (k) (t) depends on u, this in turn leads to saturations on the control input signal which allow to fulfil α(t) ∈ [α(t), α(t)] , ∀t. This is presented in the next section.

Remark 5: Note this Lemma is still valid when introducing more conservative bounds β j (t) and

β j (t) on α (j) (t), ∀1 ≤ j ≤ k: ∀t, α (j) (t) ≤ β j (t) ≤ α (j) (t) ≤ β j (t) ≤ α (j) (t) (23)
Of course, there is no more relation between these new bounds such as in Eq. ( 22). Their expressions can however be obtained using the expressions of the original bounds. This remark will be used in Sect. 3.5.1 to derive differentiable κ(t)-coefficients from differentiable -yet more conservativebounds.

Control saturations

Given Assum. 3, there is C α A k-1 B u = 0. Let also suppose that C α A k-1 B u > 0.
Considering Lemma 1 and Eq. ( 20), and supposing that the expressions of α (k) (t) and α (k) (t) are known ∀t, the saturations to apply to the control input u are obtained in Eq. ( 24). In this case, it is also supposed that d and e are known signals:

u(t) = 1 CαA k-1 Bu α (k) (t) -C α A k x(t) -E α e (k) (t) -k j=l C α A j-1 B d d (k-j) (t) u(t) = 1 CαA k-1 Bu α (k) (t) -C α A k x(t) -E α e (k) (t) -k j=l C α A j-1 B d d (k-j) (t) (24)
In case the disturbances d and e are unknown but Assum. 4 is satisfied, the following more conservative expressions are obtained:

u(t) ≤ u c (t) = 1 CαA k-1 Bu α (k) (t) -C α A k x(t) + |E α | max |e (k) (t)|, |e (k) (t)| + k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| u(t) ≥ u c (t) = 1 CαA k-1 Bu α (k) (t) -C α A k x(t) -|E α | max |e (k) (t)|, |e (k) (t)| -k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| (25)
With a slight abuse of notation, the expressions from Eq. ( 25) will be used and denoted respectively u(t) and u(t). Before concluding this section, the following remark is made where it is observed that the expressions in Eq. ( 25) may be inconsistent due to the introduced conservatism:

Remark 6: For the saturations definition in Eq. ( 25) to be consistent and to avoid the control input saturations to overlap, one must ensure that u(t) ≤ u(t), ∀t, i.e.:

α (k) (t) -α (k) (t) ≥ 2|E α | max |e (k) |, |e (k) | + 2 k j=l |C α A j-1 B d | max(|d (k-j) |, |d (k-j) |) (26)
Remark 7: In case C α A k-1 B u < 0 and to avoid loss of generality, proper re-ordering of u(t) and u(t) is required. For a given input signal u, the saturating operator can be defined as follows:

sat(u)(t) := max(min(u(t), u(t)), min(u(t), max(u(t), u(t))))

Proof. It is proved here that the operator sat(u)(t) as defined in Eq. ( 27) fulfils sat(u)(t) ∈ [u(t), u(t)], ∀t:

• suppose u(t) > u(t), ∀t. Then, min(u(t), u(t)) = u(t) and max(u(t), u(t)) = u(t). Thus, Eq. ( 27) becomes sat(u)(t) = max(u(t), min(u(t), u(t))); • Suppose u(t) < u(t), ∀t. Then, min(u(t), u(t)) = u(t) and max(u(t), u(t)) = u(t). Thus, Eq. ( 27) becomes sat(u)(t) = max(u(t), min(u(t), u(t))).

In both cases, the expected result is obtained.

Determination of bounds on the regulated variable derivatives

In this section, an expression is established of the successive bounds on the derivatives of the regulated variable α. This is done using the iterative definition given in Eq. ( 22) and supposing at first that d and e as well as their derivatives are known.

For any integer j such that 1 ≤ j ≤ k, let define the two following vectors: k+1) where u j j (t) = 1 and ∀i > j, u j i (t) = 0; -l) where ∀i > max(-1, jl -1), v j i (t) = 0 and

• U j (t) = u j 0 (t) . . . u j k (t) ∈ R 1×(
• V j (t) = v j 0 (t) . . . v j k-l-1 (t) ∈ R 1×(k
v l+1 0 (t) = u l l-1 (t)C α A l-1 B d Let A = α ˙ α . . . (α) (k) ∈ R k+1 , A = α ˙ α . . . (α) (k) and Θ = C α C α A . . . C α A k ∈ R (k+1)×n
. Supposing E(t) and D(t) are known, one can express the bounds α (j) (t) ∈ R and α (j) (t) to be satisfied by α (j) (t), ∀t, as

α (j) (t) = U j (t) {A(t) -Θx(t) -E α E(t)} + C α A j x(t) + E α e (j) (t) -V j (t)D(t) α (j) (t) = U j (t) A(t) -Θx(t) -E α E(t) + C α A j x(t) + E α e (j) (t) -V j (t)D(t) (28)
Then, using Eq. ( 22), one can obtain the coefficients vectors U j and V j iteratively. This is illustrated in Eq. ( 29) where σ U j-1 (t) is the cyclic permutation of length k + 1 on the elements of U j-1 (t), as defined in Def. 5.

U 0 (t) = 1 0 . . . 0 ∀j s.t. 1 ≤ j ≤ k, U j (t) = κ j (t)U j-1 (t) + U j-1 (t) + σ U j-1 (t) ∀j s.t. 0 ≤ j ≤ l, V j (t) = 0 ∀j s.t. l < j ≤ k, V j (t) = κ j (t) V j-1 (t) + C α A j-l-1 A l-1 B d . . . A 2l-j B d + V j-1 (t) +σ V j-1 (t) + j-l-1 w=0 u j-1 l-1+w (t)C α A l-1+w B d 0 . . . 0 (29)
Proof. Tedious rewriting of Eq. ( 28) in explicit form and using Eq. ( 22) to express α (j+1) for j ≥ 0 starting with α (0) (t) = α(t) leads to Eq. ( 29). The same calculus is performed as far as the upper bound is concerned.

Remark 8: Using the obtained iterative expressions and the fact that u 1 0 (t) = κ 1 (t), one can determine that ∀j, 0 ≤ j ≤ k, u j+1 j (t) = j+1 w=1 κ w (t).

However, E(t) and D(t) are concatenations of unknown signals. By Assum. 4, only bounding signals are known on these disturbances. As stated in Remark 5, Lemma 1 is still valid if more conservative bounds are considered for application on the derivatives α (j) (t). In case E(t) and D(t) are bounded by known bounds, the expressions in Eq. ( 28) become

β j (t) = U j (t) {A(t) -Θx(t)} + |E α | U j (t) max |E(t)| , E(t) + C α A j x(t) + |E α | max e (j) (t) , e (j) (t) + V j (t) max |D(t)| , D(t) β j (t) = U j (t) A(t) -Θx(t) -|E α | U j (t) max |E(t)| , E(t) + C α A j x(t) -|E α | max e (j) (t) , e (j) (t) -V j (t) max |D(t)| , D(t) (30)
In case differentiable bounds are expected, the expressions in Eq. ( 31) are considered, where Def. 3 is used.

γ j (t) = U j (t) {A(t) -Θx(t)} + |E α | f abs U j (t) g E(t), E(t) + C α A j x(t) + |E α | g e (j) (t), e (j) (t) + f abs V j (t) g D(t), D(t) γ j (t) = U j (t) A(t) -Θx(t) -|E α | f abs U j (t) g E(t), E(t) + C α A j x(t) -|E α | g e (j) (t), e (j) (t) -f abs V j (t) g D(t), D(t) (31)
where the following is fulfilled ∀0 ≤ j ≤ k, ∀t:

α (j) (t) ≤ β j (t) ≤ γ j (t) α (j) (t) ≥ β j (t) ≥ γ j (t) (32)
which is compatible when used along with Lemma 1 as stated in Remark 5. However, the introduction of some conservatism requires to choose the coefficients κ(t) in U j (t) and V j (t) wisely. This is considered in the next section, along with other problems.

Saturations analysis

Using the expressions in Eq. ( 30) with j = k, the objective is to ensure α (k) (t) ∈ β k (t), β k (t) , ∀t so that, after using Lemma 1, the time-domain requirement is fulfilled:

β k (t) = U k (t) {A(t) -Θx(t)} + |E α | U k (t) max |E(t)| , E(t) + C α A k x(t) + |E α | max e (k) (t) , e (k) (t) + V k (t) max |D(t)| , D(t) β k (t) = U k (t) A(t) -Θx(t) -|E α | U k (t) max |E(t)| , E(t) + C α A k x(t) -|E α | max e (k) (t) , e (k) (t) -V k (t) max |D(t)| , D(t) (33)
These expressions are then used in Eq. ( 25) in place of α (k) (t) and α (k) (t) to obtain explicit expressions of the control input saturations:

u(t) = 1 CαA k-1 Bu U k (t) {A(t) -Θx(t)} + |E α | U k (t) max |E(t)| , E(t) +2 |E α | max e (k) (t) , e (k) (t) + V k (t) max |D(t)| , D(t) + k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| u(t) = 1 CαA k-1 Bu U k (t) A(t) -Θx(t) -|E α | U k (t) max |E(t)| , E(t) -2 |E α | max e (k) (t) , e (k) (t) -V k (t) max |D(t)| , D(t) -k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| (34)
These now depend on the design parameters κ(t) from Eq. ( 21), A(t) and A(t) and on the signals describing the limited knowledge of the disturbances: D(t), D(t), E(t) and E(t). As suggested by Remark 6, some further analysis is however required. It is particularly important to detect possible saturations overlap -by choosing the κ(t)-signals wisely -or to study the reachability of the origin -a result which will be used to prove the closed-loop stability.

Saturations overlap

Let define ∀t, ∆ 0 (t) := α(t) -α(t) and, ∀1 ≤ j ≤ k, ∆ j (t) := β j (t) -β j (t). By Assumption 1, ∆ 0 (t) > 0 is satisfied ∀t. Thus,

∆ j (t) = U j (t) A(t) -A(t) -2 |E α | U j (t) max |E(t)| , E(t) -2 V j (t) max |D(t)| , D(t) -2 |E α | max e (j) (t) , e (j) (t) ( 35
)
Using Eq. ( 30) and the fact that the signals in κ(t) are chosen positive by construction (see below), the dependence on κ j (t) can be made explicit, ∀1 ≤ j ≤ k, ∀t:

∆ j (t) ≥ ∆j (t) = κ j (t) U j-1 (t) A(t) -A(t) -2 |E α | U j-1 (t) max |E(t)| , E(t) -2 V j-1 (t) + C α A j-l-1 A l-1 B d . . . A 2l-k B d max |D(t)| , D(t) + U j-1 (t) + σ U j-1 (t) A(t) -A(t) -2 |E α | U j-1 (t) + σ U j-1 (t) max |E(t)| , E(t) -2 |E α | max e (j) (t) , e (j) (t) -2 V j-1 (t) + σ V j-1 (t) + j-l-1 w=0 u j-1 l-1+w (t)C α A l-1+w B d 0 . . . 0 max |D(t)| , D(t) (36)
which is, in turn, less conservative than the differentiable approximate:

∆j (t) ≥ ∆ j (t) = κ j (t) U j-1 (t) A(t) -A(t) -2 |E α | f abs U j-1 (t) g E(t), E(t) -2f abs V j-1 (t) + C α A j-l-1 A l-1 B d . . . A 2l-k B d g D(t), D(t) + U j-1 (t) + σ U j-1 (t) A(t) -A(t) -2 |E α | f abs U j-1 (t) + σ U j-1 (t) g E(t), E(t) -2 |E α | g e (j) (t), e (j) (t) -2f abs V j-1 (t) + σ V j-1 (t) + j-l-1 w=0 u j-1 l-1+w (t)C α A l-1+w B d 0 . . . 0 g D(t), D(t) (37)
which is simplified into

∆ j (t) = κ j (t) λ d j (t) + λ n j (t) ( 38
)
where λ d j (t) and λ d j (t) only depend on κ l (t) with l < j. Considering Remark 6, the following lemma is proposed:

Lemma 2: Saturations overlap is avoided if • ∆ 0 (t) ≥ 0, ∀t, which is true by Assumption 1; • ∀1 ≤ j < k, ∆ j (t) ≥ 0, ∀t; • ∆ k (t) ≥ 2|E α | max |e (k) |, |e (k) | + 2 k j=l |C α A j-1 B d | max(|d (k-j) |, |d (k-j) |), ∀t.
The last condition ensures that u(t) -u(t) ≥ 0, ∀t. A constructive definition of the coefficients κ j (t) in Eq. ( 21) can be used to fulfil Lemma 2.

Theorem 1: Saturations overlap is avoided if

• Assumption 1 is satisfied, i.e. α(t) ≥ α(t) and λ d 1 (t) = 0, ∀t; • ∀1 ≤ j < k, one ensures ∆ j (t) ≥ 0 by choosing, ∀t, κ j (t) = κj -λ n j (t) λ d j (t) (39)
where κj > 0 is chosen such that κ

1 (t) > 1 2 , ∀j > 1, κ j (t) > 1 and λ d j+1 (t) = 0, ∀t. • for j = k, one ensures ∆ k (t) ≥ 2|E α | max |e (k) |, |e (k) | + 2 k j=l |C α A j-1 B d | max(|d (k-j) |, |d (k-j) |) by choosing, ∀t, κ k (t) = κk -λ n k (t) + 2|E α | max |e (k) |, |e (k) | + 2 k j=l |C α A j-1 B d | max(|d (k-j) |, |d (k-j) |) λ d k (t) (40)
where κk > 0 is chosen such that κ k (t) > 1 2 , ∀t. Proof. Straightforward using Lemma 2 and Eq. ( 38). As far as the minimal values for the κ j (t) are concerned, this is discussed in the proof of Prop. 2, in Appendix B.

Remark 9: Note that κ k (t) is not differentiable with respect to t by definition but this is not required contrary to the other coefficients.

By the end of this section, a vector of design parameters κ = κ1 . . . κk is obtained, ensuring no overlap and the differentiability of κ(t) (to the exception of κ k (t)).

State-independent saturations

It is interesting to note that only one term depends on the state vector x(t) in the expressions of the control input saturations obtained in Eq. ( 34). Let define:

∀t, K oist (t) := U k (t)Θ C α A k-1 B u ∈ R 1×n (41)
It is observed that introducing saturations on the control input u is equivalent to saturating the signal v defined as:

v(t) := u(t) + K oist (t)y(t) = u(t) + K oist (t)x(t) + K oist (t)D e e(t), ∀t (42)
by the following state-free saturations:

v(t) = u(t) + K oist (t)x(t) + |K oist (t)D e | max (|e(t)| , |e(t)|) v(t) = u(t) + K oist (t)x(t) -|K oist (t)D e | max (|e(t)| , |e(t)|) (43)
or, more precisely

v(t) = 1 CαA k-1 Bu U k (t)A(t) + |E α | U k (t) max |E(t)| , E(t) + 2 |E α | max e (k) (t) , e (k) (t) + V k (t) max |D(t)| , D(t) + k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| + |K oist (t)D e | max (|e(t)| , |e(t)|) v(t) = 1 CαA k-1 Bu U k (t)A(t) -|E α | U k (t) max |E(t)| , E(t) -2 |E α | max e (k) (t) , e (k) (t) -V k (t) max |D(t)| , D(t) -k j=l C α A j-1 B d max |d (k-j) (t)|, |d (k-j) (t)| -|K oist (t)D e | max (|e(t)| , |e(t)|) (44)
Remark 10: One now has to ensure that v(t) -v(t) ≥ 0, ∀t using similar considerations than in Th. 1.

Using v instead of u as the new input to system (G) in Eq. ( 12), the saturated system becomes

ẋ(t) = [A -B u K oist (t)] x(t) + B u sat v(t) v(t) (v(t)) -B u K oist (t)D e e(t) + B d d(t) (45)

Admissible asymptotic equilibrium

Using Eq. ( 34) and the definitions of the state-free saturations in Eq. ( 43) the saturations in Eq. ( 44) were obtained. Using Assum. 2, it comes that:

lim t→∞ A(t) = α 0 . . . 0 , lim t→∞ A(t) = α 0 . . . 0 (46)
Considering Th. 1 along with Assum. 5, it can be observed that the design signal κ(t) in Eq. ( 21) converges towards a constant value. Thus, this is also the case of vectors U j (t) and V j (t), ∀1 ≤ j ≤ k, and lim t→∞ K oist (t) = K oist . Consequently, as far as the saturations in Eq. ( 44) are concerned and using Assum. 4, they tend towards the following limits:

v = 1 CαA k-1 Bu u k, 0 α + |E α | u k, 0 max (|e | , |e |) + v k, 0 + C α A k-1 B d max |d |, |d | + |K oist D e | max (|e | , |e |) v = 1 CαA k-1 Bu u k, 0 α -|E α | u k, 0 max (|e | , |e |) -v k, 0 + C α A k-1 B d max |d |, |d | -|K oist D e | max (|e | , |e |) (47
) and the unsaturated state-free control becomes5 :

v = C K x K + (D K + K oist ) (x + D e e ) = 0 (48)
In the following proposition it is shown that the origin is an admissible equilibrium under some condition. This condition can be evaluated during the analysis phase of the unconstrained closedloop system.

Proposition 1: Let x ∈ R n and suppose Assum. 2, 4 and 5 are satisfied. In the non-restrictive case where x = 0, this is an admissible asymptotic equilibrium if

v = 0 ∈ [v , v ] (49)
or, more precisely, if:

1 C α A k-1 B u u k, 0 α + |E α | u k, 0 max (|e | , |e |) + v k, 0 + C α A k-1 B d max |d |, |d | + |K oist D e | max (|e | , |e |) ≤ 0 ≤ -|K oist D e | max (|e | , |e |) + 1 C α A k-1 B u u k, 0 α -|E α | u k, 0 max (|e | , |e |) -v k, 0 + C α A k-1 B d max |d |, |d | (50)
Proof. Straightforward using the asymptotic results in Eq. ( 47) and ( 48).

Using this analysis on the saturations, the stability of the system in closed-loop with the saturated nominal controller is studied in the next section.

Closed-loop stability

In this section, the stability of the system is proved in closed-loop with the saturated control signal sat

u(t) u(t) (C K x K (t) + D K y(t))
, where u(t) and u(t) have been obtained using the approach presented in Sect. 3.

Considered system

As mentioned in Sect. 3.5.2, the control signal saturations depend on the system state x(t). Changing the control signal into v(t) = u(t)+K oist (t)y(t), the system studied in this section is equivalent to the system in Eq. ( 51) where the saturations on v(t) do not depend on the state vector anymore.

(G)                    ẋ(t) = [A -B u K oist (t)] x(t) + B u sat v(t) v(t) (v(t)) -B u K oist (t)D e e(t) + B d d(t) y(t) = x(t) + D e e(t)
ẋK (t) = A K x K (t) + B K y(t) v(t) = C K x K (t) + (D K + K oist (t)) y(t) α(t) = C α x(t) + E α e(t) (51)
The following lemma will be used to demonstrate the final theorem of this paper:

Lemma 3: ∀x(t) ∈ R n , ∀t, the function -K oist (t)x(t) is Lipschitz (with respect to x). Moreover, -K oist (t)x(t) is K 1 -Lipschitz, where K 1 = max t K oist (t) ∈ R. Proof. Let define, ∀x ∈ R n , ∀t, f (x, t) = -K oist (t)x(t). Then, ∀x ∈ R n , ∂f ∂x (x, t) = -K oist (t). Since K oist (t)
is continuous ∀t by definition of the coefficients κ(t) and continuity of D(t), D(t), E(t), E(t) (see Assum. 4), the function f is continuously differentiable with respect to the state x. This implies that K oist (t)x(t) is a Lipschitz continuous function with respect to x.

Closed-loop stability using an anti-windup approach

Due to the presence of a dynamic controller and saturations, unexpected closed-loop behaviour is expected. Anti-windup techniques have been widely studied and used to avoid behaviours like controller state divergence. Some of these techniques are presented in [START_REF] Grimm | Antiwindup for Stable Linear Systems with Input Saturation: An LMI-Based Synthesis[END_REF]; [START_REF] Kapoor | An Anti-Windup Design for Linear Systems with Input Saturation[END_REF]; [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF]. The approach proposed in [START_REF] Menon | General Anti-windup synthesis for input constrained nonlinear systems controlled using nonlinear dynamic inversion[END_REF] [START_REF] Herrmann | Anti-windup synthesis for nonlinear dynamic inversion control schemes[END_REF] deals with a specific class of nonlinear systems to which the system presented in Eq. ( 51) belongs. In this article, the anti-windup framework is used to prove the closed-loop stability of the system in Eq. ( 51) where the time-varying gain K oist (t) and saturations are respectively given by Eq. ( 41) and Eq. ( 44).

Proposition 2:

The open-loop system

ẋ(t) = [A -B u K oist (t)] x(t) (52)
is GES.

The proof is inspired by [START_REF] Herrmann | Anti-windup synthesis for nonlinear dynamic inversion control schemes[END_REF].

Proof. See Appendix B.

To ensure asymptotic stability of the closed-loop containing the dynamic controller, it is necessary to introduce an anti-windup. Considering the system in Eq. ( 51), the following anti-windup with state x a ∈ R n is introduced

G(s) x d e α y = x + D e e v(t) v(t) - + sat(v) + + K(s) G a (s) + - x a + - y a = x a v K oist (t) K oist (t) x K - - u a u sat(u) Figure 8. Closed-loop illustration. (G a )              ẋa = Ax a (t) + B u u a (t) y a = x a u a (t) = -K oist (t)y a (t) -Dz v(t) v(t) (v(t)) v 1 (t) = -[K(s) + K oist (t)] y a (t) (53)
The control v is then modified into

v(t) = u(t) + K oist (t)y(t) + v 1 (t) = C K x K (t) + D K (y(t) -y a (t)) + K oist (t) (y(t) -y a (t)) (54)
The main result of this section is the following theorem which proves stability of the system in closed-loop with the saturated nominal controller. Both this theorem and its proof are inspired from [START_REF] Menon | General Anti-windup synthesis for input constrained nonlinear systems controlled using nonlinear dynamic inversion[END_REF] and [START_REF] Herrmann | Anti-windup synthesis for nonlinear dynamic inversion control schemes[END_REF].

Theorem 2: If Assum. 1 to 8 are satisfied6 (resulting in Th. 1 and Prop. 2), the origin of the closed-loop system consisting of the system in Eq. (45) -where the time-varying saturations are given in Eq. (44) -the control law in Eq. ( 54) and the anti-windup compensator given in Eq. (53) is GAS.

Proof. See Appendix C.

An illustration of the system in closed-loop with the anti-windup and saturating block is given in Fig. 8. This is the typical structure obtained when implementing OIST on a linear system.

Example

The ball and beam example which served as a case study in Sect. 2.2 is considered again for application of the method which was presented in Sect. 3.

Assumptions

In this section, the assumptions in Sect. 2.4 are reviewed in the case of the ball and beam example introduced in Sect. 2.2.

• Assum. 1 and 2 are satisfied as was already illustrated on Fig. 7;

• As far as the relative degrees are concerned, k = 2 and l = 2 which fulfils Assum. 3 and 8;

• The disturbances and bounds are represented on Fig. 9. They fulfil Assum. 4;

• Assum. 5 is satisfied;

• The state-feedback controller with integral action proposed in Eq. ( 10) asymptotically stabilises the augmented system in Eq. ( 9). Also, it is strictly stable (see Eq. ( 11)), hence Assum. 6 is satisfied; • The system is equivalent to a double-integrator with no transmission zero: T u→α (s) = -0.21 s 2 . Assum. 7 is fulfilled.

Remark 11: Note that the set-point r s = 0.6 m (and ṙs = 0 m/s) is not the origin of the system (it is still a feasible equilibrium). However, using some transformation equivalent to a translation, one can obtain a set-point on the origin of the system. Hence Th. 2 and its proof are still valid.

OIST implementation

Using results in Sect. 3 and considering α = 0.1 m, α = 0.9 m, the following expressions are obtained for the successive ∆ i (t):

∀t,        ∆ 0 (t) = 0.8 ∆ 1 (t) = κ 1 (t) λ d 1 (t) + λ n 1 (t) ∆ 2 (t) = κ 2 (t) λ d 2 (t) + λ n 2 (t) (55)
where

λ d 1 (t) = α(t) -α(t) -2g (e(t), e(t)) λ n 1 (t) = -2g e (1) (t), e (1) (t) λ d 2 (t) = κ 1 (t) 1 0 A(t) -A(t) -2f abs κ 1 (t) 1 0 g E(t), E (t)
-2g e (1) (t), e (1) (t)

λ n 2 (t) = κ1 (t) κ 1 (t) 1 A(t) -A(t) -2f abs κ1 (t) κ 1 (t) 1 g E(t), E(t) -2g e (2) (t), e (2) (t) (56)
Then, the values of the design signals κ i (t) are deduced from these expressions and Th. 1:

κ 1 (t) = κ1-λ n 1 (t) λ d 1 (t) κ 2 (t) = κ2-λ n 2 (t)+2 max(|e (2) (t)|,|e (2) (t)|)+2|CαABd| max(|d(t)|,|d(t)|) λ d 2 (t) (57)
where κ = κ1 κ2 = 0.05 2 are chosen so that the conditions in Th. 1 are satisfied. It is then possible to obtain saturations on the control signal in the form of Eq. ( 34). Note that C α AB u = -0.21 < 0 in this example so the operator in Eq. ( 27) is used to obtain the adequate saturations. As far as the anti-windup design is concerned, the time-varying coefficient K oist (t) is defined as follows:

K oist (t) = U 2 (t)Θ C α AB u ( 58
)
where

U 2 (t) = κ 2 (t)κ 1 (t) + κ1 (t) κ 1 (t) + κ 2 (t) 1 and Θ = C α C α A C α A 2 .

Simulations and results

Using the results in Sect. 5.2, simulations are performed over 100 s. The disturbance signals used in simulation are shown in Fig. 9. At time t = 65 s, the non-null disturbance input d is used to show that the nominal control law is recovered when the time-domain constraint is not violated (see results). The simulation results are represented on Fig. 10 to 12. The data are represented in blue when considering the nominal control law only (no saturations) and in orange when the saturations obtained using OIST are introduced in the closed-loop. As mentioned in Sect. 2.2, the synthesized controller is not efficient enough and the ball falls off the beam. Using OIST and the knowledge on the disturbances bounds, the time-domain constraint is satisfied and nominal performance is recovered whenever the constraint is not violated. Note that the proposed approach results in some conservatism due to the lack of knowledge on d and e, which do not necessarily equal their known bounds. Also, some conservatism is introduced by using differentiable upper-approximates of the absolute value and maximum functions. On Fig. 11, it appears that the control law variations are much sharper in the saturated case. This is a trade-off required for complying with the time-domain requirement. Optimization of the constants κ may help to obtain less demanding although satisfying control laws. This is considered future works. On Fig. 12, the coefficients κ 1 (t) and κ 2 (t) as well as the quantities ∆ i (t) are represented. This allows to check that the assumptions and conditions in Th. 1 are fulfilled.

Conclusion

In this article, the problem of keeping a linear system output -or regulated variable -in an interval has been formalized. A solution based on a transformation from the output expected "saturation" to a saturation on the existing linear control input has been proposed. A constructive method to apply this transformation has been introduced. Time-varying saturations are obtained and used in closed-loop. Special attention is paid to choose the OIST time-varying design parameters κ i (t) in Eq. ( 21) in order to avoid saturations overlap. Using results from the anti-windup design community, the stability of the system in closed-loop with the resulting non-linear control has been proved. An application to a linear ball and beam model has been proposed, showing satisfactory results.

However, throughout this article, the specific class of minimum-phase linear systems has been considered. Also, it has been supposed that the whole state is measured. Future works will be dedicated to extend the approach to non-minimum phase systems and to systems with output feedback. Optimizing the coefficients κ will also be considered. and A ZΓ is the null matrix except for the coefficient A ZΓ (nk, 1) = 1. Considering Assum. 7, A Z eigenvalues are with strictly negative real parts. As far as the dynamics of Eq. (B4) is concerned, the following candidate Lyapunov positive definite function is considered

V (Γ, Z) = 1 2 Γ Γ + 2 Z Z (B6)
where is a positive constant. Then

V (Γ, Z) = Γ A Γ Γ + Z A Z Z + Z A ZΓ Γ (B7)
where, using the logarithmic function concavity and the fact that ∀1 ≤ i ≤ k, γ i-1 γ i ≤ |γ i-1 γ i |:

Γ A Γ Γ = -k-1 i=0 κ i+1 γ 2 i + k-1 i=1 γ i-1 γ i ≤ -k-1 i=0 κ i+1 γ 2 i + 1 2 k-1 i=1 γ 2 i + 1 2 k-2 i=0 γ 2 i Γ A Γ Γ ≤ -Γ D V Γ Γ ≤ -Γ diag κ 1 -1 2 , κ 2 -1, . . . , κ k-1 -1, κ k -1 2 Γ (B8)
so that D V Γ is a positive definite diagonal matrix upon adapted selection of the positive timevarying coefficients κ i (t). In the same vein

V (x a ) ≤ -α 4 x a 2 + α 3 B u x a {[K 1 + D K ] ( e x + D e e ) + C K x K } ≤ -α 4 x a 2 + α 3 B u x a K 1 + D K C K X +α 3 B u x a K 1 + 0 D K D e W (C9)
Using k 1 > 0 and k 2 > 0 such that

k 1 ≥ α 3 B u K 1 + D K C K ∈ R k 2 ≥ α 3 B u K 1 + 0 D K D e ∈ R (C10)
then Eq. (C9) becomes

V (x a ) ≤ -α 4 x a 2 + x a [k 1 X + k 2 W ] (C11)
Applying the inequality 2 x a X ≤ 2 x a 2 + X 2 2 for > 0, Eq. ( C11) is re-written as

V (x a ) ≤ -α 4 x a 2 + 1 2 k 2 1 2 1 x a 2 + 1 2 k 2 2 2 2 x a 2 + k 2 1 2 2 1 X 2 + k 2 2 2 2 2 W 2 ≤ -α 4 + 1 2 k 2 1 2 1 + 1 2 k 2 2 2 2 x a 2 + k 2 1 2 2 1 X 2 + k 2 2 2 2 2 W 2 ≤ -α 5 x a 2 + k 2 1 2 2 1 X 2 + k 2 2 2 2 2 W 2 (C12)
where

α 5 = α 4 -1 2 k 2 1 2 2 -1 2 k 2 2 2
2 > 0 if the constants 1 and 2 are chosen small enough so that

α 4 > 1 2 k 2 1 2 1 + 1 2 k 2 2 2
2 . Using (Isidori, 1999, Lemma 10.4.2, p.21), V is thus an ISS-Lyapunov function for the system ẋa = f 1 x a , X W (C13)

where f 1 is a non-linear function adequately defined. According to (Isidori, 1999, Theorem 10.4.1, p.21), the system in Eq. ( C13) is thus ISS. At the beginning of this proof, it has been shown thatfor a specific class of bounded finite energy disturbances d and e -X 2 is finite and X converges to zero. Using a similar approach to the previous case, there exists V X and strictly positive constants β 1 , β 2 such that VX (X) ≤ -β 1 X 2 + β 2 W 2 . This function is an ISS-Lyapunov function to the following system

Ẋ = f 2 (X, W ) (C14)
where f 2 is a linear function adequately defined. Using (Isidori, 1999, Theorem 10.5.2, p.34), it is possible to conclude that the cascade of systems in Eq. (C15) is ISS. The cascade is illustrated on Fig. C1. Note that in case d = 0 and e = 0 and using (Isidori, 1999, Corollary 10.5.3, p.35), the origin (x a , X) = (0, 0) is GAS for the cascade.

Figure 1 .

 1 Figure 1. Illustration of system (G) in closed-loop with controller K. The requirement on the regulated variable α(t) is violated.

Figure 2 .

 2 Figure 2. Illustration of system (G) in closed-loop with the saturations obtained using OIST, and expected performance result.

Figure 4 .Figure 5 .

 45 Figure 4. Ball and beam system example, thick dots represent fixed axes of rotation

Figure 6 .

 6 Figure6. Simulation of the control signal u using the controller with integral action defined in Eq. (10).

Figure 7 .

 7 Figure 7. Simulation of the regulated variable α (in blue). The time-domain requirement (in red) is clearly violated. The set-point rs is represented in green.

Figure 9 .

 9 Figure 9. Disturbances d and e applied to the ball and beam system in simulation, with bounds in red.

Figure 10 .Figure 11 .

 1011 Figure10. Left: in blue (resp. orange), unsaturated (resp. saturated) system state. Right: zoom on the regulated variable in the unsaturated (resp. saturated) case, in blue (resp. orange). The time-domain constraint on α is represented in red.

Figure 12 .

 12 Figure 12. Left: time-varying coefficients κ(t). Right: checking the saturations overlap.

  by the zero dynamics as shown in(Hu et al., 2012, Chapter 4) which results in the open-loop transfer in Eq. (B1) being equivalent to the following state-

Table 1 .

 1 Acronyms.

	Acronym Definition
	CICS	Converging Input Converging State
	GAS	Globally Asymptotically Stable (system)
	GES	Globally Exponentially Stable (system)
	ISS	Input-to-state Stable (system)
	LQR	Linear-Quadratic Regulator
	LTI	Linear Time-Invariant
	MIMO	Multiple Inputs Multiple Outputs (system)
	OIST	Output to Input Saturation Transformation

For more details on how the notion is used in the following, the reader should refer to Def. 1.

In the sense: when the time-domain condition is met.

In simulation, a uniform number generator with limited rate is used.

See Sect. 3.4.

In the non-pathological case where e is a physical signal converging to zero under Assum. 5.

And in the non-pathological case where d is a non-converging finite energy distribution, which is not a realistic physical case.

Appendix A. Proof of Lemma 1

The proof is performed iteratively with fixed k. Let j such that 1 ≤ j ≤ k. Suppose that α (k) (t) ∈ α (k) (t), α (k) (t) ⇒ α (j) (t) ∈ α (j) (t), α (j) (t) , ∀t. Also, α (j-1) (0) ∈ α (j-1) (0), α (j-1) (0) . Only the lower bound is considered. The demonstration is similar in the upper bound case. Suppose ∃t 2 > 0, α (j-1) (t 2 ) < α (j-1) (t 2 ) (A1) then, since α (j-1) (0) ∈ α (j-1) (0), α (j-1) (0) and, by continuity of α (j-1) and α (j-1) ,

But, using the recurrence hypothesis, the definition of α (j) (t) and the fact that ∀t,

hence, using the property of integrals,

which contradicts Eq. (A1). In other words ∀t > 0, α (j-1) (t) ≥ α (j-1) (t) (A5)

which proves the lemma.

Appendix B. Proof of Proposition 2

Considering Eq. ( 42) and the system in Eq. ( 51), this problem is equivalent to studying the stability of the system ẋ(t) = Ax(t) + B u u(t) in closed-loop with v(t) = 0 or u(t) = -K oist (t)x(t) (d(t) = 0 and e(t) = 0). The transfer function between u and the regulated variable α is given by

Theoretically speaking, a minimum state-space representation of this transfer can be represented in the canonical form which can in turn be expressed as a chain of integrators in addition to the considered transfer zero dynamics, see (Hu, Lindquist, Mari, & Sand, 2012, Chapter 4). The chain of integrators is given by

where the last equality is obtained by observing that u = -K oist (t)x (and 22) and ( 28) with null disturbances, the chain of integrators in Eq. ( B2) can be re-written as

where ν is a positive constant. It comes that

Using the notations λ min (M) and λ max (M) to denote the minimal and maximal real parts of the eigenvalues of the matrix M , it is observed that

and

By choosing ν = -λmax(AZ ) λmax(AZΓA ZΓ ) > 0 (since A ZΓ A ZΓ is positive semi-definite) and = νλ min (D V Γ ) > 0 and by observing that the eigenvalues of A Z are with strictly negative real parts (see Assum. 7) and D V Γ is a positive definite matrix, one obtains

where k 1 > 0. Also note that V (0, 0) = 0. As a consequence, the candidate function V is a Lyapunov function and the open-loop system ẋ = [A -B u K oist (t)] x is GES.

Appendix C. Proof of Theorem 2

First, considering Prop. 1, the origin is a reachable equilibrium of the system in closed-loop with the saturated control signal. Thus, it is of some interest to study the asymptotic stability of this equilibrium. Using Eq. ( 1), the state equation in Eq. ( 45) can be re-written as

Using a similar approach to [START_REF] Kapoor | An observer-based anti-windup scheme for non-linear systems with input constraints[END_REF], let define e x (t) := x(t) -x a (t). It follows that

Under Assum. 4 and 5, W 2 is finite and W is bounded. It follows that X 2 is finite and X converges to zero. In case W = 0, the state X converges exponentially to zero. Replacing v in Eq. ( 53) by its expression in Eq. ( 54), one obtains the following equation:

Considering Prop. 2, the open-loop system ẋa = [A -B u K oist (t)] x a (t) is exponentially stable. Thus, for some positive definite function V (x a ), there exist by the converse Lyapunov theorem, see [START_REF] Khalil | Nonlinear Systems[END_REF], constants α i , 1 ≤ i ≤ 4, such that

Since, by Eq. (C4):

First, using Lemma 3 and e

Using the relation between ISS and CICS property, as stated in (Terrell, 2009, Theorem 16.4, p.373), it comes that the cascade in Eq. ( C15) is CICS. Hence, using the theorem in [START_REF] Sontag | Remarks on stabilization and input-to-state stability[END_REF] (where CIBS property is a weaker property than CICS), the origin (x a , X) = (0, 0) is GAS for the cascade. This concludes the proof.