generalized controller canonical form for linear and nonlinear dynamics a generalized controller canonical form for linear and nonlinear dynamics is proposed it is obtained using the theorem of the primitive element from differential algebra the derivation of the controller form does not apply to multivariable constant linear systems with the transformations all nonlinear dynamics can be exactly linearized by dynamic feedback the main departure from standard theory is that transformations may depend on input derivatives once differential algebraic tools are introduced the proofs of the results are easy

on minimal compensators for decoupling control abstract the general construction of minimal regular compensators for decoupling a nonlinear system is given by means of vector subspaces which are generated by differential vectors of outputs and states and the minimal dimension of a decoupling compensator is expressed in intrinsic characterization the equivalence relation of all minimal regular compensators is established though these compensators may have different dynamic behavior from each other

structure at infinity revisited abstract the structure at infinity of an ordinary differential control system is a finite sequence of increasing integers ending with the differential output rank of the system namely the number of outputs that can be given as arbitrary functions of time when the inputs are unknown its definition and construction originally done for linear systems has been extended to affine nonlinear systems and used in order to study dynamic decoupling or model matching it essentially relies on a state representation the purpose of this paper is to make a critical examination of this concept and to modify it in order to avoid the state representation at the same time we extend it to nonlinear partial differential control systems by exhibiting a link with formal integrability a highly important concept in the formal theory of systems of partial differential equations that cannot be handled by means of a transfer matrix approach many explicit examples will illustrate the main results and the possibility to use computer algebra techniques will be pointed out

existence and uniqueness results for a third order implicit differential equation the purpose of this paper is to give sufficient conditions for the existence and uniqueness of solutions to the boundary value problem of a third order implicit differential equation as follows f t u t u t u t 0 email protected email protected 1u 0 u 0 u 1 0 the lower and upper solutions method maximum principle together with an iterative technique are employed an example illustrates the application of results obtained

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

minimality of dynamic input output decoupling for nonlinear systems in this note we study the strong dynamic input output decoupling problem for nonlinear systems using an algebraic theory for nonlinear control systems we obtain for a dynamic input output decouplable nonlinear system a compensator of minimal dimension that solves the decoupling problem

nonlinear strong model matching the problem of matching a given input output behavior for systems described by general nonlinear differential equations is considered it is shown that by appropriately modifying the zero dynamics algorithm it is possible to obtain a simple necessary and sufficient condition for the solvability of the model matching problem which requires that the initial state be on an appropriate submanifold of the state space another condition necessary and sufficient for the solvability of the strong model matching problem is proposed this last condition is then related to an equality of a list of integers which under some regularity assumptions coincide with the algebraic structures at infinity of the process and of a composition of the process and the model the relation between these conditions and the equality of the algebraic structures at infinity of the process and the model is established

a nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem the nonregular dynamic disturbance decoupling problem for nonlinear control systems is introduced a local solution is given by means of a constructive algorithm that is based on singh s algorithm and the clamped dynamics algorithm further studied is the nonlinear model matching problem that is defined as follows given a nonlinear control system to be referred to as the plant and another nonlinear control system to be referred to as the model can a compensator for the plant be found in such a way that the input output behavior of the compensated plant matches that of the model by proving that the solvability of the nonlinear model matching problem is equivalent to the solvability of an associated nonregular dynamic disturbance decoupling problem a complete local solution of this problem can be established

robust output regulation for invertible nonlinear mimo systems abstract the robust output regulation problem for a class of invertible nonlinear mimo systems is addressed in this paper under a mild assumption on the high frequency gain matrix and strongly minimum phase assumption a robust regulator is proposed with the combination of nonlinear internal model and a new extended high gain observer more explicitly the robust output regulation is achieved by employing a novel extended high gain observer based stabilizer such that the regulated system augmented with the nonlinear internal model is asymptotically stabilized the novelty of this approach consists in the use of an extended observer to estimate under mild assumptions all unmeasured terms

inversion of nonlinear time varying systems a procedure for inverting nonlinear systems that presents some computational advantages when applied to flexible robot arms or more generally to mechanical structures is given the procedure is presented in a general setting for nonlinear time varying systems an iterative algorithm that computes a smooth controlled invariant time varying manifold the mathematical properties of which are illustrated and exploited is also proposed an application to a flexible two link manipulator illustrates the algorithm and its computational advantages

on implicit systems of differential equations abstract this article considers implicit systems of differential equations the implicit systems that are considered are given by polynomial relations on the coordinates of the indeterminate function and the coordinates of the time derivative of the indeterminate function for such implicit systems of differential equations we are concerned with computing algebraic constraints such that on the algebraic variety determined by the constraint equations the original implicit system of differential equations has an explicit representation our approach is algebraic although there have been a number of articles that approach implicit differential equations algebraically all such approaches have relied heavily on linear algebra the approach of this article is different we have no linearity requirements at all instead we rely on algebraic geometry in particular we use birational mappings to produce an explicit system the methods developed in this article are easily implemented using various computer algebra systems

invertibility of observable multivariable nonlinear systems based on the algorithm of hirschorn 1 for obtaining inverses sufficient conditions for invertibility of observable multivariable nonlinear systems are derived for observable and invertible systems a static left inverse system is constructed for certain noninvertible systems the sufficient condition yields a characterization of input space on which the input output map is injective

a remark on nonlinear accessibility conditions and infinite prolongations the lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order it yields to a new lie algebraic criterion and to the consideration of first integrals which apply to nonlinear systems in quite general form the latter characterization in particular is valid without distinguishing between the system variables and is independent of time scalings weak accessibility is also considered

lq control of descriptor systems by cancelling structure at infinity this paper presents results concerning lq control of arbitrary descriptor systems and a numerical algorithm for achieving it the algorithm consists of two steps the first step is using the silverman algorithm orthogonal version transformation of the problem of lq control of descriptor system into a problem of lq control of a strictly proper system with singular or non singular new matrix 22 if new 22 is singular we perform the second step i e solving the lq singular control problem using again the silverman algorithm regarding the existing results on the lq singular control problem we have new results a counter example that this problem may not have a stable solution most general solvability conditions with or without stability and a new numerical algorithm based on using orthogonal matrices when stabilizability conditions are satisfied the algorithm places simultaneously stable poles in the closed loop system we show that the traditional riccati equation associated to this proble

an extended linearization approach to nonlinear system inversion the inversion problem for nonlinear systems is considered from both the internal viewpoint state equation representations and the external viewpoint volterra series representations in order to circumvent difficulties in the known exact inversion theory a method for obtaining what will be called a partial inverse system is proposed for a given nonlinear system a cascade system typically containing differentiators is found such that the overall nonlinear system has a family of constant operating points with inputs and outputs identical furthermore the linearization of the overall system about each of these constant operating points is the identity system

generalized functional reproducibility condition for nonlinear systems a new sufficient condition for the existence of control to generate a given real analytic function as the output of a large class of multivariable nonlinear control system is derived furthermore a prefilter right inverse system is constructed to generate the required control

output input stability implies feedback stabilization abstract we study the recently introduced notion of output input stability which is a robust variant of the minimum phase property for general smooth nonlinear control systems this paper develops the theory of output input stability in the multi input multi output setting we show that output input stability is a combination of two system properties one related to detectability and the other to left invertibility for systems affine in controls we derive a necessary and sufficient condition for output input stability which relies on a global version of the nonlinear structure algorithm this condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output input stable systems

model reference adaptive and nonadaptive control of linear time varying plants we address the adaptive and nonadaptive model reference control problem for a class of multivariable linear time varying plants namely index invariant ones we show that under appropriate controllability and observability conditions this class of plants admits a fractional description in terms of polynomial differential operators and as such allows for a polynomial equation based controller design we also show that for a model reference control objective the controller can be designed by solving a set of algebraic equations further when the plant parameters are only partially known we employ a gradient based adaptive law with projection and normalization to update the controller parameters and establish the stability and tracking properties of adaptive closed loop plant finally we present a simple example to illustrate the design and realization of both the adaptive and nonadaptive control laws

local nonlinear model matching from linearity to nonlinearity the model matching problem consists of designing a compensator for a given system called the plant in such a way that the resulting input output behaviour matches that of a prespecified model in this paper a local solution of the nonlinear model matching problem is given for the case that the model is decouplable by static state feedback the main theorem states that under generic conditions on the plant the problem is solvable around an equilibrium point if and only if it is solvable for the linearization of plant and model the generic conditions are identified they naturally appear in the solution of the dynamic input output decoupling problem for the plant the theory is illustrated by means of two examples

the jarratt method in banach space setting abstract in this note we extend the jarratt method of order four into banach spaces we also establish a kantorovich type convergence theorem see the references and give an explicit expression for the error bound of the method jarratt 1966 1969

on normal forms of nonlinear systems affine in control the nonlinear equivalences of both finite and infinite zero structures of linear systems have been well understood for single input single output systems and have found many applications in nonlinear control theory the extensions of these notions to multiple input multiple output systems have proven to be highly sophisticated in this paper we propose constructive algorithms for decomposing a nonlinear system that is affine in control these algorithms require modest assumptions on the system and apply to general multiple input multiple output systems that do not necessarily have the same number of inputs and outputs they lead to various normal form representations and reveal the structure at infinity the zero dynamics and the invertibility properties all of which represent nonlinear equivalences of relevant linear system structural properties

an intrinsic sufficient condition for regular decoupling abstract this paper introduces an intrinsic regularity property namely regular invertibility which requires that a sequence of vector spaces has constant dimension after a certain order thus allowing some degeneracies in the beginning it corresponds to the system and its tangent having the same defect using the constant rank theorem this property is shown to be sufficient condition for regular decoupling

stabilization by output feedback of multivariable invertible nonlinear systems in this paper the problem of global stabilization of a rather general class of mimo nonlinear systems is addressed the systems considered in the paper are invertible have a trivial zero dynamics and possess a normal form in which certain multipliers are functions of the state vector of a special kind while special structural dependence of such multipliers on the components state vector has been exploited before in the context of achieving stabilization via full state feedback though the novelty in the approach of this paper is that a peculiar structure is identified which happens to be intimately related to the property of uniform complete observability thus making the design of observers possible and to the property of uniform invertibility relations never established before in the literature as a result for this class of mimo nonlinear systems a dynamic output feedback law can be designed yielding semiglobal and even global under appropriate assumptions asymptotic stability

necessary conditions for asymptotic tracking in nonlinear systems in the literature it has been shown that if a single input single output analytic nonlinear plant 1 has a well defined relative degree and 2 is minimum phase it is possible to achieve asymptotic tracking for an open set of output trajectories containing the origin in c sup n 0 spl infin the space of n times continuously differentiable functions taking values in r when either of these sufficient conditions is not met various authors have investigated approximate analytic solutions discontinuous solutions and solutions for restricted sets of trajectories in this paper it is shown that conditions 1 and 2 are necessary for the existence of an analytic compensator which yields asymptotic tracking for an open set of output trajectories analogous results are established for multi input multi output systems

the nonlinear interactor and its application to input output decoupling the solvability of the input output decoupling problem via pure dynamic state feedback a special type of dynamic compensator without zeros at infinity is investigated a necessary and sufficient condition for the solvability of this problem is established in terms of a nonlinear interactor a new concept in the theory of nonlinear control

approximate decoupling and asymptotic tracking for mimo systems this paper presents an algorithm for approximate input output decoupling of nonlinear mimo systems that are either numerically ill posed or exhibit nearly singular behavior in the application of decoupling algorithms although the systems considered are regular so that the exact decoupling algorithms are applicable in this case they require inversion of an ill conditioned matrix and yield high gain feedback solutions that may result in actuator saturation and cancellation of high frequency zeros the approximate algorithms of this paper are numerically robust and provide solutions that do not cancel far off right half plane zeros this latter characteristic is especially valuable when some of the far off right half plane zeros are unstable the algorithms are inspired by and are generalizations of some examples in the flight control literature

a triangular canonical form for a class of 0 flat nonlinear systems this article proposes a triangular canonical form for a class of 0 flat nonlinear systems necessary and sufficient geometrical conditions are given in order to guarantee the existence of a local diffeomorphism to transform the studied nonlinear systems into the proposed 0 flat canonical form which enables us to compute the flat output as well

a modified algorithm for invertibility in nonlinear systems the algorithm of hirschorn 1 for the construction of inverses of nonlinear systems is modified and new sufficient conditions for invertibility and inverse system for a larger class of systems are obtained for the systems which do not satisfy the invertibility conditions a characterization of input space on which the input output map is injective is derived

functional reproducibility of general multivariable analytic non linear systems in this paper the functional reproducibility of a general multivariable analytic nonlinear system of the form x f x u y h x u will be discussed where the state space is a real analytic manifold a general inversion algorithm is presented for the system a sufficient and necessary condition for the functional reproducibility of the system is derived the construction of a right inverse system is also included

notes on f0 matrices abstract the author studies f 0 matrices and shows that n n matrix a f 0 if and only if a ti b with b 0 and n 2 b t n 1 b where k b denotes the maximum of the spectral radii of all k principal submatrices of b also an inverse f 0 matrix problem is considered