lq optimal control for a class of pulse width modulated systems we consider linear quadratic optimal control for a class of pulse width modulated systems the problem is motivated from a practical application digital control of switching power converters the control synthesis problem is posed based on a sampled data model of the original switching dynamics and a linear quadratic criterion that takes the intersampling behavior into account

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

junction times in singular optimal control finding junction times on partially singular optimal control trajectories is in general difficult analytically and especially so when the optimal control is explicitly dependent on both the state and adjoint variables this paper presents a new approach for linear problems whereby variations in the final state measured from a reference optimal solution can be used to find corresponding junction times the approach is shown to be effective when applied to a problem which has an optimal solution involving a single bang singular junction for certain choices of end state

local stabilization of switched affine systems this technical note considers the local stabilization problem for the class of switched affine systems the main idea is to use an alternative representation of the switched affine system as a nonlinear system with input constraints switching laws can be derived by emulating locally classical controllers it is shown that by restricting to local stabilization the classical constraint on the existence of constant stable convex combinations may be easily avoided the approach may be interpreted as a generalization where convex combinations defined as functions of the system state are being used constructive methods for deriving switching laws are proposed

sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure e f x eta the expected value of an error weighting function f x eta where x is a random variable to be quantized where the probability density function p x describing x is continuous and positive on some finite or infinite interval and zero outside it and where eta is the quantization error the function f x eta is assumed convex and symmetric in eta and zero only for eta 0 it is shown that in the cases of f x eta eta 2 and f x eta eta the simple condition of concavity of in p x is sufficient for uniqueness of a locally optimal quantizer

stability analysis and control design of discrete time switched affine systems this technical note focuses on stability analysis and control design of switched affine systems in discrete time domain the stability conditions are obtained by taking into account that the system trajectories governed by a certain switching function converge to a set of attraction boldsymbol mathcal v containing a desired equilibrium point these conditions follow from the adoption of a general quadratic lyapunov function whose time variation is bounded above by a concave convex function with center determined by minimax theory our main contribution is to provide a stabilizing state feedback switching function and an invariant set of attraction boldsymbol mathcal v with minimum volume as far as this class of quadratic lyapunov functions is adopted the results are applied to sampled data control of continuous time switched affine systems with chattering avoidance the speed control of a dc motor is presented

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

the hybrid maximum principle is a consequence of pontryagin maximum principle we give a simple proof of the maximum principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of pontryagin type and then by using the classical pontryagin maximum principle

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

a switched lq regulator design in continuous time in this technical note the design of a lq regulator for linear switched systems in continuous time is investigated from a relaxation of the optimal control problem a lyapunov based switching law is provided even if the subsystems are all unstable the state feedback switching law can be applied subject to a positiveness condition in any cases the real cost is always upper bounded by the lyapunov function value the optimality of the switching law is also discussed and we prove that the switching conditions are optimal in some generic cases this point explains why the obtained results over examples approach finely the optimal solutions finally a design strategy is also given that extends the results to the cases where the subsystems are controlled linear systems

genericity results for singular curves let m be a smooth manifold and cal d m m geq 2 be the set of rank m distributions on m endowed with the whitney c infty topology we show the existence of an open set o m dense in cal d m so that every nontrivial singular curve of a distribution d of o m is of minimal order and of corank one in particular for m geq 3 every distribution of o m does not admit nontrivial rigid curves as a consequence for generic sub riemannian structures of rank greater than or equal to three there does not exist nontrivial minimizing singular curves

asymptotics of optimal quantizers for some scalar distributions we obtain semi closed forms for the optimal quantizers of some families of one dimensional probability distributions they yield the first examples of non log concave distributions for which uniqueness holds we give two types of applications of these results one is a fast computation of numerical approximations of one dimensional optimal quantizers and their use in a multidimensional framework the other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function laplace transform and characteristic function moreover we obtain the rate of convergence in the bucklew wise theorem and finally the asymptotic size of the voronoi tessels

a generic design methodology for sliding mode control of switched systems abstract the aim of this paper is to present a generic methodology to design sliding mode controllers for multivariable switched systems affine in control such as dc dc power converters an original formulation of the so called reachability condition suitable for this class of systems is established based on the choice of a lyapunov like function and parameterized by a single weighting matrix it allows several kinds of control strategies to be derived namely conventional piecewise continuous strategies as well as discrete boolean strategies its application to the important subclass of linear time invariant systems is investigated more specifically in the boolean case the present approach is also compared to another hybrid one called the stabilizing approach eventually its efficiency as a design methodology as well as the performance of the resulting control are shown by simulating it on non trivial examples of power converters

stability analysis and control synthesis for switched systems a switched lyapunov function approach this paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete time domain the approach followed in this paper looks at the existence of a switched quadratic lyapunov function to check asymptotic stability of the switched system under consideration two different linear matrix inequality based conditions allow to check the existence of such a lyapunov function the first one is classical while the second is new and uses a slack variable which makes it useful for design problems these two conditions are proved to be equivalent for stability analysis investigating the static output feedback control problem we show that the second condition is in this case less conservative the reduction of the conservatism is illustrated by a numerical evaluation

switched affine systems using sampled data controllers robust and guaranteed stabilization the problem of robust and guaranteed stabilization is addressed for switched affine systems using sampled state feedback controllers based on the existence of a control lyapunov function for a relaxed system we propose three sampled data controls global attracting sets computed by solving a sequence of optimization problems guarantee practical and global asymptotic stabilization for the whole system trajectories in addition robust margins with respect to parameters uncertainties and non uniform sampling are provided using input to state stability finally a buck boost converter is considered to illustrate the effectiveness of the proposed approaches

practical optimal state feedback control law for continuous time switched affine systems with cyclic steady state in this article a method for computing an optimal state feedback control law for continuous time switched affine systems exhibiting cyclic behaviour in steady state is presented the hybrid solutions are deduced from the fillipov solutions it is shown that the optimal trajectory synthesis implies to determine singular arcs algebraic conditions are given to obtain these particular arcs of the trajectory a numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two point boundary value problem occurring in optimal control synthesis the interpolation of the solutions set through a neural network yields a state feedback control law several examples in the power converters field show the feasibility and the efficiency of the method

exponential rate of convergence for lloyd s method i for a random variable with finite second moment and log concave density a unique quantizer exists which produces the minimum expected encoding error using squared error distortion an algorithm given by lloyd lloyd s method i yields a sequence of quantizers which converges to the optimum quantizer using results of fleischer it is shown that the convergence takes place exponentially fast if the logarithm of the density is not piecewise affine as a consequence the number of iterations of lloyd s algorithm needed to obtain the optimum distortion correct to n decimal places approaches infinity no faster than linearly in n another consequence is that if the output of a stationary information source at each time is distributed according to the given density the source output can be encoded at each time i using the quantizer obtained on the i th iteration of lloyd s method obtaining the same asymptotic behavior one would have obtained if the optimum quantizer had been used to encode at each time

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

on the existence of optimal quantizers corresp the existence of optimal n level quantizers is demonstrated for several classes of quantizers including block quantizers memoryless quantizers and uniform step size quantizers no assumptions whatsoever are made regarding the input probability distribution the distortion measure considered generalizes many of the commonly used error based distortion measures including mean squared error distortion and r th law distortion

piecewise quadratic stability of fuzzy systems presents an approach to stability analysis of fuzzy systems the analysis is based on lyapunov functions that are continuous and piecewise quadratic the approach exploits the gain scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities examples demonstrate the many improvements over analysis based on a single quadratic lyapunov function special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described

an optimal control approach to mode generation in hybrid systems abstract this paper presents a solution to the problem of constructing control programs i e sequences of control modes from a given motion alphabet in particular techniques are developed that enable reinforcement learning to act directly at the mode level and hence make learning applicable to continuous time control systems in a straightforward manner moreover given such a control program the issue of improving the system performance through the addition of new control laws is addressed as an optimal control problem in fact this is achieved through an optimal combination of recurring mode strings a number of examples are provided that illustrate the viability of the proposed methods