The stability analysis of H∞ and H2 control of linear systems has drawn significant research attention over the past few years. Following the pioneering work of Wilson that gave birth to the L2 – L∞ performance criterion, many other control problems based on L2 – L∞ criterion have been subsequently proposed and extensively studied. This has resulted in generalized H2 indicators for describing the system performance, stability, and ability to suppress disturbances. Typically, generalized H2 measures the disturbances input and controlled output using energy functions and corresponding peak values, respectively, which has enabled extensive studies of the generalized H2 control of time-varying and linear invariant systems. Unfortunately, the generalized H2 control problem of the continuous-time discrete-state semi-Markov jump linear system is still underexplored.
Generally, the sojourn time of most continuous-time discrete-state Markov jump linear systems follow a general distribution which is very conservative for many practical applications. As such, the transition state is always not constant because the system’s future state relies on the sojourn time and the current state. Lately, semi-Markov jump systems have been identified as promising solutions for reducing conservatism. For example, many scholars have used the semi-Markov kernel (SMK) method to study the stability of various semi-Markov systems. Nevertheless, despite the research implications of generalized H2 performance in many practical systems, the asynchronous generalized H2 control problem of these systems is sparsely reported in the literature.
On this account, Mr. Bo Xin and Professor Dianli Zhao from the University of Shanghai for Science and Technology investigated the generalized H2 control of the linear system with semi-Markov jumps. Particularly, they studied a class of asynchronous generalized H2 control problems associated with continuous-time discrete-state semi-Markov jump linear systems. This study generalized the special problems by breaking down the conservatism induced by the unchanged transition rate, unlike in the existing studies. Next, the smoothness and independent relationships between the variables were used to obtain adequate conditions to improve the system’s generalized H2 performance and stochastic stability. The effectiveness of the controller was enhanced through an unknown transition rate. Finally, simulations were carried out to demonstrate the feasibility of the proposed approach. Their work is currently published in the research journal, International Journal of Robust and Nonlinear Control.
The authors obtained sufficient conditions to improve the stochastic stability and generalized H2 performance of the system. This was attributed to the construction of the Lyapunov function, application of the smoothness expectation and the weak infinitesimal operator, and slack variables. The results were also generalized successfully for cases where the transition rates were partially unknown, enabling the design of a high-performance asynchronous controller. Furthermore, the simulation of the two specific examples of the linearized iterative algorithm confirmed the effectiveness of the proposed approach.
In summary, the study presents a comprehensive investigation of the asynchronous generalized H2 control for continuous-time discrete-state semi-Markov jump linear systems. The proposed controller mode is based on a new stochastic process related to the system through conditional probability. With the increasing applications of networked and multi-agent control systems and the inherent problem of system disturbance, the presented scheme offers a potential solution for suppressing the disturbances. In a statement to Advances in Engineering, Professor Zhao explained that the study provided useful insights to improve the stability and generalized H2 performance of different linear systems.
Xin, B., & Zhao, D. (2020). Generalized H2 control of the linear system with semi‐Markov jumps. International Journal of Robust and Nonlinear Control, 31(3), 1005-1020.