quantization of filter bank frame expansions through moving horizon optimization this paper describes a novel approach to quantization in oversampled filter banks the new technique is based on moving horizon optimization does not rely on an additive white noise quantization model and allows stability to be explicitly enforced in the associated nonlinear feedback loop moreover the quantization structure proposed here includes sigmadelta and linear predictive subband quantizers as a special case and in general outperforms them

simultaneous stabilization via static output feedback and state feedback in this paper the simultaneous stabilization problem is considered using the matrix inequality approach some necessary and sufficient conditions for simultaneous stabilizability of strictly proper multi input multi output mimo plants via static output feedback and state feedback are obtained in the form of coupled algebraic riccati inequalities it is shown that any such stabilizing feedback gain is the solution of some coupled linear quadratic control problems where every cost functional has a suitable cross term a heuristic iterative algorithm based on the linear matrix inequality technique is presented to solve the coupled matrix inequalities the effectiveness of the approach is illustrated by numerical examples

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

simultaneous stabilization using stable system inversion abstract this paper investigates the use of stable system inversion to design controllers for the simultaneous stabilization of multi input multi output systems a parametrization of all simultaneously stabilizing controllers for systems admitting a stable inverse is developed the controller is shown to have the structure of a generalized observer for strictly proper systems an intermediate step of the design requires solving a sequence of strong stabilization subproblems for systems with components having identical stable minimum phase transfer functions the design requires only a single model of the component in the controller in systems where a stable inverse fails to exist an interpolation scheme is proposed to develop a stabilizing controller

brief on simultaneous h control and strong h stabilization in this note the simultaneous h control problem is addressed it is first shown that the existence of a single controller which simultaneously stabilizes two systems with a given h norm level is equivalent to that of a strong h stabilization controller of an associated system moreover the result is extended to the case involving r 2 systems then a sufficient condition for the strong h stabilization is proposed based on the bounded real lemma finally a simple numerical example is presented to show the effectiveness of the 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

feedback system design the tracking and disturbance rejection problems the problem of designing a compensator for a specified plant which simultaneously stabilizes the resultant feedback system and causes it to track a prescribed family of inputs and or reject prescribed disturbances is considered a set of linear design equations in the space of stable systems is formulated in a general linear systems setting and an explicit parameterization of the resultant solution space is obtained for a class of generalized multivariate systems the theory is illustrated with several single and multivariate examples

observer parameterization for simultaneous observation the stable inverse approach is used to obtain the observers for the simultaneous observation of a given set of plants a parameterization in terms of a stable inverse and a stable null space is proposed for all simultaneous observers to verify the effectiveness of the proposed method a design example is also given

simultaneous stabilization of linear and nonlinear systems by means of nonlinear state feedback the simultaneous stabilization resp asymptotic stabilization of a countable family of control systems consists of finding a control which stabilizes resp asymptotically stabilizes all the systems in the family in this paper we introduce a new method which enables us to show that given any countable family of stabilizable nonlinear systems there exists a continuous state feedback law which simultaneously stabilizes not asymptotically the family then by enriching this method we prove that any finite family of stabilizable linear time invariant lti systems can be simultaneously exponentially stabilized by means of nonlinear time varying state feedback we also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of lti systems finally sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems

on bounds for the existence of a bistable controller abstract by s r we shall denote the set of polynomials nonzero on the closed disk z z r the problem to be studied is given as follows avoidance problem given polynomials a and b nonnegative integers n m and positive numbers r 1 r 2 r 3 determine if there exist p s r 1 q s r 2 with degrees n and m respectively such that ap bq is in s r 3 it is shown that the avoidance problem for r 1 r 2 r 3 1 is equivalent to the well known bistable stabilization problem for a large class of control systems also a general class of optimization problems for solving the avoidance problem is introduced the class of optimization problems includes problems with sets more general than disks s r in addition an algorithm is proposed for solving these optimization problems and numerical experience is reported as an application bistable controllers for the plant p z z z 2 are shown to exist for values of as low as 0 0246 thus improving on previously reported results by blondel et al for this difficult case furthermore it is shown how in some cases it is possible to find optimal controllers by using a combination of numerical and algebraic techniques and a variant of the algorithm already introduced as an example a low order controller is found for p for the case when 7 4 3 0 0717 this method could be applied to other problems where the plant depends on a parameter

on an open problem in bistable stabilization we propose an equivalent representation of an open problem in bistable stabilization using this new formulation we derive two simple necessary condition tests for the solution of the problem for certain special cases we also obtain the sufficient conditions

all constant gain stabilizing controllers for an interval delay system with uncertain parameters a computationally tractable design method is presented which explicitly yields the entire set of all constant gain controllers which robustly stabilize a family of systems with uncertainty a desired optimal controller may then be selected from the feasible set the method applies to the class of systems where a delay element e hs with interval finite or infinite uncertainty about h is followed by a plant characterized by a rational transfer function with independent interval coefficients illustrative numerical examples are provided

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

on the relationship between the model order reduction problem and the simultaneous stabilization problem this note points out the fact that under certain conditions the model order reduction problem is very closely tied to the simultaneous stabilization problem these conditions occur when the reduced order model is to be used to design a suboptimal controller for a high order plant under these conditions the controller must not only stabilize the low order model but must also stabilize the high order plant the result concerning the existence of a controller that will simultaneously stabilize two plants is used to conclude that the approximate inclusion of any unstable real modes of the high order plant in the low order model will guarantee the existence of such a controller

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

simultaneous and strong simultaneous stabilisation of some classes of mimo systems sufficient conditions are derived for simultaneous stabilisability of any finite number of linear time invariant multi input multi output mimo systems four general mimo plant classes with arbitrary finite number of plants are shown to be simultaneously stabilisable using simple proportional derivative controllers for some of these plant classes it is also possible to achieve simultaneous stabilisation and asymptotic tracking of step input references with zero steady state error by using integral action controllers three of these general mimo plant classes are shown to be strongly simultaneously stabilisable

simultaneous observation of linear systems this paper considers the problem of designing an observer which can observe the states for each of a given set of plants the concept of the simultaneous observation is introduced the simultaneous observation is achieved using coprime factorization techniques necessary and sufficient conditions for the existence of simultaneous observation are derived the observer design is discussed and an example is given

simultaneous stabilization for a collection of single input nonlinear systems this paper presents a novel method for designing a controller that simultaneously stabilizes a collection of single input nonlinear systems the control lyapunov function approach is used to derive necessary and sufficient conditions for the existence of time invariant simultaneously stabilizing state feedback controllers additionally a universal formula for constructing a continuous simultaneously stabilizing controller when the provided sufficient condition is satisfied is presented for any collection of second order and third order feedback linearizable systems in canonical form global simultaneous stabilization via a single state feedback controller is shown to be always possible two examples are included for illustration

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

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

infinitely many solutions for p laplacian equation involving critical sobolev growth in this paper we will prove the existence of infinitely many solutions for the following elliptic problem with critical sobolev growth pu u p 2u u p 2uin u 0on provided n p2 p where p is the p laplacian operator 1 0 and is an open bounded domain in rn

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

some new results on strong gamma_k gamma_ cl h_ infty stabilization the strong gammak gammacl hinfin stabilization problem is addressed in this paper based on the bounded real lemma an improved sufficient condition on strong hinfin stabilization as well as its dual form is derived which extends the are constraint in the former results into an inequality constraint two path following algorithms are proposed to solve the associated optimization problems with bmi constraints which lead to full order controllers a numerical example is presented to demonstrate the effectiveness of the result

simultaneous stabilization and robust control of polynomial nonlinear systems using sos techniques this technical note is concerned with the simultaneous stabilization and robust performance control for a class of polynomial nonlinear systems building on the tenets of state dependent polynomial lyapunov functions we present sufficient conditions for simultaneous stabilization with and without hinfin performance these conditions can be verified by the recently developed sum of squares technique which essentially solves a linear matrix inequality feasibility problem

brief some applications of randomized algorithms for control system design in this paper a few difficult problems related to simultaneous stabilization of three plants equivalent to a certain problem related to unit interpolation in h have been addressed through the framework of randomized algorithms these problems which were proposed by blondel simultaneous stabilization of linear systems springer berlin 1994 and blondel and gevers math control signals systems 6 1994 135 concern the existence of a controller

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 generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

robust stabilization of distributed systems abstract this paper is devoted to the problem of robust stabilization for a class of distributed plants we consider the gain margin optimization problem and the robust stabilization problem for multiplicative perturbations using techniques from interpolation theory and complex variables we obtain explicit necessary and sufficient conditions for robust stabilizability a few examples are included to illustrate the results

on the simultaneous stabilization of three or more plants in this note the problem of the simultaneous stabilization of three multivariable plants is addressed we consider the general case where the existence of a unit controller cannot be used as a sufficient condition to guarantee the existence of a simultaneous controller for three multivariable plants the sufficient conditions given in this note lead to a constructive controller design to stabilize simultaneously three multivariable plants a generalization is proposed for stabilizing simultaneously n multivariable plants

stabilization via nonsmooth nonconvex optimization nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed order controller design we support this claim by applying nonsmooth analysis and methods to a challenging belgian chocolate stabilization problem posed in 1994 find a stable minimum phase rational controller that stabilizes a specified second order plant although easily stated this particular problem remained unsolved until 2002 when a solution was found using an eleventh order controller our computational methods find a stabilizing third order controller without difficulty suggesting explicit formulas for the controller and for the closed loop system which has only one pole with multiplicity 5 furthermore our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane although the focus of the paper is stabilization once a stabilizing controller is obtained the same computational techniques can be used to optimize various measures of the closed loop system including its complex stability radius or hinfin performance

brief paper pid controller synthesis for a class of unstable mimo plants with i o delays conditions are presented for closed loop stabilizability of linear time invariant lti multi input multi output mimo plants with i o delays time delays in the input and or output channels using pid proportional integral derivative controllers we show that systems with at most two unstable poles can be stabilized by pid controllers provided a small gain condition is satisfied for systems with only one unstable pole this condition is equivalent to having sufficiently small delay unstable pole product our method of synthesis of such controllers identify some free parameters that can be used to satisfy further design criteria than stability

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

an algebraic theory for design of controllers for linear multivariable systems part i structure matrices and feedforward design in this two part paper a theory is presented in which several control problems can be solved the theory is applicable to a general class of linear multivariable systems where the measured output does not have to be equal to the controlled output or the state the system may be affected by nonmeasurable disturbances only controllers which stabilize the system and have proper transfer functions are allowed i e the controllers have to be physically realizable it is shown that the solutions to control problems of servo type e g problems of model matching decoupling and invertibility are special cases of the solution to a more general problem analogously the solutions to problems of regulator type e g disturbance decoupling output regulation and pole placement are also speclal cases of the solution to a more general problem it is shown that problems of servo type and regulator type can be solved independently of each other in part i generalized polynomials are introduced as a mathematical framework structue matrices which describe how well a system can be controlled are defined finally problems of servo type are solved part ii mainly deals with problems of regulator type

robust control of the characteristic values of systems with possible parameter variations an analytical method to control by multiparameter output feedback the characteristic values of a closed loop multi input multi output system inside or outside general aubregions of the complex plane is presented the method pertains to analogue as well as discrete systems the design method is robust in the sense that the characteristic values are designed to cluster inside the desired subregion even though any number of the specifying parameters are only known to within a given interval the interval may possibly be disjoint and possibly infinite completely unknown examples are provided which illustrate the broadness and effectiveness of the method

conditions for optimality of naive quantized finite horizon control this paper presents properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme this naive quantized control law underlies many popular algorithms such as converters and decision feedback equalities and is easily shown to be globally optimal for horizon one however the question arises as to whether it is also globally optimal for horizons greater than one i e whether it solves a finite horizon quadratic programme where decision variables are restricted to belonging to a quantized set by using dynamic programming we develop sufficient conditions for this to hold the present analysis is restricted to first order plants however this case already raises a number of highly non trivial issues the results can be applied to arbitrary horizons and quantized sets which may contain a finite or an infinite though countable number of elements

simultaneous stabilization with fixed closed loop characteristic polynomial a solution is given to the problem of simultaneously stabilizing m single input single output plants by one dynamic compensator with the constraint that each closed loop system has the same characteristic polynomial

robust stabilizability for linear systems with both parameter variation and unstructured uncertainty the authors consider the problem of finding a single compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty an uncertain system model is proposed to characterize plants containing those two different uncertainties closed loop stability criteria are then proposed for the model robust stabilizability for single input single output siso systems is studied a number of assumptions describing the set of admissable siso plants are imposed the satisfaction of these assumptions guarantees the existence of a proper stable compensator c s achieving robust stabilization the algorithm used for controller design is recursive in nature and allows the designer to select one compensator coefficient at a time also the minimum order of the desired stabilizer c s depends only on the largest relative degree of the nominal plants which often leads to a lower order compensator

synthesis of linear multivariable regulators this paper generalizes the results obtained by the authors in 1 and presents new results on the construction of proper controllers for linear multivariable regulators these results may lead to efficient computation methods that do not involve smith forms or the order relations used in 1 to construct proper controllers the methods are based on the recently developed theory of generalized polynomials 2 we discuss several numerical algorithms and how they can be applied to the regulator problem we also point out an important computational problem in need of an efficient numerical algorithm

a topological result on strong stabilization problem this note studies the robustness of nonstrong stabilizability we consider single input single output siso continuous time linear systems under graph topology a key contribution of this note is to show that there are two classes of nonstrongly stabilizable systems one which contains plants that can be rendered strongly stabilizable by an arbitrary small perturbation and the other which contains plants that are inherently nonstrongly stabilizable even if perturbed an easily implemented test referred to as weak parity interlacing property is developed to check in which class a given model lies

simultaneous stabilization of single input single output discrete time systems the problem of finding one compensator which simultaneously stabilizes a family of single input single output siso discrete time plants is considered the family of plants is described by the transfer functions pz q q in q which are generated via a z transformation a number of assumptions describing the set of allowable plants are then given these assumptions include some regularity conditions on the plants and a minimum phase requirement the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c z for simultaneous stabilization an iterative computation method is provided for control design

simultaneous stabilization of multiplicatively perturbed plants necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear time invariant multi input multi output nominal plant and a multiplicatively perturbed plant where the multiplicative perturbation is stable these conditions are derived from the general parity interlacing property applicable to any two arbitrary plants and are expressed explicitly in terms of the real axis poles and zeros of the nominal plant and the perturbation the class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions a special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation

simultaneous stabilization of linear single input single output plants the problem of designing a single compensator that simultaneously stabilizes several distinct plants is considered here using the fractional representation approach owing to saeks and murray 1982 1983 necessary and sufficient conditions for the existence of such a compensator are given and a design algorithm is proposed

p laplacian equations in rn with finite potential via truncation method the critical case abstract in this paper we consider the problem p u a x u p 2 u u p 2 u u q 2 u in r n where 1 p n max p p 1 q p n p n p 0 p is the p laplacian operator and the potential a x is positive and verifies suitable decay assumptions we prove that the problem has infinitely many solutions by using truncation method

pid stabilization of mimo plants closed loop stabilization using proportional integral derivative pid controllers is investigated for linear multiple input multiple output mimo plants general necessary conditions for existence of pid controllers are derived several plant classes that admit pid con trollers are explicitly described plants with only one or two unstable zeros at or close to the origin alternatively at or close to infinity as well as plants with only one or two unstable poles which are at or close to origin are among these classes systematic pid controller synthesis procedures are developed for these classes of plants

on minimal degree simultaneous pole assignment problems in this paper we show that a generic r tuple of m input p output linear systems is simultaneously pole assignable if r m p and the mcmillan degrees of the systems are not too different we also obtain upper bounds for the degrees of the compensators which simultaneously assign the characteristic polynomials of the r tuple of closed loop systems the upper bounds are obtained for each of the two cases r max m p and max m p r m p 2002 elsevier science inc all rights reserved

how good is quantized model predictive control with horizon one model predictive control is increasingly being used in areas where decision variables are constrained to finite or countably infinite sets well known fields include power electronics signal processing and telecommunications typically the applications utilize high speed sampling and thus there is an incentive to reduce computational burden one way of achieving this is to use small optimization horizons this raises the question as to the optimality and performance of control laws with short horizons in this paper we give necessary and sufficient conditions for horizon one quantized model predictive control to be equivalent to the use of larger horizons we also explore situations where horizon one is near optimal

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

new sufficient conditions for strong simultaneous stabilization abstract we present sufficient conditions for the existence of a single stable controller to stabilize a set of n siso plants p 1 p 2 p n strong simultaneous stabilization as is well known this is equivalent to the existence of a single controller not necessarily stable to stabilize n 1 plants simultaneous stabilization the basic assumption required in this paper is that all the plants have the same unstable zeros a practical example of where such an assumption holds is when each plant has the same pure delay which is approximated by a non minimum phase rational function the results here are applied to examples of this type

non smooth simultaneous stabilization of nonlinear systems an interpolation method abstract in this paper we introduce a method which enables us to construct a continuous simultaneous stabilizer for pairs of systems in r 2 which cannot be simultaneously stabilized by means of c 1 feedback we extend this method to higher dimensional systems and show that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized not asymptotically by means of continuous feedback

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

exact decay estimate on solutions of critical p laplacian equations in r n abstract consider the critical p laplacian equation in r n p u 0 in r n u p 2 u n u p 2 u on r n 1 where r n r n 1 0 1 p n p n 1 p n p is the critical exponent of the sobolev imbedding from w 1 p r n to l q r n 1 p q p and p is the p laplacian operator p u u p 2 u we prove polynomial decay of the solutions u x c 1 x n p p 1 for x r n the decay exponent is the best possible

algebraic theory for robust stability of interconnected systems necessary and sufficient conditions we consider an interconnected system s o made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions h given that the k th subsystem is perturbed from g k n rk d k 1 to tilde g k n rk delta n rk d k delta d k 1 and that the system s o is h stable we derive a computationally efficient necessary and sufficient condition for the h stability of the perturbed system these fractional perturbations are more general than the conventional additive and multiplicative perturbations the result is generalized to handle simultaneous perturbations of two or more subsystems

a condition for robust stabilizability a family of linear time invariant finite dimensional plants is called robustly stabilizable if there exists a single controller that stabilizes each plant in the family a condition for robust stabilizability is given that requires the plant transmission zeros to remain in the left half complex plane and moreover imposes a restriction on the high frequency behavior of the plant transfer matrix

an iterative design procedure for simultaneous stabilization of mimo systems this paper considers the problem of finding one compensator which simultaneously stabilizes a family of mimo plants the family of plants bdp s q q q represented by their transfer function matrices may be uncountable thus allowing for the possibility of a continuum of variations in plant parameters a hermite invariant factorization of the plant is introduced and shown to be quite useful as far as stabilization is concerned to obtain the desired results a number of assumptions describing the set of allowable plants are imposed these assumptions include some regularity conditions on the plant and mimo versions of minimum phase and one sign high frequency gain requirements the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c s guaranteeing simultaneous stabilization the algorithm used for controller design has one most attractive feature namely it is recursive in nature and allows the designer to select one compensator coefficient at a time

stability analysis of quadratic mpc with a discrete input alphabet we study stability of model predictive control mpc with a quadratic cost function for lti systems with a discrete input alphabet since this kind of systems may present a steady state error the focus is on practical stability i e ultimate boundedness of solutions to derive sufficient conditions for practical stability and characterize the ultimately invariant set we analyze the one step horizon solution and adapt tools used for convex mpc formulations

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

simultaneous stabilization via linear state feedback control the problem of simultaneously stabilizing a finite collection of plants via a single compensator utilizing full state information is considered the authors assume that the complete state can be measured and seek a static linear state feedback control law sufficient conditions for such full state gains to exist are given and they show via several examples how stabilizing compensators can be designed

simultaneous stabilization for a set of boolean control networks abstract this paper investigates simultaneous stabilization of a collection of boolean control networks bcns by using the semi tensor product method and presents a number of new results first an algebraic expression of the bcns is obtained by the semi tensor product based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence a state feedback control and an output feedback control respectively second using the column stacking form of matrices a new procedure is established to design both state feedback and output feedback controllers for the simultaneous stabilization problem the study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of bcns

a state space interpretation of simultaneous stabilization a state space interpretation of some previously known results on simultaneous stabilization is given the results reviewed were obtained using the yuola lemma

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

multistep detector for linear isi channels incorporating degrees of belief in past estimates this paper formulates the channel equalization problem in the framework of constrained maximum likelihood estimation this allows us to highlight key issues including the need to summarize past data and to apply a finite alphabet constraint over a sliding optimization window the approach adopted here leads to embellishments of the usual nonadaptive decision feedback equalizer and its multistep extensions it includes a provision for degrees of belief in past estimates which addresses the problem of error propagation

nonlinear unity feedback systems and q parametrization this paper concerns nonlinear systems defines a new concept of stability and extends to unity feedback systems the technique of q parametrization introduced by zames and developed by desoer chen and gustafson we specify 1 a global parametrization of all controllers that s stabilize a given s stable plant 2 a parametrization of a class of controllers that stabilize an unstable plant 3 necessary and sufficient conditions for a nonlinear controller to simultaneously stabilize two nonlinear plants

simultaneous stabilization with almost disturbance decoupling uniform rank system abstract in this paper we study the problem of stabilization with almost disturbance decoupling for a family of linear time invariant finite dimensional uniform rank plants with a fixed controller it is shown that for a family of minimum phase plants which have the same order of infinite zeros and have high frequency gain matrices satisfying a certain condition such a fixed controller exists the restriction on the high frequency gain matrices is equivalent to requiring that the sign of high frequency gain for siso plants be known

solution to the champagne problem on the simultaneous stabilisation of three plants abstract in this paper the problem proposed in blondel et al siam j optim 32 2 572 590 as an illustration of the difficulty of the simultaneous stabilization problem has been solved the same problem was also mentioned in blondel and gevers math control signals systems 6 1994 135 145 where a bottle of good french champagne was offered for its solution a new open problem is proposed based on the solution of the problem in this paper

sufficient conditions for generic simultaneous pole assignment and stabilization of linear mimo dynamical systems it has been shown that a generic r tuple of m input p output dynamical systems is simultaneously stabilizable pole assignable if r m p and the mcmillian degrees of the systems are not too different this result improves the currently known upper bound r spl les max m p on the number of plants in the case r spl les max m p an upper bound on the degree of the simultaneous pole assigning compensator has also been derived which improves the bound obtained by ghosh and byrnes 1983

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

simultaneous stabilization of nonlinear systems we study the problem of simultaneous stabilization of a given set of nonlinear plants by one nonlinear not necessarily stable compensator we obtain a necessary and sufficient condition under which there is a single compensator which stabilizes a given set of n nonlinear plants this note emphasizes the importance of the following problem when is a nonlinear unstable plant stabilizable by an incrementally stable compensator

some new results on the simultaneous stabilizability of a family of single input single output systems abstract in this paper we obtain sufficiency conditions on the problem of simultaneous stabilization of a parameterized family of single input single output plants by a non switching compensator the sufficiency conditions include simultaneous stabilization and simultaneous pole assignment of a finite number of plants

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

the solution of a transcendental problem and its applications in simultaneous stabilization problems a complete solution for a special transcendental problem in system theory is given one of the interpretations of this transcendental problem is the following simultaneous stabilization problem under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane these conditions are completely general and therefore encompass unstable non minimum phase plants as well it is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane for two plants satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is provided and illustrated a computationally checkable necessary condition for simultaneous stabilization of three plants is also given

brief simultaneously stabilizing controller design for a class of mimo systems it is shown that a class of linear time invariant multi input multi output plants can be simultaneously stabilized this class of plants all have the same number of zeros at infinity at zero or both but no other zeros in the unstable region if they have zeros at zero or infinity then their gain matrices at zero and infinity also satisfy a positive definiteness condition there is no restriction on the poles of the plants considered in this class an explicit design procedure is proposed to achieve simultaneously stabilizing controllers all simultaneously stabilizing controllers for this class of plants are also characterized in terms of a parameter matrix that satisfies a unimodularity condition

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

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

simultaneously stabilizing controllers for a class of linear plants abstract it is shown that a class of linear time invariant multi input multi output plants that all have poles at zero but do not have other unstable poles can be simultaneously stabilized a procedure is proposed to design a stable and strictly proper simultaneously stabilizing controller all simultaneously stabilizing controllers for this class are also characterized in terms of a parameter matrix that has to satisfy a unimodularity condition

some results on simultaneous stabilization with multiple domains of stability abstract in this paper the problem of simultaneous stabilization in the case where each plant has a different domain of stability associated with it is studied specifically it is supposed that one is given plants p0 p1 pr together with domains of stability d0 dr and the objective is to find a common controller c such that the poles of the closed loop transfer matrix h pi c are all inside the corresponding domain of stability di for i 0 r this formulation of the simultaneous stabilization problem generalizes earlier problem formulations wherein it was assumed that all the domains of stability were the same and is believed to be a more realistic formulation of the problem of preserving system stability in the event of sensor or actuator failure in the paper necessary and sufficient conditions for the existence of a common controller are derived in the case where one of the domains say d0 is a subset of the rest the case of two plants p0 p1 with d0 a subset of d1 is studied in great detail and the following result is established the two plants can be simultaneously stabilized in the above generalized sense if and only if they can be simultaneously stabilized with respect to the larger of the two regions more precisely there exists a controller c such that the poles of h pi c lie inside di for i 0 1 if and only if there exists a controller such that the poles of bothh p0 c and of h p1 c lie inside the larger region d1