Fully probabilistic control for stochastic nonlinear control systems with input dependent noise

Robust controllers for nonlinear stochastic systems with functional uncertainties can be consistently designed using probabilistic control methods. In this paper a generalised probabilistic controller design for the minimisation of the Kullback-Leibler divergence between the actual joint probability density function (pdf) of the closed loop control system, and an ideal joint pdf is presented emphasising how the uncertainty can be systematically incorporated in the absence of reliable systems models. To achieve this objective all probabilistic models of the system are estimated from process data using mixture density networks (MDNs) where all the parameters of the estimated pdfs are taken to be state and control input dependent. Based on this dependency of the density parameters on the input values, explicit formulations to the construction of optimal generalised probabilistic controllers are obtained through the techniques of dynamic programming and adaptive critic methods. Using the proposed generalised probabilistic controller, the conditional joint pdfs can be made to follow the ideal ones. A simulation example is used to demonstrate the implementation of the algorithm and encouraging results are obtained.

Generalised Riccati solution and pinning control of complex stochastic networks

This paper considers the global synchronisation of a stochastic version of coupled map lattices networks through an innovative stochastic adaptive linear quadratic pinning control methodology. In a stochastic network, each state receives only noisy measurement of its neighbours' states. For such networks we derive a generalised Riccati solution that quantifies and incorporates uncertainty of the forward dynamics and inverse controller in the derivation of the stochastic optimal control law. The generalised Riccati solution is derived using the Lyapunov approach. A probabilistic approximation type algorithm is employed to estimate the conditional distributions of the state and inverse controller from historical data and quantifying model uncertainties. The theoretical derivation is complemented by its validation on a set of representative examples.

Comparison between conventional and stochastic pinning control

Theoretical developments on pinning control of complex dynamical networks have mainly focused on the deterministic versions of the model dynamics. However, the dynamical behavior of most real networks is often affected by stochastic noise components. In this paper the pinning control of a stochastic version of the coupled map lattice network with spatiotemporal characteristics is studied. The control of these complex dynamical networks have functional uncertainty which should be considered when calculating stabilizing control signals. Two feedback control methods are considered: the conventional feedback control and modified stochastic feedback control. It is shown that the typically-used conventional control method suffers from the ignorance of model uncertainty leading to a reduction and potentially a collapse in the control efficiency. Numerical verification of the main result is provided for a chaotic coupled map lattice network. © 2011 IEEE.

A Bayesian perspective on stochastic neurocontrol

Control design for stochastic uncertain nonlinear systems is traditionally based on minimizing the expected value of a suitably chosen loss function. Moreover, most control methods usually assume the certainty equivalence principle to simplify the problem and make it computationally tractable. We offer an improved probabilistic framework which is not constrained by these previous assumptions, and provides a more natural framework for incorporating and dealing with uncertainty. The focus of this paper is on developing this framework to obtain an optimal control law strategy using a fully probabilistic approach for information extraction from process data, which does not require detailed knowledge of system dynamics. Moreover, the proposed control method framework allows handling the problem of input-dependent noise. A basic paradigm is proposed and the resulting algorithm is discussed. The proposed probabilistic control method is for the general nonlinear class of discrete-time systems. It is demonstrated theoretically on the affine class. A nonlinear simulation example is also provided to validate theoretical development.

Probabilistic control for uncertain systems

In this paper a new framework has been applied to the design of controllers which encompasses nonlinearity, hysteresis and arbitrary density functions of forward models and inverse controllers. Using mixture density networks, the probabilistic models of both the forward and inverse dynamics are estimated such that they are dependent on the state and the control input. The optimal control strategy is then derived which minimizes uncertainty of the closed loop system. In the absence of reliable plant models, the proposed control algorithm incorporates uncertainties in model parameters, observations, and latent processes. The local stability of the closed loop system has been established. The efficacy of the control algorithm is demonstrated on two nonlinear stochastic control examples with additive and multiplicative noise.

Multi-valued control problems and mixture density network

We have proposed a novel robust inversion-based neurocontroller that searches for the optimal control law by sampling from the estimated Gaussian distribution of the inverse plant model. However, for problems involving the prediction of continuous variables, a Gaussian model approximation provides only a very limited description of the properties of the inverse model. This is usually the case for problems in which the mapping to be learned is multi-valued or involves hysteritic transfer characteristics. This often arises in the solution of inverse plant models. In order to obtain a complete description of the inverse model, a more general multicomponent distributions must be modeled. In this paper we test whether our proposed sampling approach can be used when considering an arbitrary conditional probability distributions. These arbitrary distributions will be modeled by a mixture density network. Importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The effectiveness of the importance sampling from an arbitrary conditional probability distribution will be demonstrated using a simple single input single output static nonlinear system with hysteretic characteristics in the inverse plant model.

Distributed control of computer communication systems

Flow control in Computer Communication systems is generally a multi-layered structure, consisting of several mechanisms operating independently at different levels. Evaluation of the performance of networks in which different flow control mechanisms act simultaneously is an important area of research, and is examined in depth in this thesis. This thesis presents the modelling of a finite resource computer communication network equipped with three levels of flow control, based on closed queueing network theory. The flow control mechanisms considered are: end-to-end control of virtual circuits, network access control of external messages at the entry nodes and the hop level control between nodes. The model is solved by a heuristic technique, based on an equivalent reduced network and the heuristic extensions to the mean value analysis algorithm. The method has significant computational advantages, and overcomes the limitations of the exact methods. It can be used to solve large network models with finite buffers and many virtual circuits. The model and its heuristic solution are validated by simulation. The interaction between the three levels of flow control are investigated. A queueing model is developed for the admission delay on virtual circuits with end-to-end control, in which messages arrive from independent Poisson sources. The selection of optimum window limit is considered. Several advanced network access schemes are postulated to improve the network performance as well as that of selected traffic streams, and numerical results are presented. A model for the dynamic control of input traffic is developed. Based on Markov decision theory, an optimal control policy is formulated. Numerical results are given and throughput-delay performance is shown to be better with dynamic control than with static control.

A probabilistic indirect adaptive control for systems with input-dependent noise

A probabilistic indirect adaptive controller is proposed for the general nonlinear multivariate class of discrete time system. The proposed probabilistic framework incorporates input–dependent noise prediction parameters in the derivation of the optimal control law. Moreover, because noise can be nonstationary in practice, the proposed adaptive control algorithm provides an elegant method for estimating and tracking the noise. For illustration purposes, the developed method is applied to the affine class of nonlinear multivariate discrete time systems and the desired result is obtained: the optimal control law is determined by solving a cubic equation and the distribution of the tracking error is shown to be Gaussian with zero mean. The efficiency of the proposed scheme is demonstrated numerically through the simulation of an affine nonlinear system.

Distribution modeling of nonlinear inverse controllers under a Bayesian framework

The inverse controller is traditionally assumed to be a deterministic function. This paper presents a pedagogical methodology for estimating the stochastic model of the inverse controller. The proposed method is based on Bayes' theorem. Using Bayes' rule to obtain the stochastic model of the inverse controller allows the use of knowledge of uncertainty from both the inverse and the forward model in estimating the optimal control signal. The paper presents the methodology for general nonlinear systems and is demonstrated on nonlinear single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) examples. © 2006 IEEE.

A Bayesian approach to modeling the conditional density of the inverse controller

The inverse controller is traditionally assumed to be a deterministic function. This paper presents a pedagogical methodology for estimating the stochastic model of the inverse controller. The proposed method is based on Bayes' theorem. Using Bayes' rule to obtain the stochastic model of the inverse controller allows the use of knowledge of uncertainty from both the inverse and the forward model in estimating the optimal control signal. The paper presents the methodology for general nonlinear systems. For illustration purposes, the proposed methodology is applied to linear Gaussian systems. © 2004 IEEE.

A novel approach to modelling and exploiting uncertainty in stochastic control systems

We consider an inversion-based neurocontroller for solving control problems of uncertain nonlinear systems. Classical approaches do not use uncertainty information in the neural network models. In this paper we show how we can exploit knowledge of this uncertainty to our advantage by developing a novel robust inverse control method. Simulations on a nonlinear uncertain second order system illustrate the approach.

Improved robust control of nonlinear stochastic systems using uncertain models

We introduce a technique for quantifying and then exploiting uncertainty in nonlinear stochastic control systems. The approach is suboptimal though robust and relies upon the approximation of the forward and inverse plant models by neural networks, which also estimate the intrinsic uncertainty. Sampling from the resulting Gaussian distributions of the inversion based neurocontroller allows us to introduce a control law which is demonstrably more robust than traditional adaptive controllers.

Robust control of nonlinear stochastic systems by modelling conditional distributions of control signals

We introduce a novel inversion-based neuro-controller for solving control problems involving uncertain nonlinear systems that could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. In this work a novel robust inverse control approach is obtained based on importance sampling from these distributions. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The performance of the new algorithm is illustrated through simulations with example systems.