principle of proportional damages in a multiple criteria lqr problem a multiple criteria linear quadratic regulator lqr problem is considered the principle of proportional damages is worked out to minimize a norm of a difference between ideal and optimal values of the vector performance criteria while the distribution of the losses damages of the performance criteria are under control a multiple criteria lqr problem solution via principle of proportional damages is obtained as a linear form of state variables and is invariant to the norm which is minimized

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

a neural network modeled by an adaptive lotka volterra system in this paper the asymptotic behavior of an n dimensional lotka volterra system with nonconstant coefficients that models the dynamics of an adaptive cellular network is analyzed each cell in the network is assumed to be connected to other cells in such a way that its output level of activity eitherinhibits or excites the growth of activity in the other cells the weight of the connection between any two cells varies over time in proportion to the average level of interaction between the two cells over the recent past if all of the connections are inhibitory it is shown that the network tends to produce a spontaneous and interesting classification of its inputs

symmetric homotopy construction abstract homotopy continuation methods can be applied to compute all finite solutions to a given polynomial system computations will be performed more efficiently if the symmetric structure of the system can be exploited this paper presents the construction of a symmetric homotopy using this homotopy only the paths according to the generating solutions have to be traced during continuation

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

on input to state stability of min max nonlinear model predictive control in this paper we consider discrete time nonlinear systems that are affected possibly simultaneously by parametric uncertainties and other disturbance inputs the min max model predictive control mpc methodology is employed to obtain a controller that robustly steers the state of the system towards a desired equilibrium the aim is to provide a priori sufficient conditions for robust stability of the resulting closed loop system using the input to state stability iss framework first we show that only input to state practical stability can be ensured in general for closed loop min max mpc systems and we provide explicit bounds on the evolution of the closed loop system state then we derive new conditions for guaranteeing iss of min max mpc closed loop systems using a dual mode approach an example illustrates the presented theory

families of invariants of the motion for the lotka volterra equations the linear polynomials family the modified carleman embedding method already introduced by the authors to find first integrals invariants of the motion of polynomial form to the lotka volterra system is described in detail and its efficiency to treat the n dimensional system proved using this method an extensive investigation is performed for polynomials of the first degree which allow a classification of the integrals in three families for some systems possessing one invariant it is possible to find a second invariant using rescaling methods they represent very restrictive solutions implying that there exists a great number of conditions among the equation s coefficients to satisfy a proof is given that the volterra invariants can be deduced as a limit finally the interesting properties of the solutions of these systems are studied in detail

multiobjective output feedback control via lmi optimization this paper presents an overview of a linear matrix inequality lmi approach to the multiobjective synthesis of linear output feedback controllers the design objectives can be a mix of h sub spl infin performance h sub 2 performance passivity asymptotic disturbance rejection time domain constraints and constraints on the closed loop pole location in addition these objectives can be specified on different channels of the closed loop system when all objectives are formulated in terms of a common lyapunov function controller design amounts to solving a system of linear matrix inequalities the validity of this approach is illustrated by a realistic design example

model predictive control based on linear programming the explicit solution we study model predictive control mpc schemes for discrete time linear time invariant systems with constraints on inputs and states that can be formulated using a linear program lp in particular we focus our attention on performance criteria based on a mixed 1 norm namely 1 norm with respect to time and norm with respect to space first we provide a method to compute the terminal weight so that closed loop stability is achieved we then show that the optimal control profile is a piecewise affine and continuous function of the initial state and briefly describe the algorithm to compute it the piecewise affine form allows to eliminate online lp as the computation associated with mpc becomes a simple function evaluation besides practical advantages the availability of the explicit structure of the mpc controller provides an insight into the type of control action in different regions of the state space and highlights possible conditions of degeneracies of the lp such as multiple optima

stabilizing model predictive control of hybrid systems in this note we investigate the stability of hybrid systems in closed loop with model predictive controllers mpc a priori sufficient conditions for lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion while allowing for discontinuous system dynamics and discontinuous mpc value functions for constrained piecewise affine pwa systems as prediction models we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions for quadratic mpc costs these conditions translate into a linear matrix inequality while for mpc costs based on 1 infin norms they are obtained as norm inequalities new ways for calculating low complexity piecewise polyhedral positively invariant sets for pwa systems are also presented an example illustrates the developed theory

on stability of multiobjective nmpc with objective prioritization this paper proposes a new multiobjective model predictive control mo mpc of constrained nonlinear systems according to objective prioritization the mo mpc problem is formulated as a lexicographic optimization problem the optimal solutions are obtained by solving a hierarchy of single objective optimization problems the conditions guaranteeing the recursive feasibility of the optimization problem and stability of the closed loop system are derived which depend only on the most important objective moreover a suboptimal algorithm is presented to reduce the computational demand of mo mpc one characteristic of the proposed mo mpc is that the given objective prioritization is automatically satisfied the theoretical results are illustrated by a comparison study of an example

a generalization of volterra models with continuous time delay in population dynamics boundedness and global asymptotic stability we introduce a generalization of volterra models with continuous time delay by adding a nonnegative linear vector function of the species sufficient conditions for boundedness of solutions and global asymptotic stability of an equilibrium are provided applications from population dynamics are presented one concerns a predator prey volterra model with prey refuges and continuous time delay and another deals with a volterra system with currents of immigration for some species furthermore for this kind of model by a suitable homotopy function we provide a sufficient condition for the existence of the globally asymptotically stable equilibrium finally an example of a simple model in which two predators are competing for one prey which can take shelter is presented

the gbq algorithm for constructing start systems of homotopies for polynomial systems homotopy methods have become a standard tool for the computation of all solutions of a polynomial system this paper concerns the solution of deficient polynomial systems which appear to be typical in many engineering applications the gbq algorithm presented consists of two parts the computation of a generalized bezout number gb and the construction of a multi homogeneous start system q the approach generalizes m homogenization into multihomogenization it can also be regarded as a generalization towards the random product homotopy however without making assumptions on the coefficients of the polynomials in the system as is illustrated in the examples symmetric polynomial systems also can be solved more efficiently

on the minimax solution of multiple linear quadratic problems multiple linear quadratic problems are studied a set of coupling riccati equations is derived for vector valued cost to go a minimax solution of multiobjective convex problems is proven to be an equalizer strategy this permits the development of a new algorithm for finding the minimax solution for multiple linear quadratic problems

sweeping algebraic curves for singular solutions many problems give rise to polynomial systems these systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters using predictor corrector methods we track the solution paths a point along a solution path is critical when the jacobian matrix is rank deficient the simplest case of quadratic turning points is well understood but these methods no longer work for general types of singularities in order not to miss any singular solutions along a path we propose to monitor the determinant of the jacobian matrix we examine the operation range of deflation and relate the effectiveness of deflation to the winding number computational experiments on systems coming from different application fields are presented

finite element approximation of the p laplacian in this paper we consider the continuous piecewise linear finite el ement approximation of the following problem given p 1 oo and g find u such that v vu 2vu f iniicr2 u g on a the finite element approximation is defined over i2 a union of regular tri angles yielding a polygonal approximation to q for sufficiently regular so lutions u achievable for a subclass of data g and i2 we prove optimal error bounds for this approximation in the norm wlq q 1 q p for p 2 under the additional assumption that qh c 2 numerical results demonstrating these bounds are also presented

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

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

the linear complementarity problem with a parametric input abstract the linear complementarity problem lcp and its importance is well known over the past decade an increasing number of studies have focused on the theory of sensitivity analysis which discusses a local perturbation behavior of current solutions in response to small disturbances of data there are two questions however that remain unclear i what condition if any results in changes of nonzero variable sets ii if a nonzero variable set stays the same for certain perturbations can lcp solutions be expressed as a function of the current solution and perturbed parameters in this paper we examine the two issues for the lcp with a single parametric row and general random matrices the outcome of this paper significantly improves the results obtained in previous research in particular we establish the conditions which ensure the convexity of basis set and uniqueness of solution basis in both situations

brief paper improving performance in model predictive control switching cost functionals under average dwell time in this paper we propose a novel switched model predictive control mpc algorithm for nonlinear continuous time systems where we switch between different cost functionals in order to enhance performance thus different performance criteria can be taken into account in order to ensure stability of the resulting closed loop system we consider switching signals which exhibit a certain average dwell time when considering switching signals of this type certain assumptions are common in the switched systems literature in order to ensure stability like a matching condition for the different lyapunov functions and the possibility to find lyapunov functions with an exponential decay rate in this paper we show how these assumptions can be satisfied in the mpc context and thus stability of the proposed switched mpc algorithm can be established

lie symmetries and invariants of the lotka volterra system in this paper we use the lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3d lotka volterra system

detectability of critical points of smooth functionals from their finite dimensional approximations abstract given a critical point of a c 2 functional on a separable hilbert space we obtain sufficient conditions for it to be detectable i e visible from finite dimensional rayleigh ritz galerkin rrg approximations while examples show that even nondegenerate critical points are without any further restriction not visible we single out relevant classes of smooth functionals e g the hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations such that their nondegenerate critical points are visible from their rrg approximations

finite element error estimates for non linear elliptic equations of monotone type in this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient and the derivation of error estimates for the finite element approximations such problems arise in many practical situations for example in shock free airfoil design seepage through coarse grained porous media and in some glaciological problems by making use of certain properties of the nonlinear coefficients we shall demonstrate that the variational formulations associated with these boundary value problems are well posed we shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities with the aid of these inequalities and some standard results from approximation theory we show how one may derive error estimates for the finite element approximations in the energy norm

stability of multiobjective predictive control we propose a utopia tracking strategy to handle multiple conflicting objectives in model predictive control the controller minimizes the distance of its vector of objectives to that of the compromise solution the point along the steady state pareto front closest to the utopia point where all the objectives are independently minimized we establish conditions for asymptotic stability and propose numerical implementation variants one of the key advantages of the approach is that it avoids the computation of pareto fronts in real time environments in addition the approach can handle general objectives of different nature such as economic and regularization

on the stabilizing property of siorhc it is shown that the stabilizing property of siorhc stabilizing i o receding horizon control holds for general stabilizable discrete time linear plants irrespective of the condition that the plant has no poles at the origin

multiobjective model predictive control this paper proposes a novel model predictive control mpc scheme based on multiobjective optimization at each sampling time the mpc control action is chosen among the set of pareto optimal solutions based on a time varying state dependent decision criterion compared to standard single objective mpc formulations such a criterion allows one to take into account several often irreconcilable control specifications such as high bandwidth closed loop promptness when the state vector is far away from the equilibrium and low bandwidth good noise rejection properties near the equilibrium after recasting the optimization problem associated with the multiobjective mpc controller as a multiparametric multiobjective linear or quadratic program we show that it is possible to compute each pareto optimal solution as an explicit piecewise affine function of the state vector and of the vector of weights to be assigned to the different objectives in order to get that particular pareto optimal solution furthermore we provide conditions for selecting pareto optimal solutions so that the mpc control loop is asymptotically stable and show the effectiveness of the approach in simulation examples

brief an algorithm for multi parametric quadratic programming and explicit mpc solutions explicit solutions to constrained linear model predictive control problems can be obtained by solving multi parametric quadratic programs mp qp where the parameters are the components of the state vector we study the properties of the polyhedral partition of the state space induced by the multi parametric piecewise affine solution and propose a new mp qp solver compared to existing algorithms our approach adopts a different exploration strategy for subdividing the parameter space avoiding unnecessary partitioning and qp problem solving with a significant improvement of efficiency

stability margins of nonlinear receding horizon control via inverse optimality using the nonlinear analog of the fake riccati equation developed for linear systems we derive an inverse optimality result for several receding horizon control schemes this inverse optimality result unifies stability proofs and shows that receding horizon control possesses the stability margins of optimal control laws

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

finite dimensional approximation for a class of elliptic obstacle problems in this paper we study the finite dimensional approximation for a class of elliptic obstacle problems involving unbounded domains by using the galerkin method the convergence of the approximation is deduced

robust model predictive control of stable linear systems a new robust model predictive control mpc algorithm is presented for stable constrained linear plants that is a direct generalization of the nominally stabilizing regulator presented by rawlings and muske model uncertainty is parametrized by a list of possible plants robust stability is achieved through the addition of constraints that prevent the sequence of optimal controller costs from increasing for the true plant asymptotic stability is demonstrated through a lyapunov argument simulation experiments demonstrate the performance of the algorithm for two example processes

phase portraits of cubic polynomial vector fields of lotka volterra type having a rational first integral of degree 2 we classify all the global phase portraits of the cubic polynomial vector fields of lotka volterra type having a rational first integral of degree 2 for such vector fields there are exactly 28 different global phase portraits in the poincare disc up to a reversal of sense of all orbits