Robust observer and observer-based control designs for discrete one-sided Lipschitz systems subject to uncertainties and disturbances
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: where is the state vector, is the control input vector, is the output measurement vector, and is the exogenous disturbance vector of system (1). and 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 observerwhere is the estimate of x(k), and L is the gain matrix to be designed.
The state estimation error is defined as . The error dynamics can be computed aswhere . 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 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 can be realized to stabilize system (1), where 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-timewhereApply the Euler discretization method with the sampling time we obtain the discrete-time system (1):
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)
- et al.
Dynamical robust H∞ filtering for nonlinear uncertain systems: an LMI approach
J. Frankl. Inst.
(2010) - et al.
H∞ observers design for a class of nonlinear singular systems
Automatica
(2011) - et al.
Robust state estimation and unknown inputs reconstruction for a class of nonlinear systems: Multiobjective approach
Automatica
(2016) - et al.
State and input simultaneous estimation for a class of nonlinear systems
Automatica
(2004) - et al.
Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: an LMI approach
Appl. Math. Comput.
(2011) - et al.
On LMI conditions to design observers for Lipschitz nonlinear systems
Automatica
(2013) - et al.
Observer-based robust control of one-sided Lipschitz nonlinear systems
ISA Trans.
(2016) - et al.
Novel LMI conditions for observer-based stabilization of Lipschitzian nonlinear systems and uncertain linear systems in discrete-time
Appl. Math. Comput.
(2008) Static output feedback and guaranteed cost control of a class of discrete-time nonlinear systems with partial state measurements
Nonlinear Anal. Theory Methods Appl.
(2008)- et al.
Robust finite-time H∞ control for one-sided Lipschitz nonlinear systems via state feedback and output feedback
J. Frankl. Inst.
(2015)
On observer-based control of one-sided Lipschitz systems
J. Frankl. Inst.
Observer design for one-sided Lipschitz discrete-time systems
Syst. Control Lett.
Unknown input observer design for one-sided Lipschitz discrete-time systems subject to time-delay
Appl. Math. Comput.
Nonlinear H∞ observer design for one-sided Lipschitz systems
Neurocomputing
A note on observer design for one-sided Lipschitz nonlinear systems
Syst. Control Lett.
Robust control of a class of uncertain nonlinear systems
Syst. Control Lett.
Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback
Appl. Math. Comput.
Further results on dissipativity and stability analysis of Markov jump generalized neural networks with time-varying interval delays
Appl. Math. Comput.
LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems
Int. J. Robust Nonlinear Control
Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances
IEEE Trans. Autom. Control
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