backstepping control of coupled linear parabolic pdes with space and time dependent coefficients in this article the backstepping design of stabilizing state feedback controllers for coupled linear parabolic pdes with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented the selected target system is a cascade of exponentially stable time invariant systems with time dependent couplings that is uniformly exponentially stable with a prescribed rate of convergence to determine the state feedback controller the kernel equations are derived which results in a set of coupled pdes for a time dependent and spatially varying kernel for this the method of successive approximations is extended from the time invariant case to the present problem the applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic pdes with space and time dependent coefficients

holomorphic kernels and commuting operators necessary and sufficient conditions in terms of operator polynomials are obtained for an m tuple t tl t l of commuting bounded linear operators on a separable hilbert space x to extend to an m tuple s s1 sm of operators on some hilbert space x where each s is realized as a representation of the adjoint of a multiplication operator on the tensor product of a special type of functional hilbert spaces a1so necessary and sufficient conditions in terms of operator polynomials are obtained for t to have a commuting normal extension o introduction in this paper some results in 1 and 2 for a single bounded linear operator t on a separable hilbert space x are generalized to m commuting operators on x in 1 agler introduces a special class of functional hilbert spaces and describes conditions under which an operator t on x extends to m where m denotes the multiplication operator on x and m denotes the countable direct sum of m with itself special cases of spaces x of which the classical hardy space is the prototype are considered in 1 2 the relevant conditions for the kind of extension of t referred to above are expressed in terms of the positivity of certain operator polynomials involving t and t these conditions are closely related to the reproducing kernel associated with the space s in 2 necessary and sufficient conditions are also given for a contraction t to be subnormal these conditions are really the requirement that a certain sequence of polynomials in t and t be positive it is natural to seek generalizations of these results to m commuting operators t1 tm on x an appropriate model for this generalization is obtained by constructi ng a finite tensor product of spaces s and exploiting the well known fact that the reproducing kernel of such a tensor product is the product of the reproducing kernels of the illdividual spaces suitable modifications of the reproducing kernels which render them holomorphic on the unit polydisc can be used to describe analogous extension results for t t1 tm the question of t having a commuting normal extension n n1 nm turns out to have a direct link with the multi dimensional hausdorff moment problem from the theory of probability received by the editors march 21 1986 1980 mathematics subject classification primary 47a20 secondary 47b20 47b38

brief paper output regulation for linear distributed parameter systems using finite dimensional dual observers in this article the solution of the output regulation problem is considered for linear infinite dimensional systems where the outputs to be controlled cannot be measured it is shown that this problem can be solved by a finite dimensional dual observer that is directly implementable so that the separation principle can be applied for the stabilization as in finite dimensions a parametric design of these dual observers is proposed for riesz spectral systems that allows to achieve a low controller order and a desired control performance for the closed loop system the presented results are illustrated by determining a finite dimensional regulator for an euler bernoulli beam with kelvin voigt damping that achieves tracking for steplike reference inputs and that asymptotically rejects sinusoidal disturbances

on the local bump friedberg l function let f be a p adic field with residue field of cardinality q to each irreducible representation of gl n f we attach a local euler factor l bf q s q t pi via the rankin selberg method and show that it is equal to the expected factor l s t 1 2 phi pi l 2s lambda 2 circ phi pi of the langlands parameter phi pi of pi the corresponding local integrals were introduced in bump and friedberg the exterior square automorphic l functions on mathrm gl n 47 65 1990 and studied in matringe j reine angew math doi 10 1515 crelle 2013 0083 this work is in fact the continuation of matringe j reine angew math doi 10 1515 crelle 2013 0083 the result is a consequence of the fact that if delta is a discrete series representation of gl 2m f and chi is a character of levi subgroup l gl m f times gl m f which is trivial on gl m f embedded diagonally then delta is l chi distinguished if an only if it admits a shalika model this result was only established for chi mathbf 1 before

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

boundary feedback stabilization for an unstable time fractional reaction diffusion equation in this paper we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations new state feedback controls with actuation on one end are designed by the backstepping method for both dirichlet and neumann boundary controls by the riesz basis approach and the fractional lyapunov method we prove the existence and uniqueness and the mittag leffler stability for the closed loop systems for both cases the observers and the observer based output feedback are designed to stabilize the systems

a backstepping approach to the output regulation of boundary controlled parabolic pdes in this article the output regulation problem for boundary controlled parabolic systems with spatially varying coefficients is solved by applying the backstepping approach thereby the outputs to be controlled are not required to be measurable and can be pointwise distributed or boundary quantities whereas the measurement is located at the boundary by solving the state feedback regulator problem in the backstepping coordinates regulator equations with a simple structure result so that their analysis and solution is facilitated the output feedback regulator design is completed by determining a finite dimensional reference observer and an infinite dimensional disturbance observer for the latter a backstepping approach is presented that consists of a triangular decoupling in the backstepping coordinates this allows a systematic design and the explicit derivation of directly verifiable existence conditions for the disturbance observer it is shown that for the resulting compensator the separation principle holds implying output regulation for the exponentially stable closed loop system with a prescribed stability margin the output regulation results of the article are illustrated by means of a parabolic system with an in domain pointwise controlled output

output regulation of periodic signals for dps an infinite dimensional signal generator in this note we consider output regulation and disturbance rejection of periodic signals via state feedback in the setting of exponentially stabilizable linear infinite dimensional systems we show that if an infinite dimensional exogenous system is generating periodic reference signals solvability of the state feedback regulation problem is equivalent to solvability of the so called equations this result allows us to consider asymptotic tracking of periodic reference signals which only have absolutely summable fourier coefficients while in related existing work the reference signals are confined to be infinitely smooth we also discuss solution of the regulator equations and construct the actual feedback law to achieve output regulation in the single input single output siso case the output regulation problem is solvable if the transfer function of the stabilized plant does not have zeros at the frequencies i spl omega sub n of the periodic reference signals and if the sequence cr i spl omega sub n a bk b sup 1 spl times q spl phi sub n cr i spl omega sub n a bk p spl phi sub n sub n spl isin z spl isin l sup n2 a one dimensional heat equation is used as an illustrative example

boundary observer design for a class of semi linear hyperbolic pde systems with recycle loop abstractthe present manuscript considers the observer design problem for a class of scalar semi linear hyperbolic partial differential equation pde systems with a recycle loop through the boundar

vibration control of an euler bernoulli beam under unknown spatiotemporally varying disturbance in this article the vibration suppression of an euler bernoulli beam system is considered by using the adaptive boundary control technique the dynamics of the beam are represented by a partial differential equation and four ordinary differential equations involving functions of both space and time by using lyapunov synthesis the robust boundary control with a disturbance observer is first proposed to suppress the vibration and attenuate the effect of the external disturbances to compensate for both the system parametric uncertainties and the disturbances uncertainties the adaptive boundary control is developed with the proposed boundary control the state of the beam system is proven to be uniformly ultimately bounded and converge to a small neighbourhood of zero by appropriately choosing the design parameters the effectiveness of the proposed control is successfully verified by numerical simulations

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

real algebraic geometry for matrices over commutative rings abstract we define and study preorderings and orderings on rings of the form m n r where r is a commutative unital ring we extend the artin lang theorem and krivine stengle stellensatze both abstract and geometric from r to m n r this problem has been open since the seventies when hilbert s 17th problem was extended from usual to matrix polynomials while the orderings of m n r are in one to one correspondence with the orderings of r this is not true for preorderings therefore our theory is not morita equivalent to the classical real algebraic geometry

tracking control for boundary controlled parabolic pdes with varying parameters combining backstepping and differential flatness the combination of backstepping based state feedback control and flatness based trajectory planning and feedforward control is considered for the design of an exponentially stabilizing tracking controller for a linear diffusion convection reaction system with spatially and temporally varying parameters and nonlinear boundary input for this in a first step the backstepping transformation is utilized to determine a state feedback controller which transforms the original distributed parameter system into an appropriately chosen exponentially stable distributed parameter target system of a significantly simpler structure in a second step the flatness property of the target system is exploited in order to determine the feedforward controller which allows us to realize the tracking of suitably prescribed trajectories for the system output this results in a systematic procedure for the design of an exponentially stabilizing tracking controller for the considered general linear diffusion convection reaction system with varying parameters whose applicability and tracking performance is evaluated in simulation studies

the active disturbance rejection control to stabilization for multi dimensional wave equation with boundary control matched disturbance in this paper we consider boundary stabilization for a multi dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables the active disturbance rejection control adrc approach is adopted in investigation an extended state observer is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multi dimensional system by infinitely many test functions the disturbance is canceled in the feedback loop together with a collocated stabilizing controller all subsystems in the closed loop are shown to be asymptotically stable in particular the time varying high gain is first time applied to a system described by the partial differential equation for complete disturbance rejection purpose and the peaking value reduction caused by the constant high gain in literature the overall picture of the adrc in dealing with the disturbance for multi dimensional partial differential equation is presented through this system the numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction

output feedback boundary control of a heat pde sandwiched between two odes we present designs for exponential stabilization of an ordinary differential equation ode heat partial differential equation pde ode coupled system where the control actuation only acts in one ode the combination of pde backstepping and ode backstepping is employed in a state feedback control law and in an observer that estimates pde and two ode states using only one pde boundary measurement based on the state feedback control law and the observer the output feedback control law is then proposed the exponential stability of the closed loop system and the boundedness and exponential convergence of the control law are proved via lyapunov analysis finally numerical simulations validate the effectiveness of this method for the sandwiched system

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

control of 2 2 linear hyperbolic systems backstepping based trajectory generation and pi based tracking abstract we consider the problems of trajectory generation and tracking for general 2 2 systems of first order linear hyperbolic pdes with anti collocated boundary input and output we solve the trajectory generation problem via backstepping the reference input which generates the desired output incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by vazquez krstic and coron for the backstepping stabilization of 2 2 linear hyperbolic systems we apply our approach to a wave pde with indefinite in domain and boundary damping for tracking the desired trajectory we employ a pi control law on the tracking error of the output we prove exponential stability of the closed loop system under the proposed pi control law when the parameters of the plant and the controller satisfy certain conditions by constructing a novel non diagonal lyapunov functional we demonstrate that the proposed pi control law compensates in the output the effect of in domain and boundary disturbances we illustrate our results with numerical examples

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

compensating actuator and sensor dynamics governed by diffusion pdes for possibly unstable ode systems with actuator delay predictor based infinite dimensional feedback can compensate for actuator delay of arbitrary length and achieve stabilization we extend this concept to another class of pde ode cascades where the infinite dimensional part of the plant is of diffusive rather than convective type we derive predictor like feedback laws and observers with explicit gain kernels the gain kernels involve second order matrix exponentials of the system matrix of the ode plant which is the result of the second order in space character of the actuator sensor dynamics the construction of the kernel functions is performed using the continuum version of the backstepping method robustness to small perturbations in the diffusion coefficient is proved

existence of infinitely many periodic solutions for the radially symmetric wave equation with resonance abstract in this paper we consider the periodic dirichlet problem for a forced nonlinear wave equation with resonance u t t u u a t x u p 1 u in a n dimensional ball under some suitable assumptions on p and a t x we prove the existence of infinitely many radially symmetric time periodic solutions for the problem by variational methods

feedback and feedforward output regulation of bounded uniformly continuous signals for infinite dimensional systems this article presents a state space output regulation theory for linear infinite dimensional systems with bounded control observation and feedthrough operators in which the reference and disturbance signals are bounded and uniformly continuous output regulation problems are posed and solved for both feedforward and error feedback controllers furthermore under certain assumptions necessary conditions are obtained for those controllers which solve these output regulation problems one consequence of these necessary conditions is a generalization of the well known internal model principle a key feature in our theory is that the exogenous system generating the reference and disturbance signals is infinite dimensional

an adaptive servomechanism for a class of infinite dimensional systems a universal adaptive controller is constructed that achieves asymptotic tracking of a given class of reference signals and asymptotic rejection of a prescribed set of disturbance signals for a class of multivariable infinite dimensional systems that are stabilizable by high gain output feedback the controller does not require an explicit identification of the system parameters or the injection of a probing signal in contrast to most of the work in universal adaptive control this paper is based on an input output approach and the results do not require a state space representation of the plant the abstract input output results are applied to retarded systems and integrodifferential systems

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

robust regulation of distributed parameter systems with infinite dimensional exosystems in this paper a robust regulation problem for infinite dimensional systems with infinite dimensional exosystems is discussed the input and output spaces are also allowed to be infinite dimensional a new definition of internal model in terms of the controller parameters is given it is shown that there exists a feedback controller containing an internal model of the exosystem which robustly regulates the class of signals generated by the exosystem and strongly or weakly stabilizes the closed loop system as far as the authors know the results are new even for finite dimensional systems with infinite dimensional exosystems

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

output feedback stabilization for one dimensional wave equation subject to boundary disturbance we consider boundary output feedback stabilization for a one dimensional anti stable wave equation subject to general control matched disturbance the active disturbance rejection control adrc approach is adopted in investigation using the output of the system we first design a variable structure unknown input type state observer which is shown to be exponentially convergent the disturbance is estimated in real time through an extended state observer for an ode reduced from the pde observer the disturbance is then canceled in the feedback loop by its approximated value the stability of the resulting closed loop system is proven simulation results are presented to validate the theoretical conclusions and to exhibit the peaking value reduction by time varying gain instead of constant high gain

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

positivstellensatze for algebras of matrices abstract noncommutative positivstellensatze express positive elements of algebras in terms of sums of squares here positive elements can be defined by means of representations point evaluations or abstract orderings squares are elements of the form a a we prove various types of noncommutative positivstellensatze for matrix algebras m n a where a is an algebra with involution and subalgebras of m n a such as path algebras crossed product algebras and cyclic algebras the notion of a noncommutative sum of squares is proposed and new versions of positivstellensatze are proved

robust distributed control of quasilinear reaction diffusion equations via infinite dimensional sliding modes abstract this paper investigates the problem of robust tracking control for quasilinear reaction diffusion partial differential equations subject to external unknown perturbations the considered class of equations is quite general and includes classical equations such as the heat equation or the fisher kpp equation as special cases global practical stabilization of the tracking error system is established under mild conditions on the disturbance term using a regularized infinite dimensional sliding mode controller extensive simulations support and validate the theoretical results

tracking control of the uncertain heat and wave equation via power fractional and sliding mode techniques in the present paper preliminary results towards the generalization to the infinite dimensional setting of some well known robust finite dimensional control algorithms are illustrated more specifically we deal with the tracking problem for some classes of linear uncertain infinite dimensional systems evolving in hilbert spaces we design distributed variable structure stabilizers that are shown to be effective in the presence of external disturbances the main focus of the present paper is on the rejection of nonvanishing external disturbances the generalization to the infinite dimensional setting of the well known finite dimensional controllers namely the power fractional controller s p bhat and d s bernstein siam j control optim 38 2000 pp 751 766 and two second order sliding mode control algorithms the twisting and supertwisting algorithms l fridman and a levant higher order sliding modes as a natural phenomenon in control theory in robust control via variable structure and lyapunov techniques springer verlag berlin 1996 pp 107 133 a levant internat j control 58 1993 pp 1247 1263 is the main contribution of the present investigation first the distributed twisting control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation next the finite time state tracking of the unperturbed heat equation is provided by means of a distributed power fractional controller finally the distributed supertwisting controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances constructive proofs of stability are developed via the lyapunov functional technique which leads to simple tuning rules for the controller parameters simulation results are discussed to verify the effectiveness of the proposed schemes

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

performance output tracking for multidimensional heat equation subject to unmatched disturbance and noncollocated control this paper investigates performance output tracking for a boundary controlled multidimensional heat equation it is assumed that the so called total disturbance which is composed of internal possibly nonlinear uncertainty and external disturbance the equation is subject to is on one part of the boundary and that the control is applied on the rest of the boundary using only partial boundary measurement we first propose an extended state observer to estimate both system state and the total disturbance this allows us to design a servomechanism and then an output feedback controller under the condition that both the reference signal and the disturbance vanish or belong to spaces h 1 0 infty l 1 gamma 1 and l 2 0 infty l 2 gamma 0 respectively we show that the over all control strategy achieves three objectives on the system performance first exponentially output tracking for arbitrary given reference signal second uniformly boundedness of all internal signals and third the internal asymptotic stability of the closed loop system in addition the control strategy turns out to be robust to the measurement noise we provide numerical experiments to illustrate the effectiveness of the proposed control strategy

a finite dimensional robust controller for systems in the cd algebra robust multivariable controllers for stable infinite dimensional systems in the callier desoer cd algebra are discussed in particular the following robust regulation problem is solved given reference and disturbance signals which are linear combinations of signals of the form t sup j sin spl omega sub k t spl phi sub k j spl ges 0 k 0 spl middot spl middot spl middot n find a low order finite dimensional controller so that the outputs asymptotically track the reference signals asymptotically reject the disturbance signals and the closed loop system is stable and robust with respect to a class of perturbations in the plant the proposed controller consists of a positive scalar gain spl epsi and certain polynomial matrices k sub k s for k 0 spl middot spl middot spl middot n the main result of the paper shows that the matrices k sub k s must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies spl omega sub k for k 0 spl middot spl middot spl middot n the controller has unstable poles on the imaginary axis the behavior of these poles as a function of the scalar parameter spl epsi in the form of a puiseux series is given

finite dimensional regulators for a class of regular hyperbolic pde systems abstractin this paper the output regulation problem is addressed for a class of linear hyperbolic infinite dimensional systems with spatially varying coefficients modelling a large class of convection dominated transport reaction systems in particular distributed parameter systems with bounded input and unbounded output operators are considered first we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation based on the assumption that the hyperbolic system satisfies the exponential stability conditions the main manuscript contribution is the development of two novel finite dimensional regulators output and error feedback regulators such that the controlled output of the plant tracks a reference signal generated by a known signal process exosystem in order to guarantee the feasibility of the proposed regulators the solvability of the corresponding sylvester equations is discusse

output regulation for linear distributed parameter systems this work extends the geometric theory of output regulation to linear distributed parameter systems with bounded input and output operator in the case when the reference signal and disturbances are generated by a finite dimensional exogenous system in particular it is shown that the full state feedback and error feedback regulator problems are solvable under the standard assumptions of stabilizability and detectability if and only if a pair of regulator equations is solvable for linear distributed parameter systems this represents an extension of the geometric theory of output regulation developed in francis 1997 and isidori and byrnes 1990 we also provide simple criteria for solvability of the regulator equations based on the eigenvalues of the exosystem and the system transfer function examples are given of periodic tracking set point control and disturbance attenuation for parabolic systems and periodic tracking for a damped hyperbolic system

internal model theory for distributed parameter systems in this paper we consider robust output regulation of distributed parameter systems and the internal model principle the main purpose is to generalize the internal model principle by francis and wonham for infinite dimensional systems and clarify the relationships between different generalizations of the internal model we also construct a signal generator capable of generating infinite dimensional polynomially increasing signals

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

the internal model principle for systems with unbounded control and observation in this paper the theory of robust output regulation of distributed parameter systems with infinite dimensional exosystems is extended for plants with unbounded control and observation as the main result we present the internal model principle for linear infinite dimensional systems with unbounded input and output operators we do this for two different definitions of an internal model found in the literature namely the p copy internal model and the mathcal g conditions we also introduce a new way of defining an internal model for infinite dimensional systems the theoretic results are illustrated with an example where we consider robust output tracking for a one dimensional heat equation with boundary control and pointwise measurements

performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance in this paper we consider performance output reference tracking for a wave equation with general harmonic disturbance at one end and the control at the other end we first design a state reference system determined completely by the measured output and the reference signal where the parameter update law is presented an adaptive servomechanism output feedback control is then designed it is shown that the closed loop system is well posed three control objectives are achieved a the performance output is tracking the reference signal b all the internal loops are bounded c when the disturbance and reference are disconnected the closed loop is exponentially stable finally the state of system is shown to be tracking the reference state and the updated parameters are convergent to their true values

performance output exponential tracking for a wave equation with a general boundary disturbance abstract in this paper we consider performance output tracking for a wave equation with a general boundary disturbance the control and the disturbance are unmatched different from the existing results the hidden regularity instead of the high gain or variable structure is utilized in the adaptive servomechanism design as a result we are able to cope with more complicated and general disturbances moreover the performance output can track the reference signal exponentially and at the same time all the states of the subsystems involved are uniformly bounded more specially the overall closed loop system is exponentially stable when the disturbance and reference are disconnected to the system the numerical simulations are presented to illustrate the effect of the proposed scheme

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

on convergence of the nonlinear active disturbance rejection control for mimo systems in this paper the global and semiglobal convergence of the nonlinear active distur bance rejection control adrc for a class of multi input multi output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved as a result a class of linear systems with external disturbance that can be dealt with by the adrc is classified the adrc is then compared both analytically and numerically to the well known in ternal model principle a number of illustrative examples are presented to show the efficiency and advantage of the adrc in dealing with unknown dynamics and in achieving fast tracking with lower overstriking

transfer functions for infinite dimensional systems in this paper we study three definitions of the transfer function for an infinite dimensional system the first one defines the transfer function as the expression c si a 1 b d in the second definition the transfer function is defined as the quotient of the laplace transform of the output and input with initial condition zero in the third definition we introduce the transfer function as the quotient of the input and output when the input and output are exponentials we show that these definitions always agree on the right half plane bounded to the left by the growth bound of the underlying semigroup but that they may differ elsewhere

finsler s lemma for matrix polynomials abstract finsler s lemma characterizes all pairs of symmetric n n real matrices a and b which satisfy the property that v t a v 0 for every nonzero v r n such that v t b v 0 we extend this characterization to all symmetric matrices of real multivariate polynomials but we need an additional assumption that b is negative semidefinite outside some ball we also give two applications of this result to noncommutative real algebraic geometry which for n 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets

parameter estimation and non collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance this paper is concerned with the parameter estimation and asymptotic stabilization of a 1 d wave equation that is subject to general harmonic disturbances at the controlled end and suffers from instability at the other end first we design an adaptive observer in terms of measured position and velocity we then adopt the backstepping method for infinite dimensional systems to design an observer based output feedback law the resulting closed loop system is shown to be asymptotically stable and the estimates of the parameters converge to the unknown parameters

error bounds in differentiation of noisy signals by high gain observers the error in estimating the derivative s of a noisy signal by using a high gain observer is studied and quantified the error is bounded in terms of the infinity norms of the noise and a derivative of the signal the error bound is independent of the frequency or derivatives of the noise guidelines are presented for the observer gain design when it is used for on line differentiation analytical and simulation results are presented

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

a dynamic compensator for in domain control of burgers equation this paper addresses the problem of output regulation control of a burgers equation under pointwise in domain actuation a dynamic compensator that generates the in domain control is developed for

archimedean non vanishing cohomological test vectors and standard l functions of gl2n complex case abstract the purpose of this paper is to study the local zeta integrals of friedberg jacquet at complex place and to establish similar results to the recent work 4 joint with c chen and d jiang in this paper we will 1 give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation of gl 2 n c with a non zero shalika model 2 construct a new twisted linear period s and give a different expression of the linear model for 3 give a necessary and sufficient condition on the character such that there exists a uniform cohomological test vector v v which we construct explicitly for s as a consequence we obtain the non vanishing of local friedberg jacquet integral at complex place all of the above are essential preparations for attacking a global period relation problem in the forthcoming paper 11

output regulation for coupled linear parabolic pides abstract this paper presents a backstepping based solution of the output regulation problem for coupled parabolic partial integro differential equations pides with spatially varying coefficients and distinct diffusion coefficients the considered setup assumes in domain as well as boundary disturbances while the output to be controlled can be defined distributed in domain pointwise in domain or at the boundaries and need not be measured by assuming a finite dimensional signal model which may also be non diagonalizable a systematic solution of the output regulation problem is presented by making use of observer based feedforward control existence conditions for the corresponding regulator are formulated in terms of the plant transfer behaviour for the resulting closed loop system exponential stability with a prescribed decay rate is verified the regulator design for two unstable coupled parabolic pides demonstrates the results of the paper

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

finite dimensional regulators for a class of infinite dimensional systems abstract we show that the direct approach developed by the author for stabilization of certain classes of distributed parameter systems can be extended to cover regulation problems as well an iterative design algorithm is presented together with proof that the algorithm will converge after a finite number of steps the procedure is illustrated with an example of a constant disturbance acting on a delay system

unknown input observer design and output feedback stabilization for multi dimensional wave equation with boundary control matched uncertainty abstract in this paper we consider boundary output feedback stabilization for a multi dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance a new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance a key feature of the proposed observer in this paper is that we do not use the high gain to estimate the disturbance by the active disturbance rejection control adrc strategy the total disturbance is compensated canceled in the feedback loop which together with a collocated stabilizing controller without uncertainty leads to an output feedback stabilizing feedback control it is shown that the resulting closed loop system is well posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance the numerical experiments are carried out to show the effectiveness of the proposed scheme

backstepping based adaptive error feedback regulator design for one dimensional reaction diffusion equation abstract in this paper we consider the error feedback regulator problem efrp for one dimensional reaction diffusion equation with unknown harmonic boundary disturbance where the regulated output is anti collocated with the control some auxiliary systems are constructed in order to make the control and the disturbance become collocated or make the measured tracking error become the output then the error feedback adaptive servomechanism is presented in which the parameters of the disturbance and the reference signals are estimated our design is based on the motion planning and the backstepping approaches in addition we give a brief description that our approach is applicable to the heat equation with distributed disturbance it is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

disturbance rejection in 2 x 2 linear hyperbolic systems many interesting problems in the oil and gas industry face the challenge of responding to disturbances from afar typically the disturbance occurs at the inlet of a pipeline or the bottom of an oil well while sensing and actuation equipment is installed at the outlet only kilometers away from the disturbance the present paper develops an output feedback control law for such cases based on modelling the transport phenomenon as a 2 times 2 linear partial differential equation pde of hyperbolic type and the disturbance as a finite dimensional linear system affecting the left boundary of the pde sensing and actuation are co located at the right boundary of the pde the design provides a separation principle allowing a disturbance attenuating full state feedback control law to be combined with an observer the results are applied to a relevant problem from the oil and gas industry and demonstrated in simulations

on the internal model structure for infinite dimensional systems two common controller types and repetitive control in this paper we shall extend and amplify the recent theory of controllers employing the so called internal model structure for exponentially stable closed loop control systems this structure has been shown in another paper to be necessary and sufficient for robust output regulation also in infinite dimensional spaces here we shall derive conditions under which two controller types occurring frequently in applications have the internal model structure under these conditions robust output regulation is achieved also in infinite dimensional spaces if the closed loop system is sufficiently stable in the case that the closed loop system is only strongly but not exponentially stable it is sometimes possible to obtain conditional robustness this means that asymptotic tracking disturbance rejection is not destroyed by small perturbations so long as closed loop stability also persists our results allow for infinite dimensional plants controllers and exogenous systems and as an application of such a general setting we shall consider generalized repetitive control for exponentially stable infinite dimensional siso systems

finite time output regulation for linear 2 2 hyperbolic systems using backstepping this contribution presents the backstepping design of output feedback regulators for boundary controlled linear 2 2 hyperbolic systems that achieve regulation in finite time it is assumed that the disturbances can act in domain at both boundaries and at the output to be controlled the latter need not be available for measurement and consists of in domain pointwise distributed or boundary outputs firstly a solution of the finite time state feedback regulator problem is given on the basis of the regulator equations they are formulated in backstepping coordinates so that a solution is attainable in closed form this leads to a very straightforward regulator design for 2 2 hyperbolic systems with a general class of outputs then a finite dimensional reference observer that converges in finite time is introduced which consists of two observers and a delay this result is extended to the backstepping design of finite time disturbance observers for 2 2 hyperbolic systems with a collocated measurement in particular two backstepping disturbance observers are determined so that after introducing a delay the disturbance model and plant states can be estimated in finite time hence by combining the state feedback regulator with these observers a finite time output feedback regulator is obtained for the state feedback regulator and the disturbance observer existence conditions are derived in terms of the plant transfer behaviour a simple example with an in domain pointwise and distributed output illustrates the theoretical results

constant term of smooth h_psi spherical functions on a reductive p adic group let be a smooth character of a closed subgroup h of a reductive p adic group g if p is a parabolic subgroup of g such that ph is open in g we define the constant term of every smooth function on g which transforms by under the right action of g the example of mixed models is given it includes symmetric spaces and whittaker models in this case a notion of cuspidal function is defined and studied it leads to finiteness theorems

output and error feedback regulator designs for linear infinite dimensional systems abstract this manuscript addresses the output regulation problem for linear distributed parameter systems dpss with bounded input and unbounded output operators we introduce novel methods for the design of the output feedback and error feedback regulators in the output feedback regulator design the measurements available for the regulator do not belong to the set of controlled outputs the proposed output feedback regulator with the injection of the measurement y m t and reference y r t can realize both the plant and the exosystem states estimation disturbance rejection and reference signal tracking simultaneously moreover new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design the regulator parameters are easily configured to solve the output regulation problems and to ensure the stability of the closed loop systems the results are demonstrated via computer simulation in two types of representative systems the parabolic partial differential equation pde system and the first order hyperbolic pde system

derivatives and exceptional poles of the local exterior square l function for gl_m let pi be an irreducible admissible representation of gl m f where f is a non archimedean local field of characteristic zero in 1990 s jacquet and shalika established an integral representation for the exterior square l function we complete following the method developed by cogdell and piatetski shapiro the computation of the local exterior square l function l s pi wedge 2 via the integral representation in terms of l functions of supercuspidal representations by a purely local argument with this result we show the equality of the local analytic l functions l s pi wedge 2 via the integral representation for the irreducible admissible representation pi for gl m f and the local arithmetic l functions l s wedge 2 phi pi of its langlands parameter phi pi through local langlands correspondence

output regulation for a class of linear boundary controlled first order hyperbolic pide systems abstract this manuscript addresses the output regulation problem for a class of scalar boundary controlled first order hyperbolic partial integro differential equation pide systems with fredholm integrals in particular with the advantage of the backstepping approach simple structure systems can be obtained such that regulator equations for the state feedback regulator design are analyzed and solved in backstepping coordinates moreover the finite time output regulation is achieved in the observer based output feedback regulator design it is not necessary that the outputs to be controlled belong to the available output measurements and these outputs can be distributed point wise and or boundary in nature while the boundary placed measurements are used for regulator design for the observer gains design a transformation of the ode pde system into an ode pde cascade is considered it is also shown that the separation principle holds for the output feedback regulator design and the exponential output regulation is realized for the resulting stable closed loop system finally the output regulation results are illustrated with two numerical simulations a korteweg de vries like equation and a pde ode interconnected system

disturbance estimator and servomechanism based performance output tracking for a 1 d euler bernoulli beam equation in this paper we consider performance output tracking for a one dimensional euler bernoulli beam equation with boundary moment control and unmatched internal uncertainty and external disturbance for such system the reference feng and guo 2017 firstly studied the active disturbance rejection control adrc problem we design an infinite dimensional disturbance estimator to estimate the total disturbance internal uncertainty and external disturbance and the system state simultaneously a servomechanism generated by the reference signal is designed based on the estimated total disturbance we design an output feedback control law such that the performance output signal tracks the reference signal and all the signals of closed loop system are bounded some illustrative simulations are presented

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples