high gain feedback in non linear control systems abstract high gain state and output feedback are investigated for non linear control systems with a single additive input by using singular perturbation techniques classical approximation results tihonov like theorems in singular perturbation theory are extended to non linear control systems by defining a composite additive control strategy a control dependent fast equilibrium manifold and non linear change of coordinates those tools and an appropriate change of coordinates show that high gain state feedback and variable structure control systems can be equivalently used for approximate non linearity compensation in feedback linearizable systems next the effect of high gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility for strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input output feedback linear

improved least squares identification an improved recursive least squares algorithm for parameter estimation is presented which includes on off criteria to prevent parameter drift during periods of low excitation a variable forgetting factor which maintains the trace of the covariance matrix at a user specified value data preprocessing and normalization to improve numerical accuracy scaling of the regressor vector to minimize the condition number of the covariance matrix plus independent estimation of the mean values of the input output data which can be used to eliminate errors due to d c bias or slowly drifting elements in the regressor vector the algorithm can also include parameter projection to constrain the estimates to a priori specified regions and retains the formal properties such as convergence of a true weighted least squares algorithm the proposed algorithm is compared with other modifications suggested in the literature and its advantages are demonstrated by a simulated example

paper error propagation properties of recursive least squares adaptation algorithms the numerical properties of implementations of the recursive least squares identification algorithm are of great importance for their continuous use in various adaptive schemes here we investigate how an error that is introduced at an arbitrary point in the algorithm propagates it is shown that conventional ls algorithms including bierman s ud factorization algorithm are exponentially stable with respect to such errors i e the effect of the error decays exponentially the base of the decay is equal to the forgetting factor the same is true for fast lattice algorithms the fast least squares algorithm sometimes known as the fast kalman algorithm is however shown to be unstable with respect to such errors

grassmannian central projection and output feedback pole assignment of linear systems a sufficient rank condition for arbitrary pole assignability by static output feedback of m input p output linear systems of mcmillan degree n mp is given this condition is easy to check and is satisfied by the generic system an effective computational method to compute the feedback laws assigning any self conjugate set of closed loop poles is also given

output regulation of nonlinear systems with nonhyperbolic zero dynamics previous solvability conditions for the output relation of nonlinear systems are applicable only to those with hyperbolic zero dynamics it is shown in this note that output regulation of nonlinear systems with nonhyperbolic zero dynamics is also possible by a reduction of the plant dynamics and an augmentation of the exosystem

shuffle compatible permutation statistics abstract since the early work of richard stanley it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations we formalize this notion of a shuffle compatible permutation statistic and introduce the shuffle algebra of a shuffle compatible permutation statistic which encodes the distribution of the statistic over shuffles of permutations this paper develops a theory of shuffle compatibility for descent statistics statistics that depend only on the descent set and length which has close connections to the theory of p partitions quasisymmetric functions and noncommutative symmetric functions we use our framework to prove that many descent statistics are shuffle compatible and to give explicit descriptions of their shuffle algebras thus unifying past results of stanley gessel stembridge aguiar bergeron nyman and petersen

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

representing a nonlinear state space system as a set of higher order differential equations in the inputs and outputs an algorithm is presented for transforming a nonlinear state space system into a threefold set of equations the first subset describing the dynamics of the unobservable part of the system the second subset representing the remaining state variables as functions of inputs and outputs and their derivatives and the last subset defining the external behaviour of the system

almost disturbance decoupling with global asymptotic stability for nonlinear systems with disturbance affected unstable zero dynamics the problem of global almost disturbance decoupling problem with stability for nonlinear system is revisited and is shown to be solvable for a class of nonlinear systems whose zero dynamics contains a chain of integrators affected by disturbance this result complements some recent work on the topic where unstable zero dynamics is not allowed to be affected by the disturbances

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

the analysis of zero dynamics of nonlinear control systems with symmetries zero dynamics of nonlinear control systems with symmetries are studied in this paper it is shown that the zero dynamics of a symmetric system is also symmetric and admits a special form of cascade decomposition a local zero dynamics is always possible to be extended to the semiglobal case the relation between the zero dynamics and the quotient system is also discussed

the peak algebra and the descent algebras of types b and d we show the existence of a unital subalgebra n of the symmetric group algebra linearly spanned by sums of permutations with a common peak set which we call the peak algebra we show that n is the image of the descent algebra of type b under the map to the descent algebra of type a which forgets the signs and also the image of the descent algebra of type d the algebra n contains a two sided ideal n which is defined in terms of interior peaks this object was introduced in previous work by nyman 2003 we find that it is the image of certain ideals of the descent algebras of types b and d we derive an exact sequence of the form 0 n n n 2 0 we obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs in particular we construct two new commutative semisimple subalgebras of the descent algebra of dimensions n and n 2 1 by grouping permutations according to their number of peaks or interior peaks we discuss the hopf algebraic structures that exist on the direct sums of the spaces n and n over n 0 and explain the connection with previous work of stembridge 1997 we also obtain new properties of his descents to peaks map and construct a type b analog

an augmented ud identification algorithm an augmented ud identification audi algorithm for system identification is developed by rearranging the data vectors and augmenting the covariance matrix of bierman s ud factorization algorithm the structure of the augmented information covariance matrix is particularly easy to interpret and it is shown that the audi algorithm is a direct extension of the familiar recursive least squares rls algorithm the proposed algorithm permits simultaneous identification of the model parameters plus loss functions for all orders from 1 to n at each time step with approximately the same calculation effort as th order rls this provides a basis for simultaneous model order and parameter identification so that problems due to over and under estimation of model can be avoided based on its least squares properties numerical robustness theoretical basis and the fact that it simultaneously estimates multiple models the proposed audi algorithm is recommended for use in place of rls and bierman s ud factorizatio

global robust stabilization of minimum phase nonlinear systems with uncertainty the problem of global stabilization via state feedback is studied for a class of minimum phase nonlinear systems with unknown unmodelled time varying parameters or disturbances belonging to a given compact set sufficient conditions are presented for the existence of a robust stabilizing state feedback controller the controller is constructed explicitly by using a lyapunov based recursive design procedure the proposed feedback control law is a smooth function of the state and requires information of bounding functions on the size of uncertainties in the case where the disturbance signals represent only an unknown constant vector parameter and appear linearly in the system an adaptive controller is developed based on similar arguments achieving global stability with state regulation the main results of this paper incorporate and generalize a number of robust and adaptive control schemes for global stabilization of uncertain feedback linearizable systems without zero dynamics

stabilization of nonlinear systems via designed center manifold this paper addresses the problem of local state feedback stabilization of a class of nonlinear systems with nonminimum phase zero dynamics a new technique namely the lyapunov function with homogeneous derivative along solution curves was developed to test the approximate stability of the dynamics on the center manifold a set of convenient sufficient conditions are provided to test the negativity of the homogeneous derivatives using these conditions and assuming the zero dynamics has stable and center linear parts a method is proposed to design controls such that the dynamics on the designed center manifold of the closed loop system is approximately stable it is proved that using this method the first variables in each of the integral chains of the linearized part of the system do not affect the approximation order of the dynamics on the center manifold based on this fact the concept of injection degree is proposed according to different kinds of injection degrees certain sufficient conditions are obtained for the stabilizability of the nonminimum phase zero dynamics corresponding formulas are presented for the design of controls

bounded tracking for non minimum phase nonlinear systems with fast zero dynamics in this paper we derive tracking control laws for non minimum phase nonlinear systems with both fast and slow possibly unstable zero dynamics the fast zero dynamics arise from a perturbation of a nominal system these fast zeros can be problematic in that they may be in the right half plane and may cause large magnitude tracking control inputs in this paper we combine the ideas from some recent work of hunt meyer and su with that of devasia chen and paden on an asymptotic tracking procedure for non minimum phase nonlinear systems we give somewhat subtle conditions under which the tracking control input is bounded as the magnitude of the perturbation of the nominal system becomes zero explicit bounds on the control inputs are calculated for both siso and mimo systems using some interesting non standard singular perturbation techniques the method is applied to a suite of examples including the simplified planar dynamics of vtol and ctol aircraft

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

a bijective proof of stanley s shuffling theorem for two permutations a and w on disjoint sets of integers consider forming a permutation on the combined sets by shuffling a and w i e a and w appear as subsequences stanley 10 by considering p partitions and a q analogue of saalschutz s 3f2 summation obtained the generating function for shuffles of a and w with a given number of falls an element larger than its successor with respect to greater index sum of positions of falls it is a product of two q binomial coefficients and depends only on remarkably simple parameters namely the lengths numbers of falls and greater indexes of a and w a combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce a and w to canonical permutations for which a direct evaluation of the generating function is given

stabilization of minimum phase nonlinear systems by dynamic output feedback in this paper the stabilization problem of minimum phase nonlinear systems by dynamic output feedback is investigated it is shown that if the zero dynamics of an affine nonlinear system is asymptotically stable according to the kth approximation then the system is stabilizable by dynamic output feedback the proof is constructive which provides a method to design the stabilizer the result is applied to investigating the stabilization problem of the rotating stall in axial flow compressors

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

plethystic formulas for permutation enumeration abstract we prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions and the permutation statistics we consider include the descent number peak number left peak number and the number of up down runs we apply these results to cyclic permutations involutions and derangements and more generally to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type a number of known formulas are recovered as special cases of our results including formulas of desarmenien foata gessel reutenauer stembridge fulman petersen diaconis fulman holmes zhuang and athanasiadis

stabilization via dynamic output feedback for systems with output nonlinearities abstract the problem of asmptotically stabilizing a class of systems by means of continuous output feedback is considered these systems are characterized by nonlinear terms depending only on the ouputs it is shown that for these systems stabilization via continuous state feedback plus stabilization via output injection imply stabilization via continuous dynamic output feedback this generalizes a well knwon result for linear systems

minimum phase properties for input nonaffine nonlinear systems for input nonaffine nonlinear control systems the minimum phase property of the system in general depends on the control law switching or discontinuous controllers may offer advantages in this context in particular there may not exist a continuous control law which would keep the output identically equal to zero and for which the zero output constrained dynamics are locally stable whereas a discontinuous controller which achieves this exists for single input single output input nonaffine nonlinear systems we give sufficient conditions for existence and present a method for the design of discontinuous switching controllers which yield locally stable zero dynamics

on static output feedback stabilization of systems with stable generalized zero dynamics in this note we define generalized zero dynamics introduce a generalized zero dynamics algorithm and give a sufficient condition which ensures stabilizability of linear systems with asymptotically stable generalized zero dynamics by static output feedback