technical communique differential geometric condition for feedback complete linearization of stochastic nonlinear system necessary and sufficient condition is obtained for the feedback complete linearization problem for stochastic nonlinear systems

statistical control for performance shaping using cost cumulants the cost cumulant statistical control method optimizes the system performance by shaping the probability density function of the random cost function for a stochastic system a typical optimal control method minimizes the mean first cumulant of the cost function however there are other statistical properties of the cost function such as variance second cumulant skewness third cumulant and kurtosis fourth cumulant which affect system performance in this technical note we extend the theory of traditional stochastic control by deriving the hamilton jacobi bellman hjb equation as the necessary conditions for optimality furthermore we derived the verification theorem which is the sufficient condition for higher order statistical control and construct the optimal controller in addition we utilize neural networks to numerically solve hjb partial differential equations finally we provide simulation results for an oscillator system to demonstrate our method

linear quadratic risk sensitive and robust mean field games this paper considers two classes of large population stochastic differential games connected to optimal and robust decentralized control of large scale multiagent systems the first problem p1 is one where each agent minimizes an exponentiated cost function capturing risk sensitive behavior whereas in the second problem p2 each agent minimizes a worst case risk neutral cost function where the worst case stems from the presence of an adversary entering each agent s dynamics characterized by a stochastic differential equation in both problems the individual agents are coupled through the mean field term included in each agent s cost function which captures the average or mass behavior of the agents we solve both p1 and p2 via mean field game theory specifically we first solve a generic risk sensitive optimal control problem and a generic stochastic zero sum differential game where the corresponding optimal controllers are applied by each agent to construct the mean field systems of p1 and p2 we then characterize an approximated mass behavior effect on an individual agent via a fixed point analysis of the mean field system for each problem p1 and p2 we show that the approximated mass behavior is in fact the best estimate of the actual mass behavior in various senses as the population size n goes to infinity moreover we show that for finite n there exist epsilon nash equilibria for both p1 and p2 where the corresponding individual nash strategies are decentralized in terms of local state information and the approximated mass behavior we also show that epsilon can be taken to be arbitrarily small when n is sufficiently large we show that the epsilon nash equilibria of p1 and p2 are partially equivalent in the sense that the individual nash strategies share identical control laws but the approximated mass behaviors for p1 and p2 are different since in p2 the mass behavior is also affected by the associated worst case disturbance finally we prove that the nash equilibria for p1 and p2 both feature robustness and as the parameter characterizing this robustness becomes infinite the two nash equilibria become identical and equivalent to that of the risk neutral case as in the one agent risk sensitive and robust control theory

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

h control of linear time varying systems a state space approach in this paper the standard problem of h infty control theory for finite dimensional linear time varying continuous time plants is considered the problem is given a real number gamma 0 f

stochastic adaptive backstepping controller design by introducing dynamic signal and changing supply function a more general class of stochastic non linear systems with unmodelled dynamics and uncertain non linear functions are considered in this paper with the concept of isps being extended to stochastic case by combining changing supply function technique with backstepping design technique an adaptive output feedback controller is proposed it is shown that all the solutions of the closed loop system are uniformly bounded in probability and the output can be regulated to an arbitrarily small neighbourhood of the origin in probability a simulation example demonstrates the control scheme

stochastic nonlinear stabilization i a backstepping design while the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances the problem when the disturbance is unbounded stochastic noise has hardly been considered we present a control design which achieves global asymptotic lyapunov stability in probability for a class of strict feedback nonlinear continuous time systems driven by white noise in a companion paper we develop inverse optimal control laws for general stochastic systems affine in the noise input and for strict feedback systems a reader of this paper needs no prior familiarity with techniques of stochastic control

small parameter limit for discrete time partially observed risk sensitive control problems we show that risk sensitive control problems and deterministic dynamic games can be connected under rather mild assumptions by a small noise limit in order to control this limit new techniques are developed to study propagation of large deviations through conditional probabilities

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

optimal control of a stochastic system with an exponential of integral performance criterion abstract an approach based on the theory of positive semigroups is introduced for the analysis of the infinite horizon exponential of integral optimal control problem for a stochastic system of the ito form existence conditions for state feedback admissible controllers are formulated and optimality conditions are derived connections between the exponential of integral optimal control problem and stochastic differential games are discussed

a mechanical counterexample to kam theory with low regularity abstract we give a mechanical example concerning the fact that some regularity is necessary in kam theory we consider the model given by the vertical bouncing motion of a ball on a periodically moving plate denoting with f the motion of the plate some variants of moser invariant curve theorem apply if f is small in norm c 5 and every motion has bounded velocity this is not possible if the function f is only c 1 indeed we construct a function f c 1 with arbitrary small derivative in norm c 0 for which a motion with unbounded velocity exists

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

output feedback control design for strict feedback stochastic nonlinear systems under a risk sensitive cost in this note we study the problem of output feedback control design for a class of strict feedback stochastic nonlinear systems under an infinite horizon risk sensitive cost criterion the controller designed can guarantee an arbitrary small long term average cost for arbitrary risk sensitivity parameter and achieve boundedness in probability for the closed loop system using the integrator backstepping methodology furthermore the controller preserves the equilibrium at the origin of the nonlinear system

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

the relationship between minimum entropy control and risk sensitive control for time varying systems the connection between minimum entropy control and risk sensitive control for linear time varying systems is investigated for time invariant systems the entropy functional and the linear exponential quadratic gaussian cost are the same in this paper it is shown that this is not true for general time varying systems it does hold however when the system admits a state space representation

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

backstepping controller design for nonlinear stochastic systems under a risk sensitive cost criterion this paper develops a methodology for recursive construction of optimal and near optimal controllers for strict feedback stochastic nonlinear systems under a risk sensitive cost function criterion the design procedure follows the integrator backstepping methodology and the controllers obtained guarantee any desired achievable level of long term average cost for a given risk sensitivity parameter theta furthermore they lead to closed loop system trajectories that are bounded in probability and in some cases asymptotically stable in the large these results also generalize to nonlinear systems with strongly stabilizable zero dynamics a numerical example included in the paper illustrates the analytical results

n player statistical nash game control m th cost cumulant optimization in this note we study a statistical nash game control problem where multiple players optimize system performance by shaping the distribution of their cost functions through cost cumulants our focus is on the nonzero sum dynamic statistical nash game which generalizes the risk sensitive nash game the necessary and sufficient conditions for the nash equilibrium solution are given for the n player m th cost cumulant optimization using the hamilton jacobi bellman hjb equations in addition we derive the optimal control strategies and provide simulation results to demonstrate the performance of our method