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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 fulfilment of the relati...

A framework merging the set-membership and the stochastic paradigms is formalized and used to design an Extended Zonotopic and Gaussian Kalman Filter (EZGKF) dealing with the robust state estimation and the fault detection of uncertain discrete-time nonlinear systems. The so-called Set-membership and Gaussian Mergers (SGM) are introduced and particularized to Zonotopes (ZGM). They provide a constructive and computationally efficient solution to propagate random uncertainties with incompletely specified probability distributions combining set-based support enclosures and upper covariance matrix bounds formalized as matrix inequalities. Based on a full time-varying LPV enclosure featuring structured state matrix uncertainties, and given some confidence level expressed in probabilistic terms (maximal false alarm rate), a detection test is developed and shown to merge the usually mutually exclusive benefits granted by set-membership techniques (robustness to the worst-case within specified bounds, domain computations) and stochastic approaches (taking noise distribution into account, probabilistic evaluation of tests). A numerical example illustrates the state estimation capabilities of EZGKF and the improved tradeoff between the sensitivity to faults and the robustness to disturbances/noises.

State bounding observation based on zonotopes is the subject of this paper. Dealing with zonotopes is motivated by set operations resulting in simple matrix calculations with regard to the often huge number of facets and vertices of the equivalent polytopes. Discrete-time LTV/LPV systems with state and measurement uncertainties are considered. Based on a new zonotope size criterion called F W -radius, and by merging optimal and robust observer gain designs, a Zonotopic Kalman Filter (ZKF) is proposed with a proof of robust convergence. The notion of covariation is introduced and results in an explicit bridge between the zonotopic set-membership and the stochastic paradigms for Kalman Filtering. No intersection is used and the influence of the reduction operator limiting to a tunable maximum the size of the matrices involved in the zonotopic set computations is fully taken into account in the LMI-based robust stability analysis. A numerical example illustrates the effectiveness of the proposed ZKF.

Recent advances in the design of interval observers have made it possible to ensure the non-divergence of the computed state bounds from the stability of LTI systems under bounded inputs, with no need for additional monotony assumptions. Time-varying changes of coordinates can be used to that purpose. Most of the related works result in either continuous-time or discrete-time interval dynamics. This paper proposes a constructive algorithm to compute the exact sampled response of a linear interval predictor under bounded inputs, gives a stability equivalence result and discusses the design of interval observers. The exact sampling requires held input bounds but the uncertain input itself needs not to be held. A numerical example exhibiting an oscillatory behavior illustrates the main results.

A dynamic interval approach for the fast computation of robust adaptive thresholds for a class of uncertain linear systems is the subject of this paper. An extension of recent results about the design of stable interval observers for linear systems with additive time-varying zonotopic input bounds is proposed. It allows the explicit computation of adaptive thresholds ensuring a guaranteed robustness with respect to structured and bounded disturbances which can be not only additive but also multiplicative. Moreover, the constant term (center) of the uncertain time-varying state matrix need not be diagonalizable thanks to results based on a Jordan decomposition. A sufficient condition not only ensuring the stability of the initial uncertain model but also the nondivergence of the computed adaptive thresholds is given. A numerical example dealing with fault detection in an electrical drive illustrates the proposed scheme.

Two usually distinct paradigms exist to model uncertainties. The stochastic one deals with random uncertainties. Strong assumptions about their probability distributions are often made, especially when online computation is required. For instance, assuming independent Gaussian distributions is common practice when using standard versions of the famous Kalman Filter. Though efficient to deal with measurement noises, the assumption of known probability distributions finds its limits to characterize epistemic uncertainties. Indeed, the lack of a precise knowledge about disturbances such as the load torque of a motor may be better characterized by interval bounds with no assumption about the values distribution. This is the usual motivation for using the set-membership paradigm to model uncertainties. Though explicitly computed sets can achieve a so-called guaranteed robustness to the worse cases resulting from the specified uncertainty bounds, the bounded-error paradigm however usually suffers from a poor management of random measurement noises. In this semi-plenary talk, a joint Zonotopic and Gaussian Kalman Filter (ZGKF) will be presented. It will be shown to provide a solution for the robust fault detection of uncertain discrete-time systems simultaneously subject to bounded disturbances and Gaussian noises. The covariation of a zonotope will be introduced as a set-membership analog to covariance, making it possible to compute a time-varying optimal observer gain jointly minimizing both kinds of uncertainties: bounded/zonotopic and Gaussian. Then, given a maximal probability of false alarms, an innovation-based detection test will be shown to merge the usually mutually exclusive benefits granted by set-membership techniques (robustness to the worse cases within specified bounds, domain computations) and stochastic approaches (taking noise distribution into account, probabilistic evaluation of tests). Numerical simulations will illustrate the significantly improved tradeoff between the sensitivity to faults and the robustness to disturbances/noises, while the computations (prediction/update, optimal gain, confidence domains, adaptive thresholds, detection test) remain explicit and can be efficiently implemented. Then, the new notion of Set-membership and Probabilistic Merger (SPM) will be introduced and particularized to Zono-topic and Gaussian Mergers (ZGM). The EZGKF algorithm, an extended version of ZGKF to non-linear filtering, will be also presented and discussed as an outline to future works.

This paper deals with fault diagnosis for a class of nonlinear continuous-time systems based on an interval extension of continuous-time parity equations. The parity equations are designed based on ordinary differential equations modeling some links between the successive output and input vector derivatives. The noisy inputs and outputs are evaluated through a high-order sliding-mode differentiator. The redundancy relations are formulated as a Constraint Satisfaction Problem (CSP) with faults as unknown variables. In this case, a reliable estimation of the fault magnitude can be performed by solving the CSP using interval techniques. This technique is well suited to deal with actuator faults as well as parametric or sensor faults. However, in this paper, we choose to focus on actuator faults and the performance of the proposed approach is illustrated through simulations on oscillatory failure detection in aircraft surface servo loops. input observer, fault reconstruction, consistency techniques, nonlinear systems.

This paper deals with a set-membership approach for the fast computation of robust adaptive thresholds. Dynamic model based fault detection under bounded additive and multiplicative disturbances is considered. Based on recent results related to the design of stable interval observers for linear systems with additive time-varying zonotopic input bounds, an extension guaranteeing the robustness with respect to structured and bounded multiplicative disturbances is proposed. A sufficient condition not only ensuring the stability of the initial uncertain model but also the non divergence of the computed bounds (i.e. adaptive thresholds) is also given. Then, a numerical example illustrates the proposed scheme.

The design of interval observers with proven stability for LTI systems was often subject to additional assumptions like monotone influences in the vector field. Recently, such restrictions have been overcome by using a time-varying change of variable. In this paper, a stable interval observer for stable (or observable) LTI systems with no multiple poles is proposed by using different mathematical tools and unified notations. Based on complex numbers, they provide an alternative understanding of the problem of designing stable interval observers. The results are applied to the verified model-based design of a vehicle suspension with an oscillatory behavior and bounded inputs modeling an uncertain road profile.

The computation of envelopes enclosing the possible states and/or outputs of a class of uncertain linear dynamical systems is the subject of this paper. The resulting algorithm can be useful in several areas of control systems such as verification of safety properties and fault diagnosis (in order to choose suitable thresholds on some residuals, for instance). A particular class of polytopes, zonotopes, can be used to implicitly represent the computed sets. The related reachability algorithms have shown to be well suited to control the wrapping effect in the case of linear dynamical systems. However, parametric uncertainties, when taken into account, are often modeled by interval matrices which lead to a loss of parametric dependencies and result in the computation of rather pessimistic sets. The main contribution of this paper consists in extending an existing algorithm based on zonotopes so that it can efficiently propagate structured parametric uncertainties. A 6th order oscillating mass-spring system illustrates the resulting control of the wrapping effect by comparison with Monte-Carlo simulations.

This paper deals with the design of guaranteed set-membership fault detection tests based on a continuous time linear dynamical system with parametric uncertainties. A guaranteed discretization scheme is proposed and, then, set-membership tests are designed based on the resulting uncertain discrete time model in which affine dependencies on the initial continuous time model parameters are preserved. The proposed scheme ensures a guaranteed robustness with respect to parametric uncertainties and ensures the logical consistency of the detection test results with the initial uncertain continuous time model.

International audience; Reachability analysis of nonlinear uncertain hybrid systems, i.e., continuous-discrete dynamical systems whose continuous dynamics, guard sets and reset functions are defined by nonlinear functions, can be decomposed in three algorithmic steps: computing the reachable set when the system is in a given operation mode, computing the discrete transitions, i.e., detecting and localizing when (and where) the continuous flowpipe intersects the guard sets, and aggregating the multiple trajectories that result from an uncertain transition once the whole flow-pipe has transitioned so that the algorithm can resume. This paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem. At the core of the method is the concept of MSPB, i.e., geometrical object obtained as the Minkowski sum of a parallelotope and an axes aligned box. MSPB are a way to control the over-approximation of the Taylor's interval integration method. As they happen to be a specific type of zonotope, they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process. The method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well-chosen hybrid nonlinear examples.

Set-membership approaches generally deal with bounded uncertainties. In this context, an observer aims at computing at each sample time a domain that is guaranteed to include the set of states which are consistent both with the uncertain model and with the uncertain measurements. Given an algorithm implementing the observation of uncertain linear systems, an extended observer can be used to deal with non-linear systems and/or to adapt some parameters. This paper focuses on such an extension. An application to a generic bioreactor illustrates the proposed approach and outlines some perspectives for future work.

An interval observer for Linear Time-Varying (LTV) systems is proposed in this paper. Usually, the design of such observers is based on monotone systems theory. Monotone properties are hard to satisfy in many situations. To overcome this issue, in a recent work, it has been shown that under some restrictive conditions, the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. However, a constructive method for the construction of the transformation matrix and the observer gain, making the observation error dynamics positive and stable, is still missing and remains an open problem. In this paper, a constructive approach to obtain a time-varying change of coordinates, ensuring the cooperativity of the observer error in the new coordinates, is provided. The efficiency of the proposed approach is shown through computer simulations.

International audience; In this paper, a method for computing reachable sets of uncertain nonlinear continuous-time systems is proposed. The approach, based on Linear Parameter-Varying (LPV) inclusions of nonlinear systems, relies on a time-varying change of coordinates which ensures some monotony properties in the new base. A sufficient condition ensuring the non divergence of the enclosures is given. The efficiency of the roposed technique is illustrated through computer simulations and it is compared with simulations resulting from a method proposed in previous works relying on comparison theorems for differential inequalities using MÜller's existence theorem.

International audience; Computing the reachable set of hybrid dynamical systems in a reliable and verified way is an important step when addressing verification or synthesis tasks. This issue is still challenging for uncertain nonlinear hybrid dynamical systems. We show in this paper how to combine a method for computing continuous transitions via interval Taylor methods and a method for computing the geometrical intersection of a flowpipe with guard sets, to build an interval method for reachability computation that can be used with truly nonlinear hybrid systems. Our method for flowpipe guard set intersection has two variants. The first one relies on interval constraint propagation for solving a constraint satisfaction problem and applies in the general case. The second one computes the intersection of a zonotope and a hyperplane and applies only when the guard sets are linear. The performance of our method is illustrated on examples involving typical hybrid systems.

International audience; In this paper an interval observer for Linear-Parameter-Varying (LPV) Systems is proposed. Usually, the design of such observers is based on monotone systems theory and the observation error should have a cooperative dynamics. In many cases, such a property is hard to satisfy. In this paper, a time-varying change of coordinates is used to overcome this limitation. Two sufficient conditions, according to the center and the radius dynamics of the proposed interval observer and ensuring the stability, are given. The efficiency of the proposed observer is illustrated through computer simulations.

This technical note deals with the design of stable interval observers and estimators for continuous-time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time-varying zonotopic bounds. No monotony assumption such as cooperativity is required in the vector field: the interval observer stability directly derives from the stability of the observer state matrix, where any poles (real or complex, single or multiple) are handled in the same way.

This paper deals with the design of experiments for the validation of a class of interval dynamic models. Set-membership algorithms based on interval analysis often allow the computation of guaranteed bounds (e.g. reach tubes, bounds for some estimates) enclosing all the possible scenarios according to some model where uncertainties are specified in a bounded error context. The guarantee of inclusion is very useful to ensure a complete coverage of all the specified scenarios in verification problems (e.g. verification of safety properties). However, such a guarantee and, consequently, the verified properties hold in practice only up to the validity of the considered uncertain model. In addition, the practical validation of dynamic interval models involving bounded uncertain inputs is quite difficult since finding a relevant input excitation leading to some worst-case scenario (e.g. an output reaching its maximum or minimum admissible value at a given time instant) is not a trivial task in general. The current paper proposes a constructive method to generate such worst-case input signals based on the guaranteed sampling of linear interval predictors with non-held uncertain inputs. The results are then illustrated through the example of designing worst-case road profiles to validate the interval model of a quarter vehicle suspension.

International audience; The off-line fault detection, isolation and identification of multiplicative actuator faults using sensors providing measurements affected by unknown nonlinear distortions and bounded noises are addressed. The only assumption about the unknown functions modeling the sensor distortions is their monotonicity. Bounded disturbances are also considered as unknown inputs of the system time-varying dynamics. Polytopic faulty parameter sets resulting from the search of robust output pairs and from zonotope computations are the basis for a decision scheme which makes use of multi-scale temporal windows. A numerical example modeling part of a distillation column illustrates the off-line diagnosis of three multiplicative actuator faults from a single low-cost noisy sensor with unknown nonlinear distortion and constant bias.

The design of exponentially stable interval observers is often subject to assumptions related to monotone influences in the vector field. This may lead to significant restrictions, especially when the response of systems exhibiting an oscillatory behavior should be over-approximated under uncertain inputs. In order to overcome this issue, a stable interval observer for LTI systems with bounded inputs is proposed. It mainly relies on the assumption that the initial system is stable with no multiple poles. Any real or complex stable pole is authorized. The resulting exponentially stable interval observer is then applied to the detection of inconsistencies for the verified design of a vehicle suspension.

The paper deals with the computation of the reachable (state or output) set of a discrete-time linear dynamical system with bounded input and bounded slew rate. The dependency relations between uncertain variables are shown to be a key point in the design of set-membership algorithms and justify the use of zonotopes as domains for linear systems. It allows not only to limit the wrapping effect but also to extend the problem formulation with additional constraints such as bounded input with bounded slew-rate. Some simulations illustrate the interest of the approach to limit the pessimism of the computed sets.

The proposed fault diagnosis scheme relies on a residual generation based on an adaptive observer covering linear time varying (LTV), linear parameter varying (LPV), and state-affine non-linear systems, all with bounded uncertainties. A residual evaluation is then performed by set-membership computations based on zonotopes (polytopes defined as the image of a hypercube by a linear application). The main advantage of the approach is its rigorous computation of the propagation of pre-specified modelling uncertainty bounds. Within the assumed uncertainty bounds, fault detection is guaranteed to be free of false alarm, while not being too much conservative, as illustrated on the model of a satellite. Copyright © 2006 IFAC.

A state bounding observer aims at computing some domains which are guaranteed to contain the set of states that are consistent both with the uncertain model and with the uncertain measurements. In this paper, the estimated domains are represented by zonotopes. A zonotope is a particular polytope defined as the linear image of a unit interval vector (i.e. unit hypercube). Some results about the validated integration of ordinary differential equations are used to guarantee the inclusion of sampling errors. The main loop of the observation algorithm consists of a one step prediction with a limitation of the domain complexity and a correction using the measurements. The observer is applied to a Lotka-Volterra predator-prey model.

The proposed observer computes an outer approximation of the set of states which are consistent with a given uncertain state space model and some measurements. The uncertainties are modeled by unknown but bounded inputs. A representation of domains by zonotopes (particular polytopes) is used to reduce the computation of state bounds to rather simple matrix operations and to control the wrapping effect.