Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem

In this paper, we obtain sufficient and necessary conditions of some classical convex sets as positively invariant sets for a continuous dynamical system, namely positive invariance conditions. The approach is based on Nagumo Theorem by deriving the tangent cones of these sets. We also propose approaches using optimization theory and models to verify the existence of these sufficient and necessary conditions

Optimization Theory and Dynamical Systems: Invariant Sets and Invariance Preserving Discretization Methods

Invariant set is an important concept in the theory of dynamical systems and it has a wide range of applications in control and engineering. This thesis has four parts, each of which studies a fundamental problem arising in this field. In the first part, we propose a novel, simple, and unified approach to derive sufficient and necessary conditions, which are referred to as invariance conditions for simplicity, under which four classic families of convex sets, namely, polyhedral, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for linear discrete or continuous dynamical systems. This novel method establishes a solid connection between optimization theory and dynamical systems. In the second part, we propose novel methods to compute valid or largest uniform steplength thresholds for invariance preserving of three classic types of discretization methods, i.e., forward Euler method, Taylor type approximation, and rational function type discretization methods. These methods enable us to find a pre-specified steplength threshold which preserves invariance of a set. The identification of such steplength threshold has a significant impact in practice. In the third part, we present a novel approach to ensure positive local and uniform steplength threshold for invariance preserving on a set when a discretization method is applied to a linear or nonlinear dynamical system. Our methodology not only applies to classic sets, discretization methods, and dynamical systems, but also extends to more general sets, discretization methods, and dynamical systems. In the fourth part, we derive invariance conditions for some classic sets for nonlinear dynamical systems. This part can be considered as an extension of the first part to a more general case

Passive detection of moving aerial target based on multiple collaborative GPS satellites

Passive localization is an important part of intelligent surveillance in security and emergency applications. Nowadays, Global Navigation Satellite Systems (GNSSs) have been widely deployed. As a result, the satellite signal receiver may receive multiple GPS signals simultaneously, incurring echo signal detection failure. Therefore, in this paper, a passive method leveraging signals from multiple GPS satellites is proposed for moving aerial target detection. In passive detection, the first challenge is the interference caused by multiple GPS signals transmitted upon the same spectrum resources. To address this issue, successive interference cancellation (SIC) is utilized to separate and reconstruct multiple GPS signals on the reference channel. Moreover, on the monitoring channel, direct wave and multi-path interference are eliminated by extensive cancellation algorithm (ECA). After interference from multiple GPS signals is suppressed, the cycle cross ambiguity function (CCAF) of the signal on the monitoring channel is calculated and coordinate transformation method is adopted to map multiple groups of different time delay-Doppler spectrum into the distance−velocity spectrum. The detection statistics are calculated by the superposition of multiple groups of distance-velocity spectrum. Finally, the echo signal is detected based on a properly defined adaptive detection threshold. Simulation results demonstrate the effectiveness of our proposed method. They show that the detection probability of our proposed method can reach 99%, when the echo signal signal-to-noise ratio (SNR) is only −64 dB. Moreover, our proposed method can achieve 5 dB improvement over the detection method using a single GPS satellite

Using heterogeneous satellites for passive detection of moving aerial target

Passive detection of a moving aerial target is critical for intelligent surveillance. Its implementation can use signals transmitted from satellites. Nowadays, various types of satellites co-exist which can be used for passive detection. As a result, a satellite signal receiver may receive signals from multiple heterogeneous satellites, causing difficult in echo signal detection. In this paper, a passive moving aerial target detection method leveraging signals from multiple heterogeneous satellites is proposed. In the proposed method, a plurality of direct wave signals is separated in a reference channel first. Then, an adaptive filter with normalized least-mean-square (NLMS) is adopted to suppress direct-path interference (DPI) and multi-path interference (MPI) in a surveillance channel. Next, the maximum values of the cross ambiguity function (CAF) and the fourth order cyclic cumulants cross ambiguity function (FOCCCAF) correspond into each separated direct wave signal and echo signal will be utilized as the detection statistic of each distributed sensor. Finally, final detection probabilities are calculated by decision fusion based on results from distributed sensors. To evaluate the performance of the proposed method, extensive simulation studies are conducted. The corresponding simulation results show that the proposed fusion detection method can significantly improve the reliability of moving aerial target detection using multiple heterogeneous satellites. Moveover, we also show that the proposed detection method is able to significantly improve the detection performance by using multiple collaborative heterogeneous satellites

Modulation parameter estimation of LFM interference for direct sequence spread spectrum communication system in alpha-stable noise

The linear frequency modulation (LFM) interference is one of the typical broadband interferences in direct sequence spread spectrum (DSSS) communication system. In this article, a novel modulation parameter estimation method of LFM interference is proposed for the DSSS communication system in alpha-stable noise. To accurately estimate the modulation parameters, the alpha-stable noise should be eliminated first. Thus, we formulate a new generalized extended linear chirplet transform to suppress the alpha-stable noise, for a robust time-frequency, transformation of LFM interference is realized. Then, using the Radon transform, the maximum value after transformation and the chirp rate according to the angle related to the maximum value are estimated. In addition, a generalized Fourier transform is introduced to estimate the initial frequency of the LFM interference. For the performance analysis, the Cramér-Rao lower bounds of the estimated chirp rate and the initial frequency of the LFM interference in the presence of alpha-stable noise are derived. Moreover, the asymptotic properties of the modulation parameter estimator are analyzed. Simulation results demonstrate that the performance of the proposed parameter estimation method significantly outperforms existing methods, especially in a low SNR regime