on boundedness of q learning iterates for stochastic shortest path problems we consider a totally asynchronous stochastic approximation algorithm q learning for solving finite space stochastic shortest path ssp problems which are undiscounted total cost markov decision processes with an absorbing and cost free state for the most commonly used ssp models existing convergence proofs assume that the sequence of q learning iterates is bounded with probability one or some other condition that guarantees boundedness we prove that the sequence of iterates is naturally bounded with probability one thus furnishing the boundedness condition in the convergence proof by tsitsiklis tsitsiklis jn 1994 asynchronous stochastic approximation and q learning machine learn 16 185 202 and establishing completely the convergence of q learning for these ssp models

direct adaptive rejection of persistent disturbances given any finite set of persistent disturbances that can be represented by a finite sequence in the banach space of bounded continuous functions it is shown that stable disturbance rejection is possible for command generator tracker cgt model reference adaptive control if certain conditions are satisfied if the system in question is disturbance free the addition of disturbance accommodating adaptive gains into the control law will not impede the asymptotic convergence of the tracking error almost periodic disturbances are explored and an illustrative example of the method is given

multivariable circle criteria for multiparameter singularly perturbed systems a direct approach to the lur e problem for singularly perturbed systems sps is proposed in contrast to previous results the feedback connection between the linear and nonlinear parts of sps is allowed to depend essentially on both the slow and the fast variables the lur e problem for multiparameter sps is studied by the same framework

eigenvalue based characterization and test for positive realness of scalar transfer functions an eigenvalue based characterization of positive realness of transfer functions of a single input single output time invariant linear system is derived based on this characterization we propose an efficient computational procedure to determine if a given transfer function is positive real the input for this eigenvalue based test is any given not necessarily minimal state space representation of the linear system the test only involves standard matrix computations such as computing eigenvalues of a matrix or a matrix pencil results of numerical experiments are reported

nonholonomic versus vakonomic dynamics on a riemann cartan manifold for the chaplygin s nonholonomic constrained systems the constraint manifold can be endowed with riemann cartan geometric structure by nonholonomic mapping into a riemann manifold the two kinds of existing dynamics nonholonomic dynamics and vakonomic dynamics are compared in the framework of riemann cartan geometry it is proved that the equations of motion for nonholonomic and vakonomic dynamics are described by the equations of autoparallel and geodesic trajectories on the riemann cartan constraint manifold respectively if the metricity condition of riemann cartan connection is satisfied the torsion contorsion of the riemann cartan manifold characterizes the difference between the autoparallel and geodesic trajectories as well as the distinction between the nonholonomic and vakonomic equations

eigenvalue based algorithms for testing positive realness of siso systems a computational test for scalar positive real and strictly positive real functions is proposed in this paper the input for this algorithm is state space representation not necessarily minimal it is derived for the discrete time problem based on numerical stable eigenvalue computations the test can be used in the continuous counterpart which is in fact an improvement of an algorithm posed by bai and freund differences between the discrete and continuous time problems are also pointed out

positive real control of two dimensional systems roesser models and linear repetitive processes this paper considers the problem of positive real control for two dimensional 2 d discrete systems described by the roesser model and also discrete linear repetitive processes which are another distinct sub class of 2 d linear systems of both systems theoretic and applications interest the purpose of this paper is to design a dynamic output feedback controller such that the resulting closed loop system is asymptotically stable and the closed loop system transfer function from the disturbance to the controlled output is extended strictly positive real we first establish a version of positive realness for 2 d discrete systems described by the roesser state space model then a sufficient condition for the existence of the desired output feedback controllers is obtained in terms of four lmis when these lmis are feasible an explicit parameterization of the desired output feedback controllers is given we then apply a similar approach to discrete linear repetitive processes represented in their equivalent 1 d state space form finally we provide numerical examples to demonstrate the applicability of the approach

new characterisations of positive realness and static output feedback control of discrete time systems this article is concerned with the problems of positive real analysis and control synthesis for discrete time systems new linear matrix inequality lmi characterisations of positive realness are derived which enable one to check the positive realness by using parameter dependent lyapunov function the relationship between the proposed characterisations and the existing ones are clarified which shows that our new results are of less conservatism for characterising the positive realness of discrete time systems with polytopic uncertainty in addition sufficient conditions for static output feedback positive real controller design are given in terms of solutions to a set of linear matrix inequalities numerical examples are included for illustration

stochastic approximation for nonexpansive maps application to q learning algorithms we discuss synchronous and asynchronous iterations of the form xk 1 x k gamma k h x k w k where h is a suitable map and wk is a deterministic or stochastic sequence satisfying suitable conditions in particular in the stochastic case these are stochastic approximation iterations that can be analyzed using the ode approach based either on kushner and clark s lemma for the synchronous case or on borkar s theorem for the asynchronous case however the analysis requires that the iterates xk be bounded a fact which is usually hard to prove we develop a novel framework for proving boundedness in the deterministic framework which is also applicable to the stochastic case when the deterministic hypotheses can be verified in the almost sure sense this is based on scaling ideas and on the properties of lyapunov functions we then combine the boundedness property with borkar s stability analysis of odes involving nonexpansive mappings to prove convergence with probability 1 in the stochastic case we also apply our convergence analysis to q learning algorithms for stochastic shortest path problems and are able to relax some of the assumptions of the currently available results

simple mechanical control systems with constraints and symmetry we develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber bundle structure taking the viewpoint of affine connection control systems we derive reduced formulations of the levi civita and the nonholonomic affine connections along with the symmetric product in the presence of symmetries and nonholonomic constraints we note that there are naturally two kinds of connections to be considered here affine and principal connections leading to what we term a connection within a connection these results are then used to describe controllability tests that are specialized to simple underactuated mechanical systems on principal fiber bundles including the notion of fiber configuration controllability we present examples of the use of these tools in studying the planar rigid body with a variable direction vectored thruster and the snakeboard robot

control design for large scale lur e systems with arbitrary information structure constraints abstract in this paper an lmi based approach is proposed for the design of static output feedback for multi nonlinear lur e postnikov systems the resulting control laws ensure absolute stability and at the same time maximize the size of the nonlinear sectors the proposed method is computationally efficient and can accommodate feedback laws with arbitrary information structure constraints the effectiveness of this approach is demonstrated on a large scale example with 100 state variables

a note on spectral conditions for positive realness of single input single output systems with strictly proper transfer functions necessary and sufficient conditions for strict positive realness and positive realness of strictly proper functions are derived the conditions are expressed in terms of eigenvalues of the state matrices representation of the system previous results rendered conditions which were significantly more complex than those for proper but not strictly proper functions the present conditions for strictly proper functions are simpler than the ones for proper functions which is consistent with intuition in this case illustrative numerical examples are provided

a kyp lemma and invariance principle for systems with multiple hysteresis non linearities absolute stability criteria for systems with multiple hysteresis non linearities are given in this paper it is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem if the linear subsystem satis es a particular linear matrix inequality and a simple residue condition then as is proven the non linear system will be asymptotically stable the main stability theorem is developed using a combination of passivity lyapunov and popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non linear closed loop system the invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses the class of non linearities covered by the analysis is very general and includes multiple slope restricted memoryless non linearities as a special case simple numerical examples are used to demonstrate the e ectiveness of the new analysis in comparison to other recent results and graphically illustrate state asymptotic stability

a geometrical framework for the study of non holonomic lagrangian systems a geometrical framework is presented for the treatment of a class of dynamical systems which are modelled by a system of second order differential equations coupled with first order equations linear in the derivatives such problems in particular make their appearance in the study of lagrangian systems with non holonomic constraints among other things we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of dynamical covariant derivative and jacobi endomorphism the intrinsic tools which are being developed further allow a direct geometrical construction of the dynamics of non holonomic systems the vertically rolling disc is chosen as an illustrative example for the newly proposed formalism and results

absolute stability criteria for multiple slope restricted monotonic nonlinearities absolute stability criteria such as the classical popov criterion guarantee stability for a class of sector bounded nonlinearities although the sector restriction bounds the admissible class of the nonlinearities the local slope of the nonlinearity may be arbitrarily large in this paper the authors derive absolute stability criteria for multiple slope restricted time invariant monotonic nonlinearities like the popov criterion in the single input single output case the authors results provide a simple graphical interpretation involving a straight line in a modified popov plane

quadratic stability and singular siso switching systems in this note we consider the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices are of the form a a ght and where one of the matrices is singular a necessary and sufficient condition for the existence of such a function is given in terms of the spectrum of the product a a ght the technical note also contains a spectral characterization of strictly positive real transfer functions which are strictly proper examples are presented to illustrate our results

passivity analysis and passification for uncertain signal processing systems the problem of passivity analysis finds important applications in many signal processing systems such as digital quantizers decision feedback equalizers and digital and analog filters equally important is the problem of passification where a compensator needs to be designed for a given system to become passive this paper considers these two problems for a large class of systems that involve uncertain parameters time delays quantization errors and unmodeled high order dynamics by characterizing these and many other types of uncertainty using a general tool called integral quadratic constraints iqcs we present solutions to the problems of robust passivity analysis and robust passification more specifically for the analysis problem we determine if a given uncertain system is passive for all admissible uncertainty satisfying the iqcs similarly for the problem of robust passification we are concerned with finding a loop transformation such that a particular part of the uncertain signal processing system becomes passive for all admissible uncertainty the solutions are given in terms of the feasibility of one or more linear matrix inequalities lmis which can be solved efficiently

robust stabilization with positive real uncertainty beyond the small gain theory in many applications of feedback control phase information is available concerning the plant uncertainty for example lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions closed loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators in this paper the properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real but otherwise unknown plant uncertainty these results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty one of the principal motivations for these results is to utilize phase information in guaranteeing robust stability in this these sense results go beyond the usual limitations the small gain theorem and quadratic lyapunov functions which may be conservative when phase information is available the results of this paper are based upon a riccati equation formulation of the positive real lemma and thus are in the spirit of recent riccati based approaches to bounded real h control

new results for strict positive realness and feedback stability proposes an algorithm which enables the authors to characterize positive realness and three different classes of strict positive realness in terms of riccati equations and inherent integrations it has been an open question whether or not the feedback system consisting of a weakly strict positive realness and a nonlinear passivity is asymptotically stable the paper also shows that the statement is indeed the case

influence of nonholonomic constraints on variations symplectic structure and dynamics of mechanical systems based on a serious analysis of the frobenius integrability condition for affine differential constraints that mechanical systems are subject to the necessary and sufficient conditions for coincidence of three kinds of unfree variations the existence of simple symplectic structure of the constraint submanifold and equivalence of nonholonomic and vakonomic dynamics for the constrained systems are respectively obtained which are all related with the frobenius integrability condition in their special forms two illustrative examples are presented to verify the results

equivalence between positive real and norm bounded uncertainty this paper focuses on a linear time invariant system with positive real uncertainty and shows an equivalence between positive real and norm bounded uncertainty resulting in transformation of a proper transfer function with positive real uncertainty directly to a strictly proper transfer function with norm bounded uncertainty which is different from the result obtained by using bilinear transform this paper also shows a secondary result relations between the quadratic and robust stability of a system with positive real uncertainty

a unified theory for the cgt approach to adaptive control the command generator tracker cgt approach to model reference adaptive control of linear time invariant multi input multi output plants which are not almost strictly positive real aspr is considered three new algorithms are proposed which are characterized by the insertion of supplementary dynamics in i feedback with the plant ii in parallel with the plant and iii in cascade with the plant respectively the three different implementations are shown to be special cases of a metasystem representation for the adaptive system the new algorithms extend earlier results by introducing adaptive gains for the supplementary dynamics in addition to adaptive gains for the controller a stability proof is presented which shows that the error between the plant and model outputs is bounded in the presence of bounded input and output disturbances if the model input is constant for t t 1 and no disturbances exist then conditions are obtained for the choice of the design parameters in the parallel suppl

foundations of not necessarily rational negative imaginary systems theory relations between classes of negative imaginary and positive real systems in this technical note we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory in analogy with the theory of positive real functions in our general framework negative imaginary systems are defined in terms of a domain of analyticity of the transfer function and of a sign condition that must be satisfied in such domain in this way we do not require to restrict the attention to systems with a rational transfer function in this work we also define various grades of negative imaginary systems and aim to provide a unitary view of the different notions that have appeared so far in the literature within the framework of positive real and in the more recent theory of negative imaginary systems and to show how these notions are characterized and linked to each other

the exterior conic sector lemma the conic sector theorem is a powerful input output stability analysis tool providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis without means to systematically identify and impose exterior conic properties existing analysis and synthesis methods are limited to employing a special case of the conic sector theorem which may prove overly conservative when considering very strictly passive plants this paper removes the current limitation by establishing the exterior conic sector lemma which provides matrix inequalities equivalent to exterior conic bounds for linear time invariant systems

spectral conditions for positive realness of single input single output systems in this note we derive necessary and sufficient conditions for a single input single output system to be strictly positive real these conditions take the form of a spectral condition on a matrix product constructed from the state space representation of the linear time invariant system a b c d numerical examples are given to illustrate the usefulness of our conditions

constrained systems and analytical mechanics in spaces with torsion a system with anholonomic constraints where the trajectories of physical degrees of freedom are autoparallels on a manifold equipped with a general cartan connection is discussed a variational principle for the autoparallel trajectories is derived from the d alambert lagrange principle for anholonomic constrained systems a geometrical coordinate independent formulation of the variational principle is given its relation to sedov s anholonomic variational principle for dissipative systems and to poincar s variational principle in anholonomic reference frames is established a modification of noether s theorem due to the torsion force is studied a non local action whose extrema contain the autoparallels is proposed the action can be made local by adding auxiliary degrees of freedom coupled to the original variables in a special way

on the characterization of strict positive realness for general matrix transfer functions we present conditions which are necessary and sufficient for a transfer function or transfer function matrix to be strictly positive real a counter example is given to illustrate that the conditions presented here differ from those previously presented in the literature the proof of our results differs from previous related proofs in that it only uses properties of analytic functions and matrices and does not require state space realizations also the results are not restricted to rational transfer functions with real coefficients

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

positive real control problem for uncertain linear time invariant systems abstract this paper focuses on positive real control of linear time invariant systems which are subjected to norm bounded uncertainties in the state equation we address the problem of designing a linear dynamic output feedback controller that robustly stabilizes the uncertain system and achieves the extended strict positive realness property for a given closed loop transfer function it is shown that a solution to the above problem can be obtained by solving a scaled strict positive real control problem for which no parameter uncertainty occurs

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

a note on spectral conditions for positive realness of transfer function matrices necessary and sufficient conditions for positive realness of general transfer function matrices are derived the conditions are expressed in terms of eigenvalues of matrix functions of the state matrices representation of the lti system illustrative numerical examples are provided

an analysis of stochastic shortest path problems we consider a stochastic version of the classical shortest path problem whereby for each node of a graph we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost the costs of transition between successive nodes can be positive as well as negative we prove natural generalizations of the standard results for the deterministic shortest path problem and we extend the corresponding theory for undiscounted finite state markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive

strictly positive real transfer functions revisited the two most commonly used definitions of strictly positive real spr transfer functions are reviewed contrary to what has been suggested in the literature it is proven that the least restrictive weak definition of the two is clearly related to the yacubovich kalman lemma the relationship between time and frequency domain conditions pertaining to the weak definition of spr systems is established

a mixed value and policy iteration method for stochastic control with universally measurable policies we consider stochastic optimal control models with borel spaces and universally measurable policies for such models the standard policy iteration is known to have difficult measurability issues and cannot be carried out in general we present a mixed value and policy iteration method that circumvents this difficulty the method allows the use of stationary policies in computing the optimal cost function in a manner that resembles policy iteration it can also be used to address similar difficulties of policy iteration in the context of upper and lower semicontinuous models we analyze the convergence of the method in infinite horizon total cost problems for the discounted case where the one stage costs are bounded and for the undiscounted case where the one stage costs are nonpositive or nonnegative for undiscounted total cost problems with nonnegative one stage costs we also give a new convergence theorem for value iteration that shows that value iteration converges whenever it is initialized with a f

the octomorphic criterion for real parameter uncertainty real m bounds without circles and d n scales abstract in this paper we introduce new bounds for robust stability analysis with real parameter uncertainty the approach is based on an absolute stability criterion that excludes the nyquist plot from a paraboloidal region containing the point 1 j0 transformation of this criterion to the case of norm bounded uncertainty leads to a stability criterion in terms of the octomorphic or figure eight shaped region the requirement that the nyquist plot lie inside the octomorphic region thus yields a bound on the allowable real parameter uncertainty this stability criterion is distinct from recent bounds for real which involve frequency dependent scales having a frequency dependent off axis circle interpretation since the octomorphic region includes both upper and lower halves it is able to encompass the entire nyquist plot without using frequency dependent scales

frequency domain conditions for strictly positive real functions frequency domain conditions for strictly positive real spr functions which appear in literature are often only necessary or only sufficient this point is raised in 1 2 where necessary and sufficient conditions in the s domain are given for a transfer function to be spr in this note the points raised in 1 i2 are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be spr these frequency domain conditions are easier to test than those given in the s domain or time domain 1 2

error bounds for constant step size q learning we provide a bound on the first moment of the error in the q function estimate resulting from fixed step size algorithms applied to finite state space discounted reward markov decision problems motivated by tsitsiklis proof for the decreasing step size case we decompose the q learning update equations into a dynamical system driven by a noise sequence and another dynamical system whose state variable is the q learning error i e the difference between the true q function and the estimate a natural persistence of excitation condition allows us to sample the system periodically and derive a simple scalar difference equation from which the convergence properties and bounds on the error of the q learning algorithm can be derived

strict positive realness of descriptor systems in state space in this article we give necessary and sufficient spectral conditions for various notions of strict positive realness for single input single output descriptor systems these conditions only require calculation of eigenvalues of a single matrix a characterisation of a kalman yacubovich popov like lemma for descriptor systems is also derived and its implications for the stability of a class of switched descriptor systems are briefly discussed

low order multivariable adaptive control with application to flexible structures all adaptive controllers require a certain amount of prior information regarding the plant to be controlled to achieve the desired control objectives in the context of multivariable adaptive control not only is the prior information quite significant but also the complexity of the adaptive controller is too high in this paper we introduce a low order multivariable adaptive controller that is easy to implement and requires less prior information than most other adaptive controllers

synthesis of positive real feedback systems a simple derivation via parrott s theorem a simple derivation of the formulas for controller rendering a closed loop system strongly positive real is presented the derivation is based on parrott s theorem and the positive real lemma when the positive real lemma is applied to the closed loop system a bilinear matrix inequality results the significance of parrott s theorem in the context of a strongly positive real synthesis problem is that it allows one to convert this bilinear matrix inequality into two linear matrix inequalities

comments on a note on spectral conditions for positive realness of single input single output systems with strictly proper transfer functions the necessary and sufficient conditions for strict positive realness of strictly proper transfer functions are derived in the above mentioned paper 1 presenting a counter example we show that there are inconsistencies between definitions stated in section ii and main results proved in section iii of 1 also the required corrections in the main results of 1 are presented note however that the results in 1 are correct for the definition of spr in the circuit theory sense

necessary and sufficient conditions on negative imaginariness for interval siso transfer functions and their interconnection this article is concerned with the study of the negative imaginariness of interval transfer functions necessary and sufficient conditions for strictly negative imaginary interval transfer functions subject to parameter uncertainty are first derived based on kharitonov s theorem then certain properties of the interconnection system with negative imaginary interval transfer functions are analyzed finally an example is provided to illustrate the main results of the article

the asymptotic stability of nonlinear lur e systems with multiple slope restrictions the authors present the analysis of the asymptotic stability of multiple slope restricted nonlinear lur e systems by providing a lyapunov function they obtain a matrix language criterion in terms of algebraic riccati equations and linear matrix inequalities which are discussed at the point of computational issues additionally they consider the frequency domain interpretation of the result

extensions of mixed µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory in this paper we make explicit connections between classical absolute stability theory and modern mixed analysis and synthesis specifically using the parameter dependent lyapunov function framework of haddad and bernstein and the frequency dependent off axis circle interpretation of how and hall we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty an immediate application of this framework is the generalization and reformulation of mixed analysis and synthesis in terms of lyapunov functions and riccati equations this observation is exploited to provide robust reduced order controller synthesis while avoiding the standard d n k iteration and curve fitting procedures

stabilization of uncertain nonholonomic systems via time varying sliding mode control this note addresses the robust stabilization problem for a general class of nonholonomic systems in the presence of drift uncertainties the control approach developed is based on the combined applications of the sliding mode control technique and nonlinear time varying systems theory first some properties of nonlinear time varying systems are introduced for the purpose of designing sliding mode controller an explicit time varying feedback form is provided to guarantee the existence and uniqueness of periodic time varying solution for the corresponding linear periodic partial differential equation second an explicit discontinuous feedback control law is presented to guarantee the existence of sliding mode the first integrals obtained by the previous periodic partial differential equation are then directly used to determine the switching function the uniform asymptotic stability of the closed loop system is proved via the invariance principle of nonlinear time varying systems finally an example is given to illustrate the proposed approach

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

on a class of marginally stable positive real systems a class of marginally stable positive real systems is defined which is less restrictive than the previously published definitions of strictly positive real systems a minimal realization and state space characterization of such systems are presented and it is proven that controllers belonging to this class robustly stabilize positive real plants

jet methods in nonholonomic mechanics classical nonholonomic mechanical systems are studied whose evolution space is a fiber bundle m r the framework is that of jet spaces in which the geometrical meaning of the theory emerges clearly for systems with linear constraints a new interpretation is given of appell s equations as first order differential equations associated with a suitable vector field the d alembert principle is formulated in an appropriate way to be generalized to systems with nonlinear constraints the equivalence between the equations of motion arising from this generalization the ones set up on gauss principle of least constraint and hertz s principle of least curvature is established

on relationships among passivity positive realness and dissipativity in linear systems the notions of passivity and positive realness are fundamental concepts in classical control theory but the use of the terms has varied for lti systems these two concepts capture the same essential property of dynamical systems that is a system with this property does not generate its own energy but only stores and dissipates energy supplied by the environment this paper summarizes the connection between these two concepts for continuous and discrete time lti systems beyond that relationships are provided between classes of strictly passive systems and classes of positive real systems the more general framework of dissipativity is introduced to connect passivity and positive realness and also to survey other energy based results the frameworks of passivity indices and conic systems are discussed to connect to passivity and dissipativity after surveying relevant existing results some clarifying results are presented these involve connections between classes of passive systems and finite gain l2l2 stability as well as asymptotic stability additional results are given to clarify some of the more subtle conditions between classes of these systems and stability results this paper surveys existing connections between classes of passive and positive real systems and provides results that clarify more subtle connections between these concepts

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

higher order nonlinear complementary filtering on lie groups nonlinear observer design for systems whose state space evolves on lie groups is considered the proposed method is similar to previously developed nonlinear observers in that it involves propagating the state estimate using a process model and corrects the propagated state estimate using an innovation term on the tangent space of the lie group in the proposed method the innovation term is constructed by passing the gradient of an invariant cost function resolved in a basis of the tangent space through a linear time invariant system the introduction of the linear system completes the extension of linear complementary filters to nonlinear lie group observers by allowing higher order filtering in practice the proposed method allows for greater design freedom and with the appropriate selection of the linear filter the ability to filter bias and noise over specific bandwidths a disturbance observer that accounts for constant and harmonic disturbances in group velocity measurements is also considered local asymptotic stability about the desired equilibrium point is demonstrated a numerical example that demonstrates the desirable properties of the observer is presented in the context of pose estimation

technical communique mitigation of symmetry condition in positive realness for adaptive control feasibility of nonlinear and adaptive control methodologies in multivariable linear time invariant systems with state space realization a b c is apparently limited by the standard strictly positive realness conditions that imply that the product cb must be positive definite symmetric this paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric cb is only required to have a diagonal jordan form and positive eigenvalues the paper also shows that under the new condition any minimum phase systems can be made strictly positive real via constant output feedback the paper illustrates the usefulness of these extended properties with an adaptive control example

passivity and disturbance attenuation via output feedback for uncertain nonlinear systems the authors address the problem of disturbance attenuation with internal stability via output feedback for a class of passive systems with uncertainties the problem is approached by means of adaptive output feedback control which does not require any state observer the results obtained extend an earlier result of steinberg and corless 1985 sufficient conditions are proposed under which a nonlinear system can be made locally or globally passive via output feedback

characterizing families of positive real matrices by matrix substitutions on scalar rational functions abstract this paper concerns the characterization of positive real pr matrices generated by substitutions of the laplace variable s in scalar rational transfer functions by matrix pr functions our main results are restricted to both strongly strictly pr matrices sspr and strictly bounded real matrices as a way to illustrate our main results we also include here a partial extension of both the kalman yakubovich popov lemma for sspr systems of zero relative degree and the circle criterion for strictly pr systems of zero relative degree

h robustness properties preservation in siso systems when applying spr substitutions the preservation of both robust stability and weighted robust performance properties of controlled linear time invariant single input single output siso systems is studied when performing substitutions of the complex laplace variable s by a particular class of rational strictly positive real functions spr functions the so called strictly positive real functions of zero relative degree spr0 functions concerning weighted robust performance and model matching we provide some results on preservation of controller optimality

strictly positive real matrices and the lefschetz kalman yakubovich lemma the authors give necessary and sufficient conditions in the frequency domain for rational matrices to be strictly positive real based on this result the matrix form of the lefschetz kalman yakubovich lemma is proved which gives necessary and sufficient conditions for strictly positive real transfer matrices in the state space realization form