Observer-based H∞ control on stochastic nonlinear systems with time-delay and actuator nonlinearity

doi:10.1016/j.jfranklin.2013.02.007

Journal of the Franklin Institute

Volume 350, Issue 6, August 2013, Pages 1388-1405

https://doi.org/10.1016/j.jfranklin.2013.02.007 Get rights and content

Abstract

In this paper, an observer-based $H_{\infty}$ controller is designed for a class of extended Markov jump systems subject to time-delay and actuator saturation nonlinearity. Gradient linearization procedure is employed to describe such nonlinear systems by several linear Markov jump systems. Next, a mode-dependent Lyapunov function is constructed for these linear systems, and a sufficient condition is derived to make them stochastically stable, and then, a continuous gain-scheduled approach is applied to design a continuous nonlinear observer-based controller on the entire extended nonlinear jump system. A simulation example is given to illustrate the effectiveness of developed techniques.

Introduction

As a special kind of hybrid systems, Markov jump systems (MJSs) are appropriate and reasonable to describe systems subject to abrupt and random variation in structures or parameters. In recent years, this kind of stochastic systems has been extensively studied due to its comprehensive application in many systems, such as in manufacturing systems, economic systems, electrical systems and communication systems, and the existing results cover a large variety of problems such as stochastic stabilization [1], [2], [3], control [4], [5], [6], and filtering [7], [8], [9], etc. In spite of these developments, there is little work done on nonlinear Markov jump systems (NMJs), actually, random changes or sudden variations in structures or parameters are commonly encountered in lots of practical dynamic nonlinear systems, so the investigation of control problem on NMJs may be more reasonable, we have made some attempts to work on gain-scheduled controller design for NMJSs in [10], and this motivates us to challenge the problem of observer-based robust gain-scheduled controller design for nonlinear Markov jump systems.

It is well known that actuator saturation is probably the most dangerous nonlinearity in many manufacture systems, it is also a well known fact that this nonlinearity degrades system performance and even leads a stable system to an unstable one. It is a challenge to solve control problem in the presence of actuator saturation constraints. Actuator nonlinearity has received increasing attention in recent years, and researchers have made some attempts on control problems of linear systems and nonlinear systems with actuator saturation nonlinearity [11], [12], [13], [14], [15], [16], meanwhile, in the past decade, a great deal of work has been devoted to time delay systems (see, e.g., [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and the references therein), as we all know, time-delay plays an important role in system performance, however, to the best of our knowledge, there is little work being done on time delay MJSs with actuator nonlinearity. Hence, it is our intention in this paper to tackle such an important yet challenging problem.

On another research front line, in order to analyze the robust control problem for the aforementioned system, continuous gain-scheduled approach [28] is employed. Compared with the conventional gain-scheduled methods, continuous gain-scheduled approach has one important advantage: the designed controller is a time-varying and nonlinear one, and it varies the gain matrix with the variation of system state.

In the sequel, the notations $R^{n}$ and $R^{n \times m}$ stand for a n-dimensional Euclidean space and n×m-dimensional real matrices respectively, the transpose of a matrix is denoted by $A^{T}$ , $E {\cdot}$ denotes the mathematical statistical expectation of the stochastic process or vector, $L_{2}^{n} [0, \infty)$ stands for the space of n-dimensional square integrable function vector over $[0, \infty)$ , a positive-definite matrix is denoted by $P > 0$ , $∥$ stands for an absolute value, I is the unit matrix with appropriate dimension, and $⁎$ means the symmetric term in a symmetric matrix, $σ (\cdot)$ is the standard saturation function with appropriate dimensions, $σ (u (t)) = [σ (u_{1} (t)) σ (u_{2} (t)) \dots σ (u_{m} (t))]^{T}$ and $σ (u_{l} (t)) = sign (u_{l} (t)) \min {1, | u_{l} (t) |}$ , $l = 1 \dots m$ .

Section snippets

Problem statement and preliminaries

Consider a probability space $(M, F, P)$ where M, F and P, respectively, represent the sample space, the algebra of events and the probability measure defined on F, then the following nonlinear Markov jump systems (NMJSs) subject to time delay and actuator saturation are considered in this paper: $\{\begin{matrix} \dot{x} (t) = f_{1} (x (t), x (t - h), σ (u (t)), w (t), r_{t}) \\ y (t) = f_{2} (x (t), r_{t}) \\ z (t) = f_{3} (x (t), r_{t}) \\ x (t) = x (0), r_{t} = r_{0}, t \in [- h, 0] \end{matrix}$ where $x (t) \in R^{n}$ is the state vector of the system, $u (t) \in R^{m}$ is the input vector of the system, $y (t) \in R^{p}$ is the output

Observer-based stochastic stability

For system (2.2), observer-based controllers will be designed as the following form: $\{\begin{matrix} \dot{\bar{x}} (t) = A_{k} (i) \bar{x} (t) + B_{k} (i) σ (u (t)) + H_{k} (i) (y (t) - \bar{y} (t)) \\ \bar{y} (t) = C_{1 k} (i) \bar{x} (t) \\ u (t) = K_{k} (i) \bar{x} (t) \end{matrix}$ where $\bar{x} (t)$ and $\bar{y} (t)$ are the estimated state and output, respectively, H_k(i) are gains of the designed observer, and K_k(i) are gains of the feedback controller.

Recalling Lemma 2.1, one can denote $\bar{x} (t) \in Θ (F_{k} (i))$ , then, $σ (K_{k} (i) \bar{x} (t)) = \sum_{s = 1}^{2^{m}} θ_{s} (D_{s} K_{k} (i) + D_{s}^{-} F_{k} (i)) \bar{x} (t)$ , and the following estimation error dynamic system (3.2) can be

Observer-based $H_{\infty}$ control

Theorem 4.1

For a given matrix D_s, system (3.2) is stochastically stable (with $w (t) = 0$ , $\forall i \in Λ$ ), if there exist a set of positive definite symmetric matrices $P_{1 k} (i)$ , $P_{2 k} (i)$ and Q, and a set of matrices K_k(i) and F_k(i) such that $Φ_{1} = [\begin{matrix} b_{11} & b_{12} & b_{13} \\ ⁎ & b_{22} & b_{23} \\ ⁎ & ⁎ & b_{33} \end{matrix}] < 0 \forall s \in [1, 2^{m}]$ $ε ({\hat{P}}_{k} (i)) \in Θ (\tilde{F} (i))$

Moreover, a suitable controller u(t) for each vertex of system (3.2) can be constructed as $u (t) = (D_{s} K_{k} (i) + D_{s}^{-} F_{k} (i)) {\bar{x}}_{k}$ where $b_{11} = (A_{k} (i) - H_{k} (i) C_{1 k} (i))^{T} P_{2 k} (i) + P_{2 k} (i) (A_{k} (i) - H_{k} (i) C_{1 k} (i)) + \sum_{j = 1}^{N} π_{ij} P_{2 k} (j)$ $b_{12} = - (D_{s} K_{k} (i) + D_{s}^{-} F_{k} (i))^{T} B_{k}^{T} (i) P_{1 k} (i) + P_{2 k} ($

Simulation results

The bioreactor system [33] with Markov jump parameters and admissible disturbances is investigated to evaluate the performance of our approach. ${\dot{x}}_{1} (t) = - u_{1} (t) + x_{1} (t) (1 - x_{2} (t)) \exp (\frac{x_{2} (t)}{γ}) + ε_{1} (i) x_{1} (t - h) + 0.2 w (t) {\dot{x}}_{2} (t) = - u_{2} (t) x_{2} (t) + x_{1} (t) (1 - x_{2} (t)) \frac{1 + α (i)}{1 + α (i) - x_{2} (t)} \exp (\frac{x_{2} (t)}{γ}) z (t) = x_{1} (t) + u_{1} (t) + 0.2 w (t) y (t) = x_{1} (t) + ε_{2} (i) x_{1} (t - h)$ where $x_{1} (t)$ and $x_{2} (t)$ are system states, w(t) is the disturbance vector of the system, the Markov jump parameters are given as follows which are aggregated into three modes: $α (1) = 0.02, α (2) = 0.03$

Conclusions

In this paper, the issue on gain-scheduled $H_{\infty}$ observer-based control design for a class of extended nonlinear stochastic systems is addressed. Gradient linearization approach is applied to such systems and model-based linear stochastic error dynamic systems are obtained. In order to design continuous controller for nonlinear system, Taylor fitting approach is investigated and continuous gain-scheduled observer-based controller is designed. The simulation result shows the potential of the

Acknowledgment

This work was partially supported by the National Key Basic Research Program (973), China (2012CB215202), the 111 Project (B12018), the National Natural Science Foundation of China (61174058, 61273087), the Engineering and Physical Sciences Research Council, UK (EP/F029195), the Norwegian Centre for Offshore Wind Energy (NORCOWE) under grant 193821/S60 from Research Council of Norway (RCN).

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Observer-based H∞ control on stochastic nonlinear systems with time-delay and actuator nonlinearity

Abstract

Introduction

Section snippets

Problem statement and preliminaries

Observer-based stochastic stability