Robust observer and observer-based control designs for discrete one-sided Lipschitz systems subject to uncertainties and disturbances

https://doi.org/10.1016/j.amc.2019.01.064Get rights and content

Highlights

  • Both robust observer and observer-based control designs are studied.

  • The class of nonlinearities is broader than traditional Lipschitz nonlinearities.

  • New approach to the robust observer design problem for discrete uncertain systems.

  • Design conditions derived in terms of LMIs after linearizing the bilinear terms.

  • Advantages of new results over existing works illustrated via a numerical example.

Abstract

In this paper, we study the robust observer design and observer-based control design problems for a class of discrete one-sided Lipschitz systems subject to uncertainties and disturbances. The nonlinearities are assumed to be one-sided Lipschitz and quadratically inner-bounded. By utilizing a new approach which is an extension of the H filtering method, our robust observer design can relax some limitations in existing works. In order to derive design conditions in terms of linear matrix inequalities, several mathematical techniques are appropriately used to linearize the bilinear terms which unavoidably emerge in observer and observer-based control designs for discrete-time uncertain systems. Via a numerical example, we show that while existing works fail, our results work effectively.

Introduction

The observer design problem for nonlinear systems has been an active research area in the past few decades [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The main motivation for the study is that the full knowledge of the system state vector, which is often difficult to measure in practice, is required for many purposes including system monitoring, state feedback control, etc. [14], [15]. When the complete information of the system state vector is not available, observer-based control is implemented instead of state feedback control where an observer system is utilized to estimate the true state vector of the dynamical system [14], [15]. The observer design and observer-based control design problems have attracted lots of attention from the control society [16], [17], [18], [19], [20], [21], [22], [23].

Different from observer design and observer-based control design problems for linear systems, there is no general approach for these problems for nonlinear systems. Nonlinear dynamical systems exhibit diverse characteristics and complex behaviors. Different types of nonlinearities require different techniques to deal with. So far, the class of Lipschitz nonlinear systems, characterized by the well-known Lipschitz condition, has been commonly used in the literature thanks to the simplicity and convenience of the quadratic form of the Lipschitz condition. Due to the restrictive nature of the Lipschitz condition, however, observer and observer-based control designs based on the traditional Lipschitz condition expose some considerable conservativeness, e.g. the inability to handle nonlinear systems with large Lipschitz constants. To tackle this issue, various alternative approaches have been introduced including the linear parameter varying (LPV) approach [13], the one-sided Lipschitz condition [24]. The LPV approach reformulates the classical Lipschitz condition and thus takes into account all the characteristics of the nonlinearities. This approach can significantly relax the Lipschitz condition with regard to the Lipschitz constant. On the other hand, the class of one-sided Lipschitz nonlinearities, characterized by the one-sided Lipschitz condition, is introduced as a broader class of the traditional Lipschitz nonlinearities. While the Lipschitz constants are strictly positive, the one-sided Lipschitz constants can be positive, zero, or negative. Additionally, the one-sided Lipschitz constants are usually smaller than the Lipschitz constants. Hence, the one-sided Lipschitz condition can provide some advantages over the Lipschitz counterpart.

Designing observers for the class of one-sided Lipschitz systems has become an interesting research problem captivating researchers in control theory with many results recently reported in the literature [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. Among the works recently published for one-sided Lipschitz systems, designing observers for the class of nonlinear systems simultaneously satisfying the one-sided Lipschitz condition and the quadratically inner-bounded condition [24] has been the most popular direction. Similar to the one-sided Lipschitz condition, the quadratically inner-bounded condition also includes the Lipschitz condition as a special case. Furthermore, it is useful to derive tractable observer design conditions for one-sided Lipschitz nonlinear systems [24].

While many observer design results for one-sided Lipschitz systems have been reported in the literature, it is noticeable that there are only a limited number of works addressing the observer-based control problem of one-sided Lipschitz systems [16], [18], [23], especially in discrete-time. Observer and observer-based control designs in discrete-time are generally more difficult than those designs in continuous-time due to the inevitable emergence of bilinear terms. In addition, robustness against uncertainties and disturbances is also an interesting research topic in control theory [2], [3], [6], [22], [37], [38]. Modeling errors and disturbances are sometimes unavoidable in control systems, which can degrade the performances of systems or even lead to instability. Robust control ensures the robust performances of control systems by taking into account uncertainties and disturbances. It is also worth pointing out that the existing observer designs for uncertain systems possess some limitations which need addressing (see Remark 3 for details).

Motivated by the above works, in this paper, we consider both the problems of robust observer and observer-based control designs for discrete-time uncertain one-sided Lipschitz systems with disturbances. The nonlinearities are assumed to simultaneously satisfy the one-sided Lipschitz and quadratically inner-bounded conditions [24]. By using a new method, our robust observer design can relax some limitations in the existing related works. The bilinear terms, which naturally arise in observer and observer-based control designs for discrete-time uncertain systems, are well handled by the appropriate use of several mathematical techniques to derive design conditions in terms of linear matrix inequalities. Via a numerical example, we show that while existing works fail, our results work satisfactorily.

Standard notations are used throughout this paper.

Section snippets

Problem statement and preliminaries

Consider the following class of discrete-time nonlinear systems with uncertainties and disturbances:x(k+1)=(A+ΔA(k))x(k)+ϕ(x(k),u(k))+Wω(k)+Bu(k),kN,y(k)=(C+ΔC(k))x(k)+Dω(k), where x(k)Rn is the state vector, u(k)Rm is the control input vector, y(k)Rp is the output measurement vector, and ω(t)2q is the exogenous disturbance vector of system (1). ARn×n, WRn×q, BRn×m, CRp×n, and DRp×q are known, constant matrices. ΔA(k) and ΔC(k) are unknown matrices representing time-varying system

Robust observer design

To robustly estimate the state vector x(k) of system (1), let us consider the following Luenberger-type observerx^(k+1)=Ax^(k)+ϕ(x^(k),u(k))+Bu(k)+L(y(k)Cx^(k)),where x^(k)Rn is the estimate of x(k), and L is the gain matrix to be designed.

The state estimation error is defined as e(k)=x(k)x^(k). The error dynamics can be computed ase(k+1)=(ALC)e(k)+(ΔA(k)LΔC(k))x(k)+Δϕ(k)+(WLD)ω(k),where Δϕ(k)=ϕ(x(k),u(k))ϕ(x^(k),u(k)).

Remark 3

Because x is present in the error dynamics (9), most existing

Robust observer-based control design

If it is possible to fully measure the state vector x(k) of system (1), a state feedback controller of the form u(k)=Kx(k) can be realized to stabilize system (1). In practice, however, it may be difficult or impossible to obtain the full knowledge of x(k) [14], [15]. In this case, observer-based controller u(k)=Kx^(k) can be realized to stabilize system (1), where x^(k) is the estimate of x(k) provided by an observer system.

Consider the observer system (8), our goal in this section is to

Illustrative example

Consider the following uncertain nonlinear system with disturbances in continuous-timex˙(t)=(Ac+U1cΘ(t)Vc)x(t)+ϕc(x(t),u(t))+Wcω(t)+Bcu(t),t0,y(t)=(Cc+U2cΘ(t)Vc)x(t)+Dcω(t),whereAc=[111.10.9],Wc=[12],Bc=[11],ϕc(x,u)=[x1(x12+x22)x2(x12+x22)],U1c=[0.150.20.10.05],Vc=[0.050.10.10.2],U2c=[0.20.15],Cc=[01],Dc=1.Apply the Euler discretization method with the sampling time ts=1ms=0.001s, we obtain the discrete-time system (1): A=I+tsAc, W=tsWc, B=tsBc, ϕ(x,u)=tsϕc(x,u), U1=tsU1c, V=Vc, U2=U2c, C

Conclusion

In this paper, we have considered the robust observer design and observer-based control design problems for a class of discrete one-sided Lipschitz systems subject to uncertainties and disturbances. We assume that the nonlinearities simultaneously satisfy the one-sided Lipschitz and quadratically inner-bounded conditions. By utilizing a new approach, we are able to relax some limitations in the existing robust observer designs for uncertain systems. To derive design conditions in terms of LMIs,

References (40)

  • S. Ahmad et al.

    On observer-based control of one-sided Lipschitz systems

    J. Frankl. Inst.

    (2016)
  • M. Benallouch et al.

    Observer design for one-sided Lipschitz discrete-time systems

    Syst. Control Lett.

    (2012)
  • M.C. Nguyen et al.

    Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay

    Appl. Math. Comput.

    (2016)
  • W. Zhang et al.

    Nonlinear H observer design for one-sided Lipschitz systems

    Neurocomputing

    (2014)
  • Y. Zhao et al.

    A note on observer design for one-sided Lipschitz nonlinear systems

    Syst. Control Lett.

    (2010)
  • Y. Wang et al.

    Robust control of a class of uncertain nonlinear systems

    Syst. Control Lett.

    (1992)
  • J. Wang et al.

    Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback

    Appl. Math. Comput.

    (2018)
  • S. Jiao et al.

    Further results on dissipativity and stability analysis of Markov jump generalized neural networks with time-varying interval delays

    Appl. Math. Comput.

    (2018)
  • M. Abbaszadeh et al.

    LMI optimization approach to robust H observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems

    Int. J. Robust Nonlinear Control

    (2009)
  • M.S. Chen et al.

    Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances

    IEEE Trans. Autom. Control

    (2007)
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