system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

adaptive low gain integral control of multivariable well posed linear systems the principle of low gain integral control for finite dimensional systems is well known more recently low gain integral control results have been obtained for classes of infinite dimensional systems in this paper we show that integral control with a simple and natural adaptation of the integrator gain achieves tracking of constant reference signals for every exponentially stable multivariable well posed infinite dimensional linear system whose steady state gain matrix has its spectrum in the open right half plane our results considerably extend improve and simplify previous work by the authors siam j control optim 35 1997 pp 78 116

an adaptive tracking problem with a control input constraint abstract in this paper we consider the problem of controlling an uncertain linear time invariant lti continuous time plant when there is a control input constraint a switching controller is presented which will carry out asymptotic error regulation for constant reference and constant disturbance signals which are compatible with the control input constraint assuming that i the plant is asymptotically stable ii the plant has no transmission zeros at zero and iii the plant has as many inputs as outputs this controller is quite easy to implement especially in the single input single output siso case and is noise tolerant some examples including a model of a three input three output distillation column are included to illustrate the controller behaviour

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

brief low gain integral control of continuous time linear systems subject to input and output nonlinearities continuous time low gain integral control strategies are presented for tracking of constant reference signals for finite dimensional continuous time asymptotically stable single input single output linear systems subject to a globally lipschitz and non decreasing input nonlinearity and a locally lipschitz non decreasing and affinely sector bounded output nonlinearity both non adaptive but possibly time varying and adaptive integrator gains are considered in particular it is shown that applying error feedback using an integral controller ensures asymptotic tracking of constant reference signals provided that a the steady state gain of the linear part of the plant is positive b the positive integrator gain is ultimately sufficiently small and c the reference value is feasible in a very natural sense the classes of actuator and sensor nonlinearities under consideration contain standard nonlinearities important in control engineering such as saturation and deadzone

adaptive low gain integral control of linear systems with input and output nonlinearities abstract an adaptive low gain integral control framework is developed for tracking constant reference signals in a context of finite dimensional exponentially stable single input single output linear systems with positive steady state gain and subject to locally lipschitz monotone input and output nonlinearities of a general nature the input nonlinearity is required to satisfy an asymptotic growth condition of sufficient generality to accommodate nonlinearities ranging from saturation to exponential growth and the output nonlinearity is required to satisfy a sector constraint in those cases wherein the input nonlinearity is unbounded

integral control of infinite dimensional linear systems subject to input saturation closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller is shown to ensure asymptotic tracking of constant reference signals provided that a the steady state gain of the linear part of the plant is positive b the positive integrator gain is sufficiently small and c the reference value is feasible in a very natural sense the class of actuator nonlinearities under consideration contains standard nonlinearities important in control engineering such as saturation and deadzone

the zames falb iqc for systems with integrators a feedback interconnection of a neutrally stable linear time invariant system and a nonlinearity with 0 spl les x spl phi x spl les kx sup 2 is called critical because the worst case linearization is at best neutrally stable this characteristic makes the stability analysis of such systems particularly hard it is shown that an integrator and a sector bounded nonlinearity can be encapsulated in a bounded operator that satisfies several useful integral quadratic constraints which gives powerful tools for stability analysis of a general class of critically stable systems

multi adaptive galerkin methods for odes ii implementation and applications continuing the discussion of the multi adaptive galerkin methods mathrm mcg q and mathrm mdg q presented in a logg siam j sci comput 24 2003 pp 1879 1902 we present adaptive algorithms for global error control iterative solution methods for the discrete equations features of the implementation tanganyika and computational results for a variety of odes examples include the lorenz system the solar system and a number of time dependent pdes

stability of a heat exchanger feedback control system using the circle criterion the standard linearized model of a controlled heat exchanger in the form of a system of two first order pdes is interpreted as an abstract semigroup system on an appropriately selected hilbert state space we analyse some its basis properties exponential stability admissibility of an observation functional and regularity of the system impulse response next we give a proof of a version of the circle criterion which bases on the use of the leray schauder fixed point theorem jointly with a uniqueness result the circle criterion is then implemented to a feedback control system of the heat exchanger with a proportional controller and a non linear valve in the feedback loop all considerations are illustrated by a numerical example of the heat exchanger installed in central heating system at the agh university of science and technology cracow poland for which we estimate feasible settlements of the proportional controller

explicit time stepping for stiff odes we present a new strategy for solving stiff odes with explicit methods by adaptively taking a small number of stabilizing small explicit time steps when necessary a stiff ode system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis for many stiff problems the cost of the stabilizing small time steps is small so the improvement is large we illustrate the technique on a number of well known stiff test problems

absolute stability results in infinite dimensions we derive absolute stability results of popov and circle criterion type for infinite dimensional systems in an input output setting our results apply to feedback systems in which the linear part is the series interconnection of an input output stable linear system and an integrator and the nonlinearity satisfies a sector condition which in particular allows for saturation and deadzone effects we use the input output theory developed to derive state space results on absolute stability applying to feedback systems in which the linear part is the series interconnection of an exponentially stable well posed infinite dimensional system and an integrator

low gain control of uncertain regular linear systems it is well known that closing the loop around an exponentially stable finite dimensional linear time invariant plant with square transfer function matrix bg s compensated by a controller of the form k s gamma 0 where k in bbb r and gamma 0 in bbb r mm will result in an exponentially stable closed loop system which achieves tracking of arbitrary constant reference signals provided that i all the eigenvalues of bg 0 gamma 0 have positive real parts and ii the gain parameter k is positive and sufficiently small in this paper we consider a rather general class of infinite dimensional linear systems called regular systems for which convenient representations are known to exist both in time and in frequency domain the purpose of the paper is twofold i we extend the above result to the class of exponentially stable regular systems and ii we show how the parameters k and gamma 0 can be tuned adaptively the resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms nor is the injection of probing signals required

nonlinear krylov and moving nodes in the method of lines we report on some successes and problem areas in the method of lines from our work with moving node finite element methods first we report on our nonlinear krylov accelerator for the modified newton s method on the nonlinear equations of our stiff ode solver since 1990 it has been robust simple cheap and automatic on all our moving node computations we publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations second we discuss the need for reliable automatic choice of spatially variable time steps third we discuss the need for robust and efficient iterative solvers for the difficult linearized equations jx b of our stiff ode solver here the 1997 thesis of zulu xaba has made significant progress

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

adaptive one step ahead control subject to an input amplitude constraint this paper investigates the adaptive control of a single input single output process when there exists a constraint on the amplitude of the applied control signal in particular an adaptive control scheme is proposed and is shown to be globally convergent in the sense that the control asymptotically matches the amplitude constrained one step ahead optimal control some simulation results are presented to demonstrate the performance of the scheme

the state feedback regulator problem for regular linear systems this paper is about the state feedback regulator problem for infinite dimensional linear systems the plant assumed to be an exponentially stable regular linear system is driven by a linear possibly infinite dimensional exosystem via a disturbance signal the exosystem has its spectrum in the closed right half plane and also generates the reference signal for the plant output the regulator problem is to design a controller that while guaranteeing the stability of the closed loop system without the exosystem drives the tracking error to zero a particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller under suitable assumptions we show that the latter problem is solvable if and only if a pair of algebraic equations called the regulator equations is solvable we derive conditions in terms of the transfer function of the plant and eigenvalues of the exosystem for the solvability of the regulator equations three examples illustrating the theory are presented

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

the motion planning problem and exponential stabilisation of a heavy chain part i a model of a heavy chain system with a punctual load tip mass in the form of a system of partial differential equations is interpreted as an abstract semigroup system on a hilbert state space our aim is to solve the output motion planning problem of the same nature as in the case of an unloaded heavy chain grabowski p 2003 abstract semigroup model of heavy chain system with application to a motion planning problem in proceedings of 9th ieee international conference methods and models in automation and robotics 25 28 august miedzyzdroje poland pp 77 86 is1 2 3 pdf in order to solve this problem we first analyse its well posedness and some basic properties next we solve the output motion planning problem using a substitute of the inverse of the input output operator represented in terms of the laplace transforms a problem of exponential stabilisation is also formulated and solved using a stabiliser of the colocated type the exponential stabilisation is proved using the method of ly

a circle criterion for strong integral input to state stability abstract we present sufficient conditions for integral input to state stability iiss and strong iiss of the zero equilibrium pair of continuous time forced lur e systems where by strong iiss we mean the conjunction of iiss and small signal iss our main results are reminiscent of the complex aizerman conjecture and the well known circle criterion we derive a number of corollaries including a result on stabilisation by static feedback in the presence of input saturation in particular we identify classes of forced lur e systems with saturating nonlinearities which are strongly iiss but not iss

robust stability of systems with integral control a number of necessary and sufficient conditions are derived which must be satisfied by the plant d c gain matrix of a linear time invariant system in order for an integral controller to exist for which the closed loop system is stable based on these results the robustness of integral control systems is analyzed i e the family of plants is defined which are stable when controlled with the same integral controller conditions for actuator sensor failure tolerance of systems with integral control are also given finally parallels are drawn between the results of this paper and the bifurcation theory of nonlinear systems

universal integral controllers for minimum phase nonlinear systems we consider a single input single output siso nonlinear system that has a well defined normal form with asymptotically stable zero dynamics using only knowledge of the relative degree and the sign of the high frequency gain we design an output feedback integral controller that asymptotically regulates the output to a bounded time varying reference signal with a constant limit we give regional as well as semi global results we also show that for relative degree one and two systems the proposed integral controller reduces to the classical pi and pid controllers respectively

integral control of linear systems with actuator nonlinearities lower bounds for the maximal regulating gain closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller is known to ensure asymptotic tracking of constant reference signals provided that 1 the steady state gain of the linear part of the plant is positive 2 the positive integrator gain is sufficiently small and 3 the reference value is feasible in a very natural sense here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step response and second order systems with a time delay in the input or output the lower bounds are given in terms of open loop frequency step response data and the lipschitz constant of the nonlinearity and are hence readily obtainable

an adaptive finite element splitting method for the incompressible navier stokes equations we present an adaptive finite element method for the incompressible navier stokes equations based on a standard splitting scheme the incremental pressure correction scheme the presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity an a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization time discretization and a term that measures the deviation of the splitting scheme from a pure galerkin scheme the computational error numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices it is further demonstrated that the computational error of the navier stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

low gain integral control of well posed linear infinite dimensional systems with input and output nonlinearities abstract time varying low gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable well posed linear infinite dimensional single input single output systems subject to globally lipschitz nondecreasing input and output nonlinearities it is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the positive gain function t k t is ultimately sufficiently small and not of class l 1 under a weak restriction on the initial data it is shown that a b and c ensure asymptotic tracking if additionally the impulse response of the linear part of the system is a finite signed borel measure the global lipschitz assumption on the output nonlinearity may be considerably relaxed

time varying and adaptive integral control of infinite dimensional regular linear systems with input nonlinearities closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz nondecreasing actuator nonlinearity and compensated by an integral controller with time dependent gain k t is shown to ensure asymptotic tracking of a constant reference signal r provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the function t mapsto k t monotonically decreases to zero at a sufficiently slow rate this result forms the basis of a simple adaptive control strategy that ensures asymptotic tracking under conditions a and b

absolute stability and integral control absolute stability results of both circle criterion and popov type are derived for finite dimensional linear plants with non linearity in the feedback loop the linear plant contains an integrator and so is not asymptotically stable the possibly time varying non linearity satisfies a particular sector condition which allows for cases with zero lower gain such as saturation and deadzone the conjunction of stable but not asymptotically stable linear plants and non linearities with possibly zero lower gain is a distinguishing feature of the paper the absolute stability results are invoked in proving convergence and stability properties of low gain integral feedback control for tracking of constant reference signals in the context of exponentially stable linear systems subject to input and output non linearities

brief integral control by variable sampling based on steady state data in this paper we consider integral control algorithms with convergent adaptive sampling for multivariable infinite dimensional systems steady state gain information is used in choosing suitable integrator gains and we also consider robustness with respect to error in measuring the steady state gain

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

high gain adaptive stabilization of multivariable linear systems revisited abstract we consider the class of multi input multi output finite dimensional state space systems of the form x ax bu y cx where the state dimension is unknown the system is minumum phase and it is known that det cb 0 for this class a universal adaptive high gain controller not based on identification or estimation algorithms is presented which ensures exponential decay of the motion of the closed loop system it is shown that the controller is robust with respect to certain nonlinear perturbations

low gain integral control for multi input multioutput linear systems with input nonlinearities we consider the inclusion of a static antiwindup component in a continuous time low gain integral controller in feedback with a multi input multi output stable linear system subject to an input nonlinearity from a class of functions that includes componentwise diagonal saturation we demonstrate that the output of the closed loop system asymptotically tracks every constant reference vector which is feasible in a natural sense provided that the integrator gain is sufficiently small robustness properties of the proposed control scheme are investigated and three examples are discussed in detail

multi adaptive galerkin methods for odes i we present multi adaptive versions of the standard continuous and discontinuous galerkin methods for odes taking adaptivity one step further we allow for individual time steps order and quadrature so that in particular each individual component has its own time step sequence this paper contains a description of the methods an analysis of their basic properties and a posteriori error analysis in the accompanying paper a logg siam j sci comput submitted we present adaptive algorithms for time stepping and global error control based on the results of the current paper

tracking and disturbance rejection of mimo nonlinear systems with pi controller we study tracking and disturbance rejection of a class of mimo nonlinear systems with linear proportional plus integral pi compensator roughly speaking we show that if the given nonlinear plant is exponentially stable and has a strictly increasing dc steady state i o map then a simple pi compensator can be used to yield a stable unity feedback closed loop system which asymptotically tracks reference inputs that tend to constant vectors and asymptotically rejects disturbances that tend to constant vectors

brief circle and popov criteria as tools for nonlinear feedback design we convert classical absolute stability criteria into nonlinear design procedures which employ efficient numerical tools such as lmi s we first analyze an earlier circle criterion design and show that its feasibility is limited by conditions on the unstable part of the zero dynamics and the relative degree then we develop an extended circle criterion design which eliminates the relative degree obstacle the restrictions on the zero dynamics are relaxed by using the popov multiplier which also reduces controller complexity the results are illustrated with several design examples

adaptive sampled data integral control of stable infinite dimensional linear systems abstract in this paper adaptive discrete time low gain integral control strategies are presented for tracking constant reference signals for infinite dimensional discrete time power stable linear systems the discrete time results are applied in the development of adaptive sampled data low gain integral control of well posed infinite dimensional exponentially stable linear systems our results considerably extend improve and simplify previous work by two of the authors h logemann s townley discrete time low gain control of uncertain infinite dimensional systems ieee trans automat control 42 1997 22 37

some spectral properties of operator valued positive real functions abstract we consider operator valued positive real functions h and show that the intersections of the point continuous and residual spectra of h s with the imaginary axis do not depend on s in particular if h is positive real and h z is invertible for some z in the open right half plane then h s is invertible for all s in the open right half plane furthermore we prove that the eigenspace of h s corresponding to an imaginary eigenvalue does not depend on s it is also shown that the intersection of the numerical range of h s with the imaginary axis is independent of s finally we prove that under suitable assumptions application of a sufficiently positive real static output feedback to a positive real transfer function leads to a strictly positive real closed loop system

a riesz basis methodology for proportional and integral output regulation of a one dimensional diffusive wave equation in this article we consider a dam river system modeled by a diffusive wave equation this model is commonly used in hydraulic engineering to describe dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important in order to stabilize and regulate the system we propose a proportional and integral boundary controller contrary to many physical systems we end up with a nondissipative closed loop system with noncollocated actuators and sensors we show that the closed loop system is a riesz spectral system and generates an analytic semigroup then we shall be able to assign the spectrum of the closed loop system in the open left half plane to ensure its exponential stability as well as the output regulation independently of any known or unknown constant perturbation these results are illustrated by several numerical examples

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

word problems embeddings and free products of right ordered groups with amalgamated subgroup we use permutation groups to give necessary and sufficient conditions for the free product of right ordered groups with amalgamated subgroup to be right orderable we obtain several consequences answering previously posed problems and also prove the right orderable analogues of the higman embedding theorem and the boone higman theorem

absolute stability and integral control for infinite dimensional discrete time systems abstract we derive absolute stability results of popov and circle criterion types for infinite dimensional discrete time systems in an input output setting our results apply to feedback systems in which the linear part is the series interconnection of an l 2 stable linear system and an integrator and the nonlinearity satisfies a sector condition which allows for saturation and deadzone effects the absolute stability theory is then used to prove tracking and disturbance rejection results for integral control schemes in the presence of input and output nonlinearities applications of the input output theory to state space systems are also provided

digital variable sampling integral control of infinite dimensional systems subject to input nonlinearity in this technical note we present a novel sampled data low gain i control algorithm for infinite dimensional systems in the presence of input nonlinearity the system is assumed to be exponentially stable with invertible steady state gain we use an integral controller with fixed integrator gain chosen on the basis of state gain information and a time varying sampling period determined by the growth bound of the system we compare this new algorithm with two other algorithms one with fixed gain and sampling period the other with time varying gain

analysis and synthesis of low gain integral controllers for nonlinear systems relaxed conditions are given for stability of a feedback system consisting of an exponentially stable multi input multi output nonlinear plant and an integral controller roughly speaking it is shown that if the composition of the plant equilibrium input output map and the integral feedback gain is infinitesimally contracting then the closed loop system is exponentially stable if the integral gain is sufficiently low the main result is illustrated with an application arising in frequency control of ac power systems we demonstrate how the contraction condition can be checked computationally via semidefinite programming and how integral gain matrices can be synthesized via convex optimization to achieve robust performance in the presence of nonlinearity and uncertainty

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

positive μ modification as an anti windup mechanism abstract this paper proposes a simple anti windup mechanism for a model reference adaptive control scheme subject to saturation constraints the anti windup compensator has in essence the same structure as positive modification for the same class of systems it is shown how this structure can under certain circumstances display characteristics similar to anti windup schemes proposed for linear control systems in particular it is shown that if the unknown ideal control signal eventually lies within the control constraints then the response of the adaptive control system will converge to that of the reference system provided certain conditions are satisfied the paper illustrates the challenge of designing anti windup compensators for model reference adaptive control systems

sublattice subgroups of finitely presented lattice ordered groups graham higman proved that the finitely generated groups that occur as subgroups of finitely presented groups are precisely those that can be defined by recursively enumerable sets of relations we prove the analogue for lattice ordered groups theorem a finitely generated lattice ordered group is a sublattice subgroup of some finitely presented lattice ordered group if and only if it can be defined by a recursively enumerable set of relations consequently there is a universal finitely presented lattice ordered group

discrete time low gain control of uncertain infinite dimensional systems using a frequency domain analysis it is shown that the application of a feedback controller of the form k z 1 or kz z 1 where k spl isin r to a power stable infinite dimensional discrete time system with square transfer function matrix g z will result in a power stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals provided that i all the eigenvalues of g 1 have positive real parts and ii the gain parameter k is positive and sufficiently small moreover if g 1 is positive definite we show how the gain parameter gain x can be tuned adaptively the resulting adaptive tracking controllers are universal in the sense that they apply to any power stable system with g 1 0 in particular they are not based on system identification or plant parameter estimation algorithms nor is the injection of probing signals required finally we apply these discrete time results to obtain adaptive sampled data low gain controllers for the class of regular systems a rather general class of infinite dimensional continuous time systems for which convenient representations are known to exist both in state space and in frequency domain we emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

multiadaptive galerkin methods for odes iii a priori error estimates the multiadaptive continuous discontinuous galerkin methods mcgq and mdgq for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution in this paper we prove general order a priori error estimates for the mcgq and mdgq methods to prove the error estimates we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution the estimates then follow from interpolation estimates together with stability estimates for the discrete dual solution

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

brief input to state stability for a class of lurie systems we analyze input to state stability iss for the feedback interconnection of a linear block and a nonlinear element this study is of importance for establishing robustness against actuator nonlinearities and disturbances in the absolute stability framework we prove iss from a positive real property of the linear block by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity when this growth condition is violated examples show that the iss property is lost the result is used to give a simple proof of boundedness for negative resistance oscillators such as the van der pol oscillator in a separate application we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear