In the industrial cyber-physical environment, the performance of clock synchronization with loss of data packets should be analyzed quantitatively. The localized state estimator and controller are designed to realize the speed of convergence for exponential stability of the clock synchronization for the output feedback tubes-model predictive control (Tubes-MPC) of the state-space model. The upper and lower bounds for the variance of synchronization error under the statistical significance were quantitatively analyzed in incomplete measurement. The set of interference error in complete measurement was added into the set of additional estimated errors introduced by packet loss. Clock synchronization of the output feedback Tubes-MPC which has been constructed still has the performance of robust exponential convergence under networked packet loss, and it can be extended to an absolute state-space model with network-level in a unified approach.

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Quantitative Analysis for Networked Output Feedback MPC of Clock Synchronization in Unreliable WSN

In this paper, the distributed stochastic model predictive control (MPC) is proposed for the noncooperative game problem of the discrete-time multi-player systems (MPSs) with the undirected Markov jump graph. To reflect the reality, the state and input constraints have been considered along with the external disturbances. An iterative algorithm is designed such that model predictive noncooperative game could converge to the so-called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\varepsilon$</tex> -Nash equilibrium in a distributed manner. Sufficient conditions are established to guarantee the convergence of the proposed algorithm. In addition, a set of easy-to-check conditions are provided to ensure the mean-square uniform bounded stability of the underlying MPSs. Finally, a numerical example on a group of spacecrafts is studied to verify the effectiveness of the proposed method. 

Noncooperative Model Predictive Game With Markov Jump Graph

In this paper, the distributed stochastic model predictive control (MPC) is proposed for the noncooperative game problem of the discrete-time multi-player systems (MPSs) with the undirected Markov jump graph. To reflect the reality, the state and input constraints have been considered along with the external disturbances. An iterative algorithm is designed such that model predictive noncooperative game could converge to the so-called $\varepsilon$-Nash equilibrium in a distributed manner. Sufficient conditions are established to guarantee the convergence of the proposed algorithm. In addition, a set of easy-to-check conditions are provided to ensure the mean-square uniform bounded stability of the underlying MPSs. Finally, a numerical example on a group of spacecrafts is studied to verify the effectiveness of the proposed method.

It can be found by simulations that there exists instability in the solution of distributed Kalman filter in the state tracking problem of distributed clock. This article has analyzed that the premise of obtaining the deterministic state estimation is ensuring the observability of system. Then, an observable measurement model for constructing a new comparative transformation of minimum mean square error (MMSE) oriented to basic measurement unit is proposed. Based on the vector operation for vector space of measurement state, while ensuring the observability of system, a state uncoupled measurement equation that has approximately equivalent performance with MMSE is established, and the algorithm has the scale extensibility of MMSE measurement performance. The algorithm realized the distributed tracking of clock state by updating of measurement value oriented to the two-way message exchange.

A New Measurement Model of Clock State Tracking From Analysis of Observability

In the cyber-physical environment, the clock synchronization algorithm is required to have better expansion for network scale. In this paper, a new measurement model of observability under the equivalent transformation of minimum mean square error &#x0028 MMSE &#x0029 is constructed based on basic measurement unit &#x0028 BMU &#x0029 , which can realize the scaled expansion of MMSE measurement. Based on the state updating equation of absolute clock and the decoupled measurement model of MMSE-like equivalence, which is proposed to calculate the positive definite invariant set by using the theoretical-practical Luenberger observer as the synthetical observer, the local noncooperative optimal control problem is built, and the clock synchronization system driven by the ideal state of local clock can reach the exponential convergence for synchronization performance. Different from the problem of general linear system regulators, the state estimation error and state control error are analyzed in the established affine system based on the set-theory-in-control to achieve the quantification of state deviation caused by noise interference. Based on the BMU for isomorphic state map, the synchronization performance of clock states between multiple sets of representative nodes is evaluated, and the scale of evaluated system can be still expanded. After the synchronization is completed, the state of perturbation system remains in the maximum range of measurement accuracy, and the state of nominal system can be stabilized at the ideal state for local clock and realizes the exponential convergence of the clock synchronization system.

Xiaoming Tang

Papers

Scalable clock synchronization analysis: A symmetric noncooperative output feedback tubes-MPC approach