nonlinear time varying compensation for simultaneous performance abstract this short note considers the use of nonlinear time varying compensation for linear time invariant discrete time plants it is shown via counterexample that the problem of simultaneous performance presents fundamental limitations which cannot be overcome by nonlinear time varying compensation this result is in contrast to results on simultaneous stabilization which show that limitations due to linear compensation may be removed using nonlinear time varying compensation as a corollary of these results the conjecture that achievable disturbance rejection over stable nonlinear time varying compensation equals that for linear compensation is refuted

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

on the stabilization of nonlinear systems we extend the applicability of the global q parametrization method of controller design to a large class of unstable nonlinear plants the main result is a two step compensation theorem analogous to that of zames for unstable linear plants if p l e2 rightarrow l e1 is a nonlinear possibly unstable plant and f 0 is any incrementally stable controller such that p 1 p i f 0 p 1 is incrementally stable then the class of controllers f which yields an f g stable closed loop system in the unity feedback configuration for p is globally parametrized by finite gain stable maps q l e1 rightarrow l e2 with f f 0 q i p 1 q 1

noetherian intersections of regular local rings of dimension two abstract let d be a 2 dimensional regular local ring and let q d denote the quadratic tree of 2 dimensional regular local overrings of d we describe the desingularization of projective models over d both algebraically in terms of the saturation of complete ideals and order theoretically in terms of the quadratic tree q d we prove that this describes all the noetherian rings that are intersections of rings in q d

blowing up finitely supported complete ideals in a regular local ring let i be a finitely supported complete m primary ideal of a regular local ring r m we consider singularities of the projective models projr it and projr it over spec r where r it denotes the integral closure of the rees algebra r it a theorem of lipman implies that the ideal i has a unique factorization as a product of special simple complete ideals with possibly negative exponents for some of the factors if projr it is regular we prove that projr it is the regular model obtained by blowing up the finite set of base points of i extending work of lipman and huneke sally in dimension 2 we prove that every local ring s on projr it that is a unique factorization domain is regular moreover if dim s 2 and s dominates r then s is an infinitely near point to r that is s is obtained from r by a finite sequence of local quadratic transforms

construction of the left coprime fractional representation for a class of nonlinear control systems abstract a constructive approach to the left coprime fractional representations for a class of stable and unstable nonlinear control systems described by certain multi input multi output nonlinear ordinary differential equations is proposed the method works for the coprimeness defined by verma 15

bifurcation of limit cycles from quadratic centers abstract bifurcation of limit cycles from the class q nh 3 of quadratic systems possessing centers is investigated bifurcation diagrams for various systems in this class are constructed and are used to locate systems possessing a period annulus whose closure has cyclicity three

characterization of all nonlinear stabilizing controllers via observer based kernel representations this paper deals with the characterization of all nonlinear stabilizing controllers observer based kernel representations are introduced to avoid the difficulty which occurs in constructing the state space realizations of the parametrization of input output operators the parametrization of all stabilizing controllers for a given plant and a class of stabilizing plant controller pairs are derived by employing observer based kernel representations furthermore the relationship between the results via kernel representation approach and the existing results via state space approach is clarified

a parametrization for closed loop identification of nonlinear systems based on differentially coprime kernel representations in this paper we use the notion of a differentially coprime kernel representation to parametrize the set of all nonlinear plants stabilized by a given nonlinear controller using a so called youla parameter and to unify understanding of some stability concepts for nonlinear systems by utilizing the differential kernel representation concept we are able to convert a closed loop identification problem into one of open loop identification the main advantage of our approach using kernel representations over fractional descriptions is that we address a larger class of nonlinear systems the idea of a differential kernel representation allows us also to clarify the relationship between three different notions of internal stability available in the literature the results in the paper thus provide new insights to the stability of nonlinear feedback systems

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

analysis of time varying control strategies for optimal disturbance rejection and robustness analysis of linear time invariant systems that are controlled by time varying controllers is given the problems of disturbance rejection and robustness are formulated and analyzed using functional analytic methods which reveal the properties that are common to these problems the theoretical development presented uses the theory and techniques of nuclear operators and their relation with the duality theory of tensor products of banach spaces the authors show how time varying compensation offers no advantage over time invariant compensation for the problem of disturbance rejection over general signal spaces in both continuous and discrete time and for the problem of l sub infinity robust stabilization of time invariant plants in addition another application of the theory for the problem of norm minimization subject to a norm constraint is presented

asymptotic properties of infinite directed unions of local quadratic transforms abstract let r m be a regular local ring of dimension at least 2 for each valuation domain birationally dominating r there is an associated sequence r n of local quadratic transforms of r we consider the case where this sequence r n n 0 is infinite and examine properties of the integrally closed local domain s n 0 r n in the case where s is not a valuation domain for this sequence there is an associated boundary valuation ring v n 0 i n v i where v i is the order valuation ring of r i there exists a unique minimal proper noetherian overring t of s t is the regular noetherian ufd obtained by localizing outside the maximal ideal of s and s v t in the present paper we define functions w and e where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals we describe v explicitly in terms of w and e and prove that v is either rank 1 or rank 2 we define an invariant associated to s that is either a positive real number or if is finite then s is archimedean and t is not local in this case the function w defines the rank 1 valuation overring w of v and w dominates s the rational dependence of over w t determines whether s is completely integrally closed and whether v has rank 1 we give examples where s is completely integrally closed if is infinite then s is non archimedean and t is local in this case the function e defines the rank 1 valuation overring e of v the valuation ring e is a dvr and e dominates t and in certain cases we prove that e is the order valuation ring of t

preservation of stability under system perturbations in nonlinear control a simple characterization abstract a simple characterization of the perturbations a nonlinear closed loop control system can tolerate before losing its stability is derived the characterization is given in purely algebraic terms with the topological aspects being automatically incorporated through the theory of fraction representations of nonlinear systems the implications of system perturbations on internal stability are also discussed the presentation is for the case of discrete time nonlinear systems

non linear systems stabilization and coprimeness a theory of coprimeness is developed for non linear systems with the intention of constructing analytic tools for the solution of the problem of stabilizing a non linear system through the application of additive non linear feedback the qualitative features of the resulting theory are strikingly similar to the situation in the linear case

internally stable nonlinear systems with disturbances a parameterization this paper deals with the control of a nonlinear system whose output is subjected to an additive disturbance the main result is a simple parameterization of the set of all system responses that can be obtained through internally stable control of the given system the parameterization provides a clear indication of the effects of the disturbance on the response of the stabilized closed loop system the class of achievable responses is determined by the numerator of a right coprime fraction representation of the system being controlled

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

operator based robust nonlinear control system design for mimo nonlinear plants with unknown coupling effects in this article operator based robust nonlinear control system design for multi input multi output mimo plants with unknown coupling effects is considered that is by using operator based robust nonlinear control design coupling effects existing in the mimo nonlinear plants can be decoupled based on a feedback design and robust right coprime factorisation approach the coupling effects caused by controllers and plant outputs can be stabilised by using definition of lipschitz operator and contraction mapping theorem and output tracking performance can be realised by a tracking design scheme finally a simulation example about temperature control process of 3 input 3 output aluminum plate is given to support the theoretical analysis

on the parametrization of nonlinear finite gain over set stabilizing controllers the concepts of gain and incremental gain over set have been introduced to alleviate some of the limitations of the conventional notions of gain and incremental gain in this paper the class of all finite gain over set stabilizing controllers is parametrized in terms of a finite gain over set stable map q provided that the plant p is finite incremental gain over set stable an alternate theorem for finite gain over ball stability of the closed loop is also developed which provides a superset and a subset of the set of all finite gain over set stabilizing compensators for the feedback system

perturbation from an asymmetric cubic hamiltonian this paper concerns with the number of limit cycles from an asymmetric hamiltonian of degree three under cubic perturbation eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis two of which are new

parametrization of nonlinear stabilizing controllers the observer controller configuration the paper uses the author s 1996 previous results to develop the parametrization of nonlinear stabilizing observer controller compensators of a given nonlinear system the configuration described permits a simple parametrization of stabilizing controllers using coprime factorisation techniques

uniformly optimal control of linear time invariant plants nonlinear time varying controllers abstract it is shown that in the problems of uniformly or h optimal control of linear time invariant plants arbitrary nonlinear time varying controllers offer no advantage over linear time invariant controllers

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

right factorization of a class of time varying nonlinear systems a class of nonlinear continuous time time varying plants with a state space description which has a uniformly completely controllable linear part is considered for this class a right factorization is obtained for the case in which the state is available for feedback a normalized right coprime factorization is realized

graph topologies gap metrics and robust stability for nonlinear systems graph topologies for nonlinear operators which admit coprime factorizations are defined w r t a gain function notion of stability in a general normed signal space setting several metrics are also defined and their relationship to the graph topologies are examined in particular relationships between nonlinear generalizations of the gap and graph metrics georgiou type formulae and the graph topologies are established closed loop robustness results are given w r t the graph topology where the role of a coercivity condition on the nominal plant is emphasized

finitely supported simple complete ideals and multiplicities in a regular local ring let r m and s n be regular local rings of dim s dim r 2 such that s birationally dominates r and let v be the order valuation ring of s with corresponding valuation ords assume that is s and reessis let u t with is is where s then v w q r with w r it q s isu q where q min mr it and q min ns isu let p p be the center of w on r it and s isu respectively we prove that if sn rm 1 then r it p s isu p let i be a finitely supported complete m primary ideal of a regular local ring r m of dimension d 2 let t be a terminal base point of i and v be the mt adic order valuation of t with corresponding valuation v ordt let n 1 be an integer assume that it mtn and tmt rm 1 let p min mr it such that p q r it with v r it q q r where q min mr it we prove that the quotient ring r it p is d dimensional normal cohen macaulay standard graded domain over k with the multiplicity nd 1 in particular r it p is regular if and only if n 1 we prove that k rm is relatively algebraically closed in kv vmv also we determine the multiplicity of r it p and we prove that if it mt then r it p is regular if and only if tmt rm 1

stability of control systems and graphs of linear systems new conditions for internal stability of a closed loop control system are given in terms of the graphs of the multiplication operators induced by the transfer functions of the plant and the controller these conditions can be given a geometrical interpretation this relates closed loop stability to the minimal angle between the graph space associated with the system and the graph space associated with the controller the maximally stabilizing controller is defined as the controller that maximizes the minimum angle between the graph space associated with the system and the graph space associated with the controller it is shown that this controller can be calculated as a nehari extension of the coprime factors of the system

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

on the stabilizing conic sectors obtained using small gain techniques abstract this paper is concerned with the analysis of the scope and properties of the stabilizing conic sectors obtained using small gain and loop transformation sglt techniques first it is shown how sglt techniques are able to generate the set of all nonlinear stabilizing controllers by using transformations based on dynamical 2 2 block systems configurations then stabilizing cones obtained from static configurations are considered they induce only subsets of all stabilizing controllers however providing simplification and tractability for these cones maximality is characterized in terms of mu values and frequency responses finally an algorithm for deriving maximal cones is presented

coprimeness properties of nonlinear fractional system realizations in this paper the relationship between the bezout and the set theoretic approaches to left coprimeness is studied it is shown that left coprimeness in the set theoretic sense implies left coprimeness in the bezout sense in addition to these results we investigate whether some properties for linear left coprime realizations carry over to the nonlinear case for example we examine the relations between two left coprime realizations of the same system

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

a parametrization for the closed loop identification of nonlinear time varying systems it has recently been shown that the identification of a linear time invariant plant using closed loop measurements with a known linear controller can be effectively tackled by regarding the unknown plant as a member of the set of all plants stabilized by the known linear controller this set is parametrized by a youla kucera parameter itself a stable transfer function which it turns out can be identified using open loop techniques this paper generalizes the parametrization idea to nonlinear plants and treats identification of nonlinear plants

parametrization of all stabilizing controllers of nonlinear systems this paper develops a new approach to the parametrization of all asymptotically stabilizing controllers of nonlinear systems including time varying ones both for state feedback and output feedback assumptions of stabilizability and a kind of detectability are required the approach is very direct and simple and is not based on the so called factorization and on the h framework of nonlinear systems which are used in previous papers a key of the derivation of the parametrization in this paper is to find an explicit state space representation of free functions such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems

a note on the coprime fractional representation of nonlinear feedback systems abstract a very large infinite dimensional banach space of bounded nonlinear operators is suggested as an underlying framework for studies of nonlinear systems control meanwhile it is pointed out that all results obtained in verma 6 can be extended to this much larger family of nonlinear control systems

a new factorization of special nonlinear discrete systems and their applications in this paper a new factorization of discrete nonlinear systems is presented this factorization is based on the star product an operation corresponding to the linear connection of systems a relevant algorithm is developed based on a theoretical background finally some applications to system analysis bibo stability and feedback design are provided

right coprime factorizations and stabilization for nonlinear systems the problem of constructing right coprime factorizations which are based on the graph of an input output map instead of the bezout identity of nonlinear input output maps is considered the map is assumed to arise from a state variable realization with a fixed initial state the main result is that the existence of a stabilizing state feedback implies the existence of a right coprime factorization for the map the technique is illustrated by application to nonlinear systems which are affine in the control and have a controllable linear past and nonlinear systems which are feedback linearizable a notion of input output stability that requires a bound on the magnitude of the input signals is introduced methods of constructing such bounds are developed for a locally feedback linearizable system the problem of input bounds is transferred to the equivalent linear system this leads to a technique that allows state and input constraints for a feedback linearizable system to be mapped to the equivalent linear system

the parallel projection operators of a nonlinear feedback system abstract this paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system these operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric the use of weighted gaps in control system design and glover mcfarlane loop shaping we show that the input output l 2 stability of a feedback system amounts to a coordinatization of the input and output spaces which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively these projections are shown to have equal norms whenever one of the feedback elements is linear a bound on this norm is given in the case of passive systems with unity negative feedback

finitely supported simple complete ideals in a regular local ring abstract let i be a finitely supported complete m primary ideal of a regular local ring r m a theorem of lipman implies that i has a unique factorization as a product of special simple complete ideals with possibly negative exponents for some of the factors the existence of negative exponents occurs if dim r 3 because of the existence of finitely supported simple ideals that are not special we consider properties of special simple complete ideals such as their rees valuations and point basis let r m be a d dimensional equicharacteristic regular local ring with m x 1 x d r we define monomial quadratic transforms of r and consider transforms and inverse transforms of monomial ideals for a large class of monomial ideals i that includes complete inverse transforms we prove that the minimal number of generators of i is completely determined by the order of i we give necessary and sufficient conditions for the complete inverse transform of a product of monomial ideals to be the product of the complete inverse transforms of the factors this yields examples of finitely supported simple monomial ideals that are not special we prove that a finitely supported simple monomial ideal with linearly ordered base points is special simple

fading memory feedback systems and robust stability abstract this paper considers fading memory for nonlinear time varying systems and associated problems of robust stability we define two notions of fading memory for stable dynamical systems uniform and pointwise we then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory in particular we show that 1 all stable discrete time linear time varying ltv systems have uniform fading memory 2 all stable continuous time ltv systems have pointwise fading memory and 3 stable finite dimensional continuous time ltv systems have uniform fading memory we then show that a version of the small gain theorem which employs the asymptotic gain of a fading memory system is necessary for the stable invertibility of certain feedback operators these results are presented for both continuous time and discrete time systems using general lp or l p notions of input output stability and generalize existing results for l2 stability we further investigate fading memory in a closed loop context for linear plants we parametrize all nonlinear controllers which lead to closed loop pointwise fading memory

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

stabilization of nonlinear systems based on a generalized be zout identity the relationship between the bezout and input output approaches to right coprimeness is investigated we introduce a generalized bezout identity and analyze its relationship to right coprimeness in terms of the generalized bezout identity a right factorization of a nonlinear plant is coprime according to the input output definition if and only if it satisfies the generalized bezout identity therefore the stability of an observer controller configuration is a necessary and sufficient condition for a nonlinear plant to have a right coprime factorization finally we present generalized bezout maps for some nonlinear plants with a state space representation

operator based nonlinear feedback control design using robust right coprime factorization in this note robust stabilization and tracking performance of operator based nonlinear feedback control systems are studied by using robust right coprime factorization specifically a new condition of robust right coprime factorization of nonlinear systems with unknown bounded perturbations is derived using the new condition a broader class of nonlinear plants can be controlled robustly when the spaces of the nonlinear plant output and the reference input are different a space change filter is designed and in this case this note considers tracking controller design using the exponential iteration theorem

ideal theory of infinite directed unions of local quadratic transforms abstract let r m be a regular local ring of dimension at least 2 associated to each valuation domain birationally dominating r there exists a unique sequence r n of local quadratic transforms of r along this valuation domain we consider the situation where the sequence r n n 0 is infinite and examine ideal theoretic properties of the integrally closed local domain s n 0 r n among the set of valuation overrings of r there exists a unique limit point v for the sequence of order valuation rings of the r n we prove the existence of a unique minimal proper noetherian overring t of s and establish the decomposition s t v if s is archimedean then the complete integral closure s of s has the form s w t where w is the rank 1 valuation overring of v

optimal disturbance rejection for nonlinear control systems the problem of optimal disturbance rejection for a general class of multiinput multioutput mimo nonlinear systems achieved over a set of stabilizing compensator operators is formulated in a banach space setting conditions on the operators are imposed on the basis of the inputs as well as the outputs an existence theorem for optimal solutions is established under some realistic and verifiable conditions for solving the optimization problem a generalized recursive nonlinear programming algorithm is passed and a simple example of nonlinear optimal feedback design is given to show that the problem can be reduced to standard min max optimization and is hence solvable

directed unions of local quadratic transforms of a regular local ring abstract let r m be a d dimensional regular local domain with d 2 and let v be a valuation domain birationally dominating r such that the residue field of v is algebraic over r m let v be a valuation associated to v associated to r and v there exists an infinite directed family r n m n n 0 of d dimensional regular local rings dominated by v with r r 0 and r n 1 the local quadratic transform of r n along v let s n 0 r n abhyankar proves that s v if d 2 shannon observes that often s is properly contained in v if d 3 and granja gives necessary and sufficient conditions for s to be equal to v the directed family r n m n n 0 and the integral domain s n 0 r n may be defined without first prescribing a dominating valuation domain v if r n m n n 0 switches strongly infinitely often then s v is a rank one valuation domain and for nonzero elements f and g in m we have v f v g lim n ord r n f ord r n g if r n m n n 0 is a family of monomial local quadratic transforms we give necessary and sufficient conditions for r n m n n 0 to switch strongly infinitely often if these conditions hold then s v is a rank one valuation domain of rational rank d and v is a monomial valuation assume that v is rank one and birationally dominates s let s i 0 v m i granja martinez and rodriguez show that s implies s v we prove that s is finite if v has rational rank at least 2 in the case where v has maximal rational rank we give a sharp upper bound for s and show that s attains this bound if and only if the sequence switches strongly infinitely often

brief identification of a nonlinear plant under nonlinear feedback using left coprime fractional based representations it has been shown that the set of all nonlinear plants stabilised by a known linear controller which also stabilises a linear nominal model of the plant can be parametrised by a stable operator known as the youla kucera parameter by utilising this description it is possible to convert the closed loop plant identification problem to one of open loop identification this paper extends previous work by allowing the model of the nominal plant and the controller in the above scenario to be nonlinear the ideas rely on a concept of differential coprimeness for nonlinear fractional system descriptions

a numerical approach to bifurcations from quadratic centers abstract a combination of analytical and numerical work is done to analyze bifurcation of limit cycles from non hamiltonian codimension three quadratic centers the winding curve c 3of cyclicity three separatrix cycles qualitatively located in earlier work is determined numerically evidence is given that the 2 2 3 2 and 3 3 configurations of limit cycles do not bifurcate from this class of quadratic centers

survey paper from youla kucera to identification adaptive and nonlinear control some 20 years ago formulae were presented for the set of all linear time invariant controllers stabilizing a linear time invariant plant this paper traces the development of many ideas from these formulae covering linear h 2 and h control identification adaptive control and nonlinear systems

on graphs conic relations and input output stability of nonlinear feedback systems this paper presents necessary and sufficient conditions for the stability of general interconnected dynamical systems the results are expressed in terms of the graphs of the interconnection s subsystems conditions for robustness of stability for an interconnection with additive disturbances are also provided finally the paper gives sufficient conditions for stability using conic sector conditions that generalize earlier conditions and that incorporate a recent nonlinear small gain idea

the class of stabilizing nonlinear plant controller pairs in this paper a general approach is taken to yield a characterization of the class of stable plant controller pairs which is a generalization of the youla parameterization for linear systems this is based on the idea of representing the input output pairs of the plant and controller as elements of the kernel of some related operator which is denoted the kernel representation of the system it is demonstrated that in some sense the kernel representation is a generalization of the left coprime factorization of a general nonlinear system results giving one method of deriving a kernel representation for a nonlinear plant with a general state space description are presented