design of state detectors for nonlinear systems using symmetries and semi invariants two techniques are proposed for the design of asymptotic state detectors for nonlinear systems both in the continuous time and discrete time cases the first technique allows one to design linear state detectors for nonlinear systems belonging to a given class which is studied and characterized in the paper the second technique extends the ideas of the first by using semi invariants to compute invariant sets and study their stability

observer design for nonlinear control systems abstract this paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities a sufficient liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case

multiple augmentations of nonlinear systems and generalized minimum rank perturbations for damage detection a generalized minimum rank perturbation theory for identifying damage location and extent in nonlinear systems using a model based approach is presented this model updating method is able to handle changes in the mass damping and stiffness parameters arising from the damage the method uses a nonlinear discrete model of the system and the functional form of the nonlinearities to create an augmented linear model of the system a modal analysis technique that uses forcing that is known but not prescribed is then used to solve for the modal properties of the augmented linear system after the onset of damage the methodology has been demonstrated for cubic spring nonlinearities in this work the class of nonlinearities is expanded to include coulomb friction several new algorithms are presented including an iterative generalized minimum rank perturbation theory and a technique based on multiple augmentations to determine damages in linear and nonlinear parameters when there is an incomplete set of eigenvectors finally two eigenvector filtering algorithms which reduce the effects of measurement noise are presented

hybrid dead beat observers for a class of nonlinear systems this paper studies both the full order and the reduced order dead beat observer problem for a class of nonlinear systems linear in the unmeasured states a novel hybrid observer design strategy is proposed with the help of the notion of strong observability in finite time the proposed methodology is applied for the estimation of the frequency of a sinusoidal signal the results show that accurate estimation can be provided even if the signal is corrupted by high frequency noise a brief discussion of the robustness properties of the proposed observer with respect to measurement errors is also provided

global exponential observers for two classes of nonlinear systems abstract this paper develops sufficient conditions for the existence of global exponential observers for two classes of nonlinear systems i the class of systems with a globally asymptotically stable compact set and ii the class of systems that evolve on an open set in the first class the derived continuous time observer also leads to the construction of a robust global sampled data exponential observer under additional conditions two illustrative examples of applications of the general results are presented one is a system with monotone nonlinearities and the other is a chemostat system

the evolution of reaction diffusion waves in a class of scalar reaction diffusion equations algebraic decay rates in this paper we consider an initial boundary value problem for the generalized fisher kolmogorov petrovskii piscounov fkpp equation of order m 1 with the non negative initial data being of o x at large x dimensionless distance where 1 m 1 the corresponding problem when 1 m 1 having been considered in detail by needham and barnes nonlinearity 12 1999 41 58 using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial boundary value problem for large t time which exhibits the formation of a permanent form reaction diffusion travelling wave structure this being in contrast to the case when 1 m 1 where the large t time structure exhibits the formation of an accelerating phase wave

an existence theorem for observers of bilinear systems under suitable uniformity assumptions the existence of nonlinear state observers with exponentially decaying error for bilinear systems is proved to be equivalent to that of observers whose corresponding error dynamics is linear with respect to the error

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty this paper is a geometric study of the local observer design for a general class of nonlinear systems with real parametric uncertainty explicitly we study the observer design problem for a general class of nonlinear systems with real parametric uncertainty and with an input generator exosystem in this paper we show that for the classical case when the state equilibrium does not change with the parametric uncertainty and when the plant output is purely a function of the state there is no local asymptotic observer for the plant next we show that in sharp contrast to this case for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty there typically exist local exponential observers even when the plant output is purely a function of the state we also present a characterization and construction procedure for local exponential observers for the general class of nonlinear systems with real parametric uncertainty under some stability assumptions we also show that for the general class of nonlinear systems considered under some stability assumptions the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs finally we generalize our results to a general class of nonlinear systems with input generator and with exogenous disturbance

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

further results on the observer design problem abstract in this paper the observer design problem for nonlinear systems is considered sufficient lyapunov like conditions are presented for the existence of a nonlinear observer the theory we develop considerably improves and extends the results of our recent work 14

a geometric analysis of the lagerstrom model problem abstract lagerstrom s model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low reynolds number by the method of matched asymptotic expansions in this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems in particular invariant manifold theory as an essential part of the dynamics takes place near a line of non hyperbolic equilibria a blow up transformation is introduced to resolve these singularities this approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem in particular the source of the logarithmic switchback phenomenon is identified

input output linearization approach to state observer design for nonlinear system presents a state observer for a class of nonlinear systems based on the input output linearization while the previous result presented state observers for nonlinear systems of full relative degree we proposed a procedure fur the design of nonlinear state observers which do not require the hypothesis of full relative degree assuming that there exists a global state observer for internal dynamics and that some functions are globally lipschitz we can design a globally convergent state observer it is also shown that if the zero dynamics are locally exponentially stable then there exists a local state observer an example is given to illustrate the proposed design of nonlinear state observers

a state observer for nonlinear dynamical systems the problem of state observation for nonlinear systems is of main importance in automatic control in recent years many contributions have been presented in literature that solve the design problem for classes of nonlinear systems most works can be roughly classi ed into two categories in one the geometrical properties of vector elds de ning the system are exploited and therefore mathematical tools of the di erential geometry are extensively employed in the other one the functional properties of the input state and state output maps which de ne the system are mainly used while the geometrical methods often provide a general solution of the problem for systems that belong to restricted classes characterized by geometrical properties 1 2 3 4 5 6 7 13 14 17 18 19 the second approach gives solutions for classes of nonlinear systems whose geometrical nature is not investigated and only macroscopic parameters norms gains lipschitz constants and so on are involved in the design procedure 8 9 11 12 21 one main problem in state observation for nonlinear systems is that the property of driftobservability does not imply observability for any input function as shown in 16 19 thus it is interesting to investigate when the drift observability property allows the solution of the state observation problem nding conditions that restricts the class of input functions due to the mathematical tools employed this work may be considered as belonging to both the categories above mentioned in this paper nonlinear systems of the form

bilinearization up to output injection abstract this paper deals with nonlinear systems that can be steered to bilinear systems by output injection and nonlinear change of coordinates we give an abstract non constructive theorem and an algorithm which allows us to recognize such systems

rigorous asymptotic expansions for lagerstrom s model equation a geometric approach abstract the present work is a continuation of the geometric singular perturbation analysis of the lagerstrom model problem which was commenced in j differential equations 199 2 2004 290 325 we establish the same framework here reinterpreting lagerstrom s equation as a dynamical system which is subsequently analyzed by means of methods from dynamical systems theory as well as of the blow up technique we show how rigorous asymptotic expansions for the lagerstrom problem can be obtained using geometric methods thereby establishing a connection to the method of matched asymptotic expansions we explain the structure of these expansions and demonstrate that the occurrence of the well known logarithmic switchback terms therein is caused by a resonance phenomenon

fast solutions and asymptotic behavior in a reaction diffusion equation abstract in this paper we deal with the existence of traveling waves solutions t w s for the reaction diffusion equation u t u x x f u in a very general setting of reaction terms f with two distinguished stationary states say 0 and 1 we link the existence of some type of solutions of the second order ode u c u f u 0 with the existence of fast t w s by defining fast solutions for this ode we find a value c m 0 related to the existence of global fast solutions and determine c m through a variational formula our results allow us particularly to show that any solution u x t of the reaction diffusion equation with compactly supported initial data and 0 u x 0 1 x r satisfies lim t u x c t t 0 uniformly on compact sets for all c c c m finally we connect c m with the minimum speed of propagation of all t w s c

nonlinear observer design for state and disturbance estimation a new systematic framework for nonlinear observer design that allows the concurrent estimation of the process state variables together with key unknown process or sensor disturbances is proposed the nonlinear observer design problem is addressed within a similar methodological framework as the one introduced in n kazantzis c kravaris nonlinear observer design using lyapunov s auxiliary theorem systems control lett 34 1998 241 a j krener m xiao nonlinear observer design in the siegel domain siam j control optim 41 2002 932 for state estimation purposes only from a mathematical standpoint the problem under consideration is addressed through a system of first order singular pdes for which a rather general set of solvability conditions is derived a nonlinear observer is then designed with a state dependent gain that is computed from the solution of the system of singular pdes under the aforementioned conditions both state and disturbance estimation errors converge to zero with assignable rates the convergence properties of the proposed nonlinear observer are tested through simulation studies in an illustrative example involving a biological reactor

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

reduced order state reconstruction for nonlinear dynamical systems in the presence of model uncertainty abstract a new approach to the reduced order state reconstruction problem for nonlinear dynamical systems in the presence of model uncertainty is proposed the problem of interest is conveniently formulated and addressed within the context of singular first order non homogeneous partial differential equations pde theory leading to a reduced order nonlinear state estimator that is constructed through the solution of a system of singular pdes a set of necessary and sufficient conditions is derived that ensure the existence and uniqueness of a locally analytic solution to the above system of pdes and a series solution method is developed that is easily programmable with the aid of a symbolic software package such as maple furthermore the convergence of the estimation error or the mismatch between the actual unmeasurable states and their estimates is analyzed and characterized in the presence of model uncertainty finally the performance of the proposed reduced order state reconstruction method is evaluated in a case study involving a biological reactor that exhibits nonlinear dynamic behavior

brief paper full order observer design for a class of port hamiltonian systems we consider a special class of port hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full order observers using a passivity based approach the essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state space on which the state estimation error is zero we first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state space and then perform a partial state feedback passivation which leads to the construction of the observer we then illustrate this observer design procedure on two physical examples the magnetic levitation system and the inverted pendulum on the cart system

state observation of nonlinear uncertain dynamical systems this note proposes new types of observers for nonlinear dynamical systems subjected to bounded nonlinearities or uncertainties the design of these observers utilizes techniques related to variable structure systems theory a measure for the rate at which the estimates converge to the actual states is derived

global stabilizability of homogeneous vector fields of odd degree abstract in this paper we deal with global stabilizability at the origin of a homogeneous vector field by means of homogeneous feedbacks of the same degree we give a sufficient condition which turns out to be also necessary in the linear case for a particular class of systems this condition is implied by a local controllability assumption

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

moving horizon observers and observer based control in this paper two topics are explored a new approach to the problem of obtaining an estimate of the state of a nonlinear system is proposed the moving horizon observer produces an estimate of the state of the nonlinear system at time t either by minimizing or approximately minimizing a cost function over the preceding interval horizon t t t as t advances so does the horizon convergence of the estimator is established under the assumption that the corresponding global optimization problem can be approximately solved and a uniform reconstructability condition is satisfied the latter condition is automatically satisfied for linear observable systems the utility of the estimator for receding horizon control is explored in particular stability of a composite moving horizon system comprising a moving horizon regulator and a moving horizon observer is established

the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we study traveling wave solutions for the class of scalar reaction diffusion equations u t 2 u x 2 f m u where the family of potential functions f m is given by f m u 2 u m 1 u for each m 1 real there is a critical wave speed c crit m that separates waves of exponential structure from those which decay only algebraically we derive a rigorous asymptotic expansion for c crit m in the limit as m this expansion also seems to provide a useful approximation to c crit m over a wide range of m values moreover we prove that c crit m is c smooth as a function of m 1 our analysis relies on geometric singular perturbation theory as well as on the blow up technique and confirms the results obtained by means of asymptotic methods in d j needham a n barnes reaction diffusion and phase waves occurring in a class of scalar reaction diffusion equations nonlinearity 12 1 1999 41 58 t p witelski k ono t j kaper critical wave speeds for a family of scalar reaction diffusion equations appl math lett 14 1 2001 65 73

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

online exact differentiation and notion of asymptotic algebraic observers in recent years the availability of computer based methods has created a revival of interests in exploring algebraic methods in nonlinear context this paper proposes a new approach to algebraic nonlinear observer design after giving the notion of algebraic observability and based on a novel algorithm of exact differentiation the formulation of the nonlinear observer is realized via the construction of a set of linear time varying differentiators an example of a chemical reaction is given to show the effectiveness of our approach

nonlinear system immersion observers and finite dimensional filters abstract nonlinear deterministic and stochastic systems which can be immersed into linear observable ones are identified by necessary and sufficient conditions those systems are shown to admit observers in the deterministic case and exact finite dimensional filters in the stochastic case the sufficient conditions obtained generalize earlier results in both cases

optimization based non linear observers revisited optimization based non linear observers are those where the integrated output prediction error is used to define the dynamic of the observer s state their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold here we present some new theoretical results concerning these observers despite the title we use our observer is not based on the search for a global minimum of a cost function it is based on a descent like approach without the use of the hessian matrix a nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations besides the theoretical results some implementation issues are discussed in particular we propose to use the post stabilization approach well known when dealing with differential systems with invariant submanifolds this enables to increase the sampling period while keeping very good precision two illustrative examples are presente

gevrey properties of the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we consider front propagation in a family of scalar reaction diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity we investigate the gevrey properties of the corresponding critical propagation speed proving that the formal series expansion for that speed is gevrey 1 with respect to the inverse of the degree moreover we discuss the question of optimal truncation finally we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order and we illustrate that algorithm by calculating explicitly the first ten coefficients our analysis builds on results obtained previously in f dumortier n popovic t j kaper the asymptotic critical wave speed in a family of scalar reaction diffusion equations j math anal appl 326 2 2007 1007 1023 and makes use of the blow up technique in combination with geometric singular perturbation theory and complex analysis while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic

existence of bilinear state observers for bilinear systems a recent result by kudva et al 1 is applied to find necessary and sufficient conditions for the existence of bilinear state observers for bilinear systems

design of state observers from a drift observability property in this paper the problem of state observation with exponential error decay for nonlinear systems affine in the input is considered and an observer is proposed for such an observer the drift observability property of the system i e observability for zero input together with an assumption on the input amplitude or on the observation relative degree are sufficient conditions for exponential convergence of the observation error the existence of an exponential observer is correlated to the existence of a solution for an h sub spl infin riccati like inequality global and semiglobal convergence results are presented

on exponential observers for nonlinear systems abstract the aim of the paper is to identify the class of nonlinear systems that have exponential observers a concept introduced by previous authors it is shown that a necessary condition for the existence of an exponential observer for a nonlinear system is that the corresponding linearized system is detectable and for local exponential stabilization problems the condition is also sufficient this paper gives also a theorem on the separation property for the exponential design problem and it enables us to tell exactly to what extent the classical local linearization approach is applicable

gevrey properties of real planar singularly perturbed systems abstract by applying geometric techniques to real analytic singularly perturbed vector fields on the plane we develop a way to give a bound on the gevrey type of the taylor development of canard manifolds at degenerate planar turning points by blowing up the phase space at the turning point we find asymptotic estimates even when such expansions w r t traditional phase space variables do not exist the asymptotic estimates are then used to give a sufficient and necessary condition on the existence of local canard solutions

on nonlinear observers in this note we make some remarks about the general observer problem for nonlinear systems we show how the existence of a partial observer without stability requirements is locally equivalent to the existence of a conditioned invariant distribution we argue that by considering not only identity observers we can put the stability issue in a broader perspective

observability through delayed measurements a new approach to state observers design an original approach for the construction of state observers for dynamic systems is presented in this article new notions of observability delay observability and delay drift observability useful for the development of the observer equations and for the convergence proof are introduced and compared with other standard observability concepts in particular it is shown that delay drift observability is an extension of the well known drift observability property the construction of the proposed observer is straightforward and does not require any coordinate transformation of the system into a canonical form which is a common feature of many other approaches a numerical example is reported where a system that is not drift observable is considered in this case the proposed observer is easily constructed and performs well

well posedness and qualitative behaviour of a semi linear parabolic cauchy problem arising from a generic model for fractional order autocatalysis in this paper we examine a semi linear parabolic cauchy problem with non lipschitz nonlinearity which arises as a generic form in a significant number of applications specifically we obtain a well posedness result and examine the qualitative structure of the solution in detail the standard classical approach to establishing well posedness is precluded owing to the lack of lipschitz continuity for the nonlinearity here existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem meyer needham 2015 the cauchy problem for non lipschitz semi linear parabolic partial differential equations london mathematical society lecture note series vol 419 which examines the difference of the maximal and minimal solutions to the problem from this uniqueness result the approach of meyer needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset the comparison and continuous dependence results obtained here are novel to this class of problem this class of problem arises specifically in the study of a one step autocatalytic reaction which is schematically given by a b at rate apbq where a and b are the concentrations of a and b respectively with 0 p q 1 and well posedness for this problem has been lacking up to the present

implementation of an iterative map in the construction of quasi periodic instantons chaotic aspects and discontinuous rotation numbers an iterative map of the unit disk in the complex plane is used to explore certain aspects of self dual four dimensional gauge fields quasi periodic in the euclidean time these fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits the iterations do not correspond directly to a discretized time evolution of the gauge fields they are implemented in an indirect fashion first t r being the standard coordinates the r t half plane is mapped on the unit disk in an appropriate way this provides an r t parametrization of z0 the starting point of the iterations and makes the iterates increasingly complex functions of r and t these are then incorporated as building blocks in the generating function of the fields we explain in what sense and to what extent some remarkable features of our map indicated in the title are thus carried over into the continuous time development of the fields special features for quasiperiodicity are stu

sontag s input to state stability condition and global stabilization using state detection this paper deals with the global stabilization problem for nonlinear system using state detection sufficient algebraic conditions for stabilization and detectability concerning particular classes of systems are included these conditions are based to a version of a lyapunov like condition proposed by sontag which guarantees input to state stability

nonlinear observers a circle criterion design and robustness analysis globally convergent observers are designed for a class of systems with monotonic nonlinearities the approach is to represent the observer error system as the feedback interconnection of a linear system and a time varying multivariable sector nonlinearity using lmi software observer gain matrices are computed to satisfy the circle criterion and hence to drive the observer error to zero in output feedback design the observer is combined with control laws that ensure input to state stability with respect to the observer error robustness to unmodeled dynamics is achieved with a small gain assignment design as illustrated on a jet engine compressor example