Co-design of transition rates and sliding mode switched controller for Markovian jumping systems under intermittent DoS attacks
Introduction
Due to its powerful ability to describe structural abrupt variations, Markovian jumping systems (MJSs) have received extensive attention in the past decades and a lot of interesting results have been reported under the implicit assumption that the transition rate/probability matrix (TRM/TPM) is known/predefined. However, since the different transition of modes will have a great impact on the system dynamic performance, it should also be very possible and necessary to design the transition rate/probability for benefiting the system performance according to the practical requirement. In fact, Sehgal [1] has already carried out an interesting research on the dynamic buffer allocation for stochastic communication in networks-in-package via constructing a Markovian jump model-based communication scheme. Moreover, a few pioneer works as in [2], [3], [4], [5], [6], [7] have also been undergone on the design scheme of the transition rate/probability for MJSs. Song et al [5] investigated the hybrid design problem for output feedback exponential stabilization of MJSs. In [6], Cao et al proposed the hybrid design method of the TRM and sliding mode controller simultaneously to achieve the -EMS stability of an uncertain MJSs, meanwhile, further investigated the hybrid design problem of finite-time stochastic boundedness for the uncertain MJSs with external disturbance in [7].
On the other hand, along with the utilization of communication network in the control loops, the signal transmission among system components might inevitably encounter cyber attacks, e.g., the denial-of-service (DoS) attacks, which usually releases massive packets to block information exchange such that the data transmission is badly delayed [8], [9]. Under this cyber-attack, the system may be out of control due to the untrusted control signals. Hence, it is quite important for investigating the secure control for MJSs subject to cyber-attacks [10], [11], [12], [13], [14]. Moreover, by utilizing its stronger robustness to parameter uncertainties and external disturbance, the sliding mode control (SMC) were also introduced in [13], [14] to deal with the cyber-attacks. The key idea of SMC method is to utilize a discontinuous controller to drive the system states onto a specified sliding surface, along which tend to zero subsequently [15], [16]. In the past decades, the effective robust method has obtained successful application in the face of networked-induced detrimental phenomena including packet losses [17], channel fading [18], quantization [19], event-triggering [20], time-varying delay [21], etc. However, the above works implicitly assumed that the system mode information was available exactly at the controller side, that is, the controller is synchronous with the controlled MJSs. Apparently, this is not always true if the attack may happen in the S/C transmission channel as discussed in [14], where the signals may be attacked such that there may happen asynchronous between the designed SMC law and the controlled systems.
For practical MJSs subject to attacks, the system mode may not be simply divided into two simple cases: complete known and complete unknown. There may be an alternating transformation between known and unknown cases. For example, under the intermittent DoS (IDoS) attacks [22], the attacks happen intermittently according to the attacking frequency and duration. This may result in that the transmitted measurement signals are intermittently available for the controller as in [23], [24], which means that the system mode information of MJSs may also intermittently available or unavailable under the IDoS. Thus, an interesting issue is whether we can design some switched scheme between the synchronous controller for the system modes available and the asynchronous one for the unavailable case. Apparently, this kind of control scheme more in line with the actual situation and is more beneficial to the system performance than the simple asynchronous method, since the asynchronous controller completely ignore some available system mode information.
Motivated by the above discussions, this work focuses on the co-design issue of the transition rates and sliding mode switched controller for the MJSs subject to IDoS attacks, where the measurement output including the system mode information may lost under the attacks. In this circumstance, three key issues will be encountered, that is, how to estimate the unmeasurable state when the attack may happen or not? how to determine the lost system mode information? how to synthesize a co-design scheme of both the TRM and SMC under the IDoS attacks? Aiming at these issues, the existing conditions of the TRM are firstly established to ensure the exponential mean-square stability of the unforced MJSs. And then, according to different status of DoS attacks, a switched state estimator is constructed to estimate the unmeasured states, based on which, a synchronous/asynchronous switching SMC law is designed. Furthermore, a hybrid scheme for simultaneous synthesizing the TRM and SMC gain are given to achieve the exponential mean-square stability of the controlled MJSs under the IDoS attacks.
The main contributions are highlighted as follows: 1) It is the first attempt to cope with the exponential mean-square stabilization problem of It-type MJSs under IDoS and propose a new co-design methodology for both the TRM and SMC gain. 2) To estimate the unmeasurable state when the attack may happen or not, we design a switched state estimator according to different status of DoS attacks. That is, when the DoS attacks are absent, the estimator serves as a synchronous state observer where the system measurement output and system mode are available; when the DoS attacks are active, the measured output and the system mode cannot be received/updated, thus, the estimator serves as an asynchronous state compensator by means of a mode detector. 3) In light of different DoS status, a synchronous/asynchronous switched sliding mode controller synthesized based on the above estimator such that the specified sliding surface can be reached even in the presence of DoS attacks. 4) A switched Lyapunov function approach is introduced to obtain the stability conditions involving the DoS frequency and DoS duration and an iterative linear matrix inequality optimization algorithm is formulated to obtain the ideal TRM and SMC gain.
Notation For a symmetric matrix, () means that positive-definite (negative-definite). and denote, respectively, the Euclidean norm and 1-norm or its induced matrix norm. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
Section snippets
Problem description
Consider the uncertain It-type MJSs composed of a remote controller (shown in Fig. 1), where there may happen the IDoS attacks in S/C channel such that the controller cannot always obtain the measured output information and the system mode information. In view of this adverse situation caused by malicious attacks, a state estimator composed of an observer and a compensator will be designed later to provide the controller with the state estimation, and a mode detector is introduced to
Stabilizing TRM
In this section, we shall investigate the exponential mean-square stability problem of the unforced It-type MJS (4)-(5) () via transition rates synthesis. Theorem 1 The unforced It-type MJS (4)-(5) () is exponentially mean-square stable, if there exist matrices , , scalars and , (for ) satisfyingwith
Proof: Consider
Co-design of TRM and SMC
In the sequel, we firstly construct the SMC law, and then, we given the co-design scheme of the control gain and the TRM via analyzing the EMSS of the closed-loop system and the reachability of sliding mode dynamics.
Simulation
Example 1 In this example, all of the subsystems is not asymptotically stable, and this whole system composed of these subsystems is also unstable for a given TRM . Now, we will design a TRM to stabilize the above MJS.
Suppose that the unforced uncertain It-type MJS (1) is composed of two subsystems with
Conclusions
This paper has investigated the co-design scheme on SMC problem of It MJSs under intermittent DoS attacks. A switching synchronous/asynchronous estimator has been introduced to estimate the unmeasurable states under the measurement output is available or lost, and then a desirable sliding mode switched controller has been designed and a set of sufficient conditions for a co-design scheme has been attained. Besides, an iterative optimization algorithm has been provided for SMC gain and
Declaration of Competing Interest
All authors declare that there is no conflict of interest.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (62173222, 62073139), Shanghai Science and Technology Innovation Action Plan (22S31903700, 21S31904200) and the National Key R&D Program of China (Grant No. 2020AAA0109301).
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