Closed-loop control of stochastic non-linear systems

A new approach for optimum control of stochastic non-linear systems is developed from practical engineering assumptions. Systems amenable to this new approach include optimum guidance and navigation systems for space and terrestrial vehicles, optimum closed-loop process controllers and optimum controllers for systems with unknown parameters. The classical quadratic synthesis approach—optimization of a deterministic cost and perturbation estimation and control about that solution—is shown to give 24 per cent more cost and 97 per cent more mean-squared terminal error than the combined optimization approach presented for a sample problem involving control of a first-order system with an unknown time constant. A similar improvement in terminal error is shown for an atmospheric entry problem. Furthermore, the optimum controller automatically designs the best controller to minimize the effects of the unknown parameter without artificial augmentation of the cost function as is done in the sensitivity theory approach. The solution is obtained by expansion of the cost function in a power series around a deterministic trajectory with the assumption of linear perturbation estimation and control about that trajectory. Optimization of the expanded cost function gives necessary conditions dependent on the covariance matrices and the deterministic portion of the cost. When the necessary conditions are solved, a set of open-loop controls, perturbation controller gains, and perturbation estimator gains are obtained that can be precomputed and implemented into the system.     

Design of decoupled non-linear multivariable stochastic control systems

A technique is presented for decoupling a non-linear system with input disturbance, The class of matrices which decouple the nominal system is first obtained. Then, the free parameters in the feedback matrix are selected to minimize the effect of disturbances on the output by minimizing a performance index which involves the input-output error magnitude. An example is presented to illustrate the approach.     

Controllability of non-linear stochastic systems

Complete controllability of a semi-linear stochastic system assuming controllability of the associated linear sytem is studied. It is also shown that a non-linear stochastic system is locally null controllable provided that the corrsponding linearized system is controllable.     

Tracking control of a pendulum using a numerically efficient state estimation approach for polynomic non-linear systems

This paper reports on the tracking control of a pendulum. The particular contributions of this paper lie in the development of a numerically efficient approach for the state estimation of a class of non-linear systems which fall under the extended generalized Wiener model structure and its application to the pendulum system. It details the estimation and control aspects for achieving the tracking control of the pendulum where attention is focused on Extended Kalman Filtering (EKF) methods based on Volterra series approximations of the non-linearity for the estimation of the pendulum states. Control is then effected by the Feedback Linearization (FL) technique and the Internal Model Principle (IMP). It is argued that this offers accuracy benefits over linear techniques while substantially reducing the computational burden associated with the standard EKF approaches. The arguments are supported by evidence from a case study system. It is demonstrated that this proposed approach is significantly faster and does not demand any additional hardware configurations. Estimation error and tracking error variance results are included for substantiating the numerical efficiency of the proposed approach.     

Approximate analysis of non-linear stochastic systems

In this paper a method is developed whereby the statistical characteristics of the response of non-linear stochastic systems are calculated with good accuracy. The method consists of the formulation and solution of the differential equations for the statistical characteristics of the processes under consideration. The development is based on the theory of Markov processes. In particular, use is made of Ito's differential rule to obtain a basic result which is used for the formulation of the differential equations. For the solution of the differential equations the method employs an approximate analytic representation of the probability density function of a random process. The representation is in the form of a finite Edgeworth (asymptotic) expansion. The method is quite general and yields accurate results. The treatment is illustrated by examples.     

Controllability of non-linear impulsive stochastic systems

In this article, we consider the finite-dimensional dynamical control systems described by non-linear impulsive stochastic differential equations. Sufficient conditions for the complete and approximate controllability of non-linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable.     

On hybrid control of a class of stochastic non-linear Markovian switching systems

In this paper, we address the problem of hybrid control for a class of stochastic non-linear Markovian switching systems. First, a hybrid controller is introduced for the systems. Then under some appropriate assumptions, the stabilization condition for the systems under pure impulsive control is given. Further under impulsive control, the output feedback stabilization problem of the systems is discussed and linear output feedback controllers are designed. Finally a numerical example is provided to illustrate the effectiveness of the proposed methods.     

Robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with parametric uncertainty

The robustness of non-linear stochastic optimal control for quasi-Hamiltonian systems with uncertain parameters is studied. Based on the independence of uncertain parameters and stochastic excitations, the non-linear stochastic optimal control for the nominal quasi-Hamiltonian system with average-value parameters is first obtained by using the stochastic averaging method and stochastic dynamical programming principle. Then, the means and standard deviations of root-mean-square responses, control effectiveness and control efficiency for the uncertain quasi-Hamiltonian system are calculated by using the stochastic averaging method and the probabilistic analysis. By introducing the sensitivity of the variation coefficients of controlled root-mean-square responses, control effectiveness and control efficiency to those of uncertain parameters, the robustness of the non-linear stochastic optimal control is evaluated. Two examples are given to illustrate the proposed control procedure and its robustness.     

High-gain filters for non-linear stochastic systems

The use of high-gains in the filtering problem for non-linear stochastic systems is studied. It is shown that a high-gain filter has the same structure as a Luenberger observer. The use of high-gains is studied in the cancellation of non-linearities in order to simplify the filter design for non-linear systems. Singular perturbation techniques are used to show how the error dynamics reaches stable equilibrium in a very fast transient time, ensuring that the slow dynamics of the filter are just those of the non-linear stochastic system.     

Stabilization of non-linear differential-algebraic equation systems

In this paper, the feedback stabilization problem is investigated for non-linear differential-algebraic equation systems. The paper is composed of two algorithms, namely a regularization algorithm and a stabilization algorithm. It is shown that the feasibility of the regularization algorithm guarantees that there exists a feedback so that the closed-loop system admits a unique impulse-free solution. The application of the stabilization algorithm produces a set of new coordinates in which the original system can be expressed as a standard form. Based on the standard form, a stabilizing feedback is constructed, which guarantees that the closed-loop system admits a unique impulse-free solution and is locally asymptotically stable at the origin.     

Identification of non-linear stochastic systems by state dependent parameter estimation

The paper outlines how improved estimates of time variable parameters in models of stochastic dynamic systems can be obtained using recursive filtering and fixed interval smoothing techniques, with the associated hyper-parameters optimized by maximum likelihood based on prediction error decomposition. It then shows how, by exploiting special data re-ordering and back-fitting procedures, similar recursive parameter estimation techniques can be utilized to estimate much more rapid State Dependent Parameter (SDP) variations. In this manner, it is possible to identify and estimate a widely applicable class of nonlinear stochastic systems, as illustrated by several examples that include simulated and real data from chaotic processes. Finally, the paper points out that such SDP models can form the basis for new methods of signal processing, automatic control and state estimation for nonlinear stochastic systems.     

Modeling and control of non-linear systems using soft computing techniques

This work is an attempt to illustrate the utility and effectiveness of soft computing approaches in handling the modeling and control of complex systems. Soft computing research is concerned with the integration of artificial intelligent tools (neural networks, fuzzy technology, evolutionary algorithms, …) in a complementary hybrid framework for solving real world problems. There are several approaches to integrate neural networks and fuzzy logic to form a neuro-fuzzy system. The present work will concentrate on the pioneering neuro-fuzzy system, Adaptive Neuro-Fuzzy Inference System (ANFIS). ANFIS is first used to model non-linear knee-joint dynamics from recorded clinical data. The established model is then used to predict the behavior of the underlying system and for the design and evaluation of various intelligent control strategies.     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

Exact and approximate response of non-linear stochastic systems

The method of finding the exact solution and a new approximation method for a class of non-linear stochastic systems is presented. In both methods a non-linear transformation of the original system is applied. The idea of this method is presented for a non-linear system with stochastic parametric excitations. If the transformed system is a linear one then one can find an exact solution, otherwise the cumulant neglect closure technique is applied to this new system. The resuts obtained are illustrated by a numerical example.     

Adaptive state-feedback stabilization for high-order stochastic non-linear systems with uncertain control coefficients

This paper investigates the adaptive state-feedback stabilization problem for a class of high-order stochastic non-linear systems with unknown lower and supper bounds for uncertain control coefficients. Under some weaker and reasonable assumptions, a smooth adaptive state-feedback controller is designed, which guarantees that the closed-loop system has an almost surely unique solution on [0,∞, the equilibrium of interest is globally stable in probability and the states can be regulated to the origin almost surely. A simulation example is given to show the systematic design and effectiveness of the controller.     

Modeling and analysis of networked control systems using stochastic hybrid systems

This paper aims at familiarizing the reader with Stochastic Hybrid Systems (SHSs) and enabling her to use these systems to model and analyze Networked Control Systems (NCSs). Towards this goal, we introduce two different models of SHSs and a set of theoretical tools for their analysis. In parallel with the presentation of the mathematical models and results, we provide a few simple examples that illustrate the use of SHSs to models NCSs.     

Tracking and approximate model matching for non-linear systems

The design of non-linear tracking systems is investigated in a general setting that guarantees internal stability. It is shown that, by adopting an appropriate design methodology, tracking accuracy can be improved to any desirable level. The proposed methodology is computationally, as well as conceptually, undemanding, and it also leads to a simple solution of a class of non-linear approximate model matching problems.     

Frequency stability criteria for non-linear stochastic systems

In this paper general classes of stochastic systems described by stochastic integrals of the Volterra type are considered. Existence and uniqueness of random solutions as well as stochastic stability criteria of the Popov frequency form are established. These frequency stability criteria relate not only to the linear system dynamics (or the kernel of the Volterra integral) of the system, but also to the characteristics of the additive and multiplicative random processes that govern the system.     

Closed loop hierarchical control for non-linear systems using quasilinearisation

In this paper, the goal co-ordination method is modified in order to provide closed loop control for large-scale nonlinear dynamical systems with separable cost functions. The basis of the method is to use the last linear approximation, before overall convergence, of the quasi-linearization procedure used to solve the subsystem two point boundary value problems in order to calculate the local optimal feedback control. The method is demonstrated on a simple though interesting example of a non-linear dynamical system with two subsystems.