nonlinear state feedback for l 1 optimal control this paper considers l1 optimal control problems with full state feedback in contrast to h optimal control previous work has shown that linear l1 optimal controllers can be dynamic and arbitrarily high order however this paper shows that continuous memoryless nonlinear state feedback perform as well as dynamic linear state feedback the derivation which is nonconstructive relies on concepts from viability theory

linear nonquadratic optimal control we consider the optimization of nonquadratic measures of the transient response we present a computational implementation of dynamic programming recursions to solve finite horizon problems in the limit the finite horizon performance converges to the infinite horizon performance we provide conditions based on finite horizon computations which only assure that a receding horizon implementation of the finite horizon optimal control is stabilizing and within a specified tolerance of the infinite horizon performance

global behavior for a fourth order rational difference equation in this paper we use a method different from the known literature to investigate the global behavior of the following fourth order rational difference equation xn 1 xn 1xn 2xn 3 xn 1 xn 2 xn 3 a xn 1xn 2 xn 1xn 3 xn 2xn 3 1 a n 0 1 2

minimization of the maximum peak to peak gain the general multiblock problem a comprehensive study of the general l sub 1 optimal multiblock problem and a new linear programming algorithm for computing suboptimal controllers are presented by formulating the interpolation conditions in a concise and natural way the general theory is developed in simpler terms and with a minimum number of assumptions in addition further insight is gained into the structure of the optimal solution and different classes of multiblock problems are distinguished this leads to a conceptually attractive iterative method for finding approximate solutions

mathematical programming algorithms for regression based nonlinear filtering in r sup n this paper is concerned with regression under a sum of partial order constraints examples include locally monotonic piecewise monotonic runlength constrained and unimodal and oligomodal regression these are of interest not only in nonlinear filtering but also in density estimation and chromatographic analysis it is shown that under a least absolute error criterion these problems can be transformed into appropriate finite problems which can then be efficiently solved via dynamic programming techniques although the result does not carry over to least squares regression hybrid programming algorithms can be developed to solve least squares counterparts of certain problems in the class

computation of minimum time feedback control laws for discrete time systems with state control constraints the problem of finding a feedback law that drives the state of a linear discrete time system to the origin in minimum time subject to state control constraints is considered algorithms are given to obtain facial descriptions of the m step admissible sets these descriptions are then used to characterize the complete class of minimum time feedback laws moreover the characterization leads to a conceptually simple on line implementation the main ideas are illustrated with two simple examples

on nonlinear state feedback controllers for persistent disturbance rejection abstract an l 1 optimal linear time invariant lti compensator may have an order significantly higher than that of the plant in the state feedback case there has been recent work exploring the use of nonlinear static feedback controllers which provide near optimal performance in the case of d 12 square and invertible we show that it is enough to have the nonlinearity on a particular substate which can be used to significantly simplify the design problem

an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states an algorithm to find a polyhedral approximant to the maximal state constraint set given state and input bounds is suggested for linear discrete time dynamical systems

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

a decomposition approach to distributed control of spatially invariant systems this note considers the decentralized control of spatially invariant systems i e systems of homogeneous interacting components the main idea is for individual components to model interactions with neighbors as disturbances that satisfy certain magnitude bounds while simultaneously self imposing symmetric magnitude bounds these magnitude bounds can be interpreted as negotiated levels of interaction among components it turns out that this approach is equivalent to constructing a feedback that is robustly stabilizing with structured uncertainties

constructing simple stable descriptions for image partitioning a new formulation of the image partitioning problem is presented construct a complete and stable description of an image in terms of a specified descriptive language that is simplest in the sense of being shortest we show that a descriptive language limited to a low order polynomial description of the intensity variation within each region and a chain code like description of the region boundaries yields intuitively satisfying partitions for a wide class of images the advantage of this formulation is that it can be extended to deal with subsequent steps of the image understanding problem or to deal with other attributes such as texture in a natural way by augmenting the descriptive language experiments performed on a variety of both real and synthetic images demonstrate the superior performance of this approach over partitioning techniques based on clustering vectors of local image attributes and standard edge detection techniques

comparison principle positive invariance and constrained regulation of nonlinear systems abstract in this paper the comparison principle is applied to the design of feedback controllers for nonlinear discrete time systems subject to physical constraints first the connection between the comparison principle and the existence of positively invariant sets of a special form is established and polyhedral positively invariant sets for nonlinear systems are obtained then these results are applied to the determination of state feedback control laws that transfer to the origin all initial states of a prespecified region of the state space without violation of physical constraints on the control vector

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

parallel and deterministic algorithms from mrfs surface reconstruction deterministic approximations to markov random field mrf models are derived one of the models is shown to give in a natural way the graduated nonconvexity gnc algorithm proposed by a blake and a zisserman 1987 this model can be applied to smooth a field preserving its discontinuities a class of more complex models is then proposed in order to deal with a variety of vision problems all the theoretical results are obtained in the framework of statistical mechanics and mean field techniques a parallel iterative algorithm to solve the deterministic equations of the two models is presented together with some experiments on synthetic and real images

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

optimal h sub 2 l sub 1 control via duality theory in this paper we consider the problem of minimizing the h sub 2 norm of the closed loop map while maintaining its l sub 1 norm at a prescribed level the problem is analyzed in the case of discrete time siso closed loop maps utilizing duality theory it is shown that the optimal solution is unique and in the nontrivial case where the l sub 1 constraint is active the optimal solution has a finite impulse response a finite step procedure is given for the construction of the exact solution this procedure consists of solving a finite number of quadratic programming problems which can be performed using standard methods finally continuity properties of the optimal solution with respect to changes in the l sub 1 constraint are established

feedback control for linear time invariant systems with state and control bounds in the presence of disturbances the problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered the only information required about the disturbances is a finite convex polyhedral bound discrete and continuous time systems are considered the property of positive d invariance of a region is introduced and it is proved that a solution of the problem is achieved by the selection of a polyhedral set s and the computation of a feedback matrix k such that s is positively d invariant for the closed loop system it is shown that if polyhedral sets are considered the solution involves simple linear programming algorithms however the procedure suggested requires a great amount of computational work offline if the state space dimension is large because the feedback matrix k is obtained as a solution of a large set of linear inequalities all of the vertices of s are required

locally monotonic regression the concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal the median filter is a suboptimal smoother under this measure of smoothness since a filter pass does necessarily yield a locally monotonic output even if a locally monotonic output does result there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal locally monotonic regression is a technique for the optimal smoothing of finite length discrete real signals under such a criterion a theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given regression is considered as an approximation problem in r sub n the criterion of approximation is derived from a semimetric and the approximating set is the collection of signals sharing the property of being locally monotonic

using dynamic programming for solving variational problems in vision dynamic programming is discussed as an approach to solving variational problems in vision dynamic programming ensures global optimality of the solution is numerically stable and allows for hard constraints to be enforced on the behavior of the solution within a natural and straightforward structure as a specific example of the approach s efficacy applying dynamic programming to the energy minimizing active contours is described the optimization problem is set up as a discrete multistage decision process and is solved by a time delayed discrete dynamic programming algorithm a parallel procedure for decreasing computational costs is discussed

mean field annealing a formalism for constructing gnc like algorithms optimization problems are approached using mean field annealing mfa which is a deterministic approximation using mean field theory and based on peierls s inequality to simulated annealing the mfa mathematics are applied to three different objective function examples in each case mfa produces a minimization algorithm that is a type of graduated nonconvexity when applied to the weak membrane objective mfa results in an algorithm qualitatively identical to the published gnc algorithm one of the examples mfa applied to a piecewise constant objective function is then compared experimentally with the corresponding gnc weak membrane algorithm the mathematics of mfa are shown to provide a powerful and general tool for deriving optimization algorithms

probabilistic output admissible set for systems with time varying uncertainties this paper considers time varying uncertain constrained systems and develops a method for computing a probabilistic output admissible poa set this set consists of the initial states probabilistically assured to satisfy the constraint the time invariant counterpart has already been investigated in hatanaka and takaba computations of probabilistic output admissible set for uncertain constrained systems automatica 44 2 2008 to appear we first define the poa set for time varying uncertainties with finite dimensional probability space then we show that an algorithm similar to hatanaka and takaba computations of probabilistic output admissible set for uncertain constrained systems automatica 44 2 2008 to appear provides the poa set also in the time varying case as long as an upper bound of a what we call future output admissibility foa index is available we moreover present two methods for computing the upper bound of the foa index probabilistic and deterministic methods a numerical simulation demonstrates the effectiveness of our algorithm

the four block model matching problem in l 1 and infinite dimensional linear programming the purpose of this work is fourfold first the reader is introduced to the present state of the theory behind the solution of the general model matching problem in l 1 this solution is based on the fact that the model matching problem can be recast as an infinite dimensional linear programming problem however the transformation into linear programming form is highly nonunique which leads to the second question of discussion namely how to find a good linear programming formulation it is shown that the presently available formulations contain some redundancies that limit the applicability of the theory and lead to linear programming systems containing unnecessary degeneracies the third object of this work is to study problems related to the computational complexity of two different approximation methods for the solution of the infinite dimensional linear programming systems both of these methods are needed in order to get two sided error bounds on the cutoff error one complication is that ev

on the statistical optimality of locally monotonic regression locally monotonic regression is a recently proposed technique for the deterministic smoothing of finite length discrete signals under the smoothing criterion of local monotonicity locally monotonic regression falls within a general framework for the processing of signals that may be characterized in three ways regressions are given by projections that are determined by semimetrics the processed signals meet shape constraints that are defined at the local level and the projections are optimal statistical estimates in the maximum likelihood sense the authors explore the relationship between the geometric and deterministic concept of projection onto generally nonconvex sets and the statistical concept of likelihood with the object of characterizing projections under the family of the p semi metrics as maximum likelihood estimates of signals contaminated with noise from a well known family of exponential densities

regularization of inverse visual problems involving discontinuities inverse problems such as the reconstruction problems that arise in early vision tend to be mathematically ill posed through regularization they may be reformulated as well posed variational principles whose solutions are computable standard regularization theory employs quadratic stabilizing functionals that impose global smoothness constraints on possible solutions discontinuities present serious difficulties to standard regularization however since their reconstruction requires a precise spatial control over the smoothing properties of stabilizers this paper proposes a general class of controlled continuity stabilizers which provide the necessary control over smoothness these nonquadratic stabilizing functionals comprise multiple generalized spline kernels combined with noncontinuous continuity control functions in the context of computational vision they may be thought of as controlled continuity constraints these generic constraints are applicable to visual reconstruction problems that involve both continuous regions and discontinuities for which global smoothness constraints fail

optimization of the l sup spl infin induced norm under full state feedback this paper considers the l sup spl infin optimal control problem i e minimization of the effects of disturbances as measured by the l sup spl infin induced norm earlier work showed that even in the case of full state feedback optimal and near optimal linear controllers may be dynamic and of arbitrarily high order however previous work by the author derived the existence of near optimal nonlinear controllers which are static this paper presents a constructive algorithm for such nonlinear controllers the main idea is to construct a certain subset of the state space such that achieving disturbance rejection is equivalent to restricting the state dynamics to this set a construction of this subset requires the solution of several finite linear programs with the number of variables being at least the number of states but less than the number of states plus the number of controls the concept of controlled invariance plays a central role throughout

constrained regulation of linear systems abstract in this paper the linear constrained regulation problem without an assigned set of initial states is investigated this problem consists of the determination of a linear state feedback control law and of a set of admissible initial states d so that every trajectory of the resulting closed loop system which emanates from the set converges asymptotically to the origin without violation of the control constraints by applying well known results on the positive invariance of polyhedral sets an eigenstructure assignment approach to this problem is established if the number of unstable open loop eigenvalues does not exceed the number of control variables then this approach enables one to derive a linear state feedback stabilizing control law which makes the set of states where the control constraints are respected positively invariant this positively invariant set is the maximal admissible set of initial states for the control law under consideration for the case where the number of control variables is greater than the number of the unstable open loop eigenvalues a modified algorithm is proposed

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

controller design with multiple objectives in this paper we study multi objective control problems that give rise to equivalent convex optimization problems we develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone we present some specialized results on duality theory and we apply them to the study of three multi objective control problems the optimal l sub 1 control with time domain constraints on the response to some fixed input the mixed h sub 2 l sub 1 control problem and the l sub 1 control with magnitude constraint on the frequency response what makes these problems complicated is that they are often equivalent to infinite dimensional optimization problems the characterization of the duality relationship between the primal and dual problem allows us to derive several results these results establish connections with special convex problems linear programming or linear matrix inequality problems uncover finite dimensional structures in the optimal solution when possible and provide finite dimensional approximations to any degree of accuracy when the problem does not appear to have a finite dimensional structure to illustrate the theory and highlight its potential several numerical examples are presented

brief doubly invariant equilibria of linear discrete time games in this paper the notion of doubly invariant di equilibrium is introduced the concept extends controlled and robustly controlled invariance notions to the context of two person dynamic games each player tries to keep the state in a region of state space independently of the actions of the rival player the paper gives existence conditions criteria and algorithms for the determination of di equilibria of linear dynamic games in discrete time two examples illustrate the results the first one is in the area of fault tolerant controller synthesis the second is an application to macroeconomics

optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization the problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case the minimum value of the objective function can be obtained by solving a semi infinite linear programming problem and an iterative procedure for obtaining approximate solutions is introduced application of the l sup 1 optimal problem to robustness is discussed a mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations

brief paper computations of probabilistic output admissible set for uncertain constrained systems this paper considers uncertain constrained systems and develops two algorithms for computing a probabilistic output admissible poa set which is a set of initial states probabilistically assured to satisfy the constraint the first algorithm is inspired by an existing randomized sequential technique the second algorithm alleviates the computational effort based on heuristics the present algorithms terminate in a finite number of iterations and provide a poa set additionally we can obtain information on the size of the resulting set a posteriori a numerical simulation demonstrates the applicability of the poa set to a control system design scheme

state feedback e 1 optimal controllers can be dynamic abstract this paper considers l 1 optimal control problems given by discrete time systems with full state feedback scalar control and scalar disturbance motivation stems from the central role that this problem structure played in the development of the h 2 and h theories first systems with a scalar regulated output are studied singular problems sufficient conditions are given based on the non minimum phase zeros of the transfer function from the control to the regulated output for the existence of a static l 1 optimal controller a simple way to compute the static gain is provided using pole placement ideas it is shown however that having full state information does not prevent the l 1 optimal controller from being dynamic in general and that examples with arbitrarily high order optimal controllers can be easily constructed second problems with two regulated outputs one of them being the scalar control are considered non singular problems it is shown by means of a class of fairly general examples for which exact l 1 optimal solutions are constructed that such problems may not have static controllers that are l 1 optimal thus concluding that a separation structure does not occur in these problems in general

nonlinear l 1 optimal controllers for linear systems in this paper we study the l sub 1 optimal control problem for linear systems we show that by allowing the class of controllers to include nonlinear controllers we can make the closed loop l sub 1 norm strictly smaller than we could do using only linear controllers

worst case analysis of nonlinear systems the authors work out a framework for evaluating the performance of a continuous time nonlinear system when this is quantified as the maximal value at an output port under bounded disturbances the disturbance problem this is useful in computing gain functions and l sub spl infin induced norms which are often used to characterize performance and robustness of feedback systems the approach is variational and relies on the theory of viscosity solutions of hamilton jacobi equations convergence of euler approximation schemes via discrete dynamic programming is established the authors also provide an algorithm to compute upper bounds for value functions differences between the disturbance problem and the optimal control problem are noted and a proof of convergence of approximation schemes for the control problem is given case studies are presented which assess the robustness of a feedback system and the quality of trajectory tracking in the presence of structured uncertainty

brief optimal induced l1 norm state feedback control state feedback synthesis is considered for linear discrete time systems to minimize an upper bound of the closed loop induced l 1 norm a sufficient condition called generalized bounded real inequality gbri is presented to establish such a bound an algorithm similar to the invariant kernel algorithm shamma ieee trans automat control 41 1996 533 544 and the contractive set algorithm blanchini ieee trans automat control 39 1994 428 433 reduces the analysis and synthesis problems to linear programming if the problem is feasible our algorithm gives a polyhedral set that induces a closed loop lyapunov function and leads to feasible control laws an example is given for which the optimal induced l 1 norm achieved by a linear controller is achieved by our synthesis approach

further results on linear nonquadratic optimal control continues an investigation by the authors of minimizing the transient response of a linear system as measured by nonquadratic penalty functions in particular penalty functions which have linear growth first this note shows that the optimal state feedback which minimizes the transient response in the case of no exogenous inputs also minimizes the induced l sup 1 norm in case exogenous inputs are present second it considers the case of constrained systems and derives bounds which establish the stability and performance of receding horizon control laws finally this note illustrates the results for scheduling of reliable manufacturing systems

weak continuity with structural constraints nonlinear regression and nonlinear regularization are two powerful approaches to segmentation and nonlinear filtering in this correspondence we propose a hybrid approach that effectively combines the best of both worlds and can be efficiently implemented via the viterbi algorithm

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

computation of maximal safe sets for switching systems the problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models given the practical and theoretical relevance of the problem finding implementable procedures that could at least approximate the maximal safe set is important to this end we begin by restricting our attention to a special class of hybrid systems switching systems we exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set this procedure requires the computation of a maximal controlled invariant set we then restrict our attention to linear discrete time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets however even for this class of systems the computation may not converge in a finite number of steps we then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained a tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set our procedure is applied to the idle speed regulation problem in engine control to demonstrate its efficiency

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

boundary detection by constrained optimization a statistical framework is used for finding boundaries and for partitioning scenes into homogeneous regions the model is a joint probability distribution for the array of pixel gray levels and an array of labels in boundary finding the labels are binary zero or one representing the absence or presence of boundary elements in partitioning the label values are generic two labels are the same when the corresponding scene locations are considered to belong to the same region the distribution incorporates a measure of disparity between certain spatial features of block pairs of pixel gray levels using the kolmogorov smirnov nonparametric measures of difference between the distributions of these features the number of model parameters is minimized by forbidding label configurations which are assigned probability zero the maximum a posteriori estimator of boundary placements and partitionings is examined the forbidden states introduce constraints into the calculation of these configurations stochastic relaxation methods are extended to accommodate constrained optimization

a quadratic programming approach for solving the l sub 1 multiblock problem the authors present a new method to compute solutions to the general multiblock l sub 1 control problem the method is based on solving a standard h sub 2 problem and a finite dimensional semidefinite quadratic programming problem of appropriate dimension the new method has most of the properties that separately characterize many existing approaches in particular as the dimension of the quadratic programming problem increases this method provides converging upper and lower bounds on the optimal l sub 1 norm and for well posed multiblock problems ensures the convergence in norm of the suboptimal solutions to an optimal l sub 1 solution the new method does not require the computation of the interpolation conditions and it allows the direct computation of the suboptimal controller

l sub 2 state feedback control with a prescribed rate of exponential convergence in this paper we consider the l sub 1 state feedback problem with an internal stability constraint in particular we establish the connection between controlled invariant contractive sets and static control laws that achieve a level of l sub 1 performance as well as a desired unforced rate of convergence we outline two algorithms for computing controlled invariant contractive sets the first is a modification of standard recursive techniques used in the literature whereas the second is based on dynamic games and involves solving an appropriate discrete isaacs recursion the latter approach results in a min max characterization of l sub 1 state feedback controllers we point out that the isaacs recursion provides a one shot as opposed to iterative computation of the optimal l sub 1 performance

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a generalization of median filtering using linear combinations of order statistics we consider a class of nonlinear filters whose output is given by a linear combination of the order statistics of the input sequence assuming a constant signal in white noise the coefficients in the linear combination are chosen to minimize the output mse for several noise distributions it is shown that the optimal order statistic filter osf tends toward the median filter as the noise becomes more impulsive the optimal osf is applied to an actual noisy image and is shown to perform well combining properties of both the averaging and median filters a more general design scheme for applications involving nonconstant signals is also given

fuzzy observer based fuzzy control design for nonlinear systems with persistent bounded disturbances to date nonlinear l gain control problems have not been solved by the conventional control methods for the nonlinear dynamic systems with persistent bounded disturbances this study introduces a fuzzy control design to deal with the nonlinear l gain output feedback control problem first the takagi and sugeno t s fuzzy model is employed to approximate the nonlinear dynamic system next based on the fuzzy model a fuzzy observer based fuzzy controller is developed to minimize the upper bound of l gain of the closed loop system under some linear matrix inequality lmi constraints therefore the nonlinear l gain control problem is transformed into a suboptimal control problem i e to minimize the upper bound of the l gain of the closed loop system subject to some lmi constraints in this situation the nonlinear l gain output feedback control problem can be easily solved by a lmi based optimization method the proposed methods which efficiently attenuate the peak of perturbation due to persistent bounded disturbances extend the l gain control problems from linear dynamic systems to nonlinear dynamic systems

relatively optimal control and its linear implementation motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality roughly optimality for a specific nominal plant initial condition we consider a generic discrete time finite horizon constrained optimal control problem for linear systems and we seek for a state feedback possibly dynamic controller as a fundamental requirement we do not admit preactions or controller state initialization based on the plant initial state and we assume our controller to be time invariant in particular we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory a relatively optimal control is a stabilizing controller such that if initialized at its zero state produces the optimal constrained trajectory for the nominal initial condition of the plant we show that one of such controllers is linear dead beat and its order is equal to the length of the horizon minus the plant order thus of complexity which is known a priori some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed we show that by means of the proposed approach we can face several problems such as optimal point to point operations optimal impulse response and optimal tracking

on positively invariant sets for discrete time linear systems with disturbance an application of maximal disturbance sets this note concerns the investigation of maximal sets with respect to which a set is positively invariant and given a set a some properties of spl delta p i sets are analyzed the approach uses one step of the backward recursion introduced in d alessandro and de santis 1991 where feasibility and optimality conditions for linear dynamical input and or state constrained systems were analyzed in section ii the author gives the main definitions and some basic results in the subsequent section iii some properties of spl delta p i sets are investigated the particular case of polyhedral sets is analyzed in section iv in section v the author shows an application to linear systems with parametric uncertainties in the dynamic matrix a the author develops a technique to determine a set in the parameters space such that for any value of the parameter in the set a given set in the state space is positively invariant in the last section a numerical example is developed

on static feedback for the l 1 and other optimal control problems although e 1 optimal linear state feedback controllers are known to be dynamic it has been shown that suboptimal performance arbitrarily closed to optimal can be achieved by using a static non linear feedback law in this paper this fact is established by using a novel approach which shows that the result is a natural consequence of elementary state space theory the approach is motivated by recent works in active vision systems which have considered a state feedback problem tightly connected with e 1 optimization this problem which has independent interest is discussed in some detail the new formulation of the problem provides additional insight in e 1 state feedback in particular it leads naturally to some extensions which do not follow in a straightforward manner from previous works on the subject

the viterbi optimal runlength constrained approximation nonlinear filter simple nonlinear filters are often used to enforce hard syntactic constraints while remaining close to the observation data e g in the binary case it is common practice to employ iterations of a suitable median or a one pass recursive median openclose or closeopen filter to impose a minimum symbol runlength constraint while remaining faithful to the observation unfortunately these filters are in general suboptimal motivated by this observation we pose the following optimization given a finite alphabet sequence of finite extent y y n sub n 0 sup n 1 find a sequence x x n sub n 0 sup n 1 that minimizes d x y spl sigma sub n 0 sup n 1 d sub n y n x n subject to the following x is piecewise constant of plateau run length spl ges m we show how a suitable reformulation of the problem naturally leads to a simple and efficient viterbi type optimal algorithmic solution we call the resulting nonlinear input output operator the viterbi optimal runlength constrained approximation vorca filter the method can be easily generalized to handle a variety of local syntactic constraints the vorca is optimal computationally efficient and possesses several desirable properties e g idempotence we therefore propose it as an attractive alternative to standard median stack and morphological filtering we also discuss some applications

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

a necessary and sufficient condition for robust bibo stability abstract this paper is concerned with obtaining necessary and sufficient conditions for robustly stabilizing a class of plants characterized by a known linear shift invariant plant with additive perturbations of the form of possibly nonlinear time varying l stable operators we will show that in some sense the small gain theorem is necessary and sufficient even when the perturbations are restiricted to a class of linear operators

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

fast digital locally monotonic regression locally monotonic regression is the optimal counterpart of iterated median filtering in a previous paper restrepo and bovik see ibid vol 41 no 9 p 2796 2810 1993 developed an elegant mathematical framework in which they studied locally monotonic regressions in r sup n the drawback is that the complexity of their algorithms is exponential in n we consider digital locally monotonic regressions in which the output symbols are drawn from a finite alphabet and by making a connection to viterbi decoding provide a fast o a sup 2 spl alpha n algorithm that computes any such regression where a is the size of the digital output alphabet a stands for lomo degree and n is the sample size this is linear in n and it renders the technique applicable in practice

on polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

on the spl lscr sub 1 norm of uncertain linear systems in this paper we present an algorithm for computing the l sub spl infin induced norm of linear systems described by time varying difference equations with uncertain system matrices the approach is based on ideas from convex analysis the convergence of the algorithm is studied and vertex type results related to the stabilization and spl lscr sub 1 control of uncertain discrete time systems are established

efficient robust optimization for robust control with constraints this paper proposes an efficient computational technique for the optimal control of linear discrete time systems subject to bounded disturbances with mixed linear constraints on the states and inputs the problem of computing an optimal state feedback control policy given the current state is non convex a recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program while the reparameterized problem is theoretically tractable the number of variables is quadratic in the number of stages or horizon length n and has no apparent exploitable structure leading to computational time of mathcal o n 6 per iteration of an interior point method we focus on the case when the disturbance set is norm bounded or the linear map of a hypercube and the cost function involves the minimization of a quadratic cost here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem that is the policy optimization problem may be decomposed into a set of coupled finite horizon control problems this decomposition can then be formulated as a highly structured quadratic program solvable by primal dual interior point methods in which each iteration requires mathcal o n 3 time this cubic iteration time can be guaranteed using a riccati based block factorization technique which is standard in discrete time optimal control numerical results are presented using a standard sparse primal dual interior point solver that illustrate the efficiency of this approach

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

infinite dimensional convex optimization in optimal and robust control theory many engineering problems can be shown to be equivalent to solving semidefinite programs sps i e convex optimization problems involving linear matrix inequalities lmis powerful computation tools are available for such problems in the finite dimensional case however the problems arising in optimal and robust control theory are often infinite dimensional and so adequate computation tools are not available the key to tackling such problems with finite computation tools is to have a primal dual formulation of the problem without duality gap in this paper we study infinite dimensional sps and present a lifting technique to recast sps as parameterized linear programs lps this enables the wealth of theoretical tools available for infinite dimensional lps to be extended to infinite dimensional sps in particular we develop some new sufficient conditions for the lack of a duality gap for infinite dimensional sps and give an exact characterization of the primal and dual problems for these cases both primal and dual problems are formed as infinite dimensional sp problems with finite truncations to each giving upper and lower bounds respectively on the exact solution to the infinite dimensional problem thus these results can form the basis of practical computation schemes for infinite dimensional problems which require only finite dimensional computation tools to illustrate the power of these tools we apply the results to two previously unsolved optimization problems namely minimizing the l sub 1 norm of a closed loop system subject to bounds on the frequency response magnitude at a finite number of points and or bounds on the h sub 2 norm

brief set valued methods for linear parameter varying systems we consider stability analysis and state feedback synthesis for linear parameter varying lpv systems lpv systems are defined as linear systems whose dynamics depend on exogenous time varying parameters stability analysis of lpv systems involves testing the stability of a family of possible parameter trajectories similarly stabilization and optimal control of lpv systems involves constructing feedback for a family of possible parameter trajectories a further constraint is that the feedback depends on parameter trajectories in a causal manner previous work has considered lpv systems with unconstrained i e arbitrarily fast parameter variations we consider the utilization of known constraints on the parameter variations such as rate constraints for discrete time lpv systems whose parameters take on a finite set of values we provide constructive computational algorithms which nonconservatively assess stabilizability these algorithms lead to parameter dependent lyapunov functions for stability and parameter dependent nonlinear state feedback for stabilization we further provide algorithms to construct whenever possible nonlinear feedback to achieve a specified level of disturbance rejection as measured by signal magnitude we also consider the dynamic game of minimizing a transient response as measured by a norm summation and provide algorithms to construct the optimal cost and near optimal nonlinear feedback

minimization of the worst case peak to peak gain via dynamic programming state feedback case considers the problem of designing a controller that minimizes the worst case peak to peak gain of a closed loop system in particular we concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude we apply the principle of optimality and derive a dynamic programming formulation of the optimization problem under mild assumptions we show that at each step of the dynamic program the cost to go has the form of a gauge function and can be recursively determined through simple transformations we study both the finite horizon and the infinite horizon case under different information structures the proposed approach allows us to encompass and improve earlier results based on viability theory in particular we present a computational scheme alternative to the standard bisection algorithm or gamma iteration that allows us to compute the exact value of the worst case peak to peak gain for any finite horizon we show that the sequence of finite horizon optimal costs converges as the length of the horizon goes to infinity to the infinite horizon optimal cost the sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem we also show the existence of an optimal state feedback strategy that is globally exponentially stabilizing and derive suboptimal globally exponentially stabilizing strategies from the solutions of finite horizon problems

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

set valued observers and optimal disturbance rejection a set valued observer also called guaranteed state estimator produces a set of possible states based on output measurements and models of exogenous signals we consider the guaranteed state estimation problem for linear time varying systems with a priori magnitude bounds on exogenous signals we provide an algorithm to propagate the set of possible states based on output measurements and show that the centers of these sets provide optimal estimates in an l sup spl infin induced norm sense we then consider the utility of set valued observers for disturbance rejection with output feedback and derive the following general separation structure an optimal controller can consist of a set valued observer followed by a static nonlinear function on the observed set of possible states a general construction of this function is provided in the scalar control case furthermore in the special case of full control i e the number of control inputs equals the number of states optimal output feedback controllers can take the form of an optimal estimate of the full state feedback controller

performance robustness of discrete time systems with structured uncertainty given an interconnection of a nominal discrete time plant and a stabilizing controller together with structured norm bounded nonlinear time varying perturbations necessary and sufficient conditions for robust stability and performance of the system are provided it is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem the resulting stability robustness problem is shown to be equivalent to a simple algebraic one the solution of which provides the desired necessary and sufficient conditions for performance stability robustness these conditions provide an effective tool for robustness analysis and can be applied to a large class of problems in particular it is shown that some known results can be obtained immediately as special cases of these conditions

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

i 1 optimal control of multivariable systems with output norm constraints in this paper we consider the l1 optimal control problem for general rational plants it is shown that for plants with no poles or zeros on the unit circle an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs exactly or approximately optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed loop transfer function no assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators it is shown that constrained problems in which a set of outputs is regulated subject to l norm constraints on another set of outputs can be solved using a slight modification of the same algorithm

robust fuzzy observer based fuzzy control design for nonlinear systems with persistent bounded disturbances a novel decoupled approach unlike the l 2 gain h control case the l gain control problem is dealing with persistent bounded disturbances this study introduces l gain fuzzy observer based fuzzy control design where the premise variables depend on the state variables estimated by a fuzzy observer for nonlinear systems via t s fuzzy models the fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables do not depend on the state variables estimated by a fuzzy observer a novel decoupled method is proposed in this study to transform the non lmi conditions into some lmi forms by the proposed decoupled method and the genetic algorithm the l gain fuzzy observer based fuzzy control problem can be easily solved by an lmi based method

persistent disturbance rejection via static state feedback in contrast with spl hscr sub spl infin and spl hscr sub 2 control theories the problem of persistent disturbance rejection l sup 1 optimal control leads to dynamic controllers even when the states of the plant are available for feedback using viability theory shamma showed 1993 in a nonconstructive way that in the state feedback case the same performance achieved by any dynamic linear time invariant controller can be achieved using memoryless nonlinear state feedback in this paper we give an alternative constructive proof of these results for discrete and continuous time systems the main result of the paper shows that in both cases the l sup 1 norm achieved by any stabilizing state feedback linear dynamic controller can be also achieved using a memoryless variable structure controller

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

convergence of an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states in 1 an algorithm was presented to find an approximant to the maximal state constraint set for a linear discrete time dynamical system with polyhedral state and input hounds here it is shown that the algorithm will yield an approximant arbitrarily close to the maximal state constraint set and the number of iterations is given as a function of the prescribed precision of the approximant

the effect of median filtering on edge estimation and detection in this paper we consider the effect of median prefiltering on the subsequent estimation and detection of edges in digital images where possible a quantitative statistical comparison is made for a number of filters defined with two dimensional geometries in some cases one dimensional analyses are required to illustrate certain points noise images prefiltered by median filters defined with a variety of windowing geometries are used to support the analysis and it is found that median prefiltering improves the performance of both thresholding and zero crossing based edge detectors

two properties of l sub 1 optimal controllers the solution of the l sub 1 sensitivity minimization problem is shown to have two properties which contrast markedly with properties of the solutions to the better known h infinity sensitivity minimization problem or the lqg linear quadratic gaussian problem is given of a one parameter family of first order plants where the order of the l sub 1 optimal compensator can be arbitrarily large and thus it is impossible to bound the order of an l sub 1 optimal compensator in terms of the order of the plant a plant is considered which has a continuous one parameter family of l sub 1 optimal compensators and thus l sub 1 optimal compensators need not be unique the author s two examples are considered to answer two questions left open by m a daleh and j b pearson ibid vol 32 p 314 23 1987

robust fuzzy observer based fuzzy control design for nonlinear discrete time systems with persistent bounded disturbances to date nonlinear l infin gain control problems have not been solved by the conventional control methods for nonlinear discrete time systems with persistent bounded disturbances this study introduces fuzzy observer based fuzzy control design where the premise variables depend on the state variables estimated by a fuzzy observer to deal with the nonlinear l infin gain control problem the fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables depend on the state variables first the takagi sugeno t s fuzzy model is employed to represent the nonlinear discrete time system next based on the fuzzy model a fuzzy observer based fuzzy controller is developed to minimize the upper bound of l infin gain of the closed loop system under some nonlinear matrix inequality non lmi constraints a novel decoupled method is proposed in this study to transform the non lmi conditions into some linear matrix inequality lmi forms by the proposed decoupled method and the genetic algorithm the l infin gain fuzzy observer based fuzzy control problem for the nonlinear discrete time systems can be easily solved by an lmi based optimization method the proposed methods which efficiently attenuate the peak of perturbation due to persistent bounded disturbances extend the l infin gain control problems from linear discrete time systems to nonlinear discrete time systems a simulation example is given to illustrate the design procedures and to confirm the l infin gain performance of the proposed method

internal behaviour provided by nonlinear l1 optimal controllers abstract an l 1 optimal linear time invariant lti compensator may have an order significantly higher than that of the plant even when the state is measurable recently there has been work exploring the use of nonlinear static feedback controllers which provide near optimal performance here we consider a class of nonlinear state feedback controllers and derive superposition like bounds on both the plant state and the controlled output in the event that the plant initial condition the disturbance and the noise are all non zero

global asymptotic behavior and boundedness of positive solutions to an odd order rational difference equation in this note we consider the following high order rational difference equation x n 1 i 1k 1 x n i i 1kx n i n 0 1 where k 3 is odd number x k x k 1 x k 2 x 1 is positive numbers we obtain the boundedness of positive solutions for the above equation and with the perturbation of initial values we mainly use the transformation method to prove that the positive equilibrium point of this equation is globally asymptotically stable