on reliable stabilization via rectangular dilated lmis and dissipativity based certifications in this short note we consider the problem of reliable stabilization for multi channel systems where our main objective is to maintain the robust stability of the perturbed closed loop system when there is a single controller failure in any of the control channels specifically we present a less conservative result than results in the literature in terms of a set of dilated lmis for the reliable state feedback stabilization of the nominal system while a dissipativity based certification is used to extend the stability condition when there is an additive model perturbation in the system finally a numerical example is presented to illustrate the applicability of the proposed technique

on stabilization for discrete linear time varying systems abstract in this paper we consider the stabilization problem for discrete linear time varying systems in an operator theoretic framework by using the complete finiteness of a certain discrete nest algebra we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation this result extends the youla parametrization theorem by only requiring a strong left or a strong right representation but not both the strong stabilization problem is also discussed

on the order of stable compensators abstract by means of two typical examples it is shown that in general the minimal order of a stabilizing stable compensator may not be bounded in terms of the plant order such bounds can only be given in the special case of plants with at most one right half plane zero implications for the simultaneous stabilization problem are elucidated

transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems in this paper we investigate the existence of a compensator which simultaneously renders a given r tuple of multiinput multioutput p times m linear dynamical systems internally stable in particular we parametrize a set of simultaneously stabilizable r tuples of plants and show that provided r leqq max m p the above set is semialgebraic and dense in the space sigma of r tuples of plants furthermore we also consider an extension of the classical pole placement and stabilization problems and investigate the simultaneous partial pole placement problem consequently we consider a suitable topology in and obtain a necessary condition and a sufficient condition for the generic partial pole placement problem surprisingly enough the problem of simultaneously stabilizing a triplet of m times m plants chosen generically is shown equivalent to the problem of partially pole placing a single m times m plant by a stable minimum phase compensator on the other hand the problem of simultaneous

optimal control of hidden markov models with binary observations a hidden markov model hmm consisting of a controlled markov chain and a binary output is considered the binary output reports the occurrence of a certain state versus any other state in the markov chain the transition matrix of the markov chain depends on a control vector which is to be chosen to minimize a discounted infinite horizon cost function the information provided for decision making is the entire history of the binary output which presents an incomplete knowledge of the markov chain and as a consequence leads to a stochastic control problem with partial observations the solution to such a problem is a control policy consisting of two components an estimator to derive the posterior probability distribution of the markov chain from the observation set and a nonlinear control law to map this estimate into the control vector this paper introduces a nonlinear state space representation for the estimator develops analytical expressions for the control law and presents numerical methods for efficient computation of the optimal control application of these results to optimal management of a class of inventory systems is discussed and the performance of optimal control for this application is examined by numerical simulations

global solution branches for equations involving nonhomogeneous operators of p laplace type abstract it is shown that if is not an eigenvalue of an associated p laplacian then the equation div x u u p 2 u f x u u with nonhomogeneous which is assumed to behave asymptotically as the function generating the associated p laplacian has a global branch of solutions u also the case of modified p laplace operators and generalizations thereof are discussed

simultaneous controller design for time varying linear systems abstract in this paper we establish the parametrization of all simultaneously stabilizing controllers for two time varying linear systems a new approach to the simultaneous stabilization of three time varying linear systems is also provided

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

a dilated lmi approach to robust performance analysis of linear time invariant uncertain systems this paper studies robust performance analysis problems of linear time invariant systems affected by real parametric uncertainties in the case where the state space matrices of the system depend affinely on the uncertain parameters it is know that recently developed extended or dilated linear matrix inequalities lmis are effective to assess the robust performance in a less conservative fashion this paper further extends those preceding results and propose a unified way to obtain numerically verifiable dilated lmi conditions even in the case of rational parameter dependence in particular it turns out that the proposed dilated lmis enable us to assess the robust performance via multiaffine parameter dependent lyapunov variables so that less conservative analysis results can be achieved connections among the proposed conditions and existing results are also discussed concretely several existing results can be viewed as particular cases of the proposed conditions

on a general structure of the stabilizing controllers based on stable range in this paper we prove that some stabilizing controllers of a plant which admits a left right coprime factorization have a special form where their stable and unstable parts are separated the dimension of the unstable part depends on the algebraic concept of stable range of the ring a of siso stable plants moreover we prove that if the stable range of a is equal to 1 then every plant defined by a transfer matrix with entries in the quotient field of a and admitting a left right coprime factorization can be stabilized by a stable controller strong stabilization in particular using a result of treil proving that the stable range of h infty mathbb d is equal to 1 we show that every stabilizable plant defined by a transfer matrix with entries in the quotient field of h infty mathbb d or h infty mathbb c is strongly stabilizable and equivalently every couple of stabilizable plants can be simultaneously stabilized by a controller simultaneous stabilization finally using the fact that the topological stable range of h infty mathbb d is equal to 2 a result due to suarez we show that every unstabilizable siso plant defined by a transfer function with entries in the quotient field of h infty mathbb d is as close as we want to a stabilizable plant in the product topology

a global bifurcation for nonlinear equations with nonhomogeneous part abstract we study the following boundary value problem with a nonhomogeneous principal part div u u 0 u p 2 u q x u u in with the dirichlet boundary condition under certain assumptions on and q when 0 is not an eigenvalue of the p laplacian the aim of this paper is to show a bifurcation result on a noncompact component of solutions for the above nonlinear equation

a modification of theorem for the least bound of the minimum degree of interpolating unit abstract theorem 1 for the least bound on the degree of interpolating unit in 1 requires to check both positive definiteness and negative definiteness of the nevanlinna pick matrix

control limits for two state partially observable markov decision processes abstract a production process can be in either a good or a bad state the true state is unknown and can only be inferred from observations if the state is good during one period it may deteriorate and become bad during the next period two actions are available continue or replace for a fixed cost the objective is to maximize the expected discounted value of the total future profits we prove that dominance in expectation the expected profit is larger in the good state than in the bad state suffices for the optimal policy to be of a control limit clt type continue if and only if the good state probability exceeds the clt this condition is weaker than stochastic dominance which has been prevailing we also show that the expected profit function is convex strictly increasing

existence of radial solutions for an asymptotically linear p laplacian problem abstract we prove that an asymptotically linear dirichlet problem which involves the p laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the p derivative of the nonlinearity includes at least the first j radial eigenvalues of the p laplacian operator the main tools that we use are a uniqueness result for the p laplacian operator and bifurcation theory

some results on the problems of decentralized reliable stabilization concerns the problems of decentralized reliable control in linear time invariant multivariable control systems by the stable factorization approach decentralized reliable control is defined as the problem in which the closed loop system remains stable in spite of failure in the interconnection between the subplants based on the decentralized parametric characterization developed by desoer and gundes in 1988 a necessary and sufficient condition for the decentralized version of the strong stabilization problem is first proposed next it is shown that the decentralized reliable stabilization problem is solvable if and only if the augmented plant is decentralized strongly stabilizable moreover it is concluded that the problem of centralized reliable multiple control can be viewed as the problem of decentralized reliable stabilization of an augmented plant

partially observable markov decision processes a geometric technique and analysis this paper presents a novel framework for studying partially observable markov decision processes pomdps with finite state action observation sets and discounted rewards the new framework is solely based on future reward vectors associated with future policies which is more parsimonious than the traditional framework based on belief vectors it reveals the connection between the pomdp problem and two computational geometry problems i e finding the vertices of a convex hull and finding the minkowski sum of convex polytopes which can help solve the pomdp problem more efficiently the new framework can clarify some existing algorithms over both finite and infinite horizons and shed new light on them it also facilitates the comparison of pomdps with respect to their degree of observability as a useful structural result

simultaneous partial pole placement a new approach to multimode system design simultaneous partial pole placement of a family of single input single output plants is proposed as a generalization of the classical pole placement and stabilization problems this problem finds application in the design of a compensator for a family of linear dynamical systems in this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable minimum phase rational functions with real coefficients a necessary condition for the solvability of the associated transcendental problem is obtained finally a counterexample to the following conjecture is obtained pairs of simultaneously stabilizable plants of bounded mcmillan degree have simultaneously stabilizing compensators of bounded mcmillan degree

stabilizability and existence of system representations for discrete time time varying systems in this paper right and left representations as an alternate but equivalent framework to coprime factorizations of operators are developed the main theorem of the paper establishes that a linear time varying discrete time plant is stabilizable if and only if its graph can be represented as the range respectively kernel of a causal bounded operator which is left respectively right invertible the proof relies on certain factorization theorems of arveson for nest algebras the paper extends the youla parametrization of all stabilizing compensators to this framework also it is proven that a time invariant plant that is not stabilizable by a time invariant compensator is not stabilizable with a time varying compensator an example of a time varying plant of feintuch is considered and shown to be not stabilizable finally the continuous time case is examined and the problems encountered in extending the proof are discussed however it is shown that a stabilizable plant must have a closed graph

simultaneous stabilization of mimo systems with integral action controllers simultaneous stabilization with asymptotic tracking of step input references is explored for linear time invariant multi input multi output stable plants necessary conditions are presented for existence of simultaneous integral action controllers and existence of simultaneous pid controllers a systematic simultaneous pid controller synthesis method is proposed under a sufficient condition

a counter example for the conjecture in an algorithm for interpolation with units in h abstract it is conjectured by dorato et al automatica 25 427 430 1989 that the algorithm given by them for interpolation with units in h gives the lowest possible degree unit a counter example presented in this technical communication shows that this is not the case for real as well as complex interpolation data

global bifurcation for equations involving nonhomogeneous operators in an unbounded domain abstract we study the structure of the set of solutions of a nonlinear equation involving nonhomogeneous operators div x u u 0 g x u p 2 u f x u in r n satisfying certain conditions on g and f when 0 is not an eigenvalue of the p laplacian in some sense this is based on a bifurcation result on noncompact connected sets of solutions for nonlinear operator equations

design of stable controllers attaining low h sup infinity weighted sensitivity strong stabilization and sensitivity reduction for a linear time invariant single input single output system are considered stable controllers are practically desirable in the presence of possible actuator or sensor failures it is verified precisely that the optimal controller of h sup infinity weighted sensitivity minimization is almost always unstable when the plant has more than two zeros in the right half plane an algorithm for attaining the low sensitivity property for stable controllers is presented it is based on nevanlinna pick interpolation

the stable rank of the algebra h equals 1 abstract it is proved that if the functions f g h f 1 g 1 satisfy the carleson corona condition z inf f z g z def o then there exists a function g h such that the fucntion f gg is invertible in h and moreover g c 1 c where the constant c depends only on

on the order of simultaneously stabilizing compensators in this paper simultaneous strong stabilization problem is considered and it is shown that there is no upper bound for the minimal order of a simultaneously strongly stabilizing compensator in terms of the plant orders a similar problem was also considered by smith et al 1986 where it was shown that such a bound does not exist for the strong stabilization problem of a single plant but the examples given in the article were forcing an approximate unstable pole zero cancellation or forcing the distance between two distinct unstable zeros to go to zero in this paper it is shown that 1 if approximate unstable pole zero cancellation does not occur and the distances between distinct unstable zeros are bounded below by a positive constant then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator and 2 for the simultaneous strong stabilization problem even for the two plant case such a bound cannot be found

characterization of controllers in simultaneous stabilization this paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants the result represent a generalization of the youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function although as expected the additional constraints for simultaneous stabilization are not readily incorporated into h h2 optimization procedures as in the standard case there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants

two sided tangential interpolation with real rational units in h the problem of computing units in rh that satisfy ball gohberg rodman two sided tangential interpolation conditions over the open right half complex plane is considered we first show that every unit in rh can be written up to left multiplication by an orthogonal matrix as a finite product of positive real functions in rh with invertible value at this factorization is used to recast interpolation conditions on units as coupled positive real interpolation problems for the factors an algebraic necessary and sufficient condition for the existence of solutions and a parameterization of all solutions when one exists are given we introduce a multivariable parity interlacing property for left interpolation data and present an algorithm to compute units the algorithm involves a line search and is guaranteed to terminate in a finite number of operations when the data set has parity interlacing property as an application strong stabilization of lti plants is discussed a procedure to construct a stable stabilizing controller is given the results are illustrated with a simple mimo example a necessary and sufficient condition characterizing all real rational units that satisfy tangential interpolation conditions is given an algorithm to compute units for left interpolation is presented

on least degree unit interpolation in rh examples are presented that give insight and suggest why it is difficult to develop a unit interpolation algorithm that will yield an interpolating unit in rh of least degree

some results on simultaneous stabilization this paper considers the simultaneous stabilization problem and gives an answer to it in terms of the poles and zeros on the right half plane which is considered as an extension of the conditions previously reported

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

the optimal search for a moving target when the search path is constrained a search is conducted for a target moving in discrete time among a finite number of cells according to a known markov process the searcher must choose one cell in which to search in each time period the set of cells available for search depends upon the cell chosen in the last time period the problem is to find a searcher path i e a sequence of search cells that maximizes the probability of detecting the target in a fixed number of time periods we formulate the problem as a partially observable markov decision process and present a finite time horizon pomdp solution technique which is simpler than the standard linear programming methods

ordered solutions for dynamic programs suppose x indexes a collection of optimization problems in rn and x is an optimal program for the problem indexed by x x sufficient conditions are presented for two other families of optimization problems to possess optimal programs 0x and 0x for the respective problems when x x such that 0x x 0x for all x x this result is then applied to obtain bounds on an optimal policy in the class of dynamic programs which are bounded monotone contractions

the stabilization problem for discrete time varying linear systems in this paper we consider the strong stabilization problem for a time varying linear system which admits a coprime factorization we also study the simultaneous and robust stabilization problems for this class of systems

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

simultaneously stabilising controllers for time varying linear systems within the framework of nest algebras abstractfrom the perspective of strong transitivity a controller design method is provided to simultaneously stabilise a collection of time varying linear systems within the framework of nest algebras in particular all simultaneously stabilising controllers for a class of linear plants are characterised based on the doubly coprime factorisations these results hold as well in the time invariant case an illustrative example is given to demonstrate the validity of the method

a two state partially observable markov decision process with three actions abstract a process can be in either a stable or an unstable state interchangeably the true state is unobservable and can only be inferred from observations three actions are available continue with the process con repair the process for a certain fee bring the process to the stable state rep and obtain the state of the process for a cost ins the objective is to maximize the expected discounted value of the total future profits we formulate the problem as a discrete time partially observable markov decision process pomdp we show that the expected profit function is convex and strictly increasing and that the optimal policy has either one or two control limits also we show that dominance in expectation the expected revenue is larger in the stable state than in the unstable state suffices for a control limit structure

brief on the strong stabilization and stable h controller design problems for mimo systems a state space based design method is developed to construct strongly stabilizing controllers for a class of mimo plants the problem is reduced to solving an algebraic riccati equation are which appears in the solution of a two block h problem strongly stabilizing controllers designed here are of the same order as the plant two sets of finite dimensional strongly stabilizing controllers are parameterized in terms of linear fractional transformations further optimization guidelines are given to find stable h controllers from these parameterizations stability robustness margins with respect to coprime factor uncertainties are determined for the controllers designed from the proposed method

on the synthesis of stable spl hscr sup spl infin controllers a sufficient condition is derived for the existence of a stable suboptimal spl hscr sup spl infin controller it is shown that if a certain algebraic riccati equation determined from a parameterization of all suboptimal spl hscr sup spl infin controllers has a positive definite stabilizing solution then a stable spl hscr sup spl infin controller of order 2n can be constructed where n is the order of the generalized plant

transitivity in simultaneous stabilization for a family of time varying systems abstract this paper develops necessary and sufficient conditions for the so called transitivity and strong transitivity precise definitions later for a family of discrete time varying linear systems which when satisfied lead to characterizations for simultaneous stabilizability and simultaneously stabilizing feedback controllers some criteria for the strong transitivity are established in terms of single coprime factorizations of only one plant and one controller the resulting characterization involves coprime factorizations of fewer systems compared to previous work

a dual look at unit interpolation in h using mappings of the form 1 bf a 1 bf a and 1 bf 1 bf a we get three different unit interpolation algorithms respectively first second and third the second algorithm yields a unit in h with arbitrary specified left half j plane zeros instead of arbitrary specified left half s plane poles as in youla et al 1974 and vidyasagar 1985 with the first algorithm the third algorithm yields a unit in h with considerably lower degree in the cases where a is required to be greater than one with either the first or second algorithm however by using alternative steps from the first and second algorithms judiciously as shown by a numerical example one can reduce the order of the unit considerably compared with the third algorithm the usefulness of the second algorithm in the control system context is that the closed loop system poles can be specified arbitrarily in the lhp in a strong stabilization problem which is the opposite to the first algorithm of youla et al 1974 and vid

an algorithm for interpolation with units in h with applications to feedback stabilization abstract the problem of finding a unit in h i e an h function the inverse of which is also h which interpolates to given points in the right half plane has important applications in a number of feedback stabilization problems i e strong stabilization and simultaneous stabilization an algorithm for computing such interpolating units is presented which requires a lower degree unit than other available algorithms

indirect adaptive techniques for fixed controller performance enhancement abstract this paper develops the adaptive disturbance estimate feedback schemes of a companion paper for enhancing the performance of controllers designed by off line techniques the developments are based on the parametrization theory for the class of all stabilizing controllers for a nominal plant and the dual class of plants stabilized by a nominal controller such parametrization allows us conveniently to parametrize plant uncertainties for on line identification and control purposes minimizing the effects of unmodelled dynamics based on these parametrizations along with prefiltering which minimizes the effect of unmodelled dynamics standard adaptive stabilization adaptive pole assignment or adaptive linear quadratic schemes are shown to achieve controller enhancement the idea is to exploit a priori information about a plant and design objectives in an off line design and yet exploit the power of adaptive techniques to learn and tune on line attention is focused on techniques for fixed but u

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

unit interpolation in h bounds of norm and degree of interpolants abstract this paper studies the problem of unit interpolation in h specifically bounds of the norm and degree of interpolants are obtained using the nevanlinna pick interpolation theory also simpler bounds are given using the poincare metric these results indicate that strong stabilization and simultaneous stabilization need a cautious approach if the system is ill conditioned in fact it is demonstrated that the degree of stable compensator is not bounded for the class of plants with fixed degree also an example shows that there is a class of plants for which the sensitivity can be made small only if unstable compensators are used

on strong stabilization for linear time varying systems abstract this paper deals with the strong stabilization problem for linear time varying systems and gives a sufficient condition in terms of the coprime factors for the existence of strong stabilizers for such a system

the transitivity in simultaneous stabilization this paper deals with the transmission problem in simultaneous stabilization of time varying linear systems we establish two criteria for the transitivity in simultaneous stabilization in particular we give a necessary and sufficient condition for certain linear systems to satisfy a strengthened version of transitivity

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

a two state partially observable markov decision process with uniformly distributed observations a controller observes a production system periodically over time if the system is in the good state during one period there is a constant probability that it will deteriorate and be in the bad state during the next period and remains there the true state of the system is unobservable and can only be inferred from observations quality of output two actions are available continue or replace for a fixed cost the objective is to maximize the expected discounted value of the total future income for both the finite and infinite horizon problems the optimal policy is of a control limit clt type continue if the good state probability exceeds the clt and replace otherwise the computation of the clt involves a functional equation an analytical solution for this equation is as yet unknown for uniformly distributed observations we obtain the infinite horizon clt analytically we also show that the finite horizon clts as a function of the time remaining are not necessarily monotone which is counterintuitive