On the observer design problem for continuous-time switched linear systems with unknown switchings
Introduction
The goal of this paper is to address an observer design for Switched Linear Systems (SLS) subject to an unknown switching signal. A SLS may be viewed as a subclass of the class of Hybrid Dynamical Systems (HDS) formed by a collection of Linear Systems (LS) together with a time dependent exogenous switching signal. The system state evolution is then uniquely determined by a given initial condition and by the mentioned switching signal determining at each time instant the unique active linear system driving the state.
The study of the fundamental properties of SLS has received a great deal of attention during the last decade. In particular, SLS observability has been thoroughly analysed depending on whether the switching signal is known or unknown.
On one hand, if the switching signal is known, the focus has been on determining the continuous state after a finite number of switchings [1], [2], [3], [4]. It has been shown that the observability of each LS is not a necessary condition [1], [4]. On the other hand, if the switching signal is unknown (for instance when switchings may occur in an unpredictable way), the observability of the continuous phase and the observability of the switching signal [5], [6], [7], [8] were shown to be mutually independent properties [7]. Moreover, if the switching signal is unknown, the observability of each LS is not sufficient for the overall SLS observability [7]. As a matter of fact, the unknown switching signal computation requires the so-called distinguishability property [5] enabling to detect the current evolving LS based on the input–output information only. Last but not least, for the unknown switching signal the controlled inputs also play a central role leading to two different distinguishability notions [9]: (i) distinguishability for every nonzero state trajectory [10] and (ii) distinguishability for “almost every” input [6], [7], [8]. The latter case, as expected, requires less restrictive conditions to be fulfilled, in particular, the observability of each LS is not required.
To be more specific, the unknown switching signal is usually estimated using the so-called “location observer” based on the SLS continuous-time input–output information. In [11], a multi-observer structure based on Luenberger observers together with a residual generator (similar to those used in fault detection) is used as a location observer. The reported results, however, require a careful selection of the threshold for the residue generation. Moreover, as noticed in [11] it can occur that the residue remains true after switching to another subsystem, thus missing the detection of switching. In [12], the location observer algorithm that requires the numerical computation of derivatives of inputs and outputs has been proposed as the location observer. Nevertheless, the analysis presented there is restricted to monovariable SLS. In [13], the location observer uses a super twisting based step-by-step observer for switched nonlinear systems that can be transformed into the normal form.
However the focus is on autonomous systems and the super twisting based step-by-step observer requires the knowledge of the bound of the state velocity. In [14] the location observer is formed by a set of Luenberger observers with an associated super twisting based differentiator [15] used to obtain the exact error signal which updates the estimate. Using the results on distinguishability for every nonzero state trajectory, the authors of [14] showed the convergence of the observer. Unfortunately, this distinguishability notion requires the observability of each LS.
Furthermore, multi-observer structures have been proposed in the framework of supervisory control of a class of SLS composed of a LS with an unknown parameter [16], [17], [18] and in adaptive control [19], [20] where the aim is to decide, based on the size of the output estimation error, which candidate controller (from a bank of controllers) should be used in the feedback loop with the process. Unfortunately, as noticed in [21], this scheme cannot be used to recover the switching signal because the smallness of the output estimation error is not sufficient to infer the evolving LS [21].
The observability and the observer design problem have also been addressed in discrete-time SLS, see e.g. [9], [22], [23], [24]. Nevertheless, as pointed out in [7], continuous and discrete–time SLS have notable differences requiring these two classes of systems to be studied separately.
In this framework, the aim of this paper is twofold. First, to introduce a finite-time SLS observer guaranteeing that the observers and controllers can be designed separately. To the best of our knowledge, no result exists allowing such separation using Luenberger multi-observers. Such a task will be addressed for a rather general class of SLS for which the observability of each LS is not required. Since the observability of the switching signal is independent from the observability of the continuous state, the proposed observer is capable of estimating the continuous state and the switching signal of the SLS independently. The corresponding convergence analysis is based on the global finite-time stability [25], higher-order sliding modes techniques [26] and the SLS observability results [7], [27].
Secondly, based on the developed observers, this paper addresses the design of a synchronization scheme for several chaotic SLS. As a matter of fact, while dealing successfully with the issue of the unknown switching signal detection, the paper opens new possibilities for chaotic SLS synchronization. This is an important task enabling the possible use of SLS with chaotic behavior in various chaotic encryption schemes, such as chaos shift keying, in which different chaotic attractors are associated to different digital symbols, and two-channel transmission, in which the output of the chaotic SLS is sent to the receiver through the first channel, while the encrypted message (encrypted using the chaotic state) is conveyed by the second channel, see [28], [29] and further references within there. To the best of our knowledge, even though it has been shown that new and novel chaotic behavior can be generated by SLS (see e.g. [30], [31], [32], [33], [34], [35]), the synchronization of general classes of SLS with chaotic behavior (e.g. those presented in [32], [31], [36]) has not been reported yet in the literature.
This work is organized as follows. Section 2 introduces the basic notation and preliminaries while Section 3 reviews the observability results of SLS needed later on. The main paper results are presented in Sections 4 and 5. Section 4 is devoted to the SLS observer design and its convergence analysis while in Section 5 the later results are used it to construct an efficient synchronization scheme of SLS with chaotic behavior. The final section draws some conclusions and provides some outlooks for the related future research.
Section snippets
Linear systems
A LS or simply is represented bywhere is the state vector, is the control input, the output signal and A, B, C are constant matrices of appropriate dimensions. The input function space is denoted by and is considered to be . Through the paper stands for and for . A subspace is called A-invariant if and -invariant if there exists a state feedback such that or equivalently if .
Observability of switched linear systems
Basic results on observability of SLS needed later on are reviewed here, see [5], [6], [7], [27] for more details. These results are based on the geometric approach. An interested reader can find their extension to the perturbed case in [8] while the discrete-time case is treated in [9]. Definition 1 The continuous state trajectory x(t) (resp. the switching signal ) of the SLS (2) is said to be observable if there exists a finite-time τsuch that the knowledge on the structure (2) and the input
Observer design
The proposed SLS observer uses a bank of extended finite-observers designed one for each LS in the SLS. Thus, let us first introduce the extended finite-time observers for LS.
Synchronization of SLS with chaotic behavior
SLS can exhibit highly nonlinear behavior such as chaos when a suitable switching signal is applied. This property is exploited to create new chaotic attractors as well as to synthesize by SLS well known chaotic systems such as the Lorenz and Rössler attractors [45]. A chaotic system can be used in encryption and secure communications, where a message encrypted using the chaotic system is transmitted through an open channel which is then decrypted by the receiver by using a synchronized copy of
Conclusions
An observer design for SLS with unknown switching signal has been presented here. The corresponding observer estimates in finite-time both the state evolution during the continuous phase and the originally unknown SLS switching signal. Moreover, efficiency of these observers has been demonstrated by their application to the chaotic SLS synchronization scheme construction being an important topic of the current research with possible applications to secure encryption. Future and ongoing research
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that improved the quality of the paper.
This work was supported by CONACYT through Project 127858 and by the Czech Science Foundation through the research Grant no. 13-20433S.
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