Robust guaranteed cost state estimation for nonlinear stochastic uncertain systems via an IQC approach

doi:10.1016/j.sysconle.2009.10.006

Systems & Control Letters

Volume 58, Issue 12, December 2009, Pages 865-870

https://doi.org/10.1016/j.sysconle.2009.10.006 Get rights and content

Abstract

This paper presents an approach to robust nonlinear state estimation based on the use of Integral Quadratic Constraints and minimax LQG control. The approach involves a class of state estimators which includes copies of the system nonlinearities in the state estimator. The nonlinearities being considered are those which satisfy a certain global Lipschitz condition. The linear part of the state estimator is synthesized using minimax LQG control theory which is closely related to $H^{\infty}$ control theory and this leads to a nonlinear state estimator which gives an upper bound on the estimation error cost.

Introduction

This paper presents a new approach to robust nonlinear state estimation based on the minimax LQG control theory presented in [1], [2], [3] and the use of Integral Quadratic Constraints (IQCs) which exploit repeated nonlinearities satisfying a global Lipschitz condition. Nonlinear filtering theory has been a very active area of research in recent years motivated by the fact that in many practical state estimation problems, the dynamics of the underlying system are dominated by nonlinear effects; e.g., see [4].

The approach presented in this paper provides a systematic methodology for constructing robust nonlinear state estimators for a class of uncertain nonlinear systems. Our approach is based on the minimax LQG theory of [1], [2], [3] which is in turn based on risk-sensitive and $H^{\infty}$ control machinery. Also, our approach involves the use of IQCs to characterize the nonlinearities in the underlying system.

The fundamental idea behind our approach is to modify the standard IQC approach to robust control and state estimation by including a copy of the nonlinearity in the state estimator as shown in Fig. 1. This idea was also used in [5] to solve a problem of nonlinear output feedback guaranteed cost control and this paper is an extension of the result of [5] to consider the nonlinear robust state estimation problem. Also, a similar idea was used in [6] in a problem of robust nonlinear adaptive control. The idea of using a copy of the nonlinearity in nonlinear observer design was previously used in the paper [7]. However, in contrast to [7], we construct the linear part of the estimator using the minimax LQG control theory of [2] in order to obtain a guaranteed cost bound on the estimation error. In our case, we combine both nonlinearities into the nonlinear system model and then use an IQC which exploits the repeated nonlinearity; see Fig. 2. In this case, the nonlinear estimator design problem is converted to a linear robust controller design problem. This approach enables us to use minimax LQG control theory to construct the linear part of the estimator and then the nonlinear estimator is constructed by including a copy of the plant nonlinearity.

Section snippets

Problem statement

We consider a nonlinear stochastic uncertain system defined using a similar framework to that considered in Section 2.4.2 of [3]. Let $T > 0$ be a constant which will denote the finite time horizon. Also, let $(Ω, F, P)$ be a complete probability space on which a $p$ -dimensional standard Wiener process $W (\cdot)$ and a Gaussian random variable $x_{0} : Ω \to R^{n}$ are defined. The space $Ω$ can be thought of as the noise space $R^{n} \times R^{l} \times C ([0, T], R^{p})$ . The probability measure $P$ is defined as the product of the probability measure $μ ($

The main results

The main result of [1] and Section 8.5 of [3] solves a minimax LQG optimal control problem for an uncertain system with uncertainty described by a certain relative entropy constraint (which is equivalent to a stochastic IQC). In order to apply this result to the problem under consideration in this paper, we will show that the IQCs (21) lead to the satisfaction of such a stochastic IQC which is parameterized by a set of Lagrange multiplier parameters. We first introduce some notation: Define a

Illustrative example

To illustrate the application of our nonlinear state estimator, we consider the following uncertain nonlinear system on the finite time interval $[0 : 30]$ : $[\begin{matrix} d x_{1} (t) \\ d x_{2} (t) \end{matrix}] = [\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] d t + [\begin{matrix} 0.1 \\ 0 \end{matrix}] (ξ_{1} (t) d t + 2 μ_{1} (t) d t + d W_{1} (t))$ $w (t) = [\begin{matrix} 1 & 0 \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}]$ $ζ_{1} (t) = [\begin{matrix} 0.01 & 0 \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}]$ $ν_{1} (t) = [\begin{matrix} 0.2 & 0 \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}]$ $d y (t) = x_{2} (t) d t + 0.1 ξ_{2} (t) d t + 0.1 d W_{2} (t)$ where $μ_{1} (t) = sin (ν_{1} (t))$ . The initial conditions on the system (38) are assumed to be known to be $x_{1} (0) = 0$ and $x_{2} (0) = 0$ . In this signal model, the variable $x_{1} (t)$ is the signal to

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^☆: This work was supported by the Australian Research Council.

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Systems & Control Letters

Robust guaranteed cost state estimation for nonlinear stochastic uncertain systems via an IQC approach☆

Abstract

Introduction

Section snippets

Problem statement

The main results

Illustrative example

Finite horizon minimax optimal control of stochastic partially observed time varying uncertain systems

Mathematics of Control, Signals, and Systems

Minimax LQG control of stochastic partially observed uncertain systems

SIAM Journal on Control and Optimization

Robust Control Design using H∞ Methods

Robust Control Design using $H^{\infty}$ Methods