Email addresses: Qinghua.Zhang@inria.fr (Qinghua
Zhang), Michele.Basseville@irisa.fr (Mich` ele
application to some particular nonlinear systems. An- other approach to dealing with nonlinear systems uses linearization along the actual or nominal trajectory of the monitored system. Such a linearization generally leads to linear time-varying (LTV) systems, whereas the more classical LTI approximation is usually related to the linearization around a single working point. It is thus clear that methods for FDI in LTV systems are much more powerful than their LTI counterparts. Finally, non- linear control systems have been widely studied with the linearparametervarying (LPV) approach; see, e.g., [7, 37, 44]. Since a LPV system is essentially a particu- lar LTV system, the FDI method proposed in this paper for general LTV systems is also valid for LPV systems. FDI issues for LTV systems have been addressed using three main approaches known as fault detection filters, observers, and parity relations, as shortly recalled in the following. Even though the first and third approaches are known to be equivalent , and the third one may be seen as a particular case of the second one in the case of LTI systems (see for example ), it is useful to distinguish those three types of investigations.
III. N EW M ODEL T RANSFORMATION FOR D ELAY -D EPENDENT S TABILITY A NALYSIS
This section introduces a new model transformation which turns a time-delay system with time-varying delays into an uncertain LPV system represented in an “LFT” form. This transformation allows to use classical robust stability analysis and control synthesis on the trans- formed system, in order to derive a delay-dependent stability test ob- tained from the scaled-bounded real lemma. Similar approaches can be found for instance in  where the maximal value of the delay ap- pears explicitly in the comparison model. In this paper, the comparison model is an uncertain parametervarying system which is then studied in the robust/LPV framework. This is a real novelty in the analysis and control of time-delay systems with time-varying delays.
Keywords: Linearparameter-varyingsystems; Systems with time-delays; Robust linear matrix inequalities
Since several years, time-delay systems ([Moon et al., 2001, Niculescu, 2001, Zhang et al., 2001, Gu et al., 2003, Gouaisbaut and Peaucelle, 2006b, Fridman, 2006, Suplin et al., 2006]) have suggested more and more interest because of their destabilizing effects and performances deterioration. In high-speed systems, even a small time- delay may have a high effect and cannot be neglected, that is why specific stability tests and adapted controller designs must have been developed. Time-delay models ap- pear in various problems as chemical processes, population growth. . . Since the advent of networks and Network Con- trolled Systems appears the necessity of studying systems with time-varying delays.
Zk = J (θ0) T Σ(θ0) −1 vec K T (θ 0 )U 1,k
(14) where U1,k is the current left subspace of the Hankel ma- trix at the k-th sample data y k , obtained by an update of the SVD. Based on the recursive residual in (14) and con- sidering that the Z k ’s are independent Gaussian which de- scribe themselves a change in θ by a change in their mean, a commonly used CUSUM test (Basseville and Nikiforov ) is applied. In general, only the part Λ of θ changes with respect to time. The eigenvectors are assumed to be constant, considering the small displacement hypothesis (SDH). Now, denote ρ as the norm of one of the eigenvalues λ’s of F and notice that when one of the ρ’s increases to 1, the system goes toward instability (Basseville and Nikiforov ). The CUSUM test described below allows to detect an increase in any component of the parameter by using as Jacobian matrix the corresponding submatrix of J (θ0) : the sensitivity J (θ0) is replaced in (14) by the column J (ρ0) corresponding to the sensitivity with respect to ρ at ρ equal to ρ0: Sk(ρ0) = Σ(ρ0) −1/2 Pk j=q Zj(ρ0), Tk(ρ0) = minq≤j≤k Sj (ρ0) and gk (ρ0) = Sk(ρ0) − Tk(ρ0). The two hypotheses H 0,k and H 1,k to decide between, at each instant k, are now:
with the initial conditions
v ( 0 ) = x ( 0 ) as well as w ( 0 ) = x ( 0 ) . (11)
Transformations of linear state equations which enforce cooperativity can be classified into either point-valued approaches or into techniques that employ an interval-valued change of coordinates [ 28 ]. In general, point-valued transformations are applicable for parameter-dependent linearsystems with purely real eigenvalues. However, finding such transformations becomes more complex for an increasing degree of uncertainty in the system matrix. A failure of the similarity transformation approach by Raïssi et al. [ 1 , 2 ] can be recognized by the alternative formulation proposed by the authors in [ 29 ]. There, the transformation was cast into an optimization problem constrained by linear matrix inequalities (LMIs). Those LMIs may become infeasible (as a verification that the solution procedure is not successful) or the step size of the iteration in [ 29 ] reduces below a certain threshold leading to an excessively slow progress (a non-strict indicator that a point-valued similarity transformation may not exist). Even if interval-valued Metzler matrices can be found for the transformation of a provable asymptotically stable state equation, the transformed dynamics matrix may consist of an unstable upper bound. This commonly results from the wrapping effect in interval computations [ 30 – 32 ]. Using the aforementioned approaches, it is impossible to systematically avoid this phenomenon. However, it can be countered by the subdivision procedure introduced in this paper.
Abstract: Most subspace-based methods enabling instability monitoring are restricted to the linear time-invariant (LTI) systems. In this paper, a new subspace method of instability monitoring is proposed for the linear periodically time-varying (LPTV) case. For some LPTV systems, the system transition matrices may depend on some parameter and are also periodic in time. A certain range of values for the parameter leads to an unstable transition matrix. Early warning should be given when the system gets close to that region, taking into account the time variation of the system. Using the theory of Floquet, some symptom parameter of stability- or residual- is defined. Then, the parameter variation is tracked by performing a set of parallel cumulative sum (CUSUM) tests. Finally, the method is tested on a simulated model of a helicopter with hinged blades, for monitoring the ground resonance phenomenon.
Parameter and state estimation of switched affine systems
Jamal Daafouz, Romain Postoyan, Pierre Riedinger
Abstract— We investigate the on-line parameter and state estimation of switched affine systems. We first adapt standard techniques available for affine (time-varying) systems to the considered class of systems, which leads to new linear matrix inequality-based conditions for the design of the estimator gains. We then explain how to derive decoupled estimators in the sense that the state and the parameter estimates are generated by cascaded switched differential equations, which is numerically more efficient and thus easier to implement in practice. A first solution is presented, which generates state estimates with jumps at the switching instants. To overcome this potential issue, two “bumpless” strategies are proposed. The first one relies on a persistency of excitation (PE) condition, while the the second one applies to a given class of switching rules and avoids the PE condition. Both estimation strategies ensure the global exponential convergence to zero of the estimation errors. The results are illustrated on a model describing the behavior of a DC-DC power converter.
• l = q: the minimum-order observer or minimum observer.
With ( Roman & Bullock, 1975 ; Sirisena , 1979 ), we know that
l is a lower bound for the order of the observer (3) .
Until now the direct design of a minimal observer of a given linear functional is an open question. Since ( Fortmann & Williamson, 1972 ), design schemes have been proposed to reduce the order of the observer (3) with respect to the reduced-order observer. Mainly, these designs are based on the determination of the matrices T and F such that the Sylvester equation (4) is fulfilled ( Trinh, Nahavandi, & Tran, 2008 ; Tsui , 2004 ). Unfortunately, the problem rests in satisfying conditions (4)–(6) with the dimension
The triple (H, E, µ) is an abstract Wiener space and H is referred as Cameron-Martin space of
the process µ. We present now two results concerning abstract Wiener spaces (AWS), the first
stating that new abstract Wiener spaces can be easily constructed from others that already exist.
Theorem 2.2 ( [ 42 ]). Let H and H 0 be two separable Hilbert spaces and F a linear isometry from H to H 0 . Assume that an AWS (H, E, µ) is given. Then, there exists a Banach space E 0 ⊃ H 0 and a linear isometry ˜ F : E → E 0 whose restriction to H is F and (H 0 , E 0 , ˜ F ∗ µ) is an AWS ( ˜F ∗ µ denotes the push-forward measure of µ by ˜F).
Somecalculations point out that (5) and (4) are verified. When F is a Hurwitz matrix, the necessary and sufficient conditions for the existence of a single functional observer of the single linear functional (1) for the system (2) are fulfilled. To prove minimality let us consider that there exists a p-observer solving the same problem with p < q. Then there exist matrices such that
LIRMM, Université de Montpellier, CNRS, France email@example.com
The graph parameter of pathwidth can be seen as a measure of the topological resemblance of a graph to a path. A popular definition of pathwidth is given in terms of node search where we are given a system of tunnels (represented by a graph) that is contaminated by some infectious substance and we are looking for a search strategy that, at each step, either places a searcher on a vertex or removes a searcher from a vertex and where an edge is cleaned when both endpoints are simultaneously occupied by searchers. It was proved that the minimum number of searchers required for a successful cleaning strategy is equal to the pathwidth of the graph plus one. Two desired characteristics for a cleaning strategy is to be monotone (no recontamination occurs) and connected (clean territories always remain connected). Under these two demands, the number of searchers is equivalent to a variant of pathwidth called connected pathwidth. We prove that connected pathwidth is fixed parameter tractable, in particular we design a 2 O(k 2 ) · n time algorithm that checks whether the connected pathwidth of G is at most k. This resolves an open question by [Dereniowski, Osula, and Rzążewski, Finding small-width connected path-decompositions in polynomial time. Theor. Comput. Sci., 794:85–100, 2019 ]. For our algorithm, we enrich the typical sequence technique that is able to deal with the connectivity demand. Typical sequences have been introduced in [Bodlaender and Kloks. Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms, 21(2):358–402, 1996 ] for the design of linear parameterized algorithms for treewidth and pathwidth. While this technique has been later applied to other parameters, none of its advancements was able to deal with the connectivity demand, as it is a “global” demand that concerns an unbounded number of parts of the graph of unbounded size. The proposed extension is based on an encoding of the connectivity property that is quite versatile and may be adapted so to deliver linear parameterized algorithms for the connected variants of other width parameters as well. An immediate consequence of our result is a 2 O(k 2 ) · n time algorithm for the monotone and connected version of the edge search number.
The model with its assumptions is described in Section 3.1 , and the resolution of conflicts is addressed through the main results of this paper in Section 3.2 . Two simple conditions of robustness are given. Modeling with a log-regularly varying distribution is the first. In the second condition, the number of nonoutlying ob- servations must be larger than the maximum between the number of small and large outliers. Results of robustness are asymptotic, where the outlying observa- tions tend to −∞ or +∞. Note that the asymptotic nature is about the outliers and not the sample size, as is usually understood. Whole robustness is expressed through different types of convergence of quantities, based on the complete sam- ple, to quantities based only on the nonoutlying observations, resulting in a com- plete rejection of outliers. We obtain the uniform convergence of the posterior densities, the convergence in L 1 , the convergence in distribution and the uniform
conditions of the system. The case of a flexible beam is studied in simulation as well as in laboratory experimentation. Hence, a suitable application of the proposed LTV synthesis is done to find a compromise between the complexity of interpolation and its efficiency. Furthermore, the bad dissipation factor of flexible systems leads them to be good candidates to proof the efficiency of the proposed gain scheduling strategy. This work validates on a real system the LTV pole placement approach and opens the way to a new gain scheduling strategy for the control of LTV, nonlinear or infinite dimensional systems. Key words: linear time-varyingsystems, flexible structures, vibrations at- tenuation
Ecole Centrale de Nantes - IRCCyN UMR CNRS 6597 Nantes, France
Abstract—We provide a subclass of parametric timed au- tomata (PTA) that we can actually and efficiently analyze, and we argue that it retains most of the practical usefulness of PTA for the modeling of real-time systems. The currently most useful known subclass of PTA, L/U automata, has a strong syntactical restriction for practical purposes, and we show that the associated theoretical results are mixed. We therefore advocate for a different restriction scheme: since in classical timed automata, real-valued clocks are always compared to integers for all practical purposes, we also search for parameter values as bounded integers. We show that the problem of the existence of parameter values such that some TCTL property is satisfied is PSPACE-complete. In such a setting, we can of course synthesize all the values of parameters and we give symbolic algorithms, for reachability and unavoidability properties, to do it efficiently, i.e., without an explicit enumeration. This also has the practical advantage of giving the result as symbolic constraints between the parameters. We finally report on a few experimental results to illustrate the practical usefulness of our approach.
c Univ. de Toulouse, UPS, LAAS, F-31400, Toulouse, France
This paper is concerned with stability of a linear system with a time-varying delay. First, an optimal reciprocally convex inequality is proposed. Compared with the extended reciprocally convex inequality recently reported, the optimal reciprocally convex inequality not only provides an optimal bound for the reciprocally convex combination, but also introduces less slack matrix variables. Second, a new Lyapunov-Krasovskii functional is tailored for the use of auxiliary function-based integral inequality. Third, based on the optimal reciprocally convex inequality and the new Lyapunov-Krasovskii functional, a stability criterion is derived for the system under study. Finally, two well-studied numerical examples are given to show that the obtained stability criterion can produce a larger upper bound of the time-varying delay than some existing methods.
Keywords: Time-delay systems, integral inequalities, matrix inequality, reciprocally convex lemma.
This paper aims at providing less conservatism and com- putationally efficient stability conditions for linearsystems subject to fast-varying delays. This topic of research has attracted many researchers over the past decades. The main difficulties for the study of such a class of systems rely on two technical steps that are the derivation of efficient integral and matrix inequalities. Indeed, the differentiation of usual candidates for being Lyapunov-Krasovskii func- tionals leads to integral quadratic terms that cannot be included straightforwardly in a linear matrix inequality (LMI) setup. Including these terms requires the use of integral inequalities such as Jensen (see for instance Gu (2000)), Wirtinger-based provided in Seuret and Gouais- baut (2013), auxiliary-based from Park et al. (2015) or Bessel inequalities developed in Seuret and Gouaisbaut (2015). Althrough these inequalities have shown a great interest for constant delay systems, their application to time- or fast-varying delays leads to additional difficul- ties related to the non convexity of the resulting terms. Then, some matrix inequalities are employed to derive convex conditions. The first method corresponds to the application of Young’s inequality or Moon’s inequality, which basically results from the positivity of a square positive definite term. It can also be noted that the re- cent free-matrix inequality from Zeng et al. (2015) can be interpreted as the merge of the Wirtinger-based inequality and Moon’s inequality. Recently, the reciprocally convex lemma was proposed in Park et al. (2011). The novelty
T A − F T = HC (6) L = P T + V C (7) F is a Schur matrix. (8) Note that L can always be chosen to be a controller gain that stabilizes the closed-loop system matrix (A + BL), then the linear functional observer (3) would provide an estimate of the corresponding control signal to be directly feedback into the system. On the other hand, the designer can always chose L to represent any desired partial set of the state vector that needs to be estimated.