Article

# A note on Interval Observer Design for Unknown Input Estimation

Authors:
• University of Bordeaux -CNRS
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## Abstract

This paper addresses the problem of interval observer design for unknown input estimation in Linear Time-Invariant (LTI) systems. Although the problem of unknown input estimation has been widely studied in the literature, the design of joint state and unknown input observers has not been considered within a set-membership context. While conventional interval observers could be used to propagate, with some additional conservatism, unknown inputs by considering them as disturbances, the proposed approach allows their estimation. Under the assumption that the measurement noise and the disturbances are bounded, lower and upper bounds for the unmeasured state and unknown inputs are computed. Numerical simulations are presented to show the efficiency of the proposed approach.

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... However, the state and unknown input estimates can be very large and then useless when disturbances, uncertainties and noises are propagated without decoupling a part of their eects. To overcome such problem, two methods were proposed in [16,17] to design interval observers for LTI systems with noise and unknown input that reduce the jointly estimation conservatism for state and unknown input. ...
... In this context, this paper proposes to extend the results of [16,17]. The aim of this paper is to jointly estimate the unmeasurable states and the unknown inputs in presence of noises and uncertain parameters. ...
... Nevertheless, the method proposed in [18] does not consider uncertain parameters and it is not applied in the context of interval observer. Inspired by the methods of [17] and [18], this paper proposes a new interval observer design for a large class of uncertain LTI systems. ...
Article
Full-text available
In this paper, the problem of interval observer design for LTI uncertain systems is addressed. The aim is to estimate simultaneously the upper and lower bounds of unmeasurable states and unknown inputs. Based on a set membership approach, the proposed method interest consists in reducing the estimate conservatism assuming that all known and unknown inputs, disturbances, uncertainties and noises are bounded with a priori known bounds. For that, decoupling methods are used to reduce the latter propagation. The estimation problem is led back to a singular model representation. An example is given to illustrate the proposed design method.
... Note that in addition to noise and disturbances, real systems are often subject to unknown inputs. Such a case has been already investigated for non switched systems (the reader can for instance refer to [3], [12], [13], [23]). Furthermore, some works attempted to consider the case of continuous-time switched systems with unknown inputs [21], [22]. ...
... Remark 1. By augmenting unknown input d(k) as a part of the state vectorx(k + 1), the structural conditions for decoupling unknown input in [3], [12] are relaxed. Subsequently, the proposed method possesses a wider application scope than the above-mentioned works. ...
... In the sequel, a new framer candidate is introduced for the augmented state (12) and a sufficient condition is given such that the framer becomes an interval observer. ...
... Guerra et al. [17] have considered a continuous-time linear system affected by an UI in a specific situation, and have proposed a state observer. Gucik-Derigny et al. [18,19] have considered the more generic case of continuous-time LTI systems, and have proposed a dual state and UI observers using sliding mode. However, there can only be applied under a specific rank condition that the system's matrices have to satisfy. ...
... The state observer is constructed such that its error is independent of the UI variations. Furthermore, the rank condition needed in [18,19] is relaxed, so that the observer can be applied in a wider range of systems. ...
... Note that when r = 1 Assumption 2 corresponds to the one needed in [18,19]. Thus, the proposed assumption can be seen as a relaxation of assumptions needed in the existing works. ...
Article
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This study addresses the problem of guaranteed state and unknown input (UI) estimation for a class of linear parameter varying systems affected by UIs and other bounded perturbations. The context of interval observer in the cooperative framework is adopted. Thus, lower and upper bounds of the state and UI are provided. No assumption is made on the UI. The approach is based on a rank condition allowing to decouple the UI from the state estimation errors (so that the state estimation is guaranteed, regardless of the variation of the UI). The convergence of the proposed interval observer is studied by Lyapunov theory and stability conditions are expressed in terms of linear matrix inequalities which ease the design of the gain matrices of the observer. Finally, illustrative examples are given for discussion and comparison with respect to the existing results.
... Guerra et al. [17] have considered a continuous-time linear system affected by an UI in a specific situation, and have proposed a state observer. Gucik-Derigny et al. [18,19] have considered the more generic case of continuous-time LTI systems, and have proposed a dual state and UI observers using sliding mode. However, there can only be applied under a specific rank condition that the system's matrices have to satisfy. ...
... The state observer is constructed such that its error is independent of the UI variations. Furthermore, the rank condition needed in [18,19] is relaxed, so that the observer can be applied in a wider range of systems. ...
... Note that when r = 1 Assumption 2 corresponds to the one needed in [18,19]. Thus, the proposed assumption can be seen as a relaxation of assumptions needed in the existing works. ...
... Nous proposons ici d'expliciter cette dernière approche en présentant les travaux proposés dans [47,48]. ...
... Nous abordons ici les travaux présentés dans [48], basés sur les développements méthodologiques initalement présentés dans [53]. ...
... La structure de l'observateur intervalle proposé dans [48], est donnée par ...
Thesis
Cette thèse porte sur la conception d’une classe particulière d’estimateurs d'état, les observateurs intervalles. L’objectif est d’estimer de manière garantie, les bornes supérieure et inférieure de l’ensemble admissible de l'état d’un système, à chaque instant de temps. L’approche considérée repose sur la connaissance a priori du domaine d’appartenance, supposé borné, des incertitudes du système (incertitudes de modélisation, perturbations, bruits, etc). Une classe d'observateurs intervalles à entrées inconnues est proposée pour la classe des systèmes Linéaires à Paramètres Variants (LPV). La synthèse des paramètres de l’observateur repose sur la résolution d’un problème d’optimisation sous contraintes de type inégalités matricielles linéaires (LMI) permettant de garantir simultanément les conditions d’existence de l’observateur ainsi qu’un niveau de performance, soit dans un contexte énergie, soit dans un contexte amplitude ou soit dans un contexte mixte énergie/amplitude. Plus particulièrement, la performance de l'observateur repose sur une technique de découplage pour annuler les effets des entrées inconnues et une technique d’optimisation destinée à minimiser, au sens de critères de type gain L2et/ou gain L∞, les effets des perturbations sur la largeur totale de l’enveloppe de l'état du système LPV. La méthodologie de synthèse proposée est illustrée sur un exemple académique. Enfin, la méthodologie est appliquée au cas de la phase d’atterrissage du véhicule spatial HL20, sous des conditions de simulations réalistes.
... In the literature, several unknown input interval observers have been proposed for linear time-invariant systems [7], [16], [17], but relatively few have been proposed for LPV systems [18], the other observers being mainly based on setmembership strategies [10], [19]. These strategies are based on a change of coordinates decoupling the state from the unknown input. ...
... In [7], [16], [18] are proposed different strategies using a change of coordinates to decouple the state and the unknown inputs. Guaranteed bounds can then be computed for the state vector and this information is used to compute bounds for the unknown inputs. ...
... Moreover, the dynamics of e k is e k+1 = Πe k + ε k + β k where Π = (I 2 ⊗ T ) (I 2 ⊗ F 0 ) − LΥ and ε k = ε k ε k , with L = diag(L, L) and ε k and ε k defined in (16). Consider the candidate Lyapunov function V k = e k P e k . ...
Conference Paper
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This paper proposes a new interval observer for joint estimation of the state and unknown inputs of a discrete-time linear parameter-varying (LPV) system with an unmeasurable parameter vector. This system is assumed to be subject to unknown inputs and unknown but bounded disturbances and measurement noise, while the parameter-varying matrices are elementwise bounded. Considering the unknown inputs as auxiliary states, the dynamics are rewritten as discrete-time LPV descriptor dynamics. A new structure of interval observer is then used, providing more degrees of freedom than the classical change of coordinates-based structure. The observer gains are computed by solving linear matrix inequalities derived from cooperativity condition and $L_{\infty}$ norm. Numerical simulations are run to show the efficiency of the proposed observer.
... In the literature on set-membership estimation, only limited papers consider the systems with unknown inputs [17]- [21]. The works in [17] and [18] have considered unknown input estimation based on interval observer for linear time-invariant systems. However, the estimation performance of [17] and [18] depends on the change of coordinates and a matching condition needs to be satisfied. ...
... The works in [17] and [18] have considered unknown input estimation based on interval observer for linear time-invariant systems. However, the estimation performance of [17] and [18] depends on the change of coordinates and a matching condition needs to be satisfied. This rank condition is relaxed in [19] by utilizing relative degree technique. ...
... where (18) implies that Ψ 11 ≺ 0; define V k = (p e k ) T P d p e k , by pre-multiplying and post-multiplying Ψ 11 with (p e k ) T and p e k respectively, we have V k+1 − V k < (γ − 1)V k < 0. Hence, the center of the zonotope Z e k is stable. Using the Schur complement lemma, (18) becomes ...
Article
Full-text available
This article studies unknown input observer design for linear parameter-varying systems in a bounded error context. By presenting a novel unknown input observer structure, the output equation of linear parameter-varying systems is not restricted to be in a linear form. Moreover, by integrating decoupling technique and set-membership approach, a compromise is achieved between the restrictiveness and the conservativeness of estimation. Based on $P$ -radius criteria, sufficient design conditions are derived in terms of linear matrix inequalities. Finally, numerical simulations of a vehicle lateral dynamics system are conducted to demonstrate the validity of the presented method.
... It is known that the positivity of the interval estimation error dynamics is one of the most restrictive assumptions for the interval observer design [46][47][48]. This assumption was relaxed in [19,20,24,49,50] for linear timeinvariant (LTI) and SLS [43,45] by using similarity transformation of coordinates and extended to SLS and special types of non-linear systems [51][52][53][54][55]. In all these methods, it is shown that under some restricted conditions, a Hurwitz matrix can be transformed to a Hurwitz and Metzler one for continues-time case (Schur stable system could be transformed to a Schur and non-negative one for discrete-time systems) [51][52][53]. ...
... The solution to (49) using Lemma 8 is ...
Article
Full-text available
Interval observer design and related techniques have been researched and applied in many engineering fields and continues to be an active research area in the estimation and control society for the last two decades. Interval observer is a special class of observers that generates bounded interval vector for the real state vector in a guaranteed way under the assumption that the uncertainties are unknown but bounded. Some of the basic concepts and main developments in the designs and applications of interval observer for continuous-time, discrete-time (linear and nonlinear), fuzzy and switched systems are reviewed in this work. It also provides brief discussion of the main approaches in this area with clear descriptions of their structures and future directions.
... From the literature, few works have been carried out on interval estimation of fault or unknown input for descriptor systems. Until now, Gucik-Derigny et al. (2016) have studied interval estimation of unknown inputs in continuoustime systems via interval observer design. However, the interval estimation method in (Gucik-Derigny et al., 2016) requires an evaluation of the measurement derivatives which must be computed using a numerical differentiator. ...
... Until now, Gucik-Derigny et al. (2016) have studied interval estimation of unknown inputs in continuoustime systems via interval observer design. However, the interval estimation method in (Gucik-Derigny et al., 2016) requires an evaluation of the measurement derivatives which must be computed using a numerical differentiator. As is known, numerical differentiation is an ill-posed problem and may enlarge measurement noise. ...
Conference Paper
Full-text available
This paper considers actuator-fault estimation for discrete-time descriptor systems with unknown but bounded system disturbance and measurement noise. A zonotopic fault estimation filter is designed based on the analysis of fault detectability indexes. To ensure estimation accuracy, the filter gain in the zonotopic fault estimation filter is optimized through the zonotope minimization. The designed zonotopic filter not only can estimate fault magnitudes, but it also provides fault estimation results in an interval, i.e. the upper and lower bounds of fault magnitudes. Moreover, the proposed fault estimation filter has a non-singular structure and hence is easy to implement. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.
... In the presence of uncertainty, some conventional estimators cannot be realized, while an interval or set-membership estimation may still remain feasible [10][11]. There exist many approaches to design interval or set-membership estimators, and most of them are based on the monotone system theory [12][13][14][15][16][17][18][19]. This is also the main limitation of the interval observer design theory. ...
... Internal positive representations of systems method is also proposed [18]. As pointed out by Gucik-Derigny et al., only few works have considered unknown inputs and component faults in a set-membership framework [19][20][21]. Less attention has been paid to sensor faults. Meanwhile, most of existing results are based on systems should be observable [22]. ...
... Few works have considered unknown input interval observers. In (Gucik-Derigny et al. [2014]), (Gucik-Derigny et al. [2016]), an interval state and unknown inputs estimation for linear time-invariant continuoustime systems is studied. The work in (Xu et al. [2017]) proposes an UIO for LPV systems based on set theoretic approach. ...
... [2014]),(Gucik-Derigny et al. [2016]), an interval state and unknown inputs estimation for linear time-invariant continuoustime systems is studied. The work in(Xu et al. [2017]) proposes an UIO for LPV systems based on set theoretic approach. ...
Conference Paper
Full-text available
This work is devoted to fault estimation of discrete-time Linear Parameter-Varying (LPV) systems subject to actuator additive faults and external disturbances. Under the assumption that the measurement noises and the disturbances are unknown but bounded, an interval observer is designed, based on decoupling the fault effect, to compute a lower and upper bounds for the unmeasured state and the faults. Stability conditions are expressed in terms of matrices inequalities. A case study is used to illustrate the effectiveness of the proposed approach.
... But it is not always possible to find an observer gain such that the estimation error dynamics are nonnegative and stable. To overcome this restriction, many techniques have been proposed based on coordinates transformation [17], [26]- [28]. However, it is very difficult to find a transformation matrix and observer gain making the state estimation error dynamics positive and stable. ...
... The available information of measurement vector and the input sequence are used to calculate the state enclosure [x 2k−kn ] in (28). The generalized inverse of the observability matrix is used for the aforementioned purpose: ...
Article
Full-text available
In this paper, a novel numerical scheme to set-membership interval state estimator design is proposed for the multiple-input-multiple-output (MIMO) linear time-varying (LTV) discrete-time systems using systems observability matrix and its past input/output values. The proposed method is more simple and efficient. First, an interval state estimator is designed that will generate a tight interval vector for the real state vector in a guaranteed way by employing interval analysis and consistency techniques for the single-input-single-output (SISO) systems. The proposed interval state estimator technique is then extended easily to the MIMO systems. Secondly, the estimation errors dynamics bounds are computed a-priori to measurements for the unknown but bounded uncertainties. Finally, the convergence of the width of the interval state vector towards a known value in finite time is provided to prove the boundedness of the interval state vector and estimation error that further quantify the accuracy of the developed technique. The performance and comparison with already existing techniques are highlighted through numerical examples.
... It is known that the positivity of the interval estimation error dynamics is one of the most restrictive assumptions for the interval observer design [46][47][48]. This assumption was relaxed in [19,20,24,49,50] for linear timeinvariant (LTI) and SLS [43,45] by using similarity transformation of coordinates and extended to SLS and special types of non-linear systems [51][52][53][54][55]. In all these methods, it is shown that under some restricted conditions, a Hurwitz matrix can be transformed to a Hurwitz and Metzler one for continues-time case (Schur stable system could be transformed to a Schur and non-negative one for discrete-time systems) [51][52][53]. ...
... The solution to (49) using Lemma 8 is ...
Article
Full-text available
Interval observer design and related techniques have been researched and applied in many engineering fields and continues to be an active research area in the estimation and control society for the last two decades. Interval observer is a special class of observers that generates bounded interval vector for the real state vector in a guaranteed way under the assumption that the uncertainties are unknown but bounded. Some of the basic concepts and main developments in the designs and applications of interval observer for continuous-time, discrete-time (linear and nonlinear), fuzzy and switched systems are reviewed in this work. It also provides brief discussion of the main approaches in this area with clear descriptions of their structures and future directions.
... However, designing such estimators for SEIR models in a real scenario is challenging, especially when the uncertain parameters are not exactly known but are defined by an interval or polytope. Interval estimator techniques can solve such issues [14][15][16][17][18][19][20][21][22]. Based on the monotone system theory (MST), interval estimators are designed to estimate the real states at any time instant and generate a set of acceptable values known as the interval in [20,[23][24][25][26][27][28]. ...
... In contrast to the existing interval observer design methods [19,33], we considered a nonlinear model with unknown input affecting the output with a highly uncertain state matrix A. The bounds on the uncertain input are constructed before designing the interval estimator; 3. ...
Article
Full-text available
This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposedinfected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per the World Health Organization (WHO). Therefore, tracking and estimating the evolution of these diseases are important to make intervention strategies. We study a real case in which some uncertain variables such as model disturbances, uncertain input and output measurement noise are not exactly available but belong to an interval. Moreover, the uncertain transmission bound rate from the susceptible towards the exposed stage is not available for measurement. We designed an interval estimator using an observability matrix that generates a tight interval vector for the actual states of the SEIR model in a guaranteed way without computing the observer gain. As the developed approach is not dependent on observer gain, our method provides more freedom. The convergence of the width to a known value in finite time is investigated for the estimated state vector to prove the stability of the estimation error, significantly improving the accuracy for the proposed approach. Finally, simulation results demonstrate the satisfying performance of the proposed algorithm.
... Several unknown input interval observers have been proposed [5], [6], [19]. Such strategies often consider a change of coordinates aimed at separating the system into two subsystems, one of which is free from the influence of the unknown input. ...
... In [5], [6], [19], [22], different strategies are proposed to decouple the unknown inputs from the state with a change of coordinates in order to find an interval containing the state vector and use this information to find an interval containing the unknown input vector. In this letter, the dynamical system (5) is rewritten into a discrete-time LTI descriptor system by considering the unknown inputs as auxiliary states [20], [21] such that: ...
Article
Full-text available
This letter proposes an unknown input zonotopic Kalman filter-based interval observer for discrete-time linear time-invariant systems. In such contexts, a change of coordinates decoupling the state and the unknown inputs is often used. Here, the dynamics are rewritten into a discrete-time linear time-invariant descriptor system by augmenting the state vector with the unknown inputs. A zonotopic outer approximation of the feasible state set is then obtained with a prediction-correction strategy using the information from the system dynamics, known inputs and outputs. Bounds for both the state and unknown inputs are obtained from this zonotopic set. The efficiency of the proposed interval observer is assessed with numerical simulations.
... However, to the best of the authors knowledge, only few works have been done on interval estimation of faults or unknown inputs for control systems. In [17] , the authors have studied interval estimation of unknown inputs in continuous-time systems via interval observer design techniques. However, the method in [17] requires an evaluation of the measurement derivatives, which must be calculated by using numerical differentiator. ...
... In [17] , the authors have studied interval estimation of unknown inputs in continuous-time systems via interval observer design techniques. However, the method in [17] requires an evaluation of the measurement derivatives, which must be calculated by using numerical differentiator. As it is known, numerical differentiation is an ill-posed problem and may fail in the case of noises with large magnitude. ...
Article
This paper proposes a state augmentation approach to achieve interval fault estimation for descriptor systems with unknown but bounded disturbances and measurement noises. An augmented descriptor system, which is equivalent to the original system, is constructed while the fault vector is considered as an auxiliary state. Based on this augmented system, a robust fault estimation observer is designed to estimate the actuator faults. Under this augmented descriptor form, the conservatism of observer gain matrix can be significantly reduced by introducing more design degrees of freedom. Moreover, interval estimation of actuator faults can be achieved by using zonotopic techniques. Finally, a DC motor model simulation is used to illustrate the effectiveness of the proposed approach.
... In the set-membership framework, joint state and unknown input estimation has been considered but only for continuous-time systems [Gucik-Derigny et al. 2016b]. Standard discrete-time UIOs have already been used for state estimation in the presence of unknown inputs [Valcher 1999;Darouach 2004], and also for unknown input estimation [Maquin et al. 1994]. ...
Thesis
In this manuscript, contributions to the development of methods for on-line model-based prognosis are presented. Model-based prognosis aims at predicting the time before the monitored system reaches a failure state, using a physics-based model of the degradation. This time before failure is called the remaining useful life (RUL) of the system.Model-based prognosis is divided in two main steps: (i) current degradation state estimation and (ii) future degradation state prediction to predict the RUL. The first step, which consists in estimating the current degradation state using the measurements, is performed with filtering techniques. The second step is realized with uncertainty propagation methods. The main challenge in prognosis is to take the different uncertainty sources into account in order to obtain a measure of the RUL uncertainty. There are mainly model uncertainty, measurement uncertainty and future uncertainty (loading, operating conditions, etc.). Thus, probabilistic and set-membership methods for model-based prognosis are investigated in this thesis to tackle these uncertainties.The ability of an extended Kalman filter and a particle filter to perform RUL prognosis in presence of model and measurement uncertainty is first studied using a nonlinear fatigue crack growth model based on the Paris' law and synthetic data. Then, the particle filter combined to a detection algorithm (cumulative sum algorithm) is applied to a more realistic case study, which is fatigue crack growth prognosis in composite materials under variable amplitude loading. This time, model uncertainty, measurement uncertainty and future loading uncertainty are taken into account, and real data are used. Then, two set-membership model-based prognosis methods based on constraint satisfaction and unknown input interval observer for linear discete-time systems are presented. Finally, an extension of a reliability analysis method to model-based prognosis, namely the inverse first-order reliability method (Inverse FORM), is presented.In each case study, performance evaluation metrics (accuracy, precision and timeliness) are calculated in order to make a comparison between the proposed methods.
... Assumption 1: The following assumption is made: rank(CG) = rank(G) = q, q < n (7) Note that assumption 1 is standard in unknown input observer design [12], [13], [14]. ...
Conference Paper
Full-text available
This paper proposes a robust fault detection method based on interval observer design. This method offers the advantage of being robust against disturbances, measurement noises, uncertain parameters and unknown inputs. The optimization framework is used to reduce the conservatism of the interval-based approach. The design of the observer parameters is formulated using the Linear Matrix Inequality technique and it is shown how some extra LMI specifications can be judiciously formulated to ensure a minimum sensitivity level to the fault. Finally, the actuator fault detection of an atmospheric re-entry vehicle provides a test to demonstrate the efficiency of the proposed method.
... Due to this less complex description of the sets, they are substantially less computationally intensive while still providing a practically useful interval width of the estimates. An increase of publications in the last years can be recognized in the research area of interval observers ( [1], [6], [25], [8], [10], [17]). The aim of the design of interval observers is to get one observer for the lower bound and one for the upper bound which include the real state in a guaranteed way. ...
... A possible drawback of this approach is that it could provide quite conservative bounds that could be then uninformative for practitioners. However, in recent years, much progress has been made to improve the width of the guaranteed intervals, playing with different structures of the systems [390,391,392,393] (there exist also several results for the class of linear dynamics), changes of coordinates [394,379], considerations of bundles of observers [395,396] or with the help of purely numerical methods [397,375,374,385,377,398] based on interval analysis [399,373]. ...
Thesis
Dans le contexte de l’utilisation des microalgues pour la bioremédiation d’effluents et la production de biomasse d’intérêt industriel, la stabilité et les performances des systèmes de production sont des enjeux majeurs. Une des voies de recherche privilégiées est la mise à profit de la diversité des microalgues en assemblages (polycultures) pour l’amélioration des performances de la production. Néanmoins, l’exploitation de ces assemblages complexes en système ouvert extérieur est sujette à divers stress biotiques (microalgues indésirables ou bactéries compétitrices) et abiotiques (limitation des ressources, notammentl’azote et la lumière) qui influencent les interactions au sein des assemblages, rendant difficiles la prédictionet l’optimisation de la production. L’objectif de la présente thèse est de proposer des méthodes etdes outils permettant de comprendre, prédire et optimiser la production de biomasse d’un assemblage algal(consortium microbien naturel ou artificiellement composé) sous la fluctuation des ressources et desconditions de culture. Plus particulièrement, nous avons développé des modèles mathématiques baséssur des systèmes dynamiques, qui peuvent être confrontés à des expériences en laboratoire et à l’échellepilote.Les premiers travaux de cette thèse ont consisté à choisir une méthode expérimentale pour lacaractérisation de la vitesse de croissance spécifique des microorganismes photosynthétiques en fonctionde la ressource limitante. Par ailleurs, une nouvelle méthode d’estimation fonctionnelle est proposée produisantdeux courbes de croissance qui encadrent les données de vitesse de croissance et permettant ainsides estimations dynamiques des variables d’état par intervalles garantis. Dans un deuxième temps, la nature des interactions entre deux microalgues Chlorella sorokiniana et Scenedesmus pectinatus, qui se succèdent classiquement dans les bassins extérieurs utilisés pour le traitement des eaux usées urbaines, a été caractérisée en fonction de la fluctuation des différentes formes de l’azote (NH4+/NH3) et ensuite de la lumière disponible. Les modèles mathématiques associés aux expérimentations menées au cours de cette thèse ont permis de démontrer que le développement initial d’une espèce microalgale de type opportuniste, plus résistante aux fortes teneurs en NH4+/NH3 était nécessaire pour que puisse s’établir ensuite une espèce microalgale plus efficace vis-à-vis de la faible lumière disponible dans ces procédés très turbides. Dans un troisième temps, nous avons exploré les interactions paradoxales qui existent entre les microalgues et les bactéries hétérotrophes. En effet, les microalgues stimulent, à travers l’exsudation du carbone, leurs compétiteurs pour une ressource commune : l’azote ou le phosphore. Nous avons étudié l’influence de ce phénomène en proposant un modèle dynamique en dimension quatre. L’analyse mathématique a révélé l’unicité de l’équilibre de coexistence et la robustesse de l’installation des bactéries, car il a été démontré que l’équilibre en absence de bactéries est répulsif. Nous montrons que lorsque laconcentration de la ressource minérale alimentant en continu le bioréacteur est suffisamment grande, il existe des valeurs du taux de renouvellement du réacteur pour lesquelles il y a coexistence, alors que les bactéries ne pourraient pas se développer en l’absence de microalgues dans ces mêmes conditions opératoires.Enfin, il a été montré à l’aide de simulations que lors de la coexistence, leur biomasse respectives peut osciller.Ces résultats témoignent de la complexité des interactions biotiques, fournissent des méthodes applicables à d’autres organismes modèles permettant de les étudier, et soulèvent des possibilités d’application prometteuses pour l’optimisation et le contrôle des systèmes dynamiques de procédés pour les travaux futurs.
... Interval observers can be considered as conventional pointwise observers with a point estimate and an uncertainty quantification respectively given by the midpoint and the radius of each interval. This overview can be completed in a further paper by some results on discrete-time systems [23,24,25,26,27,28], joint unknown and input estimation [60,61] and applications of interval observers in the field of robust control [62,26,63,64,65], diagnosis [66,67,68,69], prognosis [70] and automotive [71,72]. ...
Article
Full-text available
Based on the theory of positive systems, the goal of interval observers is to compute sets of admissible values of the state vector at each instant of time for systems subject to bounded uncertainties (noises, disturbances and parameters). The size of the estimated sets, which should be minimised, has to be proportional to the model uncertainties. An interval estimation can be seen as a conventional point estimation (the centre of the interval) with an estimation error given by the interval radius. The reliable uncertainties propagation performed in this context can be useful in several fields such as robust control, diagnosis and fault-tolerant control. This paper presents some recent results on interval observers for several dynamical systems classes such as continuous-time and switched systems.
... Indeed, several works have investigated estimation problems in this alternative context for different classes of time-invariant and parameter-varying systems (Mazenc, Dinh, & Niculescu, 2014;Lamouchi, Amairi, Raïssi, & Aoun, 2016). Interval unknown input observers have received great attention see for instance (Ellero, Gucik-Derigny, & Henry, 2015;Efimov, Fridman, Raïssi, Zolghadri, & Seydou, 2012;Gucik-Derigny, Raïssi, & Zolghadri, 2016). Using the well known "relative degree" property, unknown input observers accomplish a decoupling of the unknown input from the state vector. ...
Article
Full-text available
This paper deals with unknown input estimation for switched linear systems in an Unknown But Bounded Error (UBBE) framework. Based on a known switching signal and under the fulfillment of the relative degree property by all the subsystems, a decoupling method is used to make the state partially affected by the unknown input. Assuming that the disturbances and the measurement noises are unknown but bounded with a priori known bounds, lower and upper bounds of the unmeasured state and unknown input are then computed. A numerical example illustrates the efficiency of the proposed methodology.
... where x t = S t E t I t R t T ∈ R 4 + is the state vector and ζ t := β t S t I t is an uncertain input. In contrast to [24] and [25], the matrix A t is time-varying in this paper. Moreover, it should be pointed out that only unknown inputs that have no impact on the output are considered in [25]. ...
Conference Paper
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This paper focuses on designing a state estimator for a discrete-time SEIR epidemic model of an influenza-like illness. It is assumed that only sets of admissible values are known for the model's disturbances, uncertainties and parameters, except for the time-varying transmission rate from the "susceptible" to the "exposed" stage, whose bounding values are unavailable. An interval observer is designed to estimate the set of possible values of the state, and a sufficient condition guaranteeing the asymptotic stability of the proposed estimator is formulated in terms of a linear matrix inequality. The performance of the proposed approach is demonstrated by numerical simulations.
... There have been a great number of papers concerning the problem of estimating state variables and unknown inputs for continuous-time systems using sliding mode approaches: see [1], [2], [3], [4], [5], [6], [7] and references therein. A typical property of these observers is that a sliding surface based on the output error is constructed and a nonlinear switching injection term is introduced to force the output errors to reach the sliding surface in finite time. ...
... Interval observers can be considered as conventional pointwise observers with an estimate and an uncertainty quantification respectively given by the midpoint and the radius of each interval. This overview can be completed in a further paper by some results on discrete-time systems [23,24,25,26,27,28], joint unknown and input estimation [60,61] and applications of interval observers in the field of robust control [62,26,63,64,65], diagnosis [66,67,68,69], prognosis [70] and automotive [71,72]. In addition, sliding mode control and other estimation techniques are well known for their compensation of matched disturbances and finite-time convergence. ...
... Write θ k as the form of θ k = θ 1,k θ 2,k , we can get (11) can be expressed by (12) ...
Article
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In this paper, the design of reduced-order and full-order interval observers for discrete-time system is investigated. First, the original system is transformed into a descriptor system with the ability to estimate the unknown noise of sensors. Next, a reduced-order interval observer design method for descriptor system is presented. The state matrix of the system is obtained by linear transformation, and a Sylvester equation is constructed to find the appropriate observer gain matrix. Then, an existing design method of full-order interval observer is introduced in this paper. Finally, a simulation example is given to demonstrate that the reduced-order interval observer has more accurate interval estimation results than the full-order interval observer.
... • the observer based approach through the so-called Unknown Input Observers (UIO) approaches, the eigenstructure assignment (EA) technique [34,40,41,42,43,44,45,46], the iterative learning observer (ILO) technique [47], the sliding mode observer (SMO) approach, see for instance [48,49,50] and the books [51,52] for material backgrounds about sliding mode theories and the very recent interval approaches [53,54,55,56,57]. • and the so-called norm-based approaches sometimes referred to as the approximate decoupling approach. ...
... A possible drawback of this approach is that it could provide quite conservative bounds that could be then uninformative for practitioners. However, in recent years, much progress has been made to improve the width of the guaranteed intervals, playing with different structures of the systems [39,26,14,55] (there exist also several results for the class of linear dynamics), changes of coordinates [44,8], considerations of bundles of observers [4,32] or with the help of purely numerical methods [20,21,40,41,28] based on interval analysis [35,18]. ...
Preprint
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We address the problem of determining functional framing of experimental data points in view of robust time-varying predictions, which is of crucial importance in bioprocess monitoring. We propose a method that provides guaranteed functional bounds, instead of sets of parameters values of Monod or Haldane growth functions commonly used in bioprocess modeling. We illustrate the applicability of the method with bioreactor simulations in batch and continuous mode. We also present two extensions of the method adding exibility in its application, and discuss its e�ciency in providing guaranteed state estimations.
... There is significant literature which discusses sliding mode approaches for designing observers to simultaneously estimate system states and unknown inputs for continuous-time systems: for example see Edwards et al. (2000), Tan and Edwards (2003), Alwi and Edwards (2008), Gucik-Derigny et al. (2016), Yan and Edwards (2007), Floquet et al. (2007), Bejarano and Fridman (2010), Chandra et al. (2017) and the references therein for a class of observers in which the system representing the dynamics between the disturbance inputs and the measurements is relative degree one. A common procedure in designing such observers includes (i) first selecting a sliding surface which depends on the output errors; (ii) the use of a nonlinear switching injection gain to force the output errors to reach the sliding plane in finite time. ...
Article
In this work, we consider the problem of simultaneously estimating the system states and unknown inputs in a linear sampled-data system, whose dynamics is influenced by external disturbances and uncertainties. Hardware limitations prevent an estimation scheme for a sampled-data system from achieving finite-time convergence, which is a typical property of existing sliding mode observers for dynamical continuous-time systems, because the sampling period is finite. Due to the sampling process, an approximate implementation of such an observer, designed for a continuous-time system, may not retain the desired performance in the sampled-data context. In this paper, we present an observer which takes advantage of the quasi-sliding motion concept to simultaneously estimate the state variables and the unknown input signals in a sampled-data context. A theoretical study is conducted to formally justify the convergence properties of the observer whilst simulation results are provided to show the efficiency of the proposed scheme.
... Moreover, for SEIR models, the quantity βSI represents newly exposed individuals rather than new infectives, which can be difficult to measure since these individuals might not even feel symptoms yet. Furthermore, compared to the recent literature on interval observer design, such as (Gucik-Derigny et al., 2015) and (Robinson, Marzat, & Raïssi, 2017), the matrix A governing the state dynamics is uncertain in this article. Only linear systems and unknown inputs that have no impact on the output are considered in (Robinson et al., 2017). ...
Article
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According to the World Health Organization, infectious diseases are among the top ten causes of death worldwide. To prepare intervention strategies in a timely manner, tracking the evolution of these diseases is critical. For this purpose, public health services have access to noisy counts of infected people, which we use here to design a state estimator for a nonlinear discrete-time Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model. We consider the practical case where only sets of admissible values are known for the model's disturbances, uncertainties and parameters. Moreover, no bounds are available for the uncertain transmission rate from the “susceptible” to the “exposed” stage of the illness. We estimate the set of possible values of the state using an interval observer and characterize the stability and size of the estimation errors using linear programming. Furthermore, we propose an epidemic outbreak detector that leverages these state interval estimates. We demonstrate the observer's performance in numerical simulations.
... In the set-membership framework, joint state and unknown input estimation has been considered but only for continuous-time systems (Gucik-Derigny, Raïssi, & Zolghadri, 2016). Standard discrete-time UIOs have already been used for state estimation in the presence of unknown inputs (Darouach, 2004;Valcher, 1999), and also for unknown input estimation (Maquin, Gaddouna, & Ragot, 1994). ...
Article
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In this paper, a model-based prognosis method where the degradation cannot be directly measured but only detectable through the drift of a model parameter is considered. This parameter drift is viewed as an unknown input, whose reconstruction allows the estimation of the degradation state. Model-based prognosis is divided into a filtering step where the current degradation state is estimated, and an uncertainty propagation step where the future degradation state is predicted. During these two steps, model uncertainty and measurement uncertainty are taken into account within the set-membership framework using interval techniques. The filtering step is performed with an interval unknown input observer for linear time-invariant discrete systems. Then, based on a set-membership constraint satisfaction methodology, interval propagation is performed to estimate the bounds that include the future degradation state until some failure threshold is reached, allowing the deduction of the interval including the remaining useful life. In order to demonstrate the efficiency of the proposed model-based prognosis methodology, the degradation of a suspension system is studied.
... • the observer based approach through the so-called Unknown Input Observers (UIO) approaches, the eigenstructure assignment (EA) technique [34,40,41,42,43,44,45,46], the iterative learning observer (ILO) technique [47], the sliding mode observer (SMO) approach, see for instance [48,49,50] and the books [51,52] for material backgrounds about sliding mode theories and the very recent interval approaches [53,54,55,56,57]. • and the so-called norm-based approaches sometimes referred to as the approximate decoupling approach. ...
... A possible drawback of this approach is that it could provide quite conservative bounds that could be then uninformative for practitioners. However, in recent years, much progress has been made to improve the width of the guaranteed intervals, playing with different structures of the systems [42,28,14,58] (there exist also several results for the class of linear dynamics), changes of coordinates [47,8], considerations of bundles of observers [4,34] or with the help of purely numerical methods [21,22,43,44,30] based on interval analysis [37,18]. ...
Article
We address the problem of determining functional framing from experimental data points in view of robust time-varying predictions, which is of crucial importance in bioprocess monitoring. We propose a method that provides guaranteed functional bounds, instead of sets of parameters values for growth functions such as the classical Monod or Haldane functions commonly used in bioprocess modeling. We illustrate the applicability of the method with bioreactor simulations in batch and continuous mode, as well as on real data. We also present two extensions of the method adding flexibility in its application, and discuss its efficiency in providing guaranteed state estimations.
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This paper investigates the design of interval observers for uncertain linear time invariant systems. The aim is to provide upper and lower bounds of the system state in a guaranteed way. The method is based on the monotone system theory. It provides a solution to the conditions of the existence of the interval observer under mixed performance criteria such as H∞ performance and D-stability. The core idea relies on the introduction of a new class of Unknown Input Interval Observers (UIIOs).
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This paper presents an actuator fault detection and interval reconstruction scheme based on interval observers for systems with both actuator faults and disturbances. To begin with, two full-order interval observers which are sensitive to actuator faults are designed to achieve the goal of actuator fault detection. Then, two reduced-order interval observers are constructed, and they are robust to the actuator faults in that the interval observer constructions are fulfilled without using the boundary information of the actuator faults. An interval actuator fault reconstruction method is developed based on the reduced-order interval observers. Finally, the validities of the proposed methods are verified by a numerical simulation example.
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In this paper, the problem of joint state and unknown input estimation for linear time-invariant (LTI) discrete-time systems using interval observer is addressed. This problem has already been studied in the context of continuous-time systems. To the best of our knowledge, unknown input interval-based estimation for discrete-time systems has not been considered in the litterature. Assuming that the measurement noise and disturbances are bounded, lower and upper bounds are first computed for the unmeasured state and then for the unknown inputs. The results obtained with a numerical example highlight the efficiency of the method.
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In this paper, an interval sliding mode observer design method for uncertain systems is proposed. Uncertainty is assumed between a known minimum value and a maximum value. The observer is then constructed via a convex weighted sum of an upper estimator corresponding to the maximum value of the uncertainty and a lower estimator corresponding to the minimum value of the uncertainty. The weighting factor is calculated at each time, from the different measured outputs and the bounds of the interval of the estimator.
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An interval estimation method based on reduced-order observer and peak-to-peak analysis is proposed for linear time-invariant systems with disturbance and measurement noise. The proposed method consists of two steps. First, a reduced-order observer with L∞ performance is designed to obtain point estimation. Second, interval estimation is achieved by integrating the obtained point estimation and the error interval estimation by peak-to-peak analysis. Steady-state gain optimization is used in both steps to improve the estimation accuracy of the proposed method. The superiority of the method over the reduced-order interval observer is illustrated through numerical simulations.
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The goal of this technical note is to design interval observers for a class of nonlinear continuous-time systems. The first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time-invariant changes of coordinates even if the system is not cooperative. This result is extended to a class of nonlinear systems using partial exact linearisations. The proposed observers guarantee to enclose the set of system states that is consistent with the model, the disturbances and the measurement noise. Moreover, it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity, uncorrelation or type of distribution. The proposed observer is illustrated through numerical simulations.
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This note presents a simple method to design a full-order observer for linear systems with unknown inputs. The necessary and sufficient conditions for the existence of the observer are given
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The main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises. The proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal. Such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for Lipschitz’s constant of the derivative.
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Conference Paper
This paper is a contribution to the problem of systems prognosis. More precisely, in this work, the goal is to introduce an unknown input interval observer for a class of uncertain system with an unknown input for systems prognosis. The observer synthesis is applied to unknown variables estimation of an electromechanical oscillator with a non-stationary two-well potential to illustrate the pertinence of the proposed approach.
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It is shown that, for any time-invariant exponentially stable linear system with additive disturbances, time-varying exponentially stable interval observers can be constructed. The technique of construction relies on the Jordan canonical form that any real matrix admits and on time-varying changes of coordinates for elementary Jordan blocks which lead to cooperative linear systems. The approach is applied to detectable linear systems.
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In this paper we design an interval observer for the estimation of unmeasured variables of uncertain bioreactors. The observer is based on a bounded error observer, as proposed in [Lemesle, V., & Gouzé, J.-L. (2005). Hybrid bounded error observers for uncertain bioreactor models. Bioprocess and Biosystems Engineering, 27, 311–318], that makes use of a loose approximation of the bacterial kinetics. We first show how to generate guaranteed upper and lower bounds on the state, provided that known intervals for the initial condition and the uncertainties are available. These so-called framers depend on a tuning gain. They can be run in parallel and the envelope provides the best estimate. An optimality criterion is introduced leading to the definition of an optimal observer. We show that this criterion provides directly a gain set containing the best framers. The method is applied to the estimation of the total biomass of an industrial wastewater treatment plant, demonstrating its efficiency.
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The problem of unknown input estimation and compensation is studied for actuated nonlinear systems with noisy measurements. The proposed solution is based on high-order sliding-mode differentiation and discrete-time optimization technique. Accuracy of the proposed hybrid estimation scheme is evaluated and stability conditions of the compensating mechanism are established. It is shown that the fault detection delay as well as the smallest detectable fault magnitude can be estimated. Efficiency of the proposed approach is demonstrated through oscillatory failure detection and compensation in aircraft surface servo loops.
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This paper describes a method for designing sliding mode observers for detection and reconstruction of actuator and sensor faults, that is robust against system uncertainty. The method uses ℋ∞ concepts to design the sliding motion so that an upper bound on the effect of the uncertainty on the reconstruction of the faults will be minimized. The design method is first applied to the case of actuator faults, and then by some appropriate filtering, the method is extended to the case of sensor faults. A VTOL aircraft example taken from the fault detection literature is used to demonstrate the method and its effectiveness. Copyright © 2003 John Wiley & Sons, Ltd.
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Singular value decomposition is used to formulate a technique for the design of Luenberger observers when there are completely unknown system inputs.
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This paper presents a constructive solution to the problem of designing a reduced-order Luenberger observer for linear systems subject to arbitrary unknown inputs.
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A novel state estimator design scheme for linear dynamical systems driven by partially unknown inputs is presented. It is assumed that there is no information available about the unknown inputs, and thus no prior assumption is made about the nature of these inputs. A simple approach for designing a reduced-order unknown input observer (UIO) with pole-placement capability is proposed. By carefully examining the dynamic system involved and simple algebraic manipulations, it is possible to rewrite equations eliminating the unknown inputs from part of the system and to put them into a form where it could be partitioned into two interconnected subsystems, one of which is directly driven by known inputs only. This makes it possible to use a conventional Luenberger observer with a slight modification for the purpose of estimating the state of the system. As a result, it is also possible to state similar necessary and sufficient conditions to those of a conventional observer for the existence of a stable estimator and also arbitrary placement of the eigenvalues of the observer. The design and computational complexities involved in designing UIOs are greatly reduced in the proposed approach
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A direct design procedure of a full-order observer for a linear system with unknown inputs is presented, using straightforward matrix calculations. Some examples are given; in these examples a reduced-order observer is also derived. It is shown that this may restrict the rate of convergence of some state estimates
• Edwards C.