stabilization of discrete time nonlinear uncertain systems by feedback based on ls algorithm this paper is concerned with the use of the least squares ls algorithm to design a feedback control law to stabilize a basic class of discrete time nonlinear uncertain systems the result shows that if a certain polynomial criterion is satisfied the system can be stabilized by feedback based on the ls algorithm for gaussian distributed noise and unknown parameters this result thus provides an answer to the question of what are the fundamental limitations of the discrete time adaptive nonlinear control this issue of feedback capability has been an open problem for more than ten years since it was put forward

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

on the static h control problem the problem of existence of static controller in standard h control problem or equivalently solvability of a couple of riccati inequalities with respect to the same hermitian matrix is reduced to solvability of a single riccati inequality this inequality is presented in a form which is bilinear with respect to unknown parameters in a special case the inequality is transformed to lmi

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on feedback capability in a class of nonlinearly parameterized uncertain systems in this technical note we will investigate the maximum capability and limitations of the feedback mechanism in globally stabilizing a basic class of discrete time nonlinearly parameterized dynamical systems with multiple unknown parameters both possibility and impossibility theorems together with a fairly complete characterization on the capability of feedback will be presented it will be seen that to characterize the feedback capability the growth rates of the sensitivity functions of the nonlinear dynamics with respect to the uncertain parameters play a crucial role and a suitable decomposition of the family of the nonlinearly parameterized functions in question turns out to be necessary

emptiness formation probability of the six vertex model and the sixth painleve equation we show that the emptiness formation probability of the six vertex model with domain wall boundary conditions at its free fermion point is a tau function of the sixth painleve equation using this fact we derive asymptotics of the emptiness formation probability in the thermodynamic limit

on robust stability of discrete time adaptive nonlinear control abstract we consider adaptive control of discrete time nonlinear systems with a single unknown parameter in this paper we demonstrate that the necessary and sufficient condition for the existence of a feedback stabilizer that is robust to bounded noise is that the nonlinear growth rate of the system dynamics is less than 4 this result further confirms the conclusion of guo on critical stability of discrete time adaptive nonlinear control ieee trans automat control 42 1997 1488 1499 which addresses unbounded noise in a stochastic setting also in our worst case approach we find that much simpler adaptive stabilizers can be constructed when the nonlinear growth rate is less than 4

an impossibility theorem on a class of high order discrete time nonlinear control systems a high order discrete time nonlinear system with unknown parameters and gaussian noise is studied in this paper and it is shown that under a simple algebraic condition concerning the nonlinearity of the system there does not exist any feedback controller that globally stabilizes the uncertain system this impossibility result attempts to understand the capability and limitation of feedback mechanism of a high order discrete time nonlinear system and continues a series of previous research on similar first order systems this result shows that for the high order discrete time nonlinear system studied here a polynomial corresponding to the first order principle part of the system determines the limit of the feedback mechanism

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

convergence and logarithm laws of self tuning regulators abstract this paper starts with a survey of some recent results on least squares ls and ls based self tuning regulators str several long standing problems concerning the basic properties of ls based str such as stability optimality consistency and the best convergence rate are solved within a unified framework some previously related results are also subsumed and extended various new techniques for analysing ls based adaptive tracking systems are presented which may also be useful for analysing other adaptive control problems

on critical stability of discrete time adaptive nonlinear control in this paper we examine the global stability and instability problems for a class of discrete time adaptive nonlinear stochastic control the systems to be controlled may exhibit chaotic behavior and are assumed to be linear in unknown parameters but nonlinear in output dynamics which are characterized by a nonlinear function say f x it is found and proved that in the scalar parameter case there is a critical stability phenomenon for least squares ls based adaptive control systems to be specific let the growth rate of f x be f x o spl par x spl par sup 6 with b spl ges 0 then it is found that b 4 is a critical value for global stability i e the closed loop adaptive system is globally stable if b 4 and is unstable in general if b spl ges 4 as a consequence we find an interesting phenomenon that the linear case does not have for some ls based certainty equivalence adaptive controls even if the ls parameter estimates are strongly consistent the closed loop systems may still be unstable this paper also indicates that adaptive nonlinear stochastic control that is designed based on e g taylor expansion or weierstrass approximation for nonlinear models may not be feasible in general

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

a limit to the capability of feedback feedback is ubiquitous and is a basic concept in the area of control where it is used primarily for reducing internal or external uncertainties or both in this note we will study the capability of feedback in dealing with both internal and external uncertainties for a class of pth order nonlinear autoregressive control systems the size of the uncertainty is described by the lipschitz constant say l of the uncertain nonlinear function in consideration it is shown that if p and l satisfy a certain inequality then there exists no globally stabilizing feedback for the corresponding class of uncertain systems and thus finding a quantitative limit to the capability of the feedback mechanism in dealing with structural uncertainties

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

a new critical theorem for adaptive nonlinear stabilization it is fairly well known that there are fundamental differences between adaptive control of continuous time and discrete time nonlinear systems in fact even for the seemingly simple single input single output control system y t email protected 1f y t u t w t 1 with a scalar unknown parameter q 1 and noise disturbance w t and with a known function f having possible nonlinear growth rate characterized by f x email protected x b with b 1 the necessary and sufficient condition for the system to be globally stabilizable by adaptive feedback is b 4 this was first found and proved by guo 1997 for the gaussian white noise case and then proved by li and xie 2006 for the bounded noise case subsequently a number of other types of critical values and impossibility theorems on the maximum capability of adaptive feedback were also found mainly for systems with known control parameter as in the above model in this paper we will study the above basic model again but with additional unknown control parameter q 2 i e u t is replaced by q 2u t in the above model interestingly it turns out that the system is globally stabilizable if and only ifb 3 this is a new critical theorem for adaptive nonlinear stabilization which has meaningful implications for the control of more general uncertain nonlinear systems

simultaneous identification and stabilization of nonlinearly parameterized discrete time systems by nonlinear least squares algorithm this paper addresses the challenging problem of designing an adaptive feedback strategy for simultaneous identification and stabilization for a class of nonlinearly parameterized uncertain systems in discrete time the nonlinear least squares nls algorithm is applied to estimate the unknown parameters and it turns out to be the standard least squares ls algorithm whenever the model is linearly parameterized based on this algorithm both the strong consistency of the estimator and the global stability of the system are achieved with the output feedback design for the scalar parameter case

towards understanding the capability of adaptation for time varying systems the information uncertainty coupled with the structural complexity is more pertinent than the rate of parameter changes in characterizing the capability of adaptation for time varying systems

further results on limitations to the capability of feedback in this paper we study the limitations of the capability of the feedback mechanism in dealing with structural uncertainty for a class of pth order non linear autoregressive control systems the size of the structural uncertainty is characterized by multiple lipschitz constants say l 1 l 2 lp of the uncertain non linear function it is shown that if l 1 l 2 lp satisfy certain algebraic conditions then there exists no globally stabilizing feedback control law for the corresponding uncertain systems the result here is an extension of the previous result obtained by zhang and guo 2002

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

fundamental limitations of discrete time adaptive nonlinear control a particular polynomial is introduced which can be used to determine under what conditions a typical class of discrete time nonlinear systems with uncertainties in both parameters and noises is not stabilizable by feedback thus demonstrating the fundamental limitations of discrete time adaptive nonlinear control as a consequence it is shown that for nonlinear systems with unknown parameters and noises the systems may indeed be nonstabilizable in general whenever the usual linear growth condition is relaxed and the number of unknown parameters is large even though the corresponding noise free systems are globally stabilizable