robust fault reconstruction in uncertain linear systems using multiple sliding mode observers in cascade in observer based fault reconstruction one of the necessary conditions is that the first markov parameter from the fault to the output must be full rank this paper seeks to relax that requirement by using multiple sliding mode observers in cascade signals from an observer are used as the output of a fictitious system whose input is the fault another observer is then designed and implemented for the fictitious system this process is repeated until the first markov parameter of the fictitious system with respect to the fault is full rank the result is that robust fault reconstruction can be carried out for a wider class of systems compared to other works that also seek to relax the requirement of a full rank first markov parameter in addition this paper has also investigated and presented the necessary and sufficient conditions as easily testable conditions and also the precise number of observers required a simulation example verifies the effectiveness of the scheme

robust exact differentiation via sliding mode technique abstract 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

systems of coupled generalized sylvester matrix equations this paper studies some systems of coupled generalized sylvester matrix equations we present some necessary and sufficient conditions for the solvability to these systems we give the expressions of the general solutions to the systems when their solvability conditions are satisfied

tracking nonlinear non minimum phase systems using sliding control nonlinear systems affine in the input and having a well defined vector relative degree are considered the invertibility of the input output dynamics can be used to achieve tracking of smooth desired trajectories if the associated zero dynamics are stable we consider tracking in nonlinear systems with unacceptable zero dynamics i e non minimum phase systems the desired trajectories are assumed to be generated by some exosystem we use sliding control to achieve tracking independent of disturbances entering in the channels of the input the main idea is i to do an output redefinition such that the zero dynamics with respect to this new output are acceptable ii to define a modified desired trajectory for the new output to track such that in the process the original output tracks the original desired trajectory asymptotically

on groups that have normal forms computable in logspace abstract we consider the class of finitely generated groups which have a normal form computable in logspace we prove that the class of such groups is closed under passing to finite index subgroups direct products wreath products and certain free products and infinite extensions and includes the solvable baumslag solitar groups as well as non residually finite and hence non linear examples we define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace we prove that finitely generated nilpotent groups are logspace embeddable it follows that all groups of polynomial growth are logspace embeddable

the word and geodesic problems in free solvable groups we study the computational complexity of the word problem wp in free solvable groups s r d where r 2 is the rank and d 2 is the solvability class of the group it is known that the magnus embedding of s r d into matrices provides a polynomial time decision algorithm for wp in a fixed group s r d unfortunately the degree of the polynomial grows together with d so the uniform algorithm is not polynomial in d in this paper we show that wp has time complexity o rn log 2 n in s r 2 and o n 3 rd in s r d for d 3 however it turns out that a seemingly close problem of computing the geodesic length of elements in s r 2 is np complete we prove also that one can compute fox derivatives of elements from s r d in time o n 3 rd in particular one can use efficiently the magnus embedding in computations with free solvable groups our approach is based on such classical tools as the magnus embedding and fox calculus as well as on relatively new geometric ideas in particular we establish a direct link between fox derivatives and geometric flows on cayley graphs

growth series of some wreath products the growth series of certain finitely generated groups which are wreath products are investigated these growth series are intimately related to the traveling salesman problem on certain graphs a large class of these growth series is shown to consist of irrational algebraic functions n 0 since only one generating set will be associated with each group below the generating set associated with fr x will be obvious let h be a group with a finite generating set sh these will be fixed for the rest of the paper the cayley graph of the pair h sh is as usual the directed graph whose vertices are the elements of h and there is an edge from a vertex hx xoa vertex h2 if and only if h2 hxh for some h in sh us 1 in particular the edge from hx to h2 has an opposite edge from h2 to hx let c be the graph gotten from the cayley graph of h sh simply by identifying opposite edges in other words c might be called the undirected cayley graph of h sh hex k also be a group with a finite generating set sk these will also be fixed for the rest of the paper it is possible to form what might be called a restricted direct product group p which consists of all functions p from the vertices of c to k such that there are only finitely many vertices v in c with p v t 1 this is a subgroup of the direct product group whose elements consist of all functions from the vertex set of c to k the group p admits h as a group of automorphisms by means of the action of h on c the resulting semidirect product p x h is the restricted wreath product k h

optimal differentiation based on stochastic signal models the problem of estimating the time derivative of a signal from sampled measurements is addressed the measurements may be corrupted by colored noise a key idea is to use stochastic models of the signal to be differentiated and of the measurement noise two approaches are suggested the first is based on a continuous time stochastic process as a model of the signal the second uses a discrete time arma model of the signal and a discrete time approximation of the derivative operator digital differentiators are presented in a shift operator polynomial form they minimize the mean square estimation error and are calculated from a linear polynomial equation and a polynomial spectral factorization the three obstacles to perfect differentiation namely a finite smoothing lag measurement noise and aliasing effects due to sampling are discussed

semigroup automata with rational initial and terminal sets we consider a natural extension of the usual definition of m automata also known as extended automata or valence automata which permits the automaton to utilise more of the structure of each monoid and additionally allows us to define s automata for s an arbitrary semigroup in the monoid case the resulting automata are equivalent to the valence automata with rational target sets which arise in the theory of regulated rewriting systems we study these automata in the case where the register semigroup is completely simple or completely 0 simple obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups in the process we obtain a number of results about rational subsets of rees matrix semigroups which are likely to be of independent interest

solvability conditions and general solution for mixed sylvester equations in this paper we give some necessary and sufficient solvability conditions for the mixed sylvester matrix equations and parameterize general solution when it is solvable moreover we investigate the maximal and minimal ranks of the general solution and maximal and minimal ranks and inertias of hermitian part of solution respectively

flow estimation of boundary layers using dns based wall shear information this article investigates the problem of obtaining a state space model of the disturbance evolution that precedes turbulent flow across aerodynamic surfaces this problem is challenging since the flow is governed by nonlinear partial differential algebraic equations for which there currently exists no efficient controller estimator synthesis techniques a sequence of model approximations is employed to yield a linear low order state space model to which standard tools of control theory can be applied one of the novelties of this article is the application of an algorithm that converts a system of differential algebraic equations into one of ordinary differential equations this enables straightforward satisfaction of boundary conditions whilst dispensing with the need for parallel flow approximations and velocity vorticity transformations the efficacy of the model is demonstrated by the synthesis of a kalman filter that clearly reconstructs the characteristic features of the flow using only wall vel

unknown input and state estimation for unobservable systems the concept of strong detectability and its relation with the concept of invariant zeros is reviewed for strongly detectable systems which includes the strongly observable systems it is proposed a hierarchical design of a robust observer whose trajectories converge to those of the original state vector furthermore it is shown that neither left invertibility is a sufficient condition nor strong detectability is a necessary condition to estimate the unknown inputs it is shown that the necessary and sufficient condition for estimating the unknown inputs is that the set of the invariant zeros that do not belong to the set of unobservable modes be within the interior of the left half plane of the complex space this shows that the unknown inputs could be estimated even if it is impossible to estimate the entire state vector of the system two numerical examples illustrate the effectiveness of the proposed estimation schemes

discrete differentiators based on sliding modes abstract sliding mode based differentiation of the input f t yields exact estimations of the derivatives f f n provided an upper bound l t of f n 1 t is available in real time in practice it involves discrete sampling and numerical integration of the internal variables between the measurements accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling whereas sampling and integration steps are independently variable or constant proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous time counterparts event triggered sampling is considered extensive numeric experiments are presented and analyzed

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

cone types and geodesic languages for lamplighter groups and thompson s group f we study languages of geodesics in lamplighter groups and thompson s group f we show that the lamplighter groups ln have infinitely many cone types have no regular geodesic languages and have 1 counter context free and counter geodesic languages with respect to certain generating sets we show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context free while with respect to another generating set the full language of geodesics is counter and context free in thompson s group f with respect to the standard finite generating set we show there are infinitely many cone types and that there is no regular language of geodesics we show that the existence of families of seesaw elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set

generalized conjugate direction method for solving a class of generalized coupled sylvester conjugate transpose matrix equations over generalized hamiltonian matrices abstract in this paper a generalized conjugate direction gcd method for finding the generalized hamiltonian solutions of a class of generalized coupled sylvester conjugate transpose matrix equations is proposed furthermore it is proved that the algorithm can compute the least frobenius norm generalized hamiltonian solution group of the problem by choosing a special initial matrix group within a finite number of iterations in the absence of round off errors numerical examples are also presented to illustrate the efficiency of the algorithm

on parameter and state estimation for linear differential algebraic equations the current demand for more complex models has initiated a shift away from state space models towards models described by differential algebraic equations daes these models arise as the natural product of object oriented modeling languages such as modelica however the mathematics of daes is somewhat more involved than the standard state space theory the aim of this work is to present a well posed description of a linear stochastic differential algebraic equation and more importantly explain how well posed estimation problems can be formed we will consider both the system identification problem and the state estimation problem besides providing the necessary theory we will also explain how the procedures can be implemented by means of efficient numerical methods

higher rank lamplighter groups are graph automatic abstract we show that the higher rank lamplighter groups or diestel leader groups d q for d 3 are graph automatic this introduces a new family of graph automatic groups which are not automatic

homogeneous time varying systems robustness analysis the problem of stability robustness with respect to time varying perturbations of a given frequency spectrum is studied applying homogeneity framework the notion of finite time stability over time intervals of finite length i e short finite time stability is introduced and used for that purpose the results are applied to analyze the super twisting algorithm sta behavior under time varying perturbations some simulation examples illustrate these robustness properties

an improved second order sliding mode control scheme robust against the measurement noise in this note the second order sliding mode control problem is tackled by explicitly taking into account the presence of measurement error with unknown upper bound spl delta under sensible assumptions regarding the uncertain dynamics of a broad class of nonlinear plants a new switching controller is proposed guaranteeing a sliding accuracy of order o spl delta simulations highlight the robustness and good performance of the proposed approach and confirm the expected precision order

a solution to the generalized problem of nonlinear map inversion abstract this paper deals with the generalization of some problems related to map inversion solutions of such problems are proposed by means of observers for which a structure is proposed

nonlinear extremum seeking inspired on second order sliding modes we present a new extremum seeking algorithm that uses super twisting like nonlinear terms in order to improve its precision in addition the algorithm is combined with second order sliding mode observers to build an optimization framework for uncertain dynamic systems in which the variable to optimize is not measured and needs to be estimated the relevance of the proposed framework is illustrated in the problem of biomass optimization of a biological reactor with uncertain kinetics

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

super twisting algorithm in presence of time and state dependent perturbations abstractin this paper generalized super twisting algorithm is applied to a class of systems whose perturbation and uncertain control coefficient are time and state dependent a non smooth strict lyapunov function is used to obtain the conditions for the global finite time stability and to estimate the time of convergence

finite time convergent differentiator based on singular perturbation technique a finite time convergent differentiator is presented that is based on singular perturbation technique the merits of this differentiator exist in three aspects rapidly finite time convergence compared with other typical differentiators no chattering phenomenon and besides the derivatives of the derivable signals the generalized derivatives of some classes of signals can be obtained for example the generalized derivative of a triangular wave is square wave etc the theoretical results are confirmed by computer simulations

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

a context free and a 1 counter geodesic language for a baumslag solitar group we give a language of unique geodesic normal forms for the baumslag solitar group bs 1 2 that is context free and 1 counter we discuss the classes of context free 1 counter and counter languages and explain how they are inter related

tracking controller design for a class of nonminimum phase systems via the method of system center a method of approximate output tracking for a class of causal nonminimum phase multiple input multiple output mimo systems with nonlinear time varying uncertainties and unmeasurable disturbances is considered the nonminimum phase output tracking problem is transformed into a state tracking problem the technique presented employs linear algebraic methods and the internal model principle a sliding mode control approach is used in the example to illustrate a complete constructive algorithm

a new varying gain exponent based differentiator observer an efficient balance between linear and sliding mode algorithms it is well known that the supertwisting algorithm is robust to matched perturbation but is sensitive to measurement noise contrary to this the classical linear algorithm is less sensitive to measurement noise but less robust to perturbation to combine both the good accuracy of the supertwisting algorithm with respect to perturbation and the good performance of the linear algorithm with respect to measurement noise this article proposes a new differentiator observer with a varying exponent gain alpha whose variation depends on the magnitude of measurement noise high frequency signal when the magnitude of measurement noise increases respectively decreases alpha tends to 1 respectively tends to 0 5 and the proposed differentiator observer behaves as a linear algorithm respectively as a supertwisting algorithm thus by one parameter alpha the differentiator observer can take care of high frequency noise and matched perturbations a complete stability analysis of the proposed differentiator observer is provided to highlight the applicability of the proposed methodology the dedicated differentiator observer is respectively validated on the electropneumatic actuator and electric machine test benches these experimental results are compared to those of linear and supertwisting algorithms

word problems recognisable by deterministic blind monoid automata we consider blind deterministic finite automata equipped with a register which stores an element of a given monoid and which is modified by right multiplication by monoid elements we show that for monoids m drawn from a large class including groups such an automaton accepts the word problem of a group h if and only if h has a finite index subgroup which embeds in the group of units of m in the case that m is a group this answers a question of elston and ostheimer

nonlinear output tracking in conventional and dynamic sliding manifolds nonlinear output tracking in multi input multi output mimo systems with unmatched nonlinearities and disturbances is considered in sliding modes the system is nonlinearly transformed to a form convenient for sliding mode synthesis both conventional and dynamic sliding mode controllers are designed to provide the desired nonlinear output tracking in both cases only the time dependent part of the unmatched disturbance must be estimated an example is provided to demonstrate the new design approach results show that both conventional and dynamic sliding mode control are insensitive to matched disturbances but that dynamic sliding mode control accommodates to unmatched disturbances

easy multiplications i the realm of kleene s theorem abstract we study here the properties of a family of monoids which we call the rational monoids and which are monoids with a multiplication of low complexity a monoid is rational if its multiplication may be described by a rational function from a free monoid into itself the main results are that rational monoids like finite ones have the properties that green s relations d and f are equal and that kleene s theorem holds in rational monoids as in free ones every monoid described so far in which kleene s theorem holds is a rational monoid the closure of the family of rational monoids under rees quotient direct product and free product is then studied extensions of rational monoids will be considered in a forthcoming paper

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

on groups whose word problem is solved by a counter automaton we prove that a group g has a word problem that is accepted by a deterministic counter automaton with a weak inverse property if and only if g is virtually abelian we extend this result to larger classes of groups by considering a generalization of finite state automata counter automata and pushdown automata natural corollaries of our general result include a restricted version of herbst s classification of groups for which the word problem is a one counter language and a new classification of automata that accept context free word problems

an efficient algorithm for the solution of a coupled sylvester equation appearing in descriptor systems descriptor systems consisting of a large number of differential algebraic equations daes usually arise from the discretization of partial differential algebraic equations this paper presents an efficient algorithm for solving the coupled sylvester equation that arises in converting a system of linear daes to ordinary differential equations a significant computational advantage is obtained by exploiting the structure of the involved matrices the proposed algorithm removes the need to solve a standard sylvester equation or to invert a matrix the improved performance of this new method over existing techniques is demonstrated by comparing the number of floating point operations and via numerical examples

constraint generalized sylvester matrix equations in this paper some necessary and sufficient conditions are established for the constraint generalized sylvester matrix equations to have a common solution the expression of the general common solution is also given under the solvable conditions in addition a numerical example is presented to illustrate the presented theory

necessary conditions for asymptotic tracking in nonlinear systems in the literature it has been shown that if a single input single output analytic nonlinear plant 1 has a well defined relative degree and 2 is minimum phase it is possible to achieve asymptotic tracking for an open set of output trajectories containing the origin in c sup n 0 spl infin the space of n times continuously differentiable functions taking values in r when either of these sufficient conditions is not met various authors have investigated approximate analytic solutions discontinuous solutions and solutions for restricted sets of trajectories in this paper it is shown that conditions 1 and 2 are necessary for the existence of an analytic compensator which yields asymptotic tracking for an open set of output trajectories analogous results are established for multi input multi output systems

the theory of ends pushdown automata and second order logic abstract a class of edge labeled graphs called context free are defined according to their behavior at infinity such graphs are generalizations of cayley graphs of context free groups they are also shown to be definable in a very natural way in terms of push down automata using rabin s theorem on the monadic second order theory of the finite binary tree these graphs are also shown to have a decidable monadic second order theory questions about tiling systems and cellular automata operating on these graphs are decidable even when the analogous questions are not when the systems operate on a two dimensional grid

sequential grammars and automata with valances we discuss the model of valence grammars a simple extension of context free grammars we show closure properties of context free valence languages over arbitrary monoids chomsky and greibach normal form theorems and an iteration lemma for context free valence grammars over the groups zk are proved the generative power of different control monoids is investigated in particular we show that context free valence grammars over finite monoids or commutative monoids have the same power as valence grammars over finite groups or commutative groups respectively

on the rational subset problem for groups we use language theory to study the rational subset problem for groups and monoids we show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups in particular it passes through free products amalgamated over finite subgroups and hnn extensions with finite associated subgroups we provide a simple proof of a result of grunschlag showing that the decidability of this problem is a virtual property we prove further that the problem is decidable for a direct product of a group g with a monoid m if and only if membership is uniformly decidable for g automaton subsets of m it follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership

c graph automatic groups abstract we generalize the notion of a graph automatic group introduced by kharlampovich khoussainov and miasnikov by replacing the regular languages in their definition with more powerful language classes for a fixed language class c we call the resulting groups c graph automatic we prove that the class of c graph automatic groups is closed under change of generating set direct and free product for certain classes c we show that for quasi realtime counter graph automatic groups where normal forms have length that is linear in the geodesic length there is an algorithm to compute normal forms and therefore solve the word problem in polynomial time the class of quasi realtime counter graph automatic groups includes all baumslag solitar groups and the free group of countably infinite rank context sensitive graph automatic groups are shown to be a very large class which encompasses for example groups with unsolvable conjugacy problem the grigorchuk group and thompson s groups f t and v

sliding mode based differentiation and filtering the proposed k th order filter is based on sliding modes sms it exactly estimates k input derivatives in the absence of noises and is robust to noises of small magnitudes or having small average values estimation accuracy asymptotics are calculated the filter is applied to the real time accurate estimation of the equivalent control in sm control systems