correspondence robust stabilization of uncertain time varying discrete systems and comments on an improved approach for constrained robust model predictive control this paper addresses the robust stabilizability problem of linear discrete time systems with time varying polytopic uncertainties a state feedback control law is proposed based on the construction of a parameter dependent lyapunov function also it points out that the main results of cuzzola geromel and morari automatica 38 2002 1183 1189 are incorrect and presents the corrected version of these results

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

technical communique superposition in efficient robust constrained predictive control a stabilizing control method which does not require on line optimizations is developed for linear systems with polytopic model uncertainties and hard input constraints this work is motivated by the constrained robust mpc crmpc approach ieee trans automat control 45 2000a 1765 which adopts the dual mode prediction strategy i e free control moves and invariant set and minimizes a worst case performance criterion based on the observation that a feasible control sequence for a particular state can be found as a linear combination of feasible sequences for other states we suggest a stabilizing control algorithm providing sub optimal and feasible control sequences using pre computed optimal sequences for some canonical states the on line computation of the proposed method reduces to simple matrix multiplication

correspondence comments on quasi min max mpc algorithm for lpv systems by y lu and y arkun automatica 36 2000 527 540 this note shows that the solvability of the modified quasi min max mpc algorithm of the paper lu and arkun automatica 36 2000 527 for the systems with input constraints is not guaranteed and the proof of the theorems are not sufficient via a counterexample of the remark 5

essentially compact schemes for unsteady viscous incompressible flows a new fourth order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced the scheme is based on the vorticity stream function formulation it is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions it is also very efficient at every time step or runge kutta stage only two poisson like equations have to be solved the poisson like equations are amenable to standard fast poisson solvers usually designed for second order schemes detailed comparison with the second order scheme shows the clear superiority of this new fourth order scheme in resolving both the boundary layers and the gross features of the flow this efficient fourth order scheme also made it possible to compute the driven cavity flow at reynolds number 106on a 10242grid at a reasonable cost fourth order convergence is proved under mild regularity requirements this is the first such result to our knowledge

meshless schemes for unsteady navier stokes equations in vorticity formulation using radial basis functions in this paper we provide a new scheme for unsteady incompressible flows in vorticity stream function formulation combined with the radial basis functions method it is an efficient meshless method optimal accuracy can be achieved using this method the efficiency and accuracy are demonstrated by numerical examples

robust constrained predictive control using comparison model this paper proposes a quadratic programming qp approach to robust model predictive control mpc for constrained linear systems having both model uncertainties and bounded disturbances to this end we construct an additional comparison model for worst case analysis based on a robust control lyapunov function rclf for the unconstrained system not necessarily an rclf in the presence of constraints this comparison model enables us to transform the given robust mpc problem into a nominal one without uncertain terms based on a terminal constraint obtained from the comparison model we derive a condition for initial states under which the ultimate boundedness of the closed loop is guaranteed without violating state and control constraints since this terminal condition is described by linear constraints the control optimization can be reduced to a qp problem

efficient robust constrained model predictive control with a time varying terminal constraint set abstract an efficient robust constrained model predictive control algorithm with a time varying terminal constraint set is developed for systems with model uncertainty and input constraints the approach is novel in that it off line constructs a continuum of terminal constraint sets and on line achieves robust stability by using a relatively short control horizon even n 0 with a time varying terminal constraint set this algorithm not only dramatically reduces the on line computation but also significantly enlarges the size of the allowable set of initial conditions moreover this control scheme retains the unconstrained optimal performance in the neighborhood of the equilibrium the controller design is illustrated through a benchmark problem

analysis of finite difference schemes for unsteady navier stokes equations in vorticity formulation in this paper we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible navier stokes equations in vorticity stream function formulation the no slip boundary condition for the velocity is converted into local vorticity boundary conditions thom s formula wilkes formula or other local formulas in the earlier literature can be used in the second order method while high order formulas such as briley s formula can be used in the fourth order compact difference scheme proposed by e and liu the stability analysis of these long stencil formulas cannot be directly derived from straightforward manipulations since more than one interior point is involved in the formula the main idea of the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function we choose to analyze the second order scheme with wilkes formula in detail in this case we can avoid the complicated technique necessitated by the strang type high order expansions as a consequence our analysis results in almost optimal regularity assumption for the exact solution the above methodology is very general we also give a detailed analysis for the fourth order scheme using a 1 d stokes model

brief robust receding horizon predictive control for systems with uncertain dynamics and input saturation a receding horizon predictive control method which assures stability for systems with model uncertainty and input saturation is derived by extending earlier work in two important respects i ellipsoidal invariant sets are replaced by polyhedral invariant sets and ii the constraint that the state lie in an invariant set is invoked n steps ahead the new algorithm allows for a considerable enlargement of the stabilizable set and provides extra degrees of freedom with which to improve performance

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

model predictive control for linear parameter varying constrained systems using ellipsoidal set prediction this paper proposes a new model predictive control mpc method for linear parameter varying systems with bounded parameter variation subject to input constraints the method adopts closed loop prediction and constructs ellipsoidal sets to predict the future states with reasonable computational effort then the information on the parameter variation rate is exploited to improve the accuracy of the prediction furthermore a relaxed terminal condition which guarantees the stability for infinite horizon is introduced to enlarge the stabilizable region it is shown that the feasibility of the mpc problem at the initial step ensures the stability of the closed loop system finally a simulation result illustrates the effectiveness of the proposed method

a model for non isothermal flow and mineral precipitation and dissolution in a thin strip motivated by porosity changes due to chemical reactions caused by injection of cold water in a geothermal reservoir we propose a two dimensional pore scale model of a thin strip the pore scale model includes fluid flow heat transport and reactive transport where changes in aperture is taken into account the thin strip consists of void space and grains where ions are transported in the fluid in the void space at the interface between void and grain ions are allowed to precipitate and become part of the grain or conversely minerals in the grain can dissolve and become part of the fluid flow and we honor the possible change in aperture these two processes cause we include temperature dependence and possible effects of the temperature in both fluid properties and in the mineral precipitation and dissolution reactions for the pore scale model equations we investigate the limit as the width of the thin strip approaches zero deriving upscaled one dimensional effective equations

global existence and uniqueness of solutions for a two scale reaction diffusion system with evolving pore geometry abstract we prove existence and uniqueness of weak solutions for a quasilinear parabolic system of two pdes and one ode that are coupled in a non standard way the problem results from the transformation of a two scale model for reaction and diffusion in a time dependent porous medium where the evolution of the geometry is not a priori known but is coupled to the reaction diffusion process itself the analysis is based on a comparison principle for the two scale problem and on the construction of a compact fixed point operator

brief an efficient off line formulation of robust model predictive control using linear matrix inequalities the practicality of model predictive control mpc is partially limited by its ability to solve optimization problems in real time moreover on line computational demand for synthesizing a robust mpc algorithm will likely grow significantly with the problem size in this paper we use the concept of an asymptotically stable invariant ellipsoid to develop a robust constrained mpc algorithm which gives a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off line one within another in state space this off line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed loop system and substantial reduction of the on line mpc computation the controller design is illustrated with two examples

coupled reaction diffusion processes inducing an evolution of the microstructure analysis and homogenization abstract chemical processes in porous media often cause a change of the microstructure of the porous material due to interaction with the solid matrix by reaction or adsorption e g we consider a reaction diffusion problem where a solid matrix constituent is converted into another one of different density thus the solid matrix locally grows or shrinks in volume which in turn changes the pore air volume this affects the transport of reactants in the pore air the homogenization of this problem with evolving microstructure is performed using the method of transformation to a periodic reference domain which has recently been put forward by the author the final system to be homogenized consists of three coupled partial differential equations for the concentrations coupled to one ordinary differential equation for a quantity describing the evolution of the pore air volume

a feedback min max mpc algorithm for lpv systems subject to bounded rates of change of parameters a novel closed loop model based predictive control mpc strategy for input saturated polytopic linear parameter varying lpv discrete time systems is proposed it is postulated that the plant belongs to a polytopic family of linear systems each member of which being parameterized by the value that a parameter vector assumes in the unit simplex such a parameter can be measured online and used for feedback while a bound on its rate of change is known and exploited for predictions the paper extends the mpc scheme presented by lu et al 2000 for the restricting case of 1 step long control horizons to the general case of control horizons of arbitrary length n this is done by suitably modifying the robust mpc scheme presented by casavola et al 2000 for uncertain polytopic systems the feasibility and closed loop stability of this strategy are proved and a numerical example is also presented in order to show how the freedom of extending the control horizon and knowledge of the parameter is significant in order to improve the performance of the control strategy

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

quasi min max mpc algorithms for lpv systems in this paper a new model predictive controller mpc is developed for polytopic linear parameter varying lpv systems we adopt the paradigm used in gain scheduling and assume that the time varying parameters are measured on line but their future behavior is uncertain and contained in a given polytope at each sampling time optimal control action is computed by minimizing the upper bound on the quasi worst case value of an infinite horizon quadratic objective function subject to constraints on inputs and outputs the mpc algorithm is called quasi because the first stage cost can be computed without any uncertainty this allows the inclusion of the first move u k k separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints proposed optimization problems are solved by semi definite programming involving linear matrix inequalities it is shown that closed loop stability is guaranteed by the feasibility of the linear matrix inequalities a numerical example demonstrates the unique features of the mpc design

constrained rhc for lpv systems with bounded rates of parameter variations this paper studies a stabilization problem of polytopically uncertain linear parameter varying systems with input constraints and bounded rates of parameter variations in the framework of finite receding horizon control rhc a system containing parameter uncertainties is modified into a system with parameter incremental uncertainties within each horizon for the system modified in this manner a robust rhc is designed by solving an optimization problem at each time instant based on the feasibility of the problem and the optimality of its solution the closed loop system stability is guaranteed a numerical example is included to illustrate the validity of the results

on linear programming and robust modelpredictive control using impulse responses abstract campo and morari have derived a linear programming problem with a potentially large number of constraints which is equivalent to a min max formulation for robust model predictive control of linear systems that formulation involves minimization with respect to the controls of the maximum with respect to the system s impulse response from a set of possible impulse responses of the infinity norm of the error between the predicted and required system output sequences here an alternative linear programming problem is derived which has a smaller number of constraints and is therefore potentially more convenient for on line control

technical communique a synthesis approach of on line constrained robust model predictive control a new method to address constrained robust model predictive control mpc for systems with polytopic description rpc spd is proposed this is motivated by two previous techniques one is the well known robust mpc that parameterizes the infinite horizon control moves into a single linear state feedback law the other is a nominal on line mpc that splits the infinite horizon control moves into a set of free control moves over a fixed horizon and a state feedback law in the terminal region the basic idea here is to adopt the rationale of the second technique to improve feasibility and optimality of the first one a parameter dependent lyapunov function is developed for establishing the closed loop stability the performance of the controller is demonstrated by an example

constrained mpc for uncertain linear systems with ellipsoidal target sets abstract robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain time varying linear systems with input constraints under fixed state feedback control laws with the introduction of extra degrees of freedom in the form of perturbed control laws these stabilizable regions can be enlarged this was done in lee and kouvaritakis automatica 36 2000 1497 1504 in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets we also extend the analysis to take into account both polytopic and unstructured bounded disturbances as well as unstructured uncertainties

homogenization of a nonlinear multiscale model of calcium dynamics in biological cells abstract calcium is one of the most important intracellular messengers which occurs in the cytosol and the endoplasmic reticulum of animal cells while most calcium dynamics models either do not account properly for the fact that the endoplasmic reticulum constitutes a microstructure of the cell or are infeasible by resolving the fine structure very explicitly goel et al 15 derived an effective macroscopic model by formal homogenization in this paper this approach is made rigorous using periodic homogenization techniques to upscale the nonlinear coupled system of reaction diffusion equations and moreover the appropriate scaling of the interfacial exchange term is taken into consideration

vorticity conditioning in the computation of two dimensional viscous flows abstract the problem of evaluating the boundary values of the vorticity in the calculation of two dimensional viscous flows is considered it is shown that the splitting of the fourth order equation for the stream function into two second order problems implies specific integral conditions which fix the abstract projection of the vorticity field with respect to the linear manifold of the harmonic functions these conditions are a direct consequence of the boundary conditions on the velocity and ensure satisfaction of physically essential conservation laws for the vorticity the discrete analogue of the projection conditions produces as many algebraic equations as the number of boundary points and requires the solution of an equal number of dirichlet problems in the particular case of stationary linearized equations stokes equations a direct i e noniterative method of solution is obtained steady and unsteady computational schemes relying on the projection conditions on the vorticity are introduced and extensive numerical results of finite difference calculations of the driven cavity model problem are reported and discussed

an implicit method for coupled flow body dynamics we propose an efficient method for computing coupled flow body dynamics the time stepping is implicit and uses an iterative method preconditioned gmres to solve the flow body equations the preconditioner solves a decoupled version of the equations which involves only the inversion of banded matrices and requires a small number of iterations per time step we use the method to probe the instability to horizontal motions of an elliptical body with simple vertical motions flapping and rising in both cases a linear instability to horizontal motion sets in above a critical reynolds number leading to a stable oscillatory state the pressure forces play a destabilizing role against the stabilizing viscous forces with oscillatory time scales set by either external flapping or the intrinsic flow body coupling the latter lowers the instability threshold in reynolds number

an ellipsoidal off line mpc scheme for uncertain polytopic discrete time systems an off line model predictive control mpc method based on ellipsoidal calculus and viability theory is described in order to address control problems in the presence of state and input constraints for uncertain polytopic linear plants subject to persistent disturbances in order to reduce the computational burdens and conservativeness of traditional polytopic mpc schemes the present approach carries out off line most of the computations and it makes use of closed loop predictions to improve the control performance this is done by recursively pre computing suitable ellipsoidal inner approximations of the exact controllable sets and solving on line a simple and numerically low demanding optimization problem subject to a set membership constraint comparisons with three other recent off line mpc approaches are also provided in the final example

technical communique min max predictive control strategies for input saturated polytopic uncertain systems a novel robust predictive controller for input saturated ltv discrete time systems with polytopic model uncertainties is presented the solution is based on the minimization at each time instant of an upper bound of the worst case infinite horizon quadratic cost under the constraint of steering the set valued future state evolutions emanating from the current state into a feasible and positive invariant set a condition on the initial state is given that suffices to ensure problem solvability for all subsequent time instants under the latter the proposed predictive controller is proved to robustly asymptotically stabilize the input saturated ltv polytopic system

brief an improved approach for constrained robust model predictive control in this paper we present a new technique to address constrained robust model predictive control the main advantage of this new approach with respect to other well known techniques is the reduced conservativeness specifically the technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several lyapunov functions each one corresponding to a different vertex of the uncertainty s polytope

a high order explicit method for the computation of flow about a circular cylinder in this paper a finite difference method for computing the solutions of the incompressible navier stokes equations for flow about a circular cylinder in two dimensions is presented a stream function vorticity formulation of the equation is used and the numerical method incorporates recent developments in the computation of vorticity boundary conditions as well as far field boundary conditions three schemes are described one of second order accuracy one of fourth order accuracy and a hybrid method which is second order accurate in the computation of the vorticity transport and fourth order accurate in the determination of the stream function fully resolved solutions for flow past a cylinder have been computed over a range of reynolds numbers from 1000 to 9500 comparisons are made between the results obtained with methods of different orders of accuracy as well as of the effectiveness of the vorticity and far field boundary conditions

compact h4 finite difference approximations to operators of navier stokes type a method of obtaining compact finite difference approximations of h4 accuracy to operators of navier stokes type is considered the basic procedure is developed for operators in one space dimension and subsequently applied to problems in more space dimensions and in time four illustrative numerical examples are given which indicate clearly in various cases that excellent accuracy may be obtained using the methods comparisons with previous results and with the results of h2 accurate computations are made