further results on stabilizing controllers for discrete non linear stochastic systems more relationships that exist among stabilizability conditions of controllers designed for a class of discrete non linear stochastic systems are described the conditions are in terms of both mean square and almost sure asymptotic stability of the controlled systems and provide converse results to those presented earlier

optimal stochastic control for performance and stability robustness a design method for discrete time control systems that maintain stability with minimum performance degradation in the presence of a variety of deterministic and stochastic perturbation disturbances is presented deterministic perturbation and modeling errors are represented by maximum deviations from a nominal linear discrete time system and stochastic perturbations are modeled by generalization of the sector bound nonlinearity concept to the stochastic case based on the given bounds on deterministic and stochastic perturbations an upper bound on the quadratic performance index which is useful in assessing the performance deterioration according to the feedback gain used is found then the optimal feedback gain which minimizes this bound on the performance index is derived in the time invariant case the existence of a stabilizing property of the bound optimal controller which is obtained by minimizing an average cost per stage is shown

brownian time processes the pde connection ii and the corresponding feynman kac formula we delve deeper into our study of the connection of brownian time processes btps to fourth order parabolic pdes which we introduced in a recent joint article with w zheng probabilistically btps and their cousins btps with excursions form a unifying class of interesting stochastic processes that includes the celebrated ibm of burdzy and other new intriguing processes and is also connected to the markov snake of le gall btps also offer a new connection of probability to pdes that is fundamentally different from the markovian one they solve fourth order pdes in which the initial function plays an important role in the pde itself not only as initial data we connect two such types of interesting and new pdes to btps the first is obtained by running the btp and then integrating along its path and the second type of pdes is related to what we call the feynman kac formula for btps a special case of the second type is a step towards a probabilistic solution to linearized cahn hilliard and kuramoto sivashinsky type pdes which we tackle in an upcoming paper

a stability criterion for a class of discrete nonlinear systems with random parameters abstract in this paper discrete nonlinear models with random parameters are studied a new frame to classify and analyze discrete stochastic nonlinear systems has been developed from deterministic nonlinear systems to stochastic nonlinear systems this frame is broad and includes a large class of stochastic nonlinear systems a stability criterion developed for this frame is a non lyapunov method of stability analysis and is easily applied in addition this derived sufficient condition of stability is obtained without the assumption of stationarity for the random noise as frequently assumed in the literature

a control scheme for a class of discrete nonlinear stochastic systems a control scheme is presented for effective stabilization of discrete time nonlinear stochastic systems where the nonlinearity involves a zero mean independent random sequence the control is of a constant feedback type and makes use of finite time solutions of a riccati like matrix difference equation stability results are both in terms of mean square and almost sure stochastic stability moreover a matrix inequality is given to check the existence of weighting matrices which would result in a stable regulator problem

robust design of stochastic controllers for nonlinear systems the previous results are generalized on the stabilizing property of a control scheme designed for a class of discrete time nonlinear stochastic systems first the existing controller is made more robust with a prescribed degree of stability by proper modifications next the inherent robustness is illustrated by a design utilizing erroneous noise characteristics reconsideration of the stability analysis used allows one to treat a larger class of nonlinear stochastic systems with more general structures

robustness of discrete time systems for unstructured stochastic perturbations several stochastic stability robustness measures are presented for nominally exponentially stable linear discrete time systems with unstructured perturbations having second moment bounds dependence of these measures on the stability degree of the nominal system and other parameters used in the procedure is illustrated by using the time evolution of the second moment of the system state and stochastic lyapunov stability results positive super martingale convergence theorems the ability of nominally exponentially stable systems to maintain stability in the presence of unstructured stochastic linear and nonlinear perturbations is demonstrated quantitative results are given to determine the maximum modeling uncertainty which can be tolerated in design upper bounds on the second moments of stochastic perturbations to maintain the mean square and almost sure stability of these systems in the presence of unstructured perturbations are obtained

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

a note on the receding horizon control method in this correspondence the relation of receding horizon control method with the state deadbeat control is brought out it is shown for single input systems that the receding horizon control is a state deadbeat control if the horizon length is taken to be equal to the state dimension

hilbert s tenth problem and mazur s conjecture for large subrings of mathbb q we give the first examples of infinite sets of primes s such that hilbert s tenth problem over z s 1 has a negative answer in fact we can take s to be a density 1 set of primes we show also that for some such s there is a punctured elliptic curve e over z s 1 such that the topological closure of e z s 1 in e r has infinitely many connected components

parametrization of all linear compensators for discrete time stochastic parameter systems abstract for discrete time stochastic parameter systems this paper presents a characterization of all state covariances assignable by a linear controller and a parametrization of all controllers that achieves a desired covariance these results indirectly provide the parametrization of all linear fixed order compensators which are mean square stabilizing for this class of systems the paper also includes robustification of the derived compensators and an example to illustrate the results

new robustness bounds for discrete systems with random perturbations some new stochastic stability robustness bounds for state space models are reported nominally exponentially stable discrete time systems are assumed to be subject to random parameter perturbations and novel bounds are obtained on the maximum variances of these random perturbations to maintain stability robustness the methods employed in the analysis are the existing ones used in the deterministic framework after transforming the stochastic robustness problem to a deterministic one the results are compared with each other and with the exact stability region in an example

stabilization analysis of discrete nonlinear systems in this paper we set a new framework for the analysis of nonlinear discrete systems to carry out the stabilization theorem and to design stable controllers of the systems within that framework this framework includes many practically important systems described for example by lipschitz functions functions with arbitrary order bounded derivatives bilinear functions and all polynomial functions an important existence theorem for stabilizing has been proven for this framework the theorem basically states that under definite rather broad conditions the zero solution of the plants can be made stable by a nonlinear output feedback in addition this nonlinear feedback is from a larger class of functions than these classes given by current literature following these stabilization theorems a new design procedure for the output feedback control law is presented all results presented are easy to use and convenient to apply the application is illustrated by several examples including the design of stabilizing control for a bilinear control system

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

absolute stabilization and minimax optimal control of uncertain systems with stochastic uncertainty this paper is concerned with existence and optimality properties of so called guaranteed cost controllers for an uncertain system subject to structured uncertainty the uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints it is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system for each initial state of the system this controller can be found by parametric optimization of solutions of a parameter dependent generalized matrix riccati equation arising in stochastic h infty theory

a suboptimal terminal controller for linear discrete time systems abstract in this work we consider what happens when the receding horizon control laws are designed with finite final state penalization matrices a sufficient condition in terms of system and performance index parameters is given for the discrete receding horizon control rhc laws to be stabilizing when used for suboptimal regulation this condition enables the weighting matrices in the quadratic criterion to be pro assigned so that stability of design is secured the exact value of rhc cost in terms of optimal cost and an easily computable upper bound on its deviation from the optimal cost is found fast algorithms are included to calculate the particular rhc feedback gains introduced dual results in estimation theory are pointed out in this way rhc is proposed as an attractive suboptimal alternative to the optimal terminal control of linear discrete time systems

stability robustness of linear discrete time systems in the presence of uncertainty the stability robustness problem is considered for nominally stable linear discrete time systems using time domain analysis methods and lyapunov theory bounds on the norms of the time varying non linear perturbations are given to maintain the asymptotic stability of these systems in the presence of such perturbations

robustness of feedback stabilized systems in the presence of non linear and random perturbations the general robustness problem is discussed for a controlled discrete time system subject to a lur e type feedback non linearity together with random multiplicative noises effective on state input and the non linearity a sufficient condition is given utilizing the discrete stochastic version of the second method of lyapunov to guarantee that any stabilizing controller designed for the deterministic linear part of the system renders the closed loop system with perturbations asymptotically stable with probability one then specializations to systems given in controllability and controller canonical forms are made independent noise signals are assumed to be effective on individual parameters of these canonical forms whose inputs are designed to assign the closed loop poles to zero in these cases stronger results are obtained in terms of system parameters and known characteristics of perturbations

absolute stability approach to stochastic stability of infinite dimensional nonlinear systems recent applications of absolute stability methods to robustness analysis have led to a rebirth of interest in this topic this paper is concerned with the robust stability of feedback distributed parameter systems against nonlinear and random disturbances our analysis is based on the lyapunov direct method to derive a stochastic lyapunov function we introduce the special stochastic infinite dimensional counterparts of the kalman yakubovich lemma they are proved by use of dynamic programming methods in combination with the properties of hilbert space valued wiener processes examples exhibit extensions of the popov and circle criteria to the feedback stochastic heat and delay equations respectively

relationships between several novel control schemes proposed for a class of non linear stochastic systems several different novel controllers are proposed to stabilize a class of discrete time non linear stochaslic systems where the non linearity involves the state and input of the system as well as a white noise sequence vector it is shown that the conditions of stabilizability by these controllers are closely related

deterministic and stochastic robustness measures for discrete systems deterministic and stochastic lyapunov theorems are used to demonstrate the robustness of a stable linear time variant discrete time nominal system to both unknown deterministic and stochastic perturbations time domain conditions are presented on the appropriate deterministic or random characteristics of perturbations to maintain the proper stability behavior of the overall system it is concluded that the novel robustness conditions proposed can find application in the feedback design of control systems where the closed loop system is known to be stable with a certain degree e g as in the case of linear quadratic optimal control with alpha degree of prescribed stability

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

stabilizing compensator design for discrete time non linear stochastic systems with incomplete state information in this work an approach to stabilizing compensator design is presented that is suitable for a large class of discrete time non linear stochastic systems with incomplete state information the non linearity is contained either in the state or the measurement equation and it involves a white noise vector sequence the proposed compensator is a combination of a constant gain observer and a constant feedback control gain the stabilizability conditions depend upon the system and chosen design parameter characteristics and are given for both mean square and almost sure stochastic boundedness and stability of the controlled system a specific subclass of these systems subject to certain control laws is shown to possess stabilizability conditions that can be tested a priori