a spectral radius formula for the fourier transform on compact groups and applications to random walks abstract let g be a compact group not necessarily abelian let be its unitary dual and for f l1 g let f n f f denote n fold convolution of f with itself and f the fourier transform of f in this paper we derive the following spectral radius formula f n 1 1 n max r g f r where f r is the spectral radius of f r thereby extending the well known beurling gelfand spectral radius formula for the fourier transform on a locally compact abelian group we also establish a partial result in this direction for arbitrary regular borel measures on g as applications we give conceptual and rather short proofs of existing results concerning convergence in the total variation norm of random walks on compact groups and uniform convergence of the corresponding densities we also describe how these results may be transferred to homogeneous spaces

constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

norm convergence of random walks on compact hypergroups where ig is the order of g and p is the nth convolut ion power of p they show that this is a natural tool for studying strong uniform times for r a n d o m walks on g and they show in theorem 7 of 1 stated as theorem 5 in 7 chap 4c that if i p ml o as n then s n o as n at a rate that depends on how fast l ip m converges to 0 but does not depend on the size of the group g in particular if j p m is small then s 2n is small

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

lq control for constrained continuous time systems a design method of lq optimal control law is considered for constrained continuous time systems by introducing singular value decomposition for finite time horizon linear systems the sequence of lq sub optimal control laws which converges to the exact solution is obtained based on quadratic programming it is also shown that the nonlinear sub optimal feedback gain is found by the union of affine state functions

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

whither discrete time model predictive control this note proposes an efficient computational procedure for the continuous time input constrained infinite horizon linear quadratic regulator problem clqr to ensure satisfaction of the constraints the input is approximated as a piecewise linear function on a finite time discretization the solution of this approximate problem is a standard quadratic program a novel lower bound on the infinite dimensional clqr problem is developed and the discretization is adaptively refined until a user supplied error tolerance on the clqr cost is achieved the offline storage of the required quadrature matrices at several levels of discretization tailors the method for online use as required in model predictive control mpc the performance of the proposed algorithm is then compared with the standard discrete time mpc algorithms the proposed method is shown to be significantly more efficient than standard discrete time mpc that uses a sample time short enough to generate a cost close to the clqr solution

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

performance of discontinuous galerkin methods for elliptic pdes in this paper we compare the performance of several discontinuous galerkin dg methods for elliptic partial differential equations pdes on a model problem theoretical estimates of the condition number of the stiffness matrix are given for dg methods whose bilinear form is symmetric and which are shown to be numerically sharp then the efficiency of the methods in the computation of both the potential and its gradient is tested on unstructured triangular meshes

multi resolution explicit model predictive control delta model formulation and approximation this paper deals with the explicit solution and approximation of the constrained linear finite time optimal control problem for systems with fast sampling rates to this aim the recently developed explicit model predictive control empc is reformulated and characterized using the operator to enjoy its promising advantages compared to the time shift operator using the proposed model empc formulation a systematic method is proposed for first designing a low complexity approximate empc solution and then improving its closed loop action without first determining an exact optimal solution that might be of prohibitive complexity it is shown that the stability and feasibility of the proposed sub optimal solution is guaranteed

a parametrization of the solutions of the finite horizon lq problem with general cost and boundary conditions a generalization of the finite horizon linear quadratic regulator problem is proposed for lti continuous time controllable systems in particular a formulation of the linear quadratic lq problem is considered with affine constraints on the initial and the terminal states and with general quadratic costs in the initial and terminal states the solution presented is simple and attractive from a computational point of view and is based on the solutions of an algebraic riccati equation and of a lyapunov equation that enable all the solutions of the hamiltonian differential equation to be parametrized in closed form

error analysis of a modified discontinuous galerkin recovery scheme for diffusion problems abstract a theoretical error analysis using standard sobolev space energy arguments is furnished for a class of discontinuous galerkin dg schemes that are modified versions of one of those introduced by van leer and nomura these schemes which use discontinuous piecewise polynomials of degree q are applied to a family of one dimensional elliptic boundary value problems the modifications to the original method include definition of a recovery flux function via a symmetric l 2 projection and the addition of a penalty or stabilization term the method is found to have a convergence rate of o h q for the approximation of the first derivative and o h q 1 for the solution computational results for the original and modified dg recovery schemes are provided contrasting them as far as complexity and cost numerical examples are given which exhibit sub optimal convergence rates when the stabilization terms are omitted

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

continuous time linear quadratic regulator with control constraints via convex duality a continuous time infinite horizon linear quadratic regulator with input constraints is studied optimality conditions both in the open loop and feedback form and continuity and differentiability properties of the optimal value function and of the optimal feedback are shown arguments rely on basic ideas of convex conjugacy and in particular use a dual optimal control problem

singular value decomposition for a class of linear time varying systems with application to switched linear systems abstract the present paper considers singular value decomposition svd for a class of linear time varying systems the class considered herein describes time driven switched linear systems based on an appropriate input output description the calculation method of singular values and singular vectors is derived the svd enables us to characterize the dominant input output signals using singular vectors which form orthogonal systems in input and output spaces the svd is then applied to switched linear systems to improve the transient response a numerical example is provided to demonstrate the proposed method

asymptotically exact a posteriori ldg error estimates for one dimensional transient convection diffusion problems in this paper new a posteriori error estimates for the local discontinuous galerkin ldg formulation applied to transient convection diffusion problems in one space dimension are presented and analyzed these error estimates are computationally simple and are computed by solving a local steady problem with no boundary conditions on each element we first show that the leading error term on each element for the solution is proportional to a p 1 degree right radau polynomial while the leading error term for the solution s derivative is proportional to a p 1 degree left radau polynomial when polynomials of degree at most p are used these results are used to prove that for smooth solutions these error estimates at a fixed time converge to the true spatial errors in the l 2 norm under mesh refinement more precisely we prove that our ldg error estimates converge to the true spatial errors at o h p 5 4 rate finally we prove that the global effectivity indices in the l 2 norm converge to unity at o h 1 2 rate our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates

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

random rotations characters and random walks on so n we analyze a random walk on the orthogonal group so n given by repeatedly rotating by a fixed angle through randomly chosen planes of r we derive estimates of the rate at which this random walk will converge to haar measure on so n using character theory and the upper bound lemma of diaconis and shashahani in some cases we are able to establish the existence of a cut off phenomenon for the random walk this is the first interesting such result on a non finite group