nonlinear finite dimensional control of a class of nonlinear distributed parameter systems using residual mode filters a proof of local exponential stability abstract in any attempt to control an infinite dimensional distributed parameter system dps with a finite dimensional controller some compensation for deleterious controller interaction with unmodeled modes is needed we introduce the concepts of both a finite dimensional nonlinear controller based on any reasonable nonlinear reduced order model of the dps and stability compensation by a finite dimensional nonlinear residual mode filter under appropriate lipschitz continuity conditions on the nonlinearities this new pair of interconnected finite dimensional systems will produce closed loop exponentially stable control of the nonlinear dps

quasi equilibrium grid algorithm geometric construction for model reduction the method of invariant grid mig is an iterative procedure for model reduction in chemical kinetics which is based on the notion of slow invariant manifold sim a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 a n gorban i v karlin a y zinovyev invariant grids for reaction kinetics physica a 333 2004 106 154 a n gorban i v karlin invariant manifolds for physical and chemical kinetics lecture notes physics 660 springer berlin heidelberg 2005 doi 10 1007 b98103 important role in that method is played by the initial grid which once refined gives a description of the invariant manifold the invariant grid a convenient way to get a first approximation of the sim is given by the spectral quasi equilibrium manifold sqem a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 in the present paper a flexible numerical method to construct the discrete analog of a quasi equilibrium manifold in any dimension is presented that object is named quasi equilibrium grid qeg while the procedure quasi equilibrium grid algorithm qega extensions of the qem notion are also suggested the qega is a numerical tool which can be used to find a grid based approximation for the locus of minima of a convex function under some linear constraints the method is validated by construction of one and two dimensional grids for a model of hydrogen oxidation reaction

robust feed back control of travelling waves in a class of reaction diffusion distributed biological systems abstract reaction diffusion rd mechanisms can describe many biological phenomena such as neuron firing in the brain the heartbeat cellular organization activities or even biological disorders such as fibrillation the fitzhugh nagumo fhn model is a particular case of rd systems it is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena related to a travelling plane wave as well as biological disorders associated with spiral or irregular fronts the objective of this work is to design a control law that is able to stabilize complex behaviors travelling plane wave in biological systems using the fhn model as a case study since in biological systems there usually exists a lack of detailed information on the system structure our control law will be designed to be robust i e it must be able to reach the predefined reference regardless the presence of structural uncertainties to this purpose we will extend some classical results on the finite dimensional robust control theory to rd systems by means of order reduction techniques in particular the proper orthogonal decomposition method

feedback control of the kuramoto sivashinsky equation abstract this work focuses on linear finite dimensional output feedback control of the kuramoto sivashinsky equation kse with periodic boundary conditions under the assumption that the linearization of the kse around the zero solution is controllable and observable linear finite dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution for any value of the instability parameter the controllers are synthesized on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the performance of the controllers is successfully tested through computer simulations

a galerkin neural network based design of guaranteed cost control for nonlinear distributed parameter systems this paper presents a galerkin neural network based guaranteed cost control gcc design for a class of parabolic partial differential equation pde systems with unknown nonlinearities a parabolic pde system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement motivated by this in the proposed control scheme galerkin method is initially applied to the pde system to derive an ordinary differential equation ode system with unknown nonlinearities which accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently parameterized by a multilayer neural network mnn with one hidden layer and zero bias terms then based on the neural model and a lure type lyapunov function a linear modal feedback controller is developed to stabilize the closed loop pde system and provide an upper bound for the quadratic cost function associated with the finite dimensional slow system for all admissible approximation errors of the network the outcome of the gcc problem is formulated as a linear matrix inequality lmi problem moreover by using the existing lmi optimization technique a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained finally the proposed design method is applied to the control of the temperature profile of a catalytic rod

h _ infty fuzzy observer based control for a class of nonlinear distributed parameter systems with control constraints an hinfin fuzzy observer based control design is proposed for a class of nonlinear parabolic partial differential equation pde systems with control constraints for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement in the proposed control scheme galerkin s method is initially applied to the pde system to derive a nonlinear ordinary differential equation ode system that accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently represented by the takagi sugeno t s fuzzy model then based on the t s fuzzy model a fuzzy observer based controller is developed to stabilize the nonlinear pde system and achieve an optimized hinfin disturbance attenuation performance for the finite dimensional slow system while control constraints are respected the outcome of the hinfin fuzzy observer based control problem is formulated as a bilinear matrix inequality bmi optimization problem a local optimization algorithm that treats the bmi as a double linear matrix inequality is presented to solve this bmi optimization problem finally the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness

robust stabilization of the distributed parameter system with time delay via fuzzy control in this paper stabilization of the distributed parameter system dps with time delay is studied using galerkin s method and fuzzy control with the help of galerkin s method the dynamics of dps with time delay can be first converted into a group of low order functional ordinary differential equations which will be used for design of the robust fuzzy controller the fuzzy controller designed can guarantee exponential stability of the closed loop dps some sufficient conditions are derived for the stabilization together with the linear matrix inequality design approach the effectiveness of the proposed control design methodology is demonstrated in numerical simulations

lq optimal boundary control of infinite dimensional systems with yosida type approximate boundary observation a class of boundary control systems with boundary observation is considered for which the unbounded operators often lead to technical difficulties an extended model for this class of systems is described and analyzed which involves no unbounded operator except for the dynamics generator a method for the resolution of the lq optimal control problem for this model is described and the solution provides a stabilizing feedback for the nominal system with unbounded operators in the sense that in closed loop the state trajectories converge to zero exponentially fast the model consists of an extended abstract differential equation whose state components are the boundary input the state up to an affine transformation and a yosida type approximation of the output of the nominal system it is shown that under suitable conditions the model is well posed and in particular that the dynamics operator is the generator of a c 0 semigroup moreover the model is shown to be observable and to carry controllability stabilizability and detectability properties from the nominal system a general method of resolution based on the problem of spectral factorization of a multi dimensional operator valued spectral density is described in order to solve a lq optimal control problem for this model it is expected that this approach will lead hopefully to a good trade off between the cost of modeling and the efficiency of methods of resolution of control problems for such systems

non linear feedback control of parabolic partial differential difference equation systems this paper proposes a general method for the synthesis of non linear output feedback controllers for single input singleoutput quasi linear parabolic partial differential difference equation pdde systems for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement initially a non linear model reduction scheme which is based on combination of galerkin s method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation dde systems that describe the dominant dynamics of the pdde system then these dde systems are used as the basis for the explicit construction of non linear output feedback controllers through combination of geometric and lyapunov techniques the controllers guarantee stability and enforce output tracking in the closed loop parabolic pdde system independently of the size of the state delay provided that the separation o

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

a stone type duality for st0 stratified alexandrov l topological spaces abstract we show that for every complete lattice a both the set of completely prime elements and the set of completely coprime elements are one to one corresponding to ch a the set of complete lattice homomorphisms from a to the two element lattice 2 a complete lattice is called completely generated a cg lattice for short if it is generated by the set of completely prime elements or equivalently by the set of completely coprime elements then we restudy the duality between the category of t 0 alexandrov topological spaces and the category of cg lattices by means of ch a with these preparations for a frame l as the truth value table we introduce s t 0 separation axiom for stratified alexandrov l topological spaces and finally establish a duality between the category of s t 0 stratified alexandrov l topological spaces and the category of completely generated complete l ordered sets we also investigate some properties of the s t 0 axiom for example it is hereditary by closed subspaces and productive

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

nonlinear feedback control of parabolic partial differential equation systems with time dependent spatial domains partial differential equations pdes with time dependent spatial domains whose dynamics can be partitioned into slow and fast ones initially a nonlinear model reduction scheme similar to the one introduced in christofides and daoutidis j math anal appl 216 1997 398 420 which is based on combinations of galerkin s method with the concept of approximate inertial manifold is employed for the derivation of low order ordinary differential equation ode systems that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times then these ode systems are used as the basis for the explicit construction of nonlinear time varying output feedback controllers via geometric control methods the controllers guarantee stability and enforce the output of the closed loop parabolic pde system to follow up to a desired accuracy a prespecified response for almost all times provided that the separation of the slow and fast dynamics is sufficiently large differences in the nature of the model reduction and control problems between parabolic pde systems with fixed and moving spatial domains are identified and discussed the proposed control method is used to stabilize an unstable steady state of a diffusion reaction process whose spatial domain changes with time it is shown to lead to a significant reduction on the order of the stabilizing nonlinear output feedback controller and outperform a nonlinear controller synthesis method that does not account for the variation of the spatial domain q 1999 academic press

brief analysis and control of parabolic pde systems with input constraints this paper develops a general framework for the analysis and control of parabolic partial differential equations pde systems with input constraints initially galerkin s method is used for the derivation of ordinary differential equation ode system that capture the dominant dynamics of the pde system this ode systems are then used as the basis for the synthesis via lyapunov techniques of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed loop stability in the presence of constraints precise conditions that guarantee stability of the constrained closed loop parabolic pde system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator the theoretical results are used to stabilize an unstable steady state of a diffusion reaction process using constrained control action

separation axioms in l fuzzy topological spaces i t0 and t1 abstract using the notion of remote neighborhood we define the separation axioms t 0 and t 1 in l fuzzy topological spaces l fts the relations between our definitions hutton and reilly s and wang s are discussed and the separations of hutton s fuzzy unit interval and gantner s fuzzy real line are examined characterizations of these l fts are proved and a series of properties of them are investigated moreover some results on minimal t 0 l fts and t 1 l fts are established which will be used immediately in our subsequent discussion

stability of distributed parameter systems with finite dimensional controller compensators using singular perturbations abstract using a singular perturbation formulation of the linear time invariant distributed parameter system we develop a method to design finite dimensional feedback compensators of any fixed order which will stabilize the infinite dimensional distributed parameter system the synthesis conditions are given entirely in terms of a finite dimensional reduced order model the stability results depend on an infinite dimensional version of the klimushchev krasovskii lemma presented here

an application of inertial manifold theory to boundary stabilization of semilinear diffusion systems this paper deals with an application of inertial manifold theory to boundary stabilization of semilinear diffusion systems on a one dimensional domain it is shown that a sakawa type of controller plays an important role in the construction of an inertial manifold for the closed loop system with a given semilinear diffusion system this means that the use of a sakawa type of controller reduces the stabilization problem for the closed loop system to the one on the inertial manifold

finite dimensional control of parabolic pde systems using approximate inertial manifolds abstract this paper introduces a methodology for the synthesis of nonlinear finite dimensional output feedback controllers for systems of quasi linear parabolic partial differential equations pdes for which the eigenspectrum of the spatial differential operator can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement combination of galerkin s method with a novel procedure for the construction of approximate inertial manifolds for the pde system is employed for the derivation of ordinary differential equation ode systems whose dimension is equal to the number of slow modes that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times these ode systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow up to a desired accuracy a prespecified response for almost all times

model reduction of parabolic pdes using multivariate splines abstracta new methodology is presented for model reduction of linear parabolic partial differential equations pdes on general geometries using multivariate splines on triangulations state space descriptions are derived that can be used for control design this method uses galerkin projection with b splines to derive a finite set of ordinary differential equations odes any desired smoothness conditions between elements as well as the boundary conditions are flexibly imposed as a system of side constraints on the set of odes projection of the set of odes on the null space of the system of side constraints naturally produces a reduced order model that satisfies these constraints this method can be applied for both in domain control and boundary control of parabolic pdes with spatially varying coefficients on general geometries the reduction method is applied to design and implement feedback controllers for stabilisation of a 1 d unstable heat equation and a more challenging 2 d reaction convection d

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

a sampled data singularly perturbed boundary control for a heat conduction system with noncollocated observation this note presents a sampled data strategy for a boundary control problem of a heat conduction system modeled by a parabolic partial differential equation pde using the zero order hold the control law becomes a piecewise constant signal in which a step change of value occurs at each sampling instant through the dasialiftingpsila technique the pde is converted into a sequence of constant input problems to be solved individually for a sampled data formulation the eigenspectrum of the parabolic system can be partitioned into two groups a finite number of slow modes and an infinite number of fast modes which is studied via the theory of singular perturbations controllability and observability of the sampled data system are preserved irrelevant to the sampling period a noncollocated output feedback design based upon the state observer is employed for set point regulation the state observer serves as an output feedback compensator with no static feedback directly from the output satisfying the so called dasialow pass propertypsila the feedback controller is thus robust against the observation error due to the neglected fast modes

distributed fuzzy control design of nonlinear hyperbolic pde systems with application to nonisothermal plug flow reactor this paper considers the problem of fuzzy control design for a class of nonlinear distributed parameter systems that is described by first order hyperbolic partial differential equations pdes where the control actuators are continuously distributed in space the goal of this paper is to develop a fuzzy state feedback control design methodology for these systems by employing a combination of pde theory and concepts from takagi sugeno t s fuzzy control first the t s fuzzy hyperbolic pde model is proposed to accurately represent the nonlinear first order hyperbolic pde system subsequently based on the t s fuzzy pde model a lyapunov technique is used to analyze the closed loop exponential stability with a given decay rate then a fuzzy state feedback control design procedure is developed in terms of a set of spatial differential linear matrix inequalities sdlmis from the resulting stability conditions furthermore utilizing the finite difference approximation method with a backward difference for the spatial derivative a recursive algorithm is presented to solve the sdlmis via the existing lmi optimization techniques finally the developed design methodology is successfully applied to the control of a nonisothermal plug flow reactor

neumann boundary stabilization of one dimensional linear parabolic systems with input delay this paper addresses the boundary stabilization problem of one dimensional linear parabolic systems with input delay especially the case of neumann boundary control is studied the purpose is to derive the stabilizing controller of predictor type in a hilbert space by using a backstepping method combined with the semigroup theory the use of the semigroup theory makes the proof of continuity of the inverse transformation easy also it is shown that the implementation of the abstract controller is feasible by using a finite number of eigenvalues and eigenfunctions of the system operator

global stabilization of the kuramoto sivashinsky equation via distributed output feedback control abstract this work addresses the problem of global exponential stabilization of the kuramoto sivashinsky equation kse subject to periodic boundary conditions via distributed static output feedback control under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the kse and a necessary and sufficient stability condition is satisfied linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the kse the controllers are designed on the basis of finite dimensional approximations of the kse which are obtained through galerkin s method the theoretical results are confirmed by computer simulations of the closed loop system

finite dimensional approximation and control of non linear parabolic pde systems this article proposes a rigorous and practical methodology for the derivation of accurate finite dimensional approximations and the synthesis of non linear output feedback controllers for non linear parabolic pde systems for which the manipulated inputs the controlled and measured outputs are distributed in space the method consists of three steps first the karhunen loeve expansion is used to derive empirical eigenfunctions of the non linear parabolic pde system then the empirical eigenfunctions are used as basis functions within a galerkin s and approximate inertial manifold model reduction framework to derive low order ode systems that accurately describe the dominant dynamics of the pde system and finally these ode systems are used for the synthesis of non linear output feedback controllers that guarantee stability and enforce output tracking in the closed loop system the proposed method is used to perform model reduction and synthesize a non linear dynamic output feedback controller for a rapi

predictive control with guaranteed stability for water hammer equations we study the application of the receding horizon optimal control rhoc for hydraulic pipeline systems described by the so called water hammer equations sufficient conditions to guarantee an asymptotic stability to an equilibrium state are first introduced and then integrated in the rhoc scheme for the implementation calculus of variations is employed to characterize the optimal solution in terms of the adjoint state and the recently proposed lattice boltzmann method is used to solve both direct and adjoint partial differential equations this approach is finally validated in simulation

nonlinear control of incompressible fluid flow application to burgers equation and 2d channel flow abstract this paper proposes a methodology for the synthesis of nonlinear finite dimensional feedback controllers for incompressible newtonian fluid flows described by two dimensional navier stokes equations combination of galerkin s method with approximate inertial manifolds is employed for the derivation of low order ordinary differential equation ode systems that accurately describe the dominant dynamics of the flow these ode systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow the reference input asymptotically the method is successfully used to synthesize nonlinear finite dimensional output feedback controllers for the burgers equation and the two dimensional channel flow that enhance the convergence rate to the spatially uniform steady state and the parabolic velocity profile respectively the performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers

distributed parameter thermal controllability a numerical method for solving the inverse heat conduction problem this paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes a numerical control algorithm is developed for distributed parameter conduction systems based on galerkin optimization of an energy index employing green s functions various temperature profiles of variable complexity are studied using the proposed technique in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body furthermore the effect of altering the iterative time step and duration of processing time on the convergence of the solution generated by the aforementioned method is investigated it is proved that despite the variations in numerical processing the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials copyright 2004 john wiley sons ltd

combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials in this paper we present a combinatorial proof of the inversion formula on the kazhdan lusztig r polynomials this problem was raised by brenti as a consequence we obtain a combinatorial interpretation of the equidistribution property due to verma stating that any nontrivial interval of a coxeter group in the bruhat order has as many elements of even length as elements of odd length the same argument leads to a combinatorial proof of an extension of verma s equidistribution to the parabolic quotients of a coxeter group obtained by deodhar as another application we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element

stabilization of a coupled transport diffusion system with boundary input we consider the problem of stabilizing a coupled transport diffusion system with boundary input the system is described by two linear transport diffusion equations and is not asymptotically stable in order to stabilize the system with boundary input sensor influence functions are assumed to be located at interior of the domain first we formulate the system as an evolution equation with unbounded output operators in a hilbert space using variable transformation next we derive a reduced order model with a finite dimensional state variable for the infinite dimensional system then a stabilizing controller is constructed for the reduced order model under an additional assumption it is shown that the finite dimensional controller together with a residual mode filter plays a role of a finite dimensional stabilizing controller for the original infinite dimensional system if the order of the residual mode filter is chosen sufficiently large finally the validity of the design method is demonstrated through a numerical simulation

robust output feedback control of quasi linear parabolic pde systems abstract this paper proposes a methodology for the synthesis of nonlinear robust output feedback controllers for systems of quasi linear parabolic partial differential equations with time varying uncertain variables the method is successfully applied to a typical diffusion reaction process with uncertainty